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+{"text":"\\section{Introduction}\n\\section{Introduction}\nQuantum mechanics is one of the most transformative physical theories of the 20th century. However, while the evolution of the quantum wave function is deterministically described by Sch\\\"odinger's equation, the outcome of a measurement is probabilistic, given by Born's rule. Despite recent progress \\cite{Ozawa_Bayes_1997,Zurek_Darwinism_2009,Shrapnel_Born_2018}, there is no consensus on how to reconcile these two viewpoints, as illustrated by the measurement paradox \\cite{Schroedinger_Cat_1935}. There are two conceptually distinct approaches: either the interpretative postulates must be modified \\cite{Adler_Quantum_2009,Zeh_Measurement_1970,Everett_Relative_1957,Heisenberg_Prinzipien_1930,Bohm_Hidden_1952}, or quantum mechanics approximates a deeper theory yet to be discovered. The later approach gives rise to collapse models \\cite{Bassi_Review_2012,Bassi_Gravitational_2017}, postulating a stochastic nonlinear modification to Schr\\\"odinger's equation. Irrespective of whether they successfully allow the reconciliation of quantum evolution and measurement theory, these collapse models are considered the only mathematically consistent, phenomenological modifications against which quantum theory can be tested \\cite{Adler_Quantum_2009,Ferialdi_Dissipative_2012}.\n\nThe most universal and well studied collapse model is Continuous Spontaneous Localization (CSL) \\cite{Pearle_Localization_1989,Ghirardi_Markov_1990},\nwhich serves as a framework to describe a variety of collapse mechanisms \\cite{Bassi_Review_2012,Bassi_Gravitational_2017,Ghirardi_gravity_1990,Adler_Nonwhite_2007,Adler_Nonwhite_2008,Smirne_Dissipative_2014,Smirne_Dissipative_2015,Bassi_Breaking_2010}.\nIn CSL, a collapse noise field is introduced which couples nonlinearly to the local mass density. In its simplest form this noise is white, and the model has two parameters --- the collapse rate $\\lambda_c$, which determines the interaction strength with the collapse noise field, and the correlation length $r_c$, which determines the spatial resolution of the collapse process \\cite{Ghirardi_Markov_1990,Adler_Quantum_2009}.\nThe correlation length is expected to be $\\sim100$~nm \\cite{Adler_Bounds_2006}, since the behaviour of larger systems is generally adequately described by classical theories, whereas quantum mechanics appears to apply on smaller scales.\nRefined dissipative and coloured models introduce two additional parameters, associating a temperature and high-frequency cut-off to the collapse noise field to ensure energy conservation and permit an identifiable physical origin of collapse \\cite{Adler_Nonwhite_2007,Adler_Nonwhite_2008,Smirne_Dissipative_2014,Smirne_Dissipative_2015,Bassi_Breaking_2010}. \nBased on the assumption that the origin is of cosmological nature, and thermalised to the photon-, neutrino-, or gravitational wave background,\nthe high-frequency cut-off is estimated to occur at $\\Omega_{\\rm csl}\/2\\pi\\sim 10^{10}-10^{11}$~Hz \\cite{Bassi_Breaking_2010}.\n\nTo date, the most stringent unambiguous upper bounds on the collapse rate at the expected correlation length are based on mechanical resonators, with signatures of spontaneous collapse expected to manifest as an anomalous temperature increase.\nHowever, the suggested lower bounds to the collapse-induced heating are lower than one phonon per day \\cite{Adler_Bounds_2006,Bassi_Breaking_2010,Ghirardi_Unified_1986}.\nThe challenge of resolving these exquisitely small collapse signatures over a large thermal noise background has precluded conclusive tests of CSL, and has also introduced significant challenges for data interpretation \\cite{Vinante_Improved_2017}.\nEven were these issues resolved, quantum backaction heating \\cite{Khalili_backaction_2012,Nimmrichter_Optomechanical_2014} would remain orders of magnitude larger than the predicted collapse signatures.\nMoreover, with micron- \\cite{Vinante_Cantilever_2015,Vinante_Improved_2017} to meter-sizes \\cite{Carlesso_Gravitational_2016,Helou_LISA_2017}, the resonators employed to-date are larger than the anticipated correlation length and have frequencies far below the expected high-frequency cut-off. As such they are unable to provide insight into the physical origin of collapse \\cite{Adler_Nonwhite_2007,Adler_Nonwhite_2008,Smirne_Dissipative_2014,Smirne_Dissipative_2015,Bassi_Breaking_2010}.\n\n  \\begin{figure}[h!]\n  \\includegraphics{Fig_Setup_nc_1.pdf}\n\\caption{{\\bf Illustration of protocol.}\nTop left: array of optomechanical cavities.\nTop right: nonlinear pair production from a signal and pump photon (frequency $\\omega_{\\rm pump}$).\nBottom: Energy level diagram for scattering of a probe photon with a phonon.\n$n_b$:  phonon number.\n}\n  \\label{Fig_Setup}\n    \\end{figure}\n\nIn this work we propose to test collapse theories with high frequency nanomechanical resonators. This offers the advantages of miniaturisation to match the expected collapse correlation length, resonance frequency around the high-frequency cut-off, and the abilities to both exponentially suppress thermal phonons via passive cryogenic cooling and apply quantum measurement techniques to improve performance.\nTo assess the approach, we develop a specific experimental implementation that makes use of phonon counting in a nanoscale mechanical resonator. \nOur proposal includes new mitigation strategies for optical, thermal, electrical and quantum back-action noise that, for the first time, provide a way to bring each of these noise sources below the expected lower bounds for collapse-induced heating. We conclude that with challenging but plausible improvements in the state-of-the-art our approach could conclusively test CSL, closing the gap between measured upper bounds and predicted lower bounds on the collapse rate, and could also potentially identify the physical mechanism underlying the collapse process.\nThis provides an experimental pathway to answer one of the longest standing questions in physics, and  \nalso opens up possibilities for laboratory tests of astrophysical models of dark matter \\cite{Riedel_undetectable_2013,Riedel_collisions_2017}, \nand other exotic particles \\cite{Riedel_diffusion_2015}.\n\n\n\n\\section*{Results}\n\n\\subsection*{Basic protocol}\n\n  \n    \nOur protocol is illustrated in Fig. \\ref{Fig_Setup}, and is based on a gigahertz nanomechanical resonator, or array thereof,\nwithin a millikelvin environment.\nAs opposed to standard optomechanical measurement, consisting of an optical cavity linearly coupled to a mechanical resonator \\cite{Aspelmeyer_Review_2014}, we propose to perform phonon counting in a three-mode optomechanical system where two optical modes\nare coupled via a mechanical resonator with resonance frequency $\\Omega$.\nThis allows collapse signatures to be spectrally distinguished from most noise sources.\nOne mode, the {\\it probe mode}, is excited by a continuous weak laser at its resonance frequency $\\omega_p$.\nIn the ideal case, the other, the {\\it signal mode} at frequency $\\omega_s=\\omega_p+\\Omega$, is only excited by resonant anti-Stokes Raman scattering between collapse induced phonons and probe photons.\nA single-photon readout scheme minimises both absorption heating  \\cite{Meenehan_millikelvin_2014} and quantum back-action heating.\nSignal photons are spectrally separated from probe photons by a filter cavity, while dark counts are suppressed by nonlinearly downconverting signal photons to pairs and performing coincidence detection.\n\n\n\nAs a concrete example, we consider using a three-mode photonic-phononic crystal optomechanical system, such as proposed in \\cite{Chang_Array_2011,BasiriEsfahani_Phonon_2012,Ludwig_TwoMode_2012,Safavi_Naeini_traveling_2011}. \nWe choose most parameters based on those achieved in \\cite{MacCabe_Ultralong_2019}, with a mechanical resonance frequency $\\Omega\/2\\pi=5.3$~GHz, a mechanical damping rate $\\Gamma\/2\\pi=108$~mHz, an effective mass $m_{\\rm eff}=136$~fg, and thermalisation to the base temperature of a dilution refrigerator ($T=10$~mK).\nWe use the theoretical scattering-limited intrinsic decay rate of $\\kappa_{p,0}=\\kappa_{s,0}=2\\pi\\cdot 9.2$~MHz calculated for these devices \\cite{Ren_Crystal_2019} for both optical modes, where the subscripts `$p$' and `$s$' distinguish the probe and signal mode throughout \\cite{Aspelmeyer_Review_2014}.\nFinally, we assume a tenfold improved single-photon optomechanical coupling rate of $g_0\/2\\pi=11.5$~MHz, as predicted to be feasible with optimized designs \\cite{Matheny_coupling_2018}.\n\n\n\n\n\\subsection*{Phonon flux induced by CSL}\n\nThe CSL phonon flux is $\\dot n_c= \\lambda_c D$, where $D$ is a geometrical factor that quantifies the susceptibility of the resonator to spontaneous collapse.\nThe requirement that CSL should resolve the measurement problem introduces lower bounds on $\\lambda_c$, and therefore on the phonon flux. \nAdler proposed  $\\lambda_c\\geq 10^{-8\\pm 2}$~s$^{-1}$ from the postulate that collapse should account for latent image formation in photography \\cite{Adler_Bounds_2006}, while Bassi {\\it et al.} proposed $\\lambda_c\\geq 10^{-10\\pm 2}$~s$^{-1}$ from the presumption that collapse should occur in the human eye \\cite{Bassi_Breaking_2010}.\nWe estimate  $D=5.1\\cdot 10^{5}$ for our proposed device \\cite{Vinante_Cantilever_2015} (see Supplemental Material \\cite{Supp}), which combined with these bounds implies minimum CSL induced phonon fluxes of $\\dot n_c=5.1\\cdot10^{-3\\pm 2}$~s$^{-1}$ and $\\dot n_c=5.1\\cdot10^{-5\\pm 2}$~s$^{-1}$, respectively. \n\n\n\n\n\\subsection*{Optomechanical dynamics and conversion efficiency}\n\nIf the oscillator is initially in its ground state,\nwith one photon in the probe mode, a phonon introduced by spontaneous collapse\nprepares the state $\\ket{n_b n_p  n_s}=\\ket{110}$, where $n_b$ is the phonon number in the mechanical resonator, while $n_p$ and $n_s$ are the photon numbers in probe- and signal-mode, respectively. \nThe optomechanical conversion efficiency $\\eta_{\\rm om}$ for this state to emit a signal photon at frequency $\\omega_s$\nis obtained by numerically solving the Born-Markov master equation taking into account that $\\Gamma\\ll\\kappa_p,\\kappa_s$ (see Methods).\nWe choose the external probe decay rate $\\kappa_{p,\\rm ex}\/2\\pi=2.2$~MHz \\cite{Aspelmeyer_Review_2014}, allowing operation at the threshold of strong coupling with $g_0\\approx\\kappa_p$. \nThis is advantageous for efficient conversion of collapse-induced phonons to signal photons and ensures low occupancy, minimising noise, as discussed later.\nWe choose the signal mode to be significantly overcoupled ($\\kappa_{s,\\rm ex}\/2\\pi=0.7\\kappa_{s}=21$~MHz) in a trade-off between optimising the conversion efficiency and suppressing noise from direct occupancy of the signal mode (see later).\nTogether, these external decay rates result in relatively high conversion efficiency of $\\eta_{\\rm om}=0.32$.\n\n \\subsection*{Noise sources}\n\n\\begin{figure}[!htbp]\n\t \\includegraphics{Fig_heating_nc_1.pdf}\n\\caption{{\\bf Heating rates of a $Q=\\Omega\/\\Gamma=10^7$ silica sphere resonator vs. mechanical frequency and sphere diameter}. Red traces: heating due to coupling to the thermal environment at temperatures 300,1 and 0.01 K. Gray shaded: Lower bounds on CSL heating rates for a sphere, according to  Adler \\cite{Adler_Bounds_2006}, Bassi {\\it et al.} \\cite{Bassi_Breaking_2010} and GRW \\cite{Ghirardi_Unified_1986}, assuming the fundamental mechanical breathing mode frequency $\\Omega=c\/R$, with sphere radius $R$ and speed of sound $c=3000$ m\/s. CSL heating rates drop once the resonator becomes smaller than the noise correlation length, which is set to $r_c=10^{-7}$~m. Green: lower bound on heating rate predicted from classical channel gravity \\cite{Kafri_Classical_2014,Khosla_Classical_2018,Altamirano__Pairwise_2018}. At high frequencies and low temperatures, collapse signatures exceed the thermal heating. Blue shaded: proposed range of $\\Omega_{\\rm csl}$.}\n  \\label{Fig_heating} \n\\end{figure}\n\nFour classes of noise can potentially imitate a collapse signal:  thermal phonons, probe photons that leak through the system, phonons introduced by the measurement process, and detector dark counts.\nPhotons that leak through the system can be efficiently filtered using a standard laser stabilisation reference cavity \\cite{Kessler_Laser_2011}\n(see Table \\ref{tab:noise comparison} and Methods), and will not be considered further here.\n\n\n\n\n{\\it Thermal phonons.}\n A collapse signature is resolvable in a thermal noise background if $\\dot n_c\/\\dot n_{\\rm th}>1$, where $\\dot n_{\\rm th}=\\Gamma (e^{\\hbar \\Omega\/k_BT}-1)^{-1}$ is the thermal phonon flux.\nThis gives a minimum testable collapse rate $\\lambda_{c,\\rm th}=\\dot n_{\\rm th}\/D$. \nExisting experiments have operated with comparatively low frequency oscillators in the high temperature limit, $k_BT \\gg \\hbar\\Omega$ \\cite{Vinante_Cantilever_2015,Vinante_Improved_2017,Carlesso_Gravitational_2016,Helou_LISA_2017}, with thermal phonon flux significantly larger than Bassi {\\it et al.}'s lower bound,\nand have sought to resolve small collapse signatures on top of this large thermal noise background.\nA significant advantage of our approach is that miniaturisation and cryogenic cooling allow access to the regime where $k_BT \\ll \\hbar\\Omega$.\nThe average thermal phonon occupation is then exponentially suppressed due to Bose-statistics, $\\dot{n}_{\\rm th} \\approx \\Gamma e^{-\\frac{\\hbar\\Omega}{k_ BT}}$.\nFig. \\ref{Fig_heating} shows this exponential suppression as a function of resonator size, and in comparison to the CSL signal, for the simple example of the fundamental breathing mode of a silica sphere (see Supplemental Material \\cite{Supp} for calculation).\nAs can be seen, for gigahertz resonators at millikelvin temperatures the exponential suppression allows thermal phonon fluxes beneath both Adler and Bassi {\\it et al.}'s lower bounds.\nFor the proposed photonic crystal device, we find $\\lambda_{c,\\rm th}=1.2\\cdot 10^{-17}$~s$^{-1}$,\nalso well beneath both bounds.\nThis provides the potential for\nunambiguous tests of collapse models.\n\n  \\begin{figure}[!htbp]\n\\includegraphics{Fig_Energy_nc_1.pdf}\n\\caption{{\\bf Signal pathways due to measurement-induced phonons.}\na) Phonon created after direct excitation of signal mode.\nb) Phonon due to counter-rotating transition.\nc) Two phonons created by counter-rotating transition followed by resonant transition. \n}\n\\label{Fig_Energy}\n      \\end{figure}\n\n{\\it Measurement-induced phonons.}\nPhonons introduced by the optomechanical measurement can imitate collapse signatures. These phonons are created by non-resonant scattering processes between the signal and probe modes, the three lowest order of which are represented in Fig. \\ref{Fig_Energy}.\nWe calculate the probability of phonon occupancy due to these processes numerically by solving the Born-Markov master equation (see Methods).\nWe find that each of these processes is suppressed by the square of the resolved sideband ratio $\\Omega\/\\kappa_p$, with predicted phonon occupancies shown in Fig. \\ref{figSOM} (a) and (b).\nPhotoabsorptive heating can also introduce phonons. \nHowever, it only adds a negligible contribution to the measurement-induced phonon occupancy (see Table \\ref{tab:noise comparison} and Methods).\n\n  \n\n\n\\begin{figure}[!htbp]\n\t\\includegraphics{Fig_OM_nc_2.pdf}%\n\n\n\n\\caption{{\\bf Numerical calculations of noise magnitude.}\na) Blue (orange): occupancy of density matrix elements containing one (two) phonon(s);\ndashed blue: contribution from direct signal mode excitation.\nFast oscillations on timescale $\\Omega^{-1}$ correspond to the counter-rotating transition $\\ket{010}\\leftrightarrow\\ket{101}$; \nslow oscillations to the resonant process $\\ket{101}\\leftrightarrow\\ket{210}$, with period $g_0^{-1}=\\kappa_p^{-1}$. \nDotted lines: asymptotic values for $\\kappa_p^{-1},\\kappa_s^{-1}\\ll t\\ll\\Gamma^{-1}$.\nb) Same as (a) for $t\\sim\\Gamma^{-1}\\gg\\kappa_p^{-1}, \\kappa_s^{-1}$. \nc) Cumulative probability $p_{\\rm om}(t)$ of a probe photon creating a photon at frequency $\\omega_s$. \nDotted line: asymptotic value for $t\\gg\\Gamma^{-1}$.\n}\n\\label{figSOM}\n\\end{figure} \n \nA measurement-induced phonon can only be converted to a collapse-imitating photon at frequency $\\omega_s$ if it scatters with a second photon entering the probe mode within the lifetime $\\Gamma^{-1}$ of the mechanical resonator (see Fig. \\ref{Fig_Energy}).\nIt is therefore possible to suppress these photons by operating with a low average photon occupancy $\\bar n_p$. Here, we choose the photon occupancy so that the probability of a photon entering the probe mode during one mechanical oscillation lifetime is $\\eta_p = \\bar n_p \\kappa_p\/\\Gamma\\sim$~1\\%.\nThis reduces the rate of measurement-induced photons by a factor of a hundred.\nThe cumulative probability of a probe photon generating a phonon, and a second probe photon then causing emission of a photon at frequency $\\omega_s$, is shown in Fig. \\ref{figSOM} (c).\nThe asymptotic probability is $p_{\\rm om}(t\\rightarrow \\infty)=8.4\\cdot 10^{-8}$ (see Supplemental Material \\cite{Supp}).\n\n\n{\\it Coincidence dark counts.}\n Detecting collapse induced phonons at the predicted rate of less than one per day necessitates very high suppression of photon dark counts, which typically occur at hertz to kilohertz rates.\nOne way to  achieve this is to nonlinearly downconvert signal photons to pairs (bottom right inset, Fig. \\ref{Fig_Setup}) \nusing a bright pump beam in a third-order nonlinear medium.\nIt has been shown that this process can convert single photons to pairs with near-unit efficiency $\\eta_\\chi$ \\cite{Langford_Conversion_2011} (see Supplemental Material \\cite{Supp}). \nA signal is recorded only if a coincidence detection event is registered.\nThe coincidence dark count rate is suppressed as the square of the single-detector dark count rate $R_{d,1}$, $R_{d,2}=R_{d,1}^2 \\cdot \\tau_c$, where $\\tau_c$ is the coincidence timing resolution. \nFor commercially available photon counters with $R_{d,1}=3.5$~s$^{-1}$ and $\\tau_c=30$~ps \\cite{Photonspot_private}, we predict $R_{d,2}\\sim 3.7\\cdot10^{-10}$~s$^{-1}$.\n\n \n \\subsection*{Minimum testable collapse rate}\n \n For $r_c=10^{-7}$~m, the rate of coincidence counts attributed to collapse is  $R_c=\\lambda_c D \\eta=5.5\\cdot 10^{2}\\lambda_c$, where the efficiency  $\\eta=\\eta_p\\eta_{\\rm om}\\eta_{\\chi}\\eta_d\\eta_{\\rm f}=1.1\\cdot10^{-3}$ quantifies the fraction of phonons in the mechanical resonator that result in a coincidence count, $\\eta_{\\chi}=0.95$, and $\\eta_d=0.64$ is the coincidence detection efficiency (see Supplemental Material \\cite{Supp}).\nThis rate must exceed the sum of the noise rates, setting the limit to the minimum observable collapse rate $\\lambda_c$.\nFor optomechanically induced phonons, probe photons leaking through the system, and thermal phonons, $R_{\\rm om}=\\kappa_{p,\\rm ex} \\bar n_pp_{\\rm om}(t\\rightarrow \\infty)\\eta_f\\eta_\\chi\\eta_d$, $R_{\\rm phot}=\\kappa_{p,\\rm ex} \\bar n_pp_f\\eta_\\chi\\eta_d$, and $R_{\\rm th}=\\eta\\dot n_{\\rm th}$, respectively, where $\\eta_f=0.56$ is the transduction efficiency through the filter and $p_f=3.5\\cdot 10^{-10}$ the probability of a probe photon leaking through the filter (see Supplemental Material \\cite{Supp}).\nThe numerical values and corresponding minimum testable collapse rates are given in Table \\ref{tab:noise comparison} (see also Supplemental Material \\cite{Supp}).\nOptomechanical measurement-induced phonons and coincidence dark counts set comparable limits on $\\lambda_c$, with negligible contributions from leaked probe photons, photoabsorption, and thermally excited phonons.  \nThe minimum testable collapse rate limited by all noise sources is $\\lambda_c=\\sum_i\\lambda_{\\rm c,i}=1.0\\cdot10^{-12}$~s$^{-1}$, sufficient to test both Bassi {\\it et al.}'s and Adler's proposals.\n\n\\begin{table}\n\\center\n\\begin{tabular}{c | c | c | c} \n\nnoise type & scaling & signal rate~[s$^{-1}$]   &$\\lambda_{c,\\rm min}$~[s$^{-1}$] \\\\\n\n\\hhline{=|=|=|=}\n\\rule{0pt}{9pt} collapse noise & $\\eta l^2\\rho^2x_0^2$\\cite{Supp} & $5.5\\cdot 10^{2}\\lambda_c$ & --- \\\\\n\\hline\n\\rule{0pt}{12pt} thermal & $e^{-\\frac{\\hbar\\Omega}{k_BT}}$ &  $6.7 \\cdot 10^{-15}$ & $1.2\\cdot 10^{-17}$ \\\\\n\\hline\n\\rule{0pt}{10pt} optom. phonons &  $\\eta_p\\kappa^2\/\\Omega^2$ & $1.9 \\cdot 10^{-10}$  & $3.5 \\cdot 10^{-13}$  \\\\\n\\hline\n\\rule{0pt}{10pt} abs. heating & $l\/(m^{\\frac{1}{4}}\\Omega)$ \\cite{Supp} & $1.4 \\cdot 10^{-14}$  & $ 2.6 \\cdot 10^{-17}$ \\\\ %\n\\hline\n\\rule{0pt}{10pt} probe photons & see \\cite{Supp}& $1.4\\cdot 10^{-12}$  &$2.6\\cdot 10^{-15}$\\\\\n\\hline\n\\rule{0pt}{9pt} dark counts & $R_{d,1}^2 \\cdot \\tau_c\/N$ &$3.7\\cdot 10^{-10}$ & $ 6.7\\cdot 10^{-13}\/N$ \\\\\n\\hline\n\\rule{0pt}{9pt} all noise & --- & $5.6\\cdot 10^{-10} $ & $ 1.0\\cdot 10^{-12}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{{\\bf Comparison of noise sources and respective testable CSL parameter $\\lambda_c$.} $m$ is the oscillator mass, $l$ its linear size, $\\rho$ its density, and $x_0$ its zero point motion.}\n\\label{tab:noise comparison}\n\\end{table}\n\n\n\n \\subsection*{Signal rate and measurement time}\nThe predicted average time required to observe one signal due to CSL-collapse is $t_{\\rm meas}=(\\lambda_c D\\eta)^{-1}$. \nFully probing both Adler's and Bassi {\\it et al.}'s proposals with a single optomechanical resonator, including their respective uncertainties, would require $t_{\\rm meas}>57$~years.\nFabricating an array of $N$ optomechanical cavities on a silicon wafer \\cite{Bekker_tuning_2018,Xu_cascaded_2006,GilSantos_cascaded_2016,Zhang_synchronisation_2011} (see Fig. \\ref{Fig_Setup}), coupled to a single filter cavity, nonlinear medium and detector  (or a small number of such elements)\ncould significantly reduce this time to $t_{\\rm meas}^{(N)}=t_{\\rm meas}\/N$, and also the dark count-limited testable collapse rate to $\\lambda_{c,\\rm det}^{(N)}=\\lambda_{c,\\rm det}\/N$.\nWe estimate that $N\\sim10^{4}$ may be feasible (see Supplemental Material \\cite{Supp}), essentially eliminating detector dark counts as a limit, and allowing a reduction of the measurement time to about two days.\n\n\n \\subsection*{Feasibility and alternative parameter regimes}\nThe only optomechanical parameters that must be improved from the current state-of-the-art \\cite{MacCabe_Ultralong_2019} to realise our protocol, are a reduced optical linewidth (by a factor of $\\sim50$), as predicted by theoretical modelling based on the device realized in \\cite{Ren_Crystal_2019}, and an enhanced single-photon coupling rate (by a factor of $\\sim10$), based on theoretical modelling in \\cite{Matheny_coupling_2018}. \nAlternatively, effective enhanced single-photon coupling could be achieved by coupling to a qubit or other highly nonlinear system, e.g. as demonstrated in \\cite{Pirkkalainen_hybrid_2013}.\nGiven the trajectory of the field,\nwe estimate these requirements to be likely achievable in the intermediate future. \nNevertheless, it is also useful to consider alternative realizations of the method.\n\n \n\n{\\it Quadratic coupling.}\nWhile here we consider phonon-counting via an optomechanical Raman interaction,\nin principle the method could be implemented with any low-noise phonon-counting method applied to a high-frequency oscillator \\cite{Oconnell_quantum_2010,Dellantonio_nondemolition_2018,Sletten_Fock_2019,Cohen_counting_2014,ArrangoizArriola_nanomechanical_2019}. \nOne promising approach may be quantum non-demolition measurement of phonon number using non-linear optomechanics \\cite{Thompson_strong_2008}. \nIn the regime of quadratic optomechanical coupling and resolved mechanical sidebands \\cite{Aspelmeyer_Review_2014}, a collapse-induced phonon imparts a frequency shift $2\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}$ on the optical resonance at frequency $\\omega$, where $\\bar n_{\\rm cav}=\\langle a^\\dagger a \\rangle$ is the average intracavity photon number with $a$ the annihilation operator for the optical cavity field, and $g_0^{(2)}$ the zero-point quadratic coupling rate \\cite{Thompson_strong_2008,Paraiso_Squared_2015,Hauer_nondemolition_2018}.\nThe shift is detectable if it is larger than the significant noise sources, which are random fluctuations in the probe frequency, absorption heating, and quantum-backaction from spurious linear coupling.\n\nConsidering Bassi {\\it et al.}'s proposed mechanism \\cite{Bassi_Breaking_2010}, taking $\\bar n_{\\rm cav}=10^2$ and assuming that the probe is shot noise limited, we find that a zero-point quadratic coupling rate of  $g_0^{(2)}\\gtrsim3.5\\sqrt{\\kappa\\Gamma}\\bar n_{\\rm cav}^{-3\/2}\\gtrsim 2\\pi\\cdot 28$~Hz would be sufficient for\nthe weakest possible collapse signal to exceed the probe frequency noise using the photonic-phononic crystal considered in the protocol above \\cite{MacCabe_Ultralong_2019}\n(see Methods).\nThis is well within experimentally achieved values in optomechanical photonic crystals (e.g. $g_0^{(2)}\/2\\pi=245$~Hz in \\cite{Paraiso_Squared_2015}).\n\nPerhaps the most significant challenge in this approach would be to engineer a strong suppression of linear optomechanical coupling, so that the phonon flux due to quantum back-action does not exceed the predicted CSL signature. \nIf using standard architectures, there is a fundamental limit to this suppression of linear coupling \\cite{Miao_Limit_2009}. Hence, either a different architecture would have to be employed \\cite{Kaviani_paddle_2015,Dellantonio_nondemolition_2018,Hauer_nondemolition_2018}, or the substantially more stringent condition $g_0^{(2)}\\geq\\kappa$ would have to be realized.\nThe phonon flux due to quantum back-action is given by  $\\dot{n}_{\\rm ba}=4g_0^2\\bar n_{\\rm cav}\/\\kappa=4g^2\/\\kappa$. \nTo resolve a potential CSL signature, $\\dot{n}_{\\rm c}$ must be greater than $\\dot{n}_{\\rm ba}$.\nAs a result, the linear optomechanical coupling would need to be suppressed to $g\\leq\\sqrt{\\lambda_c D \\kappa \/4}$. To test $\\lambda_c=10^{-12}$, we find the condition $g\/2\\pi\\lesssim 10^{-1}$~Hz, about seven orders of magnitude lower than typical linear coupling rates in photonic-phononic crystal structures  \\cite{MacCabe_Ultralong_2019}.\nWhile some architectures may in principle allow for vanishing linear coupling $g$, achieving the required suppression in practice may be challenging  \\cite{Dellantonio_nondemolition_2018,Kaviani_paddle_2015,Anetsberger_Near-Field_2009}.\n\nIn continuous operation, with currently available technology \\cite{MacCabe_Ultralong_2019}, absorption heating would exceed the expected heating from collapse by about seven orders of magnitude.\nEven with this very large heating in the continuous domain, it may be possible to resolve the problem by operating in a pulsed regime, so that each optomechanical measurement process is completed in a timescale much shorter than the time required for absorption events to create phonons.\nIn this case, the measurements would need to be sufficiently temporally spaced to allow for phonons to fully dissipate.\n\n\n \n\n\n\\subsection*{Discussion}\n\n\\begin{figure}[!htbp]\n  \\includegraphics{Fig_CSL_nc_2.pdf}\n\\caption{{\\bf Parameter diagram for CSL-model.}\nExcluded upper bounds: gravitational wave detectors (yellow shaded); cold atoms (gray shaded); microcantilevers (dashed blue line); KDTL-interferometry (dashed black); Excluded for simple CSL only: neutron stars (dashed black) and X-ray (dotted black).\nProposed lower bounds: Adler (vertical blue bars and dotted blue line) and Bassi {\\it et al.} (vertical black bar).\nRed: predicted testable parameter space using our protocol.\n}\n\n\n\\label{figCSL}\n\\label{stats}\n\\end{figure}\n\n\nFig. \\ref{figCSL} compares the predicted upper bound on the collapse rate $\\lambda_c$ from our protocol to those of existing experiments,\ntogether with Adler's and Bassi {\\it et al.}'s lower bounds, and their uncertainties.\nExisting upper bounds  are provided by the motional stability of gravitational wave interferometers  \\cite{Carlesso_Gravitational_2016,Helou_LISA_2017,Nobakht_Unraveling_2018} (yellow region); the thermalisation of ultracold cantilevers \\cite{Vinante_Cantilever_2015,Vinante_Improved_2017} (blue outlined region); Kapitza-Dirac-Talbot-Lau (KDTL)-Interferometry \\cite{Fein_25kDa_2019,Nimmrichter_Matterwave_2011,Toros_Colored_2017,Feldmann_Parameter_2012} (dashed black); spontaneous X-ray emission from Germanium \\cite{Curceanu_Xray_2016,Curceanu_LNGS_2020,Fu_Radiation_1997} (dotted black) and the observed temperature of neutron stars \\cite{Adler_Fermi_2019,Stace_Neutron_2019} (dashed black), which are valid however only for white noise CSL; and cold atom interference \\cite{Toros_Colored_2017} (gray region), though we note the controversy \\cite{StamperKurn_Comment_2016} on the actual size of the superposition reported in \\cite{Kovachy_Superposition_2015}.\n\nThe red shaded region in Fig. \\ref{figCSL} could be tested by our protocol\nas discussed above. In the case of white-noise collapse, the protocol could for the first time fully test Bassi {\\it et al.}'s proposal. \nIf collapse noise has one of the proposed physical origins \\cite{Bassi_Breaking_2010,Smirne_Dissipative_2014,Smirne_Dissipative_2015}, the envisaged protocol would also for the first time probe Adler's prediction, which is in this case not tested by X-ray emission (black dotted line in Fig. \\ref{figCSL}).\nThe resonance frequency is close to the frequency range in which a drastic frequency-dependent reduction of the collapse noise stemming from a physical origin is expected \\cite{Bassi_Breaking_2010}. \nTo identify the physical origin of collapse, and to differentiate between collapse-induced signal and technical noise, we suggest employing a number of mechanical resonators of slightly different frequencies, or one frequency-tunable resonator \\cite{Pfeifer_tunable_2016}, at frequencies around $\\Omega\/2\\pi\\sim10$~GHz, such as reported in \\cite{Ren_Crystal_2019}.\n\nWe also evaluate the capability of the protocol to constrain parameters in gravitational collapse models.\nWhile for the Di\\'osi-Penrose model \\cite{Diosi_Gravitation_1984,Diosi_Models_1989,Penrose_Reduction_1996} we find that it cannot exceed existing bounds, for the classical channel gravity model in a typical parameter range \\cite{Altamirano__Pairwise_2018,Kafri_Classical_2014,Khosla_Classical_2018} we predict about a one order-of-magnitude stronger bound than previously achieved \\cite{tbp}  (see Supplemental Material \\cite{Supp}).\n\n\nIn summary, we have proposed the concept of testing quantum linearity using high-frequency mechanical oscillators. \nThis offers the advantages of thermal noise suppression to well below expected collapse signatures, and the potential for identification of the physical origin of collapse. \nAs a possible implementation we suggest a protocol based on a dual-cavity high-frequency optomechanical device passively ground-state-cooled and operating in the strong coupling regime. This design, combined with nonlinear optical techniques to reduce dark counts, is predicted to allow measurement of the minuscule phonon-flux generated by collapse-induced heating.  While challenging, the protocol has the potential to conclusively test CSL, and thus whether collapse mechanisms can be invoked to resolve the measurement paradox. Unlike previous proposals and experiments, it is designed to allow for identification of the physical noise field underlying CSL, and for differentiation between excess technical noise and signatures of collapse.\n      \n\n\n\\section*{Methods}  \n\n\\subsection*{Born-Markov master equation}  \n\nTo model the dynamics of the three-mode optomechanical system we employ the Born-Markov framework for open quantum systems \\cite{Nunnenkamp_Single_2011,Liu_Strong_2013,BasiriEsfahani_control_2016}.\n The interaction picture Hamiltonian for our system is \\cite{BasiriEsfahani_Phonon_2012,BasiriEsfahani_control_2016,Chang_Array_2011}\n\\begin{equation}\nH_{\\rm int}=\\hbar g_0(b^{\\dagger}e^{-i\\Omega t}+be^{i\\Omega t})(a_p^{\\dagger}a_se^{i\\Omega t}+a_pa_s^{\\dagger}e^{-i\\Omega t})+\\hbar\\sqrt{\\kappa_{p,\\rm ex}}(a_p^{\\dagger}a_{\\rm in}+a_{\\rm in}^{\\dagger}a_p),\n\\end{equation}\nwhere $b$, $a_p$ and $a_s$ are annihilation operators for the mechanical mode and optical modes, respectively, and $a_{\\rm in}$ is the coherent input field. The first term describes the mechanically mediated cross-coupling of the optical modes, while the second term describes the coherent excitation \\cite{Aspelmeyer_Review_2014}. In the parameter regime of this work where $g_0\\ll \\Omega$ and $\\Gamma \\ll \\kappa_p,\\kappa_s,g_0$, the dynamics of the system can be described by the\nBorn-Markov master equation as \\cite{Nunnenkamp_Single_2011,Liu_Strong_2013}\n\\begin{equation}\n\\frac{d\\hat \\rho}{dt}=-\\frac{i}{\\hbar}\\big[H_{\\rm int},\\hat \\rho\\big] + \\kappa_p \\mathcal{D}[a_p]\\hat \\rho + \\kappa_s \\mathcal{D}[a_s]\\hat \\rho + \\Gamma(1+\\bar{n}_{\\rm th}) \\mathcal{D}[b]\\hat \\rho + (\\Gamma \\bar{n}_{\\rm th} + \\dot n_c)\\mathcal{D}[b^{\\dagger}]\\hat \\rho,\n\\label{eq: master_equation}\n\\end{equation}\nwhere $\\hat \\rho$ is the density matrix, $\\bar n_{\\rm th}$ the mechanical mean thermal occupancy, \nand $\\mathcal{D}$ the dissipating superoperator, $\\mathcal{D}[A]\\hat \\rho=A\\hat \\rho A^{\\dagger}-\\frac{1}{2}(A^{\\dagger}A\\hat\\rho+\\hat\\rho A^{\\dagger}A)$. \nA weak phonon flux due to spontaneous collapse is described by $\\dot n_c=\\lambda_c D$, independent of its origin.\nThis allows us to model the conversion of a signal phonon to a signal photon, as well as creation of noise phonons introduced by measurement (see Supplemental Material \\cite{Supp}).\n\n\n\n\n\\subsection*{Negligible sources of noise}\n\n{\\it Probe photons leaking though the system.}\n Probe photons passing directly from the laser through the optomechanical system, without a scattering event, could in principle imitate a signal, obfuscating collapse signatures.\nWe find that, using a standard laser stabilisation reference cavity as a filter \\cite{Kessler_Laser_2011}, this noise can be suppressed well below both Adler's and Bassi {\\it et al.}'s lower bounds.\nSimilarly, if a photon is created in an optomechanical conversion process and subsequently outcoupled into the signal mode, due to energy conservation it either remains at frequency $\\omega_p$, or has a frequency reduced by integer multiples $n$ of the mechanical resonance frequency, $\\omega_p-n\\Omega$.  In both cases, this noise is doubly suppressed --- first by the suppression of the direct occupation pathway, and second by the filter. \nThis makes probe photons that leak through the system a negligible source of noise (see Supplemental Material \\cite{Supp} for details).\n\n{\\it Optical absorption heating.}\nTo estimate the phonon occupancy due to optical absorption,\nwe use the model for absorption heating in silicon optomechanical crystals outlined in \\cite{Meenehan_millikelvin_2014,Ren_Crystal_2019}.\nPhotoabsorption creates an electronic excitation, which is then transferred to terahertz-frequency phonons. While radiating from the resonator to the environment with a geometry- and material-dependent rate $\\gamma_{\\rm THz}$, they also couple to lower energy phonons with a generally longer timescale, potentially exciting the mechanical resonator. \nIn \\cite{Meenehan_millikelvin_2014,Ren_Crystal_2019}, the average phonon number $\\bar n_b$ is related to the average intracavity photon number $\\bar n_{\\rm cav}$ via $\\bar n_b\\propto \\bar n_{\\rm cav}^{1\/3}$. \nWe expect this relationship to break down when the time between photoabsorption events is long enough for the generated heat to fully dissipate, $\\bar n_{\\rm cav} \\cdot \\gamma_{\\rm THz}\/\\kappa \\lesssim1$, where $\\kappa$ is the loaded optical decay rate, as any discrete photon absorption event is expected to create a fixed amount of heat.\nIn this case $\\bar n_{\\rm cav}$ determines the frequency of these events, but not the magnitude of dissipated heat.\nWe compute the average phonon number excited by of one probe photon in the mechanical resonator, $\\bar n_{\\gamma}$, due to photoabsorption for time $t_{\\rm abs}$, at which the oscillator is in thermal equilibrium with the material, but not yet with the environment, $\\Gamma^{-1}\\gg t_{\\rm abs} \\gg \\gamma_{\\rm THz}^{-1}$.\nFor the proposed setup we find $p_{\\rm abs}(t\\rightarrow\\infty)=6.1\\cdot 10^{-12}$ and  $R_{\\rm abs}=1.4\\cdot10^{-14}$~s$^{-1}$ (see Supplemental Material \\cite{Supp} for calculation details).\n\n\\subsection*{Measurement-induced phonons.}\nA probe photon can create a noise phonon by coupling directly into the signal mode instead of the probe mode (Fig. \\ref{Fig_Energy} (a)). \nThis process is suppressed by the square of the resolved-sideband ratio $\\Omega\/\\kappa_s$.\nThe corresponding occupancy is calculated by numerically solving the Born-Markov master equation (see Methods and Supplemental Material \\cite{Supp}) and shown by the dashed blue line in Fig. \\ref{figSOM} (a).\n\nA photon that does enter the probe mode, corresponding to the state $\\ket{ n_b  n_p  n_s}=\\ket{010}$, can introduce noise by undergoing the non-resonant phonon-creating transition $\\ket{010}\\rightarrow\\ket{101}$ (see Fig. \\ref{Fig_Energy} (b)).\nThe resulting state can also resonantly transition to a two-phonon state, $\\ket{101}\\rightarrow\\ket{210}$, as shown in Fig. \\ref{Fig_Energy} (c).\nSimilarly to above, noise phonons from this process are suppressed by $\\sim(\\Omega\/\\kappa_p)^2$.\nPredicted phonon occupancies are shown in Fig. \\ref{figSOM} (a) and (b).\n\n\\subsection*{Zero-point quadratic coupling rate required to fully test Bassi et al.'s lower bound}\n\nThe linearised quadratic part of the optomechanical interaction Hamiltonian is $H_{\\rm int}^{(2)}=\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}(a^{\\dagger}+a)(2b^{\\dagger}b+b^{\\dagger}b^{\\dagger}+bb)$ \\cite{Hauer_nondemolition_2018}. \nThe term proportional to $b^{\\dagger}b$ yields a per-phonon optical resonance frequency shift of $2\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}$, which is the signature of a collapse-induced phonon.\nA random fluctuation $\\delta \\omega$ in the frequency of the probe can imitate a signal if it is larger or equal to this frequency shift, and sustained over a time comparable to the phonon lifetime $\\Gamma^{-1}$. \nFor shot noise limited probe, the probability of a fluctuation larger than $2\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}$ is given by an error function of a Gaussian distribution\n\\begin{equation}\np(\\delta \\omega)=\\bigg(\\frac{1}{\\sqrt{2\\pi \\sigma^2}}\\int^\\infty_{\\delta \\omega}e^{-\\omega^2\/2 \\sigma^2}d\\omega\\bigg),\n\\label{eq:Gauss}\n\\end{equation}\nwith standard deviation of\n$\\sigma\\approx\\kappa\/\\sqrt{N}$, where $N$ is\nthe number of photons interacting with a phonon within the mechanical lifetime $\\Gamma^{-1}$, and is related to the average intracavity photon number via $N=\\bar n_{\\rm cav} \\cdot \\kappa\/\\Gamma$ for a continuous measurement. \nThe rate of spurious signals due to such fluctuations is $R_{\\delta\\omega}=\\Gamma p(\\delta\\omega)$.\nTo test a collapse-induced phonon flux of $\\dot n_{\\rm c}$, we require $\\dot n_{\\rm c}\\geq R_{\\rm \\delta\\omega}$.\nFrom Eq. \\ref{eq:Gauss} we find that, to fully exclude Bassi {\\it et. al's} lower bound using the photonic-phononic crystal considered in the protocol above \\cite{MacCabe_Ultralong_2019}, requires $\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}\\gtrsim 3.5\\sigma$.\nAssuming an average intracavity photon number of $\\bar n_{\\rm cav}=10^2$, with $\\kappa\/2\\pi=575$~MHz \\cite{MacCabe_Ultralong_2019}, leads to the condition $g_0^{(2)}\\gtrsim3.5\\sqrt{\\kappa\\Gamma}\\bar n_{\\rm cav}^{-3\/2}\\gtrsim 2\\pi\\cdot 28$~Hz.\n\nThe term proportional to $b^{\\dagger}b^{\\dagger}a$ converts a probe photon to two phonons, potentially imitating a collapse-signature. \nHowever, the shift induced by two phonons is $4\\bar n_{\\rm cav}^{1\/2}g_0^{(2)}$ and can be clearly distinguished from the collapse-induced shift caused by one phonon.\nTherefore, two-phonon creation can only imitate a collapse signal if it coincides with a frequency fluctuation of the probe mode $-\\delta \\omega\\geq2 n_{\\rm cav}^{1\/2}g_0^{(2)}$, sustained at least over the two-phonon lifetime $(\\sqrt2\\Gamma)^{-1}$. The low probability of such a fluctuation, together with suppression on the order of $(2\\Omega\/\\kappa)^2$ due to the non-resonant nature of the interaction, make this source of noise negligible.\n\n\n\\section*{Data availability}\nThe data that support the findings of this study are available within the paper and its Supplemental Material.\nCodes for the numerical simulations are available on request from the corresponding author.\n\n\\section*{Author contributions}\nWPB provided overall leadership for the project. SF and WPB conceptualized the idea. SF, SB, MZ and KK developed the theoretical model. SF and SB performed numerical simulations.  All co-authors contributed in the development of the manuscript which was drafted by SF and WPB.\n\n\n\n\\section*{Acknowledgements}\nThe authors thank Gerard Milburn, Nathan McMahon and James Bennett for helpful discussions, and Nicolas Mauranyapin for preparing Figure 1. We also acknowledge funding by Australian Research Council grants (EQUS, CE170100009, DE180101443) and the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 663830.\n\n\n\n\\section*{Additional information}\n\n\n{\\bf Competing interests:} The Authors declare no competing interests.\n\n\n\n\\newpage\n\\footnotesize \\renewcommand{\\refname}{\\vspace*{-30pt}}  \n\\bibliographystyle{apsrev4-2} \n\n\\section*{Supplementary Note 1. Signal rates from spontaneous collapse}\n\\label{sec:collapserates}\nWhile there exists a plethora of collapse models \\cite{Bassi_Review_2012,Bassi_Gravitational_2017}, they can all be formulated in terms of a stochastic nonlinear modification to Schr\\\"odinger's equation of the general form \\cite{Ferialdi_Dissipative_2012,Adler_stochastic_2001}: \n\\begin{equation}\n\\frac{d\\Psi}{dt} = \\bigg[-\\frac{i}{\\hbar}H + \\sqrt{\\lambda}A\\xi(t)-\\frac{\\lambda}{2}(A^{\\dagger}A+A^2) \\bigg]\\Psi,\n\\label{stochastic} \n\\end{equation}\nwhere the operator $H$ is related to the standard Hamiltonian of the system, $\\xi(t)$ is defined in terms of an increment $dW(t)$ of a stochastic Wiener process $W(t)$ through\n$\\xi(t) dt = dW(t)$; \n$\\lambda$ is a coupling that sets the strength of the collapse and $A$ is the reduction operator, which is specific to the particular realization of the collapse mechanism. The requirements of norm-preservation of the state evolution and {of} no-superluminal signalling make these collapse models the only mathematically consistent, phenomenological modifications against which quantum theory can be tested in this context \\cite{Ferialdi_Dissipative_2012}.\n\n\nIn the following we give expressions for the  \ndecoherence rates due to two commonly studied collapse mechanisms. Collapse models \ndiffer in the properties and nature of the mechanisms purported to cause the collapse and can thus be classified according to the basis in which decoherence occurs, the mathematical properties of the stochastic mechanism and whether the mechanism has a quantum mechanical origin, or is due to a modification to Sch\\\"odinger's equation from a deeper-level theory. In the models discussed here decoherence acts in the position basis with Gaussian correlations in space, see \\cite{Bassi_Review_2012} and references therein. \nA generic expression for the diffusion term $\\sqrt{\\lambda}A\\xi(t)dt$ in these models is $\\sqrt{\\lambda}\\int d\\vec x \\left(\\rho(\\vec x)-\\langle{\\rho(x)}\\rangle\\right) dW(\\vec x,t)$ with  $\\rho(\\vec x)$ being the mass density, $W(\\vec x,t)$ the ensemble of Wiener processes (for different points $\\vec x$ in space) which can in general be correlated. The noise correlation length $r_c$ and the coupling rate $\\lambda$ to the noise field  are the model parameters. The mass density typically takes the form $\\rho(\\vec x) = \\sum_jm_j\\int dy G(y-x)\\psi_j^\\dagger(y)\\psi_j(y)$ with $G(y)$ a  ``smearing function'' and $\\psi_j^\\dagger(y), \\psi_j(y)$ creation and annihilation operators of the particle with mass $m_j$. \nThe stochastic Sch\\\"odinger equation Eq. \\eqref{stochastic} and the above form of the diffusion term result in a master equation for the density operator $\\hat\\rho$ where the off-diagonal elements $\\vec x', \\vec x''$ evolve as $\\frac{\\partial}{\\partial t}\\bra{\\vec x'} \\hat\\rho \\ket{\\vec x''} = -\\tilde{D}( x',  x'')\\bra{\\vec x'} \\hat\\rho \\ket{\\vec x''}$ where \n $\\tilde{D}(\\vec x', \\vec x'') = \\sum_{i,j}\\frac{\\lambda}{2}\\left[G(\\vec x_i' -\\vec x_j')+G(\\vec x_i'' -\\vec x_j'')-2G(\\vec x_i' - \\vec x_j'')\\right]$\nand describes the rate at which the  spatial coherence is suppressed. Taking as an example the typically used Gaussian smearing function of the mass density $G(\\vec x)=(4\\pi r^2_c)^{-3\/2}e^{-\\vec x^2\/4r_c^2}$, the decoherence rate reads $\\tilde{D}(\\vec x', \\vec x'')=\\lambda\\cdot(4\\pi r^2_c)^{-3\/2}\\cdot(1-e^{-|\\vec x'-\\vec x''|^2\/4 r^2_c})$. We note that this is exactly the one-particle decoherence rate of the Ghirardi-Rimini-Weber (GRW) \\cite{Ghirardi_Unified_1986} and in the Continuous Spontaneous Localization (CSL) models \\cite{Pearle_Localization_1989,Ghirardi_Markov_1990}. Furthermore, for small superpositions sizes (as compared to the correlation length $r_c$) $|\\vec x'-\\vec x''|\\ll r_c$ the decoherence rate becomes approximately $\\tilde{D}(\\vec x', \\vec x'')\\propto |\\vec x'-\\vec x''|^2\/4 r^2_c$. In this limit one obtains as expected that the collapse rate increases with the size of superposition and the corresponding Lindblad term in the master equation takes a simple form $\\propto - [\\vec x, [\\vec x, \\hat\\rho]]$. \n\nPosition-basis decoherence yields localization of macroscopic superposition states. The same Lindblad term $-\\tilde{D}[\\vec x,[\\vec x,  \\hat\\rho]]$ results in momentum diffusion of the system (this can be seen as a direct consequence of the Heisenberg Uncertainty \nPrinciple and the fact that the above decoherence process can equivalently be seen as a reduction of the position uncertainty of the system). In turn, the momentum diffusion can be interpreted as heating with the collapse-induced heating rate being directly proportional to the decoherence rate $\\tilde{D}$. \nThis allows testing the collapse models also by monitoring spontaneous heating of even isolated systems.\nFor a quantum oscillator the heating leads to an increase of the phonon number expectation value. \nThe equivalent Lindblad term reads $-\\tilde{D}(b)[ b,[ b, \\hat\\rho]]$, with $\\tilde{D}(b)=x_0^2\\tilde{D}(\\vec x', \\vec x'')$, where $x_0$ is the zero-point motion (or equivalently, for spatial superpositions of massive particles, the superposition size $\\Delta x$). \n\n\nThe most studied collapse model is the\nCSL model \\cite{Pearle_Localization_1989,Ghirardi_Markov_1990}.\nIt considers second-quantized (albeit non-relativistic) indistinguishable particles where the collapse occurs in the particle number (Fock) basis. The key consequence of the indistinguishability of the particles is that for multi-particle systems the model predicts quadratic dependence of the decoherence rate on the number of particles that are within the cutoff distance $r_c$. For a comparison, the GRW model postulates discrete in time collapse events of the wave-function of individual (and also distinguishable) particles in the positions basis which yields linear dependence of the decoherence rate on the particle number \\cite{Ghirardi_Unified_1986}.\nThe stochastic process in the CSL model is introduced in terms of a time-dependent Wiener noise at each point in space, coupling to mass-density smeared over some length scale $r_c$. As CSL is linear in the coupling rate $\\lambda_c$ of matter to the collapse noise field, we define a dimensionless decoherence operator $D$ by $\\tilde{D}_{\\rm csl}(b)=\\lambda_c D$.\nFor an oscillator with mass density $\\rho(\\vec x)$ and direction of motion along the $z$-axis, the decoherence operator in the CSL-model reads \\cite{Nimmrichter_Optomechanical_2014,Vinante_Cantilever_2015}\n\n\\begin{equation}\nD_{\\rm csl}=\\frac{(4\\pi)^{3\/2} r_c^3 x_0^2}{u^2}\\int \\frac{d^3\\vec{k}}{(2\\pi)^3}k^2_z e^{-k^2r_c^2}\\vert\\tilde{\\rho}(\\vec{k})\\vert^2,\n\\end{equation}\nwhere $\\tilde{\\rho}(\\vec{k})=\\int d^3r\\rho(\\vec{r})e^{-i\\vec{k}\\cdot\\vec{x}}$ is the Fourier transform of the mass density and $u=1.66\\cdot 10^{-27}$~kg is the atomic mass unit\n\nCSL in its original form predicts infinite energy increase as time goes to infinity \\cite{Ghirardi_Markov_1990,Bassi_Dynamical_2003,Bassi_energy_2005}. This problem can be solved by postulating a finite temperature of the noise \\cite{Smirne_Dissipative_2015}. Furthermore, the model assumes a white noise spectrum, which cannot be identified with any physical origin \\cite{Ferialdi_Dissipative_2012}. In order to generalise the model to become compatible with relativity and with observations, the noise field should have a more general, i.e.~non-white spectrum. In such a case, however,  the model becomes non-Markovian~\\cite{Adler_Nonwhite_2007, Adler_Nonwhite_2008}. While such models are difficult to study in full generality, it has been demonstrated~\\cite{Adler_Nonwhite_2007, Adler_Nonwhite_2008} that to lowest order in $\\lambda$ the qualitative features  of the model are the same as for the white-noise model.  A common assumption that helps to lift ambiguities in defining the model is that the  field underlying the collapse process has a cosmological origin. This allows one to introduce a high-frequency cutoff of the order of $\\Omega_{\\rm csl}\/2\\pi\\approx 10^{10}-10^{11}$ Hz \\cite{Bassi_Breaking_2010,Smirne_Dissipative_2015} which ensure the collapse rate is essentially as in a white-noise CSL model, but which changes the relaxation behaviour: in the coloured-noise model the system in the limit of long times thermalises to the temperature of the noise field, while in the white-noise model the system energy keeps growing with time \\cite{Pearle_Collapse_1996}.\n\n$D$ can be calculated analytically for simple geometries of composite test-systems \\cite{Nimmrichter_Optomechanical_2014,Vinante_Cantilever_2015}.\nFor a sphere of radius $R$  the decoherence operator reads\n\n\\begin{equation}\nD_{\\rm sphere}=\\frac{14\\pi^2 R^2 r_c^2\\rho^2 x_0^2}{3 u^2} \\big(1-\\frac{2r_c^2}{R^2}+e^{-R^2\/r_c^2}\\big(1+\\frac{2r_c^2}{R^2}\\big)\\big),\n\\end{equation} \nand for the case of a cuboid with constant density $\\rho$ and sidelengths $L_1$, $L_2$ and $L_3$, where $L_3$ is the direction of motion \n\\begin{equation}\nD_{\\rm cuboid}= \\frac{32 r_c^4\\rho^2 x_0^2}{u^2}\\big(1-e^{-\\frac{L_3^2}{4r_c^2}}\\big) \\big(e^{-\\frac{L_2^2}{4r_c^2}}-\\frac{\\sqrt{\\pi}L_2}{2r_c}{\\rm Erf}(\\frac{L_2}{2r_c}) -1\\big)  \\big(e^{-\\frac{L_1^2}{4r_c^2}}-\\frac{\\sqrt{\\pi}L_1}{2r_c}{\\rm Erf}(\\frac{L_1}{2r_c}) -1\\big).\n\\label{eq:dcuboid}\n\\end{equation}\nFor our proposed experiment, we estimate length, height and width of the photonic crystal beam to $L_1,L_2,L_3=1.21,0.22,0.22~\\mu$m, respectively, reproducing the effective motional mass of the relevant mechanical mode of  136~fg \\cite{MacCabe_Ultralong_2019}, where we used the density of silicon, $\\rho=2.33\\cdot10^{3}$~kg\/m$^3$.  We choose the cuboid approximation for the modeshape following \\cite{Vinante_Cantilever_2015}. From Eq. \\eqref{eq:dcuboid} we find $D=5.1\\cdot10^{5}$.  \n\n\nFor the models where the collapse is assumed to have a gravitational origin, two main types of theories can be distinguished: where decoherence arises due to an intrinsic uncertainty in the local value of the gravitational field \\cite{Diosi_Gravitation_1984,Diosi_Models_1989} or, equivalently, gravitational self-interaction \\cite{Penrose_Reduction_1996}; and where decoherence is a consequence of the assumption that gravity is fundamentally a classical channel \\cite{Kafri_Classical_2014,Khosla_Classical_2018,Altamirano_Pairwise_2018}. In both cases for small superposition size, the resulting effect has the same general form as the corresponding regime of the CSL and GRW models: For an oscillator it is proportional to the square of the zero point motion, for spatial superpositions of massive particles the effect is proportional the square of the superposition size.  Gravity-based decoherence has also been described within the framework of CSL in \\cite{Ghirardi_gravity_1990}.\n\nIn the Di\\'osi-Penrose (DP) model the decoherence rate is quantified by gravitational potential evaluated between superposed amplitudes of the system: \n$\\frac{G}{2\\hbar}[U(XX)+U(YY)-2U(XY)]$, where $U(XY)=-G\\int d^3r\\int d^3r'\\frac{\\rho_X(\\vec x)\\rho_Y(\\vec x')}{|\\vec x-\\vec x'|}$ is the gravitational interaction between mass-densities $\\rho_X, \\rho_Y$ associated with the superposed configurations $X, Y$ \\cite{Diosi_bulk_2014}, with $G=6.67\\cdot10^{-11}$~m$^3$kg$^{-1}$s$^{-2}$ being Newton's gravitational constant. For point particles the above expression gives divergent decoherence rate and thus a short-distance cutoff $r_{\\rm DP}$ is needed. The decoherence operator reads \\cite{Nimmrichter_Optomechanical_2014}\n\n\\begin{equation}\n\\tilde{D}_{\\rm DP}=\\frac{x_0^2 G}{6\\sqrt{\\pi}\\hbar}\\big(\\frac{a}{r_{\\rm DP}}\\big)m\\rho(\\vec x),\n\\label{eq:DPheating}\n\\end{equation}\nwhere $a$ is the lattice constant of the composite object.\nComparing the heating rate expected from Eq. \\eqref{eq:DPheating} to measurement-induced spurious phonons, which constitute the strongest noise source in our proposed experiment (see main text), we find that short-distance cutoffs up to $r_{\\rm DP}\\approx 3.9$~fm can be excluded. For a discussion of experimental tests of classical channel gravity \\cite{Kafri_Classical_2014,Khosla_Classical_2018,Altamirano_Pairwise_2018}, see \\cite{tbp}.\n\n\n\n\n\n\n\n\\section*{Supplementary Note 2. Calculation details: efficiency and noise levels of the optomechanical system}\n\\label{sec:optomechanics}\nThe optomechanical system is probed by a strongly attenuated coherent source, such as described in \\cite{Kessler_Laser_2011}, which can be stabilized to a sub-Hz linewidth $\\kappa_L$. Because $\\kappa_L$ is much smaller than $\\kappa_s$, $\\kappa_p$, and $\\kappa_f$, which are the linewidths of the {\\it probe mode}, the {\\it signal mode}, and the filter cavity, respectively, the field $a_{\\rm in}$ of the incoming probe laser is well approximated with a $\\delta$ - function: $a_{\\rm in}(\\omega)=a_{\\rm in}\\delta(\\omega_L)$, where $\\omega_{\\rm L}$ is the laser frequency. We set $\\omega_L=\\omega_p$ to maximize coupling into the probe mode $a_p$. \nThe mechanical decay rate $\\Gamma$, as well as the collapse rate are slow compared to timescales of the optomechanical interaction:  $D,\\Gamma \\ll g_0,\\kappa_{p\/s}$, where $g_0$ is the single photon optomechanical coupling rate \\cite{Aspelmeyer_Review_2014}. \nIn this limit, the conversion dynamics can be modelled by the reduced master equation:\n\\begin{equation}\n\\frac{d\\hat \\rho}{dt}=-\\frac{i}{\\hbar}\\big[H_{\\rm int},\\hat\\rho\\big] +\\kappa_p \\mathcal{D}[a_p]\\hat\\rho + \\kappa_s \\mathcal{D}[a_s]\\hat\\rho,\n\\label{eq:reducedME}\n\\end{equation}\nwhere $\\hat \\rho$ is the density matrix, the operators $a_p$ and $a_s$ correspond to annihilation operators for the optical probe- and signal-modes, respectively, and $H_{\\rm int}$ is the interaction Hamiltonian given in the main text.\n\n{\\it Resonant anti-Stokes scattering.} \nIf a phonon is introduced into the ground-state cooled oscillator, and a probe pulse is incident within the lifetime of the mechanical excitation, the system is in the initial state $\\ket{ n_b n_p  n_s}=\\ket{110}$. \nThe optomechanical conversion efficiency $\\eta_{\\rm om}$ is the probability of one photon in the probe mode $a_p$ scattering with one phonon in the mechanical resonator, creating a photon in the signal mode $a_s$ ($\\ket{110}\\rightarrow\\ket{001}$) via a anti-Stokes Raman process, and this photon being outcoupled to create a signal photon at frequency $\\omega_s$. \n$\\eta_{\\rm om}$ is obtained by numerically solving Eq. (\\ref{eq:reducedME}) and time-integrating over the emission from the signal mode, $\\eta_{\\rm om}=\\kappa_{s,\\rm ex}\\int_0^{\\infty}\\langle a_s^{\\dagger}(t)a_s(t)\\rangle dt$, where $\\kappa_{s,\\rm ex}=\\kappa_s-\\kappa_{s,0}$ is the external decay rate of the signal mode due to coupling, with $\\kappa_{s,0}$ and $\\kappa_s$ its intrinsic and loaded decay rates, respectively.\nSupplementary Figure \\ref{fig_hOM} (a) shows $\\eta_{\\rm om}$ as a function of the effective coupling strength $g_0\/\\kappa_p$ for critical coupling of the probe mode $\\kappa_{p,0}=\\kappa_{p,\\rm ex}$ and equal intrinsic couplings $\\kappa_{s,0}=\\kappa_{p,0}$, for critically ($\\kappa_{s,0}=\\kappa_{s,\\rm ex}$) and overcoupled ($\\kappa_{s,\\rm ex}\/\\kappa_p=2$ and $4$) signal mode.\nThe efficiency $\\eta_{\\rm om}$ is sensitive to the ratio $g_0\/\\kappa_p$, as $g_0$ sets the rate for anti-Stokes scattering, and $\\kappa_p$ sets the rate for the competing optical decay.\nFurthermore, it depends on the ratio $\\kappa_{s,\\rm ex}\/\\kappa_p$ as higher values favour outcoupling of the signal photon.\nFor our proposed experiment, $g_0=\\kappa_p$ and $\\kappa_{s,\\rm ex}\/\\kappa_p=1.8$.\n\n\n \n\n \\begin{figure}[h!]\n\n  \\includegraphics[width=\\textwidth]{Fig_Supp_hOM5.pdf}\n\\caption{Efficiencies of optomechanical transitions.\na) Cumulative probability for the output of a signal photon due to anti-Stokes scattering from the initial state $\\ket{ n_bn_p n_s}=\\ket{110}$ as a function of $g_0\/\\kappa_p$, for $\\kappa_{s,\\rm ext}\/\\kappa_p=1,2,$ and $4$.\nb) Efficiency of spurious Stokes scattering, corresponding to the phonon number occupancy expectation value $\\eta_{\\rm Stokes}=\\langle b^{\\dagger}(t)b(t)\\rangle$ after an incident photon in the signal mode, for times $\\Gamma^{-1}\\gg t \\gg \\kappa_p^{-1},\\kappa_s^{-1}$.\nc) Cumulative probability for the output of a signal photon due to anti-Stokes scattering from the two-phonon state $\\ket{n_b n_p n_s}=\\ket{210}$.\n}\n\n\\label{fig_hOM}\n\\end{figure}\n\n\nThe probability of a phonon in the mechanical resonator translating to a coincidence count, imitating a signal, is given by $\\eta=\\eta_p\\eta_{\\rm om}\\eta_{\\rm f}\\eta_{\\chi}\\eta_d=1.1\\cdot10^{-3}$, where $\\eta_p$ is the probability of a photon entering the probe mode during the mechanical excitation lifetime, $\\eta_f=0.56$ is the transduction efficiency through the filter for a signal photon at frequency $\\omega_s$, $\\eta_{\\chi}=0.95$ and $\\eta_d=0.64$ are downconversion and coincidence detection efficiencies, respectively. These efficiencies are analysed in more detail in the following paragraphs. As the rate of phonons created in the resonator due to spontaneous collapse is given by the collapse rate $\\lambda_cD$, the rate of registered collapse signatures is $R_c=\\lambda_cD\\eta=5.5\\cdot 10^{2}\\lambda_c$.\n\n{\\it Probe field occupancy.}\nThe average number of photons encountered by one phonon is given by\n\\begin{equation}\n\\bar n_{\\rm ph}=\\int_0^\\infty e^{-\\Gamma t}\\kappa_p\\bar n_p dt,\n\\label{eq:etap}\n\\end{equation}\nwhere $\\bar n_p$ is the average photon occupancy of the probe mode and $e^{-\\Gamma t}$ is the phonon occupancy at time $t$ after one phonon is created in the mechanical resonator at time $t=0$. \nIn the limit $\\bar n_p \\ll \\Gamma\/\\kappa_p$, Eq. (\\ref{eq:etap}) also quantifies the probability $\\eta_p$ of a phonon in the mechanical resonator encountering a probe photon, $\\eta_p=\\bar n_{\\rm ph}$. \nFurthermore, in this limit, $\\eta_p$ asymptotes to $\\eta_p=\\bar n_p\\kappa_p\/\\Gamma$. \nIn our protocol, the probe laser power is adjusted so that $\\eta_p=0.01$, corresponding to an average of 0.01 photons per mechanical oscillator lifetime.\nIn the steady state, the average intracavity photon number is given by \n$\\bar n_p=4\\kappa_{p,\\rm ex}\\bar n_{\\rm in}\/\\kappa_p^2$ \\cite{Aspelmeyer_Review_2014},\nand hence, to achieve a given $\\eta_p$ the input field occupancy is adjusted to $\\bar n_{\\rm in}(\\eta_p)=(\\eta_p\\Gamma\\kappa_p)\/(4\\kappa_{p,\\rm ex})$. \nBecause the signal is proportional to $\\eta_p$, and the noise background from measurement-induced noise phonons is proportional to $\\eta_p^2$, the input field occupancy gives a handle to lower the noise at the cost of longer measurement time, or vice versa.\n\n{\\it Transmission through the filter cavity.} \nThe efficiency  $\\eta_f$ of the signal passing through the filter, assuming equal input- and output coupling strengths $\\kappa_{f,\\rm ex}$, is given by \\cite{Hecht_Optics_2002}\n\\begin{equation}\n\\eta_f=\\big(1-\\frac{\\kappa_{f,0}}{\\kappa_f}\\big)^2,\n\\end{equation}\nwhere $\\kappa_{f,0}$ is the intrinsic filter linewidth and $\\kappa_f=\\kappa_{f,0}+\\kappa_{f,\\rm in}+\\kappa_{f,\\rm out}$ is the loaded filter linewidth, with $\\kappa_{f,\\rm in}$ and $\\kappa_{f,\\rm out}$ the in- and output coupling, respectively.\nUsing a typical a laser stabilisation filter cavity \\cite{Kessler_Laser_2011},  $\\kappa_{f,0}=30$~kHz, and overcoupling both at the input and output $\\kappa_{f,\\rm in}=\\kappa_{f,\\rm out}=1.5\\cdot\\kappa_{f,0}$, a transmission efficiency of $\\eta_f=0.56$ is achieved.\n\n{\\it Nonlinear downconversion.} \nAfter separation from probe light at frequency $\\omega_p$, signal photons at frequency $\\omega_s$ are downconverted to photon pairs in a nonlinear medium.\nIn order to minimize detector dark counts, we propose a nonlinear conversion process to convert signal photons to pairs. A bright classical pump beam with electric field amplitude $E$ is coupled into a medium exhibiting a third order $\\chi^{(3)}$ optical nonlinearity. This yields an effective second order interaction \n$H_{\\chi,\\rm eff}=\\gamma E a_{f,\\rm out}d_1^{\\dagger}d_2^{\\dagger}+\\gamma^* E a_{f,\\rm out}^{\\dagger}d_1d_2$, where $\\gamma$ is the nonlinear coupling strength \\cite{Langford_Conversion_2011}, $a_{f,\\rm out}$ is the mode of the signal transmitted through cavity and filter, and $d_{1\/2}$ are the modes coupled to the detectors. \nGiven the input state $\\ket{n_{f,\\rm out}n_{d1}n_{d2}}=\\ket{100}$, the time evolution in the nonlinear medium is \\cite{Langford_Conversion_2011}\n\\begin{equation}\n\\ket{\\Psi(t)}=\\cos{(\\abs{\\gamma}t\/\\hbar)}\\ket{100}+i \\sin{(\\abs{\\gamma}t\/\\hbar)}\\ket{011}.\n\\end{equation}\n \nBy setting the length of the nonlinear medium to $L=\\frac{1}{2}\\pi\\hbar c_n\\abs{\\gamma}^{-1}$, where $c_n$ is the speed of sound in the medium, the output state is $\\ket{\\Psi(t_{\\rm final})}=\\ket{011}$, corresponding to a photon in each detector mode $d_{1\/2}$.\nIt has been shown that this conversion can be performed with near-unit efficiency \\cite{Langford_Conversion_2011}, hence we assume $\\eta_\\chi=0.95$.\n\n{\\it Coincidence detection.}  \nAssuming a detection efficiency of $80\\%$ for a single detector \\cite{Photonspot_private}, the efficiency for coincidence detection is $\\eta_d=(0.80)^2=0.64$. \nThe coincidence dark count rate is $R_{\\rm coincidence}=R_{\\rm d}^2 \\cdot \\tau_c$, where $R_{\\rm d}$ is the dark count rate of a single detector and $\\tau_c$ is the coincidence timing resolution (time jitter). \nThis allows a suppression of dark counts with the square of the single-detector dark count rate. For $R_{\\rm d}=3.5$~Hz and $\\tau_c=30$~ps \\cite{Photonspot_private}, the predicted coincidence dark count rate is $R_{\\rm coincidence}=3.7\\cdot10^{-10}$~s$^{-1}$.\n \n{\\it Probe photons leaking through the system.}\nThere are two ways in which probe photons can leak through the system and potentially imitate a collapse signature.\nFirstly, optomechanical conversion processes can create photons at frequency $\\omega_p$, or at a frequency reduced by multiple integers $n$ of the mechanical resonance frequency, $\\omega_p-n\\Omega$.\nDiscussions of these processes are included in the following paragraphs on noise phonons.\nThe amplitudes of these processes are negligible due to strong suppression (see main text).\nSecondly, measurement noise is introduced by probe photons transmitted through the filter cavity and downconverted to a pair of photons in the nonlinear medium, imitating a signal. \nThe probability of a probe photon with detuning $\\Delta=\\omega_s-\\omega_p=\\Omega$ transmitting through both the near-critically coupled optomechanical system and a subsequent filter cavity of linewidth $\\kappa_f$ and free spectral range $\\omega_{\\rm fsr}$ is given by \\cite{Hecht_Optics_2002}\n\n\\begin{equation}\np_f(\\Omega)=\\eta_f\\big[ 1 + (\\frac{4}{\\kappa_f})^2 \\cdot {\\rm sin}^2{(\\frac{\\pi\\Delta}{\\omega_{\\rm fsr}})}\\big]^{-1}.\n\\label{eqFilter}\n\\end{equation}\nFor the frequency of the proposed mechanical resonator of $\\Omega\/2\\pi=5.3$~GHz \\cite{Chan_Nanomechanical_2011}, a finesse of $3.16\\cdot10^{5}$ as in a standard laser stabilisation reference cavity \\cite{Kessler_Laser_2011} and a cavity length $L$ of one centimeter, we find $\\omega_{\\rm fsr}\/2\\pi=c\/2L=15$~GHz and $p_f=3.5\\cdot 10^{-10}$.\nA noise photon is then downconverted to a photon pair and registered as a coincidence count with efficiency $\\eta_\\chi\\eta_d$. \nAs the rate of incoming probe photons coupled into the probe mode in the steady state is $4\\kappa_{p,\\rm ex}\\bar n_{\\rm in}\/\\kappa_{p}=\\bar n_p\\kappa_p=\\eta_p\\Gamma$, the rate of coincidence counts due to noise photons, imitating a signal, is \n\\begin{equation}\nR_{\\rm phot}=\\eta_p\\Gamma p_f \\eta_\\chi\\eta_d=1.4\\cdot 10^{-12}\\,{\\rm s}^{-1}.\n\\label{eqRf}\n\\end{equation}\n\n\n{\\it Noise phonons from direct occupation of the signal mode.}\nA photon from the coherent probe laser can create a phonon by coupling into the signal mode $a_s$ instead of the probe mode $a_p$, which results in a direct occupation of the signal mode $\\ket{ n_b  n_p  n_s}=\\ket{001}$, as shown in Fig. 3 (a) in the main text. \nThis process is suppressed due to the small spatiotemporal overlap $\\Theta$ of the signal mode with the coherent laser beam, $\\Theta=\\kappa_s^2\/(\\kappa_s^2+\\Omega^2)\\approx(\\kappa_s\/\\Omega)^2=3.2\\cdot 10^{-5}$, where $\\kappa_s\/2\\pi=30$~MHz is the loaded decay rate of the signal mode.\nIf the photon is outcoupled, it has a frequency of $\\omega_p$ due to energy conservation and can be efficiently filtered from a signal at frequency $\\omega_s$.\nHowever, a scattering process to the probe mode can create a noise phonon in mode $b$, imitating a decoherence-signature.\nThe efficiency of this Stokes scattering process is given by the probability of the initial state $\\ket{001}$ causing the mechanical oscillator to be in the excited state, which equals the phonon occupancy after optical decay, $\\eta_{\\rm Stokes}=\\langle b^{\\dagger}(t)b(t)\\rangle$, for times $t\\gg\\kappa_p^{-1},\\kappa_{s}^{-1}$.\n$\\eta_{\\rm Stokes}$ is shown as a function of $g_0\/\\kappa_p$, for $\\kappa_{s,\\rm ex}\/\\kappa_p =1,2,$ and $4$ in Supplementary Figure \\ref{fig_hOM} (b).\nHigher values of $\\kappa_{s,\\rm ex}$ correspond to lower probabilities for this type of noise, as the decay process of rate $\\kappa_{s,\\rm ex}$ competes with the Stokes scattering of rate $g_0$.\nFor the proposed experiment with $\\kappa_{s,\\rm ex}\/\\kappa_p=1.8$ we find $\\eta_{\\rm Stokes}=0.17$.\nThe probability of phonon creation, due to this process, \nat time $t_0$ after incidence of the probe photon is $p_{\\rm direct}=(\\kappa_p\/\\kappa_{p,\\rm ex})\\cdot\\Theta\\eta_{\\rm Stokes}=2.7\\cdot10^{-5}$.\n\n\n{\\it Noise phonons from counterrotating optomechanical processes.}\nAnother mechanism for noise phonon creation is given by probe photons anti-Stokes scattering into the signal mode, corresponding to the resonantly suppressed (counterrotating) transition   $b^{\\dagger}a_p a_s^{\\dagger}e^{-2i\\Omega t}\\ket{010}\\rightarrow\\ket{101}$. \nThe resulting state contains a photon in the signal mode as well as a phonon in the mechanical resonator, which could in principle both imitate a signal. \nHowever, the outcoupled photon has a frequency of $\\omega_p-\\Omega$ and can, therefore, be filtered from the signal at frequency $\\omega_s$.\nThe probability of phonon creation due to this process, caused by a single incident probe photon, is obtained by numerically solving Eq. (\\ref{eq:reducedME}), is $p_{\\rm counterrot,1}=1.7\\cdot 10^{-5}$, where $p_{\\rm counterrot,1}=\\bra{001}\\hat \\rho(t\\gg\\kappa_p^{-1},\\kappa_s^{-1}) \\ket{100}+\\bra{101}\\hat \\rho(t\\gg\\kappa_p^{-1},\\kappa_s^{-1}) \\ket{101}$, with $\\hat \\rho(t)$ the density matrix initialized in the state $\\hat\\rho(t=0)=\\ket{010}\\bra{010}$  (see Fig. 4 (a) and (b) in the main text).\nFurther, the state $\\ket{101}$ can resonantly transition to a state with two phonons in the mechanical resonator and one photon in the probe mode: $b^{\\dagger}a_p^{\\dagger}a_s\\ket{101}\\rightarrow\\ket{210}$.\nThe probability of one incident probe photon preparing the system in this two-phonon state $p_{\\rm counterrot,2}=$\\mbox{$\\bra{002}\\hat \\rho(t\\gg\\kappa_{p}^{-1},\\kappa_s^{-1}) \\ket{200}$}$+\\linebreak$\\mbox{$\\bra{012}\\hat \\rho(t\\gg\\kappa_p^{-1},\\kappa_s^{-1}) \\ket{210}$} (with $\\hat\\rho(t=0)=\\ket{010}\\bra{010}$) numerically obtained by solving Eq. (\\ref{eq:reducedME}) (see Fig. 4 (b) in the main text), finding $p_{\\rm counterrot,2}=1.6\\cdot 10^{-6}$.\nWe find that the occupancies $p_{\\rm counterrot,1}$ and $p_{\\rm counterrot,2}$ scale with $(\\kappa_p\/\\Omega)^2$ in the limit $g_0=\\kappa_p$, with higher order transitions suppressed by $(\\kappa_p\/\\Omega)^4$ and therefore negligible.\n\n\n{\\it Scattering of noise phonons to signal photons.}\nA spurious signal photon at frequency $\\omega_s$ is created if a noise phonon scatters with a second photon entering the probe mode within the lifetime of the mechanical excitation. \nFor timescales of the mechanical excitation lifetime,  $t \\sim \\Gamma^{-1}\\gg \\kappa_{p}^{-1},\\kappa_{s}^{-1}$, after incidence of a probe photon, it is convenient to define the `occupancy probabilities' of the one- and two-phonon states from the optomechanical processes described above: $p_{n_b=1}(t_0)=p_{\\rm direct}+p_{\\rm counterrot,1}$ and $p_{n_b=2}(t_0)=p_{\\rm counterrot,2}$, where $t_0$ is long compared to timescales of the optical decay, so that all optomechanical conversions have concluded, but short compared to the mechanical decay ($\\Gamma^{-1}\\gg t_0\\gg\\kappa_{p}^{-1},\\kappa_{s}^{-1}$). The time dynamics of these one- and two phonon occupancy probabilities are described by\n\\begin{equation}\np_{n_b=1}(t)=n_{n_b=1}(t_0) e^{-\\Gamma t}+2p_{n_b=2}(t_0)\\big( e^{-\\Gamma t}-e^{-2\\Gamma t} \\big)\n\\end{equation} \nand \n\\begin{equation}\np_{n_b=2}(t) =p_{n_b=2}(t_0)  e^{-2\\Gamma t},\n\\end{equation}\nrespectively.\nThe probability of these measurement-induced phononic states encountering a second probe photon is given by $\\eta_p\\Gamma\\int_0^\\infty p_{n_b=1}(t) dt$ and $\\eta_p\\Gamma\\int_0^\\infty p_{n_b=2}(t) dt$, respectively. The conversion efficiency of the one- and two-phonon state to a spurious photon outcoupled from the cavity is then given by $\\eta_{\\rm om}$ and $ \\eta_{\\rm om,2}$, respectively, where latter is numerically calculated, and plotted as a function of $g_0\/\\kappa_p$ for different values of $\\kappa_{s,\\rm ex}\/\\kappa_p =1,2,$ and $4$, as shown in Supplementary Figure \\ref{fig_hOM} (c).\nThe resulting probability of a noise phonon of frequency $\\omega_s$ coupled out of the cavity, due to optomechanical processes, potentially imitating a signal, is\n\\begin{equation}\np_{\\rm om}(t)= \\eta_p\\Gamma\\int_0^t p_{n_b=1}(t)\\eta_{\\rm om}+ p_{n_b=2}(t)\\eta_{\\rm om,2} dt,\n\\label{eq:pom} \n\\end{equation}\nas shown in the main text in Fig. 4 (c). \nWe find the asymptotic value $p_{\\rm om}(t\\rightarrow\\infty)=8.4\\cdot 10^{-8}$.\nIn analogy to Eq. (\\ref{eqRf}), the rate of coincidence counts due to noise is given by $R_{\\rm om}=\\eta_p\\Gamma p_{\\rm om}(t\\rightarrow\\infty)\\eta_f \\eta_\\chi\\eta_d=1.9\\cdot 10^{-10}$~s$^{-1}$.\n\n\n\n{\\it Noise phonons due to photoabsorptive heating.}\nThe number of noise phonons originating from absorption heating is estimated following \\cite{Meenehan_millikelvin_2014,Ren_Crystal_2019}.\nWe compute the average phonon number excited by one probe photon in the mechanical resonator, $\\bar n_{\\rm abs,1}$, due to photoabsorption for time $t_{\\rm abs}$, at which the oscillator is in thermal equilibrium with the material, but not yet with the environment  ( $\\Gamma^{-1}\\gg t_{\\rm abs} \\gg \\gamma_{\\rm THz}^{-1}$, with $\\gamma_{\\rm THz}$ the rate at which THz-frequency phonons radiate to the environment, see also Methods).\nWe approximate $\\bar{n}_{\\rm abs,1}$ by extrapolating from \\cite{MacCabe_Ultralong_2019,Ren_Crystal_2019}.\nIn this case, an average intracavity photon number $ \\bar n_{\\rm cav}=1$ yields an average phonon number of $\\bar n_{\\rm abs}=10$ in the mechanical resonator.\n$\\bar n_{\\rm cav}=1$ is equivalent to $\\kappa\/\\Gamma$ photons passing through the cavity within the lifetime $\\Gamma^{-1}$ of the mechanical resonator, creating in total 10 phonons.\nThe intrinsic optical decay is limited by photon absorption, $\\kappa_0\\approx\\kappa_{\\rm abs}$ and  $\\gamma_{\\rm THz} \\approx \\kappa_0\/2\\pi$, within the errors given.\nTherefore, from one photon we expect the induced occupancy $\\bar{n}_{\\rm abs,1}=10\\frac{\\Gamma}{\\kappa}=10\\cdot\\frac{108\\  \\rm mHz\/2\\pi}{575\\ \\rm MHz\/2\\pi}\\approx 1.9\\cdot 10^{-9}$.\n\n\nAs with the optomechanical heating rates, a spurious signal will only occur if the phonons resulting from absorption heating interact with another probe photon. In analogy to Eq. (\\ref{eq:pom}), the probability of a probe photon creating a spurious signal photon at frequency $\\omega_s$ due to absorption heating is \n\\begin{equation}\np_{\\rm abs}(t)=\\eta_p\\Gamma \\int_0^t\\bar n_{\\rm abs,1} e^{-\\Gamma t'}\\eta_{\\rm om} dt'.\n\\label{eq:pabs}\n\\end{equation}\nFor the proposed setup we find $p_{\\rm abs}(t\\rightarrow\\infty)=6.1\\cdot 10^{-12}$ and  $R_{\\rm abs}=1.4\\cdot10^{-14}$~s$^{-1}$.\n\nFor the quadratic coupling approach, $\\bar n_{\\rm cav}=10^2$ (see main text), and the relation $\\bar n_{\\rm abs} \\propto \\bar n_{\\rm cav}^{1\/3}$ holds \\cite{Meenehan_millikelvin_2014,Ren_Crystal_2019}, and we find an average intracavity phonon number of $\\bar n_{\\rm abs} \\approx 5$. This corresponds to a noise phonon rate of $\\dot n=\\Gamma\\bar n_{\\rm abs} \\approx 3$~s$^{-1}$, close to seven orders of magnitude higher than the lowest possible phonon flux expected from Bassi {\\it et al.}'s proposal of $\\dot n_c=5.1\\cdot10^{-7}$~s$^{-1}$.\n\n\n\n{\\it Multiplexing.}\nA photonic-phononic crystal, including suspension and phononic shield, requires an area of about 1000 $\\mu$m$^2$ \\cite{MacCabe_Ultralong_2019}. Thus it would be conceivable to fabricate a high number of them on a 4-inch-wafer with an area of $\\sim 2\\cdot 10^{9}$ $\\mu$m$^2$. We assume $N\\sim10^{4}$, allowing more than 99$\\%$ of the wafer to be reserved for waveguide coupling, fabrication tolerances, etc. In principle, all these devices to may be connected to the same filter cavity, nonlinear medium, and detector pair, or to a small number of such elements. \n\n\n\n\n\\section*{References}\n\n\\footnotesize \\renewcommand{\\refname}{\\vspace*{-30pt}}  \n\\bibliographystyle{apsrev4-2}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nComputational fluid dynamics (CFD) is a branch of fluid mechanics that deals with numerically solving and analyzing fluid flow problems such as those found in aerodynamics, geology, biology, etc. CFD simulations are known for their high computational requirements, memory usage, and run times. Because of this, there is an ever growing body of work using simulation data to create reduced order models or surrogate models that can be evaluated with significantly less resources. Towards this end, we develop a neural network approach that both compresses the computation time and memory usage of fluid simulations.\n\nWe investigate fluid simulations that contain complex time dependent turbulence. Simulations of this form are difficult because they require fluid solver to have high resolution and small times steps. Never the less, they frequently occur in nature and are an important area of study. Motivated by need for these simulations and the recent success of neural network based models in related areas \\cite{tompson2016accelerating} \\cite{guo2016convolutional} \\cite{yang2016data}, we choice this setting to test our model.\n\nThe most popular approach to modeling fluid flow is with the Navier stokes equation. The solution to this partial differential equation gives the flow velocity field for a given domain. Recently, a new method for simulating fluid flow has emerged named the Lattice Boltzmann Method (LBM). It is derived from the Boltzmann equation and grew out of Lattice Gas Automaton (LGA) in the late 80s \\cite{mcnamara1988use}. The main advantages of the LBM are its ability to run on complex geometries, its scalability to parallel architectures (particularly GPUs) and applicability to complex flows that contain phenomena such as heat transfer and chemical reactions. Our method is centered around this method of simulating flow.\n\nLat-Net works by compressing the state of a simulation while learning the dynamics of the simulation on this compressed form. The model can be broken up into three pieces, an encoder, compression mapping, and decoder. The encoder compresses both the state of the simulation as well as the given boundary conditions. The compression mapping learns the dynamics on the compressed state that correspond to the dynamics in the fluid simulation. The decoder decompresses the compressed state allowing either the whole simulation state or desired pieces to be extracted.\n\nWe focus the content of this paper on LBM fluid simulations because this is the most popular use of the LBM, however, this method of simulation is known to be able to solve a large set of partial differential equations \\cite{galindolattice}. In fact, LBM can simulate many physical systems of interest such as Electromagnetism, Plasma, Multiphase flow, Schr\u00f6dinger equation etc. \\cite{mendoza2010three} \\cite{kim2008wavelet} \\cite{zhong2006lattice} \\cite{shan1993lattice}. With this in mind, we keep our method general and show evidence our method works equally well on Electromagnetic simulations. However, because the dominate use of LBM is on fluid flow problems we center discussion on this subject.\n\nOur work has the following contributions.\n\\begin{itemize}\n  \\item It allows for simulations to be generated with less memory then the original flow solver. There is a crucial need for such methods because memory requirements grow cubic to grid size in 3D simulations. In practice, this quickly results in the need for large GPU clusters \\cite{onodera2013large} \\cite{xian2011multi}.\n  \\item Once our model is trained, it can be used to generate significantly larger simulations. This allows the model to learn from a training set of small simulations and then generate simulations as much as 16 times bigger with little effect in accuracy.\n  \\item Our method is directly applicable to a variety of physics simulations, not just fluid flow. We show this with our electromagnetic example and note that the changes to our model are trivial.\n\\end{itemize}\n\n\\section{Related Work}\n\nRecently, there have been several papers applying neural networks to fluid flow problems. Guo etc. \\cite{guo2016convolutional} proposed to use a neural network to learn a mapping from boundary conditions to steady state flow. Most related to our own work, Yang etc. \\cite{yang2016data} and Tompson etc. \\cite{tompson2016accelerating} use a neural network to solve the Poisson equation in order to accelerate Eulerian fluid simulations. The key difference between this and Lat-Net is its ability to compress the memory usage and the generality of our method to other physics simulations.\n\nThere has also been an increasing body of work applying neural networks to other physics modeling problems. For example, neural networks have been readily adopted in many chemistry applications such as predicting molecular properties from descriptors, protein contact prediction and computational material design \\cite{goh2017deep}. Very recently, neural networks have been applied to quantum mechanics problems as seen in Mills etc. \\cite{mills2017deep} and Giuseppe etc. \\cite{carleo2017solving} where neural networks are used to approximate solutions to the Schr\u00f6dinger equation. In high energy Physics, Paganini etc. \\cite{2017arXiv170502355P} uses a generative adversarial networks (GAN)\\cite{goodfellow2014generative} to model electromagnetic showers in a longitudinally segmented calorimeter. Many of these applications are relatively recent and indicate a resurgence of interest in applications of neural networks to modeling physics.\n\nReduced order Modeling is an area of research that focuses on techniques to reduce the dimensionality and computational complexity of mathematical models. A Reduced order model (ROM) is constructed from high-fidelity simulations and can subsequently be used to generate simulations for lower computation. The most popular ROM method for fluid dynamics is Galerikin projection \\cite{rowley2004model} \\cite{barone2009reduced}. This method uses Proper Orthogonal Decomposition to reduce the dimensionality of flow simulations and then finds the dynamics on this reduced space. There are other methods that build on this such as reduced basis methods and balanced truncation \\cite{veroy2005certified} \\cite{rowley2005model}. While these approaches are centered around the Navier stokes equation and thus not directly comparable to our own, we note that the compression mapping present in these methods is typically quite simple. Given the recent success neural networks have had in creating well structured encodings (such as Variational Autoencoders \\cite{kingma2013auto} \\cite{watter2015embed}), we feel our approach is well justified.\n\n\\section{Deep Neural Networks for Compressed Lattice Boltzmann}\n\nIn this section, we present our model for compressing Lattice Boltzmann simulations.\n\n\\subsection{Review: The Lattice Boltzmann Method}\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.19]{.\/figs\/lattice_boltzmann.pdf}}\n\\caption{Illustration of the Lattice Boltzmann update steps}\n\\label{lattice_boltzmann}\n\\end{figure}\n\nIn a Lattice Boltzmann simulation, the domain is discretized into an equal sized Cartesian grid. Each cell of this grid contains a velocity distribution function $f_i$ that describes the velocity of flow at that point. $f_i$ has values ranging over $i$ that correspond to the $\\{ \\vec{c}_i \\}$ directions of flow. In our 2 dimensional simulations, there are 9 such directions (D2Q9 scheme). A figure showing this grid structure is seen in \\ref{lattice_boltzmann}. From the distribution function $f_i$, one can calculate the density ($p$) and velocity ($\\vec{u}$) of the fluid flow with the following equations.\n\n\\begin{equation}\n  p = \\sum_i f_i  \\qquad\\text{and}\\qquad \\vec{u} = \\sum_i \\vec{c}_i f_i \n\\end{equation}\n\nThe lattice states are updated with two separate steps, the collision step and the streaming step. The collision step mimics the flow interacting with itself and is updated in the following way,\n\n\\begin{equation}\n  f^t_i(x, t + \\delta_t) = f_i(x,t) + \\frac{1}{\\tau} (f_i^{eq} - f_i)\n\\end{equation}\n\nwhere $\\tau$ is the relaxation constant and $f_i^{eq}$ is the flow equilibrium. For our simulations, we use the $f_i^{eq}$ from the Lattice Bhatnagar-Gros-Krook (LBGK) scheme \\cite{guo2013lattice}. After the collision step is applied, the flow propagates to adjacent cells following the streaming step. \n\n\\begin{equation}\n  f_i(x + c_i, t + \\delta_t) = f^t_i(x,t + \\delta_t)\n\\end{equation}\n\nThis step will contain bounce back if one of the adjacent cells is a boundary. Figure \\ref{lattice_boltzmann} illustrates these steps for the 2 dimensional case.\n\nIt is interesting to note the simplicity of this method and its similarity to convolutional neural networks. In fact, if we treat the lattice state $f$ as a tensor of size ($n_x$,$n_y$,9), as we do for the remainder of this paper, the streaming operator can be mimicked with a 3 by 3 convolution and the collision step can be performed with a 1 by 1 convolution (D2Q9). This offers a unique way to interpret our method. In some sense, we are taking a large convolutional neural network and compressing it onto a much smaller and more memory efficient network. With this mental picture in mind, we now describe our approach.\n\n\\subsection{Proposed Architecture}\n\nFigure \\ref{fig_1} shows a sketch of the model. The figure can be understood by following the arrows starting from the flow state $f_t$ and the boundary $b$. We treat $f_t$ as a tensor with shape ($n_x,n_y,9$) for the 2D case and ($n_x,n_y,n_z,15$) for the 3D case. The boundary is treated as a binary tensor of shape ($n_x,n_y,1$) and ($n_x,n_y,n_z,1$) with the value being 1 if the cell is solid. Bellow we walk through each step of our method.\n\nFirst, we compress both the state of the fluid simulation $f_t$ and the boundary conditions $b$ using two separate neural networks $\\phi_{enc}$ and $\\phi'_{enc}$ respectively. The result from $\\phi_{enc}$ is a compressed representation of the flow $g_t$ and the result of $\\phi'_{enc}$ are two tensors $b_{mul}$ and $b_{add}$ of equal size to $g_t$. These three tensors represent the entirety of the compressed state of the simulation.\n\nIn a Lattice Boltzmann solver, the boundary conditions are used at each time-step to add bounce back to the streaming step. In a similar way, our model applies the compressed boundary to the compressed state every time-step. We do this in the following way,\n\\begin{equation}\n  g_t = (g_t \\odot b_{mul}) + b_{add}\n\\end{equation}\nThis method proved extremely successful at keeping the boundary information firmly planted through the duration of the simulation. This method of applying boundary conditions was inspired by \\cite{vondrick2016generating} where they use a similar method to combine foreground and background information in video prediction. After the boundary is applied to $g_t$, we can run the state through another neural network to emulate the dynamics, i.e. $\\phi_{comp}:g_{t} \\rightarrow g_{t+1}$. Each step of $\\phi_{comp}$ is equivalent to $n$ time-steps of the Lattice Boltzmann solver. For example, in the 2 dimensional simulation, each step of $\\phi_{comp}$ mimics 120 steps of the Lattice Boltzmann solver. Once $g_t$ is computed, we can extract out the generated state of the simulation with a decoder network $\\phi_{dec}$. \n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.28]{.\/figs\/fig_1.png}}\n\\caption{Illustration of the Lat-Net architecture}\n\\label{fig_1}\n\\end{figure}\n\n\\subsection{Network Implementation Details}\n\nWe implement 2 networks trained on 2 dimensional and 3 dimensional lattice simulations. The encoder, compression mapping and decoder pieces of the 2D network are each a series of 3 by 3 residual blocks with the sequences 4x(down res-res)-res, 4x(res), and 3x(transpose conv-res-res)-transpose conv, respectively \\cite{he2016deep}. For the 3D network, the sequences are 2x(down res)-res, 3x(res), and (transpose conv-res-transpose conv) where 3 by 3 by 3 convolutions are used. The down residual blocks are created by changing the first convolution to have kernel size 4 by 4, stride 2 and double the filter size. The up sampling is achieved with transpose convolutions of kernel size 4 by 4, stride 2 and half the filter size. For the last residual block on the 3D network, the filter size is halved once again.\n\nAs mentioned above, each network is kept entirely convolutional. Fluid flow is inherently spatially correlated so using convlutional layers allows this spatial information to be preserved. Keeping the network convolutional also allows different input sizes to be used. This is how our model is able to train on small simulations and then generate larger simulations.\n\nResidual connections have been used in many deep learning architectures with much success. Adding residual connections allows for much deeper networks to be trained, often resulting in improved results \\cite{he2016deep}. When training our model, it is necessary to unroll the compression network over several time-steps. This has the same effect as making the network deeper. For this reason, it seems logical to take advantage of this network architecture. We have seen that removing these residual connections results in much slower convergence and worse accuracy.\n\n\\subsection{Training Details}\n\nLat-Net is trained by unrolling the network and comparing the generated flow with the true. Our loss function is Mean Squared Error (MSE) with Image Gradient Difference Loss (GDL) \\cite{mathieu2015deep}. The GDL is multiplied by $\\lambda_{GDL}$ and then added to the MES. In all our experiments, $\\lambda_{GDL}$ is set to $0.2$. Removing the GDL tended to produce less accurate models. Lat-Net is unrolled 5 time-steps and then trained with the Adam optimizer \\cite{kingma2014adam}.\n\n\\section{Experiments}\n\nIn this section, we describe our experiments testing Lat-Net on a variety of problems. Our experiments are designed to test our models ability to generate large simulations as well as its transferability to new boundary geometries. We also explore computation time and working memory usage. Finally, we briefly show results applying this method to electromagnetic simulations.\n\n\\subsection{Dataset Generation}\nIn order to train and test our model, we generate sets of fluid and electromagnetic simulations. All simulations were generated with the MechSys library \\cite{mechsys}.\n\nThe train set for the 2D fluid simulations consists of 50 runs of grid size 256 by 256 and 9 directional flows in the lattice Boltzmann solver (D2Q9 scheme)\\cite{guo2013lattice}. The simulations use periodic boundary conditions on top and bottom as well as uniform inlet flow and outlet flow of 0.04 from the left and right. 8 Objects are placed randomly with height and width sizes ranging from 140 to 20 cells. The test set for the 2 dimensional simulations consists of 10 runs of size 256 by 256, and 5 runs of size 1024 by 1024 with the same boundary conditions and object densities. We also generate a test set of size 256 by 512 with vehicle cross sections as objects. There are 28 cross sections used ranging from trucks to minivans. For all 2 dimensional simulations, the ratio of network steps to Lattice Boltzmann steps is 1 to 120.\n\nThe train set for the 3D fluid simulations consists of 50 runs of grid size 40 by 40 by 160 and 15 directional flows in the lattice Boltzmann solver (D3Q15 scheme)\\cite{guo2013lattice}. Similar to the 2D simulations, periodic boundary conditions are used with same inlet and outlet flow. 4 spheres are randomly placed with height and width 24. The reason different object geometries and sizes were not explored was due to the fact that smaller objects or objects with complex geometries tended to have too course a resolution for the lattice Boltzmann solver and larger objects required too large a simulation size. The test set comprises 10 runs of 40 by 40 by 160 and 5 runs of 160 by 160 by 160 simulations with the same object density. The ratio of network steps to Lattice Boltzmann steps is 1 to 60.\n\nThe train set for the electromagnetic simulations are grid size 256 by 256 with periodic boundaries. An electromagnetic wave is initialized in the top of the simulations and proceeds to interact with randomly placed objects of different dielectric constants. When the wave hits these objects, the reflection and refraction phenomenon is seen. The test set consists of simulations of size 512 by 512 with the same object density.\n\n\\subsection{Generating Simulations}\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.302]{.\/figs\/256x256_2d_flow_image.png}}\n\\subfigure{\\includegraphics[scale=0.302]{.\/figs\/1024x1024_2d_flow_image.png}}\n\\subfigure{\\includegraphics[scale=0.302]{.\/figs\/160x160x160_3d_flow_image.png}}\n\\caption{A visual comparison of flows generated by Lat-Net and the Lattice Boltzmann method. Each figure shows the Generated, True, and Difference of the flow for various time-steps.}\n\\label{2d_image_plot}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.280]{.\/figs\/256x256_2d_error_plot.png}}\n\\subfigure{\\includegraphics[scale=0.280]{.\/figs\/1024x1024_2d_error_plot.png}}\n\\subfigure{\\includegraphics[scale=0.280]{.\/figs\/40x40x160_3d_error_plot.png}}\n\\subfigure{\\includegraphics[scale=0.280]{.\/figs\/160x160x160_3d_error_plot.png}}\n\\caption{ Comparison plot of the flows generated by Lat-Net and the Lattice Boltzmann method. Each plot shows the average mean squared error of the true and generated generated along with the average divergence of the velocity vector field for both simulations. The standard deviation is also displayed. In addition, the calculated values for drag and flux are displayed for a single simulation run. For the 1024 by 1024 simulation, the flow produced by the Lattice Boltzmann solver tended to produce instabilities resulting in the chaotic divergence observed.}\n\\label{2d_error_plot}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.74]{.\/figs\/car_2d_flow_image.png}}\n\\subfigure{\\includegraphics[scale=0.28]{.\/figs\/256x512_2d_error_plot.png}}\n\\caption{Comparison of generated flows for the vehicle cross section dataset. The images show the generated and true flow at step 100 for 3 different cars in the dataset. The plot shows the same values as described in \\ref{2d_error_plot}.}\n\\label{car_dataset}\n\\end{figure}\n\n\nA key component of our model is its ability to generate larger simulations then those trained on. To test its effectiveness in doing so, we compare the accuracy of generated 2D and 3D simulations to ground truth simulations.\n\nComparing the accuracy of our simulations require some consideration. The naive approach is to compare the MSE between the generated and true simulation at various time-steps. The problem with this approach is that fluid flow is a chaotic dynamical system and small perturbations in flow quickly compound leading to dramatic differences at latter times. For this reason, we compare a variety of metrics in evaluating the generated flows accuracy. Similar to \\cite{tompson2016accelerating}, we compare the divergence of the generated and true velocity vector field to test our models stability. We also compare the computed values of drag and flux. Flow simulations are often run to calculate such values so comparing this is a strong indicator of our models real world applicability. The drag is calculated directly from the lattice state via the momentum transfer method \\cite{guo2013lattice}. The flux value is the average of the flux in each non-boundary cell. These values can be used to calculate important quantities such as the drag coefficient and Reynolds number. Lastly, we visually inspect the produced flow to check for instabilities and blurring effects.\n\nIn figure \\ref{2d_error_plot}, we see the predicted values for different grid sizes in 2D and 3D simulations. In the 2D simulations, Lat-Net is able to effectively transfer to larger domain sizes with very similar calculated values of drag and flux. The generated flow also maintains its stability even after hundreds of steps. In the 3D simulation we see that, while our model predicts realistic values for the 40x40x160 simulation, it tends to have a slight bias in the direction of flow that manifests itself in the 160x160x160 simulation.\n\nWhen visually inspecting our produced flows (figure \\ref{2d_image_plot}), we see a slight blurring effect but overall similar structure in the 2D flows. We attribute this blur effect to the dimensionality reduction and use of MSE. This can possibly be overcome with the use of generative adversarial network \\cite{goodfellow2014generative} where the loss is derived from a discriminator network. Another solution may be to craft a loss function that takes advantage of the statistical properties of flow \\cite{kim2008wavelet}. We leave these pursuits to future work.\n\nThere is a distinct difference in the generated and true flow for the 3D 160x160x160 simulation. While the generated flow appears accurate close to the objects, in regions far between objects the network tends to underestimate the flow velocity. We believe this is due to these types of regions not being present in the train set and is the probable cause for the biases seen in the drag and flux. As mentioned above, our 3D train set is limited due to memory constraints and so developing a diverse train set to overcome this proved difficult.\n\nThe boundaries used in the above evaluation are drawn from the same distribution as the train set. This motivates the question of how our model performs on drastically different geometries. To test this, we apply our model to predicting flow around vehicle cross sections. Surprisingly, even though are model is only trained on flows around simple shapes (ovals and rectangles) it can effectively generalize to this distinctly different domain. In figure \\ref{car_dataset}, we see the predicted flows are quite similar but with the same blurring effect. Calculating the same values as above, we see the flow is stable and produces similar drag and flux.\n\n\\subsection{Computation and Memory Compression}\n\n\\begin{table}[]\n\\small\n\\caption{Computation Time of Network} \\label{compute_times}\n\\centering\n\\begin{tabular}{|c|cccccc|}\n\\hline\nSimulation    & Comp. Size       & Comp. Mapping       & Full State  & Plane      & Line       & Point \\\\ \\hline\n(1024, 1024, 9)  & (64, 64, 128)                 & 2.7 ms            & 36.2 ms   & NA         & 6.7 ms   & 6.6 ms \\\\\n(160, 160, 160, 15) & (40, 40, 40, 64)                 & 23.1 ms           & 272.1 ms  & 38.2 ms  & 25.6 ms  & 24.1 ms  \n\\\\ \\hline\n\\end{tabular}\n\\label{computation_table}\n\\end{table}\n\n\nIn this section, we investigate the computational speed-up of our model. The standard performance metric for Lattice Boltzmann Codes is Million Lattice Updates per Second (MLUPs). This metric is calculated by the following equation,\n\\begin{equation}\n  MLUP = \\frac{n_x \\times n_y \\times n_z \\times 10^{-6}}{Compute \\ Time}\n\\end{equation}\n where $n_x$, $n_y$, and $n_z$ are the dimensions of the simulation. For 3 dimensional simulations like the ones seen in this paper, a speed of 1,200 MLUPS can be achieved with a Nvidia K20 GPU and single precision floats \\cite{januszewski2014sailfish}. We use this as our benchmark value to compare against.\n\nThe computation time and memory usage of the encoder can be neglected because this is a one time cost for the simulation. In addition, if the simulation is started with uniformly initialized flow as seen in our experiments, the computation to compress the flow is extremely redundant and can easily be optimized.\n\nAs seen in table \\ref{computation_table}, the computation time of the compression mapping is 23.1 ms for a 3D simulation of grid size 160 by 160 by 160. Because each step of the compression mapping is equivalent to 60 Lattice Boltzmann steps, this equates to 10,600 MLUPS and a roughly 9x speed increase (a similar speed-up is seen with the 2D simulation). This does not give a complete picture though. Once the compressed states have been generated, the flow must be extracted with the decoder. Unfortunately, this requires considerable amounts of computation and memory because it involves applying convolutions to the full state size. Fortunately, there are ways around this. In many applications of CFD, it is not necessary to to have the full state information of the flow at each time-step. For example, calculating the drag only requires integrating over the surface of the object. By using the convolutional nature of the decoder, we can extract specif pieces of the flow without needing to compute the full state. In table \\ref{computation_table}, we show computation times for extracting flow information of a plane, line and single point. While these computations can still be somewhat expensive, they do not necessarily need to be performed at every time-step and require very little working memory. \n\nUnfortunately, there are some measurements that do require the full state information to compute such as the average flux seen in our tests. Our method is currently unable to handle these without requiring high run-times and large working memory. A possible solution is training a separate neural network that takes in the compressed state and predicts the desired measurement. This would negate the need to extract out the full state and keep memory usage low. We leave this and similar ideas for future work.\n\nWhile Lat-Net compresses the simulation state size by more then an order of magnitude, the working memory requirements for the compression network must be considered. A typical GPU based Lattice Boltzmann solver requires around 1.5 times as much working GPU memory as the memory size of the lattice \\cite{januszewski2014sailfish}. For example, the maximum sized D3Q15 lattice that can fit on a GPU with 8 Gigabytes is $446^3$. We have observed that in our implementation of Lat-Net, the maximum 3D compression network we can run with an 8 Gigabyte GPU corresponds to a lattice size of $672^3$. This represents a 3.4x efficiency gain in working memory usage. While this is certainly a nontrivial gain, we feel that further improvements can be realized with a more memory efficient implementation of the compression mapping.\n\n\n\\subsection{Electromagnetic Results}\n\nFinally, we illustrate the generality of our method by applying it to electromagnetic simulations. The same neural network architecture is used as in the 2D flow simulations with the only difference being the filter size on the compression is half of that in the flow network. The loss is kept identical however the lattice values are scaled up by a factor of 10 so they are on the same range as the flow lattice values. In figure \\ref{em_dataset}, we see very similar waves formed with the same reflection and refraction.\n\n\\begin{figure}[!t]\n\\centering\n\\subfigure{\\includegraphics[scale=0.28]{.\/figs\/512x512_2d_em_image.png}}\n\\subfigure{\\includegraphics[scale=0.14]{.\/figs\/512x512_2d_em_error_plot.png}}\n\\caption{ Comparison of generated Electromagnetic fields. The images show the true and generated magnetic field a various time-steps. The reflection and refraction phenomena can clearly be seen in both. The plot shows the mean squared error of the true and generated simulation.}\n\\label{em_dataset}\n\\end{figure}\n\n\n\\section{Conclusion}\n\nFluid Simulations are incredibly important for a variety of tasks however they are extremely computation and memory intensive. In this work, we have developed a unique method to overcome this using deep neural networks. We have demonstrated it is capable of accurately reconstructing a variety of simulations under different conditions with significantly less computation and memory. We have also shown that our method can be readily applied to other physics simulations such as electromagnetic simulations. While our method has proved successful on the problems in this paper, there is still significant room for improvement. A loss function that either takes into account the statistical nature of the flow or uses recent advances in GANs could produce shaper, more realistic flow. Training a network to extract desired measurements from the compressed state such as average flux would overcome the current memory limitation for such a task. We leave these and other improvements for future work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Light-cone Co-ordinates}\\label{appendixA}\nTo perform the light-cone expansion one relate the meson's 4-momentum $P_\\mu$, polarisation vector $e^{(\\lambda)}$ and the coordinate $x_\\mu$ to two light-like vectors $p_\\mu$ and $z_\\mu$. We have the usual relations\n\\begin{equation}\np^2=0, \\hspace{1in} z^2=0\\,,\n\\end{equation}\nand\n\\begin{equation}\nP^2=m_{K^*}^2, \\hspace{1in} e^{(\\lambda)}\\cdot e^{(\\lambda)} =-1,\\hspace{1in} P\\cdot e^{(\\lambda)} =0,\n\\end{equation}\nso that the limit $m_{K^*}^2\\to 0$ gives $p \\to P$ and $x^2 \\to 0$ gives $z \\to x$. From this it follows that\n\\begin{eqnarray}\\label{lccoords1}\n  z_\\mu &=& x_\\mu-P_\\mu\\,\\frac{1}{m_{K^*}^2}\\left[x\\cdot P -\\sqrt{(x\\cdot P)^2-x^2m^2_{K^*}}\\,\\right]\n= x_\\mu\\left[1-\\frac{x^2m_{{K^*}}^2}{4(z\\cdot p)^2}\\right]\n-\\frac{1}{2}p_\\mu\\,\\frac{x^2}{p\\cdot z}+ \\mbox{\\cal O}(x^4)\\,,\\nonumber\\\\\np_\\mu &=& P_\\mu-\\frac{1}{2}\\,z_\\mu\\, \\frac{m^2_{K^*}}{p\\cdot z}\\,.\n\\end{eqnarray}\nThe meson's polarization vector $e^{(\\lambda)}$ can be decomposed into projections onto the two light-like vectors and the orthogonal plane\n\\begin{eqnarray}\\label{lccoords2}\n e^{(\\lambda)}_\\mu &=& \\frac{e^{(\\lambda)}z}{p\\cdot z}\\, p_\\mu + \\frac{e^{(\\lambda)} p}{p\\cdot z}\\, z_\\mu +\n                     e^{(\\lambda)}_{\\perp\\mu} =  \\frac{e^{(\\lambda)} z}{p\\cdot z}\\left( p_\\mu -\\frac{m^2_{K^*}}{2p\\cdot z}\\, z_\\mu \\right)+e^{(\\lambda)}_{\\perp\\mu}\\,,\\nonumber\\\\\n&=&(e^{(\\lambda)} \\cdot x)\\frac{P_\\mu (x\\cdot P)-x_\\mu m^2_{K^*}}{(x\\cdot P)^2 -x^2 m^2_{K^*}}+ e^{(\\lambda)}_{\\perp\\mu}\\,.\n\\end{eqnarray}\nWe also need the projector $g_{\\mu\\nu}^\\perp$ onto the directions orthogonal to $p$ and $z$\n\\begin{equation}\ng^\\perp_{\\mu\\nu} = g_{\\mu\\nu} -\\frac{1}{p\\cdot z}(p_\\mu z_\\nu+ p_\\nu z_\\mu)\\,.\n\\end{equation}\nSome useful scalar products are\n\\begin{eqnarray}\nz\\cdot P = z\\cdot p &=& \\sqrt{(x \\cdot P)^2 - x^2 m^2_{K^*}}\\,,\\nonumber\\\\\np \\cdot e^{(\\lambda)}&=& -\\frac{m^2_{K^*}}{2 pz} z \\cdot e^{(\\lambda)}\\,,\\nonumber\\\\\nz \\cdot e^{(\\lambda)}&=&x \\cdot e^{(\\lambda)}\\,.\n\\end{eqnarray}\nWill use the notations\n\\begin{equation}\\label{note1}\na_z\\equiv a_\\mu z^\\mu, \\qquad b_p\\equiv b_\\mu p^\\mu,\\qquad \\slash{c}\\equiv \\gamma_\\mu c^\\mu,\\qquad d_\\mu^\\perp\\equiv g_{\\mu \\nu}^\\perp d^\\nu,\n\\end{equation}\nfor  arbitrary Lorentz vectors $a_\\mu$, $b_\\mu$, $c_\\mu$ and $d_\\mu$ and \n\\begin{equation}\nx^\\mu = x_- n^\\mu + x_+ \\bar{n}^\\mu +x^\\mu_\\perp\\,,\n\\end{equation}\nfor null unit vectors $n^2=\\bar{n}^2=0$ and $n \\cdot \\bar{n} =1$. The following notation is also used:\\begin{equation}\na_+=a\\cdot z\\,,\\qquad a_- =\\frac{a\\cdot p}{p\\cdot z}\\,,\\qquad a^{\\perp}_\\mu\n= a_\\mu - \\frac{a_- p_\\mu}{p\\cdot z}-a_+ z_\\mu\\,.\n\\end{equation}\n\n\n\n\n\\chapter{Useful formulas for sum rule determinations}\\label{appendixB}\n\\section{Loop Integrals}\nHere we summarise the loop integrals needed for calculating the twist-3 correlation\nfunctions in Chapter~\\ref{chapter4_det}.  At one loop, one has ($z^2=0$)\\cite{Ball:2003sc}\n\\begin{eqnarray}\n\\int \\left[d^L k\\right]  e^{i f_k k\\cdot z} \\,\\frac{(k\\cdot z)^n}{(k^2)^a\n  ((k-p)^2)^b} & = & (-1)^{a+b} \\left(-p^2\\right)^{D\/2-a-b} (p\\cdot z)^n\n  \\,\\frac{\\Gamma(a+b-D\/2)}{\\Gamma(a)\\Gamma(b)}\n\\nonumber\\\\\n&& \\times\\int_0^1 dw\\,\n  e^{i(1-w) f_k p\\cdot z}\\, w^{D\/2-1-b} (1-w)^{D\/2+n-1-a}\\,,\\nonumber\\\\ \\label{E.1}\n\\end{eqnarray}\nwhere the integration measure is defined as  $d^D k = i\/(4\\pi)^2 \\left[d^L k\\right]$ and $f_k$ is an arbitrary numerical factor, which in the cases considered in Chapter~\\ref{chapter4_det} is either $v$ or $\\bar v$. One also needs the integral\n\\begin{eqnarray}\n\\lefteqn{\\int \\left[d^L l\\right]  e^{i f_l l\\cdot z} \\,\\frac{(l\\cdot p)(l\\cdot z)^j}{(l^2)^c\n  ((l-k)^2)^d}}\n\\nonumber\\\\\n& = & (-1)^{\\frac{D-4}{2}} \\left(k^2\\right)^{D\/2-c-d} (k\\cdot p) (k\\cdot z)^j\n  \\,\\frac{\\Gamma(c+d-D\/2)}{\\Gamma(c)\\Gamma(d)}\\int_0^1 du\\,\n  e^{i(1-u) f_l k\\cdot z}\\, u^{D\/2-1-d} (1-u)^{D\/2+j-c}\n\\nonumber\\\\\n&&{}+(-1)^{\\frac{D-4}{2}} \\left(k^2\\right)^{D\/2+1-c-d} (p\\cdot z)(k\\cdot z)^{j-1}\n  \\,\\frac{\\Gamma(c+d-D\/2-1)}{2\\Gamma(c)\\Gamma(d)}\n\\nonumber\\\\\n&&{}\\times\\int_0^1 du\\,\n  e^{i(1-u) f_l k\\cdot z}\\, u^{D\/2-d} (1-u)^{D\/2-1+j-c}\\left( j + i f_l\n  (1-u) (k\\cdot z)\\right)\\,.\\label{E.2}\n\\end{eqnarray}\nTwo-loop integrals are obtained by combining the above one-loop integrals.\n\n\\section{Borel Subtraction}\nTo derive the sum rules from $\\widetilde{\\pi}_{3;V}^\\parallel$, $\\pi_{3;V}^\\parallel$ and $\\pi_{3;V}^\\perp$ we use the relation\n\\begin{equation}\n\\frac{1}{\\pi}\\textrm{Im}_s \\left[-q^2-i0\\right]^\\alpha=\\frac{s^\\alpha}{\\Gamma(-\\alpha)\\Gamma(1+\\alpha)}\\Theta(s)\\,,\n\\end{equation}\nwhere $s=-q^2$, to find the imaginary part. Using the following notation for the Borelisation and continuum subtraction procedure\n \\begin{equation}\n\\hat{\\mathcal{B}}_{sub}\\left[X\\right]=\\int^{s_0}_0ds\\,e^{-s\/M^2}\\frac{1}{\\pi} \\textrm{Im}_s X\\,,\n\\end{equation}\nand the definition of the Borel transform (\\ref{borel1}) allows one to write the required results as\n\\begin{eqnarray}\n\\qquad\\hat{\\mathcal{B}}_{sub}\\left[\\frac{1}{(q^2)^\\alpha}\\right]&=&\\frac{(-1)^\\alpha}{(\\alpha-1)! (M^2)^{\\alpha-1}} \\,,\\qquad\n\\hat{\\mathcal{B}}_{sub}\\left[\\ln(-q^2)\\right]=-M^2+\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}\\,,\\nonumber\\\\\n\\hat{\\mathcal{B}}_{sub}\\left[q^2\\ln(-q^2)\\right]&=&-M^4+\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}s\\,,\\nonumber\\\\\n\\hat{\\mathcal{B}}_{sub}\\left[\\frac{\\ln(-q^2)}{q^2}\\right]&=&\\gamma_{E}-\\ln M^2+\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}\\frac{1}{s}\\,,\\nonumber\\\\\n\\hat{\\mathcal{B}}_{sub}\\left[\\frac{\\ln(-q^2)}{q^4}\\right]&=&\\frac{1}{M^2}\\left(1-\\gamma_E +\\ln M^2\\right)+\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}\\frac{1}{s^2}\\,,\\nonumber\\\\\n\\hat{\\mathcal{B}}_{sub}\\left[\\ln(-q^2)^2\\right]&=&2M^2\\left(\\gamma_E-\\ln M^2\\right) +2\\int^\\infty_{s_0}ds\\,e^{-s\/M^2}\\ln s\\,,\n\\end{eqnarray}\nwhere $\\gamma_E$ is Euler's constant. \n\n\\section{Input Parameters}\nFor the twist-2 and twist-3 DA parameter sum rule determinations of Chapter~\\ref{chapter4_det} we use the following input parameters:\n\\begin{table}[ht]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{|r@{\\:=\\:}l||r@{\\:=\\:}l|}\n\\hline\n\\langle \\bar q q\\rangle & (-0.24\\pm0.01)^3\\,\\mbox{GeV}^3 & \\langle \\bar s s\\rangle & (1-\\delta_3)\\,\\langle \\bar q q\\rangle\\\\\n\\langle \\bar q \\sigma g_sG q\\rangle & m_0^2\\,\\langle \\bar q q\\rangle &  \\langle \\bar s \\sigma g_sG s\\rangle & (1-\\delta_5)\\langle \\bar q \\sigma g_sG q\\rangle\\\\[6pt]\n\\displaystyle \\left\\langle \\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle & (0.012\\pm 0.003)\\, \n{\\rm GeV}^4 & \\multicolumn{2}{l|}{}\\\\[6pt]\\hline\n\\multicolumn{4}{|c|}{m_0^2 = (0.8\\pm 0.1)\\,{\\rm GeV}^2~,\\qquad \\delta_3\n  = 0.2\\pm 0.2, \\qquad \\delta_5 = 0.2\\pm 0.2}\\\\\\hline\n\\multicolumn{4}{|c|}{\\overline{m}_s(2\\,\\mbox{GeV}) = (100\\pm\n20)\\,\\mbox{MeV}\\qquad\\longleftrightarrow\\qquad\\overline{m}_s(1\\,\\mbox{GeV})\n= (133\\pm 27)\\,\\mbox{MeV}}\\\\\n\\multicolumn{4}{|c|}{\\overline{m}_q(\\mu) = \\overline{m}_s(\\mu)\/R\\,,\n  \\qquad R = 24.6\\pm 1.2}\\\\\\hline\n\\multicolumn{4}{|c|}{\\alpha_s(M_Z) = 0.1176\\pm 0.002  ~\\longleftrightarrow~ \n\\alpha_s(1\\,\\mbox{GeV}) = 0.497\\pm 0.005}\\\\\\hline\n\\end{array}\n$$\n\\renewcommand{\\arraystretch}{1}\n\\addtolength{\\arraycolsep}{-3pt}\n\\caption[Summary of input parameters for Chapter 4.]{\\small Input parameters for sum rules at the renormalisation scale $\\mu=1\\,$GeV. The value of $m_s$ is obtained from unquenched lattice calculations with $N_f=2$ flavours  as summarised in \\cite{mslatt}, which agrees with the results from QCD sum rule calculations \\cite{jamin}. $\\overline{m}_q$ is taken from chiral perturbation theory \\cite{chPT}. $\\alpha_s(M_Z)$ is the PDG  average \\cite{Yao:2006px}.}\n\\label{QCDSRinput}\n\\end{table}\n\nTo evaluate the sum rules for the three-particle twist-3 DA parameters we use the following values of the continuum threshold $s_0$\n\\begin{eqnarray}\ns_0^\\parallel (\\rho) &=& (1.3\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad \ns_0^\\parallel (K^*) = (1.3\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad\ns_0^\\parallel (\\phi) = (1.4\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad \\nonumber\\\\\ns_0^\\perp (\\rho) &=& (1.5\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad \ns_0^\\perp (K^*) = (1.6\\pm 0.3)\\,{\\rm GeV}^2\\,,\\quad\ns_0^\\perp (\\phi) = (1.7\\pm 0.3)\\,{\\rm GeV}^2\\,. \\nonumber\\\\\n\\end{eqnarray}\nThe threshold for the $\\rho$ channel is from \\cite{Shifman:1978bz}. \n\n\n\\chapter*{Introduction}\\label{chapter0_intro}\\addcontentsline{toc}{chapter}{\\protect\\numberline{Introduction\\hspace{-96pt}}} \nOne only has to ask the question ``why?'' a handful of times before one reaches the answer ``I don't know'', regardless of the topic considered and regardless of the person asked.  It is safe to say, however, almost all questions of the structure of matter at the smallest of distances leads one directly to, or at least through, the field of modern particle physics. The beginnings of our understanding of the physical world harks back to the dawn of scientific reasoning in the ancient world; logic and reasoning were applied with the aim of describing the behaviour of physical systems in terms of simple universal axioms, a philosophy which still holds strong today. Through experimentation and the language of mathematics the scientific method has driven back the edge of ignorance to frontiers unimaginable to those physicists of 100 years ago, let alone the natural philosophers of millennia ago. The present ``coal face'' is known as the Standard Model \\cite{weak, QCD} which describes three of the four known forces of nature -- electromagnetism, and the weak and strong nuclear forces -- in one unifying framework. \n\nFrustratingly, the Standard Model does not explain many of the things which it encompasses; it does not provide an origin for CP violation but only gives a parameterisation, nor does it explain why there are three generations of quarks and leptons, or their hierarchy of masses.  All attempts to bring gravity into the fold have so far failed, however, whatever theory lies beyond must yield the Standard Model as some limiting case.\n\nThe Standard Model has been scrutinised relentlessly since its inception. Remarkably, nearly without fail it has held its ground over the entire breadth of its theoretical reach and so the task of finding new ways to probe its structure requires ever more the creativity and ingenuity of both theorists and experimentalists alike.  Novel experimental signatures, against which to pit theory, must be used to maximum potential. From a theoretical standpoint there are still many challenges to be met, especially in preparation for the next generation of collider experiments now just round the corner. Particularly, the control and reduction of the theoretical uncertainty of Standard Model predictions is of paramount importance as only then can one hope to be in a position to discern signs of new physics from that of the Standard Model background.\n\nSome of its most challenging tests of the Standard Model fall in the field of heavy-flavour physics, within which $B$ physics has proven itself to be rich and fertile. Today it is an area of high activity with many success stories, including the recent measurement of the $B_s^0$-$\\bar{B}_s^0$ mass difference $\\Delta m_s$ at the Tevatron \\cite{Abulencia:2006ze}. Moreover, two dedicated ``$B$-factories'', Belle at KEK \\cite{:2000cg} and \\textsc{BaBar} at SLAC \\cite{Aubert:2001tu}, have measured a range of observables, such as branching fractions and CP asymmetries, of a vast number of $B$ decay modes. Looking to the future, the $B$ physics community eagerly await the forthcoming LHCb experiment, and beyond that so-called ``superflavour factories'' \\cite{superB} have been championed with the aim of probing rare $B$ decays to extract CP violation parameters to much higher levels of accuracy. It is imperative to find tests of the Standard Model  which may be observed in these up-and-coming experiments \\cite{Gershon:2006mt} and promising modes include the rare decays $B\\to V \\gamma$ and $B\\to K \\mu^+\\mu^-$. \n\nThe strict pattern of CP violation of the Standard Model finds its origin in the Cabbibo-Kobayashi-Maskawa (CKM) matrix \\cite{Cabibbo:1963yz,Kobayashi:1973fv}. CP violation was discovered in $B$ physics via the decay mode $B_d^0\\to J\/\\psi K_S^0$ and found to be large, in contrast to $K$ decays where the violation is tiny. The possible largeness of CP violation in $B$ decays offers promising ways to detect new physics indirectly via CP violating observables testing the CKM paradigm. \n\nTheoretically, central to the description of $B$ decays is the disentanglement of the weak decay process from strong interaction effects leading to a low-energy effective Hamiltonian in which the physics at a scale $\\mathcal{O}(M_W)$ is well under control. Achieving this goal for the wide range of $B$ decays of interest has only been possible through huge calculational effort; the availability in the literature of Wilson coefficients at next-to-leading-order, and in some cases next-to-next-to-leading-order, is testament to this.  Furthermore, the theoretical description of the matrix elements of effective $B$ decay operators has been hugely improved through QCD factorisation methods. We discuss and make use of one such framework, namely that introduced by Beneke, Buchalla, Neubert and Sachrajda \\cite{Beneke:1999br, Beneke:2000ry,Beneke:2001ev}. The so-called BBNS approach showed, to leading-order in a $1\/m_b$ expansion, that the $\\alpha_s$ corrections beyond naive-factorisation of a large class of non-leptonic $B$ decay matrix elements are calculable in terms of $B$ transition form factors and meson light-cone distribution amplitudes. Armed with the corresponding amplitudes the phenomenologist may construct observables, such as branching ratios, CP asymmetries and isospin symmetries, which may then be compared to experiment. The predictive power of the QCD factorisation framework is jeopardised by a poor understanding of both these non-perturbative QCD quantities and the impact of the generally unknown power-suppressed contributions $\\mathcal{O}(1\/m_b)$; this in part motivates the work of this thesis.\n\nIn this thesis we investigate $\\rm SU(3)_F$-breaking effects in vector meson distribution amplitudes which are crucial in differentiating between the particles $\\rho$, $K^*$ and $\\phi$. The leading non-perturbative DA parameters are determined via the method of QCD sum rules introduced by Shifman, Vainshtein and Zakharov  \\cite{Shifman:1978bz, Shifman:1978by, Shifman:1978bx}. The method provides a prescription for the systematic calculation of non-perturbative QCD parameters,  albeit with an irreducible error $\\sim 20-30\\%$, and constitutes an extremely useful theoretical tool. \n\nThe sum rule results have a direct application in the QCD factorisation description of $B$ decays to $\\rho$, $K^*$ and $\\phi$ mesons. In particular, radiative $B$ decays to vector mesons $B\\to V\\gamma$, are an excellent example of a process potentially sensitive to new physics contributions, as at leading order the decays are mediated at loop level in the Standard Model. We perform a phenomenological analysis of these decays using the QCD factorisation framework of Bosch and Buchalla \\cite{Bosch:2001gv,Bosch:2002bw} including leading power-suppressed corrections for which the updated non-perturbative distribution amplitude parameters find use.  The impact of the power-suppressed corrections on the key decay observables is discussed and leads to a better understanding of the theoretical uncertainty of the QCD factorisation predictions. \n\nAlso, we calculate important contributions to the $B\\to\\eta^{(\\prime)}$ transition form factors via a variant sum rule approach, known as light-cone sum rules, for which distribution amplitudes play a crucial role. The result of the analysis elucidates a major source of theoretical uncertainty of the $B\\to \\eta^{(\\prime)}$ form factor. The result impacts $B\\to K^* \\eta^{(\\prime)}$, for example, where the experimental data and QCD factorisation predictions of the branching ratios are inconsistent.\n\nThe thesis is structured as follows: \n\\begin{itemize}\n\\item{Chapter~\\ref{chapter1_basics} introduces some of the fundamentals of the Standard Model and its application to $B$ physics. We define the QCD Lagrangian and the CKM matrix, introduce CP violation in Standard Model $B$ decays, and briefly discuss the structure of the $\\Delta B =1$ weak effective Hamiltonian. }\n\\item{Chapter~\\ref{chapter2_DAs} covers the definitions of the light-cone distribution amplitudes of the light vector mesons $\\rho$, $K^*$ and $\\phi$. We determine their structure up to twist-3 accuracy and using the conformal expansion and QCD equations of motion express the distribution amplitudes in terms of a finite set of non-perturbative parameters. We extend previous determinations in order to fully differentiate between the three particles by including all G-parity violating contributions and $\\rm{SU}(3)_F$-breaking effects. }\n\\item{Chapter~\\ref{chapter3_SR}  discusses the QCD sum rule method and its extension light-cone sum rules.  The methods allow, amongst other things, the determination of the non-perturbative distribution amplitude parameters and transition form factors respectively, and are very widely applicable in and beyond $B$ physics.}\n\\item{In Chapter~\\ref{chapter4_det} we apply QCD sum rules to determine the leading non-perturbative distribution parameters defined in Chapter~\\ref{chapter2_DAs}. Consistency requires the inclusion of all G-parity violating contributions and $\\rm{SU}(3)_F$-breaking effects to the sum rules, and we extend previous determinations by including higher-order strange quark mass effects and $\\mathcal{O}(\\alpha_s)$ contributions to the quark condensates. We analyse the resulting sum rules and provide updated numerical results for all parameters. The results of this section find immediate application in QCD factorisation and light-cone sum rule descriptions of processes involving these vector mesons.}\n\\item{In Chapter~\\ref{chapter5_eta} we calculate the gluonic flavour-singlet contribution to the semileptonic $B\\to \\eta^{(\\prime)}$ transition form factor in the framework of light-cone sum rules. In doing so we discuss pseudoscalar meson and two-gluon distribution amplitudes. The new contribution is combined with the previous determination of the quark contribution, to complete the theoretical treatment of these form factors. The $\\eta^{(\\prime)}$ system is complicated due to large mixing effects via the $\\rm U(1)_A$ anomaly. We introduce the phenomenological framework of $\\eta$-$\\eta^{\\prime}$ mixing and connect it to the form factor calculation in a consistent manner. The results of this chapter find immediate application in the QCD factorisation description of $B\\to \\eta^{(\\prime)}$ transitions, which in turn, in principle, allow a determination of the CKM matrix element $|V_{ub}|$ from $B\\to\\eta^{(\\prime)} l \\nu$.}\n\\item{Chapter~\\ref{chapter6_QCDF}  introduces the framework of QCD factorisation, which is an important application of meson distribution amplitudes and transition form factors. We briefly discuss the BBNS approach and then go on to discuss the leading contributions to QCD factorisation in the context of $B\\to V \\gamma$ decays.}\n\\item{In Chapter~\\ref{chapter7_rad} we investigate the impact of the relevant, power-suppressed contributions to $B\\to V \\gamma$ beyond the QCD factorisation formula. We include long-distance photon emission from weak annihilation diagrams and soft gluon emission from quark loops. The non-perturbative distribution amplitude parameters determined in Chapter~\\ref{chapter4_det} find use in a light-cone sum rule estimation of the latter. The key observables are the branching ratios, isospin asymmetries and the indirect time-dependent CP asymmetry $S(V\\gamma)$  which, as has been know for some time, forms the basis of a ``null test'' of the Standard Model. Assuming no new physics contributions, we extract the ratio of CKM matrix parameters $\\left|V_{td}\/V_{td}\\right|$ to a competitive degree of accuracy.}\n\\item{We summarise and conclude in Chapter~\\ref{chapter8_conc}.}\n\\end{itemize}\nThe material of Chapters~\\ref{chapter2_DAs} and \\ref{chapter4_det} follows Ref.~\\cite{Ball:2007rt} and the material of Chapters~\\ref{chapter5_eta} and \\ref{chapter7_rad} follows Refs.~\\cite{Ball:2007hb} and \\cite{Ball:2006eu}, respectively. Some of the more bulky equations, and material not necessary in the general flow of reading the thesis, are given in two appendices. \n\\chapter{Fundamentals Of $B$ Physics}\\label{chapter1_basics}\nIn this chapter we begin with the basics of the Standard Model and then go on to  discuss two concepts which are central to the investigations of $B$ physics, and those of this thesis:\n\\begin{itemize}\n\\item{CP violation in the flavour sector, which follows a strict pattern in the Standard Model and can readily be sensitive to new physics;}\n\\item{the $\\Delta B=1$ effective weak Hamiltonian, which we briefly discuss as it is the starting point of many phenomenological studies in $B$ physics.}\n\\end{itemize}\n\n\n\\section{The Standard Model}\nThe \\textit{Standard Model} (SM) \\cite{weak,QCD} is a model of great scope and predictive power. Despite its successes, however, we know it to be incomplete; for example, the recent discovery of neutrino oscillation and the lack of conclusive evidence for the Higgs particle providing two areas of intense theoretical and experimental effort.  The SM describes three of the four known fundamental forces of nature;  the strong force, the weak force and electromagnetism. \\textit{Quantum Chromodynamics} (QCD) is a Yang-Mills gauge theory based on the gauge group $\\rm SU(3)$ and describes the fundamental interactions of the strong interaction as  interactions between quarks and gluons \\cite{Yang:1954ek,Gell-Mann:1964nj,FGM,Fritzsch:1973pi}. The basic QCD Lagrangian is\n\\begin{equation}\n\\mathcal{L}_{\\rm QCD}=\\sum_q \\bar{q}^i \\left(i \\gamma_\\mu \\left(D^\\mu\\right)_{ij}-m_q \\delta_{ij}\\right)q^j -\\frac{1}{4} G^a_{\\mu\\nu} G^{a \\mu\\nu}\\,,\n\\label{basics_eq1}\n\\end{equation}\nwith\n\\begin{equation}\n(D_\\mu)_{ij} = \\delta_{ij} \\partial_\\mu -i g_s (t^a)_{ij} A^a_\\mu\\,, \\qquad G^a_{\\mu\\nu}=\\partial_\\mu A_\\nu^a-\\partial_\\nu A_\\mu^a + g_s f^{abc}A^b_\\mu A^c_\\nu\\,,\n\\label{basics_eq2}\n\\end{equation}\nwhere the sum is over all quark flavours $q$, $i,j=\\{1,2,3\\}$ are colour indices, the $t^d$ are the $3\\times3$ colour matrices with $d=\\{1,\\dots,8\\}$ and $f^{abc}$ are the structure constants.  $G^a_{\\mu\\nu}$ is the gluonic field strength tensor, and $A^a_\\mu$ is the gluon field. We will make use of the notation $(G_{\\mu\\nu})_{ij}=G^a_{\\mu\\nu} (t^a)_{ij}$ and the relation $g_s^2=4 \\pi \\alpha_s$ (and $e^2=4 \\pi \\alpha_{\\rm QED}$). The Lagrangian can alternatively be defined with the replacement $g_s\\to-g_s$ and the sign convention matters for the applications in Chapters~\\ref{chapter4_det} and \\ref{chapter7_rad}.\n\nThe non-Abelian nature of QCD leads to the possibility of gluon self-interaction and the celebrated \\textit{asymptotic freedom} property of QCD \\cite{FGM,Fritzsch:1973pi,Politzer:1973fx,Gross:1973id1}. The coupling tends to zero, giving a theory of free quarks, at asymptotically high energy. On the other hand, at low energy, or large distances, the coupling increases.  At energies for which $\\alpha_s\\gtrsim1$ perturbation theory is not applicable, and one has to resort to \\textit{non-perturbative} methods to determine the effects of QCD. Despite the simplicity of the QCD Lagrangian (\\ref{basics_eq1}) an accurate determination of non-perturbative QCD from first principles, and hence  \\textit{confinement}, poses a major challenge. One such method, based on ideas of Wilson \\cite{Wilson:1974sk}, is that of Lattice QCD, which aims to calculate the QCD action computationally on a grid of discretised spacetime points.  An altogether different, and less rigourous, method is that of QCD sum rules, which encodes non-perturbative effects in terms of non-vanishing vacuum expectation values of operators with the quantum numbers of the vacuum. This method is central to the work in this thesis, and shall be discussed in Chapter~\\ref{chapter3_SR}.\n\nThe electroweak force is the unification of the weak nuclear force and electromagnetism given by the \\textit{Glashow-Salam-Weinberg model}. The model is based on the gauge group $\\rm SU(2)_L \\otimes U(1)_Y$, which is broken by  \\textit{spontaneous symmetry breaking} to yield $\\rm U(1)_Q$ - the gauge group corresponding to \\textit{Quantum Elecrodynamics} (QED).  The weak interaction is mediated by three massive gauge bosons $W^\\pm$ and $Z^0$ and occurs between quarks and leptons. The quarks and leptons are arranged, within the three generations, into left-handed doublets and right-handed singlets under $\\text{SU(2)}_{\\rm L}$\n\\begin{eqnarray}\n Q_{\\rm L}=  \\left(\\begin{array}{c} \n               U\\\\ \n                D \\\\ \n         \\end{array}\\right)_{\\rm L}\\,,\\quad\n          E_{\\rm L}=  \\left(\\begin{array}{c} \n               \\nu_l \\\\ \n                 l^- \\\\ \n         \\end{array}\\right)_{\\rm L} \\,;\\quad\n         U_{\\rm R}\\,,D_{\\rm R}\\,,l^-_{\\rm R}\\,,\n         \\label{basics_eq5}\n\\end{eqnarray}\nwhere the \\textit{weak eigenstates} $U=\\{u,c,t\\}$, $D=\\{d,s,b\\}$ and $l^-=\\{e^-,\\mu^-,\\tau^-\\}$ are the up-type quarks,  down-type quarks and charged leptons respectively. The subscript L (R) represents the left (right)-handed projectors $q_{\\rm L (R)}=\\frac{1}{2}(1\\mp \\gamma_5)q$ which reflect the chiral nature of the weak  interaction. The neutrinos are massless in the SM, and the right handed neutrino does not exist.  The electroweak interactions of the quarks are described by the following Lagrangian, which consists of a \\textit{charged current} ($CC$) and a \\textit{neutral current} ($NC$)\n\\begin{eqnarray}\n   \\mathcal{L}^{\\textrm{ew}} &=& \\mathcal{L}_{CC} +\n   \\mathcal{L}_{NC}\\,,\\nonumber \\\\\n    &=& \\frac{g}{\\sqrt{2}}\\left[J_\\mu^+W^{+\\mu} + J_\\mu^-W^{-\\mu}\\right]\\,,\\nonumber \\\\\n    & +&\n   e\\,\\left[J^{\\textrm{em}}_\\mu A^\\mu  \\right]+ \\frac{g}{\\cos{\\theta_W}}\\left[ \\left(J^3_\\mu - \\sin^2{\\theta_W}J^{em}_\\mu\\right) Z^\\mu   \\right]\\,.\n\\end{eqnarray}\nThe neutral current part of the Lagrangian is made up of the electromagnetic  current $J^{\\textrm{em}}_\\mu$ and neutral weak current $J^3_\\mu$:\n\\begin{equation}\n   J^{\\textrm{em}}_\\mu = Q_U\\,\\bar{U}_{\\rm L}\\gamma_\\mu U_{\\rm L}+Q_D\\,\\bar{D}_{\\rm L}\\gamma_\\mu D_{\\rm L}\\,, \\qquad\n   J^3_\\mu= \\frac{1}{2}(\\bar{U}_{\\rm L} \\gamma_\\mu U_{\\rm L}-\\bar{D}_{\\rm L}\\gamma_\\mu D_{\\rm L})\\,,\n\\end{equation}\nwhere $Q_{U(D)}=2\/3\\,(-1\/3)$ is the electric charge of the $U$ $(D)$ quarks, $\\theta_W$ is the weak mixing angle and $g$ is the electroweak coupling related to the electromagnetic coupling by $e=g \\sin \\theta_W$. Rotating to the basis of \\textit{mass eigenstates} modifies the charged current in the quark sector to\n\\begin{equation}\n   J^+_\\mu =  \\bar{U}_{\\rm L}^m \\gamma_\\mu \\,\\hat{V}_{\\rm{CKM}}\\, D_{\\rm L}^m\\,,\n\\end{equation}\nwhere $\\hat{V}_{\\mathrm{CKM}}$ is the \\textit{Cabbibo-Kobayashi-Maskawa} matrix \\cite{Cabibbo:1963yz,Kobayashi:1973fv} and the superscript $m$ denotes mass eigenstates. The CKM matrix is $3\\times3$ (for three quark generations), unitary, and its off-diagonal entries allow for transitions between the quark generations.  There are no flavour-changing neutral-currents (FCNC) at tree-level in the SM as the neutral currents $ J^{\\textrm{em}}_\\mu$ and $J^3_\\mu$ are invariant under the transformation to the mass eigenbasis, which is known as  the \\textit{Glashow-Iliopoulos-Maiani (GIM) mechanism} \\cite{Glashow:1970gm}. The entries of the CKM matrix are written as\n\\begin{equation}\n   \\hat{V}_{\\mathrm{CKM}} = \\left(\\begin{array}{ccc}\n                    V_{ud} & V_{us} & V_{ub} \\\\\n                    V_{cd} & V_{cs} & V_{cb} \\\\\n                    V_{td} & V_{ts} & V_{tb}  \n              \\end{array}\\right)\\,,\n\\label{basics_eq6}\n\\end{equation}\nand are fundamental parameters of the SM that have to be determined from experiment. Evidently, the matrix has $n^2=9$ parameters $n (n-1)\/2=3$ of which are rotation angles due to its unitarity. The six quark fields in Eq.~(\\ref{basics_eq5}) can be re-phased, up to an overall phase, leaving the Lagrangian invariant and therefore $9-5-3=1$ phase remains giving rise to complex entries -- complex coupling constants. This is the origin of \\textit{CP violation} in the quark sector of the weak interaction. The leptonic sector is described by an analogous mixing matrix which, in the absence of neutrino masses, is given by the unit matrix because all phases can be rotated away.\n\nThe CKM matrix (\\ref{basics_eq6}) is often parameterised to incorporate the constraints of unitarity.\\footnote{The ``standard'' parameterisation of the CKM matrix is in terms of the three mixing angles $\\theta_{ij}$ $(i,j=1,2,3)$ and the CP violating phase $\\delta$ \\cite{Yao:2006px}.} A very useful and convenient parameterisation is the \\textit{Wolfenstein parameterisation} \\cite{Wolfenstein:1983yz} which, along with unitarity, incorporates the experimental observations $|V_{us}|\\ll 1$, $|V_{cb}|\\sim|V_{us}|^2$ and $|V_{ub}| \\ll |V_{cb}|$. It is an expansion in $\\lambda=|V_{us}| \\approx 0.22$, and as such is only approximately unitary at a given order in $\\lambda$:\n\\begin{equation}\n  \\hat{V}_\\mathrm{CKM}=\\left(\n  \\begin{array}{ccc} \n     1-\\frac{\\lambda^2}{2} & \\lambda & A\\lambda^3 (\\rho-i\\eta) \\\\\n     -\\lambda & 1-\\frac{\\lambda^2}{2} & A\\lambda^2 \\\\\n     A \\lambda^3 (1-\\rho-i\\eta) & -A\\lambda^2 & 1\n   \\end{array}\\right)+\\mathcal{O}(\\lambda^4)\\,.\n   \\label{basics_eq7}\n\\end{equation}\nThe matrix is given in terms of the four parameters ($A,\\lambda,\\rho,\\eta$); $A$ and $\\rho^2+\\eta^2$ are order unity and the hierarchy of sizes of elements can be infered from the powers of $\\lambda$. The smallness of $V_{cb}$ and $V_{ub}$ are responsible for the relatively long lifetime of $B$ mesons (and baryons), which facilitates their experimental detection. The unitarity of the CKM matrix gives six equations that equal zero and can be represented as triangles in the complex plane. The most widely used of these relations in $B$ physics  is\n\\begin{equation}\nV_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0\\,,\n\\label{basics_eq8}\n\\end{equation}\nwhich is invariant under phase transformations and is an observable. The above relation is divided by $V_{cd}V_{cb}^*$ to give a triangle in the complex plane with a base of unit length and upper apex at the point $(\\bar\\rho,\\bar\\eta)$\\footnote{The following rescaling proves convenient to the definition of the UT: $\\rho\\to\\bar\\rho=\\rho\\,(1-\\lambda^2\/2)$ and $\\eta\\to\\bar\\eta=\\eta\\,(1-\\lambda^2\/2)$.} known as \\textit{The Unitary Triangle} (UT), see Figs.~\\ref{basics_fig1} and \\ref{basics_fig2}. The sides of the UT are given by\n\\begin{eqnarray}\nR_b &\\equiv& \\frac{|V_{ud}V_{ub}^*|}{|V_{cd}V_{cb}^*|} = \\sqrt{\\bar\\rho^2+ \\bar\\eta^2}=\n\\left(1-\\frac{\\lambda^2}{2}\\right)\\frac{1}{\\lambda}\n\\left|\\frac{V_{ub}}{V_{cb}}\\right|\\,,\\label{basics_eq88}\\\\ \nR_t &\\equiv& \\frac{|V_{td}V_{tb}^*|}{|V_{cd}V_{cb}^*|} = \\sqrt{(1-\\bar\\rho)^2+ \\bar\\eta^2}\\,.\n\\label{basics_eq9}\n\\end{eqnarray}\nThe angles are given by\n\\begin{equation}\n\\alpha\\equiv\\arg\\left(-V_{td}V_{ub}V^*_{tb}V_{ud}^*\\right)\\,,\\quad\\beta\\equiv\\arg\\left(-V_{cd}V_{tb}V^*_{cb}V_{td}^*\\right)\\,,\\quad\\gamma\\equiv\\arg\\left(-V_{ud}V_{cb}V^*_{ub}V_{cd}^*\\right)\\,.\n\\end{equation}\nThe (over) determination of the sides and angles of the UT is a major quest in understanding the SM.   To achieve this goal one must construct decay observables, which can then be matched to experimental results in order to extract values for the desired CKM (or equivalently UT) parameters. Such observables include branching ratios, which may appear simply proportional to a CKM matrix element, and CP asymmetries, which encode the effects of the SM predictions of CP violation, and can also be measured experimentally.\n\\begin{figure}[h]\n$$  \\epsfxsize=0.5\\textwidth\\epsffile{UT.eps}$$\n\\caption[The Unitary Triangle.]{\\small The Unitary Triangle. The determination of the sides $R_b$ and $R_t$ and the angles $\\alpha$, $\\beta$ and $\\gamma$ lead to stringent tests of the Standard Model.} \n\\label{basics_fig1}\n$$  \\epsfxsize=0.5\\textwidth\\epsffile{CKMfitterUT.eps}$$\n\\caption[Constraints on the angles and sides of the Unitarity Triangle.]{\\small Constraints on the angles $\\alpha$, $\\beta$, and $\\gamma$ and sides $R_b$ and $R_t$ of the Unitarity Triangle as imposed from numerous experimental sources. Complied by the CKM fitter group \\cite{global}.} \n\\label{basics_fig2}\n\\end{figure}\n\n\\section{CP Violation In $B$ Decays}\nDoes  the CKM matrix (\\ref{basics_eq6}) account for the CP violation observed in nature? Examining CP violation in $B$ decays allows one to probe the structure of the CKM matrix and is a very promising way to detect the effects of new physics, which many not be expressed through other decay observables. Consequently, the CP properties of  FCNC processes, which are characterised by their potential sensitivity to new physics effects, have been under intense theoretical and experimental investigation for many years. Prime examples of such processes include $B^0$-$\\bar B^0$ mixing (see for example Ref.~\\cite{Ball:2006xx}) and radiative $B$ decays, see Chapter~\\ref{chapter7_rad}.\n\nThe idea that the weak interaction may violate parity was first suggested many years ago by Lee and Yang \\cite{Lee:1956qn}, and quickly confirmed in the $\\beta$ decay of $^{60}$Co by Wu \\textit{et al.} \\cite{Wu:1957my}. The violation of the combined CP symmetry was first observed in the context of $K$ decays in 1964 \\cite{Christenson:1964fg} and it was not until 2001 that it was first observed outside the $K$ system in $B^0_d \\to J\/ \\psi \\,K^0_S$ decays \\cite{Aubert:2001nu,Abe:2001xe}; in both cases the CKM paradigm was upheld.  Recently discoveries in $B$ physics include the measurement by CDF of the mass difference $\\Delta m_s$ \\cite{Abulencia:2006ze}. Some of the most important sources of information about the UT from $B$ physics include: the determination of $\\sin 2 \\beta$ from the ``gold-plated'' decay $B\\to J\/\\psi \\,K_S$; the  extraction of $\\alpha$ from non-leptonic $B$ decays such as $B \\to \\pi\\pi$; the extraction of $|V_{td}|\/|V_{ts}|$ from $B$ mixing and radiative $B$ decays, such as $B\\to V \\gamma$; and the determination of $|V_{ub}|$ from $B\\to\\pi l\\nu$.\n\nThe $B^0_q$-$\\bar B^0_q$ systems, where $q=\\{d,s\\}$, exhibit the phenomenon of particle-antiparticle mixing, which, in the SM is mediated by so-called \\textit{box diagrams} whose amplitudes are $\\sim G_F^2$ and therefore very small.  We do not go into any detail about the theory of neutral state mixing and we restrict ourselves to only the formulas required in this thesis; for more information see Refs.~\\cite{Buras:1998ra,CPV}. State mixing causes, for example, an initially pure beam of $B^0$ mesons to evolve into a time-dependent linear combination of $B^0$ and $\\bar B^0$ mesons. There are four main quantities that describe the $B^0_q$-$\\bar B^0_q$ system and its decays: the width difference $\\Delta \\Gamma_q$, the mass difference $\\Delta m_q$, the CP violating mixing phase $\\phi_q$ and $\\lambda_f$ (not to be confused with the Wolfenstein CKM parameter $\\lambda\\approx 0.22$). One begins by writing the  heavy (H) or light (L) eigenstates of evolution in terms of the flavour states:\n\\begin{equation}\n\\ket{B_{\\rm H}}=p \\ket{B^0}-q \\ket{\\bar{B}^0}\\,,\\qquad \\ket{B_{\\rm L}}=p \\ket{B^0}+q \\ket{\\bar{B}^0}\\,,\n\\label{basics_eq10}\n\\end{equation}\nwith $|p|^2+|q|^2=1$. The ratio $q\/p$ is given in terms of the $B^0_q$-$\\bar B^0_q$  mixing matrix $M_{12}^q$,  by\n\\begin{equation}\n\\left.\\frac{q}{p}\\right|_{q} = \\sqrt{\\frac{(M_{12}^{q})^*}{M_{12}^{q}}} = e^{-i\\phi_{q}}\\,,\n\\label{basics_eq11}\n\\end{equation}\nunder the condition $\\Delta \\Gamma_q \\ll \\Delta m_q$.  Experimentally, there is no evidence for \\textit{mixing-indiced} CP violation in the $B^0_q$-$\\bar B^0_q$ systems, i.e. $\\left|q\/p\\right|_{d,s}\\approx 1$ \\cite{Barberio:2007cr}.   The CP violating mixing phase is given by $ \\phi_q={\\rm arg}\\left[M_{12}^q\\right]$ which in the SM and the Wolfenstein parametrisation of the CKM matrix can be written in terms of the UT angles as\n\\begin{equation}\n\\phi_d \\equiv {\\rm arg}[(V_{td}^* V_{tb})^2] = 2 \\beta\\,,\\qquad \n\\phi_s \\equiv {\\rm arg}[(V_{ts}^* V_{tb})^2] = -2 \\lambda\n\\left|\\frac{V_{ub}}{V_{cb}}\\right|  \\sin\\gamma\\,.\n\\label{basics_eq12}\n\\end{equation}\nBesides mixing-induced CP violation  there also exists \\textit{direct} and \\textit{indirect} CP violation for $B$ and $\\bar B$ decays to a common CP eigenstate $f$. The corresponding time-dependent CP asymmetry is given by\n\\begin{eqnarray}\nA_{CP}(t) &=& \\frac{\\Gamma(\\bar B^0_q(t)\\to f) - \\Gamma( B^0_q(t)\\to   \\bar   f)}{\\Gamma(\\bar B^0_q(t)\\to  f) + \\Gamma( B^0_q(t)\\to \\bar     f)} \\nonumber\\\\\n&=&\\underbrace{S(f)}_{\\rm indirect} \\sin(\\Delta m_q\\, t )-\\underbrace{C(f)}_{\\rm direct}\\cos(\\Delta m_q\\, t)\\,,\n\\label{basics_eq13}\n\\end{eqnarray}\nwhere we have neglected the width difference $\\Delta\\Gamma_q=2 {\\rm Re}\\left[M^q_{12} \\Gamma^{q*}_{12}\\right]\/|M^q_{12}|$. The oscillation frequency is set by the mass difference between the heavy and light states\n\\begin{equation}\n\\Delta m_q = m_H^q-m_L^q=2|M_{12}^{q}|\\,,\n\\label{basics_eq14}\n\\end{equation}\nand the current world averages are \\cite{Barberio:2007cr}:  \n\\begin{equation}\n\\Delta m_d =0.507\\pm 0.004 \\,{\\rm ps}^{-1}\\,,\\qquad\\Delta m_s=17.77\\pm \\overbrace{0.10}^{stat.}\\pm \\overbrace{0.07}^{sys.} {\\rm ps}^{-1}\\,.\n\\label{basics_eq15}\n\\end{equation}\nFinally, if  we define the observable quantity \n\\begin{equation}\n\\lambda_f =\\frac{q}{p} \\frac{\\bar A}{A}\\,,\n\\label{basics_eq16}\n\\end{equation}\nwhere $A$ denotes the decay amplitude, then the two CP asymmetries can be written as\n\\begin{equation}\nC(f)=\\frac{1-|\\lambda_f|^2}{1+|\\lambda_f|^2}\\,,\\qquad S(f)=\\frac{2 \\,{\\rm Im}\\left[ \\lambda_f\\right]}{1+|\\lambda_f|^2}\\,.\n\\label{basics_eq17}\n\\end{equation}\n\n\n\\section{Effective Field Theories Of Weak Decays}\\label{basics_eftowd}\nA very widely used tool in the theoretical description of $B$ decay processes is the framework of \\textit{effective field theories} \\cite{Gilman:1979bc,Buras:1998ra}. The framework simplifies the dynamics of the weak decay by relying on an \\textit{operator product expansion}  (OPE) \\cite{Wilson:1969zs} of the weak vertices to separate the short and long distance physics. The OPE yields a concise \\textit{effective Hamiltonian} $\\mathcal{H}^{eff}$ built from a set of local effective operators $Q_i$ multiplied by renormalisation-scale dependent perturbatively calculable \\textit{Wilson coefficient functions} $C_i(\\mu)$:\n\\begin{equation}\n\\left<\\mathcal{H}\\right> \\stackrel{\\rm OPE}{\\longrightarrow}\\left<\\mathcal{H}^{eff}\\right> \\sim \\sum_i C_i(\\mu) \\left<Q_i\\right>+\\mathcal{O}(k^2\/M_W^2)\\,,\\\n\\label{basics_eq18}\n\\end{equation}\nwhere $k$ is the momentum flowing through the $W$ boson propagator. The separation of energy scales stems naturally from the fact that the weak decay of the $B$ meson is governed by physics originating at well separated scales: $m_t,\\,M_W\\gg m_{b,c}\\gg \\Lambda_{\\rm QCD} \\gg m_{u,d,s}$. It is the interplay of weak and strong effects that complicates the treatment of these decays, and must be dealt with appropriately.  By taking into account radiative corrections to tree-level and penguin diagrams, ultimately one obtains the effective Hamiltonian in terms of the set of all relevant local operators, which is closed under renormalisation. The full $\\Delta B=1$ effective Hamiltonian is, for a final state containing a $D$ quark\n\\begin{equation}\n\\mathcal{H}^{eff}=\\frac{G_f}{\\sqrt{2}}\\sum_{U=u,\\,c}\\lambda_U^{(D)} \\left[C_1 Q_1^U+C_2 Q_2^U+C_{7\\gamma} Q_{7\\gamma}+C_{8g} Q_{8g}+\\sum_{i=3,\\dots,10} C_i Q_i\\right]\\,,\n\\label{basics_eq20}\n\\end{equation}\nwhere make use of the standard short-hand notation for the product of CKM matrix elements $\\lambda_U^{(D)}\\equiv V^*_{UD} V_{Ub}$.  The form of Eq.~(\\ref{basics_eq20}) is chosen by assuming the unitarity of the CKM matrix (\\ref{basics_eq8}) to explicitly remove the dependence of the top quark CKM matrix elements which originate from penguin loops.  The effective operators are\n\\begin{eqnarray}\n\\lefteqn{\\bf{Current-Current\\footnote{The literature is not consistent concerning the labelling of the two operators $Q_{1,2}$ and one should be aware that the practice of swapping of these two operators is commonplace. We use the convention that the larger Wilson coefficient belongs to $Q_2$; that is, $Q_1$ is the new operator.}:}}\\hspace{4cm}\\nonumber\\\\\nQ^U_1 &=& (\\bar D_i U_j)_{V-A}(\\bar U_j b_i)_{V-A}\\,,\n\\qquad Q^U_2 = (\\bar DU)_{V-A}(\\bar Ub)_{V-A}\\,,\\nonumber\\\\\n\\lefteqn{\\rm\\bf{QCD~Penguin:}}\\hspace{4cm}\\nonumber\\\\\nQ_3 &=& (\\bar Db)_{V-A} \\sum_q (\\bar qq)_{V-A}\\,,\n\\qquad Q_4 = (\\bar D_i b_j)_{V-A} \\sum_q (\\bar q_j q_i)_{V-A}\\,,\\nonumber\\\\\nQ_5 &=& (\\bar Db)_{V-A} \\sum_q (\\bar qq)_{V+A}\\,, \n\\qquad Q_6 = (\\bar D_i b_j)_{V-A} \\sum_q (\\bar q_j q_i)_{V+A}\\,,\\nonumber\\\\\n\\lefteqn{\\rm\\bf{Electroweak~Penguin:}}\\hspace{4cm}\\nonumber\\\\\nQ_7 &=& (\\bar Db)_{V-A} \\sum_q \\frac{3}{2} e_q (\\bar qq)_{V+A}\\,,\n\\qquad Q_8 = (\\bar D_i b_j)_{V-A} \\sum_q  \\frac{3}{2} e_q (\\bar q_j q_i)_{V+A}\\,,\\nonumber\\\\\nQ_9 &=& (\\bar Db)_{V-A} \\sum_q \\frac{3}{2} e_q  (\\bar qq)_{V-A}\\,, \n\\qquad Q_{10} = (\\bar D_i b_j)_{V-A} \\sum_q  \\frac{3}{2} e_q (\\bar q_j q_i)_{V-A}\\,,\\nonumber\\\\\n\\lefteqn{\\rm\\bf{Electromagnetic~Dipole:}}\\hspace{4cm}\\nonumber\\\\\nQ_{7\\gamma} &=& \\frac{e}{8\\pi^2}m_b\\, \n        \\bar D\\sigma^{\\mu\\nu}(1+\\gamma_5)F_{\\mu\\nu}\\,b\n         + \\frac{e}{8\\pi^2}m_D\\, \n        \\bar D\\sigma^{\\mu\\nu}(1-\\gamma_5)F_{\\mu\\nu}\\,b\\,, \\nonumber\\\\\n\\lefteqn{\\rm\\bf{Chromomagnetic~Dipole:}}\\hspace{4cm}\\nonumber\\\\\nQ_{8g} &=& \\frac{g_s}{8\\pi^2}m_b\\, \n        \\bar D\\sigma^{\\mu\\nu}(1+\\gamma_5)G_{\\mu\\nu}\\, b\n        + \\frac{g_s}{8\\pi^2}m_D\\, \n        \\bar D\\sigma^{\\mu\\nu}(1-\\gamma_5)G_{\\mu\\nu}\\, b\\,,\n        \\label{basics_eq21}\n\\end{eqnarray}\nwhere $e_q$ is the electric charge of the quark $q$ in units of $|e|$ and $F_{\\mu\\nu}$ is the photonic field strength tensor. The Wilson coefficients entering the effective Hamiltonian are essentially effective coupling constants of the local effective operators. One can view the renormalisation of the matrix elements as an equivalent renormalisation of their Wilson coefficients. One makes use of renormalisation-group techniques to sum the potentially large logarithms $\\sim \\ln M_W^2\/\\mu^2$ that appear naturally in the evolution from weak scales $\\mathcal{O}(M_W)$ to hadronic scales, such as $\\mu\\sim m_b$. The operators (\\ref{basics_eq21}) mix with each other under evolution and from the renormalisation-scale invariance of $\\mathcal{H}^{eff}$ one finds\n\\begin{equation}\n\\mu \\frac{d}{d \\mu} C_i (\\mu)=\\gamma_{ji}(\\mu)\\, C_j(\\mu)\\,, \n\\label{basics_eq22}\n\\end{equation}\nwhere $\\hat \\gamma$ is the \\textit{anomalous dimension} matrix, which can be given as an expansion in the strong coupling via the renormalisation constant $\\hat Z$\n\\begin{equation}\n\\gamma_{ji}(\\mu)= Z^{-1}_{ik}\\frac{d Z_{kj}}{d \\ln \\mu}\\,,\\qquad \\hat{\\gamma}=\\left(\\frac{\\alpha_s}{4\\pi}\\right)\\hat{\\gamma}^{(0)}+\\left(\\frac{\\alpha_s}{4\\pi}\\right)^2\\hat{\\gamma}^{(1)}+\\mathcal{O}(\\alpha_s^3)\\,.\n\\label{basics_eq23}\n\\end{equation}\nSolving  Eq.~(\\ref{basics_eq22}) yields the evolution of the Wilson coefficients via the evolution matrix $\\hat U(\\mu,\\mu_0)$\n\\begin{equation}\n C_i(\\mu)= U_{ij}(\\mu,\\mu_0)\\, C_j(\\mu_0)\\,,\\qquad\\hat{U}(\\mu,\\mu_0)=\\exp \\int^{g(\\mu)}_{g(\\mu_0)}dg^\\prime\\,\\frac{\\hat{\\gamma}^{T}(g^\\prime)}{\\beta(g^\\prime)}\\,,\n\\label{basics_eq24}\n\\end{equation}\nwhere $\\beta(g)$ is the QCD $\\beta$-function. To leading order one has\n\\begin{equation}\n\\hat{U}^{\\rm LO}(\\mu,\\mu_0)=\\left(\\frac{\\alpha_s(\\mu_0)}{\\alpha_s(\\mu)}\\right)^{\\frac{\\hat{\\gamma}^{(0)T}}{2 \\beta_0}}= \\hat V\\left[\\left(\\frac{\\alpha_s(\\mu_0)}{\\alpha_s(\\mu)}\\right)^{\\frac{\\overrightarrow{\\gamma}^{\\left(0\\right)}}{2 \\beta_0}}\\right]_D\\hat V^{-1}\\,,\n\\label{basics_eq25}\n\\end{equation}\nwhere $V$ is the matrix that diagonalises $\\hat{\\gamma}^{(0)T}$ and $\\overrightarrow{\\gamma}^{\\left(0\\right)}$ is a vector of the eigenvalues of the leading order anomalous dimension matrix $\\hat{\\gamma}^{(0)}=\\hat V\\hat{\\gamma}^{(0)T}_D\\hat V^{-1}$. At NLO we have\n\\begin{equation}\nC_i(\\mu)=C_i^{(0)}(\\mu)+\\frac{\\alpha_s(\\mu)}{4\\pi}C_i^{(1)}(\\mu)\\,,\n\\label{basics_eq26}\n\\end{equation}\nand the evolution is a bit more complicated:\n\\begin{equation}\n\\hat{U}^{\\rm NLO}(\\mu,\\mu_0)=\\left[1+\\frac{\\alpha_s(\\mu)}{4\\pi}\\hat{J}\\right]\\hat{U}^{\\rm LO}(\\mu,\\mu_0)\\left[1-\\frac{\\alpha_s(\\mu_0)}{4\\pi}\\hat{J}\\right]\\,,\n\\label{basics_eq27}\n\\end{equation}\nwith\n\\begin{equation}\n\\hat{J}=V \\hat{S}V^{-1}\\,,\\qquad S_{ij}=\\delta_{ij}\\gamma_{i}^{(0)}\\frac{\\beta_1}{2\\beta_0^2}-\\frac{G_{ij}}{2\\beta_0+\\gamma_i^{(0)}-\\gamma_j^{(0)}}\\,,\\qquad\\hat{G}=V^{-1} \\hat{\\gamma}^{(1)T}V\\,.\n\\label{basics_eq28}\n\\end{equation}\nTo NLO the required $\\beta$-function coefficients are $\\beta_1=\\frac{34}{3}N_c^2-\\frac{10}{3}N_c N_f -2 C_F N_f$ and $\\beta_0=\\frac{11}{3}N_c  -\\frac{2}{3} N_f$ with $N_f$ is the number of active flavours, $C_F=(N_c^2-1)\/(2N_c)$ and $N_c$ the number of colours. Care must be taken in evolving through ``thresholds'' where the number of active flavours $N_f$ changes; the evolution must then be taken in stages, as a change in $N_f$ changes the $\\beta$-function coefficients and the anomalous dimension matrices. If there is a flavour threshold $\\mu_{\\rm th}$  between $\\mu_0$ and $\\mu$, which changes the number of active flavours from $N_f$ to $N_f+1$, then one has to make the replacement\n\\begin{equation}\n\\hat U(\\mu,\\mu_0)\\to \\left.\\hat U(\\mu,\\mu_{\\rm th})\\right|_{N_f +1}\\cdot \\left.\\hat U(\\mu_{\\rm th},\\mu_0) \\right|_{N_f}.\n\\label{basics_eq29}\n\\end{equation} \nThe effective Hamiltonian, combined with the renormalisation-group improvement of the perturbative series forms an exceptionally powerful framework. The matrix elements of the local operators $\\left<Q_i\\right>$ are the subject of QCD factorisation theorems, such as that discussed in Chapter~\\ref{chapter6_QCDF}, which allow the calculation of $B$ decay amplitudes.  From these amplitudes one can construct observables such as branching fractions, CP asymmetries, and isospin asymmetries which can be investigated phenomenologically.\n\\chapter{Vector Meson Light-Cone Distribution Amplitudes}\\label{chapter2_DAs}\nIn this chapter we discuss light vector meson light-cone distribution amplitudes and via the (approximate) conformal symmetry of QCD present expressions for the distribution amplitudes up to twist-3. The method introduces a set of non-perturbative parameters which is reduced in size by invoking the QCD equations of motion to relate the two-particle twist-3 distribution amplitudes to the three-particle twist-3 and two-particle twist-2 distribution amplitudes. In our analysis we include all $\\rm SU(3)_F$-breaking effects and G-parity violating terms thus allowing one to fully differentiate between $\\rho$, $K^*$ and $\\phi$ mesons. Moreover, a non-zero quark mass induces a mixing between twist-2 and twist-3 parameters under a change of renormalisation scale $\\mu$. To simplify notation we explicitly consider the $K^*$ meson, with quark composition $s\\bar{q}$ where $q=\\{u,d\\}$.\\footnote{The notation in this thesis, $K^*$ being a $(s\\bar q)$   bound state, is in contrast to the standard labelling, according to  which $K^{*0}=(d\\bar s)$ and $\\bar K^{*0}=(s\\bar d)$. This is the standard notation used for light-cone distribution amplitudes where $K^*$ always contains an $s$ quark, and $\\bar K^*$ an $\\bar s$  quark. This distinction is relevant because of  a sign change of G-odd matrix elements under $(s\\bar  q)\\leftrightarrow (q\\bar s)$. This notation also applies to calculations of form factors and other matrix  elements which involve light-cone distribution amplitudes.}\n\nThere are two main applications of meson distribution amplitudes that motivate their study:\n\\begin{itemize}\n\\item{they are directly applicable to the theoretical description of exclusive decay processes via QCD factorisation theorems, which require the distribution amplitudes as a non-perturbative input, see Chapter~\\ref{chapter6_QCDF}.}\n\\item{they are also applicable to the determination of transition form factors from the light-cone sum rule approach and as such are indirectly applicable to the same QCD factorisation theorems for which the transition form factors are also required,  see Chapters~\\ref{chapter3_SR} and \\ref{chapter5_eta}.}\n\\end{itemize}\nIn Chapter~\\ref{chapter4_det} we calculate,  from QCD sum rules,  numerical values for the leading twist-2 and twist-3 distribution amplitude  parameters defined here. Standard notations used, such as the light-cone coordinates, are given in Appendix~\\ref{appendixA}. The material covered in this chapter partially follows that of Ref.~\\cite{Ball:2007rt}.\n\n\\section{Introduction}\nHadronic light-cone distribution amplitudes (DAs) of light mesons were first discussed in the ground-breaking papers of Brodsky, Lepage, and others, see Refs.~\\cite{Chernyak:1977fk, Chernyak:1980dk, Lepage:1980fj, Efremov:1979qk, Efremov:1978rn,Chernyak:1977as, Lepage:1979zb,Chernyak:1980dj} and play an essential role in the QCD description of hard exclusive processes \\cite{Chernyak:1981zz,Brodsky:1989pv}.  The amplitudes that describe such processes factorise in the asymptotic limit $Q^2\\sim 1\/x^2 \\to \\infty$ -- where $Q^2$ is the momentum transfer and $x$ the transverse separation of the partons -- and are dominated by contributions from near the light-cone. The factorisation is given by the convolution of a hard-scattering kernel, calculable in perturbation theory, and process-independent, universal, non-perturbative DAs. \n\nThe study of hadronic DAs has a long history. The simplest and first to be investigated were the twist-2 DA of the $\\pi$ \\cite{Lepage:1980fj,Efremov:1979qk,Chernyak:1977as,Lepage:1979zb}. Higher twist DAs of the $\\pi$, alongside those of the other pseudoscalar mesons followed \\cite{Ball:1998je}.   For vector mesons, the leading-twist DAs of the $\\rho$ were first investigated by Chernyak and Zhitnitsky in Ref.~\\cite{Chernyak:1983ej} and later in Refs.~\\cite{Ali:1993vd, Ball:1996tb}. The formalism of higher twist-3 and twist-4 contributions, including meson mass corrections, was investigated by Ball \\textit{et al.} in Refs.~\\cite{Ball:1998sk,Ball:1998ff,Ball:2006wn,Ball:2007zt}. \n\nThe DAs of the $K^*$  ($K$) differ to those of the $\\rho$ ($\\pi$) due to the non-zero strange quark mass which yields $\\rm SU(3)_F$-breaking and G-parity violating corrections from a number of different sources.\\footnote{Perfect $\\rm SU(3)_F$ symmetry is realised for equal $u,d,$ and $s$ quark masses.} The study of the various contributions span many publications:\n\\begin{itemize}\n\\item{explicit quark mass corrections to DAs and evolution equations are generated by the QCD equations of motion (EOM) and only affect higher twist DAs. The contributions for vector mesons were calculated in Ref.~\\cite{Ball:1998sk} up to twist-3, and those to the evolution equations for vector mesons in Ref.~\\cite{Ball:2007rt} and flavour-octet pseudoscalar mesons Ref.~\\cite{Ball:2006wn}.}\n\\item{G-parity violating contributions, which are proportional to $m_s-m_q$ and hence vanish for equal quark masses, i.e. for $\\rho$ and $\\phi$, were investigated for twist-2 DAs in Refs.~\\cite{Chernyak:1983ej,Ball:1998sk,Ball:2003sc,Braun:2004vf,Ball:2005vx,Ball:2006fz} and for twist-3 DAs in Ref.~\\cite{Ball:2007rt}.}\n\\item{$\\rm SU(3)_F$-breaking of non-perturbative hadronic parameters entering the DAs.\nThe effects for the twist-2 parameters are known from\nRefs.~\\cite{Chernyak:1983ej,Ball:1998sk,Ball:2003sc}, twist-3 from Ref.~\\cite{Ball:2007rt} and twist-4 from Ref.~\\cite{Ball:2007zt}. The twist-3 vector meson parameters are discussed in Chapter~\\ref{chapter4_det} where we include all these effects in a determination of numerical values using QCD sum rules.}\n\\end{itemize}\nThe objects which define the DAs are vacuum-to-meson matrix elements of non-local operators at strictly light-like separations $z^2=0$ \\cite{Chernyak:1983ej}. Two examples we shall encounter are\n\\begin{eqnarray}\n\\bra{0}\\bar{q}(z)\\Gamma [z,-z] s(-z) \\ket{K^*(p,\\lambda)}\\,,\\qquad \\bra{0}\\bar{q}(z) [z,v z] g_s G_{\\mu \\nu}(vz)\\Gamma [v z,-z] s(-z) \\ket{K^*(p,\\lambda)}\\,,\\nonumber\\\\\n\\label{das_eq1}\n\\end{eqnarray}\nwhere $\\Gamma$ is a general Dirac matrix, $\\lambda=\\{\\parallel,\\perp\\}$ is the polarisation of the $K^*$ meson and the quark fields are taken at symmetric separation for simplicity.\\footnote{The Dirac matrices  $\\Gamma =\\{ \\sigma_{\\mu \\nu},\\,i \\gamma_5,\\, \\bf 1\\}$ give rise to so-called \\textit{chiral-odd} distributions because they are chirality-violating. Likewise, distributions generated from $\\Gamma =\\{ \\gamma_{\\mu},\\,\\gamma_{\\mu} \\gamma_5\\}$ are \\textit{chiral-even}.}  The first (second) matrix element above corresponds to a two- (three-) particle Fock state. To render the matrix element gauge invariant the path-ordered gauge factor is included\n\\begin{equation}\n[x,y]=\\textrm{P}\\, \\exp \\left[ i g_s \\int^1_0 dt\\, (x-y)_\\mu A^\\mu (t x+(1-t)y)\\right].\n\\end{equation}\nFor convenience we work in the \\textit{fixed-point} gauge\\footnote{also known as the \\textit{Fock-Schwinger} gauge.}\n\\begin{equation}\n(x-x_0)^\\mu A_\\mu^a (x)=0\\,,\n\\label{das_eq2}\n\\end{equation}\nand by choosing $x_0=0$ we have $[x,-x]=1$. The gauge factor will be implied unless otherwise stated. The DAs are dimensionless functions of  the collinear momentum fractions of a fixed number of constituents within a meson, at zero transverse separation. For two-particle DAs the constituent strange quark and antiquark  ($\\bar{q}$) share $u$ and $\\bar{u}=1-u$ of the meson momentum $p$ respectively.  For three-particle DAs we have $\\underline{\\alpha} = (\\alpha_1, \\alpha_2, \\alpha_3 )$ corresponding to the momentum fractions carried by the strange quark, antiquark ($\\bar{q}$) and gluon, respectively.  For a minimum number of constituents, the DAs are related to the \\textit{Bethe-Salpeter wavefunction} $\\phi_{BS}$ by integration over the transverse momenta\n\\begin{equation}\n\\phi(u, \\mu) \\sim \\int^{|k_{\\perp}|<\\mu}d^2 k_{\\perp}\\,\\phi_{BS}(u,k_{\\perp})\\,,\n\\label{das_eq3}\n\\end{equation}\nwhere  $\\mu$ is the renormalisation scale.  The price to pay for integrating out $k_{\\perp}$ below $\\mu$ is a renormalisation-scale dependence of the DAs governed by renormalisation-group equations.  The DAs have to be evaluated at the scale $\\mu^2 \\sim x^{-2}$ i.e. of the order of the deviation from the light-cone \\cite{Balitsky:1987bk}.\n\nNon-local operators that appear at finite $Q^2$ or mass scales are expanded near the light-cone $x^2 \\neq 0$ as an OPE in terms of the renormalised  non-local operators on the light-cone - the \\textit{light-cone expansion} \\cite{Balitsky:1987bk}.\\footnote{The expansion is facilitated by using light-cone coordinates which are given in Appendix~\\ref{appendixA}.} After taking matrix elements the resulting Lorentz-invariant amplitudes are matched to the definitions of the DAs with the coefficient functions of the expansion taken at tree-level, to leading logarithmic accuracy.  \n\nThe structure of vector meson DAs follows the same pattern as the nucleon structure functions and can be classified in the same way \\cite{Jaffe:1991ra}. They are described by separate DAs for each polarisation and thus there are more vector meson DAs than pseudoscalar DAs. \n\nLastly, we briefly mention some other DAs. Flavour-singlet pseudoscalar meson DAs\nare complicated by the $\\rm U(1)_A$ anomaly of QCD and are discussed in Chapter~\\ref{chapter5_eta} in the context of the $B \\to \\eta^{(\\prime)}$ transition form factor \\cite{Ball:2007hb}. Much work has been done concerning the DAs of heavy mesons, such as the $B$ meson \\cite{Szczepaniak:1990dt ,Braun:2003wx}; indeed, the DAs of $B$ mesons enter the QCD factorisation framework of radiative and non-leptonic $B$ decays, as discussed in Chapter~\\ref{chapter6_QCDF}, and a variant light-cone sum rule method devised in Ref.~\\cite{Khodjamirian:2006st}. There also exist DAs of the photon which describe its ``soft'' hadronic components, along with the usual ``hard'' electromagnetic components \\cite{Ball:2002ps}. The photonic DAs can be important in, for example, $B \\to V \\gamma$ decays \\cite{Ball:2006eu} as investigated in Chapter~\\ref{chapter7_rad}, and $B \\to \\gamma e \\nu$ \\cite{Descotes-Genon:2002mw,Ball:2003fq}. Finally, the field of baryon DAs is also active and many of the tools and concepts we cover in this thesis find application there, see for example Ref.~\\cite{Braun:2006hn} for a review.\n\n\\section{The Conformal Expansion}\nThe standard determination of meson DAs  proceeds by making use of the conformal symmetry of massless QCD at tree-level.  The conformal expansion is analogous to the partial wave expansion of wave functions in quantum mechanics in spherical harmonics $\\psi(r,\\theta,\\phi) \\to R(r) \\sum_{m,l} Y^l_m (\\theta,\\phi)$. The expansion uncovers a simple multiplicative renormalisation at leading-order, and as such different partial waves, with different \\textit{conformal spin}, do not mix under  a change of renormalisation scale.  At next-to-leading-order this is not the case, because strictly speaking the conformal symmetry of a quantum theory requires its $\\beta$ function to vanish. Proximity to the conformal limit in QCD is therefore governed by the value of the strong coupling constant, becoming true as $\\alpha_s \\to 0$ and we pass to the free theory.\\footnote{It must be noted that mass terms break the conformal expansion immediately at the classical level. This does not upset the conformal expansion, however. See Ref.~\\cite{Braun:2003rp} for details.} Using the QCD equations of motion we can elucidate this mixing order-by-order in the conformal expansion. \n\nThe application of conformal symmetry to exclusive processes has recieved a lot of attention in the literature, see Refs.~\\cite{Brodsky:1980ny,Brodsky:1985ve,Ohrndorf:1981qv,Braun:1989iv,Makeenko:1980bh}. The main benefit of the conformal expansion is the systematic separation of the longitudinal and transverse degrees of freedom in meson DAs. The former correspond to the longitudinal momentum fractions and is given by irreducible representations of the relevant symmetry group, SL(2,$\\mathbb R$).  The latter are integrated out to yield a renormalisation-scale dependence of the DAs, described by renormalisation-group equations.  Here we focus on the most important points, see Ref.~\\cite{Braun:2003rp} for a detailed review. \n\n\\subsection{Conformal Group}\nThe conformal group is defined as all transformations that change only  the scale of the metric and as such preserve angles and leave the light-cone invariant $g_{\\mu\\nu}^\\prime(x^\\prime)=\\omega(x) g_{\\mu\\nu}(x)$; the spacetime interval $ds^2=g_{\\mu\\nu}(x) \\,dx_\\mu dx_\\nu$ is conserved up to scaling. These transformations form a generalisation of the Poincar\\'e group. The full conformal algebra in 4 dimensions includes fifteen generators\n\\begin{eqnarray}\n\\textbf{P}_\\mu &\\to& \\rm 4~ Translations, \\nonumber\\\\\n\\textbf{M}_{\\mu \\nu}&\\to& \\rm 6 ~Lorentz ~rotations, \\nonumber\\\\\n\\textbf{D} &\\to& \\rm 1 ~Dilatation,\\nonumber\\\\\n\\textbf{K}_\\mu &\\to& \\rm 4 ~Special ~conformal ~translations.\n\\label{das_eq4}\n\\end{eqnarray}\nOur hadronic picture is of partons moving collinearly in, say the $p_{\\mu}$ direction, existing near the light-cone. We therefore restrict the \\textit{fundamental fields} of the conformal group to the light-cone\n$\\Phi(x) \\to \\Phi(\\alpha z)$, where $\\alpha $ is a real number, and we assume fields to be eigenstates of the spin operator \n\\begin{equation}\n\\Sigma^{\\mu\\nu} \\psi = \\frac{i}{2}\\sigma^{\\mu\\nu} \\psi\\,,\n\\label{das_eq5}\n\\end{equation}\nso as to have a fixed Lorentz-spin projection $s$ in the $z_{\\mu}$ (``plus'') direction $\\Sigma_{+-} \\Phi(\\alpha z) = s \\,\\Phi(\\alpha z)$. For leading-twist operators this is automatically satisfied and for higher-twist operators projections are used to separate different spin states, as we shall discuss shortly. The full conformal symmetry (\\ref{das_eq4}) is now modified  and it turns out that the resulting group of transformations form the special linear group SL(2,$\\mathbb R$), or so-called \\textit{collinear conformal group}, given by just four generators. They are written in standard form by constructing the following linear combinations\n\\begin{eqnarray}\n\\textbf{L}_+= \\textbf{L}_1+i \\textbf{L}_2=-i \\textbf{P}_+\\,,& \\qquad& \\textbf{L}_-= \\textbf{L}_1-i \\textbf{L}_2= \\frac{i}{2} \\textbf{K}_-\\,,\\nonumber\\\\\n\\textbf{L}_0= \\frac{i}{2}(\\textbf{D}+\\textbf{M}_{+-})\\,,& \\qquad& \\textbf{E}=\\frac{i}{2}(\\textbf{D}-\\textbf{M}_{+-})\\,.\n\\end{eqnarray}\nwhich leads to the familiar relations\n\\begin{equation}\n[\\textbf{L}_0,\\,\\textbf{L}_\\mp]=\\mp \\textbf{L}_\\mp\\,, \\qquad [\\textbf{L}_-,\\,\\textbf{L}_+]=-2 \\textbf{L}_0\\,.\n\\end{equation}\nThe operators act on the fundamental fields as\n\\begin{eqnarray}\n\\left[\\textbf{L}_+,\\Phi(\\alpha z) \\right]&=&-\\partial_{\\alpha} \\Phi(\\alpha n)\\,,\n\\label{das_eq6}\\\\\n\\left[\\textbf{L}_{-},\\Phi(\\alpha z)\\right]&=&(\\alpha^2 \\partial_{\\alpha} +2 j \\alpha)\\Phi(\\alpha n)\\,,\n\\label{lower}\\\\\n\\left[\\textbf{L}_0,\\Phi(\\alpha z) \\right]&=&(\\alpha \\partial_{\\alpha}+j) \\Phi(\\alpha n)\\,,\n\\label{das_eq7}\\\\\n\\left[\\textbf{E},\\Phi(\\alpha z)\\right]&=&\\frac{1}{2} (l-s) \\Phi(\\alpha n)\\,,\n\\label{das_eq8}\n\\end{eqnarray}\nwhere $t=l-s$ is the \\textit{twist},\\footnote{strictly it is the \\textit{collinear twist} which is defined as ``dimension minus spin projection on the positive direction''. There also exists \\textit{geometric twist} which is defined as ``dimension minus spin''.} $l$ is the canonical mass dimension,\\footnote{For example, $l=3\/2$ for quarks and $l=2$ for gluons.} $s$ the Lorentz-spin projection, and $j=\\frac{1}{2}(l+s)$ the \\textit{conformal spin} of the field $\\Phi$. The conformal spin specifies the representation of the collinear conformal group. The operator $\\textbf{E}$ commutes with all $\\textbf{L}_i$ and therefore twist is a good quantum number for each conformal field. The Casimir operator commutes with all $\\textbf{L}_i$ and is given by\n\\begin{equation}\n\\sum_{i=0,1,2}[\\textbf{L}_i,[\\textbf{L}_i,\\Phi(\\alpha z)]]=j(j-1)\\Phi(\\alpha z) = \\textbf{L}^2 \\Phi(\\alpha z)\\,.\n\\end{equation}\nAt the origin of the light-cone $\\alpha=0$ and the field $\\Phi(0)$ is killed by the lowering operator $\\textbf{L}_-$ and as such has the minimum spin projection $j_{\\rm min}$ of states of conformal spin $j$. One can define a \\textit{conformal operator} $\\mathbb{O}_n=\\Phi(0)$ by requiring that it transforms just as the fundamental field, Eqs.~(\\ref{lower} - \\ref{das_eq8}), and is killed by the lowering operator $\\textbf{L}_-$. The raising operator $\\textbf{L}_+$ can be repeatedly applied to $\\Phi(0)$ to give\n\\begin{equation}\n\\mathbb{O}_{n,n+k}=\\underbrace{\\left[ \\textbf{L}_+,...,\\left[\\textbf{L}_+,\\left[\\textbf{L}_+\\right.\\right.\\right.}_{k},\\left.\\left.\\left.\\Phi(0)\\right]\\right]\\right]=\\left(-i \\partial_+\\right)^k \\mathbb{O}_n\\,,\n\\end{equation}\nwhere $\\mathbb{O}_{n,n}=\\mathbb{O}_n$ and the subscript $n$ defines the \\textit{conformal tower} of states, of conformal spin $j_{\\textrm{min}}<j_{\\textrm{min}}+k<\\infty$, generated by the collinear conformal algebra. This is an infinite dimensional representation of the collinear conformal group.\n\n\\subsection{States of Definite Spin}\nNow the main language of the collinear conformal group is defined, if one can relate the fundamental fields $\\Phi$ to the operators of hard processes in QCD one can export all the machinery above and immediately reap the benefits. To this end,  consider the non-local two-particle operator at light-like separation (\\ref{das_eq1}) and expand at small distances\n\\begin{equation}\n\\bar{q}(z)\\Gamma s(-z)= \\sum_k \\frac{1}{k!}\\, \\bar{q}(0) (\\stackrel{\\leftrightarrow}{D} \\cdot z)^k \\Gamma s(0)\\,,\n\\label{das_eq9}\n\\end{equation}\nwhere $\\stackrel{\\leftrightarrow}{D}_\\mu= \\stackrel{\\rightarrow}{D}_\\mu- \\stackrel{\\leftarrow}{D}_\\mu$. The question is, how does one express these local operators in terms of conformal operators and thus separate all the different twist contributions?  To proceed one decomposes the quark fields into definite Lorentz-spin components using the projection operators\n\\begin{equation}\n\\Pi_+ = \\frac{1}{2} \\frac{\\slash{p}\\slash{z}}{ p\\cdot z}\\,, \\qquad \\Pi_- = \\frac{1}{2} \\frac{\\slash{z}\\slash{p}}{ p\\cdot z}\\,, \\qquad \\Pi_+ +\\Pi_- =1\\,.\n\\end{equation}\nwhich project onto the ``plus'' and ``minus'' components of the spinor respectively. Using the generator of the spin rotations of a spinor field (\\ref{das_eq5}) one can show\n\\begin{equation}\n\\underbrace{\\psi_+ = \\Pi_+ \\psi}_{ \\shortstack{\\footnotesize{$s=+1\/2$} \\\\ \\footnotesize{$j=1$}}}\\,, \\qquad \\underbrace{\\psi_- = \\Pi_- \\psi}_{ \\shortstack{\\footnotesize{$s=-1\/2$} \\\\ \\footnotesize{$j=1\/2$}}}\\,.\n\\end{equation}\nThe composite operators (\\ref{das_eq9}) have  conformal spin projection $j=j_q+j_s+k$  where the subscripts correspond to the separate quark fields. The composite operators are ordered by increasing twist\n\\begin{equation}\n\\underbrace{\\bar{q}_{+} \\Gamma s_{+}}_{ \\rm t=2}\\,, \\qquad \\underbrace{\\bar{q}_{+} \\Gamma s_{-} + \\bar{q}_{-} \\Gamma s_{+}}_{ \\rm t=3}\\,, \\qquad \\underbrace{\\bar{q}_{-} \\Gamma s_{-}}_{ \\rm t=4}\\,. \n\\label{das_eq10}\n\\end{equation}\nIt can be shown that the corresponding local conformal operators are\n\\begin{eqnarray}\n\\mathbb{Q}_n^{\\rm t=2}(x) &=& (i\\partial_+)^n \\left[\\bar{q}(x) \\gamma_+ C^{3\/2}_{n} \\left(\\stackrel{\\leftrightarrow}{D}_+ \/ \\partial_+\\right) s(x) \\right]\\,,\n\\label{das_eq11}\\\\\n\\mathbb{Q}_n^{\\rm t=3}(x) &=& (i\\partial_+)^n \\left[\\bar{q}(x)\\gamma_+ \\gamma_\\perp \\gamma_-P^{(1,0)}_{n} \\left(\\stackrel{\\leftrightarrow}{D}_+ \/ \\partial_+\\right) s(x) \\right]\\,,\\label{das_eq12}\\\\\n\\mathbb{Q}_n^{\\rm t=4}(x) &=& (i\\partial_+)^n \\left[\\bar{q}(x) \\gamma_- C^{1\/2}_{n} \\left(\\stackrel{\\leftrightarrow}{D}_+ \/ \\partial_+\\right) s(x) \\right]\\,,\n\\label{das_eq13}\n\\end{eqnarray}\nwhere $\\partial_+ =\\stackrel{\\leftarrow}{D}_+ +\\stackrel{\\rightarrow}{D}_+$, $C^m_n(x)$ are \\textit{Gegenbauer polynomials} and $P^{(r,s)}_n(x)$ are \\textit{Jacobi polynomials}. There is another twist-3 operator corresponding to Eq.~(\\ref{das_eq12}) with the replacement $P^{(1,0)}_{n}(x) \\to P^{(0,1)}_{n}(x)$. One can now connect firmly to QCD with the specific example of the leading-twist operator; consider again Eq.~(\\ref{das_eq1}) and specify the twist-2 Dirac matrix $\\Gamma \\to \\gamma_\\mu$ (projected onto $z_\\mu$) and define the DA as\n \\begin{equation}\n \\bra{0} \\bar{q}(z)\\gamma_z s(-z) \\ket{K^*(p,\\lambda)}= f_{K^*}^\\parallel m_{K^*} (e^{(\\lambda)}\\cdot z) \\int^1_0 du\\, e^{-i [\\bar{u} z+ u(-z)]\\cdot  p} \\phi^{\\parallel}_{2;K^*}(u,\\mu)\\,.\n \\label{das_eq14}\n\\end{equation}\nThen, using Eq.~(\\ref{das_eq9}) and comparing the result to Eq.~(\\ref{das_eq11}), one finds\n\\begin{equation}\n\\int^1_0 du\\,  C^{3\/2}_{n}(\\xi)\\,\\phi^{\\parallel}_{2;K^*}(u,\\mu) = \\langle\\langle \\mathbb{Q}_n^{\\rm t=2}\\rangle\\rangle\\,,\n\\label{das_eq15}\n\\end{equation}\nwhere we introduce the shorthand $\\xi = u-\\bar{u} = 2u-1$, and $\\langle\\langle \\mathbb{Q}_n^{\\rm t=2}\\rangle\\rangle$ are the reduced matrix elements of the operator $\\mathbb{Q}_n^{\\rm t=2}$. The Gegenbauer polynomials form a complete set of orthogonal functions over the weight function $6 u (1-u)$ in the interval $0<u<1$\n\\begin{equation}\n\\int^1_0 du\\,  u \\bar u \\,C^{3\/2}_{n}(\\xi)\\, C^{3\/2}_{m}(\\xi)=\\delta_{m n}\\frac{(n+1)(n+2)}{4(2n+3)}\\,,\n\\end{equation}\nand so one can invert (\\ref{das_eq15}) to find\n\\begin{equation}\n\\phi^{\\parallel}_{2;K^*}(u,\\mu) = 6 u (1-u) \\left\\{1+ \\sum_{n=1}^{\\infty} a_n^{\\parallel}(\\mu) \\,C^{3\/2}_{n}(\\xi)\\right\\}\\,,\n\\label{das_eq16}\n\\end{equation}\nwhere the \\textit{Gegenbauer coefficients} $a_n^{\\parallel}(\\mu)$ are related to the reduced matrix elements, Eq.~(\\ref{das_eq15}), as\n\\begin{equation}\na_n^{\\parallel}(\\mu) =\\frac{2(2n+3)}{3(n+1)(n+2)}\\langle\\langle \\mathbb{Q}_n^{\\rm t=2}\\rangle\\rangle\\,, \\qquad a_0^{\\parallel} = 1\\,.\n\\label{das_eq17}\n\\end{equation}\nThe result is that the conformal symmetry has separated the longitudinal degrees of freedom -- as contained in the orthogonal Gegenbauer polynomials which are function of the momentum fraction $u$ for the  twist-2 distribution -- from the transverse degrees of freedom, which now show up as the renormalisation-scale dependence of the Gegenbauer coefficients $a_n^{\\parallel}$. The Gegenbauer coefficients  contain the non-perturbative information of the DA and, for the leading-twist DA, are of conformal spin $n+2$.  The higher-twist two-particle DAs are expanded analogously in $P^{(1,0)}_{n}$ and $P^{(0,1)}_{n}$ for twist-3 and $C^{1\/2}_{n}$ for twist-4, see Eqs. (\\ref{das_eq12})  and (\\ref{das_eq13}) respectively.  The explicit expression for the DA of an $m$-particle state with the lowest possible conformal spin $j=j_1+\\dots+j_m$, the so-called \\textit{asymptotic distribution amplitude}, is given by\n\\begin{equation}\n\\phi_{as}(\\alpha_1,\\alpha_2,\\cdots,\\alpha_m) =\n\\frac{\\Gamma(2j_1+\\cdots +2j_m)}{\\Gamma(2j_1)\\cdots \\Gamma(2j_m)}\\,\n\\alpha_1^{2j_1-1}\\alpha_2^{2j_2-1}\\ldots \\alpha_m^{2j_m-1}\\,,\n\\label{das_eq18}\n\\end{equation}\nwhere $\\sum_{k=1}^m \\alpha_k =1$ \\cite{Braun:1989iv}. For the twist-2 two-particle DA considered $j=1$ and we recover the weight function $6 u (1-u)$. Analogously, for a multi-particle DA, states higher in conformal spin are multiplied by polynomials orthogonal over the weight function Eq.~(\\ref{das_eq18}). The matrix element Eq.~(\\ref{das_eq14}) with the chiral-odd Dirac matrix $\\Gamma \\to \\sigma_{z p}$ starts at twist-2 also, and gives rise to the second two-particle twist-2 DA\n\\begin{equation}\n\\phi^{\\perp}_{2;K^*}(u,\\mu) = 6 u (1-u)\\left\\{1+ \\sum_{n=1}^{\\infty} a_n^{\\perp}(\\mu) \\,C^{3\/2}_{n}(\\xi)\\right\\}\\,.\n\\label{das_eq19}\n\\end{equation}\nThe expansion of DAs in terms of an infinite sum of partial waves, as in Eq. (\\ref{das_eq19}), is very general, and is valid at the level of operators. In practice the concept of \\textit{G-parity} allows one to classify which Gegenbauer coefficients contribute for specific matrix elements of those operators. The G-parity operator $\\cal{G}$ is defined as $\\mathcal{G}=\\mathcal{C} e^{-i\\pi T_2}$ where $T_2$ is the isospin generator of the $2$ axis and $\\cal{C}$ is the charge conjugation operator.  G-parity is the generalisation of charge conjugation to particle multiplets, for example, $\\mathcal{G}\\ket{\\pi^{\\pm,0}}=-\\ket{\\pi^{\\pm,0}}$ and is conserved in QCD. It effectively swaps quarks for anti-quarks and therefore for equal mass quarks $u \\leftrightarrow \\bar{u}$ and consequently, for $\\pi,\\,\\rho,\\,\\omega$ and $\\phi$, the odd Gegenbauer coefficients $a_{2n+1}$ vanish. \n\n\\section{Two-Particle Twist-2 Distribution Amplitudes}\\label{das_sec1.3}\nAs mentioned in the last section, there are two two-particle matrix elements that begin at twist-2 \\cite{Ball:1998sk}:\\footnote{The vacuum-vector meson matrix elements vanish for $\\Gamma =i \\gamma_5$ because it is impossible to construct a pseudoscalar quantity from the three available 4-vectors $p_\\mu,\\,z_\\mu$ and $e^{(\\lambda)}_\\mu$.}\n\\begin{eqnarray}\n\\lefteqn{\n\\bra{0} \\bar q(x) \\gamma_\\mu s(-x) \\ket{K^*(P,\\lambda)} = }\\hspace*{1cm}\n\\nonumber\\\\\n&&{}f_{K^*}^\\parallel m_{K^*} \\left\\{ \\frac{e^{(\\lambda)}x}{P\\cdot x}\\,P_\\mu \\int_0^1 du\\, e^{i\\xi P\\cdot x} \\left[ \\phi_{2;K^*}^\\parallel(u) +   \\frac{1}{4}\\, m_{K^*}^2 x^2  \\phi^\\parallel_{4;K^*}(u)\\right]\\right.\\nonumber\\\\\n&&{} + \\left( e^{(\\lambda)}_\\mu -\nP_\\mu\\,\\frac{e^{(\\lambda)}x}{P\\cdot x}\\right) \\int_0^1 du\\,e^{i\\xi  P\\cdot x}\\,\\phi_{3;K^*}^\\perp(u) \\nonumber\\\\\n&&\\left. - \\frac{1}{2}\\,x_\\mu \\,\\frac{e^{(\\lambda)}x}{(P\\cdot x)^2} \\, m_{K^*}^2 \\int_0^1 du\\,e^{i\\xi P\\cdot x}\\,\n\\left[ \\psi_{4;K^*}^\\parallel(u) + \\phi_{2;K^*}^\\parallel(u) - 2   \\phi_{3;K^*}^\\perp(u)\\right]\\right\\}\\,,\n  \\label{das_eq20}\\\\[10pt]\n\\lefteqn{\n\\bra{0} \\bar q(x) \\sigma_{\\mu\\nu} s(-x) \\ket{ K^*(P,\\lambda)} =}\\hspace*{1cm} \\nonumber\\\\\n&&{}i f_{K^*}^\\perp \\left\\{ (e_\\mu^{(\\lambda)} P_\\nu - e_\\nu^{(\\lambda)}   P_\\mu) \\int_0^1 du \\, e^{i\\xi P\\cdot x} \\left[ \\phi_{2;K^*}^\\perp(u) +   \\frac{1}{4}\\, m_{K^*}^2 x^2  \\phi^\\perp_{4;K^*}(u)\\right]\\right. \\nonumber\\\\\n&&{}\n+ (P_\\mu x_\\nu - P_\\nu x_\\mu)\\,\\frac{e^{(\\lambda)}x}{(P\\cdot x)^2}\\,m_{K^*}^2 \\int_0^1 du \\,\ne^{i\\xi P\\cdot x} \\left[ \\phi_{3;K^*}^\\parallel(u) -   \\frac{1}{2}\\phi_{2;K^*}^\\perp(u) -   \\frac{1}{2}\\psi_{4;K^*}^\\perp(u)\\right] \\nonumber\\\\\n&&{}\\left.\n+ \\frac{1}{2}\\,(e_\\mu^{(\\lambda)} x_\\nu - e_\\nu^{(\\lambda)}x_\\mu)\\, \\frac{m_{K^*}^2}{P\\cdot x} \\int_0^1 du \\, e^{i\\xi P\\cdot x} \\left[ \\psi_{4;K^*}^\\perp(u)-\\phi_{2;K^*}^\\perp(u)  \\right]\\right\\}\\,.\n\\label{das_eq21}\n\\end{eqnarray}\nAll other DAs in the above relations are of twist-3 or -4 and all terms in the light-cone expansion of twist-5 and higher are neglected. The twist-4 DAs are shown for completeness. The normalisation of all DAs is given by\n\\begin{equation}\n\\int_0^1 du\\, \\phi(u) = 1\\,.\n\\end{equation}\nThe conformal expansions of the leading-twist DAs $\\phi_{2;K^*}^\\parallel$ and $\\phi_{2;K^*}^\\perp$ are given by Eqs.~(\\ref{das_eq16}) and (\\ref{das_eq19}) respectively. In the local limit $x^\\mu \\to 0$ the matrix elements (\\ref{das_eq20}) and (\\ref{das_eq21}) reduce to the longitudinal $f^{\\parallel}_{K^*}$ and transverse $f^{\\perp}_{K^*}$ decay constants;\n\\begin{eqnarray}\n\\bra{0} \\bar q(0) \\gamma_\\mu s(0) \\ket{K^*(P,\\lambda)} &=&  e_{\\mu}^{(\\lambda)}  m_{K^*} f^{\\parallel}_{K^*}\\,,\n\\label{das_eq22}\\\\\n\\bra{0} \\bar q(0) \\sigma_{\\mu\\nu} s(0) \\ket{ K^*(P,\\lambda)}&=& i \\left(e_{\\mu}^{(\\lambda)} P_\\nu-e_{\\nu}^{(\\lambda)}\nP_{\\mu}\\right)f^{\\perp}_{K^*}(\\mu)\\,.\n\\label{das_eq23}\n\\end{eqnarray}\nNote that $f^\\perp_{K^*}(\\mu)$ is scale-dependent because the tensor current is not conserved. Numerical values of the decay constants are discussed in Chapter~\\ref{chapter4_det}. The above DAs are related to those defined in Refs.~\\cite{Ball:1998sk,Ball:1998ff} by\n\\begin{equation}\n\\addtolength{\\arraycolsep}{5pt}\n\\begin{array}[b]{r@{\\ =\\ }l@{\\qquad}r@{\\ =\\ }l@{\\qquad}r@{\\ =\\ }l}\n\\phi_{2;K^*}^{\\parallel(\\perp)} & \\phi_{\\parallel(\\perp)}\\,, &\n\\phi_{3;K^*}^\\parallel & h_\\parallel^{(t)}\\,, & \\psi_{4;K^*}^\\parallel &\ng_3\\,,\\\\[10pt]\n\\phi_{4;K^*}^{\\parallel(\\perp)} & {\\mathbb A}_{(T)}\\,, &\n\\phi_{3;K^*}^\\perp & g_\\perp^{(v)}\\,, & \\psi_{4;K^*}^\\perp &\nh_3\\,.\n\\end{array}\n\\addtolength{\\arraycolsep}{-5pt}\n\\label{das_eq24}\n\\end{equation}\n\n\\section{Two-Particle Twist-3 Distribution Amplitudes}\\label{das_sec1.4}\nThe two-particle twist-3 DAs $\\phi_{3;K^*}^{\\perp,\\parallel}$ have already been defined in\nEqs.~(\\ref{das_eq20}) and (\\ref{das_eq21}). There are two more two-particle DAs,\n$\\psi_{3;K^*}^{\\perp,\\parallel}$, defined as:\\footnote{In the notations of Ref.~\\cite{Ball:1998sk}, $\\psi_{3;K^*}^\\perp =   \\{1-(f_{K^*}^\\perp\/f_{K^*}^\\parallel) (m_s+m_q)\/m_{K^*}\\} g_\\perp^{(a)}$, $\\psi_{3;K^*}^\\parallel =   \\{1-(f_{K^*}^\\parallel\/f_{K^*}^\\perp) (m_s+m_q)\/m_{K^*}\\} h_\\parallel^{(s)}$.}\n\\begin{eqnarray}\n\\bra{0}\\bar q(z) \\gamma_{\\mu} \\gamma_{5}s(-z)\\ket{K^*(P,\\lambda)}\n&=& \\frac{1}{2}\\,f_{K^*}^\\parallel m_{K^*} \\epsilon_{\\mu}^{\\phantom{\\mu}\\nu \\alpha \\beta}\ne^{(\\lambda)}_{\\nu} p_{\\alpha} z_{\\beta}\\int_{0}^{1} \\!du\\, e^{i \\xi p \\cdot z} \\psi_{3;K^*}^\\perp(u)\\,,\\\\\n\\bra{0}\\bar q(z)s(-z)\\ket{K^*(P,\\lambda)} & = & {} -i f_{K^*}^\\perp(e^{(\\lambda)} \\cdot z) m_{K^*}^{2}\n\\int_{0}^{1} \\!du\\, e^{i \\xi p \\cdot z} \\psi_{3;K^*}^\\parallel(u)\\,.\n\\end{eqnarray}\nThe normalisation is given by\n\\begin{equation}\n\\int_0^1 du\\, \\psi_{3;K^*}^{\\parallel(\\perp)}(u) = 1-\\frac{f_{K^*}^{\\parallel(\\perp)}}{f_{K^*}^{\\perp(\\parallel)}}\\,\\frac{m_s+m_q}{m_{K^*}}\\,,\n\\end{equation}\nwhich differs from Ref.~\\cite{Ball:1998sk}, where all DAs were normalised to 1; here we keep the full dependence on the quark masses. \n\n\\section{Three-Particle Twist-3 Distribution Amplitudes}\\label{das_sec1.5}\nThere are also three three-particle DAs of twist-3:\n\\begin{eqnarray}\n\\bra{0}\\bar q(z) g_s \\widetilde{G}_{\\beta z}(vz)\\gamma_{z}\\gamma_5 s(-z) \\ket{K^*(P,\\lambda)} & = & f_{K^*}^\\parallel m_{K^*} (p\\cdot z)^2\ne^{(\\lambda)}_{\\perp\\beta}\n\\widetilde\\Phi_{3;K}^\\parallel(v,p\\cdot z)+\\dots\\,,\\\\\n\\bra{0}\\bar q(z) g_s G_{\\beta z}(vz)i\\gamma_{z} s(-z) \\ket{K^*(P,\\lambda)} & = & \nf_{K^*}^\\parallel m_{K^*} (p\\cdot z)^2\ne^{(\\lambda)}_{\\perp\\beta} \\Phi_{3;K}^\\parallel(v,p\\cdot z)+\\dots\\,,\\nonumber\\\\\n\\bra{0}\n\\bar q(z) g_s G_{z\\beta}(vz)\\sigma_{z\\beta} s(-z) \\ket{K^*(P,\\lambda)} & = & f_{K^*}^\\perp m_{K^*}^2\n(e^{(\\lambda)}\\cdot z)(p\\cdot z) \\Phi_{3;K^*}^\\perp(v,p\\cdot z)+\\dots\\,,\\nonumber\n\\label{das_eq25}\n\\end{eqnarray}\nwhere the dots denote terms of higher twist and we use the short-hand notation\n\\begin{equation}\n{\\cal F}(v,p\\cdot z) = \\int {\\cal D}\\underline{\\alpha}\\,e^{-ip\\cdot z(\\alpha_2-\\alpha_1+v\\alpha_3)} {\\cal F}(\\underline{\\alpha}) \n\\end{equation}\nwith ${\\cal F}(\\underline{\\alpha})$ being a three-particle DA  and the integration measure ${\\cal D}\\underline{\\alpha}$ is defined as\n\\begin{equation}\n\\int{\\cal D}\\underline{\\alpha} \\equiv \\int_0^1 d\\alpha_1 d\\alpha_2 d\\alpha_3\\, \\delta(1-\\sum \\alpha_i)\\,.\n\\label{das_eq26}\n\\end{equation}\nThe twist-3 three-particle DAs correspond to the light-cone projection $\\gamma_z G_{z\\perp}$ and $\\sigma_{\\perp z} G_{\\perp z}$, respectively, which picks up the $s=\\frac{1}{2}$ component of the\nquark fields and the $s=1$ component of the gluonic field strength tensor. According to Eq.~(\\ref{das_eq18}), the (normalised) asymptotic DA is then given by $360\\,\\alpha_1\\alpha_2\\alpha_3^2$. To NLO in the conformal expansion, each three-particle twist-3 DA involves three hadronic parameters, which we label in the following way: $\\zeta,\\kappa$ are LO and $\\omega,\\lambda$ NLO parameters.  $\\zeta$ and $\\omega$ are G-parity conserving, whereas \n$\\kappa$ and $\\lambda$ violate G-parity and hence vanish for mesons with quarks of equal mass, i.e.\\  $\\rho$, $\\omega$ and $\\phi$. We then have\n\\begin{eqnarray}\n\\Phi_{3;K^*}^\\parallel(\\underline{\\alpha})\n & = & 360\\alpha_1\\alpha_2\\alpha_3^2 \\left\\{\n\\kappa_{3K^*}^\\parallel + \\omega_{3K^*}^\\parallel (\\alpha_1-\\alpha_2) +\n\\lambda_{3K^*}^\\parallel \\frac{1}{2}\\,(7\\alpha_3 -\n3)\\right\\},\\nonumber\\\\\n\\widetilde\\Phi_{3;K^*}^\\parallel(\\underline{\\alpha}) & = & \n360\\alpha_1\\alpha_2\\alpha_3^2 \\left\\{\n\\zeta_{3K^*}^\\parallel + \\widetilde\\lambda_{3K^*}^\\parallel (\\alpha_1-\\alpha_2) +\n\\widetilde\\omega_{3K^*}^\\parallel \\frac{1}{2}\\,(7\\alpha_3 -\n3)\\right\\},\\nonumber\\\\\n\\Phi_{3;K^*}^\\perp(\\underline{\\alpha}) \n& = & 360\\alpha_1\\alpha_2\\alpha_3^2 \\left\\{\n\\kappa_{3K^*}^\\perp + \\omega_{3K^*}^\\perp (\\alpha_1-\\alpha_2) +\n\\lambda_{3K^*}^\\perp \\frac{1}{2}\\,(7\\alpha_3 -\n3)\\right\\}\\,.\n\\label{das_eq27}\n\\end{eqnarray}\nThe parameters defined above are related to those of Ref.~\\cite{Ball:1998sk} by:\n\\begin{eqnarray}\n\\zeta_{3}^A = \\zeta^\\parallel_{3},&\\qquad & \\zeta_{3}^V =\\omega_{3}^\\parallel\/14,  \\nonumber\\\\[10pt]\n\\zeta_{3}^T = \\omega_{3}^\\perp\/14,&\\qquad &\\zeta_3^\\parallel\\,\\omega^A_{1,0} =\n\\widetilde\\omega_{3}^\\parallel\\,.\n\\label{das_eq28}\n\\end{eqnarray}\nG-parity breaking terms were not considered in Ref.~\\cite{Ball:1998sk}. For equal mass quarks, $\\Phi^{\\perp,\\parallel}_{3;K^*}$ are antisymmetric under $\\alpha_1\\leftrightarrow \\alpha_2$, whereas $\\widetilde\\Phi^{\\parallel}_{3;K^*}$ is symmetric. All these parameters can be defined in terms of matrix elements of local twist-3 operators. For chiral-odd operators, for instance, one has\n\\begin{eqnarray}\n\\bra{0}\\bar q \\sigma_{z\\xi} g_sG_{z\\xi}\ns\\ket{K^*(P,\\lambda)} & = & f_{K^*}^\\perp m_{K^*}^2 \n(e^{(\\lambda)}\\cdot z)(p\\cdot z)\\kappa_{3K^*}^\\perp\\,,\\nonumber\\\\\n\\bra{0}\\bar q \\sigma_{z\\xi}  [iD_z,g_sG_{z\\xi}] s\n- \\frac{3}{7}\\, i\\partial_z \\bar q \\sigma_{z\\xi} \ng_sG_{z\\xi} s \\ket{K^*(P,\\lambda)} \n& = & f_{K^*}^\\perp m_{K^*}^2 (e^{(\\lambda)}\\cdot z)(p\\cdot z)^2\n\\,\\frac{3}{28}\\, \\lambda_{3K^*}^\\perp\\,,\n\\nonumber\\\\\n\\bra{0} \\bar q i \\!\\stackrel{\\leftarrow}{D}_z\\!\n\\sigma_{z\\xi}  g_sG_{z\\xi} s - \\bar q\n\\sigma_{z\\xi}  g_sG_{z\\xi} i\\! \\stackrel{\\rightarrow}{D}_z\\! s\n \\ket{K^*(P,\\lambda)}  & = & f_{K^*}^\\perp m_{K^*}^2 (e^{(\\lambda)}\\cdot z)(p\\cdot z)^2\n\\,\\frac{1}{14}\\, \\omega_{3K^*}^\\perp\\,;\\nonumber\n\\label{das_eq29}\n\\end{eqnarray}\nthe formulas for chiral-even operators are analogous.  In Chapter~\\ref{chapter4_det} we calculate numerical values for all the parameters in Eq.~(\\ref{das_eq27}) from QCD sum rules.\n\n\\section{Relations Between Distribution Amplitudes}\nThe QCD EOM are a crucial ingredient in simplifying the kinematic contributions of different operators, a task which is facilitated by the fact that they are preserved to all orders in the conformal expansion. The EOM relate via integral equations the two-particle twist-3 DAs, defined in Sections~\\ref{das_sec1.3} and \\ref{das_sec1.4}, to the two-particle twist-2 DAs, Eqs.~(\\ref{das_eq16}) and (\\ref{das_eq19}), and three-particle twist-3 DAs, defined in Section~\\ref{das_sec1.5}. We do not quote the EOM themselves for which we refer the reader to the literature. The framework for the procedure was developed in Ref.~\\cite{Braun:1989iv} based on deriving the EOM for non-local light-ray operators \\cite{Balitsky:1987bk}. The operator relations are then sandwiched between the vacuum and meson states and the definitions of the DAs used to convert them into integral equations, making use of partial integration to remove explicit dependence on co-ordinates and momentum 4-vectors. The resulting expressions are then solved order-by-order in the conformal expansion, see Ref.~\\cite{Braun:2003rp} for an overview. The EOM contain mass dependent contributions $\\propto m_s\\pm m_q$ that were calculated in Ref.~\\cite{Ball:1998sk}. In the present analysis,  G-parity breaking terms of the three-particle twist-3 DAs are included, which then, via the EOM, impact on the two-particle DAs \\cite{Ball:2007rt}. The resulting integral equations are:\n\\begin{eqnarray}\n\\psi_{3;K^*}^\\parallel(u) & = & \\bar{u}\\int_0^u dv\\,\\frac{1}{\\bar v}\\,\n\\Upsilon(v) + u \\int_u^1 dv\\,\\frac{1}{v}\\, \\Upsilon(v)\\,,\\nonumber\\\\\n\\phi_{3;K^*}^\\parallel(u) \n& = & \\frac{1}{2}\\,\\xi\\left[\\int_0^u dv\\,\\frac{1}{\\bar v}\\,\n\\Upsilon(v) - \\int_u^1 dv\\,\\frac{1}{v}\\, \\Upsilon(v)\\right] + \n\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\,\\frac{m_s+m_q}{m_{K^*}}\\,\n\\phi_{2;K^*}^\\parallel(u)\\nonumber\\\\\n&& {}+\\frac{d}{du}\\,\\int_0^u d\\alpha_1\\int_0^{\\bar u} d\\alpha_2\n\\frac{1}{\\alpha_3}\n\\,{\\Phi}^\\perp_{3;K^*}(\\underline{\\alpha})\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\Upsilon(u) & = & 2\\phi_{2;K^*}^\\perp(u) - \n\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\,\\frac{m_s+m_q}{m_{K^*}} \\left[\n1 - \\frac{1}{2}\\,\n\\xi\\frac{d}{du}\\right]\\phi_{2;K^*}^\\parallel(u) -\n\\frac{1}{2}\\,\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\,\n\\frac{m_s-m_q}{m_{K^*}}\\,\\frac{d}{du}\\,\\phi_{2;K^*}^\\parallel(u)\\nonumber\\\\\n& & {}+\\frac{d}{du}\\,\\int_0^u\\!\\!d\\alpha_1\\int_0^{\\bar\nu}\\!\\! d\\alpha_2\n\\,\\frac{1}{\\alpha_3}\\left(\\alpha_1\\,\\frac{d}{d\\alpha_1} +\n\\alpha_2\\,\\frac{d}{d\\alpha_2}\\, - 1\\right) \n\\Phi_{3;K^*}^\\perp(\\underline{\\alpha})\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\psi_{3;K^*}^\\perp(u)& = & \\bar{u}\\int_0^u dv\\,\\frac{1}{\\bar v}\\,\n  \\Omega(v) + u \\int_u^1 dv\\,\\frac{1}{v}\\, \\Omega(v)\\,,\\nonumber\\\\\n\\phi_{3;K^*}^\\perp(u) \n& = & \\frac{1}{4}\\left[\\int_0^u dv\\,\\frac{1}{\\bar v}\\,\n\\Omega(v) + \\int_u^1 dv\\,\\frac{1}{v}\\, \\Omega(v)\\right] + \n\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\,\\frac{m_s+m_q}{m_{K^*}}\\,\n\\phi_{2;K^*}^\\perp(u)\\nonumber\\\\\n&& {}+\\frac{d}{du}\\,\\int_0^u d\\alpha_1\\int_0^{\\bar u} d\\alpha_2\n\\frac{1}{\\alpha_3}\n\\,{\\Phi}^\\parallel_{3;K^*}(\\underline{\\alpha})\\nonumber\\\\\n&&{} + \\int_0^u d\\alpha_1\\int_0^{\\bar u}\nd\\alpha_2\\,\\frac{1}{\\alpha_3}\\left( \\frac{d}{d\\alpha_1} +\n\\frac{d}{d\\alpha_2} \\right) \n\\widetilde{\\Phi}_{3;K^*}^\\parallel(\\underline{\\alpha})\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\Omega(u)  &=&  2\\phi_{2;K^*}^\\parallel(u) + \n\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\,\\frac{m_s+m_q}{m_{K^*}}\\, \n\\xi\\,\\frac{d}{du}\\,\\phi_{2;K^*}^\\perp(u)-\n\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\,\\frac{m_s-m_q}{m_{K^*}}\\, \n\\frac{d}{du}\\,\\phi_{2;K^*}^\\perp(u)\\nonumber\\\\\n& & {}+2\\frac{d}{du}\\,\\int_0^u\\!\\!d\\alpha_1\n\\int_0^{\\bar u}\\!\\! d\\alpha_2\n\\,\\frac{1}{\\alpha_3}\\left(\\alpha_1\\,\\frac{d}{d\\alpha_1} +\n\\alpha_2\\,\\frac{d}{d\\alpha_2}\\right)\n\\Phi_{3;K^*}^\\parallel(\\underline{\\alpha})\\nonumber\\\\\n&&{} +2\\,\\frac{d}{du}\\,\\int_0^u\\!\\!d\\alpha_1\n\\int_0^{\\bar u}\\!\\! d\\alpha_2\n\\,\\frac{1}{\\alpha_3}\\left(\\alpha_1\\,\\frac{d}{d\\alpha_1} -\n\\alpha_2\\,\\frac{d}{d\\alpha_2}\\right) \n\\widetilde\\Phi_{3;K^*}^\\parallel(\\underline{\\alpha})\\,.\n\\end{eqnarray}\nUsing Eq.~(\\ref{das_eq27}), and the corresponding relations for twist-2 DAs, one obtains expressions for the twist-3 two-particle DAs, which are valid to NLO in the conformal expansion. As discussed in Ref.~\\cite{Ball:1998sk}, the structure of this expansion is complicated by the fact that  these DAs do not correspond to a fixed Lorentz-spin projection $s$ of the quark fields.  The resulting expansion is in $C^{3\/2}_n(\\xi)$ for $\\psi^{\\perp,\\parallel}_{3;K^*}$ and $C^{1\/2}_n(\\xi)$ for $\\phi^{\\perp,\\parallel}_{3;K^*}$:\n\\begin{eqnarray}\n\\phi_{3;K^*}^\\parallel(u) & = & 3\\xi^2 + \\frac{3}{2}\\,\\xi(3\\xi^2-1)\na_1^\\perp + \\frac{3}{2}\\,\\xi^2 (5\\xi^2-3)a_2^\\perp\\nonumber\\\\\n&&{} + \\left(\\frac{15}{2}\\,\\kappa_{3K^*}^\\perp -\n\\frac{3}{4}\\,\\lambda_{3K^*}^\\perp\\right) \\xi (5\\xi^2-3) +\n\\frac{5}{8}\\,\\omega_{3K^*}^\\perp (3-30\\xi^2 + 35\\xi^4) \\nonumber\\\\\n&&{}+\\frac{3}{2}\\,\\frac{m_s+m_q}{m_{K^*}}\\,\n\\frac{f^\\parallel_{K^*}}{f_{K^*}^\\perp}\n\\left\\{ 1 + 8\\xi a_1^\\parallel + 3 (7-30 u\\bar{u} )a_2^\\parallel + \\xi \\ln\n\\bar{u} (1+3 a_1^\\parallel + 6 a_2^\\parallel)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.6cm} - \\xi \\ln u (1-3\na_1^\\parallel + 6 a_2^\\parallel)\\right\\}\\nonumber\\\\\n&&{}-\\frac{3}{2}\\,\\frac{m_s-m_q}{m_{K^*}}\\,\n\\frac{f^\\parallel_{K^*}}{f_{K^*}^\\perp}\\,\n\\xi \\left\\{ 2 + 9\\xi a_1^\\parallel + 2 (11-30 u\\bar{u} )a_2^\\parallel + \\ln\n\\bar{u} (1+3 a_1^\\parallel + 6 a_2^\\parallel)\\right.\\nonumber\\\\\n&&{}\\hspace*{3.9cm}\\left. + \\ln u (1-3\na_1^\\parallel + 6 a_2^\\parallel)\\right\\},\n\\label{das_eq30}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\psi_{3;K^*}^\\parallel(u) & = & 6 u\\bar u \\left\\{ 1 + \\left(\n\\frac{a_1^\\perp}{3} + \\frac{5}{3} \\kappa_{3K^*}^\\perp\\!\\right)\nC_1^{3\/2}(\\xi) + \\left( \\frac{a_2^\\perp}{6}  + \\frac{5}{18}\n\\omega_{3K^*}^\\perp\\right) C_2^{3\/2}(\\xi) -\n\\frac{1}{20}\\lambda_{3K^*}^\\perp C_3^{3\/2}(\\xi)\\right\\}\\nonumber\\\\\n&&{}+ 3\\,\n\\frac{m_s+m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\left\\{\nu\\bar u (1 + 2 \\xi a_1^\\parallel + 3 (7-5u\\bar u) a_2^\\parallel) + \\bar\nu \\ln \\bar u (1+3 a_1^\\parallel + 6 a_2^\\parallel)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.2cm} + u \\ln u \n(1-3 a_1^\\parallel + 6 a_2^\\parallel)\\right\\}\\nonumber\\\\\n&&- 3\\,\n\\frac{m_s-m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\parallel}{f_{K^*}^\\perp}\\left\\{\nu\\bar u (9 a_1^\\parallel + 10\\xi a_2^\\parallel) + \\bar\nu \\ln \\bar u (1+3 a_1^\\parallel + 6 a_2^\\parallel)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.2cm} - u \\ln u \n(1-3 a_1^\\parallel + 6 a_2^\\parallel)\\right\\},\n\\label{das_eq31}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\psi_{3;K^*}^\\perp(u) & = & 6 u\\bar u \\left\\{ 1 + \\left(\n\\frac{1}{3} a_1^\\parallel + \\frac{20}{9} \\kappa_{3K^*}^\\parallel\\right)\nC_1^{3\/2}(\\xi) \n\\right.\\nonumber\\\\\n&&\\left.\\hspace*{0pt} + \\left( \\frac{1}{6} a_2^\\parallel +\n\\frac{10}{9}\\zeta_{3K^*}^\\parallel + \\frac{5}{12}\\,\n\\omega_{3K^*}^\\parallel-\\frac{5}{24}\\,\n\\widetilde\\omega_{3K^*}^\\parallel\\!\\right)\nC_2^{3\/2}(\\xi) +\n\\left(\\frac{1}{4}\\widetilde\\lambda_{3K^*}^\\parallel - \\frac{1}{8}\n\\lambda_{3K^*}^\\parallel\\right) C_3^{3\/2}(\\xi)\\right\\}\\nonumber\\\\\n&&{}+ 6\\,\n\\frac{m_s+m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\left\\{\nu\\bar u (2 + 3 \\xi a_1^\\perp + 2 (11-10u\\bar u) a_2^\\perp) + \\bar\nu \\ln \\bar u (1+3 a_1^\\perp + 6 a_2^\\perp)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.2cm} + u \\ln u \n(1-3 a_1^\\perp + 6 a_2^\\perp)\\right\\}\\nonumber\\\\\n&&- 6\\,\n\\frac{m_s-m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\perp}{f_{K^*}^\\parallel}\\left\\{\nu\\bar u (9 a_1^\\perp + 10\\xi a_2^\\perp) + \\bar\nu \\ln \\bar u (1+3 a_1^\\perp + 6 a_2^\\perp)\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.2cm} - u \\ln u \n(1-3 a_1^\\perp + 6 a_2^\\perp)\\right\\},\n\\label{das_eq32}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\phi_{3;K^*}^\\perp(u) & = & \\frac{3}{4}\\,(1+\\xi^2) + \\frac{3}{2}\\,\\xi^3\na_1^\\parallel + \\left\\{ \\frac{3}{7}\\,a_2^\\parallel + 5\n\\zeta_{3K^*}^\\parallel \\right\\} (3\\xi^2-1) + \\left\\{5\\kappa_{3K^*}^\\parallel\n- \\frac{15}{16}\\,\\lambda_{3K^*}^\\parallel \\right.\\nonumber\\\\\n&&{} \\left. +\n\\frac{15}{8}\\,\\widetilde{\\lambda}_{3K^*}^\\parallel \\right\\} \\xi\n(5\\xi^2-3)+ \\left\\{ \\frac{9}{112}\\,a_2^\\parallel +\n\\frac{15}{32}\\,\\omega_{3K^*}^\\parallel -\n\\frac{15}{64}\\,\\widetilde\\omega_{3K^*}^\\parallel \\right\\} (35\\xi^4-30\n\\xi^2+3)\\nonumber\\\\\n&&{}+\\frac{3}{2}\\,\\frac{m_s+m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\perp}{\nf_{K^*}^\\parallel} \\left\\{ 2 + 9 \\xi a_1^\\perp + 2 (11-30 u \\bar{u})\n    a_2^\\perp\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.5cm} + \n(1-3 a_1^\\perp + 6 a_2^\\perp) \\ln u + (1+3 a_1^\\perp + 6 a_2^\\perp) \n\\ln \\bar{u} \\right\\}\\nonumber\\\\\n&&{}-\\frac{3}{2}\\,\\frac{m_s-m_q}{m_{K^*}}\\,\\frac{f_{K^*}^\\perp}{\nf_{K^*}^\\parallel} \\left\\{ 2\\xi + 9 (1-2u \\bar{u}) a_1^\\perp + 2 \\xi (11-20 u \\bar{u})\n    a_2^\\perp\\right.\\nonumber\\\\\n&&{}\\left.\\hspace*{3.5cm} + \n(1+3 a_1^\\perp + 6 a_2^\\perp) \\ln \\bar{u} - (1-3 a_1^\\perp + 6 a_2^\\perp) \n\\ln u \\right\\}\\,.\n\\label{das_eq33}\n\\end{eqnarray}\nThese expressions  supersede those given in Ref.~\\cite{Ball:1998sk} where G-parity violating terms in $\\kappa_3$ and $\\lambda_3$ were not included. The DAs given above now contain a minimum number of parameters which can be determined from one's favourite method, such as QCD sum rules or Lattice QCD.  We briefly mention that the matrix elements of QCD operator identities can also be used to relate twist-2 and twist-4 DA parameters to each other; such investigations were performed for the G-partiy violating twist-2  $K^*$ parameters $a_1^{\\parallel,\\perp} (K^*)$ in Refs.~\\cite{Braun:2004vf,Ball:2006fz}.\n\n\\section{Evolution Equations}\\label{evolution}\nThe scale dependence of the leading-twist DAs of Eqs.~(\\ref{das_eq16}) and (\\ref{das_eq19}) can be investigated using perturbation theory. The resulting renormalisation-group equation is the \\textit{Efremov-Radyushkin-Brodsky-Lepage (ER-BL) evolution equation} \\cite{Lepage:1980fj,Efremov:1979qk,Efremov:1978rn,Lepage:1979zb}\n\\begin{equation}\n\\mu^2 \\frac{d}{d \\mu^2}\\, \\phi(u,\\mu)=\\int^1_0dv\\,V(u,v;\\alpha_s(\\mu))\\,\\phi(v,\\mu)\\,,\n\\label{das_eq34}\n\\end{equation}\nwhich completely specifies $\\phi(u,\\mu)$ given $\\phi(u,\\mu_0)$. The kernel is given by an expansion in $\\alpha_s$\n\\begin{equation}\nV(u,v;\\alpha_s(\\mu)) = \\frac{\\alpha_s(\\mu)}{4 \\pi} V^{(0)}(u,v)+ \\left(\\frac{\\alpha_s(\\mu)}{4 \\pi}\\right)^2 V^{(1)}(u,v)+\\dots \\,.\n\\end{equation}\nThe evolution equation (\\ref{das_eq34}) can be solved readily at leading-order using the conformal expansion \\cite{Brodsky:1980ny, Ohrndorf:1981qv, Makeenko:1980bh, Brodsky:1980XX}. This amounts to finding its eigenfunctions, which we already know to be Gegenbauer polynomials, and using this fact it can be shown that the leading-order kernel can be written as\n\\begin{equation}\nV^{(0)}(u,v)=-6u \\bar{u} \\sum^\\infty_{n=0} \\frac{4(2n+3)}{(n+1)(n+2)}\\gamma_n^{(0)}C_n^{3\/2}(2u-1)C_n^{3\/2}(2v-1)\\,,\n\\end{equation}\ngiving the LO anomalous dimensions of the Gegenbauer coefficients $\\gamma_n$. The renormalisation is hence multiplicative at leading-order\n\\begin{equation}\na^{\\rm LO}_n(\\mu^2)=a_n(\\mu^2_0)L^{\\gamma^{(0)}_n \/(2 \\beta_0)}\\,,\n\\end{equation}\nwhere $L=\\alpha_s(\\mu^2)\/\\alpha_s(\\mu^2_0)$. The one-loop anomalous dimensions of the twist-2 Gegenbauer coefficients are \\cite{Gross:1973id1,Shifman:1980dk}\n\\begin{eqnarray}\n\\gamma^{\\parallel(0)}_{(n)}&=&8 C_F \\left(\\sum_{k=1}^{n+1}\\frac{1}{k} -\\frac{3}{4}-\\frac{1}{2(n+1)(n+2)}\\right)\\,,\n\\label{das_eq35}\\\\\n\\gamma^{\\perp(0)}_{(n)}&=&8 C_F\\left(\\sum_{k=1}^{n+1} \\frac{1}{k}-\\frac{3}{4}\\right)\\,.\n\\label{das_eq36}\n\\end{eqnarray}\nAt next-to-leading-order the scale dependence is more complicated \\cite{Mueller:1993hg,Mueller:1994cn}\n\\begin{equation}\na_n^{\\rm  NLO}(\\mu^2)=a_n(\\mu^2_0) E_n^{\\rm NLO}+\\frac{\\alpha_s}{4 \\pi}\\sum_{k=0}^{n-2} a_k(\\mu^2_0)L^{\n\\gamma^{(0)}_k \/(2 \\beta_0)} d_{nk}^{(1)}\\,,\n\\label{evo}\n\\end{equation}\nwhere\n\\begin{equation}\nE_n^{\\rm  NLO}=L^{\\gamma^{(0)}_k \/(2 \\beta_0)}\\left\\{1+\\frac{\\gamma_n^{(1)} \\beta_0 - \\gamma_n^{(0)} \\beta_1}{8 \\pi \\beta_0^2}\\left[\\alpha_s(\\mu^2)-\\alpha_s(\\mu^2_0)\\right]\\right\\}\\,.\n\\end{equation}\n$\\gamma_n^{(1)}$ are the diagonal two-loop anomalous dimensions which are available for the vector current \\cite{Floratos}, and the tensor current \\cite{Haya}. The mixing coefficients $d^{(1)}_{nk}$ are given in closed form in Refs.~\\cite{Mueller:1993hg,Mueller:1994cn} where the formulas are valid for arbitrary currents by substitution of the corresponding one-loop anomalous dimension. For the lowest moments $n=\\{0,1,2\\}$ one has\n\\begin{eqnarray}\n\\gamma_0^{\\parallel(1)} =0\\,, \\qquad \n\\gamma_1^{\\parallel(1)} = \\frac{23110}{243} - \\frac{512}{81}\\, N_f\\,,\\qquad  \n\\gamma_2^{\\parallel(1)} =  \\frac{34072}{243}-\\frac{830}{81}\\, N_f\\,,\\\\[10pt]\n\\gamma_0^{\\perp(1)} =\\frac{724}{9} - \\frac{104}{27}\\,N_f\\,, \\qquad \n\\gamma_1^{\\perp(1)} = 124 - 8 N_f\\,,\\qquad  \n\\gamma_2^{\\perp(1)} =  \\frac{38044}{243}-\\frac{904}{81}\\, N_f\\,,\n\\end{eqnarray}  \nand\n\\begin{equation}\n  d^{\\parallel(1)}_{20}  = \n  \\frac{35}{9}\\,\\frac{20-3\\beta_0}{50-9\\beta_0} \\left(1-L^{50\/(9\\beta_0)-1}\\right)\\,, \\quad  d^{\\perp(1)}_{20}  =   \\frac{28}{9}\\,\\frac{16-3\\beta_0}{40-9\\beta_0}  \\left(1-L^{40\/(9\\beta_0)-1}\\right)\\,.\n\\end{equation}\nIt is evident from Eqs.~(\\ref{das_eq35}) and  (\\ref{das_eq36}) that the anomalous dimensions of the Gegenbauer coefficients increase logarithmically  with conformal spin. This implies that as $\\mu \\to \\infty$ the coefficients higher in conformal spin are damped and the DA approaches its asymptotic form (\\ref{das_eq18})\n\\begin{equation}\n\\lim_{\\mu \\to \\infty} \\phi_{2;K^*}(u,\\mu^2) \\to 6 u \\bar{u}\\,.\n\\end{equation}\nThis limit offers a great simplification in that if it can be verified, experimentally of otherwise, that a given process is well described by the asymptotic form of the DA at hadronic scales, there are no non-perturbative parameters to be determined. The convergence of the conformal expansion in general has to be verified case by case and there is no \\textit{a priori} reason why it should do so. In practice one has to truncate the expansion at some order in conformal spin, usually $n=2$, and as this constitutes an approximation it thus introduces a model dependent assumption. In Ref.~\\cite{Ball:2005ei} for example,  these issues are discussed and an alternative method suggested.\n\nThe scale dependence of the three-particle twist-3 DAs can in principle be deduced from evolution equations for the non-local operators in (\\ref{das_eq25}) using the techniques of Ref.~\\cite{Balitsky:1987bk}. The evolution of the parameters in (\\ref{das_eq27}) could then be found by projecting out the desired conformal spin, as  for the leading-twist DA. Another approach is to consult the literature of results for the corresponding nucleon structure functions \\cite{andim}. The three-particle twist-3 parameters $\\zeta_{3K^*}^\\parallel$, $\\kappa_{3K^*}^{\\perp,\\parallel}$, $\\omega_{3K^*}^\\perp$ and $\\lambda_{3K^*}^\\perp$ renormalise multiplicatively in the chiral limit, and the others mix with each other. For non-zero strange quark mass, there is additional mixing with twist-2 parameters with the mass corrections featuring as $m_s \\pm m_q$ depending on the G-parity of the parameter. Here, we write down explicitly only the renormalisation-group improved relations for the above  5 parameters.   The relations can be written in compact form as\n\\begin{equation}\nP_i(\\mu^2) = L^{(\\gamma_P)_i\/\\beta_0}\\, P_i(\\mu_0^2) + \\sum_{j=1}^3\nC_{ij} \\left( L^{(\\gamma_Q)_{ij}\/\\beta_0} -\nL^{(\\gamma_P)_i\/\\beta_0}\\right) Q_{ij}(\\mu_0^2)\\,,\n\\label{das_eq37}\n\\end{equation}\nwhere the parameters are given by:\n\\begin{eqnarray}\nP & = & \\{f_{K^*}^\\parallel \\zeta_{3K^*}^\\parallel,\\,f_{K^*}^\\parallel\n\\kappa_{3K^*}^\\parallel,\\,f_{K^*}^\\perp \\kappa_{3K^*}^\\perp,\\,f_{K^*}^\\perp\n\\omega_{3K^*}^\\perp,\\,f_{K^*}^\\perp \\lambda_{3K^*}^\\perp\\}\\,,\\nonumber\\\\\nQ_{1(2)} &=& \\frac{f_{K^*}^\\perp}{m_{K^*}}\\,\\{\nm_s\\pm m_q,\\,(m_s\\mp m_q) a_1^\\perp,\\,(m_s\\pm m_q)a_2^\\perp\\}\\,,\\nonumber\\\\\nQ_{3,5} &=& \\frac{f_{K^*}^\\parallel}{m_{K^*}}\\,\\{\nm_s-m_q,\\,(m_s+m_q) a_1^\\parallel,\\,(m_s-m_q) a_2^\\parallel\\}\\,,\\nonumber\\\\\nQ_{4}& =& \\frac{f_{K^*}^\\parallel}{m_{K^*}}\\,\\{\nm_s+m_q,\\,(m_s-m_q) a_1^\\parallel,\\,(m_s+m_q) a_2^\\parallel\\}\\,,\n\\nonumber\\\\\n\\gamma_P & = & \\left\\{\n\\frac{77}{9},\\,\\frac{77}{9},\\,\\frac{55}{9},\\,\\frac{73}{9},\\,\n\\frac{104}{9}\\right\\}, \\nonumber\\\\\n(\\gamma_Q)_{1,2} & = & \\left\\{ \\frac{16}{3},\\, 8,\\,\n\\frac{88}{9} \\right\\},\\qquad\n(\\gamma_Q)_{3,4,5}  =  \\left\\{ 4,\\, \\frac{68}{9},\\,\n\\frac{86}{9} \\right\\}\\,,\n\\end{eqnarray}\n\\begin{eqnarray}\nC & = & \\left(\n\\begin{array}{r@{\\hskip10pt}r@{\\hskip10pt}r}\n\\displaystyle \\frac{2}{29} & \\displaystyle \\frac{6}{25} & \\displaystyle 0\\\\[10pt]\n\\displaystyle -\\frac{2}{29} & \\displaystyle-\\frac{6}{25} & 0\\\\[10pt]\n\\displaystyle-\\frac{4}{19} & \\displaystyle \\frac{12}{65} & 0\\\\[10pt]\n\\displaystyle \\frac{14}{37} & \\displaystyle-\\frac{42}{25} & \\displaystyle\\frac{12}{13}\\\\[10pt]\n\\displaystyle -\\frac{1}{85} & \\displaystyle -\\frac{1}{5} & \\displaystyle-\\frac{4}{15}\n\\end{array} \\right)\\,.\n\\end{eqnarray}\nWe refrain from delving into a full discussion here of the mixing including the remaining parameters $\\lambda_3^\\parallel$, $\\omega_3^\\parallel$, and guide the reader to Appendix A of Ref.~\\cite{Ball:2007rt} for details. \n\\chapter{QCD Sum Rules}\\label{chapter3_SR}\nThe original QCD sum rule approach was introduced by the revolutionary work of Shifman, Vainshtein and Zakharov in Refs.~\\cite{Shifman:1978bz, Shifman:1978by, Shifman:1978bx}, and has proven itself to be one  of the most effective tools for determining non-perturbative parameters of low-lying hadronic states. It does so in terms of a finite number of  universal non-perturbative input parameters, and as such has great predictive power. The approach has been massively successful in ascertaining  a wide range of phenomena of non-perturbative origin.  QCD sum rules are particularly advantageous for $B$ physics because the presence of an  intrinsic heavy-quark mass scale $m_b$ provides the necessary conditions required for the application of the short-distance OPE or light-cone  expansion of relevant correlation functions, from which the relevant quantities can be extracted. The heavy-quark limit $m_b\\to\\infty$  is not necessary and sum rules can be derived in full QCD for finite $m_b$. Despite its successes, the  method is limited by an inherent  irreducible systematic uncertainty of $20 - 30\\%$. However, such is the relative ease of the QCD sum rule method, as compared to, for example, Lattice QCD, that its place in the tool-box of the QCD practitioner is ensured.\n\nFirstly we  discuss step-by-step the methodology of the original QCD sum rule approach. Secondly we discuss its modification to accommodate non-local correlation functions which aides the extraction of DA parameters of beyond leading-order in conformal spin. Thirdly, we outline an extension of the original\napproach; light-cone sum rules. All three methods find application in this thesis:\n\\begin{itemize}\n\\item{in Chapter~\\ref{chapter4_det} we make use of the non-local formalism to extract numerical values for the leading twist-2 and twist-3 DA parameters defined in the Chapter~\\ref{chapter2_DAs}.}\n\\item{In Chapter~\\ref{chapter5_eta} we calculate important contributions to the semileptonic $B\\to \\eta^{(\\prime)}$ transition form factors in light-cone sum rules.}\n\\item{An example of the original method is presented in the last section of this chapter, section \\ref{example}, where we calculate the $\\alpha_s$ corrections to the gluon condensate contribution from a local correlation function.}\n\\end{itemize}\n We focus solely on the points required for future chapters. For more information on sum rules, see for example Refs.~\\cite{Colangelo:2000dp,Narison:1989aq,Shifman:1992xu,Reinders:1984sr}.\n\n\\section{SVZ Sum Rules}\nThe original sum rule method, which we refer to as \\textit{SVZ sum rules}, parameterises unknown non-perturbative QCD vacuum effects in terms of the so-called universal \\textit{vacuum condensates}. These quantities are vacuum expectation values of local operators $O_i$ that vanish in perturbation theory by definition and are ordered by their dimension $D$.\n\nThe calculation of a QCD sum rule starts from the calculation in QCD of a suitable correlation function in which the mesons are represented by interpolating currents possessing the correct quantum numbers. The method proceeds by equating two different representations of the correlation function. The first is obtained by performing a short-distance OPE, the result of which is matched to a second representation, in terms of a dispersion relation over physical hadronic states, leading to a sum rule from which various properties of the hadronic states can be extracted.\n\nThe SVZ sum rules find an important application in determining the universal hadronic parameters that appear in meson DAs.  Indeed, some of the first SVZ sum rule calculations were performed to extract decay constants $f_{\\pi, \\rho}$ \\cite{Shifman:1978bx}, and Gegenbauer coefficients  $a_n$ \\cite{Zhitnitsky:1982dd,Zhitnitsky:1983dd,Zhitnitsky:1985dd} of light meson DAs. The method can only be applied to parameters of the lowest few orders in conformal spin; parameters higher in conformal spin must be determined from other methods because the sum rules become unreliable.  One such method is that of \\textit{non-local} condensates, see for example Ref.~\\cite{Mikhailov:1986be}.\n\n\\subsection{Correlator}\nThe following \\textit{two-point correlation function} describes the propagation of a quark-antiquark pair\n\\begin{equation}\n\\Pi(q) = i \\int d^4 x \\,e^{i q \\cdot x} \\bra{0} T J_{1}(x) J_{2}(0) \\ket{0}\\,,\n\\label{sr_eq1}\n\\end{equation}\nwhere $q$ is the incoming momentum and possible Lorentz indices are omitted for simplicity. The local currents $J_i$  are chosen to have the correct quantum numbers and particle content corresponding to the particular hadronic parameters under investigation.  The physical picture of a hadron is of quarks, antiquarks and gluons confined within a typical hadronic size $R$  which is large when compared to the scale associated with perturbative effects. If one can show, however, that the correlation function is dominated by small spacial distances and time intervals\n\\begin{equation}\n|\\vec{x}| \\sim x^0 \\sim 1\/\\sqrt{Q} \\ll R\\,,\n\\label{sr_eq2}\n\\end{equation}\nfor a certain momentum configuration, then one has ensured the small size of the strong coupling $\\alpha_s$ and hence the use of perturbation theory in our calculation. One begins by noting that after contracting any Lorentz indices which may appear in the currents $J_i$ in Eq.~(\\ref{sr_eq1}), the correlation function can only depend on the interval $x^2=x^2_0-\\vec{x}^2$. By taking the Fourier transform, completing the square, and shifting the variable $x$ one finds\n\\begin{equation}\n\\Pi(q^2) = i \\int d\\kappa \\,\\int d^4 x \\,e^{i \\kappa x^2}\\,e^{i Q^2\/4 \\kappa} f(\\kappa)\\,.\n\\label{sr_eq3}\n\\end{equation}\nThe integral is dominated by the region where the arguments of the exponential vary slowly. This condition requires  $\\kappa \\sim 1\/x^2$ and $\\kappa \\sim Q^2$ which are both  fulfilled for $x^2 \\sim 1\/Q^2$; for large momentum transfer the quarks propagate near the light-cone.\\footnote{An expansion round $x^2 \\to 0$ is the basis of QCD sum rules on the light-cone -- see Section \\ref{LCSR}.} To find the true short-distance dominance one needs to dig a little further and by choosing the Lorentz frame $q_0=0$ one finds $\\vec{x}^{2} \\sim 1\/Q^2$ as required (\\ref{sr_eq2}). In the case of light quarks one needs the momentum transfer to the quarks to be large, $Q^2 \\equiv -q^2\\gg \\Lambda^2_{\\rm QCD}$. In the case of heavy quarks a large energy scale is introduced through the quark mass, for example $m_b$, which then serves to set the characteristic distances for the correlation function $|\\vec{x}|\\sim x_0\\sim 1\/(2 m_b)$; one is thus automatically in the perturbative regime.\n\n\\subsection{Short-Distance OPE}\nThe first of the two representations of the correlation function is obtained by performing the QCD calculation, valid for $Q^2 \\equiv -q^2 \\gg  \\Lambda^2_{\\rm QCD}$, using the short-distance OPE\n\\begin{equation}\n\\Pi (q^2) \\stackrel{\\rm{OPE}}{\\longrightarrow}\\sum_{i} C_i(q^2) \\left< O_i\\right> \\equiv \\Pi^{\\rm{OPE}}\\,,\n\\label{sr_eq4}\n\\end{equation}\nwhere the non-perturbative long distance effects of QCD are encoded in the condensates $\\left< O_i\\right>$ and the short-distance effects are included in the Wilson coefficient functions $C_i$ which are calculable in perturbation theory. Both the condensates and their coefficients are in general renormalisation scale dependent. Perturbative corrections to the condensates are calculated when necessary.  The perturbation theory contribution to Eq.~(\\ref{sr_eq4}) has $D=0$ and corresponds to the unit operator $\\left<O_{\\rm PT}\\right>=\\bf{1}$. The condensates play the role of power-corrections and are suppressed by inverse powers of the hard scale as $(Q^2)^{-D\/2}$. In the asymptotic limit $Q^2\\to\\infty$ only the unit operator survives, corresponding to asymptotic freedom. \n\n\\subsection{Condensates}\nThe condensates represent the effects of non-perturbative QCD and they cannot be determined from first principles due to the unknown nature of the QCD vacuum. The determination of the condensates is an industry in itself. The light quark condensate $\\bra{0} \\bar{q} q\\ket{0}$ has been known for a long time \\cite{Gell-Mann:1968rz} and it drives the breakdown of the chiral symmetry of the light quarks $q=\\{u,d\\}$ and its value can be extracted from experiment:\n\\begin{equation}\nm_\\pi^2 f_\\pi^2 \\approx-(m_u+m_d)\\left<\\bar q q\\right>\\,,\n\\end{equation}\nwhere we use the notation $\\left<O_i\\right>\\equiv\\bra{0}O_i\\ket{0}$. To define other condensates, one notes that the only vacuum expectation values of operators that can survive are those which are Lorentz invariant, spin zero, colour and flavour-singlets i.e. possess the quantum numbers of the vacuum. The complete set of condensates $\\left<O_i\\right>$ that contribute with $D\\le 6$ are\n\\begin{eqnarray}\n\\underbrace{\\left<\\textbf{1}\\right>}_{D=0}\\,, \\hspace{1cm}&\\underbrace{m_q\\left< \\bar{q} q\\right>}_{D=4}\\,,& \\hspace{1cm} \\underbrace{\\left<\\frac{\\alpha_s}{\\pi}G^2\\right>}_{D=4}\\,,\\nonumber \\\\\n\\underbrace{m_q\\left<  \\bar{q}\\sigma g_s G q \\right>}_{D=6}\\,, \\hspace{1cm}&\\underbrace{\\left<\\bar{q} \\Gamma_1q\\,\\bar{q}\\Gamma_2 q\\right>}_{D=6}\\,,&\\hspace{1cm} \\underbrace{\\left<g_s^3f G^3\\right>}_{D=6}\\,,\\nonumber\n\\end{eqnarray}\nwhere $q=\\{u,d,s\\}$ is a light quark spinor and all indices are contracted.\\footnote{The heavy quarks $c$, $b$ and $t$ do not form condensates because they are too massive to interact non-perturbatively with the QCD vacuum.}  We assume isospin symmetry  for $q=\\{u,d\\}$ and one must differentiate $q=s$ when $\\rm SU(3)_F$-breaking effects are taken into account.  Higher dimensional condensates $D> 6$ are not very well determined and generally unknown. If required, however, they can be estimated by employing the \\textit{vacuum saturation hypothesis} whereby the operator fields are simply split to form products of known condensates; for example, the quark-antiquark  $D=6$ operator can be simplified to the product of two $\\bar{q} q$ operators \\cite{PT:84,Shifman:1978by}. In practice, the OPE is truncated to a given order, and is usually justified by the stability of the resulting sum rule. The series, Eq.~(\\ref{sr_eq4}), is then given in terms of a limited number of condensates allowing sum rules to be written in terms of a small set of parameters incorporating the general features of non-perturbative QCD, while retaining its predictive power.\n\nThe procedure works in reverse, of course, where the values of condensates are deduced from sum rules for which the hadronic parameters are known from other methods;  two-point correlation functions featuring $\\bar{b} \\gamma_\\mu b$ or $\\bar{c} \\gamma_\\mu c$ currents correspond to the $\\Upsilon$ and $J\/\\Psi$ resonances respectively, of which the decay constants and masses are known. Values for the condensates are given, along with other input parameters, in Appendix~\\ref{appendixB}. Uncertainties in the values of the condensates and other input parameters constitute part of the reducible theoretical uncertainty of the sum rule approach. \n\n\\subsection{Dispersion Relation}\nTo proceed we need to relate the result of the OPE to a second representation of the correlation function which is obtained in terms of the spectrum of hadronic states in the physical region  $q^2>0$.   This is done via a \\textit{dispersion relation}, which is derived from the analytic properties of the correlation function as follows. The function $\\Pi(q^2)$ is analytic in all $q^2$ except on the real axis starting at a pole corresponding to the ground state particle. At higher energy  higher mass excited states and a continuum of many-particle states also feature. The higher mass resonances give poles above the ground state, the details of which depend on the physical spectrum of particles which possess the correct quantum numbers to couple to $\\Pi$. The continuum of many-particle states, correspond to a continuous cut, see Fig.~\\ref{sr_fig1}.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.5\\textwidth\\epsffile{analytic.eps}$$\n  \\caption[The spectral density function in the complex plane.]{\\small The general features of a spectral density function $\\rho^{\\rm had}(s)$ in the complex plane. The blob represents the pole due to the ground-state, the cross possible poles due to  higher mass resonances, and the thick line the cut due to the continuum of multi-particle states. The dotted line is the integration contour.} \n  \\label{sr_fig1}\n\\end{figure}\n\nUsing Cauchy's formula we can write\n\\begin{equation}\n\\Pi(q^2) =\\frac{1}{2 \\pi i} \\oint\\limits_{|z|=R} dz\\, \\frac{\\Pi(z)}{z-q^2}+\\frac{1}{2 \\pi i} \\int^R_0 dz\\, \\frac{\\Pi(z+i \\epsilon)-\\Pi(z- i \\epsilon)}{z-q^2}\\,,\n\\end{equation}\nwhere the region of integration is split into the parts just above and below the positive real axis and the circle of radius $R$. Provided that the correlation function vanishes at least as quickly as $q^{-2}$ as $|q^2| \\sim R \\to \\infty$ then the integral over the circle at radius $R$ goes to zero.\\footnote{If $\\Pi$ does not vanish quickly enough we subtract the first few terms in its Taylor expansion as required. We shall see that this does not matter in the end, due to the Borel transformation.} The remaining integral can be simplified using the fact that below the first pole at $q^2=s_{\\rm min}$,  $\\Pi(q^2)$ is real and above this point, according to the Schwarz reflection principle, $\\Pi(z+i \\epsilon)-\\Pi(z- i \\epsilon)= 2 i\\, \\textrm{Im}\\,\\Pi(z+ i \\epsilon)$. Hence\n\\begin{equation}\\label{disp}\n\\Pi(q^2) = \\int_{s_{\\rm min}}^{\\infty} ds\\, \\frac{\\rho(s)}{s-q^2- i \\epsilon}\\,,\n\\end{equation}\nwhere the function $\\rho(s)=\\frac{1}{\\pi} \\textrm{Im}\\, \\Pi(s)$ is the \\textit{spectral density} and describes the physical particle spectrum as a function of energy $s$. \n\n\\subsection{Unitarity Relation}\nAs we have seen, for large negative $q^2$ our correlation function is dominated by short-distance physics.  As $q^2$  becomes more positive the separation of the quarks increases. For large enough positive values of $q^2$ long-distance QCD interactions become more important and the correlation function then describes the creation of hadrons, which is the basis of its second representation. As discussed in the last section, $\\Pi$ uncovers a very complicated spectrum of states for $q^2>0$. We describe this situation by using the \\textit{unitarity relation}, which allows in insertion of a complete set of states into the correlation function\n\\begin{equation}\\label{complete}\n\\textbf{1}= \\sum_{n} \\int d \\Omega_n\\, \\ket{n(p)}\\bra{n(p)}\\,,\n\\end{equation}\nwhere $d \\Omega_n$ includes all phase-space factors and momentum conservation and the sum runs over all possible particles and polarisations, starting from the ground state $M$ of mass $m_M$. Inserting (\\ref{complete}) between the currents of our original correlation function (\\ref{sr_eq1}) yields an expression which we can relate to the hadronic spectral density\n\\begin{equation}\\label{unitarity}\n \\Pi^{\\rm had}(q^2) =   \\int \\frac{d^4 p}{(2 \\pi)^4} \\frac{1}{m^2_{M}-p^2} \\int d^4 x \\,e^{i q \\cdot x} \\bra{0} J_{1}(x)   \\ket{M(p)}\\bra{M(p)}J_{2}(0) \\ket{0} + \\dots\\,,\n\\end{equation}\nwhere the dots denote higher mass states which contribute to the continuum. We are usually interested in the ground state, and can insert the expressions for the matrix elements on the right hand side. The local matrix elements considered here can be used to extract vacuum-meson decay constants, for example.  Using the unitarity relation (\\ref{unitarity}) one can single out the ground state $M$ by comparing it to (\\ref{disp}) and writing the hadronic spectral density  as:\n\\begin{equation}\\label{gs}\n\\rho^{\\rm had}(s)=f_{M} \\,\\delta(s-m_{M}^2)+\\rho^{\\textrm{cont}}(s),\n\\end{equation}\nwhere $f_{M}$  is directly related to the matrix elements of the currents $J_1$ and $J_2$ in Eq.~(\\ref{unitarity}). For example, one could choose $J_1=J_2^\\dagger= \\bar q \\gamma_{z} s$ to extract $(f^\\parallel_{K^*})^2$ c.f. Eq.~(\\ref{das_eq22}). The exact form of the spectral density beyond the ground state is mostly unknown and the higher mass states and continuum contributions are usually lumped together in one function $\\rho^{\\textrm{cont}}(s)$. If the next highest particle above the ground state occurs at an energy not very much higher than $m_{M}$ then it is possible to explicitly include this particle as another delta-function term, analogously to the ground state. This procedure was used, for example, while investigating the leading-twist $K^*$ and $\\rho$ DA parameters for which the relevant correlators couple to the  $K_1$ and $b_1$ resonances respectively \\cite{Ball:1996tb,Ball:2005vx}. \n\n\n\\subsection{Quark-Hadron Duality}\nIt is possible to write the result of the OPE as a dispersion relation, with spectral density $\\rho^{\\textrm{OPE}}(s)$. As $\\rho^{\\textrm{cont}}(s)$ is mostly unknown  we replace it by $\\rho^{\\textrm{OPE}}(s)$ above a certain energy $s_0$\n\\begin{equation}\\label{qhd}\n\\rho^{\\textrm{cont}}(s) \\to \\rho^{\\textrm{OPE}}(s)\\,\\Theta(s-s_0)\\,.\n\\end{equation}\nThis assumption relies on the validity of  the hadronic  representation  being approximated by the partonic representation at higher energies. Inserting Eqs.~(\\ref{gs}) and  (\\ref{qhd}) into Eq.~(\\ref{disp}) one finds\n\\begin{equation}\n\\Pi^{\\rm had}(q^2)=\\frac{f_M}{m_M^2-q^2}+\\int_{s_{0}}^{\\infty} ds\\, \\frac{\\rho^{\\rm OPE}(s)}{s-q^2- i \\epsilon}\\,.\n\\end{equation}\nNow the assumption is not so strict because we only require a duality between the integrated spectral densities, not the spectral densities themselves. This is called \\textit{semi-global quark-hadron duality}. The parameter $s_0$ is called the  \\textit{continuum threshold} and its value  is specific for each particle spectrum being roughly equal to the energy of the next highest resonance above the ground state: $s_0 \\sim (m_M + \\Delta)^2$ where $\\Delta \\sim \\mathcal{O}(\\Lambda_{\\textrm{QCD}})$.  Ultimately it must be determined from the sum rule itself by requiring the numerical value of the determined quantity to be largely insensitive to its variation and this introduces the first source of systematic uncertainty to the sum rule method. We are now in a position to equate both representations\n\\begin{equation}\n\\Pi^{\\rm{had}}=\\Pi^{\\rm{OPE}}\\,, \n\\end{equation}\nto derive our sought after sum rule, however, before we do so, there is one last procedure to discuss, which greatly improves the behaviour of the sum rule.\n\n\n\\subsection{Borel Transformation And The Sum Rule}\nThe sum rule can be improved by suppressing the continuum contribution, which we have assumed to be well described by $ \\rho^{\\textrm{OPE}}(s>s_0)$ and the possible detrimental impact of this assumption is thus reduced.  We do this by performing a \\textit{Borel transformation} to both sides of the sum rule. The transformation is obtained by applying the operator\n\\begin{equation}\\label{borel1}\n\\mathcal{\\hat{B}} = \n\\lim_{\\stackrel{-q^2,n \\to \\infty}{-q^2\/n=M^2}}\\frac{(-q^2)^{(n+1)}}{n!}\\left(\\frac{d\\phantom{q^2}}{dq^2}\\right)^{n+1},\n\\end{equation}\nwhich takes a function of $q^2$ and produces a new function of  the \\textit{Borel parameter} $M^2$. One frequently encountered example is\n\\begin{equation}\n\\mathcal{\\hat{B}} \\frac{1}{(m^2-q^2)^k}= \\frac{1}{(k-1)!}\\frac{e^{-m^2\/M^2}}{(M^2)^{k}}\\,,\n\\end{equation}\nproviding an exponential suppression of the unknown continuum contributions, and a suppression of the power-corrections by factorials thus reducing the impact of neglected higher dimensional condensates. Also, as $\\mathcal{\\hat{B}} (q^2)^k=0$, any subtraction terms introduced to Eq.~(\\ref{disp}), which can only appear as polynomials in $q^2$, are killed off. The Borel transformation improves the stability and accuracy of the sum rule.\n\nThe Borel parameter $M^2$ is the second and last sum rule specific parameter to be introduced; along with $s_0$ it is required to impact very little, when varied, on the numerical value of the quantity being determined. The variation of $M^2$ changes the relative impact of the power-corrections and perturbation theory contributions.  In evaluating sum rules one looks for a \\textit{Borel window} which is  usually in the range $1\\,\\rm GeV^2\\leqslant M^2\\leqslant 2 \\,\\rm GeV^2$ for a typical mesonic DA parameter.  The sum rule should be reliable if a weak dependence (a plateau) is found, the contribution from the continuum is small, and there are no unnatural numerical cancellations.\n\nWe now equate Eqs.~(\\ref{disp}) and (\\ref{sr_eq4}) to reach the sum rule\n\\begin{equation}\\label{sr1}\nf_M\\,e^{-m_M^2\/M^2}= \\int^{s_0}_0 ds\\,e^{-s\/M^2} \\,\\rho^{\\textrm{OPE}}(s)\\,,\n\\end{equation}\nwhere the hadronic quantity $f_M$ is given as a function of the universal non-perturbative condensates, the perturbative short-distance coefficients as calculated from QCD, and the sum rule parameters $s_0$ and $M^2$. The sum rule is saturated by the ground state and higher mass states are suppressed. As the correlation function (\\ref{sr_eq1}) does not depend on the renormalisation scale, the $\\mu$ dependence of the condensates, when multiplied by their coefficient functions, must cancel in the sum of (\\ref{sr_eq4}). The sum is always truncated, however, and the residual $\\mu$ dependence will be a source of theoretical uncertainty.\n\n\\subsection{Non-local Formalism}\nOne way to gain access to parameters higher in conformal spin is to calculate sum rules involving operators  which are related to moments of DAs\n\\begin{equation}\n\\bra{0}\\bar{q}(0) (\\stackrel{\\leftrightarrow}{D} \\cdot z)^k \\Gamma s(0)\\ket{V}\\sim \\int^1_0 du\\, (2u-1)^k \\phi(u)\\equiv \\left<\\xi^k\\right>\\,.\n\\end{equation}\nFor the $K^*$ for example the first few moments of both the leading-twist DAs are $\\left<\\xi^0\\right>=1$, $\\left<\\xi^1\\right>=\\frac{3}{5} a_1(K^*)$, $\\left<\\xi^2\\right>=\\frac{1}{35}(7+12 a_2(K^*))$ and $\\left<\\xi^3\\right>=\\frac{1}{105}(27 a_1(K^*)+20 a_3(K^*))$. A more elegant method, enabling the  DA parameters to be extracted individually, relies on calculating a correlator of two currents, one of which is non-local, with fields at light-like separations ($z^2=0$) \\cite{Ball:2003sc}. Consider the following\n\\begin{equation}\n\\Pi(q\\cdot z) = i \\int d^4 x \\,e^{i q \\cdot x} \\bra{0} T J(x) \\bar{s}(0) \\gamma_{z} q(z) \\ket{0}\\,,\n\\label{nonlocalCF}\n\\end{equation}\nwhere $J(x)$ is local, and the non-local current yields the leading-twist DA (\\ref{das_eq16}). The sum rule (\\ref{sr1}) then reads\n\\begin{equation}\\label{sr2}\nf_J f^\\parallel_{K^*}\\,e^{-m_M^2\/M^2}\\int^1_0 du\\,e^{-i \\bar{u}q\\cdot z} \\phi_{2;K^*}^\\parallel= \\int^{s_0}_0 ds\\,e^{-s\/M^2} \\,\\int^1_0 du\\,e^{-i \\bar{u}q\\cdot z}\\rho^{\\textrm{OPE}}(s,u)\\,.\n\\end{equation}\nThe integration over $u$ on the right hand side naturally arises via the Feynman parameterisation used in the calculation. At this point one can exploit the orthogonality of the Gegenbauer polynomials by replacing the exponential weight function $e^{-i\\xi q\\cdot z} \\to C_n^{3\/2}(\\xi)$ on both sides to project out $a_n^\\parallel(K^*)$ via Eqs.~(\\ref{das_eq15}) and (\\ref{das_eq17}). \n\\begin{figure}[h]\n$$ \\epsfxsize=0.3\\textwidth\\epsffile{generalnonlocal.eps}$$\n  \\caption[A generic diagram for a non-local sum rule.]{\\small A generic non-local diagram. The dotted line denotes the path ordered gauge factor $[z,-z]$ between the two quark fields. The momentum $q$ is injected at point $y$ - the vertex on the right hand side.} \n  \\label{sr_fig2}\n\\end{figure}\nIn Fig.~\\ref{sr_fig2} we show the leading diagram of the non-local correlation function (\\ref{nonlocalCF}). The dotted line denotes the path ordered gauge factor $[z,-z]$ between the two quark fields. The non-local formalism allows, in principle, an extraction of  parameters of arbitrary order $n$. In practice, however, only the parameters of the lowest few orders $n$ are accessible due to instability of the resulting sum rules. One finds that the power-corrections in $\\rho^{\\textrm{OPE}}$ grow with positive powers of $n$ compared to the perturbative contribution. For high enough $n$ this behaviour upsets the hierarchy of contributions to the OPE, where non-perturbative terms are expected to be moderately sized corrections to the leading term. Hence the method is justified for low-order coefficients $n\\leq2$ where the non-perturbative coefficients describe the general features of the DA. It breaks down for higher-order coefficients $n>3$ because the local vacuum condensates appear with $\\delta$-functions which cannot accommodate the information needed to describe the more detailed shape of the DA, see Refs.~\\cite{Ball:1997rj,Ball:2003sc}. \n\n\\section{QCD Sum Rules On The Light-Cone}\\label{LCSR}\nA modification of the QCD sum rule method known as QCD sum rules on the light-cone, or \\textit{light-cone sum rules} (LCSRs)  \\cite{Balitsky:1989ry,Braun:1988qv,Chernyak:1990ag}, was developed to overcome difficulties encountered when calculating transition and electromagnetic form factors in the SVZ method.\\footnote{The term ``light-cone sum rules\" first appears in Ref.~\\cite{Ball:1991bs}.} The problems are related to the asymptotic scaling behaviour of the form factors in the heavy-quark limit $m_b \\to \\infty$. LCSRs rely on the use of DAs as their universal non-perturbative hadronic input and lead to the correct asymptotic behaviour in the heavy-quark limit. The DAs represent a  partial re-summation of the operators appearing in the condensates and appear ordered in contributions of increasing twist \\cite{Ball:1997rj}. We can view LCSRs as a marriage of the SVZ technique and the theory of hard exclusive processes \\cite{Chernyak:1977fk,Lepage:1980fj,Efremov:1979qk,Chernyak:1977as}. In the case of the ``heavy-to-light'' $B\\to M$ transition form factors, LCSRs have been applied successfully to pseudoscalar transition form factors \\cite{oldpseudo,Ball:1998tj,Ball:2001fp,Ball:2004ye} and vector transition form factors \\cite{Ball:1998kk,Ball:2004rg}. \n\nFor LCSRs to become competitive with the SVZ sum rules, a good knowledge of higher-twist DAs is required. This motivates the determination of the non-perturbative DA parameters via SVZ sum rules and via LCSR, the DAs themselves to determine other non-perturbative parameters, such as transition form factors. As with SVZ sum rules, the starting point of LCSRs is with a suitable correlation function. For the extraction of $B\\to M$ transition form factors we require a two-point correlator, this time sandwiched between the vacuum and the meson state $M$, which is the example considered in this section. One employs much of the same methodology as in the last section, although now one requires the correlation function to be expanded in an OPE on the light-cone. In doing so one finds that the correlation function factorises and can be written in terms of a convolution of hard scattering kernels and the universal non-perturbative DAs of the light-meson.   To that end, consider a correlation function of two quark currents taken between the vacuum and an on-shell meson $M$\n\\begin{equation}\\label{CFLCSR}\n\\Pi(q,p_B)=i\\int d^4x \\,e^{i q\\cdot x} \\bra{M(p)}T J_1(x) j_B^{\\dagger}(0)\\ket{0}\\,,\n\\end{equation}\nwhere $j_B = m_b\\, \\bar{q}i \\gamma_5 b$ is the \\textit{interpolating current} of the $B$ meson which defines the $B$ meson decay constant\n\\begin{equation}\n\\bra{B(p_B)}j_B\\ket{0}=m^2_B f_B\\,.\n\\label{bdecayconstant}\n\\end{equation}\nThe current $J_1(x)$ is chosen to project out the form factor of interest. The  momentum $q$ is  injected into the weak vertex, $p_B$ is the momentum of the $B$ meson and $p$ is the  momentum of $M$ with $q+p=p_B$. The correlation function is dominated by light-like distances for virtualities\n\\begin{equation}\nm_b^2 - p_B^2 \\geq \\mathcal{O}(\\Lambda_{\\rm QCD} m_b)\\,, \\qquad m_b^2 - q^2 \\geq \\mathcal{O}(\\Lambda_{\\rm QCD} m_b)\\,,\n\\end{equation}\nwhich ensures the slow variation of the exponential in Eq.~(\\ref{CFLCSR}) and its suitability for an expansion around the light-cone. The light-cone expansion results in the transverse and ``minus'' degrees of freedom being integrated out, leaving the longitudinal momenta of the partons as the relevant degrees of freedom. As a result a cutoff $\\mu$ is introduced below which the transverse momenta are included in the resulting light-mesons DAs. The contributions from momenta above this cutoff are calculable in perturbation theory. The procedure yields the \\textit{collinear factorisation} of the correlation function\n\\begin{equation}\\label{LCSR:fac}\n\\Pi(q^2,p_B^2)=\\sum_n \\int_0^1 \\,du\\, T^{(n)}(u,q^2,p_B^2,\\mu)\\,\\phi_{n;M}(u,\\mu)\\,,\n\\end{equation}\nwhere $u$ $(1-u)$ denotes the momentum fraction of the outgoing quark (antiquark) and the sum is over all twist and possible polarisation contributions. The scale dependence of the hard scattering kernels $T^{(n)}$ must cancel that of the DAs $\\phi_{n;M}$. The factorisation formula has to be verified by direct calculation and a proof to all orders in  $\\alpha_s$ does not exist. The verification relies on the cancelation of divergences, of which there are two types: the IR and UV singluarities arising from loop calculations and so-called soft singularities which appear when the convolution over $u$ does not converge at the end-point regions ($u \\sim 0~\\textrm{or}~1$) i.e. when one of the quarks is soft. In terms of kinematics there are two main contributing processes: the hard-scattering mechanism and the soft contribution or Feynman mechanism. Both mechanisms are included in the LCSR approach for which there are no soft divergences and the IR\/UV divergences can be treated in dimensional regularisation. \n\nOne can write the result of the light-cone expansion (\\ref{LCSR:fac}) as a dispersion relation in $p_B^2$\n\\begin{equation}\n\\Pi^{\\rm  LC} (p_B^2, q^2) = \\int_{m_b^2}^{\\infty} ds\\,\\frac{\\rho^{\\rm  LC} (s, q^2)}{s-p_B^2}\\,.\n\\end{equation}\nTaking the imaginary part, to obtain $\\rho^{\\rm  LC} $, is straight forward after integration over the momentum fraction $u$ is performed. The correlation function has a cut in $p_B^2$ starting at $m_b^2$ over the physical region. One now matches this calculation to the hadronic representation of the correlation function, which can also be written as a dispersion relation\n\\begin{equation}\n\\Pi^{\\rm  had} (p_B^2, q^2) = \\int_{m_B^2}^{\\infty} ds\\,\\frac{\\rho^{\\rm  had} (s, q^2)}{s-p_B^2}\\,,\n\\end{equation}\nwhere the physical spectral density is given by the ground state $B$ meson plus higher mass states forming a continuum as\n\\begin{equation}\n\\rho^{\\rm had} (s, q^2)= F_M\\, \\delta(s-m_B^2) +  \\rho^{\\rm LC } (s, q^2) \\,\\Theta(s-s_0)\\,.\n\\end{equation}\nThe quantity $F_M$ will contain the form factor we require. We perform the Borel transformation to arrive at the LCSR\n\\begin{equation}\nF_M\\, e^{-m_B^2\/M^2}  =\\int_{m_b^2}^{s_0} ds \\, e^{-s\/M^2} \\rho^{\\rm LC} (s, q^2)\\,.\n\\end{equation}\nTo extract the form factor we need to find a sets of parameters $M^2$ and $s_0$ such that the form factor is largely insensitive to their variation. As with SVZ sum rules, there is no rigourous way to do this and so the procedure introduces the irreducible source of uncertainty to the method. \n\n\\section{Example Calculation - The Gluon Condensate}\\label{example}\nHere we present an example calculation within the SVZ sum rule framework. The result of the calculation is used in the sum rule for the G-even $K$ meson three-particle twist-3 DA parameter $f_{3K}$, see Ref.~\\cite{Ball:2006wn}. We calculate the $\\alpha_s$ correction to the gluon condensate $\\left< \\frac{\\alpha_s}{\\pi} G^2 \\right>$ which proceeds from the following local correlation function\n\\begin{equation}\\label{cf2}\n\\Pi^{(G^2)}   =  i \\int d^4 y \\, e^{i q\\cdot y}\\bra{0} T \\bar q(0) \\sigma^{\\mu z}  g_sG_{\\mu  z}(0) s(0) \\bar s(y) \n\\sigma^{\\nu z} g_sG_{\\nu  z}(y) q(y)\\ket{0}\\,,\n\\end{equation}\nfor which the leading-order contribution  vanishes. A convienient way of extracting the gluon condensate is to make use of the \\textit{back-ground field technique} in which the fixed-point gauge allows the Taylor expansion of quark and gluon fields to be written in a gauge-covariant form, see  Ref.~\\cite{Novikov:1983gd} for details. The gluon field in the QCD Lagrangian (\\ref{basics_eq1}) is split into ``quantum''  and  ``classical''  (background) fields\n\\begin{equation}\\label{split}\nA^a_\\mu\\to a^a_\\mu+\\mathcal{A}^a_\\mu\\,,\n\\end{equation}\nwhere the background field $\\mathcal{A}^a_\\mu$ is taken in the fixed-point gauge at $x_0=0$. The quantum field $a^a_\\mu$ is taken to be in the Feynman gauge, thus requiring the gauge fixing term ($\\xi=1$)\n\\begin{equation}\n\\mathcal{L}^{\\rm fix}=-\\frac{1}{2}(\\partial^\\mu a_\\mu^a+g_s f^{abc}\\mathcal{A}^{b \\mu} a_\\mu^c)^2\\,,\n\\end{equation}\nto be added to the QCD Lagrangian. The quantum field propagates perturbatively and we may use the standard expression\n\\begin{equation}\n\\begC1{a^a_\\mu}\\conC{(x)\\,}\\endC1{a^b_\\nu}(y)=i \\delta^{ab}\\int \\frac{d^4 l}{(2 \\pi)^4}D_{\\mu\\nu}(l)e^{-i l\\cdot (x-y)}\\,,\\qquad D_{\\mu\\nu}(l)=\\frac{-g_{\\mu\\nu}}{l^2},\n\\end{equation}\nThe background field does not propagate perturbatively, and is the field that goes to form the condensate; it represents the low-energy, long distance modes of the gluon field that probe the non-perturbative structure of the QCD vacuum. The fixed-point gauge condition allows $\\mathcal{A}_\\mu^a(x)$ to be expressed in terms of the gluonic field strength tensor as\n\\begin{equation}\n\\mathcal{A}_\\mu^a(x)=\\sum^\\infty_{n=0}\\frac{1}{n!(n+2)} x^\\omega x^{\\omega_1}... x^{\\omega_n}\\left[ D_{\\omega_1}(0),\\left[D_{\\omega_2}(0),\\left[...\\left[D_{\\omega_n}(0),G_{\\omega\\mu}^a(0)\\right]...\\right]\\right]\\right]\\,,\n\\end{equation}\nand translating to momentum space one finds\n\\begin{equation}\\label{condfield}\n\\mathcal{A}_\\mu^a(k)=-\\frac{i}{2} G^a_{\\omega\\mu}(0)\\left[(2\\pi)^4 \\delta^{(4)}(k)\\right]\\frac{\\partial\\phantom{ k^\\omega}}{\\partial k^\\omega}+\\dots\\,,\n\\end{equation}\nwhere we only require the first term; higher order terms give rise to higher dimensional condensates which we do not consider. As we have to introduce two condensate gluons to construct $\\left<G^2\\right>$ we introduce two auxiliary vacuum momenta $k$ and $k^\\prime$ for which the fixed-point $x_0=0$ is a sink. After integration over coordinates these momenta appear in the quark and gluon propagators. The two corresponding derivatives are then taken, and the vacuum momenta set to zero. The following expression proves very useful in managing derivatives of quark propagators\n\\begin{equation}\\label{quarkderiv}\n\\frac{\\partial}{\\partial p_\\mu} S^{(q)}(p)=-S^{(q)}(p)\\gamma^\\mu S^{(q)}(p)\\,,\\qquad S^{(q)}(p)=\\frac{\\slash{p}+m_q}{p^2-m_q^2}\\,,\n\\end{equation}\nfor arbitrary quark flavour $q$. The gluon condensate is finally extracted using\n\\begin{equation}\\label{vacav}\nG^a_{\\omega\\mu}(0)G^b_{\\omega^{\\prime}\\nu}(0)=\\frac{1}{D(D-1)}\\frac{\\delta^{ab}}{N_c^2-1}\\left(g_{\\omega\\omega^{\\prime}}g_{\\mu \\nu}-g_{\\omega\\nu}g_{\\omega^{\\prime}\\mu}\\right)\\left<G^2\\right>\\,,\n\\end{equation}\nwhere $D$ is the spacetime dimension. Due to Eq.~(\\ref{split}) the expansion of $\\mathcal{L}_{\\rm QCD}$  yields ``interaction'' terms in which background fields are radiated from the propagating gluons at single or double vertices, both of which contribute to the $\\mathcal{O}(\\alpha_s)$ correction to the gluon condensate. These vertices are shown in Fig.~\\ref{sr_fig3} and the corresponding terms are \n\\begin{eqnarray}\\label{glueint}\n \\mathcal{L}^{\\mathcal{A}aa}_{int}&=&-\\frac{1}{2}g_s f^{abc}\\left[\\left(\\partial^\\mu\\mathcal{A}^{a\\nu}-\\partial^\\nu\\mathcal{A}^{a\\mu}\\right)a_\\mu^b a_\\nu^c\\right.\\nonumber\\\\\n &+&\\left.(\\partial^\\mu a^{a\\nu}-\\partial^\\nu a^{a\\mu})(\\mathcal{A}^b_\\mu a^c_\\nu+a^b_\\mu\\mathcal{A}^c_\\nu)+2(\\partial_\\mu a^{a\\mu})\\mathcal{A}^{b\\nu}a_\\nu^c\\right]\\,,\\nonumber\\\\\n \\mathcal{L}^{\\mathcal{AA}aa}_{int}&=&-\\frac{1}{2} g_s^2 f^{abc} f^{ade}\\left[\\mathcal{A}^b_\\mu \\mathcal{A}^{d\\mu} a_\\nu^e a^{c\\nu} +\\mathcal{A}^b_\\mu  a^{d\\mu} \\mathcal{A}^e_\\nu a^{c\\nu}+\\mathcal{A}^b_\\mu  a^{c\\mu} \\mathcal{A}^d_\\nu a^{e\\nu}\\right]\\,,\n\\end{eqnarray}\nwhere terms which vanish eventually via Eq.~(\\ref{vacav}) due to $f^{abc}\\delta^{bc}=0$ are omitted.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.8\\textwidth\\epsffile{gluonvertices.eps}$$\n\\caption[Interactions of the background field $\\mathcal{A}^a_\\mu$ with the quantum field $a^a_\\mu$.]{\\small  The interactions of the background field $\\mathcal{A}^a_\\mu$ (denoted by a cross) with the quantum field $a^a_\\mu$ corresponding to $\\mathcal{L}^{\\mathcal{A}aa}_{int}$ and $\\mathcal{L}^{\\mathcal{AA}aa}_{int}$ respectively.} \n\\label{sr_fig3}\n\\end{figure}\nContributions also stem directly from the gluonic field strength tensors in Eq.~(\\ref{cf2}) which give rise to gluon emission of either one or two fields from the vertices at co-ordinates $0$ and $y$. Due to the gauge condition there is no ``left-right'' symmetry and all diagrams with two gluons, of which at least one is a condensate gluon, emerging from the vertex at $x=0$ vanish due to $A_\\mu(0)=0$.  Diagrams with two condensate gluons at point $y$, which originate from the non-linear part of the gluonic field strength tensor,  also give zero due to $f^{abc}\\delta^{bc}=0$. There is an ``up-down'' symmetry where diagrams related by a  reflection in the central horizontal axis are equal. Overall we find there to be 10 distinct non-zero diagrams to be calculated which are shown in Fig.~\\ref{sr_fig4}. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.7\\textwidth\\epsffile{cond1.eps}$$\n$$\\epsfxsize=0.5\\textwidth\\epsffile{cond2.eps}$$\n\\caption[Diagrams contributing to the gluon condensate at $\\mathcal{O}(\\alpha_s)$.]{\\small The diagrams contributing to the gluon condensate at $\\mathcal{O}(\\alpha_s)$ for the SVZ sum rule of the $K$ twist-3 DA parameter $f_{3K}$ -- see Ref.~\\cite{Ball:2006wn}. For each diagram the fixed-point $x_0=0$ is at the left most vertex and the right most is at $y$.} \n\\label{sr_fig4}\n\\end{figure}\n\nSome of the diagrams are divergent, however, all divergences cancel in the sum of all diagrams.\\footnote{We use dimensional regularisation and the $\\overline{MS}$ renormalisation scheme throughout this thesis.} For an explicit example consider the last diagram in the second line of Fig.~\\ref{sr_fig4}. It is evident that we require $\\mathcal{L}^{\\mathcal{A}aa}_{int}$ to be contracted in all possible ways with quantum fields originating from the linear part of the gluonic field strength tensors at points $0$ and $y$. This, multiplied by the condensate field originating from the quark loop yields the gluonic part of the calculation\n\\begin{equation}\\label{allcon}\n\\sim\\mathcal{A}^d_\\delta(v) \\left.(\\partial_\\mu a^a_z(0)-\\partial_z a^a_\\mu(0))\\, \\mathcal{L}^{\\mathcal{A}aa}_{int}(w) \\,(\\partial_\\nu a^b_z(y)-\\partial_z a^b_\\nu(y))\\right|_{\\rm all\\,contractions}\\,,\n\\end{equation}\nwhich is eventually given in momentum space by (omitting Lorentz indices)\n\\begin{equation}\\label{gluonicpart}\n\\sim \\frac{\\partial}{\\partial k}\\frac{\\partial}{\\partial k^\\prime}\\frac{f(l,k^\\prime)}{l^2 (l-k^\\prime)^2} \\left<G^2\\right>\\,,\n\\end{equation}\nwhere the condensate gluon within $\\mathcal{L}^{\\mathcal{A}aa}_{int}(w)$ is expressed by Eq.~(\\ref{condfield}) with momentum $k^\\prime$ and $f(l,k^\\prime)$ is a function of the loop momentum $l$ and the vacuum momentum $k^\\prime$. The  quark loop yields a usual trace \n\\begin{equation}\\label{quarkpart}\n\\sim \\frac{\\textrm{tr}\\left[(\\slash{p}+\\slash{q}-\\slash{l})\\sigma^{\\mu z}(\\slash{p}+\\slash{k})\\gamma^\\delta\\slash{p}\\,\\sigma^{\\nu z}\\right]}{(p+q-l)^2(p+k)^2 p^2}\\,,\n\\end{equation}\nand after multiplying together Eqs.~(\\ref{gluonicpart}) and (\\ref{quarkpart}), performing the derivatives in $k$ and $k^\\prime$ and integrating over the momenta $p$ and $l$ we find\n\\begin{equation}\n\\Pi^{(G^2)}_{\\rm example}  = \\frac{1}{384}\\frac{\\alpha_s}{\\pi}  \\left\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle \\frac{(q\\cdot z)^4}{q^2}\\,.\n\\end{equation}\nIn this way we can include all the other diagrams shown in Fig.~\\ref{sr_fig4} to obtain the contribution to the sum rule\n\\begin{equation}\n\\Pi^{(G^2)}  = \n-\\frac{89}{5184}\\,\\frac{\\alpha_s}{\\pi}\\,\\left\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle \\,\\frac{(q \\cdot z)^4}{q^2}\\,,\n\\label{last}\n\\end{equation}\nwhich differs from the result obtained in Ref.~\\cite{Zhitnitsky:1985dd}; the logarithmic term is not reproduced:\n\\begin{equation}\n\\sim \\log \\left(\\frac{M^2}{\\mu^2}\\right) \\left< \\frac{\\alpha_s}{\\pi} \\,G^2 \\right>\\,.\n\\label{least}\n\\end{equation}\n\n\\chapter{The Determination Of Vector Meson Twist-2 And Twist-3  Parameters}\\label{chapter4_det}\nIn this chapter we determine the leading twist-2 and twist-3 two- and three-particle vector meson DA parameters using the non-local modification of SVZ sum rules. The parameters are defined in Chapter~\\ref{chapter2_DAs} and the sum rule method is outlined in Chapter~\\ref{chapter3_SR}. We express the relevant correlation functions, via the OPE, in terms of the perturbative and condensate  contributions.  Key to the analysis is the inclusion of all G-parity and  $\\rm SU(3)_F$-breaking effects which, as discussed in Chapter~\\ref{chapter2_DAs}, come from a variety of sources, and allow a consistent determination of the parameters for the $\\rho$, $K^*$, and $\\phi$. Motivation for the present analysis comes from various sources, including:\n\\begin{itemize}\n\\item{values for the decay constants and leading-twist DA Gegenbauer moments are required as input for QCD factorisation frameworks which provide a systematic method for the calculation of $B$ decay matrix elements. We discuss one such framework in Chapter~\\ref{chapter6_QCDF}.}\n\\item{Twist-2 and twist-3 DAs provide the leading non-perturbative input within the method of LCSR, as discussed in Chapter~\\ref{chapter3_SR}, and as such are applied to many problems in heavy-flavour physics, such as the calculation of $B$ transition form factors and the estimation of $B$ decay matrix elements including power-suppressed contributions to QCD factorisation frameworks, see Chapter~\\ref{chapter7_rad}.}\n\\item{A full determination of the twist-3 DA parameters, including $\\rm SU(3)_{F}$-breaking and G-parity violating effects, and the inclusion of $\\mathcal{O}(\\alpha_s)$ and $\\mathcal{O}(m_s^2)$ corrections to the quark condensate contributions to the twist-2 DA parameter sum rules are new to the present analysis, allowing $a_2^{\\parallel,\\perp}(\\phi)$ to be determined, for the first time, to the same accuracy as $a_2^{\\parallel,\\perp}(\\rho,K^*)$.}\n\\end{itemize}\nAll input parameters for the sum rules, and useful formulas, such as those required to take the imaginary parts of intermediate results, and various relevant integrals, are given in Appendix~\\ref{appendixB}. In performing the calculations we find Refs.~\\cite{PT:84,Borodulin:1995xd} very useful. The material covered in this chapter partially follows that of Ref.~\\cite{Ball:2007rt}.\n\n\\section{Twist-2}\nIn this section we focus on the determination of the twist-2 DA Gegenbauer coefficients $a_{2}^{\\parallel,\\perp}$ defined by Eqs.~(\\ref{das_eq16}) and (\\ref{das_eq19}).  The sum rules for $f_{K^*}^{\\parallel,\\perp}$, including $\\rm SU(3)_F$-breaking corrections, were calculated in Refs.~\\cite{Govaerts:1986ua,Ball:2005vx,Ball:2006fz}. Those for the G-parity violating $a_1^{\\parallel,\\perp}(K^*)$ in Refs.\\cite{Ball:2005vx,Ball:2006fz} and those for $a_2^{\\parallel,\\perp}(K^*)$ in \\cite{Ball:2003sc} apart from perturbative terms in $m_s^2$ and the $\\mathcal{O}(\\alpha_s)$ and $\\mathcal{O}(m_s^2)$ corrections to the quark condensate, which are new to the present analysis.  Motivation for including these corrections is found by examining the individual contributions to the sum rules for $a_2^{\\parallel,\\perp}(K^*)$ given in Ref.~\\cite{Ball:2003sc}. They are found to be dominated by $\\left<\\bar s s\\right>$ as we can see from the following explicit break down of contributions:\n\\begin{eqnarray}\na_2^\\parallel(K^*)&=&\\overbrace{0.05}^{\\textrm{PT}}+\\overbrace{0.08}^{\\left<\\frac{\\alpha_s}{\\pi} G^2\\right>}+\\overbrace{0.11}^{\\left<\\bar s g_s \\sigma G s\\right>}+\\overbrace{0.04}^{\\left<\\bar q q\\right>^2}-\\overbrace{0.16}^{\\left<\\bar s s\\right>}+\\overbrace{0.02}^{\\left<\\bar s s\\right>^2}-\\overbrace{0.05}^{\\left<\\bar q q\\right>\\left<\\bar s s\\right>}\\nonumber\\\\\na_2^\\perp(K^*)&=&0.06+\\,\\,0.10\\,\\,+\\,\\,\\,0.25\\,\\,\\,+0.03-0.27+0.02\\,-0\\,,\n\\end{eqnarray}\nfor the reference point $s_0=1.2\\,\\textrm{GeV}^2$, $M^2=1\\,\\textrm{GeV}^2$ and $\\mu=1\\,\\textrm{GeV}$. Moreover, for the  $\\phi$ the impact of a finite strange quark mass may be even more pronounced with respect to perturbation theory and the gluon condensate. \n\nFirstly, we give an overview of the calculation of the $\\mathcal{O}(\\alpha_s)$ and $\\mathcal{O}(m_{s}^2)$ corrections to the quark condensate $\\left<\\bar s s\\right>$; the calculations for $\\left<\\bar q q\\right>$ are analogous. We only need extract terms proportional to $m_s$  as the contributions proportional to $m_q$ are identical; we can find the contributions for $\\phi$ by simply replacing $\\left<\\bar q q\\right>\\to\\left<\\bar s s\\right>$ and doubling the terms in $m_{s} \\left<\\bar s s\\right>$, $m_{s} \\left<\\bar q q\\right>$ and $m_{s} \\left<\\bar s g_s G s\\right>$. Contributions for $\\rho$ are found by setting $m_s\\to0$. Secondly, we go on to analyse the sum rules for $a_2^{\\parallel,\\perp}(\\phi)$. We end this section by presenting the results.\n\n\\subsection{Calculation}\nFor both polarisations we begin from the diagonal correlation function\n\\begin{equation}\n\\Pi_{2;K^*}(q\\cdot z) = i \\int d^4y \\,e^{-iq\\cdot y} \\bra{0} T \\bar q(y) \\Gamma s(y) \\bar s(0) \\Gamma [0,z]q(z) \\ket{0}\\,\n\\label{C.0}\n\\end{equation}\nwhere $\\Gamma^{\\parallel}=\\gamma_z$ and $\\Gamma^{\\perp}=\\sigma_{\\mu z}$. For the longitudinal parameters the sum rule is exactly that given by Eq.~(\\ref{sr2}) with $f_J\\to f^\\parallel_{K^*}$ and for the transverse parameters the sum rule is analogous. Both polarisations have the same projections onto the DA parameters\n\\begin{eqnarray}\n\\left(f_{K^*}\\right)^2e^{-m_{K^*}^2\/M^2}\\left[1\\right]\n & = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int^1_0 du\\,\\left[1\\right]\n      \\frac{1}{\\pi}\\, {\\rm Im}_{s}\n\\pi_{2;K^*}(u)\\,,\\label{C0}\n\\\\\n\\left(f_{K^*}\\right)^2  e^{-m_{K^*}^2\/M^2}\n      \\,\\left[\\frac{9}{5}\\,a_{1}(K^*)\\right]\n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int^1_0 du\\,\n     \\left[ 3 \\xi \\right]\\frac{1}{\\pi}\\, {\\rm Im}_{s}\n \\pi_{2;K^*}(u)\\,,\n\\nonumber\\\\\n\\left(f_{K^*}\\right)^2 e^{-m_{K^*}^2\/M^2}\n      \\,\\left[\\frac{18}{7}\\,a_{2}(K^*)\\right]\n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int^1_0 du\\,\n    \\left[ \\frac{1}{2}(  15 \\xi^2-3)\\right]\\frac{1}{\\pi}\\, {\\rm Im}_{s}\n    \\pi_{2;K^*}(u)\\,, \\nonumber\n\\end{eqnarray}\nwhere ${\\rm Im}_s$ denotes taking the imaginary part with respect to $s$.  The fact that we are dealing with non-local correlation functions means that we do not integrate over the co-ordinate $z$. The resulting residual exponential function remains throughout the calculation and can contribute to the momentum integrals yielding powers of $i c (q\\cdot z)$, where $c$ is a constant. Ultimately the exponential functions can be cast into  the ``canonical form'' set by the exponential appearing in front of the leading-twist DA i.e. $e^{- i \\bar{u}q\\cdot z}$ -- see Eq.~(\\ref{sr2}). \n\\subsection*{Quark Condensate}\n\\begin{figure}[h]\n$$ \\epsfxsize=0.25\\textwidth\\epsffile{tw2_quark0_sr.eps}$$\n\\caption[Diagram contributing to the quark condensate $\\left<\\bar s s \\right>$ at leading-order.]{\\small The leading-order diagram contributing to the quark condensate $\\left<\\bar s s \\right>$.} \n\\label{det_fig1}\n\\end{figure}\nThe tree-level diagram is shown in Fig.~\\ref{det_fig1}. To extract the quark condensates to $\\mathcal{O}(m_s^2)$ we use the following expansion of the quark fields (for general quark flavour $q$)\n\\begin{eqnarray}\n\\bra{0}\\!:\\! \\bar q^i_\\alpha(x_1)\\, q^j_\\beta(x_2)\\!:\\!\\ket{0}&=&\\delta^{ij}\\frac{\\left<\\bar q q\\right>}{12}\\left\\{\\delta_{\\beta\\alpha}\\left(1-\\frac{\\Delta^2}{2D}m_q^2\\right)\\right.\\nonumber\\\\\n&-&\\left.m_q\\frac{i }{D}(\\gamma_\\lambda)_{\\beta\\alpha}\\Delta^\\lambda\\left(1-\\frac{\\Delta^2 }{2(2+D)}m_q^2\\right)\\right\\}\\,,\n\\label{quarkextract}\n\\end{eqnarray}\nwhere  $\\Delta_\\mu=(x_2-x_1)_\\mu$ and $i,j$ are colour and $\\alpha,\\beta$ spinor indices. One can deal with the co-ordinate $\\Delta_\\mu$ by trading it, via partial integration (PI), for a derivative of the trace that arises from the quark loop. A convenient way to do so is via an auxiliary momentum $Q$\n\\begin{equation}\n\\Delta_\\kappa \\stackrel{\\textrm{PI}}{\\longrightarrow} ie^{-i\\Delta\\cdot Q}\\frac{\\partial}{\\partial Q_\\kappa}\\Big|_{Q\\to 0}\\,.\n\\label{deriv1}\n\\end{equation}\n\\begin{figure}[h]\n$$ \\epsfxsize=0.8\\textwidth\\epsffile{tw2_quark1_sr.eps}$$\n\\caption[Diagrams contributing to the quark condensate $\\left<\\bar s s \\right>$ at $\\mathcal{O}(\\alpha_s)$.]{\\small Diagrams contributing to the quark condensate $\\left<\\bar s s \\right>$ at $\\mathcal{O}(\\alpha_s)$. The crossed circle $\\otimes$ depicts the emission of a gluon from the non-local gauge factor -- see Eq.~(\\ref{emissiongf}). The corresponding diagrams for $\\left<\\bar q q\\right>$ are identical but reflected top to bottom.} \n\\label{det_fig2}\n\\end{figure}\nDiagrams for the $\\mathcal{O}(\\alpha_s)$ corrections to the strange quark condensate are shown in Fig.~\\ref{det_fig2}.  Importantly there are contributions from the gauge-factor which need to be included\n\\begin{equation}\n[0,z]=\\textrm{P} \\exp{\\left\\{-i g_s \\int^1_0 dt\\,z^\\mu A_\\mu(\\bar{t}z)\\right\\}}=1-i g_s \\int^1_0 dt\\,z^\\mu A_\\mu(\\bar{t}z)+\\dots\\,.\n\\label{emissiongf}\n\\end{equation}\nCalculating $\\mathcal{O}(\\alpha_s)$ corrections leads to divergent diagrams and the dependence of the condensate on the spacetime dimension $D$ leads to $\\mathcal{O}(\\epsilon)$ contributions at tree level, that then cause finite counter-terms upon renormalisation. Also, the derivative with respect to $Q_\\kappa$ in Eq.~(\\ref{deriv1}) yields $\\gamma_\\kappa$ in the trace via Eq.~(\\ref{quarkderiv}) which can also give a finite counter-term. This happens for the vertex correction diagrams.\n\n\n\\subsection{Evaluation of The Sum Rules}\nThe new quark condensate contributions are added to the results presented in the literature, see Refs.~\\cite{Ball:2005vx,Ball:2007rt}.  For $f_{K^*}^{\\parallel,\\perp}$ the sum rules read\n\\begin{eqnarray}\n\\lefteqn{(f_{K^*}^\\parallel)^2 e^{-m_{K^*}^2\/M^2}  = \n\\frac{1}{4\\pi^2}\\int\\limits_{m_s^2}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{(s-m_s^2)^2 (s+2m_s^2)}{s^3} +\n\\frac{\\alpha_s}{\\pi}\\, \\frac{M^2}{4\\pi^2}\\left( 1 -\ne^{-s_0\/M^2}\\right)}\\nonumber\\\\[5pt]\n&&{} +\\frac{m_s\\langle \\bar s s\\rangle}{M^2}\\left(1 +\n\\frac{m_s^2}{3M^2} - \n\\frac{13}{9}\\,\\frac{\\alpha_s}{\\pi}\\right)+\n\\frac{4}{3}\\,\\frac{\\alpha_s}{\\pi} \\, \\frac{m_s\\langle \\bar q\n  q\\rangle}{M^2}+\\frac{1}{12M^2}\\,\n\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\rangle \\left(1+\\frac{1}{3}\\frac{m_s^2}{M^2}\\right)\\label{tw2sr1}\n\\nonumber\\\\[5pt]\n&&{}  -\\frac{16\\pi\\alpha_s}{9M^4}\\,\n\\langle \\bar q q\\rangle\\langle \\bar s s\\rangle +\n\\frac{16\\pi\\alpha_s}{81M^4}\\,\\left( \\langle \\bar q q\\rangle^2 +\n\\langle \\bar s s\\rangle^2 \\right),\\label{eq:fKP}\\\\ \\nonumber \\\\\n\\lefteqn{(f_{K^*}^\\perp)^2 e^{-m_{K^*}^2\/M^2}\n= \\frac{1}{4\\pi^2}\\int\\limits_{m_s^2}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{(s-m_s^2)^2 (s+2m_s^2)}{s^3} }\\nonumber\\\\[5pt]\n&&{}+ \\frac{1}{4\\pi^2}\\int\\limits_{0}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{\\alpha_s}{\\pi}\\left( \\frac{7}{9} +\n\\frac{2}{3}\\,\\ln \\frac{s}{\\mu^2}\\right) \n-\\frac{1}{12M^2}\\,\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\rangle \n\\nonumber\\\\[5pt]\n&&{}\n\\times\\left\\{ 1-\\frac{2m_s^2}{M^2}\\left( \\frac{7}{6}-\\gamma_E + {\\rm   Ei}\\left(\n-\\frac{s_0}{M^2}\\right) - \\ln\\,\\frac{\\mu^2}{M^2} +\n\\frac{M^2}{s_0}\\left( 1 - \\frac{M^2}{s_0}\\right) e^{-s_0\/M^2}\n\\right)\\right\\} \\nonumber\\\\[5pt]\n&&{} +\\frac{m_s\\langle \\bar s\n  s\\rangle}{M^2}\\left\\{1+\\frac{m_s^2}{3M^2}+\n\\frac{\\alpha_s}{\\pi}\\left(-\\frac{22}{9} + \\frac{2}{3}\n\\left[ 1-\\gamma_E + \\ln\\,\\frac{M^2}{\\mu^2} +\n  \\frac{M^2}{s_0}\\,e^{-s_0\/M^2} + {\\rm Ei}\\left(-\\frac{s_0}{M^2}\\right)\\right]\n\\right)\\right\\}\\nonumber\\\\[5pt]\n&&{} \n-\\frac{1}{3M^4}\\,m_s\\langle \\bar s\\sigma gGs\\rangle -\n\\frac{32\\pi\\alpha_s}{81M^4}\\,\\left( \\langle \\bar q q\\rangle^2 +\n\\langle \\bar s s\\rangle^2 \\right)\\,,\\label{eq:fKT}\n\\end{eqnarray}\nand for $a_2^{\\parallel,\\perp}(K^*)$\n\\begin{eqnarray}\n\\lefteqn{\na_2^\\parallel(K^*) (f_{K^*}^\\parallel)^2 e^{-m_{K^*}^2\/M^2} = }\\nonumber\\\\\n&&{}\\frac{7}{4\\pi^2}\\,m_s^4\\int\\limits_{m_s^2}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{(s-m_s^2)^2(2m_s^2-s)}{s^5} +\n\\frac{7}{72\\pi^2}\\,\\frac{\\alpha_s}{\\pi} \\,M^2 (1-e^{-s_0\/M^2})\n+\\frac{7}{36M^2}\\,\\left\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle \n\\nonumber\\\\[5pt]\n&&{}+\\frac{7}{3}\\,\\frac{m_s\\langle \\bar s\n  s\\rangle}{M^2}\\left\\{1+ \n\\frac{\\alpha_s}{\\pi}\\left[ -\\frac{184}{27} +\n\\frac{25}{18} \\left(1-\\gamma_E + \\ln\\,\\frac{M^2}{\\mu^2} +\n\\frac{M^2}{s_0}\\,e^{-s_0\/M^2} + {\\rm Ei}\\left(-\\frac{s_0}{M^2}\\right)\n\\right)\\right]\\right\\}  \\nonumber\\\\[5pt]\n&&{}+\\frac{49}{27}\\,\\frac{\\alpha_s}{\\pi} \\, \\frac{m_s\\langle \\bar q\n q\\rangle}{M^2} - \\frac{35}{18}\\, \n\\frac{m_s\\langle\\bar s \\sigma g Gs\\rangle}{M^4}\n+\\frac{224\\pi\\alpha_s}{81M^4}\\,\\left( \\langle \\bar q q\\rangle^2 +\n\\langle \\bar s s\\rangle^2 \\right) -\n\\frac{112\\pi\\alpha_s}{27M^4}\\,\\langle \\bar q q\\rangle \\langle \\bar s\ns\\rangle\\,,\\label{tw2sr3}\\\\ \\nonumber \\\\[5pt]\n\\lefteqn{\na_2^\\perp(K^*) (f_{K^*}^\\perp)^2 e^{-m_{K^*}^2\/M^2} = }\\nonumber\\\\[5pt]\n&&{}\\frac{7}{4\\pi^2}\\,m_s^4\\int\\limits_{m_s^2}^{s_0}\nds\\,e^{-s\/M^2} \\,\\frac{(s-m_s^2)^2(2m_s^2-s)}{s^5} +\n\\frac{7}{90\\pi^2}\\,\\frac{\\alpha_s}{\\pi} \\,M^2 (1-e^{-s_0\/M^2})\n+\\frac{7}{54M^2}\\,\\left\\langle\\frac{\\alpha_s}{\\pi}\\,G^2\\right\\rangle \n\\nonumber\\\\[5pt]\n&&{}+\\frac{7}{3}\\,\\frac{m_s\\langle \\bar s\n  s\\rangle}{M^2}\\left\\{1+ \n\\frac{\\alpha_s}{\\pi}\\left[ -\\frac{206}{27} +\n\\frac{16}{9} \\left(1-\\gamma_E + \\ln\\,\\frac{M^2}{\\mu^2} +\n\\frac{M^2}{s_0}\\,e^{-s_0\/M^2} + {\\rm Ei}\\left(-\\frac{s_0}{M^2}\\right)\n\\right)\\right]\\right\\}  \\nonumber\\\\[5pt]\n&& - \\frac{49}{18}\\, \n\\frac{m_s\\langle\\bar s \\sigma g Gs\\rangle}{M^4}\n+\\frac{112\\pi\\alpha_s}{81M^4}\\,\\left( \\langle \\bar q q\\rangle^2 +\n\\langle \\bar s s\\rangle^2 \\right)\\,.\\label{tw2sr4}\n\\end{eqnarray}\nTo obtain the sum rules for $f_{\\phi}^{\\parallel,\\perp}$ and\n$a_2^{\\parallel,\\perp}(\\phi)$, one has to substitute $\\langle \\bar qq\\rangle\\to \\langle\n\\bar s s\\rangle$ and to double the terms in $m_s\\langle\n\\bar s s\\rangle$, $m_s\\langle \\bar q q\\rangle$ and $m_s \\langle \\bar s\n\\sigma g G s \\rangle$, and replace the perturbative  contribution by\n\\begin{eqnarray}\n\\mbox{for $(f_{\\phi}^{\\parallel,\\perp})^2$:~~}&&\n \\frac{1}{4\\pi^2}\\int_{4m_s^2}^{s_0} ds \\,e^{-s\/M^2} \\frac{(s+2 m_s^2)\n \\sqrt{ 1-4 m_s^2\/s}}{s}\\,,\\nonumber\\\\\n\\mbox{for $a_2^{\\parallel,\\perp}(\\phi)(f_{\\phi}^{\\parallel,\\perp})^2$:~~}&&\n -\\frac{7}{2\\pi^2}\\int_{4m_s^2}^{s_0} ds \\,e^{-s\/M^2} \\frac{m_s^4\n \\sqrt{ 1-4 m_s^2\/s}}{s^2}\\,.\n\\end{eqnarray}\nWe have derived sum rules for the decay constants $f^{\\parallel,\\perp}_V$, however, numerical values can be extracted from experiment for the longitudinal decay constants.  The perpendicular decay constants, on the other hand, must be determined from non-perturbative methods; results are available from Lattice QCD calculations and previous QCD sum rule determinations.  A detailed discussion of the latest numerical values of the decay constants can be found in Ref.~\\cite{Ball:2006eu} from which we just quote the following\n\\begin{eqnarray}\nf_{\\phi}^\\parallel=(215\\pm5)\\,\\textrm{MeV}\\,,\\qquad f_{\\phi}^\\perp =(186\\pm9)\\,\\textrm{MeV}\\,,\n\\label{decayresults}\n\\end{eqnarray}\nwhere $f_{\\phi}^\\parallel$ is an experimental result, and $f_{\\phi}^\\perp$ is from Lattice QCD \\cite{Becirevic:2003pn}. We can compare these results to the sum rules of Eqs.~(\\ref{eq:fKP}) and (\\ref{eq:fKT}) which are plotted in the upper row of Fig.~\\ref{aaa}. The sum rule determinations of  $a_2^{\\parallel,\\perp}(\\phi)$ are plotted in the lower row.  \n\\begin{figure}[h]\n$$ \\epsfxsize=0.5\\textwidth\\epsffile{fparaphi.eps}\\quad \\epsfxsize=0.5\\textwidth\\epsffile{fperpphi.eps}$$\n$$ \\epsfxsize=0.5\\textwidth\\epsffile{a2paraphi.eps}\\quad \\epsfxsize=0.5\\textwidth\\epsffile{a2perpphi.eps}$$\n  \\caption[The hadronic parameters $f^{\\parallel,\\perp}_\\phi$ and $a_2^{\\parallel,\\perp}(\\phi)$ as a function of $M^2$.]{\\small  The decay constants $f^\\parallel_\\phi$ (upper left) and $f^\\perp_\\phi$ (upper right) and the  Gegenbauer coefficients $a_2^\\parallel(\\phi)$ (lower left) and $a_2^\\perp(\\phi)$ (lower right) plotted as a function of $M^2$. The continuum thresholds are $s_0^\\parallel =1.85\\pm 0.05\\,{\\rm GeV}^2$ and $s_0^\\perp =1.40\\pm 0.05\\,{\\rm GeV}^2$ -- see text.   Solid line: central input parameters of Tab.~\\ref{QCDSRinput}. Dashed lines: variation due to the uncertainties of $m_s$ and the gluon condensate. All quantities are evaluated at $\\mu=1\\,{\\rm GeV}$.} \n  \\label{aaa}\n\\end{figure}\n\nIn all the plots the dashed line and shaded region represent the central value and uncertainty of the parameter in question. To evaluate the sum rules we use the input parameters of Tab.~\\ref{QCDSRinput}. For the continuum threshold we note that for the sum rule determination of $f_{K^*}^\\parallel$ in Ref.~\\cite{Ball:2005vx} it is taken to be $s_0^\\parallel(K^*)=1.7\\,{\\rm GeV}^2$, and we expect for $\\phi$ it to be slightly larger. Indeed, by taking $s_0^\\parallel(\\phi)=1.85\\pm 0.05\\,{\\rm GeV}^2$ we find a stable  plateau and excellent agreement with the experimental result for $f_{\\phi}^{\\parallel}$ (upper left plot). Likewise, guided by $s_0^\\perp(K^*)=1.3\\,{\\rm GeV}^2$ \\cite{Ball:2005vx} we find $s_0^\\perp(\\phi)=1.40\\pm 0.05\\,{\\rm GeV}^2$ yields a result consistent with that from Lattice QCD (upper right plot). We use these thresholds in evaluating the sum rules for $a_2^{\\parallel,\\perp}(\\phi)$ and also replace the decay constants by their sum rules, which helps reduce dependence on the Borel parameters. The results are plotted for $a_2^{\\parallel}(\\phi)$ (lower left plot) and $a_2^{\\perp}(\\phi)$ (lower right plot). It is found that the longitudinal parameters exhibit a stronger continuum threshold dependence, which is reflected in the larger uncertainty of the determined value of $a_2^\\parallel(\\phi)$.  The sum rule determinations of the other particle DA parameters follow analogously and all the numerical results  are given in Tab.~\\ref{det_tab1}.\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{| l || c | c ||  l | l || c | c |}\n\\hline\n& \\multicolumn{2}{c||}{\\rho} & \\multicolumn{2}{c||}{K^*}  &  \n\\multicolumn{2}{c|}{\\phi}\\\\\n\\cline{2-7}\n& \\mu = 1\\,{\\rm GeV} & \\mu = 2\\,{\\rm GeV} & \\mu = 1\\,{\\rm GeV} & \\mu =\n2\\,{\\rm GeV} & \\mu = 1\\,{\\rm GeV} & \\mu = 2\\,{\\rm GeV}\\\\\n\\hline\na_1^\\parallel & 0 & 0 & \\phantom{-}0.03(2) & \\phantom{-}0.02(2) & 0 & 0 \n\\\\\na_1^\\perp & 0 & 0 & \\phantom{-}0.04(3) & \\phantom{-}0.03(3) & 0 & 0\n\\\\\na_2^\\parallel & 0.15(7) & 0.10(5) & \\phantom{-}0.11(9) & \n\\phantom{-}0.08(6) & 0.18(8) & 0.13(6)\n\\\\\na_2^\\perp & 0.14(6) & 0.11(5) & \\phantom{-}0.10(8) &\n\\phantom{-}0.08(6) & 0.14(7) & 0.11(5)\n\\\\\\hline\n\\end{array}\n$$\n\\renewcommand{\\arraystretch}{1}\n\\addtolength{\\arraycolsep}{-3pt}\n\\caption[Results for the leading twist-2 distribution amplitude parameters.]{\\small Results for the twist-2 hadronic DA parameters  at the scale  $\\mu= 1\\,{\\rm GeV}$ and scaled up to $\\mu= 2\\,{\\rm GeV}$\nusing the evolution equations (\\ref{evo}). Note that  $a_1^{\\parallel,\\perp}({K^*})$ refers to a $(s\\bar q)$ bound state; for a $(q\\bar  s)$ state it changes sign.}\n\\label{det_tab1}\n\\end{table}\n\n\\section{Twist-3}\nIn this section we determine the twist-3 three-particle parameters of the DAs $\\Phi_{3;K^*}^\\perp$, $\\Phi_{3;K^*}^\\parallel$ and $\\widetilde\\Phi_{3;K^*}^\\parallel$ as defined by Eq.~(\\ref{das_eq27}).  Previous determinations of these parameters are rather few and far between, thus motivating the present analysis. The chiral-even $\\rho$ parameters $\\zeta^\\parallel_{3\\rho}$, $\\omega^\\parallel_{3\\rho}$, and $\\widetilde{\\omega}^\\parallel_{3\\rho}$ were obtained in Ref.~\\cite{Zhitnitsky:1985dd}, and  $\\omega^\\perp_{3\\rho}$ was obtained in Ref.~\\cite{Ball:1998sk}. We make a comparison with these results in Section~\\ref{section_evaluation}.\n\nFirstly, we outline the calculation of the three functions $\\pi_{3;K^*}$ which all proceed in a similar manner, and secondly we explicitly discuss the sum rules for $\\widetilde\\Phi_{3;K^*}^\\parallel$ and present the results. In the diagrams that follow, $q$ is the upper line and $s$ is the lower line.\n\n\\subsection{Calculation}\nEach DA is accessed via a correlation function featuring its defining current. The chiral-even twist-3 parameters $\\zeta_{3K^*}^\\parallel$, $\\widetilde\\omega_{3K^*}^\\parallel$, $\\widetilde\\lambda_{3K^*}^\\parallel$\ncan be determined from\n\\begin{equation}\n\\widetilde\\Pi^\\parallel_{3;K^*}(v,q\\cdot z) = \\frac{i g_{\\alpha\\mu}^{\\perp}}{(q\\cdot z)^2 (2-D)} \\int d^4y \\,e^{-iq\\cdot y} \\bra{0} T \\bar q(z) g_s\n\\widetilde G^{\\alpha z} (vz) \\gamma_z \\gamma_5 s(0) \\bar s(y)\n\\gamma^\\mu q(y) \\ket{0}\\,,\n\\label{C.1}\n\\end{equation}\nwhere the definition of $g^\\perp_{\\mu\\nu}$ is given in Appendix~\\ref{appendixA}.\\footnote{We also make use of the relation $\\gamma_\\mu\\gamma_5=\\frac{i}{6}\\epsilon_{\\mu\\lambda\\nu\\pi}\\gamma^\\lambda\\gamma^\\nu\\gamma^\\pi$ defined in $D$ dimensions.} The parameters $\\kappa_{3K^*}^\\parallel$, $\\omega_{3K^*}^\\parallel$ and $\\lambda_{3K^*}^\\parallel$ can be obtained from the correlation\nfunction $\\Pi_{3;K^*}^\\parallel$ obtained from $\\widetilde\\Pi^\\parallel_{3;K^*}$ by making the replacement\n\\begin{equation}\ng_s\\widetilde G_{\\alpha z} \\gamma_z\\gamma_5 \\to g_s G_{\\alpha z} i \\gamma_z\\,.\n\\end{equation}\nLastly for the chiral-odd operator \n\\begin{equation}\n\\Pi^\\perp_{3;K^*}(v,q\\cdot z)= \\frac{1}{ (q\\cdot z)^3 }\\int d^4y e^{-iq\\cdot y} \\bra{0}T \\bar q(z) \\sigma_{z\\mu} g_s\nG_{z\\mu}(vz) s(0) \\bar s(y) \\sigma_{qz} q(y) \\ket{0}\\,.\n\\label{C.3} \n\\end{equation}\nAll three correlation functions $\\Pi$ can be written as\n\\begin{equation}\n\\Pi_{3;K^*}(v,q\\cdot z) = \\int {\\cal D}\\underline{\\alpha}\\,e^{-i q\\cdot z (\\alpha_2 + v \\alpha_3)}\\pi_{3;K^*}(\\underline{\\alpha})\\,,\n\\label{C.4}\n\\end{equation}\nwhere the exponential function is due to the fact that we keep the correlation functions non-local. The calculation proceeds for each correlation function analogously.  Considering Eq.~(\\ref{C.1}) for instance, firstly we express it in terms of hadronic contributions\n\\begin{equation}\n\\widetilde{\\Pi}_{3;K^*}^\\parallel(v,q\\cdot z) =\n\\frac{(f_{K^*}^\\parallel)^2\n  m_{K^*}^2}{m_{K^*}^2-q^2} \\int{\\cal\n  D}(\\underline{\\alpha})\\,e^{-iq\\cdot z(\\alpha_2+v\\alpha_3)}\\,\n\\widetilde{\\Phi}_{3;K^*}^ \\parallel(\\underline{\\alpha}) + \\dots\\, ,\n\\label{C.5}\n\\end{equation}\nwhere the dots denote contributions from higher-mass states. To derive the sum rule we tread down a well worn path;  express Eq.~(\\ref{C.4}) as a dispersion relation and equate to\nEq.~(\\ref{C.5}), subtract the continuum contribution for $s>s_0$, perform the Borel transformation and project out the desired DA parameter by substitution of the relevant polynomial.  The three hadronic parameters $\\zeta_{3K^*}^\\parallel$, $\\widetilde\\omega_{3K^*}^\\parallel$, $\\widetilde\\lambda_{3K^*}^\\parallel$ are projected out like so:\n\\begin{eqnarray}\n\\left(f_{K^*}^ \\parallel\\right)^2 m_{K^*}^2 e^{-m_{K^*}^2\/M^2}\n      \\left[  \\zeta_{3K^*}^\\parallel\\right]  \n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int {\\cal D}\\underline{\\alpha}\\,\n    \\left[1\\right]  \\frac{1}{\\pi}\\, {\\rm Im}_{s}\n    \\widetilde{\\pi}^\\parallel_{3;K^*}(\\underline{\\alpha})\\,,\n\\\\\n\\left(f_{K^*}^ \\parallel\\right)^2 m_{K^*}^2 e^{-m_{K^*}^2\/M^2}\n      \\,    \\left[\\frac{1}{14}\\,\\widetilde\\lambda_{3K^*}^\\parallel\\right]  \n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int {\\cal D}\\underline{\\alpha}\\,\n         \\left[ \\alpha_1-\\alpha_2\\right]  \\frac{1}{\\pi}\\, {\\rm Im}_{s}\n      \\widetilde{\\pi}_{3;K^*}^\\parallel(\\underline{\\alpha})\\,,\n\\nonumber\\\\\n\\left(f_{K^*}^ \\parallel\\right)^2 m_{K^*}^2 e^{-m_{K^*}^2\/M^2}\n      \\,    \\left[\\frac{3}{28}\\,\\widetilde\\omega_{3K^*}^\\parallel\\right]  \n& = & \\int_0^{s_0}ds\\, e^{-s\/M^2} \\int {\\cal D}\\underline{\\alpha}\\,\n         \\left[\\alpha_3-\\frac{3}{7}\\right]  \\frac{1}{\\pi}\\, {\\rm Im}_{s}\n      \\widetilde{\\pi}_{3;K^*}^\\parallel(\\underline{\\alpha})\\,.\\nonumber\n      \\label{C.6}\n\\end{eqnarray}\nThe formulas for the other parameters are analogous. In calculating  the functions  $\\pi_{3;K^*}$ we keep explicit mass corrections $\\mathcal{O}(m_s^2,m_q^2,m_s m_q)$ and all operators up to $D=6$ except  the triple gluon condensate $\\left<g_s^3 f G^3\\right>$ which is expected to yield a negligible contribution. By retaining all mass terms the resulting formulas for $\\pi_{3;K^*}$ can be used to derive sum rules for all the DA parameters for $K^*$, $\\rho$ and $\\phi$ by setting  $m_q=0$, $m_q= m_s=0$ and $m_q= m_s$ respectively. For $\\rho$ and $\\phi$ expressions for the three-particle twist-3 DAs are analogous to Eq.~(\\ref{das_eq27}), except that the G-parity violating parameters $\\kappa$ and $\\lambda$ vanish.\n\n\\subsection*{Perturbation Theory}\nThe perturbation theory calculation is given by the two diagrams shown in Fig.~\\ref{a}. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{pt_sr.eps}$$\n  \\caption[Diagrams contributing to perturbation theory.]{\\small Diagrams contributing to perturbation theory.} \n  \\label{a}\n\\end{figure}\nAs an example, consider the first diagram, which up to an overall factor can be written generally as\n\\begin{eqnarray}\n\\Pi^{(\\rm{PT}_1)}&=&g_s^2\\int\\frac{d^D p}{(4 \\pi)^D}\\int\\frac{d^D l}{(4 \\pi)^D}\\,\\textrm{Tr}[\\Gamma_1 S^{(s)}(\\slash{p}+\\slash{q})\\Gamma_2 S^{(q)}(\\slash{p})\\gamma^{\\beta}S^{(q)}(\\slash{p}+\\slash{l})] \\nonumber\\\\\n&&\\cdot\\,[l_\\mu D_{\\nu\\beta}(l)-l_\\nu D_{\\mu\\beta}(l)]\\,e^{iz \\cdot(l \\bar{v}+p)}\\,.\\label{part1}\n\\end{eqnarray}\nwhere the Dirac matrices $\\Gamma_{1,2}$ depend on the correlation function. In performing the two successive integrations over $l$ and $p$, Feynman parameterisation leads to shifting the variables $l\\to l-p\\bar x$ and $p\\to p-q\\bar y$ respectively. Each time the exponential in (\\ref{part1}) is also shifted. In expanding the part of the exponential that contributes to the integral, for example, for $l$ we have $e^{i  l  \\cdot z \\bar{v}}=1+i (l  \\cdot z)\\bar{v} +\\dots$, only the first two terms contribute; higher order terms are killed off either via $z^2=0$ or because integrals with odd numbers of open indices, for example $l^{\\mu_1}l^{\\mu_2}l^{\\mu_3}$, in the numerator vanish due to symmetry. After the integrations any terms $(\\mathcal{T})$ including factors of $i( q  \\cdot z)\\bar{v}$ are dealt with by trading them for derivatives of $\\mathcal{T}$ by using partial integration of the final exponential\n\\begin{equation}\ni (q\\cdot z) \\bar{v}=\\frac{1}{\\bar{y}}\\frac{\\partial}{\\partial\\bar{x}}e^{-i q  \\cdot z \\bar{y}(1-\\bar{x}\\bar{v})} \\qquad\\Rightarrow\\qquad (q\\cdot z) \\bar{v}\\, \\mathcal{T} \\stackrel{\\textrm{PI}}{\\longrightarrow} \\frac{i}{\\bar{y}}\\frac{\\partial }{\\partial\\bar{x}}\\mathcal{T} \\,,\\label{PI}\n\\end{equation}\nwhere surface terms do not contribute as they vanish for $x=\\{1,0\\}$. The exponential can be matched to the ``canonical form'' by writing\n\\begin{equation}\n\\int^1_0dx\\,\\int^1_0dy\\, e^{-i q  \\cdot z \\bar{y}(1-\\bar{x}\\bar{v})} = \\int^1_0dx\\,\\int^1_0dy\\,  \\int {\\cal D}\\underline{\\alpha}\\, \\delta(\\alpha_1-y)\\delta(\\alpha_2-\\bar{x}\\bar{y})\\delta(\\alpha_3- x\\bar{y})\\,e^{-iq\\cdot z(\\bar\\alpha_1-\\bar{v}\\alpha_3)}\\,.\n\\end{equation}\nPerforming the $x$ and $y$ integration of the whole expression gives the desired result\n\\begin{equation}\n\\int {\\cal D}\\underline{\\alpha}\\, e^{-iq\\cdot z(\\alpha_2+v\\alpha_3)}\\pi^{(\\textrm{PT}_1)}(\\underline{\\alpha})\\,.\n\\end{equation}\nThe second diagram follows analogously. Both diagrams are divergent and need to be renormalised separately. We find finite counter terms which are proportional to the quark masses. \n\n\\subsection*{Gluon Condensate}\nThe leading order contribution to the gluon condensate $\\left<\\frac{\\alpha_s}{\\pi} G^2\\right>$ is found using the background field method as outlined in Section~\\ref{example}. There are only two diagrams contributing  as depicted in Fig.~\\ref{b}. One vacuum momentum $k$, from the gluon attached to the quark line, is introduced and hence one derivative is taken. As the gluon emerging from the non-local vertex $G(vz)$ carries no momentum these diagrams are proportional to $\\delta(\\alpha_3)$ and the remaining momentum fractions are related by $1-\\alpha_1=\\alpha_2$; the identification of the momentum fractions with the Feynman parameters is therefore straightforward. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{g2_sr.eps}$$\n  \\caption[Diagrams contributing to the gluon condensate $\\left<\\frac{\\alpha_s}{\\pi} G^2\\right>$.]{\\small Diagrams contributing to the gluon condensate $\\left<\\frac{\\alpha_s}{\\pi} G^2\\right>$.}\n  \\label{b} \n\\end{figure}\nThe calculation requires the integration over one momentum $p$ and the result can simply be written unexpanded in the quark masses. \n\n\n\\subsection*{Mixed Condensate}\nThe mixed condensates $\\left<\\bar q \\sigma g_s G q \\right>$ and $\\left<\\bar s \\sigma g_s G s \\right>$ originate from the diagrams shown in Fig.~\\ref{f}. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{mixed_sr.eps}$$\n  \\caption[Diagrams contributing to the mixed condensates $\\left<\\bar q \\sigma g_s G q \\right>$ and $\\left<\\bar s \\sigma g_s G s \\right>$.]{\\small Diagrams contributing to the mixed condensates $\\left<\\bar q \\sigma g_s G q \\right>$ and $\\left<\\bar s \\sigma g_s G s \\right>$.}\n  \\label{f} \n\\end{figure}\nTo extract the mixed condensates one uses the first non-local term in the expansion  $(D=4)$ \\cite{PT:84}\n\\begin{eqnarray}\n\\lefteqn{\\bra{0}\\!:\\!\\bar{q}^i_\\alpha(x_1) g_s (G_{\\mu\\nu})_{ij}(y)q^j_\\beta(x_2)\\!:\\!\\ket{0}=}\\hspace{1.2in}\\\\\n&&\\delta^{ij}\\left[\\frac{\\left<\\bar{q} g_s \\sigma G q\\right>}{144}\\left\\{\\sigma_{\\mu\\nu}+\\frac{m_q}{2}\\left[\\Delta_\\mu \\gamma_\\nu-\\Delta_\\nu \\gamma_\\mu -i(\\Delta^\\lambda \\gamma_\\lambda)\\sigma_{\\mu\\nu}\\right]\\right\\}\\right.\\nonumber\\\\\n&&\\left.+g_s^2 \\left<6\\right>\\left\\{\\frac{i}{288}(x_2^\\xi \\sigma_{\\mu\\nu}\\gamma_\\xi-x_1^\\xi \\gamma_\\xi \\sigma_{\\mu\\nu})-\\frac{1}{216}(y_\\mu\\gamma_\\nu-y_\\nu\\gamma_\\mu)\\right\\}\\right]_{\\beta\\alpha}\\,. \\nonumber\n\\label{mixed}\n\\end{eqnarray}\nThe first $\\sigma_{\\mu\\nu}$ does not contribute, but the term $\\sim m_q$ does. The $\\Delta_\\mu$s can be expressed as derivatives of the trace via partial integration which is dealt with simply by using Eq.~(\\ref{quarkderiv}). Along with the condensate gluon, the quark condensate lines carry no momentum. There is therefore no loop integration to perform and the results are proportional to $\\delta(\\alpha_3)\\delta(\\alpha_{1,2})$.\n\n\\subsection*{Quark Condensates}\nThe diagrams of Fig.~\\ref{c} generate the condensates $m_{q,s}\\left<\\bar q q\\right>$ and $m_{q,s}\\left<\\bar s s \\right>$. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.8\\textwidth\\epsffile{quark1_sr.eps}$$\n  \\caption[Diagrams contributing to the quark  condensates $\\left<\\bar q q\\right>$ and $\\left<\\bar s s \\right>$.]{\\small Diagrams contributing to the quark  condensates $\\left<\\bar q q\\right>$ and $\\left<\\bar s s \\right>$.}\n  \\label{c} \n\\end{figure}\nWe do not consider $\\mathcal{O}(m_{q,s}^2)$ corrections, which are however of dimension six, as they are very well suppressed with respect to the other contributions. To extract all $\\mathcal{O}(m_{q,s})$ mass corrections the first non-local term in the expansion of the quark fields, given by Eq.~(\\ref{quarkextract}), is needed. There is one loop momentum to integrate over and one finds contributions from the exponential which can be dealt with via partial integration in the same way as with the perturbation theory calculation, see Eq.~(\\ref{PI}). The results are proportional to $\\delta(\\alpha_{1,2})$.  The diagrams in Fig.~\\ref{d} generate the condensate $\\left<\\bar q q\\right>\\left<\\bar s s \\right>$ which is already of dimension six, so we do not require mass corrections. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{quark2_sr.eps}$$\n  \\caption[Diagrams contributing to the quark  condensate $\\left<\\bar q q\\right>\\left<\\bar s s \\right>$.]{\\small  Diagrams contributing to the quark  condensate $\\left<\\bar q q\\right>\\left<\\bar s s \\right>$.}\\label{d} \n\\end{figure}\nThe two diagrams are of equal magnitude and cancel, however only for $\\widetilde{\\pi}^{\\parallel}_{3;K^{*}}$ they add. There is no loop integral to perform and the result is proportional to $\\delta(\\alpha_1)\\delta(\\alpha_2)$. The four quark condensate is simplified via the vacuum saturation hypothesis (VSH) \\cite{PT:84,Shifman:1978by}\n\\begin{equation}\n\\bra{0}\\!:\\!\\bar{q}_\\alpha^i (x_1) q_\\beta^j(x_2)\\bar{s}_\\gamma^k (x_3) s_\\delta^l(x_4)\\!:\\!\\ket{0} \\stackrel{\\textrm{VSH}}{\\longrightarrow}\\bra{0}\\!:\\!\\bar{q}_\\alpha^i (x_1) q_\\beta^j(x_2)\\!:\\!\\ket{0}\\bra{0}\\!:\\!\\bar{s}_\\gamma^k (x_3) s_\\delta^l(x_4)\\!:\\!\\ket{0}\\,.\n\\end{equation}\nThe diagrams in Fig.~\\ref{e} generate the condensates $\\left<\\bar q q\\right>^2$ and $\\left<\\bar s s \\right>^2$. They stem from the operator $\\left<6\\right>$ appearing in the expansion of the mixed condensate, Eq.~(\\ref{mixed}), which simplifies as\n\\begin{equation}\n \\left<6\\right>=\\langle\\bar q \\gamma_\\kappa t^a q \\sum_{u,d,s}\\bar{q} \\gamma^\\kappa t^a q\\rangle\\stackrel{\\textrm{VSH}}{\\longrightarrow} -\\frac{4}{9}\\left<\\bar q q\\right>^2\\,;\n \\end{equation}\nthus at higher order the mixed condensate also contributes to the quark condensates. The light-like co-ordinate of the gluonic field strength tensor $v z_\\mu$ simplifies the resulting trace via $z^2=0$ from Eq.~(\\ref{mixed}) and the other co-ordinates are dealt with as before. \n \\begin{figure}[h]\n$$ \\epsfxsize=0.4\\textwidth\\epsffile{quark3_sr.eps}$$\n  \\caption[Diagrams contributing to the quark  condensates $\\left<\\bar q q\\right>^2$ and $\\left<\\bar s s \\right>^2$.]{\\small Diagrams contributing to the quark  condensates $\\left<\\bar q q\\right>^2$ and $\\left<\\bar s s \\right>^2$ from the expansion of the mixed condensate -- see Eq.~(\\ref{mixed}).}\n  \\label{e} \n\\end{figure}\n\\subsection*{Results}\nFor the functions $\\pi_{3;K^*}$, given by Eq.~(\\ref{C.6}), we find (dropping all terms that vanish upon taking the imaginary part):\n\\begin{eqnarray}\n\\pi^{\\perp }_{3;K^{*}}\\left(\\underline{\\alpha}\\right)\n&=&\n\\frac{\\alpha_{s}}{2\\pi^{3}}\\ln\\frac{-q^2}{\\mu^{2}}\n\\left[q^2\\alpha_{1}\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{2}}-\n\\frac{1}{\\bar{\\alpha}_{1}}\\right)\\right.\n\\nonumber \\\\\n&+&m_{s}m_{q}\\frac{\\alpha_{3}^{2}}{\\bar{\\alpha}_{1}\\bar{\\alpha}_{2}}\n\\left[\\bar{\\alpha}_{2}\\left(\\ln\\frac{\\alpha_{2}\\alpha_{3}}{\\bar{\\alpha}_{1}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)-\\left\\{\\alpha_{1}\n\\leftrightarrow\\alpha_{2}\\right\\}\\right]\n\\nonumber \\\\\n&+&m_{s}^{2}\\left\\{-\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{2}}-\n\\frac{1}{\\bar{\\alpha}_{1}}\\right)-\\frac{\\alpha_{2}\\alpha_{3}^{2}}{\n\\bar{\\alpha}_{2}^{2}}\\left(\\ln\\frac{\\alpha_{1}\\alpha_{3}}{\\bar{\\alpha}_{2}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right\\}\n-m_{q}^{2}\\left\\{\\alpha_{1}\\leftrightarrow\\alpha_{2}\\right\\}] \n\\nonumber \\\\\n&+&\\frac{1}{12}\\langle\\frac{\\alpha_{s}}{\\pi}G^{2}\\rangle\n\\frac{\\alpha_{1}\\alpha_{2}\\left(\\alpha_{1}-\\alpha_{2}\\right)\\delta\n\\left(\\alpha_{3}\\right)}{\\alpha_{1}m_{q}^{2}+\\alpha_{2}m_{s}^{2}-\n\\alpha_{1}\\alpha_{2}q^2}\n\\nonumber \\\\\n&+&\\frac{2}{3q^2}\\frac{\\alpha_{s}}{\\pi}\\left\\{\\right.\n\\frac{\\bar{\\alpha}_{3}}{2}\\left(1+\\alpha_{3}\\right)\\left(m_{q}\n\\langle\\bar{q}q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{s}\\langle\\bar{s}s\n\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right) \n\\nonumber \\\\\n&+&\\alpha_{3}\\left[1+\\alpha_{3}\\left(1+\\ln\\left(\\alpha_{3}\n\\bar{\\alpha}_{3}\\right)+\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right]\\left(m_{s}\n\\langle\\bar{q}q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{q}\\langle\\bar{s}s\n\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right)\\left.\\right\\} \n\\nonumber \\\\\n&+&\\frac{1}{6q^4}\\delta\\!\\left(\\alpha_{3}\\right)\\left\\{m_{q}\\langle\\bar{q}\n\\sigma g_s G q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{s}\\langle\\bar{s}\n\\sigma g_s G s\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right\\}\n\\nonumber \\\\\n&+&\\frac{16}{27q^4} \\pi \\alpha_{s} \\delta\\!\\left( \\alpha_{3}\\right) \n\\left\\{\\langle\\bar{q}\nq\\rangle^{2}\\delta\\!\\left(\\alpha_{2}\\right)-\\langle\\bar{s}s\\rangle^{2}\n\\delta\\!\\left(\\alpha_{1}\\right)\\right\\},\\label{corresult1}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\pi^{\\parallel}_{3;K^{*}}\\left(\\underline{\\alpha}\\right)\n&=&\n\\frac{\\alpha_{s}}{4\\pi^{3}}\\ln\\frac{-q^2}{\\mu^{2}}\\left[\\right.q^2\n\\alpha_{1}\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{2}}-\n\\frac{1}{\\bar{\\alpha}_{1}}\\right)\n\\nonumber \\\\\n&+&m_{s}m_{q}\\frac{\\alpha_{3}^{2}}{\\bar{\\alpha}_{1}\\bar{\\alpha}_{2}}\n\\left\\{\\bar{\\alpha}_{2}\\left(\\ln\\frac{\\alpha_{2}\\alpha_{3}}{\\bar{\\alpha}_{1}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)-\\left\\{\\alpha_{1}\n\\leftrightarrow\\alpha_{2}\\right\\}\\right\\}\n\\nonumber \\\\\n&+&m_{s}^{2}\\left\\{-\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{2}}-\n\\frac{1}{\\bar{\\alpha}_{1}}\\right)-\\frac{\\alpha_{2}\\alpha_{3}^{2}}{\n\\bar{\\alpha}_{2}^{2}}\\left(\\ln\\frac{\\alpha_{1}\\alpha_{3}}{\\bar{\\alpha}_{2}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right\\}\n-m_{q}^{2}\\left\\{\\alpha_{1}\\leftrightarrow\\alpha_{2}\\right\\}\\left.\\right]\n\\nonumber \\\\\n&+&\\frac{1}{24}\\langle\\frac{\\alpha_{s}}{\\pi}G^{2}\\rangle\\frac{\\alpha_{1}\n\\alpha_{2}\\left(\\alpha_{1}-\\alpha_{2}\\right)\\delta\\!\\left(\\alpha_{3}\\right)}{\n\\alpha_{2}m_{s}^{2}+\\alpha_{1}m_{q}^{2}-\\alpha_{1}\\alpha_{2}q^2}\n\\nonumber\\\\\n&+&\\frac{1}{3q^2}\\frac{\\alpha_{s}}{\\pi}\\left\\{\\right.\\frac{\n\\bar{\\alpha}_{3}}{2}\\left(1+\\alpha_{3}\\right)\\left(m_{q}\\langle\\bar{q}q\n\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{s}\\langle\\bar{s}s\\rangle\\delta\n\\left(\\alpha_{1}\\right)\\right) \n\\nonumber \\\\\n&+&\\alpha_{3}\\left[1+\\alpha_{3}\\left(\\ln\\left(\\alpha_{3}\\bar{\\alpha}_{3}\n\\right)+\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right]\\left(m_{s}\\langle\\bar{q}q\n\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{q}\\langle\\bar{s}s\\rangle\\delta\n\\left(\\alpha_{1}\\right)\\right)\\left.\\right\\}\n\\nonumber \\\\\n&+&\\frac{1}{12 q^4}\\delta\\!\\left(\\alpha_{3}\\right)\\left\\{\nm_{q}\\langle\\bar{q}\n\\sigma g_s G q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)-m_{s}\\langle\\bar{s}\n\\sigma g_s G s\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right\\}\n\\nonumber \\\\\n&+&\\frac{8}{27 q^4}\\alpha_{s}\\pi \\delta\\!\\left( \\alpha_{3}\\right)\n\\left(\\langle\\bar{q}q\\rangle^{2}\\delta\\!\\left(\\alpha_{2}\\right) -\n\\langle\\bar{s}s\\rangle^{2}\\delta\\!\\left(\\alpha_{1}\\right) \\right),\\label{corresult2}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\widetilde{\\pi}^{\\parallel}_{3;K^{*}}\\left(\\underline{\\alpha}\\right)\n&=&\\frac{\\alpha_{s}}{4\\pi^{3}}\\ln\\frac{-q^2}{\\mu^{2}}\\left[\\right.-\nq^2\\alpha_{1}\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{1}}+\n\\frac{1}{\\bar{\\alpha}_{2}}\\right)\n\\nonumber \\\\\n&+&m_{s}m_{q}\\frac{\\alpha_{3}^{2}}{\\bar{\\alpha}_{1}\\bar{\\alpha}_{2}}\n\\left\\{\\bar{\\alpha}_{1}\\left(\\ln\\frac{\\alpha_{1}\\alpha_{3}}{\\bar{\\alpha}_{2}}-\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)+\\left\\{\\alpha_{1}\n\\leftrightarrow\\alpha_{2}\\right\\}\\right\\}\n\\nonumber \\\\\n&+&m_{s}^{2}\\left\\{\\alpha_{2}\\alpha_{3}\\left(\\frac{1}{\\bar{\\alpha}_{1}}+\n\\frac{1}{\\bar{\\alpha}_{2}}\\right)+\\frac{\\alpha_{2}\\alpha_{3}^{2}}{\n\\bar{\\alpha}_{2}^{2}}\\left(\\ln\\frac{\\alpha_{1}\\alpha_{3}}{\\bar{\\alpha}_{2}}+\n\\frac{1}{2}\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right\\}\n+m_{q}^{2}\\left\\{\\alpha_{1}\\leftrightarrow\\alpha_{2}\\right\\}\\left.\\right]\n\\nonumber \\\\\n&+&\\frac{1}{24}\\langle\\frac{\\alpha_{s}}{\\pi}G^{2}\\rangle\\frac{\\alpha_{1}\n\\alpha_{2}\\delta\\!\\left(\\alpha_{3}\\right)}{\\alpha_{2}m_{s}^{2}+\\alpha_{1}\nm_{q}^{2}-\\alpha_{1}\\alpha_{2}q^2}\n\\nonumber\\\\\n&+&\\frac{1}{3q^2}\\frac{\\alpha_{s}}{\\pi}\\left\\{\\right.\\frac{\n\\bar{\\alpha}_{3}^{2}}{2}\\left(m_{s}\\langle\\bar{s}s\\rangle\\delta\\!\\left(\n\\alpha_{1}\\right)+m_{q}\\langle\\bar{q}q\\rangle\\delta\\!\\left(\\alpha_{2}\\right)\n\\right) \n\\nonumber \\\\\n&+&\\alpha_{3}\\left[1-\\alpha_{3}\\left(2+\\ln\\left(\\alpha_{3}\\bar{\\alpha}_{3}\n\\right)+\\ln\\frac{-q^2}{\\mu^{2}}\\right)\\right]\\left(m_{s}\\langle\\bar{q}q\n\\rangle\\delta\\!\\left(\\alpha_{2}\\right)+m_{q}\\langle\\bar{s}s\\rangle\\delta\n\\left(\\alpha_{1}\\right)\\right)\\left.\\right\\}\n\\nonumber \\\\\n&+&\\frac{1}{12 q^4}\\delta\\!\\left(\\alpha_{3}\\right)\\left\\{ m_{q}\\langle\n\\bar{q}\\sigma g_s\nGq\\rangle\\delta\\!\\left(\\alpha_{2}\\right)+m_{s}\\langle\\bar{s}\n\\sigma g_s G s\\rangle\\delta\\!\\left(\\alpha_{1}\\right)\\right\\}\n\\nonumber \\\\\n&+&\\frac{8}{27 q^4}\\alpha_{s}\\pi \\delta\\!\\left( \\alpha_{3}\\right)\n\\left(\\langle\\bar{q}q\\rangle^{2}\\delta\\!\\left(\\alpha_{2}\\right)+\n\\langle\\bar{s}s\\rangle^{2}\\delta\\!\\left(\\alpha_{1}\\right) \\right)\n\\nonumber\\\\\n&+&\\frac{2}{3 q^4}\\alpha_{s}\\pi \\delta\\!\\left( \\alpha_{1}\\right) \n\\delta\\!\\left( \\alpha_{2}\\right)\\langle\\bar{q}q\\rangle\\langle\\bar{s}s\n\\rangle.\\label{corresult3}\n\\end{eqnarray}\n\n\n\\subsection{Evaluation of The Sum Rules}\\label{section_evaluation}\nIn the following we consider $\\widetilde{\\pi}_{3;K^*}^\\parallel$; the sum rules for the other DA parameters and particles $\\rho$ and $\\phi$ follow similarly. The values of the input parameters  and the continuum thresholds used for all sum rules are given in Appendix~\\ref{appendixB}. \n\nOne subtlety must be noted: upon integration over $\\alpha_i$ and subsequent\nexpansion in powers of the quark masses, the gluon condensate contribution yields\nterms in $m_{q,s}^2 \\ln (m_{q,s}^2\/(-q^2))$, which are long-distance\neffects and must not appear in the short-distance OPE of the correlation functions of Eqs~(\\ref{C.1}) and (\\ref{C.3}). The appearance of these \nlogarithmic terms is due to the fact that the expressions of Eqs.~(\\ref{corresult1}-\\ref{corresult3}) are\nobtained using Wick's theorem which implies that the condensates are normal-ordered: \n$\\langle O \\rangle =\\bra{0}\\!:\\!O\\!:\\! \\ket{0}$ \\cite{logms}. Rewriting the OPE in terms of\nnon-normal-ordered operators, all infrared sensitive terms can be\nabsorbed into the corresponding condensates. Indeed, using, \n\\begin{equation}\n\\bra{0}\\bar s g_s G s \\ket{0} = \\bra{0}\\!:\\!\\bar s g_s G s\\!:\\!\\ket{0} +\n\\frac{m_s}{2}\\, \\ln\\,\\frac{m_s^2}{\\mu^2} \\,\\bra{0}\\! :\\!\n\\frac{\\alpha_s}{\\pi}\\, G^2\\!:\\!\\ket{0}\\,,\n\\end{equation}\nand the corresponding formula for $q$ quarks, all terms in $\\ln m_{q,s}^2$ can be absorbed into the mixed quark-quark-gluon condensate and the resulting short-distance coefficients\ncan be expanded in powers of $m_{q,s}^2$. \n\\begin{figure}[h]\n$$\n\\epsfxsize=0.6\\textwidth\\epsffile{ltileven.eps}\n$$\n$$\n\\epsfxsize=0.6\\textwidth\\epsffile{otileven.eps}\n$$\n$$\n\\epsfxsize=0.6\\textwidth\\epsffile{zeven.eps}\n$$\n\\caption[Hadronic parameters of $\\widetilde\\Phi_{3;K^*}^\\parallel$ as functions of $M^2$.]{\\small  Hadronic parameters of the twist-3 distribution amplitude $\\widetilde\\Phi_{3;K^*}^\\parallel$ as functions of $M^2$.  Upper: $\\widetilde\\lambda_{3K^*}^\\parallel$, middle:  $\\widetilde\\omega_{3K^*}^\\parallel$, and lower:  $\\zeta_{3K^*}^\\parallel$. The solid curve is for central input values for $\\mu=1\\,{\\rm GeV}$ and outer curves take into consideration their uncertainties -- see Tab.~\\ref{QCDSRinput}. Horizontal dashed line is the extracted DA parameter value and shaded region its uncertainty -- see Tab.~\\ref{det_tab2}.}\n  \\label{g} \n\\end{figure}\n\nIn Fig.~\\ref{g} we plot the sum rules  for $\\widetilde\\lambda_{3K^*}^\\parallel$,  $\\widetilde\\omega_{3K^*}^\\parallel$ and $\\zeta_{3K^*}^\\parallel$, given by Eqs.~(\\ref{C.6}), which are evaluated for the central input parameters of Tab.~\\ref{QCDSRinput} and at a scale $\\mu=1\\,{\\rm GeV}$. The parameters unfortunately exhibit  very strong $M^2$ dependence, which leads to increased uncertainty of their values; we do not find a stable plateau in the region $ 1\\,\\textrm{GeV}^2\\leqslant M^2\\leqslant  2.5\\,\\textrm{GeV}^2$. On the other hand, there is only a very small $s_0$ dependence $\\approx 1 \\%$ over the range $s_0^\\parallel (K^*) = (1.3\\pm 0.3)\\,{\\rm GeV}^2$. The curves flatten at high $M^2$ which is expected, as the power corrections become negligible compared to the perturbative contribution.\\footnote{The quark condensates survive as $M^2\\to\\infty$ as $\\hat{\\mathcal{B}}\\left[q^{-2}\\right]=-1$ but perturbation theory $\\sim M^4$ -- see Appendix~\\ref{appendixB}.} The sum rules for the other parameters and particles show the same general behaviour which is fairly typical of non-diagonal correlation functions. If one were to use diagonal correlation functions then it is possible that the sum rules would be better behaved and thus the uncertainties would be reduced somewhat. The calculation of diagonal correlation functions of three-particle operators, as we saw with the gluon condensate in Chapter~\\ref{chapter3_SR}, is rather more involved, especially when calculating radiative corrections, which may very well be necessary in this case. \n\nAll the numerical results, including the uncertainties from the variation of $M^2$, $s_0$, and input parameters,  are given in Tab.~\\ref{det_tab2}.  The results are presented at the scale  $\\mu= 1\\,{\\rm GeV}$ and scaled up to $\\mu= 2\\,{\\rm GeV}$, \nusing the evolution equations, Eq.~(\\ref{evo}).  The only previous determination for  comparison is for the chiral-even $\\rho$ parameters, $\\zeta^\\parallel_{3\\rho}(1\\,\\rm{GeV})=0.033\\pm0.003$, $\\omega^\\parallel_{3\\rho}(1\\,\\rm{GeV})=0.2$, and \n$\\widetilde{\\omega}^\\parallel_{3\\rho}(1\\,\\rm{GeV})=-0.1$ \\cite{Zhitnitsky:1985dd}\nand  $\\omega^\\perp_{3\\rho}(1\\,\\rm{GeV})=0.3\\pm0.3$ \\cite{Ball:1998sk}. These results agree with ours, although we consider the uncertainty of $\\zeta^\\parallel_{3\\rho}$ to be optimistic.\n\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{| l || c | c ||  l | l || c | c |}\n\\hline\n& \\multicolumn{2}{c||}{\\rho} & \\multicolumn{2}{c||}{K^*}  &  \n\\multicolumn{2}{c|}{\\phi}\\\\\n\\cline{2-7}\n& \\mu = 1\\,{\\rm GeV} & \\mu = 2\\,{\\rm GeV} & \\mu = 1\\,{\\rm GeV} & \\mu =\n2\\,{\\rm GeV} & \\mu = 1\\,{\\rm GeV} & \\mu = 2\\,{\\rm GeV}\\\\\n\\hline\n\\zeta_{3V}^\\parallel \n& 0.030(10) & 0.020(9) & \\phantom{-}0.023(8) & \\phantom{-}0.015(6) & \n0.024(8) & 0.017(6)\n\\\\\n\\widetilde\\lambda_{3V}^\\parallel \n& 0 & 0 & \\phantom{-}0.035(15)& \\phantom{-}0.017(8) & 0 & 0\n\\\\\n\\widetilde\\omega_{3V}^\\parallel \n& -0.09(3) & -0.04(2) & -0.07(3) &  -0.03(2) & -0.045(15) & -0.022(8)\n\\\\\n\\kappa_{3V}^\\parallel \n& 0 & 0 & \\phantom{-}0.000(1) & -0.001(2) & 0 & 0\n\\\\\n\\omega_{3V}^\\parallel \n& 0.15(5) & 0.09(3) & \\phantom{-}0.10(4) & \\phantom{-}0.06(3) &\n0.09(3) & 0.06(2)\n\\\\\n\\lambda_{3V}^\\parallel \n& 0 & 0 & -0.008(4) & -0.004(2) & 0 & 0\n\\\\\n\\kappa_{3V}^\\perp \n& 0 & 0 & \\phantom{-}0.003(3) & -0.001(2) & 0 & 0 \n\\\\\n\\omega_{3V}^\\perp \n& 0.55(25) & 0.37(19) & \\phantom{-}0.3(1) & \\phantom{-}0.2(1) &  \n0.20(8) & 0.15(7)\n\\\\\n\\lambda_{3V}^\\perp \n& 0 & 0 & -0.025(20) & -0.015(10) & 0 & 0\\\\\\hline \n\\end{array}\n$$\n\\renewcommand{\\arraystretch}{1}\n\\addtolength{\\arraycolsep}{-3pt}\n\\caption[Results for the leading twist-3 distribution amplitude parameters.]{\\small Results for the leading three-particle twist-3 hadronic parameters of the DAs of Eq.~(\\ref{das_eq27}). The results are presented at the scale  $\\mu= 1\\,{\\rm GeV}$ and scaled up to $\\mu= 2\\,{\\rm GeV}$ using the evolution equations (\\ref{evo}). The sign of the  parameters corresponds to the sign convention for the strong coupling defined by the covariant derivative $D_\\mu = \\partial_\\mu - i g_s A^a_\\mu t^a$; they change sign if $g_s$ is fixed by $D_\\mu = \\partial_\\mu + i g_s A^a_\\mu t^a$.}\n\\label{det_tab2}\n\\end{table}\n\nIn Fig.~\\ref{graphs} we plot the two-particle twist-3 DAs as defined by Eqs.~(\\ref{das_eq30}- \\ref{das_eq33}). G-parity violating effects cause the small asymmetry of the $K^*$ curves. The effects of $\\rm SU(3)_F$-breaking are larger and cause the pronounced difference between $\\phi_{3}^\\parallel$ and $\\phi_{3}^\\perp$  for the $\\rho$ and $\\phi$. We notice in particular the end-point behaviour of the DAs is greatly modified. The fact that both $\\phi_{3;\\rho}^{\\parallel\\perp}$ and $\\phi_{3;K^*}^{\\parallel\\perp}$ diverge as $u\\to1$ and $\\phi_{3;\\rho}^{\\parallel\\perp}$ for $u\\to0$ is in itself not  a problem. It is only the leading-twist DA that can be considered a probability distribution and likewise there is no cause for concern that $\\phi_{3;\\rho}^{\\parallel}$ takes negative values. Moreover, in practical calculations we are only interested in convolutions of the DAs with hard scattering kernels, which are generally finite. If not, this signals a problem with the hard scattering kernel, rather than the DA, as happens with end-point divergences within the QCD factorisation framework for non-leptonic $B$ decays, see Chapter~\\ref{chapter6_QCDF}. \n\\begin{figure}[t]\n$$\n\\epsfxsize=0.5\\textwidth\\epsffile{fig1a.eps}\\quad\n\\epsfxsize=0.5\\textwidth\\epsffile{fig1b.eps}\n$$\n\\caption[The  distribution amplitudes $\\phi^\\parallel_{3;V}$ and $\\psi^\\parallel_{3;V}$ as a function of  $u$.]{\\small Left: $\\phi^\\parallel_{3}$ as a function of  $u$ for the central values of hadronic parameters, for $\\mu=1\\,$GeV.   Red line: $\\phi_{3;\\rho}^\\parallel$, green:   $\\phi_{3;K^*}^\\parallel$, blue: $\\phi_{3;\\phi}^\\parallel$. Right: same for $\\psi^\\parallel_{3}$.}\n$$\n\\epsfxsize=0.5\\textwidth\\epsffile{fig2a.eps}\\quad\n\\epsfxsize=0.5\\textwidth\\epsffile{fig2b.eps}\n$$\n\\caption[The distribution amplitudes $\\phi^\\perp_{3;V}$ and $\\psi^\\perp_{3;V}$ as a function of  $u$.]{\\small Left: $\\phi^\\perp_{3}$ as a function of $u$ for the central values of hadronic parameters, for $\\mu=1\\,$GeV.   Red line: $\\phi_{3;\\rho}^\\perp$, green:   $\\phi_{3;K^*}^\\perp$, blue: $\\phi_{3;\\phi}^\\perp$. Right: same for $\\psi^\\perp_{3}$.}\n  \\label{graphs}\n\\end{figure}\n\n\\chapter{ $B \\to \\eta^{(\\prime)}$ Form Factors in QCD}\\label{chapter5_eta}\nIn this chapter we discuss the semileptonic $B\\to\\eta^{(\\prime)}$ form factors $f_+^{B\\to\\eta^{(\\prime)}}$ in the LCSR approach. The previous LCSR determination of the $B\\to\\eta^{(\\prime)}$ form factors  presented in Ref.~\\cite{Ball:2004ye} is completed by calculating the gluonic contribution, the mechanism for which involves the annihilation of the $B$ meson to two gluons. The $\\eta^{(\\prime)}$ particles undergo pronounced mixing with each other due to the $\\rm U(1)_A$ anomaly of QCD and the $\\eta$-$\\eta^{\\prime}$ system, after many years of investigation, has succumbed to the phenomenologically motivated mixing scheme proposed by Feldmann, Kroll and Stech \\cite{Feldmann:1998vh,Feldmann:1998sh}. The consideration of this mixing scheme is central to the correct description of the $B\\to\\eta^{(\\prime)}$ form factors.\n\nMotivation to complete the calculation of  $f_+^{B\\to\\eta^{(\\prime)}}$ comes from a variety of sources, with probably the most prominent being:\n\\begin{itemize}\n\\item{the flavour-singlet contributions to the QCD factorisation framework to be discussed in Chapter~\\ref{chapter6_QCDF} were added by Beneke and Neubert in Ref.~\\cite{Beneke:2002jn}. It is found that the branching ratios of $B \\to \\eta^{\\prime} (V,P)$ are very sensitive to $f_+^{B\\to\\eta^{(\\prime)}}$  as the leading-order annihilation diagrams can be interpreted as a  gluon contribution to the $B \\to \\eta^{(\\prime)}$ form factors \\cite{Beneke:2003zv}.  Therefore a consistent estimation of the annihilation diagrams necessitates the inclusion of the gluonic contributions to the form factor.}\n\\item{There exists a ``tension'' in the determinations of $|V_{ub}|$ from  inclusive semileptonic decays $B\\to X_u l\\nu$ and their exclusive counterparts, namely from $B\\to\\pi l \\nu$. The former have led to larger values than the latter, and the reason for the discrepancy is unclear. $B\\to \\eta^{(\\prime)}$ transitions are at leading order a $b\\to u$ transition and so sensitive to $|V_{ub}|$ which can, in principle, be extracted from $B\\to \\eta^{(\\prime)} l \\nu$.  An improved calculation of $f_+^{B\\to\\eta^{(\\prime)}}$ would reduce the theoretical uncertainty of the result.}\n\\item{Finally, the observation that exclusive $B\\to \\eta^{\\prime} K$ and inclusive $B\\to \\eta^{\\prime} X$ decays have shown unexpectedly large branching ratios with respect to $B\\to\\pi$ transitions, for example, is an unresolved issue which an improved calculation of $f_+^{B\\to\\eta^{(\\prime)}}$ may help clarify.}\n\\end{itemize}\nWe begin by introducing the $\\eta^{(\\prime)}$ system and define two closely related $\\eta$-$\\eta^{\\prime}$ mixing schemes. We then discuss the calculation of the flavour-singlet contribution to the form factor before lastly we discuss the results of the LCSR analysis, the framework for which was covered in Chapter~\\ref{chapter3_SR}.  The material presented in this chapter follows that of Ref.~\\cite{Ball:2007hb}.\n\n\\section{The $\\eta$-$\\eta^{\\prime}$ System}\nThe approximate chiral symmetry of light quarks $u,d$ and $s$ in QCD seems to be broken by Nature to reveal the pseudoscalar mesons $(\\pi^0,\\pi^+,\\pi^-, K^+ ,K^-, K^0 ,\\bar{K}^0, \\eta)$ as the corresponding octet of Goldstone bosons (all massless in the \\textit{chiral limit} $m_{u,d,s}\\to 0$) of the broken $\\rm SU(3)\\otimes SU(3)$ symmetry. There is another symmetry of the QCD Lagrangian (\\ref{basics_eq1}); a global $\\rm U(1)_A$ symmetry which exists at the classical level in the chiral limit. Due to non-vanishing quark masses,  the broken $\\rm U(1)_A$ symmetry creates a Goldstone boson, but such a light particle does not appear in the physical spectrum and this embodies the \\textit{$\\rm U(1)_A$ problem}. At the quantum level, however, the $\\rm U(1)_A$ symmetry in the massless limit is broken due to the QCD anomaly and so was not present in the first place; thus a ninth state, the  $\\eta^{\\prime}$, exists as a singlet and only becomes massless in the chiral limit \\textit{and} as $N_c\\to\\infty$, causing the effects of anomaly to vanish. The situation is complicated by instanton effects, but was ultimately resolved by 't Hooft  with the same conclusion \\cite{Hooft:1976up,Hooft:1986nc}.  It has been known for a while that the $\\rm U(1)_A$ anomaly plays a decisive role in the $\\eta^{(\\prime)}$ system with the $\\eta^{\\prime}$ consisting of a large gluonic component \\cite{Witten:1978bc,Ball:1995zv}. The large mass of the $\\eta^{\\prime}$ is mostly generated by the anomaly and $\\rm SU(3)_F$-breaking effects.\\footnote{The particles $\\eta^{(\\prime)}$ have masses $m_{\\eta}=547.51 \\pm 0.18 ~\\textrm{MeV}$ and $m_{\\eta^{\\prime}}=957.78\\pm 0.14 ~\\textrm{MeV}$  and quantum numbers  $J^{PC}=0^{-+}$ \\cite{Yao:2006px}.}\n\nThe $\\eta$-$\\eta^{\\prime}$ system has been of considerable interest for a number of years \\cite{Fritzsch:1976qc,Isgur:1976qg,Novikov:1979ux}. Vast simplifications can be made in studying the low-energy particle spectrum of QCD by employing  \\textit{Chiral Perturbation Theory} (ChPT) which is an effective theory in which the heavy quarks are integrated out and the dynamically relevant  light quarks remain at a scale $\\mu\\sim\\Lambda_{\\rm QCD}$ after an expansion in powers of energies, momenta and quark masses. Alongside the $1\/N_c$ expansion, ChPT is the method of choice for analysing the light pseudoscalar mesons.\\footnote{Another interesting approach to understanding the $\\eta^{(\\prime)}$ system was given in Ref.~\\cite{Katz:2007tf}.} We do not discuss ChPT in any detail although we do quote a few of its constraints; for more details see for example \\cite{Weinberg:1978kz,Gasser:1984gg,Leutwyler:2001hn} \n\nConcerning $\\eta$-$\\eta^{\\prime}$  mixing, ChPT requires a description in terms of two mixing angles  beyond leading-order \\cite{Leutwyler:1997yr,Feldmann:1997vc}. How this is implemented in practice has caused some confusion in the past but a consistent picture has emerged \\cite{Feldmann:1998vh,Feldmann:1998sh}. Key to the phenomenological picture of the $\\eta$-$\\eta^{\\prime}$ system is the understanding that the main contributions to the mixing are due to the $\\rm U(1)_A$ anomaly of QCD, and so-called \\textit{OZI-rule violating} processes. Named after Okubo, Zweig and Iizuka the OZI-rule states that strong interaction processes that must proceed via the annihilation of all initial state quarks to gluons are suppressed \\cite{Okubo:1963fa,Iizuka:1966fk,Zweig:1964wu}. In Fig.~\\ref{eta_ozi} we show the unsuppressed process $\\phi \\to K^+ K^-$ (left) alongside the suppressed process $\\phi \\to \\pi^+\\pi^-\\pi^0$ (right) for which the rule was originally formulated.  Such processes are shown to be $\\mathcal{O}(1\/N_c)$ in a $1\/N_c$ expansion and phenomenologically they are found to be small $\\approx 10\\%$; they can be safely neglected, leaving the $\\rm U(1)_A$ anomaly as the only mixing mechanism. For the mixing schemes we discuss in the next section, this assumption has been confronted with experimental data and holds to the expected accuracy. \n\\begin{figure}[h]\n$$ \\epsfxsize=0.6\\textwidth\\epsffile{ozi.eps}$$\n\\caption[Examples of an OZI-rule suppressed and allowed strong decays.]{\\small Examples of strong interaction decays. Left: $\\phi \\to K^+ K^-$, right: $\\phi \\to \\pi^+\\pi^-\\pi^0$. The former occurs preferentially over the latter due to the fact that the annihilation of the $\\phi$ requires all gluons to be hard, yielding a suppression via a small $\\alpha_s$ which need not be the case for the first decay. This forms the basis of the OZI-rule.}\n\\label{eta_ozi}\n\\end{figure}\n\nA schematic picture of the $\\rm U(1)_A$ anomaly at work for $B\\to\\eta^{(\\prime)}$ is shown in Fig.~\\ref{eta_u1a}., where the flavour-singlet contribution is defined as the amplitude for producing either a quark-antiquark pair in a singlet state which does not contain the $B$'s spectator quark, or two gluons, which then hadronise into an $\\eta^{(\\prime)}$.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.3\\textwidth\\epsffile{eta_schem.eps}$$\n\\caption[$B\\to \\eta^{(\\prime)}$ via the $\\rm U(1)_A$ anomaly.]{\\small $B\\to \\eta^{(\\prime)}$ via the $\\rm U(1)_A$ anomaly. The $b\\to u$ transition allows for an annihilation of the $B$ meson's quarks to two gluons, thus probing the gluonic content of $\\eta^{(\\prime)}$.}\n\\label{eta_u1a}\n\\end{figure}\n\nWhat about mixing between other pseudoscalar mesons? In $\\eta$ - $\\eta^{\\prime}$ - $\\pi^{0}$ mixing the  gluonic component present in the $\\pi^0$ is found to be at the level of a few percent  and so can be neglected \\cite{Feldmann:1998sh,Feldmann:1999uf,Kroll:2004rs}. There also exists a $c \\bar{c}$ component to $\\eta^{(\\prime)}$ ($\\eta_{c}$) which is considered in Ref.~\\cite{Feldmann:1998vh} and found to be small with the conclusion that it is not the solution to the abnormally large $B \\to K \\eta^{\\prime}$ branching ratio. Sometimes other particles are included as possible glueball candidates produced via OZI-rule suppressed processes in $J\/\\psi$ decay, see for example Refs.~\\cite{Ball:1995zv,Li:2007ky}.  Although it is unclear whether pseudoscalar mesons contain pure glueball properties, Ref.~\\cite{Kroll:2003yi}  concludes that it is unlikely. Thus the $\\eta$-$\\eta^{\\prime}$  system stands out on its own. \n\nPhenomenologically, the semileptonic decay $B \\to \\eta^{(\\prime)} l \\nu_l$ can be used to determine the size of  the CKM matrix element $|V_{ub}|$ from the spectrum\n\\begin{equation}\\label{eq:spectrum}\n\\frac{d\\Gamma}{dq^2}(B \\to P l \\nu_l) = \\frac{G_F^2 |V_{ub}|^2}{\n192\\pi^3m_B^3}\\lambda^{3\/2}_P(q^2) |f^P_+(q^2)|^2 \\,,\n\\end{equation}\nwhere $P=\\{\\eta,\\eta^{\\prime}\\}$ and $\\lambda_P(x) = (m_B^2+m_{P}^2-x)^2-4m_B^2m_{P}^2$. Alternatively,\nas we shall see, the ratio of branching ratios ${\\cal B}(B\\to\\eta^{\\prime}\n\\ell\\nu)\/{\\cal B}(B\\to \\eta\\ell\\nu)$ can be used to constrain the\ngluonic Gegenbauer moment $B_2^g$.\n\n\\section{State Mixing}\nThe first step in describing $\\eta$-$\\eta^{\\prime}$ mixing is to decompose the two physical states $\\ket{\\eta^{(\\prime)}}$ into other, more convenient orthogonal states. As proposed in Refs.~\\cite{Feldmann:1998vh,Feldmann:1998sh}  one can proceed in two ways; either by employing the singlet-octet scheme (SO) or the quark-flavour scheme (QF). The SO axial-vector currents are respectively\n\\begin{equation}\n J^{0}_{\\mu 5}=\\frac{1}{\\sqrt{3}}\\left(\\bar u \\gamma_\\mu\\gamma_5 u+\\bar d \\gamma_\\mu\\gamma_5  d +\\bar s \\gamma_\\mu\\gamma_5  s\\right)\\,,\\quad  J^{8}_{\\mu 5}=\\frac{1}{\\sqrt{6}}\\left(\\bar u \\gamma_\\mu\\gamma_5  u+\\bar d \\gamma_\\mu\\gamma_5  d -2\\bar s \\gamma_\\mu\\gamma_5  s\\right)\\,,\n\\end{equation}\nand their couplings are given by\n\\begin{equation}\n\\bra{0} J^{i}_{\\mu 5}\\ket{P(p)} = i f^{i}_{P} p_{\\mu} \\qquad (i=0,8)\\,,\n\\label{SOdc}\n\\end{equation}\nwhere $J^{8}_{\\mu 5}$ denotes the $\\rm SU(3)_F$-octet and $J^{0}_{\\mu 5}$ the $\\rm SU(3)_F$-singlet axial-vector current. The four quantities are related to the decay constants of a pure singlet or octet state $\\ket{\\eta_i}$ by two mixing angles  $\\theta_i$\n\\begin{equation}\n\\left(\n\\begin{array}{cc}\nf_\\eta ^8 & f_\\eta^0 \\\\\nf_{\\eta^{\\prime}} ^8 & f_{\\eta^{\\prime}}^0\n\\end{array}\\right)\n= \n\\left(\n\\begin{array}{cc}\n\\cos\\theta_8 & -\\sin\\theta_0 \\\\\n\\sin\\theta_8 & \\phantom{-}\\cos\\theta_0 \n\\end{array}\\right)\n\\left(\n\\begin{array}{cc}\nf_8 & 0 \\\\\n0 & f_0\n\\end{array}\n\\right).\n\\label{corr}\n\\end{equation}\nEvidently $\\rm SU(3)_F$-breaking effects cause $\\theta_i\\neq 0$ and $f_8\\neq f_\\pi$, and as such the SO scheme is very natural. In fact, at leading-order in ChPT an expansion in quark masses  and $1\/N_{c}$ gives \\cite{Leutwyler:1997yr}\n\\begin{equation}\n\\sin (\\theta_{0}-\\theta_{8})=\\frac{2 \\sqrt{2} (f_{K}^{2}-f_{\\pi}^{2})}{4f_{K}^{2}-f_{\\pi}^{2}}+\\dots\\,,\n\\label{su3}\n\\end{equation}\nwhere the dots denote neglected higher-order terms which are required to match phenomenology \\cite{Kaiser:1998ds}. The impact of the U(1)$_{\\rm A}$ anomaly is plainly localised in $f_0$ via the divergence of the singlet current $J_{\\mu 5}^0$ which can be written\n\\begin{equation}\n\\partial^\\mu J^{a}_{\\mu 5} = 2\\,\\bar{q} \\left[ t^a \\hat{m} i \\gamma_5 \\right]q + \\delta^{a 0}\\,\\frac{\\alpha_{s}}{4 \\pi} G \\widetilde{G}\\,,\n\\label{anomaly}\n\\end{equation} \nwhere $a=\\{0,1,\\dots,8\\}$, $\\textrm{Tr}[t^a t^b ] = \\frac{1}{2}\\delta^{a b}$, $t^0 = \\textbf{1}\/\\sqrt{3}$ and the mass matrix $\\hat{m}=\\textrm{diag}[m_u,m_d,m_s]$. The SO scheme diagonalises the renormalisation-scale dependence of parameters; $f_8$ and $\\theta_i$ are\nscale-independent, whereas $f_0$ renormalises multiplicatively\n\\begin{equation}\n\\mu\\,\\frac{d f_0}{d \\mu} = - N_f \\left(\\frac{\\alpha_s}{\\pi}\\right)^2 f_0 + O(\\alpha_s^3)\\,.\n\\label{scaledep}\n\\end{equation}\nIn the QF mixing scheme, on the other hand, the basic axial-vector currents are\n\\begin{equation}\nJ^q_{\\mu 5} = \\frac{1}{\\sqrt{2}} \\left(\\bar u \\gamma_\\mu \\gamma_5 u + \n\\bar d \\gamma_\\mu \\gamma_5 d \\right),\\qquad\nJ^s_{\\mu 5} = \\bar s \\gamma_\\mu \\gamma_5 s\\,,\n\\end{equation}\nand the corresponding couplings are\n\\begin{equation}\\label{6}\n\\bra{0} J^r_{\\mu 5}\\ket{P(p)} = i f_P^r p_\\mu \\quad (r=q,s)\\,.\n\\end{equation}\nThe mixing is analogous to (\\ref{corr}) with\n\\begin{equation}\n\\left(\n\\begin{array}{cc}\nf_\\eta ^q & f_\\eta^s \\\\\nf_{\\eta^{\\prime}} ^q & f_{\\eta^{\\prime}}^s \n\\end{array}\\right)\n= \n\\left(\n\\begin{array}{cc}\n\\cos\\phi_q & -\\sin\\phi_s \\\\\n\\sin\\phi_q & \\phantom{-}\\cos\\phi_s \n\\end{array}\\right)\n\\left(\n\\begin{array}{cc}\nf_q & 0 \\\\\n0 & f_s\n\\end{array}\n\\right).\n\\label{corr2}\n\\end{equation}\nBoth quark flavour states $\\ket{\\eta_{q,s}}$ have vanishing vacuum-particle matrix elements with the opposite currents\n\\begin{equation}\n\\bra{0}J^s_{\\mu 5}\\ket{\\eta_q}=\\bra{0}J^q_{\\mu 5}\\ket{\\eta_s}=0\\,,\n\\end{equation}\nwhich is an assumption that has been tested. It is in part motivated by the observation  of near ideal mixing in vector and tensor mesons. It implies that the mixing of states is the same as that of the decay constants and moreover leads to the diagonalisation of the mass matrix, which we come back to shortly. This hypothesis does not hold for the SO basis. It is found by Refs.~\\cite{Feldmann:1997vc,Feldmann:1999uf} that the difference  between the two mixing angles of the QF scheme $\\phi_q-\\phi_s$ is generated by OZI-rule suppressed processes and is not caused by SU(3)$_{\\rm F}$-breaking effects, as for the SO scheme (\\ref{su3}). While the numerical values of $\\theta_i$ differ largely, with typical values $\\theta_8\\approx -20^\\circ$ and $\\theta_0\\approx - 5^\\circ$, one finds $\\phi_s-\\phi_q\\, \\lesssim\\, 5^\\circ$, with $\\phi_q\\approx\n\\phi_s \\approx 40^\\circ$ \\cite{Feldmann:1998vh,Feldmann:1998sh,Feldmann:1997vc}. This observation led the authors of Refs.~\\cite{Feldmann:1998vh,Feldmann:1998sh} to suggest the QF scheme as an approximation to\ndescribe $\\eta$-$\\eta^{\\prime}$ mixing, based on neglecting the difference $\\phi_q-\\phi_s$ (and all other OZI-breaking effects):\n\\begin{equation}\n\\phi\\equiv \\phi_{q,s},\\qquad \\phi_q-\\phi_s\\equiv 0\\,.\n\\end{equation}\nThe state mixing is then given by\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\\ket{\\eta} \\\\ \\ket{\\eta^{\\prime}}\n\\end{array}\n\\right) \n= \n\\left(\n\\begin{array}{ll}\n\\cos\\phi & -\\sin\\phi\\\\\n\\sin\\phi & \\phantom{-}\\cos\\phi\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\n\\ket{\\eta_q}\\\\ \\ket{\\eta_s}\n\\end{array}\n\\right)\\,.\n\\label{8}\n\\end{equation}\nThe re\\-nor\\-ma\\-li\\-sa\\-tion-scale dependence of $f_0$ given by Eq.~(\\ref{scaledep}) is not reproduced as it is induced precisely by neglected OZI-breaking terms \\cite{Feldmann:1999uf}. Numerically, this is not a problem as the scale-dependence of $f_0$ is a two-loop effect. In the case of non-local matrix elements, the DAs, this lack of scale dependence of the QF scheme is somewhat problematic. We come back to this point in the next section.\n\nReturning to the diagonalisation of the mass matrix; from Eq.~(\\ref{SOdc}) one finds the quadratic diagonal mass matrix, for example\n\\begin{equation}\\label{div}\n\\bra{0} \\partial^{\\mu}J^{s}_{\\mu 5}\\ket{\\eta (p)} = M^{2}_{\\eta} f^{s}_{\\eta}\\,,\n\\end{equation}\nwhich, via Eq.~(\\ref{anomaly}), gives the mass matrix in QF basis\n\\begin{equation}\n\\mathcal{M}_{\\rm QF}^2=\\left(\n\\begin{array}{cc}\nm_{qq}^2 +\\frac{\\sqrt{2}}{f_q}\\bra{0}\\frac{\\alpha_s}{4\\pi}G\\widetilde{G}\\ket{\\eta_q} & \\frac{1}{f_s}\\bra{0}\\frac{\\alpha_s}{4\\pi}G\\widetilde{G}\\ket{\\eta_q} \\\\\n\\frac{\\sqrt{2}}{f_q}\\bra{0}\\frac{\\alpha_s}{4\\pi}G\\widetilde{G}\\ket{\\eta_s}&m_{ss}^2+\\frac{1}{f_s}\\bra{0}\\frac{\\alpha_s}{4\\pi}G\\widetilde{G}\\ket{\\eta_s}\n\\end{array}\\right)\\,,\n\\label{mass}\n\\end{equation}\nwith the short-hand notation\n\\begin{equation}\nm_{qq}^2=\\frac{\\sqrt{2}}{f_q}\\bra{0}m_u \\bar{u}i\\gamma_5 u+m_d \\bar{d} i \\gamma_5 d\\ket{\\eta_q}\\,,\\qquad m_{ss}^2=\\frac{2}{f_s}\\bra{0}m_s \\bar{s} i \\gamma_5 s\\ket{\\eta_s}\\,.\n\\end{equation}\nFrom Eq.~(\\ref{mass}) the crucial impact of the anomaly, as the only term in the off-diagonal elements, is evident. To first order in $\\rm SU(3)_F$-breaking, the decay constants and quantities $m_{qq,ss}^2$ are fixed giving the theoretical estimate\n\\begin{eqnarray}\nf_q=f_\\pi\\,,&\\quad& f_s =\\sqrt{2 f_K^2-f^2_\\pi}\\,,\\nonumber\\\\\n m_{qq}^2=M_\\pi^2\\,,&\\quad& m_{ss}^2=2 M_K^2-M_\\pi^2\\,,\n\\end{eqnarray}\nwhich also leads to a fixed value of $\\phi$; there is no free parameter left and thus the QF scheme is totally determined \\cite{Feldmann:1998vh}. We do not work in this limit, however, and take numerical values of the decay constants and mixing angle from phenomenology. Given enough data to fix all independent parameters, there is no reason to prefer the QF over the SO scheme. The QF scheme is beneficial when considering DAs as the SO scheme leads to a proliferation of unknown parameters. For this reason we decide to use the QF scheme for the analysis. Its basic parameters have been determined as \\cite{Feldmann:1998vh,Feldmann:1998sh}\n\\begin{equation}\nf_q  =  (1.07\\pm 0.02)f_\\pi,\\qquad f_s = (1.34\\pm\n0.06)f_\\pi\\,,\\qquad\n\\phi =  39.3^\\circ\\pm 1.0^\\circ\\,.\n\\end{equation}\nThis can be translated into values for the SO parameters as\n\\begin{eqnarray}\nf_8 & = & \\sqrt{\\frac{1}{3}\\,f_q^2 + \\frac{2}{3} f_s^2} = (1.26\\pm\n0.04) f_\\pi\\,,\\nonumber\\\\\nf_0  &=&  \\sqrt{\\frac{2}{3}\\,f_q^2 + \\frac{1}{3} f_s^2} = (1.17\\pm\n0.03) f_\\pi\\,,\\nonumber\\\\\n\\theta_8 & = & \\phi-{\\rm arctan}[\\sqrt{2} f_s\/f_q] = (-21.2 \\pm\n1.6)^\\circ\\,,\\nonumber\\\\\n\\theta_0  &=&  \\phi-{\\rm arctan}[\\sqrt{2} f_q\/f_s] = (-9.2 \\pm\n1.7)^\\circ\\,,\n\\end{eqnarray}\nNote that in the QF scheme $f_{q,s}$ are scale-independent parameters, and so is $f_0$ as obtained from the above relations. The SO decay constants are related to those of the QF scheme by a change of basis\n\\begin{equation}\\label{11}\n\\left(\n\\begin{array}{cc}\nf_\\eta ^8 & f_\\eta^0 \\\\\nf_{\\eta^{\\prime}} ^8 & f_{\\eta^{\\prime}}^0 \n\\end{array}\\right)\n= \n\\left(\n\\begin{array}{cc}\n\\cos\\phi & -\\sin\\phi \\\\\n\\sin\\phi & \\phantom{-}\\cos\\phi \n\\end{array}\\right)\n\\left(\n\\begin{array}{cc}\nf_q & 0 \\\\\n0 & f_s\n\\end{array}\\right)\n\\left(\n\\begin{array}{cc}\n\\phantom{-}\\sqrt{\\frac{1}{3}} & \\sqrt{\\frac{2}{3}}\\\\\n-\\sqrt{\\frac{2}{3}} & \\sqrt{\\frac{1}{3}}\n\\end{array}\\right).\n\\end{equation}\nThe last matrix originates from the ideal mixing angle $\\theta_{\\textrm{ideal}}=\\arctan{\\sqrt{2}}$ which rotates from the QF basis to the SO basis.\n\n\\section{Pseudoscalar Meson Distribution Amplitudes}\nAs discussed in Chapter~\\ref{chapter3_SR}, the method of LCSRs relies on the non-perturbative universal light-cone DAs; specifically here we require pseudoscalar meson DAs including the two-gluon DA. At leading-twist both these DAs contribute and indeed mix with each other under renormalisation. The quark-antiquark DAs are extensions of the matrix elements given by Eqs.~(\\ref{SOdc}) and (\\ref{6}) to those of non-local operators on the light-cone. Pseudoscalar mesons' quark-antiquark DAs have been investigated previously in  Refs.~\\cite{Ball:1998je,Ball:2006wn,Braun:1989iv}. The two-gluon DAs of leading and higher twist have been investigated in Ref.~\\cite{AP03}. In this analysis we only include the effects of the leading-twist two-gluon DA, which is justified as its effects turn out to be fairly small and higher-twist DAs are estimated to have even smaller impact. Following Ref.~\\cite{Kroll:2002nt}, the twist-2 two-quark DAs of $\\eta^{(\\prime)}$ are defined as\n\\begin{equation}\n\\bra{0} \\bar\\Psi(z) {\\cal C}^i\\gamma_z \\gamma_5 [z,-z] \\Psi(-z) \\ket{P(p)} = i (p\\cdot z) f_P^i \\int_0^1 du\\, e^{i \\xi p \\cdot  z} \\phi_{2;P}^i(u) \\,.\n\\end{equation}\n$\\phi_{2;P}^i(u)$ is the twist-2 DA of the meson $P$ with respect to the current whose flavour content is given  by ${\\cal C}^i$, with $\\Psi = (u,d,s)$ the triplet of light-quark fields in flavour space. For the SO currents, one has ${\\cal C}^0 = \\mbox{\\boldmath $1$}\/\\sqrt{3}$ and ${\\cal C}^8 = \\sqrt{2}\\, t^8$, while for the QF currents ${\\cal C}^q = (\\sqrt{2} {\\cal C}^0 + {\\cal C}^8)\/\\sqrt{3}$ and  ${\\cal C}^s = ({\\cal C}^0 - \\sqrt{2} {\\cal C}^8)\/\\sqrt{3}$. Due to the positive G-parity of $\\eta$ and $\\eta^{\\prime}$, the two-quark DAs are symmetric under $u\\leftrightarrow 1-u$, and hence all odd Gegenbauer moments vanish:\n\\begin{equation}\n\\phi_{2;P}^i(u) = \\phi_{2;P}^i(1-u)\\,,\n\\end{equation}\nand the DAs are expanded in terms of Gegenbauer polynomials in exactly the same way as for the vector mesons\n\\begin{equation}\n\\phi_{2;P}^i(u) = 6 u (1-u) \\left( 1 + \\sum_{n=2,4,\\dots} a_n^{P,i}(\\mu)\nC^{3\/2}_n(\\xi) \\right) \\,\\quad (i=1,8,q,s)\\,,\n\\label{expansion}\n\\end{equation}\nwhere $a_n^{P,i}$ are the quark Gegenbauer moments. The gluonic twist-2 DA is defined as\\footnote{This definition refers to the ``$\\sigma$-rescaled'' DA $\\phi^\\sigma_g$ in Ref.~\\cite{Kroll:2002nt} with  $\\sigma = \\sqrt{3}\/C_F$. It agrees with that used in Refs.~\\cite{AP03,Charng:2006zj}, which means that we can use their results for the two-gluon Gegenbauer moment\n$B^g_2$ without rescaling.}\n\\begin{equation}\n\\bra{0} G_{\\mu z}(z) [z,-z] \\widetilde G^{\\mu z}(-z) \\ket{P(p)} = \\frac{1}{2}\\,(p\\cdot z)^2 \\frac{C_F}{\\sqrt{3}} f_P^0 \\int_0^1 du\\, e^{i\\xi p\\cdot z} \\psi_{2;P}^g(u)\\,.\n\\end{equation}\nIn order to perform the calculation of the correlation function defined in the next section, we also need the matrix element of the meson $P$ over two gluon fields. Dropping the gauge factor $[z,-z]$ one has\n\\begin{equation}\n\\bra{0} A^a_\\alpha(z) A^b_\\beta(-z)\\ket{P(p)} =\n\\frac{1}{4}\\,\\epsilon_{\\alpha\\beta\\rho\\sigma} \\,\\frac{z^\\rho\n  p^\\sigma}{p\\cdot z} \\,\\frac{C_F}{\\sqrt{3}}\\, f_P^0 \\,\\frac{\\delta^{ab}}{8}\n\\int_0^1 du\\, e^{i\\xi p\\cdot z}\\,\\frac{\\psi_{2;P}^g(u)}{u(1-u)}\\,.\n\\label{gluefields}\n\\end{equation}\nThe two-gluon asymptotic DA is $u^{2j-1}(1-u)^{2j-1}$ with $j=3\/2$ the lowest conformal spin of the operator $G_{\\mu z}$ and the expansion goes in terms of Gegenbauer polynomials $C^{5\/2}_n$, see Eq.~(\\ref{basics_eq18}). One can show that $\\psi_{2;P}^g$ is antisymmetric:\n\\begin{equation}\n\\psi_{2;P}^g(u) = - \\psi_{2:P}^g(1-u)\\,,\n\\end{equation}\nand in particular $\\int_0^1 du\\, \\psi_{2;P}^g(u) = 0$ and the local twist-2 matrix element $\\bra{0} G_{\\mu z} \\widetilde G^{\\mu z}\\ket{P}$ vanishes. The non-vanishing coupling $\\bra{0}G_{\\alpha\\beta}  \\widetilde G^{\\alpha\\beta}\\ket{P}$ induced by the U(1)$_{\\rm A}$ anomaly is a twist-4 effect. The corresponding matrix elements are discussed in Refs.~\\cite{Feldmann:1998vh,Feldmann:1998sh} and are given, in the QF scheme, by:\n\\begin{eqnarray}\n\\bra{0}\\frac{\\alpha_s}{4\\pi} G\\widetilde{G} \\ket{\\eta_q} & = & \nf_s (m_\\eta^2-m_{\\eta^{\\prime}}^2) \\sin\\phi \\cos\\phi\\,,\\nonumber\\\\\n\\bra{0}\\frac{ \\alpha_s}{4\\pi} G\\widetilde{G} \\ket{\\eta_s} & = & \nf_q (m_\\eta^2-m_{\\eta^{\\prime}}^2)\/\\sqrt{2} \\sin\\phi \\cos\\phi\\,.\n\\label{extra}\n\\end{eqnarray}\nIn taking the ratios of both sides of the above relations one can see that $\\rm SU(3)_F$-breaking in the decay constants $f_q\/f_s$ is driven by the anomaly. There are no twist-3 two-gluon DAs and the remaining twist-4 DAs also have vanishing normalisation \\cite{AP03}.  The conformal expansion of the twist-2 two-gluon DA reads\n\\begin{equation}\n\\psi_{2;P}^g(u,\\mu) = u^2 (1-u)^2 \\sum_{n=2,4,\\dots} B^{P,g}_n(\\mu)\nC^{5\/2}_{n-1}(\\xi)\\,,\n\\end{equation}\nwith the gluonic Gegenbauer moments $B^{P,g}_n$. In this analysis, we truncate both $\\phi^i_{2;P}$ and $\\psi^g_{2;P}$ at $n=2$. An estimate of the effect of higher Gegenbauer moments in $\\phi_{2;\\pi}$ on the $B\\to\\pi$ form factor $f_+^\\pi$ has been given in Ref.~\\cite{Ball:2005ei}, based on a\ncertain class of models for the full DA beyond conformal expansion. The effect of neglecting $a_{n\\geq 4}^\\pi$ was found to be very small $\\approx 2\\%$ hence we expect the truncation error from neglecing $B^g_{n\\geq 4}$ to be of similar size.\n\n$\\phi_{2;P}^0$ and $\\psi_{2;P}^g$ mix upon a change of scale $\\mu$ and as discussed in  Refs.~\\cite{Baier:1981pm,Kroll:2002nt} this amounts to a mixing of $a_2^{P,0}$ and $B^{P,g}_2$, resulting in the renormalisation-group equation to LO accuracy\n\\begin{equation}\n\\mu\\,\\frac{d}{d\\mu}\\left(\\begin{array}{c} a_2^0\\\\ B^g_2\n\\end{array}\\right)\n=\n-\\frac{\\alpha_s}{4\\pi} \\left(\\begin{array}{cc} \\displaystyle\\frac{100}{9} &\n\\displaystyle  -\\frac{10}{81}\\\\\\vphantom{\\displaystyle\\frac{100}{9}}\n -36 & 22\\end{array}\\right)\n\\left(\\begin{array}{c} a_2^0\\\\ B^g_2\n\\end{array}\\right),\n\\label{20}\n\\end{equation}\nwhere for simplicity we have dropped the superscript $P$. The solution for $a_2^0$ reads\n\\begin{eqnarray}\na_2^0(\\mu^2) & = & \\left[ \\left(\\frac{1}{2} -\n  \\frac{49}{2\\sqrt{2761}}\\right) L^{\\gamma_2^+\/(2\\beta_0)} + \n\\left(\\frac{1}{2} +\n  \\frac{49}{2\\sqrt{2761}}\\right) L^{\\gamma_2^-\/(2\\beta_0)}\\right]\n  a_2^0(\\mu_0^2) \\nonumber\\\\\n&&{}+ \\frac{5}{9\\sqrt{2761}}\\left[L^{\\gamma_2^-\/(2\\beta_0)}-\nL^{\\gamma_2^+\/(2\\beta_0)}\\right] B_2^g(\\mu_0^2)\n\\label{21}\n\\end{eqnarray}\nwith the anomalous dimensions $\\gamma_2^\\pm = (149\\pm \\sqrt{2761})\/9$. The octet Gegenbauer moment does not have another DA with which it can mix and so its evolution is simpler\n\\begin{equation}\na_2^8(\\mu^2) = L^{50\/(9\\beta_0)} a_2^8(\\mu_0^2)\\,.\n\\label{22}\n\\end{equation}\nThe mixing amongst the DAs complicates matters; as the scale dependence of the decay constants is lost in the QF scheme, one expects to have to lose scale dependence in the DAs too, and we must be careful to be consistent. The verification of the anomalous dimensions in Eq.~(\\ref{20}) from the singlet and octet parts of the form factor calculations is a crucial test of the LCSR analysis. For this reason, we discuss the implications of mixing on the twist-2 DA parameters, and only briefly cover higher-twist quark DAs which are included in the octet part; for a detailed discussion one is referred to Ref.~\\cite{Ball:2007hb}. Following Ref.~\\cite{Kroll:2002nt}, for the DAs introduced by Eq.~(\\ref{expansion}) we have, in terms of the quark valence Fock states $\\ket{q\\bar q}$ and $\\ket{s\\bar s}$\n\\begin{equation}\n\\ket{\\eta_q} \\sim \\phi_2^q (u) \\ket{q\\bar q} + \\phi_2^{\\rm\n OZI}(u) \\ket{s\\bar s}\\,,\\quad\n\\ket{\\eta_s} \\sim  \\phi_2^{\\rm OZI} (u) \\ket{q\\bar q} + \n\\phi_2^s(u) \\ket{s\\bar s}\\,,\n\\end{equation}\nwhere $q\\bar q$ is shorthand for $(u\\bar u + d\\bar d)\/\\sqrt{2}$ and \n\\begin{equation}\n\\phi_2^q = \\frac{1}{3}\\,(\\phi_2^8 + 2\\phi_2^0)\\,,\\quad\n\\phi_2^s = \\frac{1}{3}\\,(2\\phi_2^8 + \\phi_2^0)\\,,\\quad\n\\phi_2^{\\rm OZI} = \\frac{\\sqrt{2}}{3} (\\phi_2^0-\\phi_2^8)\\,.\n\\label{darel}\n\\end{equation}\nIn the QF scheme, the ``wrong-flavour'' DA  $\\phi_2^{\\rm OZI}$, which is generated by OZI-violating interactions,  is set to 0. Once this is done at a certain scale, however, the different evolution of $a_n^0$ and $a_n^8$  will generate a non-zero $\\phi_2^{\\rm OZI}$ already to LO accuracy. A consistent implementation of the QF scheme hence requires one to either set $a_n^{0,8}\\equiv 0$ and also $B^g_n\\equiv 0$, or to set $a_n^8\\equiv a_n^0$ and neglect the different \nscale-dependence of these parameters. The induced non-zero DA $\\phi_2^{\\rm OZI}$ is numerically very small for the scales relevant for our calculation, $\\mu=1\\,$GeV and $2.4\\,$GeV.\\footnote{$2.4\\,$GeV is  a typical scale in the calculation of form factors from LCSRs: $\\mu=\\sqrt{m_B^2-m_b^2}$ is chosen as an intermediate scale between $m_b$ and the typical hadronic scale $1\\,{\\rm GeV}$.} The left panel of Fig.~\\ref{eta_fig2} shows a plot of $\\Delta=100\\,| (a^0_2(\\mu)-a^8_2(\\mu))\/a^0_2(\\mu)|$ as a function of scale $\\mu$, according to Eqs.~(\\ref{21}) and (\\ref{22}),  for $a_2^8(1\\,{\\rm GeV})\\equiv a_2^0(1\\,{\\rm GeV})$ and $B_2^g=0$.  We see that $\\Delta$ is less than $0.25\\,\\%$ over the range $1\\,\\textrm{GeV}<\\mu<2.4\\,\\textrm{GeV}$.  Choosing $a_2^8(1\\,{\\rm GeV})=0.25\\pm0.15$, guided by our knowledge of twist-2 DAs of the $\\pi$; we have $a_2^8(2.4\\,{\\rm   GeV}) = 0.171$ from Eq.~(\\ref{22}), and $a_2^0(2.4\\,{\\rm GeV}) = 0.171$ for $B^g_2=0$, from Eq.~(\\ref{21}). Evidently, the impact of the different anomalous dimensions of $a_2^0$ and $a_2^8$ is negligible.\n\\begin{figure}[h]\n$$\\epsfxsize=0.45\\textwidth\\epsffile{scaling1.eps}\\qquad \\qquad\\epsfxsize=0.45\\textwidth\\epsffile{scaling2.eps}$$\n\\caption[Scale dependence of the twist-2 distribution amplitude parameters.]{\\small  Left: $\\Delta=100\\,| (a^0_2(\\mu)-a^8_2(\\mu))\/a^0_2(\\mu)|$ as a function of scale $\\mu$, according to Eqs.~(\\ref{21}) and (\\ref{22}) with $B_2^g=0$. Right: dependence of $a^0_2(2.4\\,{\\rm GeV})$ on $B^g_2(1\\,{\\rm GeV})$ for  $a^0_2(1\\,{\\rm GeV})=0.25$ according to Eq.\\,(\\ref{21})}\n\\label{eta_fig2}\n\\end{figure}\nAlso, the evolution of $a_2^0$ is not hugely different to that of $a_2^8$, for a wide range of values of $B^2_g$.  The right panel of Fig.~\\ref{eta_fig2} shows the evolution of the singlet Gegenbauer moment $a_2^0$ from $\\mu=1\\,{\\rm GeV}$ - $2.4\\, {\\rm GeV}$, from  Eq.~(\\ref{21}), for the range of gluon Gegenbauer moments $|B_2^g(1\\,{\\rm  GeV})|<20$, which is a \\textit{very} conservative estimated range, as discussed below. The mixing of $B_2^g$ into $a_2^0$ is up to $20\\%$ for $B_2^g=20$ and $40\\%$ for $B_2^g=-20$.\n\nFrom the conclusions of the above discussion we are justified in implementing the QF scheme for DAs as follows: we set $\\phi_2^0\\equiv \\phi_2^8$ at the scale $\\mu=1\\,$GeV, which, by virtue of Eq.~(\\ref{darel}), implies $\\phi_2^q\\equiv\\phi_2^s$ at the same scale. We then evolve $a_2$ according to the scaling-law for the octet Gegenbauer moment (\\ref{22}).\\footnote{This is equivalent to imposing the QF-scheme  relation $a_2^0=a_2^8$ as the scale $\\mu=2.4\\,$GeV and defining $B^g_2$ as $B^g_2(2.4\\,{\\rm GeV})$.}  We also set $\\psi_{2;\\eta}^g=\\psi_{2;\\eta^{\\prime}}^g$; again any SU(3)$_{\\rm F}$-breaking of this relation is expected to have only very small impact on $f_+^{B\\to\\eta^{(\\prime)}}$. The twist-2 parameters used in our calculation are then reduced to two: $a_2$ and $B^g_2$.  \n\nConcerning numerical values, we assume that the bulk of  SU(3)$_{\\rm F}$-breaking effects is described by the decay constants via $f_q\\neq f_\\pi$, and that SU(3)$_{\\rm F}$-breaking in Gegenbauer moments is sub-leading \\cite{Ball:2006wn}. Sum rules for $a_2^\\pi$ and $a_2^q$ would essentially be the same, with $f_\\pi\\neq f_q$ driving the SU(3)$_{\\rm F}$-breaking and any small differences in $s_0$ and $M^2$ being negligible. This motivates setting $a_2^q = a_2^\\pi$,  with $a_2^\\pi(1\\,{\\rm GeV}) = 0.25\\pm 0.15$ as an average over a large number of  calculations and fits to experimental data \\cite{Ball:2006wn}. \n\nFor $B_2^g$, however, no direct calculation is available.  Results from fits to data have been obtained from the $\\eta^{\\prime}\\gamma$ transition form factor, yielding $B_2^g(1\\,{\\rm GeV}) = 9\\pm 12$ \\cite{Kroll:2002nt}, and the combined analysis of this form factor and the inclusive decay $\\Upsilon(1S)\\to \\eta^{\\prime} X$ yielding $B_2^g(1.4\\,{\\rm GeV}) = 4.6\\pm 2.5$ \\cite{AP03}.  Caution must be taken when considering these results as they are highly correlated with the simultaneous determination of $a_2^0$ and $a_2^8$ from the same data, yielding $a_2^0(1\\,{\\rm GeV}) = -0.08\\pm 0.04$, $a_2^8(1\\,{\\rm GeV}) =\n-0.04\\pm 0.04$ and $a_2^0(1.4\\,{\\rm GeV}) = a_2^8(1.4\\,{\\rm GeV}) = -0.054\\pm 0.029$, respectively. The same analysis, applied to the $\\pi\\gamma$ form factor, returns $a_2^\\pi (1\\,{\\rm GeV}) =-0.06\\pm 0.03$ \\cite{vogt}. These results are not really compatible with those from the direct calculation of $a_2^\\pi$ from Lattice QCD and QCD sum rules; in particular the sign of $a_2^\\pi$ is unambiguously fixed as being positive. A possible reason for this discrepancy is the neglection of higher-order terms in the light-cone expansion and that, in addition, as one of the photons in the process is nearly real with virtuality $q^2\\approx 0$, one also has to take into account long-distance photon interactions, of order $1 \\sqrt{q^2}$, as discussed in Ref.~\\cite{rady}. For this reason, we assume the very conservative range $B_2^g(2.4\\,{\\rm GeV}) = 0\\pm20$ in the analysis.\n\nAs far as higher-twist quark DAs are concerned, we only need those involving currents with flavour content $\\bar q q = (\\bar u u + \\bar d d)\/\\sqrt{2}$. In line with the implementation of the QF scheme\nfor twist-2 DAs, we include SU(3)$_{\\rm F}$-breaking only via the decay constants. The precise definitions of all twist-3 and 4 DAs, as well as up-to-date numerical values of the $\\pi$'s hadronic parameters can be found in Ref.~\\cite{Ball:2006wn}. A discussion of the correct treatment of these DAs\nwithin LCSR, as modified to describe $\\eta^{(\\prime)}$, can be found in Ref.~\\cite{Ball:2007hb}. \n\n\\section{Calculation}\nWe define the $B\\to P$ form factors analogously to those of other pseudoscalar mesons as  \\cite{Ball:2004ye}\n\\begin{equation}\n\\bra{P(p)} \\bar u \\gamma_\\mu b \\ket{B(p+q)} = \\left\\{\n(2p+q)_\\mu - \\frac{m_B^2-m_P^2}{q^2}\\,q_\\mu\\right\\} \\frac{f_+^P(q^2)}{\\sqrt{2}} + \\frac{m_B^2-m_P^2}{q^2}\\,q_\\mu\\,\\frac{f_0^P(q^2)}{\\sqrt{2}}\\,.\n\\label{FF}\n\\end{equation}\nwhere the factor of $1\/\\sqrt{2}$ on the right-hand side is to ensure that in the SU(3)$_{\\rm F}$ symmetry limit, without $\\eta$-$\\eta^{\\prime}$ mixing, $f_+^\\eta = f_+^\\pi$. For semileptonic decays $B\\to \\eta^{(\\prime)} l \\nu_l$ the form factor $f_0^P$ appears proportional to $q^2\\approx m_l^2$ which is negligible for light leptons $l=\\{e,\\mu\\}$ for which only $f_+^P$ is required.  Using the LCSR method outlined in Chapter~\\ref{chapter3_SR} we extract the semileptonic form factor $f_{+}^P$  from the following correlation function\n\\begin{eqnarray}\n\\Pi^P_{\\mu}(p,q) &=& i\\int d^4x\\,e^{i q\\cdot x} \\bra{P(p)}  T [\\bar u \\gamma_\\mu b](x) j_B^{\\dagger}(0)\\ket{0}\\\\\n&=& \\Pi_+^P(q^2,p_B^2) (2p+q)_\\mu + \\dots\\,,\\nonumber\n\\label{eq:corr}\n\\end{eqnarray}\nwhere $j_B= m_b \\bar u i\\gamma_5 b$ is the interpolating current for the $B$ meson and $p_B^2=(p+q)^2$ its virtuality. In calculating the correlation function, we use Eq.~(\\ref{8}) which relates the physical states $\\ket{\\eta^{(\\prime)}}$ and the QF basis states  $\\ket{\\eta_{q,s}}$ so that\n\\begin{equation}\n\\Pi^\\eta_\\mu = \\frac{1}{\\sqrt{2}}\\left(\\Pi^q_{\\mu} \\cos\\phi - \\Pi^s_{\\mu} \\sin\\phi\\right),\\quad\n\\Pi^{\\eta^{\\prime}}_\\mu = \\frac{1}{\\sqrt{2}}\\left(\\Pi^q_{\\mu} \\sin\\phi + \\Pi^s_{\\mu} \\cos\\phi\\right)\\,.\n\\label{32}\n\\end{equation}\nThe interpolating current $\\bar u \\gamma_\\mu b$ only probes the $\\bar u u $ quark component of the $\\eta^{(\\prime)}$ so $\\Pi^s_\\mu$ vanishes to leading order in $\\alpha_s$ and at $O(\\alpha_s)$ is due only to gluonic Fock states of the meson. $\\Pi^q_\\mu$, on the other hand, receives contributions from both quark and gluon states. The final LCSR for $f_+^P$ then reads\n\\begin{equation}\\label{35}\ne^{-m_B^2\/M^2}\\,m_B^2 f_B\\, \\frac{f_+^P(q^2)}{\\sqrt{2}} = \\int_{m_b^2}^{s_0}\nds\\,e^{-s\/M^2}\\, \\frac{1}{\\pi}\\,{\\rm Im}_s\\,\\Pi^P_+(s,q^2)\\,,\n\\end{equation}\nwith the usual sum rule specific parameters $M^2$, the Borel parameter, and $s_0$, the continuum threshold.\n\\subsection*{Quark Contribution}\nThe quark contributions follow from the studies already undertaken for the $\\pi$, for more details see Ref.~\\cite{Ball:2004ye}. We briefly cover the general features of the calculation to put the singlet contribution in context. The leading quark contributions to $\\Pi_+^P$ originate from the diagrams of Fig.~\\ref{eta_fig1}, where first order $\\mathcal{O}(\\alpha_s)$ corrections are shown. The external quarks have momentum fractions $u p$ and $(1-u)p$ and are on-shell; $p^2=m_P^2$.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.85\\textwidth\\epsffile{eta_diags1.eps}$$\n\\caption[The quark contributions to $f_+^{\\eta^{(\\prime)}}(q^2)$ to $\\mathcal{O}(\\alpha_s)$.]{\\small The quark-antiquark contributions to the semileptonic $B\\to\\eta^{(\\prime)}$ form factors $f_+^{\\eta^{(\\prime)}}(q^2)$ from light-cone sum rules. The top left diagram is the leading one, the others are $\\mathcal{O}(\\alpha_s)$. The double line corresponds to the $b$ quark and the dashed lines the injection of the weak vertex momentum $q$, and the momentum of the $B$ meson $p_B$.} \n\\label{eta_fig1}\n\\end{figure}\nThe two-particle DAs are projected out by using the general spinor decomposition of quark fields\n\\begin{eqnarray}\n\\bar{q}_a q_b^\\prime &=& \\frac{1}{4} (\\textbf{1})_{ba}(\\bar q q^\\prime)-\\frac{1}{4} (i \\gamma_5)_{ba}(\\bar q i \\gamma_5  q^\\prime)+\\frac{1}{4} (\\gamma_\\mu)_{ba}(\\bar q \\gamma^\\mu q^\\prime)-\\frac{1}{4} (\\gamma_\\mu \\gamma_5)_{ba}(\\bar q\\gamma^\\mu \\gamma_5  q^\\prime)\\nonumber\\\\\n&-&\\frac{1}{8} (i \\sigma_{\\mu\\nu}\\gamma_5)_{ba}(\\bar q i \\sigma^{\\mu\\nu} \\gamma_5 q^\\prime)\\,.\n\\end{eqnarray}\nThe vacuum-meson matrix elements of each term above either vanish or yield a DA depending on the quantum numbers of the meson in question. For pseudoscalar mesons the leading-twist contribution comes from $\\gamma_\\mu\\gamma_5$, whereas $i\\gamma_5$ and $i\\sigma_{\\mu\\nu}\\gamma_5$ give two-particle twist-3 contributions, and although the two-particle twist-3 contributions appear in the sum rules as formally $1\/m_b$, they are \\textit{chirally enhanced} by numerically large factors \\cite{Ball:1998je} and so are included in typical LCSR analyses \\cite{Ball:2004ye}. Three-particle twist-3 and two- and three-particle twist-4 DAs are also included; all twist-2 and -3 contributions include $O(\\alpha_s)$ corrections twist-4 contributions are to tree level accuracy. The corresponding expressions yield $\\Pi^q_+$, with the replacement $f_\\pi\\to f_q$.  \n\n\\subsection*{Gluonic Contribution}\nIn order to obtain the gluonic contribution to $\\Pi_+^P$, one needs to calculate the diagrams shown in Fig.~\\ref{eta_diags}. The last diagram is divergent and the other two are finite.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.85\\textwidth\\epsffile{eta.eps}$$\n\\caption[The leading diagrams for the flavour-singlet contribution to $f_+^{\\eta^{(\\prime)}}(q^2)$.]{\\small The leading diagrams for the flavour-singlet contribution to the semileptonic $B\\to\\eta^{(\\prime)}$ form factors  from light-cone sum rules. The double line corresponds to the $b$ quark. The dashed lines the injection of weak vertex momentum $q$, and momentum of the $B$ meson  interpolating current $p_B$.} \n\\label{eta_diags}\n\\end{figure}\nThe gluon fields are introduced in the standard way\n\\begin{eqnarray}\n\\left.\\Pi^{P}_{\\mu}\\right|_{\\textrm{gluon}}=i\\int d^4[x,w,y] \\, e^{i q\\cdot x} \\bra{P(p)} T [\\bar{u}\\gamma_{\\mu} b](x) [m_b \\bar{b} i\\gamma_5 u](0) \\, S \\mathcal{L}^{q_1}_{g}(w)\\mathcal{L}^{q_2}_{g}(y)\\ket{0}\\,,\\nonumber\n\\end{eqnarray}\nwith the usual interaction Lagrangian $\\mathcal{L}^{q_{i}}_{g}(x) = ig_{s} [\\bar{q_i} \\gamma^{\\alpha} A_{\\alpha}^a t^a q_i](x)$ with $q_{i} = \\{u,b\\}$ and the statistical factor $S$ takes values $1$ if $q_1\\neq q_2$ and  $1\/2$ if $q_1=q_2$. The integral is over each co-ordinate separately. To extract the gluon contribution we need the projection onto the twist-2 two-gluon DA, which can be read off Eq.~(\\ref{gluefields}), which amounts to the following replacement of the gluon fields (up to the numerical factor)\n\\begin{equation}\nA_{\\alpha}^a(w) A_{\\beta}^b(y) \\stackrel{\\textrm{twist-2}}{\\longrightarrow}\\delta^{ab} \\epsilon_{\\alpha \\beta \\rho\\sigma} \\,\\frac{\\tilde{z}^\\rho p^\\sigma }{p \\cdot \\tilde{z}}\\,\\,\\int^1_0 du\\,\\frac{\\psi^g_{2;P}(u)}{u \\bar{u}} \\,e^{i p \\cdot (u w +  \\bar{u} y)}\\,,\n\\end{equation}\nwhere the separation $\\tilde{z}$ is light-like i.e. $\\tilde{z}^2 = (w-y)^2=0$. Via partial integration we can simplify the resulting expression for $\\left.\\Pi^{P}_{+}\\right|_{\\textrm{gluon}}$; the co-ordinate $\\tilde{z}$ is traded for a derivative of the hard scattering kernel with respect to the momentum of one of the emitted gluons; and the dot product $1\/ (p\\cdot \\tilde{z})$ can be traded for an integral with respect to the DA momentum fraction. As the boundary terms vanish due to the leading-twist gluon DA being antisymmetric, the calculation takes a rather simple form:\n\\begin{equation}\n\\left.\\Pi^{P}_{+}\\right|_{\\textrm{gluon}}=\\left.\\int^1_0 du  \\, \\left[\\frac{\\partial \\,T_{\\mu }^\\rho(u p) }{\\partial(u p)^\\rho}\\right] \\int^u_0 dv \\,\\frac{\\psi_{2;P}^g(v)}{v\\bar{v}}\\right|_{p_\\mu \\to \\frac{1}{2},\\,q_\\mu \\to 0}\\,,\n\\end{equation}\nwhere $T_{\\mu }^\\rho(u p) $ is the hard scattering kernel.  Both the gluonic and quark contributions are renormalisation scale dependent. The relevant term concerning the quark Gegenbauer moment $a_2$ is\n\\begin{equation}\n\\Pi^q_+ \\sim 18 f_q  a_2 \\left( 1 + \\frac{\\alpha_s}{4\\pi}\n\\,\\frac{50}{9}\\,\\ln\\,\\frac{\\mu^2}{m_b^2}\\right)F(p_B^2,q^2)\\,,\n\\label{33}\n\\end{equation}\nwhere $F(p_B^2,q^2)$ is a function of $p_B^2$ and $q^2$. The logarithmic terms in the convolution of the gluonic diagrams of Fig.~\\ref{eta_diags} with $\\psi_{2;P}^g$ are\n\\begin{equation}\n\\Pi^P_+ \\sim -\\frac{10}{9\\sqrt{3}}\\,\\frac{\\alpha_s}{4\\pi}\\, B_2^g f^0_P\n\\ln\\,\\frac{\\mu^2}{m_b^2}\\, F(p_B^2,q^2) \\,.\n\\end{equation}\nBy expressing $f_q$ via Eq.~(\\ref{11}) in terms of $f^0_\\eta$ and $f^0_{\\eta^{\\prime}}$, respectively, and inserting Eq.~(\\ref{33}) into Eq.~(\\ref{32}), one verifies that the renormalisation-group equation, Eq.~(\\ref{20}), is fulfilled.  The twist-2 two-gluon contribution to the correlation functions $\\Pi_+^P$, Eq.~(\\ref{32}), is given in terms of a spectral density as\n\\begin{equation}\n\\left.\\Pi_{+}^{P}\\right|_{\\textrm{gluon}} = \\int_{m_b^2}^\\infty ds\\,\\frac{\\rho^P_{\\textrm{gluon}}(s)}{s-p_B^2}\n\\end{equation}\nwith the result being\n\\begin{eqnarray}\n\\rho^P_{\\textrm{gluon}}(s) \n& = & B_2^g \\alpha_s f_0^P m_b\\, \\frac{5}{36\\sqrt{3}}\\, \\frac{m_b^2-s}{(s-q^2)^5} \\, \\left\\{ 59 m_b^6 + 21 q^6 - 63 q^4 s - 19 q^2 s^2 + 2 s^3\\right. \\nonumber\\\\\n&& \\hspace*{3cm}\\left. + \\,m_b^2 s (164 q^2 + 13 s) - m_b^4 (82 q^2 + 95s)\\right\\}\n\\nonumber\\\\\n&& {} + B_2^g \\alpha_s f_0^P m_b\\, \\frac{5}{6 \\sqrt{3}}\\, \\frac{(m_b^2-q^2)(s-m_b^2)}{(s-q^2)^5} \\,\\{ 5 m_b^4 + q^4 + 3 q^2 s + s^2 - 5m_b^2 (q^2+s)\\} \\nonumber\\\\\n&& \\hspace*{3cm} \\times\\left\\{ 2 \\ln\\,\\frac{s-m_b^2}{m_b^2} - \\ln\\,\\frac{\\mu^2}{m_b^2} \\right\\}.\n\\end{eqnarray}\n\n\\section{Discussion}\nFor the evaluation of the LCSR, Eq.~(\\ref{35}), as with any sum rule, optimum values of  $M^2$ and $s_0$ need to be found. The standard procedure \\cite{Ball:2004ye} is to replace $f_B$ by its sum rule, derived via SVZ sum rules, thus reducing the dependence of the LCSR on $m_b$ for which we use the one-loop pole mass $m_b=4.80\\pm0.05\\,\\rm{GeV}$ \\cite{Colangelo:2000dp}. From the $f_B$ sum rule the optimum threshold parameter  $s_0=34.2 \\pm 0.7\\,{\\rm GeV}^2$ is found, and this value is taken over to the LCSR. As mentioned before $\\mu=2.4\\,{\\rm GeV}$ is chosen as an intermediate scale between $m_b$ and $1\\,{\\rm GeV}$. The Borel parameter is taken to be $M^2>6\\,{\\rm GeV}^2$ and is varied in the range $6\\,{\\rm GeV}^2<M^2<14\\,{\\rm GeV}^2$ to reflect the corresponding uncertainty. In Fig.~\\ref{eta_diags3} we plot $f^{B\\to \\eta}_+(0)$ and $f^{B\\to \\eta^{\\prime}}_+(0)$ respectively as functions of $M^2$, making explicit the result of varying $s_0$ by $\\pm0.7\\,{\\rm GeV}^2$, $a_2$ by $\\pm0.15$ and $B_2^g$ by $\\pm10$.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.45\\textwidth\\epsffile{eta_fig4a.eps}\\qquad\\qquad \\epsfxsize=0.45\\textwidth\\epsffile{eta_fig4b.eps}$$\n\\caption[$f_+^\\eta(0)$  and $f_+^{\\eta^{\\prime}}(0)$ as a\n  function of the Borel parameter $M^2$.]{\\small $f_+^\\eta(0)$ (left) and $f_+^{\\eta^{\\prime}}(0)$ (right) as a\n  function of the Borel parameter $M^2$ and various choices of input\n  parameters. Solid curves: central values of input parameters and\n  $s_0=34.2\\,{\\rm GeV}^2$. Long-dashed curves: $s_0$ varied by\n  $\\pm 0.7\\,{\\rm GeV}^2$. Short-dashed curves: $a_2(1\\,{\\rm\n  GeV})$ varied by $\\pm 0.15$. Dash-dotted curves: $B_2^g$ varied\n  by $\\pm 10$.} \n\\label{eta_diags3}\n\\end{figure}\nAs expected, $f_+^\\eta(0)$ is not very sensitive to the gluonic twist-2 DA parameter $B^g_2$ (dashed-dotted curves), but is quite sensitive to the Gegenbauer moment $a_2$ (short-dashed\ncurves). For  $f_+^{\\eta^{\\prime}}(0)$, on the other hand, the dependence on $B^g_2$ is more pronounced than that of $a_2$. Varying all relevant parameters within their respective ranges, i.e.\\ $\\Delta m_b = \\pm\n0.05\\,{\\rm GeV}$, $\\Delta a_2(1\\,{\\rm GeV})=\\pm0.15$ and $\\Delta B^g_2=\\pm 20$, as well as all twist-3 and twist-4 parameters within the ranges given in Ref.~\\cite{Ball:2006wn}, we find\n\\begin{eqnarray}\\label{fp1}\nf_+^\\eta(0) & = & 0.229\\pm \\overbrace{0.005}^{M^2}\\pm \\overbrace{0.006}^{s_0}\\pm\n\\overbrace{0.016}^{a_2^\\eta}\\pm \\overbrace{0.007}^{B^g_2}\\pm \\overbrace{0.005}^{f_q,\\phi}\\pm \\overbrace{0.011}^{{\\rm T3}}\\pm \\overbrace{0.001}^{{\\rm T4}} \\pm \\overbrace{0.007}^{f_B,m_b}\\nonumber\\\\\n& = &\n0.229\\pm \\underbrace{0.024}_{{\\rm param.}}\\pm \\underbrace{0.011}_{{\\rm syst.}}\\,,\\\\\nf_+^{\\eta^{\\prime}}(0) & = & 0.188\\pm \\overbrace{0.004}^{M^2}\\pm \\overbrace{0.005}^{s_0}\\pm\n\\overbrace{0.013}^{a_2^{\\eta^{\\prime}}}\\pm \\overbrace{0.043}^{B^g_2}\\pm \\overbrace{0.005}^{f_q,\\phi}\n\\pm \\overbrace{0.009}^{{\\rm T3}}\\pm \\overbrace{0.005}^{{\\rm T4}} \\pm \\overbrace{0.006}^{f_B,m_b}\\nonumber\\\\\n& =&\n0.188\\pm \\underbrace{0.043}_{B^g_2} \\pm \\underbrace{0.019}_{{\\rm param.}}\\pm \\underbrace{0.009}_{{\\rm syst.}}\\,.\n\\label{fp0}\n\\end{eqnarray}\nThe entry labelled ``T4'' also contains an estimate of the possible impact\nof the local twist-4 two-gluon matrix elements (\\ref{extra}). For\nthis estimate, we exploit the fact that the asymptotic DA of the\nnon-local generalisation of Eq.~(\\ref{extra}) is the same as for the\ntwist-2 two-quark DA: $6 u (1-u)$.\\footnote{This follows from Eq.~(\\ref{das_eq18}). For\n$G_{\\perp\\perp}$, one has $l=2$ and $s=0$.} We then\nassume that the corresponding correlation function is the same as that\nfor the leading conformal wave in the two-quark twist-2 contribution,\ni.e.\\ the coefficient in the Gegenbauer moment $a_0=1$, and replace\n$a_0$ by $\\bra{0} \\alpha_s\nG\\tilde{G}\/(4\\pi)\\ket{\\eta_{q,s}}\/(f_{q,s}m_b^2)$. The factor\n$1\/m_b^2$ comes from the fact that this is a twist-4 effect and hence\nsuppressed by two powers of $m_b$ with respect to the twist-2\ncontribution. This is only a\nrough estimate, of course, as the true spectral density will be\ndifferent. The results (\\ref{fp0}) show that for small $B^g_2\\approx 2$ both twist-2 and -4\ntwo-gluon effects can indeed be of similar size. In this case,\nhowever, the total flavour-singlet contribution to $f_+^{\\eta^{\\prime}}$ will\nalso be small, $\\sim 0.008$.\nIn the third lines, we have added all uncertainties from the input\nparameters (param.) in quadrature and the sum-rule specific\nuncertainties from $M^2$ and $s_0$ (syst.) linearly. For\n$f_+^{\\eta^{\\prime}}(0)$, we have displayed the dependence on $B^g_2$\nseparately. The new result for $f_+^\\eta(0)$ is, within errors, in agreement with\nthe previous one from LCSR, $f_+^\\eta(0) = 0.275\\pm 0.036$, obtained in\nRef.~\\cite{Ball:2004ye}. That for $f_+^{\\eta^{\\prime}}(0)$ is new to the present analysis.\nThe results agree well with those obtained in Ref.~\\cite{Charng:2006zj},\nfrom pQCD,\n$f_+^\\eta(0) = 0.208$ and $f_+^{\\eta^{\\prime}}(0) = 0.171$, including a\nrescaling by a factor $\\sqrt{2}$ to bring their definition of the form\nfactors into agreement with Eq.~(\\ref{FF}). We confirm the finding of\nRef.~\\cite{Charng:2006zj} that the range of the singlet contribution to the form\nfactor estimated in Ref.~\\cite{Beneke:2002jn} is likely to be too large, unless\n$B_2^g$ assumes extreme values $\\sim 40$.\n\\begin{figure}[h]\n$$ \\epsfxsize=0.55\\textwidth\\epsffile{eta_fig5.eps}$$\n\\caption[$f_+^{\\eta^{\\prime}}(0)\/f_+^\\eta(0)$ as a\n  function of the Borel parameter $M^2$.]{\\small $f_+^{\\eta^{\\prime}}(0)\/f_+^\\eta(0)$ as a\n  function of the Borel parameter $M^2$ and various choices of input\n  parameters. Solid line: central values of input parameters,\n  which corresponds to $f_+^{\\eta^{\\prime}}(0)\/f_+^\\eta(0)\\equiv \\tan\\phi =\n  0.814$. Dot-dashed curves: $B_2^g$ varied\n  by $\\pm 10$. Dotted curves: $a_2^{\\eta,\\eta^{\\prime}}(1\\,{\\rm GeV})$\n  varied independently: $a_2^\\eta=0.1$, $a_2^{\\eta^{\\prime}}=0.4$ and\n  $a_2^{\\eta^{\\prime}}=0.4$, $a_2^\\eta=0.1$.} \n\\label{eta_fig5}\n\\end{figure}\n\nIn Fig.~\\ref{eta_fig5} we plot the ratio $f^{\\eta^{\\prime}}_+(0)\/f^{\\eta}_+(0)$\nas a function of the Borel parameter. In the ratio, many uncertainties\ncancel, in particular that on $f_B$. As we have chosen $B_2^g=0$ as\ncentral value, $f^{\\eta^{\\prime}}_+(0)\/f^{\\eta}_+(0)\\equiv \\tan\\phi = 0.814$ exactly,\nsee Eq.~(\\ref{32}). The figure also illustrates the change of the\nresult upon inclusion of a non-zero $B^g_2$ (dot-dashed curves). The ratio is actually rather sensitive to that parameter. While the dependence on $a_2$ largely cancels when $a_2^\\eta$ and\n$a_2^{\\eta^{\\prime}}$ are set equal, there is a considerable residual dependence on\n$a_2^\\eta - a_2^{\\eta^{\\prime}}\\neq 0$ (dotted curves). While\n$|a_2^\\eta - a_2^{\\eta^{\\prime}}|=0.3$ as illustrated by these curves is\nrather unlikely, and would signal very large OZI-breaking\ncontributions (recall that $a_2^\\eta\\neq a_2^{\\eta^{\\prime}}$ or,\nequivalently, $a_2^1\\neq a_2^8$ signals the presence of\n``wrong-flavour'' contributions to the $\\eta_{q,s}$ DAs and is set to 0 in\nthe QF mixing scheme), one should nonetheless keep in mind that moderate\ncorrections of this type are not excluded and compete with the\nOZI-allowed corrections in $B^g_2$.\n\\begin{figure}\n$$\\epsfxsize=0.55\\textwidth\\epsffile{eta_fig8.eps}$$\n\\caption[${\\cal    B}(B\\to\\eta^{\\prime} e\\nu)\/{\\cal B}(B\\to\\eta e\\nu)$ as a function of the    singlet-parameter $B_2^g$.]{\\small The ratio of branching ratios    $R_{\\eta\\eta^{\\prime}} = {\\cal    B}(B\\to\\eta^{\\prime} e\\nu)\/{\\cal B}(B\\to\\eta e\\nu)$ as a function of the    singlet-parameter $B_2^g$. Solid curve: central values of    input parameters; Long-dashed curves: including variation of extrapolation model parameters, See Ref.~\\cite{Ball:2007hb}. Short-dashed curves: theoretical uncertainty of $R_{\\eta\\eta^{\\prime}}$ for $B^g_2=0$, for $a_2^{\\eta,\\eta^{\\prime}}(1\\,{\\rm GeV})$ varied independently, as in Fig.~\\ref{eta_fig5}.\n    }\\label{eta_fig8}\n\\end{figure}\n\nFinally, in Fig.~\\ref{eta_fig8} we show the dependence of the ratio of\nbranching ratios $R_{\\eta\\eta^{\\prime}} = {\\cal\n    B}(B\\to\\eta^{\\prime} e\\nu)\/{\\cal B}(B\\to\\eta e\\nu)$ on\n$B_2^g$. The advantage of this observable is that all hadronic effects\nare encoded in the form factors and that $|V_{ub}|$ cancels.\nThe solid curve corresponds to the branching ratios obtained\nfrom the central values of input parameters; the long-dashed curves illustrate the dependence on parameters originating from the model  used to extrapolate the $q^2$ dependence of the form factor from beyond the limit of the LCSR approach, in this case $q^2=16\\,{\\rm GeV}^2$, to the maximum possible value $q_{\\rm max}^2=(m_B-m_{\\eta^{(\\prime)}})^2$. It may be noted that the dependence on these parameters is very small. We do not go into detail about the extrapolation procedure and refer the reader to Ref.~\\cite{Ball:2007hb}.  On the other hand, $R_{\\eta\\eta^{\\prime}}$ also depends on $a_2^\\eta\\neq\na_2^{\\eta^{\\prime}}$. This dependence is shown by the short-dashed\ncurves. The conclusion is that large values of $B^g_2$, $|B^g_2|>5$,\ncan be distinguished from the OZI-breaking parameter\n$|a_2^\\eta-a_2^{\\eta^{\\prime}}|$, once an accurate experimental value of\n$R_{\\eta\\eta^{\\prime}}$ is available, but that for smallish $B^g_2$ and\nunknown $|a_2^\\eta-a_2^{\\eta^{\\prime}}|$ only mutual constraints on these\nparameters can be extracted from the data. In this case  also twist-4 gluonic DAs can become important. \n\n\n\n\n\\chapter{QCD Factorisation}\\label{chapter6_QCDF}\nIn this chapter we discuss the framework of QCD factorisation which was introduced  in the context of exclusive two-body non-leptonic $B$ decays by Beneke, Buchalla, Neubert and Sachrajda in Refs.~\\cite{Beneke:1999br, Beneke:2000ry}. We shall refer to the the original implementation of the framework as the BBNS approach.  We also focus on its application to the radiative $B$ decays $B \\to V \\gamma$, as presented by Bosch and Buchalla in Refs.~\\cite{Bosch:2001gv,Bosch:2002bw}.  \n\nQCD factorisation allows a rigourous determination of the $B$ decay matrix elements of the weak effective Hamiltonian (\\ref{basics_eq20}) to leading order in the heavy-quark limit of QCD  $m_b\\gg \\Lambda_{\\rm QCD}$, and yields a neat factorisation formula. It relies on the factorisation of hadronic matrix elements into universal non-perturbative hadronic parameters, given by transition form factors and meson light-cone DAs,  and process dependent hard-scattering kernels, calculable in perturbation theory. The validity of the QCD factorisation formula, to all orders in $\\alpha_s$, and the impact of generally unknown power corrections, formally suppressed by powers of $1\/m_b$, must be addressed case by case. The introduction of the QCD factorisation framework has made more discerning phenomenological studies of exclusive $B$ decays possible whereby key observables, such as branching ratios, CP and isospin asymmetries, can be calculated and confronted with experimental data.\n\nThe dependence of the factorisation formula on meson DAs, either directly or via LCSR calculations of the transition form factors, greatly motivates their study, with their better determination reducing the theoretical uncertainty of the QCD factorisation predictions, and aiding the quest to discover new physics effects from decay observables. \n\nWe begin with a short introduction, in the context of $B\\to M_1 M_2$ decays, of the general features of QCD factorisation, and in particular, discuss the appearance of meson DAs. We then discuss the framework as applied to the radiative $B$ decays $B \\to V \\gamma$. We postpone all discussions of phenomenology to Chapter~\\ref{chapter7_rad} in which we perform an analysis of the decays $B_{u,d} \\to (\\rho, \\omega, K^*)\\gamma$ and $B_{s} \\to (\\bar{K}^*,\\phi)\\gamma$ using QCD factorisation, augmented by the inclusion of the dominant power-suppressed corrections.\n\n\\section{Introduction}\n\\textit{QCD factorisation} (QCDF) \\cite{Beneke:1999br, Beneke:2000ry} was introduced in the context of the ``heavy-to-light'' decays $B \\to \\pi\\pi$ where the factorisation of the relevant QCD matrix elements was shown to apply, to leading order in a $1\/m_b$ expansion, to a large class of non-leptonic $B$ decays.   Consequently, QCDF has opened up the rich and varied landscape of $B$ decays to a more complete quantitative analysis. The existence of factorisation in non-leptonic decays is non-trivial and complicated by the possible gluonic interactions amongst the initial and final states. Conversly, leptonic and semi-leptonic decays factorise much more easily into the product of a quark current and a leptonic current, which cannot interact via gluon exchange.\n\nPhenomenologically,  QCDF has been remarkably successful, especially given the range of processes for which the method holds. After its introduction, it was swiftly generalised to encompass $\\pi K$ final states \\cite{Beneke:2001ev}, pseudoscalar-vector final states \\cite{Beneke:2003zv} and vector-vector meson final states \\cite{Kagan:2004uw}. The gluonic flavour-singlet contributions to $B \\to K^{(*)} \\eta^{(\\prime)} $ decays were added by Ref.~\\cite{Beneke:2002jn}. To date, the framework has been extended to many other processes, including for example, (double) radiative $B$ decays $B \\to \\gamma (\\gamma, V)$ \\cite{Bosch:2002bv,Bosch:2002bw} and $B \\to \\gamma l \\nu$ \\cite{Descotes-Genon:2002mw}.  Also,  other factorisation frameworks have since been developed and applied to the same problems:\n\\begin{itemize}\n\\item{\\textit{Soft Collinear Effective Theory} (SCET) \\cite{SCET1, SCET2, SCET3, SCET4}  makes a careful distinction between a hierarchy of ``hard'' $(m_b)$, ``hard-collinear'' ($\\sqrt{\\Lambda_{\\rm QCD} m_b}$) and ``collinear'' ($\\Lambda_{\\rm QCD}$) scales via contributions of internal quark and gluon lines. Details of the differences between the SCET and BBNS approaches to QCD factorisation can be found in Refs.~\\cite{Bauer:2004tj,Beneke:2004bn,Bauer:2005wb}.}\n\\item{The \\textit{Perturbative QCD}  (pQCD) approach \\cite{PQCD}, which yields a factorisation formula that depends on the mesons' transverse momenta.}\n\\item{The method of LCSRs, although having existed before the advent of QCDF, was applied to $B \\to \\pi \\pi$, both to the matrix elements which exhibit factorisation and also a class of power corrections, providing some useful complementary insights, see Refs.~\\cite{Khodjamirian:2000mi, Khodjamirian:2005wn}.}\n\\end{itemize}\nWe now go on to discuss the general features of QCDF. \n\\section{General Structure}\nConsider the case of non-leptonic decays where the $B$ meson decays into two mesons. The simplest way of dealing with the resulting matrix elements is to employ \\textit{naive factorisation} \\cite{Fakirov:1977ta,Cabibbo:1977zv}. Simply put, naive factorisation splits each local operator $Q_i$ of the effective Hamiltonian into two colour-singlet currents, whose matrix elements are proportional to a decay constant and a transition form factor respectively. For example, consider the four-quark operator $Q_2^U =(\\bar{D} U)_{\\rm V-A}(\\bar U b)_{\\rm V-A}$ then\n \\begin{equation}\n\\bra{M_1 M_2}(\\bar{D} U)_{\\rm V-A}(\\bar U b)_{\\rm V-A}\\ket{B} \\,\\stackrel{\\rm{NF}}{\\longrightarrow}\\,\\underbrace{\\bra{M_2}(\\bar{D} U)_{\\rm V-A}\\ket{0}}_{f_{M_2}}\\underbrace{\\bra{M_1}(\\bar U b)_{\\rm V-A}\\ket{B}}_{F^{B \\to M_1}}\\,.\n\\label{qcdf_1}\n\\end{equation}\nThe motivation for factorising in this way comes from the \\textit{colour transparency} argument \\cite{Bjorken:1988kk}. It follows that a major shortcoming of naive factorisation is that it assumes the exchange of gluons of virtualites $\\mu \\lesssim m_b$ to be negligible and hence rescattering between the decay products is not considered; there is then no mechanism for the generation of strong phase effects between different amplitudes. Also, the matrix elements (\\ref{qcdf_1})  do not display the correct renormalisation-scale dependence.\n\nThe framework of QCDF allows the calculation of $\\mathcal{O}(\\alpha_s)$ corrections to naive factorisation, which occur at scales $\\mu\\lesssim m_b$. It is constructed by observing the cancelation of infrared (IR) and collinear divergences, via consistent power-counting arguments, allowing the use of perturbation theory to describe the hard-gluon exchanges. The resulting intuitive factorisation formula thus presents a massive simplification of the long-distance QCD effects, with QCDF recovering naive factorisation in the limit $m_b\\to \\infty$. In terms of two-body non-leptonic $B$ decays to light pseudoscalar mesons $B\\to M_1 M_2$ the factorisation formula, as presented in Ref.~\\cite{Beneke:1999br}, reads schematically as\n\\begin{eqnarray}\n\\bra{M_1 M_2} Q_i\\ket{B}&=& F^{B \\to M_1}\\int^1_0 du\\,T^{I}_i (u)\\, \\phi_{2;M_2}(u) + (M_1 \\leftrightarrow M_2)\\nonumber \\\\\n&+&\\int^1_0 d\\xi\\,du\\,dv\\,T^{II}_i (\\xi,u,v)\\,\\phi_B(\\xi) \\,\\phi_{2;M_1}(v)\\,\\phi_{2;M_2}(u) \\nonumber \\\\\n&+&\\mathcal{O}(\\Lambda_{\\rm QCD}\/m_b)\\,\n\\label{qcdf_2}\n\\end{eqnarray}\nwhere $F^{B \\to M_1}$ is the relevant form factor, $T^{I,II}_i$ are the hard-scattering kernels, $\\phi_{B}$ is one of the leading-twist DAs of the $B$ meson and $\\phi_{2;P}$ the leading-twist DA of the final state meson $P$, and the $Q_i$ are the operators of the effective Hamiltonian. The matrix elements are given as the convolution of the universal DAs and the process dependent hard-scattering kernels, with respect to the meson momentum fractions. Since the transition form factor and the DAs are real functions, all strong phases are generated by the hard-scattering kernels and are suppressed by powers of $\\alpha_s$. Factorisation has be proven to one-loop for ``light-light'' final states and two-loop for ``heavy-light'' final states \\cite{Beneke:2000ry}. It has be proven to all orders in $\\alpha_s$ for $B\\to D \\pi $ using SCET \\cite{SCET2}.\n\nThe ability of QCDF to accurately describe $B$ decay processes is limited by two main considerations; firstly, by the nature of the factorisation formula itself, which is valid up to power corrections $\\mathcal{O}(1\/m_b)$ and to a given order in $\\alpha_s$; and secondly by uncertainties of the necessary input parameters, such as the DAs, the transition form factors, the strange quark mass, the $B$ meson decay constant $f_B$ etc.  Whether a discrepancy between experiment and QCDF predictions can be put down to new physics, or not, requires an estimation of neglected power corrections; certainly the $b$ quark mass is not asymptotically large $m_b\\sim 5 \\,{\\rm GeV}$ and power corrections are therefore expected to feature at the level of $\\mathcal{O}(\\Lambda_{\\rm QCD}\/m_b) \\sim 10\\%$. The size and nature of power corrections can be probed via phenomenology, however, the task is not straight forward; even the initial focus of the approach, the decays $B \\to \\pi (K,\\pi)$, which stands as a crucial test, has not been resolved satisfactorily, see for example Ref.~\\cite{Feldmann:2004mg} and Refs.~\\cite{Fleischer:2005vz,Fleischer:2007wd}. Better determined input parameters will nevertheless shed light, case by case, on whether power corrections are important, and the QCDF predictions must be used to determine or constrain CKM matrix elements (UT angles), or detect signs of new physics, with that in mind.  \n\n\n\\section{Light-Cone Distribution Amplitudes}\nTo leading-order in the heavy-quark limit the leading-twist final state meson DAs contribute to the factorisation formula and can be safely truncated after the second Gegenbauer moment $a_2$. For pseudoscalar meson final states the two-particle twist-3 DAs come with large normalisation factors $r^P_\\chi$ and are said to be \\textit{chirally enhanced}, and are therefore included even though they are formally $1\/m_b$ suppressed. The vector mesons  do not have the same large normalisation factors but their two-particle twist-3 DAs are included in the BBNS approach for consistency. For a pseudoscalar  or vector meson, with valence quark content $\\bar q q^{\\prime}$, the normalisation factors are respectively\n\\begin{equation}\nr_\\chi^P(\\mu) = \\frac{2 m_P^2}{m_b(\\mu)(m_q+m_{q^\\prime})(\\mu)} \\sim\\frac{\\Lambda_{\\rm QCD}}{m_b}\\,,\\qquad r^V_\\chi(\\mu)=\\frac{2 m_V}{m_b(\\mu)}\\frac{f_V^\\perp(\\mu)}{f_V^\\parallel}\\,.\n\\label{qcdf_3}\n\\end{equation}\nThree-particle twist-3 DAs are neglected because they do not come with large normalisations.  The inclusion of the chirally enhanced DAs leads to end-point divergences from the convolutions of the two-particle twist-3 pseudoscalar DAs with the corresponding hard-scattering kernels originating from both the hard-spectator scattering and annihilation contributions. The resulting divergent integrals signal the breakdown of factorisation and are parameterised by two universal unknown parameters $X_{H,A}$, introducing a source of theoretical uncertainty to the BBNS approach \\cite{Beneke:1999br}. \n\nAt leading-twist the $B$ meson is described by two DAs, only one of which is required as input for Eq.~(\\ref{qcdf_2}) and appears in the hard-spectator diagrams contributing to $T^{II}_i$. The DAs of the $B$ mesons are complicated by the fact that the momentum of the meson is shared in a highly antisymmetric way: the $b$ quark has most of it. The $B$ meson DAs are given, at leading-order in $1\/m_b$, by\n\\begin{equation}\n\\bra{0}\\bar{q}_\\alpha(0) b_{\\beta}(z)\\ket{B(p_B)}=i \\frac{f_B}{4} \\left[(\\slash{p}_B+m_b) \\gamma^5\\right]_{\\beta\\gamma}\\int^1_0 d\\xi\\,e^{-i\\xi (p_B)_+  z_-}\\left[\\Phi_{B1}(\\xi)+\\slash{n}_- \\Phi_{B2}(\\xi)\\right]_{\\gamma \\alpha}\\,,\n\\label{qcdf_4}\n\\end{equation}\nwith the decay constant $f_B$ given by Eq.~(\\ref{bdecayconstant}). With a careful choice of $n_-=(1,0,0,-1)$ only the following normalisation conditions are required\n\\begin{equation}\n\\int^1_0d\\xi\\, \\Phi_{B1}(\\xi)=1\\,,\\qquad\\int^1_0d\\xi\\, \\Phi_{B2}(\\xi)=0\\,,\n\\label{qcdf_5}\n\\end{equation}\nalong with the first inverse moment of $ \\Phi_{B1}$ which is parameterised as\n\\begin{equation}\n\\int^1_0 d\\xi\\,\\frac{\\Phi_{B1}(\\xi)}{\\xi}\\equiv \\frac{m_B}{\\lambda_B}\\,,\n\\label{qcdf_6}\n\\end{equation}\nand the numerical value of $\\lambda_B$ is a source of uncertainty in the QCDF framework for both $B\\to M_1 M_2$ and $B\\to V \\gamma$. We now discuss the radiative decays $B\\to V \\gamma$ within QCDF.\n\n\\section{Radiative $B$ decays to Vector Mesons}\\label{qcdf_rad}\nWe consider the leading contributions to the $B\\to V \\gamma$ QCDF factorisation formula  as of Refs.~\\cite{Bosch:2001gv,Bosch:2002bw,Bosch:2004nd,Beneke:2001at} in which a model independent framework is presented.  Contributions that are power-suppressed by one power of $1\/m_b$ or more \\textit{and} are $\\mathcal{O}(\\alpha_s)$ are not considered. At the quark level the decays are $b\\to D \\gamma$ transitions, where $D=\\{s,d\\}$. If otherwise not stated, in the following we refer to $\\bar{B}\\to V\\gamma$ decays where $\\bar{B}$ ($V$) denotes a $b \\bar{q}$ ($D \\bar{q}$)  bound state. For $B\\to V \\gamma$ decays the matrix element of each relevant local operator in the effective Hamiltonian factorises as\n\\begin{equation}\n\\bra{V\\gamma}Q_i\\ket{B}=e^* \\cdot \\left[ T_1^{B\\to V} (0) \\,T^{I}_i+\\int^1_0 d\\xi du\\,T^{II}_i(\\xi,u)\\phi_B(\\xi) \\phi_{2;V}^{\\perp}(u)\\right]+\\mathcal{O}(1\/m_b)\\,,\n\\label{qcdf_7}\n\\end{equation}\nwhere $e_\\mu$ is the photon polarisation vector and $T_1^{B \\to V}(0)$ is the relevant form factor. $\\phi_{2;V}^{\\perp}$ the leading-twist DA of the perpendicularly polarised final state vector meson (\\ref{das_eq19});  contributions from $\\phi^{\\parallel}_{2;V}$ are power-suppressed in the heavy-quark limit. Problems of end-point divergences are not encountered in $B\\to V\\gamma$ decays and the twist-3 vector meson DA does not feature -- the $B$ meson DAs (\\ref{qcdf_6}) do however. The factorisation formula is accurate up to corrections suppressed by powers of $1\/m_b$, as shown, and was proven to hold to all orders in $\\alpha_s$ in SCET \\cite{Becher:2005fg}.  The form factor $T_1^{B \\to V}(0)$ has been calculated, for example, from LCSR in Ref.~\\cite{Ball:2004rg}. \n\nThe $B\\to V \\gamma$ decay produces either left- or right-handed photons, which therefore constitute, in principle,  two separate observable processes. In practise the direct measurement of the photon's helicity is very difficult; indirectly, however, it can be accessed by measurement of the time-dependent CP asymmetry in $\\bar B^0\\to V^0\\gamma$, which vanishes if one of them is absent, see Chapter~\\ref{chapter7_rad}.  We define the two amplitudes as\n\\begin{equation}\n\\bar{\\cal A}_{L(R)} = {\\cal A}(\\bar B\\to V\\gamma_{L(R)})\\,,  \\qquad\n{\\cal A}_{L(R)} = {\\cal A}(B\\to \\bar V \\gamma_{L(R)})\\,.\n\\label{qcdf_8}\n\\end{equation}\nFor ($B$) $\\bar B$  decays the production of the (left-) right-handed photon is suppressed by $1\/m_b$ with respect to the opposite helicity. The decays are dominated by the electromagnetic dipole operator $Q_{7\\gamma}$, and as such are penguin mediated and so loop-suppressed. The operators $Q_{7\\gamma}^{L(R)}$ are given by\n\\begin{equation}\nQ_{7\\gamma}^{L(R)} = \\frac{e}{8\\pi^2}\\, m_b \\bar D \\sigma_{\\mu\\nu}\\left(1 \\pm \\gamma_5\\right)b F^{\\mu\\nu}\\,,\n\\label{qcdf_9}\n\\end{equation} \nand generate left- (right-) handed photons.  Their matrix elements can be parameterised in terms of the form factor $T_1^{B\\to V}$ as\n\\begin{eqnarray}\n\\lefteqn{\\bra{V(p,\\eta) \\gamma_{L(R)}(q,e)} Q_{7\\gamma}^{L(R)} \\ket{\\bar\nB}}\\hspace*{1cm}\\nonumber\\\\\n&=& -\\frac{e}{2\\pi^2}\\, m_b T_1^{B\\to V}(0) \\left[\n\\epsilon^{\\mu\\nu\\rho\\sigma} e_\\mu^* \\eta_\\nu^* p_\\rho q_\\sigma \\pm i\n\\{ (e^* \\cdot \\eta^*) (p \\cdot  q) - (e^* \\cdot p)(\\eta^* \\cdot q)\\}\\right]\n\\nonumber\\\\\n&\\equiv& -\\frac{e}{2\\pi^2}\\, m_b T_1^{B\\to V}(0) S_{L(R)}\\,,\n\\label{qcdf_10}\n\\end{eqnarray}\nwhere $S_{L,R}$ are the helicity amplitudes corresponding to left- and right-handed photons, respectively, and $e_\\mu$ $(\\eta_\\mu)$ is the polarisation four-vector of the photon (vector meson).   The leading-order diagram is given in Fig.~\\ref{qcdf_fig1} which is also the leading diagram for the form factor $T_1^{B\\to V}$.\n\\begin{figure}[h]\n$$\\epsfxsize=0.2\\textwidth\\epsffile{rad_leading.eps}$$\n\\caption[The leading contribution to $B \\to V\\gamma$.]{\\small The leading contribution to $B \\to V\\gamma$ due to the  electromagnetic dipole operator $Q_{7\\gamma}$. }\\label{qcdf_fig1}\n\\end{figure}\nThe factorisation formula (\\ref{qcdf_10}) is therefore trivial to leading order in $\\alpha_s$ and the heavy-quark limit; the matrix element given by the standard form factor, the scattering kernel $T^I_7$ by a purely kinematical function and $T^{II}_7$ does not feature.  The electroweak penguin operators $Q_{7,\\dots,10}$ appear at higher-order and safely neglected in the analysis. All other operators begin to contribute at $\\mathcal{O}(\\alpha_s)$. The hard-vertex corrections contribute to $T^I_i$ yielding functions of $m_{u,c}^2\/m_b^2$ and originate from penguin contractions of the operators $Q_{1,\\dots,6}$ and the chromomagnetic operator $Q_{8g}$ as shown in Fig.~\\ref{qcdf_fig2}. \n\\begin{figure}[h]\n$$\\epsfxsize=0.5\\textwidth\\epsffile{rad_vertexcorr.eps}$$\n\\caption[Contributions to the hard-scattering kernel $T^{I}_i$ for $B\\to V \\gamma$ decays.]{\\small Penguin contractions of $Q_{1,\\dots,6}$ (top line) and the chromomagnetic dipole operator $Q_8$ (bottom line) contributing to the hard-vertex corrections of $T^{I}_i$ at $\\mathcal{O}(\\alpha_s)$. Crosses denote possible photon emission vertices.}\n\\label{qcdf_fig2}\n\\end{figure}\nThe hard-spectator scattering diagrams of Fig.~\\ref{qcdf_fig3}, in which the spectator quark of the $B$ meson participates, contribute to  $T^{II}_i$ and involve the same operators as the hard-vertex corrections. The hard-gluon exchange probes the momentum distribution of the $B$ and vector mesons and so requires the introduction of the mesons' light-cone DAs, as suggested by the factorisation formula; it is in these contributions that the $B$ meson DA parameter $\\lambda_B$ and decay constants $f_B$ and $f_V^\\perp$ appear. \n\\begin{figure}[h]\n$$\\epsfxsize=0.5\\textwidth\\epsffile{rad_spec.eps}$$\n\\caption[Contributions to the hard-scattering kernel $T^{II}_i$ for $B\\to V \\gamma$ decays.]{\\small Penguin contractions of $Q_{1,\\dots,6}$ (left) and the chromomagnetic dipole operator $Q_{8g}$ (right) contributing to the hard-scattering kernel  $T^{II}_i$ at $\\mathcal{O}(\\alpha_s)$. Crosses denote possible photon emission vertices at leading order. Photon emission from the other quark lines power-suppressed. Photon emission from the final state meson for $Q_{8g}$ breaks factorisation.}\n\\label{qcdf_fig3}\n\\end{figure}\nAlso, the dominant power-suppressed  \\textit{weak annihilation} (WA) contributions, shown in Fig.~\\ref{qcdf_fig4}, are calculable in the QCDF approach, and involve the operators $Q_{1,\\dots,6}$. WA contributions are\n$O(1\/m_b)$; photon emission from the $b$ quark and the quarks in the vector meson is further suppressed and $O(1\/m_b^2)$ -- unless the weak interaction operator is $Q_{5,6}$, which can be Fierz transformed into $(\\bar D (1+\\gamma_5) q) (\\bar q (1-\\gamma_5) b)$ and picks up an additional factor $m_B$ from the projection onto the $B$ meson DA thus resulting in this contribution being $O(1\/m_b)$. Consequently, due to the large Wilson coefficients $C_{1,2}$ these contributions are sizeable and important phenomenologically, see Chapter~\\ref{chapter7_rad}. \n\\begin{figure}[h]\n$$\\epsfxsize=0.2\\textwidth\\epsffile{rad_ann.eps}$$\n\\caption[Weak annihilation contributions to $B\\to V \\gamma$.]{\\small Weak annihilation contributions, which are suppressed by one power of $1\/m_b$. Crosses denote possible photon emission vertices at leading order. The dominant mechanism for $Q_{1,\\dots,4}$ is the emission of the photon from the light quark in the $B$ meson and for $Q_{5,6}$ it is the emission from the final state vector meson quarks. Other possible emissions are either vanishing or more strongly suppressed.}\n\\label{qcdf_fig4}\n\\end{figure}\n\nThe decay amplitude is then given by\n\\begin{equation}\n\\mathcal{A}(\\bar{B}\\to V \\gamma_{L(R)})=\\frac{G_F}{\\sqrt{2}}\\left(\\lambda_u^D a_{7 L(R)}^u(V)+ \\lambda_c^D a_{7 L(R)}^c(V)\\right)\\bra{V\\gamma_{L(R)}}Q_{7\\gamma}^{L(R)}\\ket{\\bar{B}}\\,,\n\\label{qcdf_11}\n\\end{equation}\nwhere the left-handed coefficients are given, to leading order in QCDF, by\n\\begin{equation}\na_{7L}^{U,{\\rm QCDF}}(V)=C_7+\\mathcal{O}(\\alpha_s,1\/m_b)\\,,\n\\label{qcdf_12}\n\\end{equation} \nand the right-handed parameters, for a $b\\to D$ transition, by \\cite{Ball:2006cv}\n\\begin{equation}\na_{7R}^{U,{\\rm QCDF}} = C_7\\,\\frac{m_D}{m_b}+\\mathcal{O}(1\/m_b,\\alpha_s\/m_b)\\,.\n\\label{qcdf_13}\n\\end{equation}\nExplicit expressions for the $\\mathcal{O}(\\alpha_s)$ corrections to the left-handed coefficients can be found in Refs.~\\cite{Bosch:2001gv,Bosch:2002bw} and will be considered in Chapter~\\ref{chapter7_rad}, alongside the dominant power-suppressed corrections.\n\\chapter{$B \\to V \\gamma$ Beyond QCD Factorisation}\\label{chapter7_rad}\nIn this chapter we perform a phenomenological analysis of the exclusive radiative $B$ decays to vector mesons. We make use of the QCDF framework outlined in Chapter~\\ref{chapter6_QCDF} and investigate the impact of the leading power-corrections on the branching ratios, CP asymmetries and isospin asymmetries for all $b\\to D$ transitions; $B_{u,d} \\to (\\rho, \\omega, K^*)\\gamma$ and  $B_{s} \\to (\\phi, \\bar{K}^*)\\gamma$.  Weak annihilation effects, although power-suppressed, are calculable in QCDF, and are included for all decay modes in this analysis.  The other power-suppressed contributions ``beyond QCDF''  considered are; soft photon emission from the soft $B$ spectator quark \\cite{Ball:2003fq}; and long-distance contributions from heavy quark loops \\cite{Ball:2006cv} and light quark loops \\cite{Ball:2006eu} which have been estimated from LCSR.  The estimation of the light quark loop contribution is new to the present analysis. Whereas the branching ratios are generally dominated by the leading contributions, and power-suppressed contributions play a minor role, the same cannot be said for the CP and isospin asymmetries for which the impact of power-corrections is in fact crucial.\n\nThe motivation to study radiative $B$ decays stems from a variety of sources:\n\\begin{itemize}\n\\item{as loop-induced, penguin mediated decays, they allow the extraction of the CKM matrix element $|V_{t,(d,s)}|$ complimentarily to the determination from $B$ mixing and also that from the SM UT analysis based on the tree-level observables $|V_{ub}\/V_{cb}|$ and the angle $\\gamma$.}\n\\item{They are sensitive to new physics contributions, which may occur within the penguin loops, with the time-dependent CP asymmetry a very promising avenue of investigation. They are also subject to large short-distance QCD corrections, which now approach next-to-next-to-leading-order accuracy, see Refs.~\\cite{misiak,NNLObsgamma}.}\n\\item{The decay rates are of order $G_F^2 \\alpha_{\\rm QED}$ and are enhanced with respect to other loop-induced non-radiative rare decays which are of order $G_F^2 \\alpha_{\\rm QED}^2$. Also, the $b\\to s$ modes are CKM-favoured. Consequently there exist good experimental results for the exclusive branching ratios; $B\\to K^* \\gamma$ is known to 5\\%, but the $b\\to d$ transitions are not so well known.}\n\\end{itemize}\n\nAs discussed in Chapter~\\ref{chapter6_QCDF}, the QCDF framework for $B\\to V\\gamma$ relies on the leading-twist vector meson DA $\\phi_{2;V}^\\perp$. Moreover, the LCSR calculations of the form factors $T_1^{B\\to V}$ and the parameters entering expressions for the soft-quark contributions rely also on the higher-twist DAs of the vector mesons and thus we find immediate use for the results of the twist-2 and twist-3 DA parameters of Chapter~\\ref{chapter4_det}, as presented in Tab.~\\ref{det_tab1} and Tab.~\\ref{det_tab2}.\\footnote{The analysis presented in Ref.~\\cite{Ball:2006eu} used preliminary input for the DA parameters, values for which were later finalised in Ref.~\\cite{Ball:2007rt}. The conclusions and numerics of the analysis are unaffected, due somewhat to the large errors attributed to the soft quark loop calculations in which the twist-3 DA parameters feature.}\n\nWe begin with an introduction, and then go on to discuss the power-suppressed contributions and investigate their impact on the decay observables. We extract the CKM parameter $|V_{t,d}\/V_{ts}|$ from the branching ratio results, assuming no new physics contributions, and discuss possible new physics contributions to the CP and isospin asymmetries. The material covered in this chapter follows that of Ref.~\\cite{Ball:2006eu}.\n\n\\section{Introduction}\n$B \\to V \\gamma$ decays are a very rich and promising probe of flavour physics. Both the inclusive decay $B\\to X_s \\gamma$ and the exclusive decays $B\\to (K^*,\\rho)\\gamma$ have been under scrutiny for many years, see for example Refs.~\\cite{Neubert:2002ku,Kagan:1998bh}.  The experimental results for $B\\to (\\rho,\\omega,K^*)\\gamma$ are shown in Tab.~\\ref{rad_tab1}.  For $B_s\\to\\phi\\gamma$  only an upper bound ${\\cal B}(B_s\\to\\phi\\gamma)<120\\times 10^{-6}$ exists and no experimental information is available for $B_s\\to \\bar K^*\\gamma$ \\cite{Yao:2006px}. \n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.4}\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l||c|c||l|c}\n\\hline\n{\\cal B} \\times 10^6 & \\mbox{{\\sc BaBar} \\cite{babar_rad}} & \\mbox{Belle\n  \\cite{belle_rad}} & {\\cal B} \\times 10^6 & \\mbox{HFAG \\cite{Barberio:2007cr}}\n\\\\\\hline\nB\\to (\\rho,\\omega)\\gamma & 1.25^{+0.25}_{-0.24}\\pm 0.09 &\n                           1.32^{+0.34}_{-0.31}{}^{+0.10}_{-0.09} \n                         &\nB^+\\to K^{*+}\\gamma & 40.3\\pm 2.6\n                          \n\\\\\nB^+\\to \\rho^+\\gamma &      1.10^{+0.37}_{-0.33}\\pm 0.09 &\n                           0.55^{+0.42}_{-0.36}{}^{+0.09}_{-0.08} \n                          \n                    &\nB^0\\to K^{*0}\\gamma &  40.1\\pm 2.0\n\\\\\nB^0\\to\\rho^0\\gamma  &      0.79^{+0.22}_{-0.20}\\pm0.06 &\n                           1.25^{+0.37}_{-0.33}{}^{+0.07}_{-0.06} \n                          \n\\\\\nB^0\\to\\omega\\gamma  &      <0.78 &\n                           0.96^{+0.34}_{-0.27}{}^{+0.05}_{-0.10}\n                          \n\\\\\\hline\n\\end{array}\n$$\n\\caption[Experimental branching ratios of exclusive $b\\to (d,s)\\gamma$\n  transitions.]{\\small Experimental branching ratios of exclusive $b\\to (d,s)\\gamma$\n  transitions. All entries are CP averaged. The first error is statistical, the second\n  systematic. $B\\to (\\rho,\\omega)\\gamma$ is the CP   average of the\n  isospin average over $\\rho$ and $\\omega$ channels:\\\\  $\\overline{\\cal B}(B\\to\n  (\\rho,\\omega)\\gamma) = \\frac{1}{2}  \\left\\{ \\overline{\\cal B}(B^\\pm\\to \\rho^\\pm\\gamma) +\n  \\frac{\\tau_{B^\\pm}}{\\tau_{B^0}} \\left[ \\overline{\\cal B}(B^0\\to \n  \\rho^0\\gamma) +  \\overline{\\cal B}(B^0\\to \\omega \\gamma)\\right]\\right\\}$.}\n  \\label{rad_tab1}\n\\end{table}\n\nIn the SM the decays are flavour-changing-neutral-current (FCNC) $b\\to D\\gamma$ transitions, mediated by penguin diagrams; they are therefore loop-suppressed and potentially very sensitive to new physics. To determine the relative sizes of contributions to the decays one must consider the following points:\n\\begin{itemize}\n\\item{the leading term is loop-suppressed $\\sim 1\/(4 \\pi)^2$ and proportional to $C_7\\sim-0.3$.}\n\\item{Evidently from Eq.~(\\ref{qcdf_11}) for each mode there are two amplitudes proportional to different CKM factors $\\lambda^{(D)}_{u,c}$. For $b\\to d$ transitions both $\\lambda^{(d)}_u$ and $\\lambda^{(d)}_c$ are $\\sim\\lambda^3$, however, for $b\\to s$ transitions $\\lambda^{(s)}_u\\sim\\lambda^4$ and $\\lambda^{(s)}_c\\sim\\lambda^2$; there is a relative CKM suppression of the up-quark contribution.}\n\\item{Power suppressed corrections from WA are formally $\\sim 1\/m_b$ although come with  large Wilson coefficients $C_1\\sim-0.3$ and $C_2\\sim 1$ and are not loop suppressed. The WA contributions drive the isospin asymmetries.}\n\\item{The production of ``wrong'' helicity photons is suppressed by $m_D\/m_b$ (\\ref{qcdf_13}). The interplay of both helicity amplitudes generates the time-dependent CP asymmetries, which are small in the SM due to this suppression.}\n\\end{itemize}\n\n\\section{Wilson Coefficients}\nConsiderable effort has gone into calculating the Wilson coefficients to NLO accuracy. Using the expressions for the NLO anomalous dimension matrices available in the literature we employ the renormalisation techniques of Eqs.~(\\ref{basics_eq22}-\\ref{basics_eq29}) to calculate the Wilson coefficients at the required scales. Numerical values of all the NLO Wilson coefficients $C_i$ used in the analysis are given in Tab.~\\ref{rad_tab2}. The situation is complicated by the fact that the QCDF results of Ref.~\\cite{Bosch:2002bw} are given in terms of two bases. The first, the so-called BBL basis named after the authors of Ref.~\\cite{Buchalla:1995vs}, is that of Eqs.~(\\ref{basics_eq20}) and (\\ref{basics_eq21}) except with $Q_1$ and $Q_2$ exchanged with respect to the basis of Ref.~\\cite{Bosch:2001gv}. The second is the so-called CMM basis of Ref.~\\cite{munz, buras}. The two bases differ except for $Q_{7(8)}^{\\rm BBL}=Q_{7(8)}^{\\rm CMM}$. Following Ref.~\\cite{Bosch:2002bw}, the CMM set is used for calculating hard-vertex  corrections to the QCDF formulas and the BBL set at the lower scale $\\mu_h\\sim \\sqrt{\\Lambda_{h} \\,\\mu}$ (with $\\lambda_h\\sim 0.5\\,{\\rm GeV}$ and $\\mu= \\mathcal{O}(m_b)$)  is used to calculate hard-spectator corrections. Power corrections are calculated from the BBL set at scale $m_b$. \n\nNLO accuracy is mandatory only for $C_7$, as it is for this term only that the hadronic matrix element is also known to NLO accuracy. We evaluate all $\\mathcal{O}(\\alpha_s)$ and power-suppressed corrections using both LO and NLO scaling for Wilson coefficients and hadronic matrix elements and include the resulting scale dependence in the theoretical uncertainty.\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{2pt}\n$$\n\\begin{array}{c|c|c|c|c|c|c}\nC^{\\rm CMM}_1(m_b) & C^{\\rm CMM}_2(m_b) & C^{\\rm CMM}_3(m_b) & \nC^{\\rm CMM}_4(m_b) & C^{\\rm CMM}_5(m_b) & C^{\\rm CMM}_6(m_b) &\nC^{\\rm CMM}_7(m_b)\n\\\\\\hline\n-0.322 & 1.009 & -0.005 & -0.087 & 0.0004 & -0.001 & -0.309 \n\\\\\\hline\\hline\nC^{\\rm BBL}_1(m_b) & C^{\\rm BBL}_2(m_b) & C^{\\rm BBL}_3(m_b) & \nC^{\\rm BBL}_4(m_b) & C^{\\rm BBL}_5(m_b) & C^{\\rm BBL}_6(m_b) \n& C^{\\rm CMM}_8(m_b) \n\\\\\\hline\n-0.189 & 1.081  & 0.014 & -0.036 & 0.009  & -0.042 & -0.170\n\\\\\\hline\\hline\nC^{\\rm BBL}_1(\\mu_h) & C^{\\rm BBL}_2(\\mu_h) & C^{\\rm BBL}_3(\\mu_h) & \nC^{\\rm BBL}_4(\\mu_h) & C^{\\rm BBL}_5(\\mu_h) & C^{\\rm BBL}_6(\\mu_h) &\nC^{\\rm CMM}_8(\\mu_h)\n\\\\\\hline\n-0.288 & 1.133 & 0.021 & -0.051 & 0.010 & -0.065 & -0.191\n\\end{array}\n$$\n\\caption[Numerical values of the next-to-leading-order Wilson coefficients.]{\\small NLO Wilson coefficients to be used in the analysis, at the scales $m_b=4.2\\,$GeV and $\\mu_h=2.2\\,$GeV. The coefficients labelled BBL correspond to the operator  basis of Ref.~\\cite{Buchalla:1995vs} and given in Eq.~(\\ref{basics_eq21}),   whereas CMM denotes the basis of Ref.~\\cite{munz}. We use   $\\alpha_s(m_Z) = 0.1176$ \\cite{Yao:2006px} and ${m}_t({m}_t) =   163.6\\,$GeV \\cite{mt}. Note that $C_1^{\\rm BBL}$ and $C_2^{\\rm BBL}$   are exchanged with respect to the basis of Ref.~\\cite{Bosch:2001gv} and  that $C_{7(8)}^{\\rm BBL}=C_{7(8)}^{\\rm CMM}$. Following Ref.~\\cite{ Bosch:2002bw}, the CMM set is used for calculating hard-vertex  corrections to the QCDF formulas and the BBL set at the lower scale  $\\mu_h$ is used to calculate hard-spectator corrections. The BBL set at  scale $m_b$ is used for the calculation of power-corrections.}\n\\label{rad_tab2}\n\\end{table}\n\n\n\n\\section{Leading and Power Suppressed Contributions}\nIt proves convenient to split to the coefficients in Eq.~(\\ref{qcdf_11}) into three contributions which we will investigate separately:\n\\begin{eqnarray}\na_{7L}^U( V) &=& a_{7L}^{U,{\\rm QCDF}}( V) + a_{7L}^{U,{\\rm ann}}(  V) + a_{7L}^{U,{\\rm soft}}( V)+\\dots\\,,\\nonumber\\\\\na_{7R}^U( V) &=& a_{7R}^{U,{\\rm QCDF}}( V) + a_{7R}^{U,{\\rm ann}}(V)+  a_{7R}^{U,{\\rm soft}}( V)+\\dots\\,,\n\\label{asplit}\n\\end{eqnarray} \nwhere the leading term in the $1\/m_b$ expansion is given by Eq.~(\\ref{qcdf_12}) and all other terms are suppressed by at least one power of $m_b$. The dots denote terms of higher order in $\\alpha_s$ and further $1\/m_b$ corrections to QCDF, most of which are incalculable. We only include those power-suppressed terms that are either numerically large or relevant for isospin and CP asymmetries.  \n\n\n\n\\subsection{Leading Contributions}\nThe diagrams giving the leading QCDF contributions are given in Chapter~\\ref{chapter6_QCDF}. It turns out that, at the level of two decimal places, all $a_{7L}^{c,{\\rm QCDF}}$ are equal and so are\n$a_{7L}^{u,{\\rm QCDF}}$.\\footnote{Explicit formulas for  $a_{7L}^{U,{\\rm QCDF}}$, complete to $\\mathcal{O}(\\alpha_s)$, can be found in Ref.~\\cite{Bosch:2002bw}.} For central values of the input parameters of Tab.~\\ref{rad_tab8} we obtain\n\\begin{eqnarray}\na^{c,{\\rm QCDF}}_{7L}(V) & = & -\\overbrace{(0.41+0.03i)}^{\\shortstack{{\\rm \\footnotesize Vertex}\\\\{\\rm \\footnotesize Corrections}}} - \n\\overbrace{(0.01+0.01i)}^{\\shortstack{{\\rm \\footnotesize Hard-Spectator}\\\\{\\rm \\footnotesize Corrections}}}\\,,\\nonumber\\\\\na^{u,{\\rm QCDF}}_{7L}(V) & = & -(0.45+0.07i) + (0.02-0i)\\,.\n\\label{10}\n\\end{eqnarray}\nThe size of the hard-spectator corrections is set by the factor\n\\begin{equation}\nh_V = \\frac{2 \\pi^2}{9}\\,\\frac{f_B f_V^\\perp}{m_B  \nT_1^{B\\to V}(0)\\lambda_B}\\,.\n\\end{equation}\nFor $B_s$ decays one has to set $f_B\\to f_{B_s}$ and correspondingly for the other $B$ meson parameters. We estimate the value of  $\\lambda_{B_s}$, the first inverse moment of the twist-2 $B$-meson light-cone DA, from $\\lambda_{B_d}$ by a simple scaling argument:\n\\begin{equation}\n\\frac{m_{B_s}}{\\lambda_{B_s}}\\,(\\Lambda_{\\rm QCD}+m_s) = \n\\frac{m_{B_q}}{\\lambda_{B_q}}\\,\\Lambda_{\\rm QCD}\\,,\n\\label{rad_bs}\n\\end{equation}\nwhich follows from the assumption that the $B_q$ DA peaks at the spectator momentum $k_+ = \\Lambda_{\\rm QCD}$, whereas that of $B_s$ peaks at $\\Lambda_{\\rm QCD}+m_s$. Its numerical value is given, along with all the other input parameters, in Tab.~\\ref{rad_tab8}.\n\n\n\n\\subsection{Weak Annihilation}\n $a_{7L}^{U,{\\rm ann}}$ encodes the $\\mathcal{O}(1\/m_b)$ contribution of the WA diagram of Fig.~\\ref{rad_fig1}(a) which drives  the isospin asymmetries and has been  calculated in QCDF in  Ref.~\\cite{Bosch:2002bw} with $\\alpha_s$ corrections given in Ref.~\\cite{Kagan:2001zk} for $\\rho$ and $K^*$  and in Ref.~\\cite{Bosch:2004nd} for $\\omega$. WA receives contributions from the current-current operator $Q_2^u$, which for $b\\to s$ transitions is doubly CKM suppressed, and QCD penguin operators $Q_{3,\\dots,6}$, which are not CKM suppressed. Formulas for $a_{7L}^{U,{\\rm ann}}(\\rho,K^*)$ in QCDF can be found in Refs.~\\cite{Bosch:2002bw,Bosch:2004nd};  in this approximation, there is no contribution to  $a_{7R}^{U,{\\rm ann}}$. \n\\begin{figure}[h]\n$$\\epsfxsize=0.6\\textwidth\\epsffile{rad_power.eps}$$\n\\caption[Diagrams for weak annihilation and soft-gluon emission from a quark loop.]{\\small (a) Weak annihilation diagram where photon emission from the $B$ meson spectator quark is power-suppressed. The crosses denote possible photon emission vertices for $Q_{5,6}$ only. (b) soft-gluon emission from a quark loop, where there is also a second diagram in which the gluon is picked up by the $B$ meson.}\n\\label{rad_fig1}\n\\end{figure}\n\nPreliminary results for the $\\mathcal{O}(\\alpha_s)$ corrections to WA in $B\\to\\rho \\gamma$ were presented in Ref.~\\cite{chamonix}.   In QCDF,  the $a_{7L}^{U,{\\rm ann}}$  are expressed in terms of the hadronic quantities\n\\begin{equation}\nb^V = \\frac{2\\pi^2}{T_1^{B\\to V}(0)} \\,\\frac{f_B m_V f_V}{m_B m_b\n  \\lambda_B}\\,, \\qquad\nd^V_{v} = -\\frac{4\\pi^2}{T_1^{B\\to V}(0)} \\,\\frac{f_B f_V^\\perp}{m_B\n  m_b} \\,\\int_0^1 dv\\,\\frac{\\phi_{2;V}^\\perp(v)}{v}\n\\end{equation}\nand $d^V_{\\bar v}$, obtained by replacing $1\/v\\to 1\/\\bar v$ in the integrand; $\\phi_{2;V}^\\perp$ is the twist-2 DA of a transversely polarised vector meson, (\\ref{das_eq19}). Numerically, one finds, for instance for the $\\rho$, $b^\\rho = 0.22$ and $d^\\rho = -0.59$, at the scale $\\mu = 4.2\\,$GeV. As $T_1\\sim 1\/m_b^{3\/2}$ and $f_B\\sim m_b^{-1\/2}$ in the heavy-quark limit, these terms are $\\mathcal{O}(1\/m_b)$, but not numerically small because of the tree-enhancement factors of $\\pi^2$.\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l||c|c|c|c|c}\n\\mbox{WA} &  \nB^-\\to K^{*-} & \\bar B^0\\to K^{*0} & B\\to (\\rho,\\omega) & B_s\\to\\phi & \nB_s\\to \\bar  K^*\\\\\\hline\n\\mbox{induced by} & \\mbox{C (and P)} & \\mbox{P} & \\mbox{C and P} &\n\\mbox{P} & \\mbox{P}\\\\\n\\mbox{CKM} & \\lambda^2 \\mbox{~(and 1)} & 1 & 1 & 1 & 1\n\\end{array}\n$$\n\\caption[Parametric size of the weak annihilation contributions.]{\\small Parametric size of WA  contributions to $B\\to V\\gamma$. C denotes the charged-current operators $Q_{1,2}$, P  the penguin operators $Q_{3,\\dots,6}$; their Wilson coefficients are   small -- see Tab.~\\ref{rad_tab2}. CKM denotes the order in  the Wolfenstein parameter $\\lambda$ with respect to the dominant amplitude induced by $Q_7$.}\n\\label{rad_tab3}\n\\end{table}\n\nFor $\\omega$, $\\bar K^*$ and $\\phi$ we obtain\n\\begin{eqnarray}\n\\left. a_{7L}^{u,{\\rm ann}}(\\omega)\\right|_{\\rm QCDF}\n & = & Q_d b^\\omega (a_1 + 2 (a_3+a_5)\n+ a_4) + Q_d (d^\\omega_v + d^\\omega_{\\bar v}) a_6\\,,\\nonumber\\\\\n\\left. a_{7L}^{c,{\\rm ann}}(\\omega)\\right|_{\\rm QCDF} \n& = & Q_d b^\\omega (2 (a_3+a_5)\n+ a_4) + Q_d (d^\\omega_v + d^\\omega_{\\bar v}) a_6\\,,\\nonumber\\\\\n\\left. a_{7L}^{U,{\\rm ann}}(\\phi)\\right|_{\\rm QCDF} \n& = & Q_s b^\\phi (a_3+a_5) + \n         Q_s (d^\\phi_v + d^\\phi_{\\bar v}) a_6\\,,\\nonumber\\\\\n\\left. a_{7L}^{U,{\\rm ann}}(\\bar K^*)\\right|_{\\rm QCDF} \n& = & Q_s b^{\\bar K^*} a_4 + \n         Q_s (d^{\\bar K^*}_v Q_d\/Q_s + d^{\\bar K^*}_{\\bar v}) a_6\\,,\n\\label{15}\n\\end{eqnarray}\nwith $a_1 = C_1+C_2\/3$, $a_3 = C_3+C_4\/3$, $a_4 = C_4+C_3\/3$, $a_5 = C_5+C_6\/3$, $a_6 = C_6+C_5\/3$.\\footnote{Note that $a_1\\leftrightarrow a_2$ as compared to \\cite{Bosch:2002bw} as in our operator basis  (i.e.\\ the BBL basis) $Q_1$  and $Q_2$ are exchanged.} The expressions for $\\phi$ and $\\bar K^*$ are new; for $\\omega$, we do not agree with \\cite{Bosch:2004nd}.  Apart from for $\\rho$ and $\\omega$, all the WA coefficients are numerically small and do not change the branching ratio significantly; the terms in $a_6$, however, are relevant  for the isospin asymmetries. \n\nIn Tab.~\\ref{rad_tab3} we show the relative weights of these diagrams in terms of CKM factors and Wilson coefficients. The numerically largest contribution occurs for $B^\\pm\\to \\rho^\\pm\\gamma$: it comes with the large combination of  Wilson coefficients $C_2+C_1\/3=1.02$ and is not CKM suppressed. For $B^0\\to (\\rho^0,\\omega)\\gamma$ it comes with the factor $C_1+C_2\/3 = 0.17$\ninstead and an additional suppression factor $1\/2$ from the electric charge of the spectator quark ($d$ instead of $u$). For all other decays, WA is suppressed by small (penguin) Wilson coefficients. Apart from $B\\to(\\rho,\\omega)\\gamma$, WA is not relevant so much for the total values of $a_{7L}$, but rather for isospin breaking, which is set by photon emission from the spectator quark. WA is the only mechanism to contribute to isospin asymmetries at tree-level; see Ref.~\\cite{Kagan:2001zk} for $\\mathcal{O}(\\alpha_s)$ contributions.\n\nIn view of the large size of $a_{7L}^{u,{\\rm ann}}(\\rho)$ it is appropriate to have a look at further  corrections. The most  obvious ones are $\\mathcal{O}(\\alpha_s)$ corrections to the QCDF expressions, shown in Fig.~\\ref{rad_fig2}.\n\\begin{figure}\n$$\\epsfxsize=\\textwidth\\epsffile{rad_fig2.eps}$$\n\\caption[Example radiative corrections to \nweak annihilation.]{\\small Example radiative corrections to \nweak annihilation. The corrections to the $B$ vertex in (a) are known \n\\cite{bellnu,Descotes-Genon:2002mw} and those to the $V$ vertex in (b) are included in $f_V$. \nFor the non-factorisable corrections in (c) only preliminary results\nare available \\cite{chamonix}.}\n\\label{rad_fig2}\n\\end{figure}\nAs it turns out, the corrections to the $B$ vertex in Fig.~\\ref{rad_fig2}(a) are known: they also enter the decay $B\\to\\gamma \\ell\\nu$ and were calculated in Ref.~\\cite{bellnu,Descotes-Genon:2002mw}. Numerically, they are at the level of 10\\%.  Fig.~\\ref{rad_fig2}(b) shows the vertex corrections to the $V$ vertex, which are actually included in the decay constant $f_V$. For the non-factorisable corrections shown in Fig.~\\ref{rad_fig2}(c) preliminary results have been reported in Ref.~\\cite{chamonix} according to which these corrections are of a size similar to the $B$ vertex corrections. \n\nIn Ref.~\\cite{Kagan:2001zk} also another class of $1\/m_b$ corrections to $B\\to K^*\\gamma$ was calculated, namely $\\mathcal{O}(\\alpha_s)$  corrections to the isospin asymmetry in this decay. As these corrections break factorisation (require an infra-red cut-off in the  momentum  distribution of the valence quarks in the $K^*$ meson) and are numerically small, we do not include them in our analysis. \n\n\\subsection{Long-Distance Photon Emission}\nAnother class of corrections is suppressed by one power of $m_b$ with respect to the QCDF contributions and is  due to  long-distance photon emission from the soft $B$ spectator quark. A first calculation of this effect was attempted in Ref.~\\cite{WA} and was corrected and extended in Ref.~\\cite{Ball:2003fq}. The long-distance photon emission from a soft-quark line requires the inclusion of higher-twist terms in the expansion of the quark propagator in a photon background field, beyond the leading-twist (perturbative) contribution; a comprehensive discussion of this topic can be found in Ref.~\\cite{Ball:2002ps}. The quantity calculated in Ref.~\\cite{Ball:2003fq} is\n\\begin{eqnarray}\n\\lefteqn{\n\\bra{\\rho^-(p)\\gamma(q)} (\\bar d u)_{V-A} (\\bar u b)_{V-A}\\ket{B^-(p+q)} =}\\hspace*{3cm}\\nonumber\\\\\n& = & e\\,\\frac{m_\\rho f_\\rho}{m_B} \\eta^*_\\mu\n\\left\\{F_V \\epsilon^{\\mu\\nu\\rho\\sigma} e^*_\\nu p_{\\rho}\n  q_\\sigma - i F_A [e^{*\\mu} (p \\cdot q) - q^\\mu\n    (e^* \\cdot p)]\\right\\}\\nonumber\\\\\n& = & -e \\,\\frac{m_\\rho f_\\rho}{m_B} \\left\\{ \\frac{1}{2}\\,\n  F_V (S_L+S_R) + \\frac{1}{2}\\, F_A (S_L-S_R)\\right\\}\n  \\label{problem}\n\\end{eqnarray}\nin terms of the photon-helicity amplitudes $S_{L,R}$.\\footnote{Eq.~(\\ref{problem}) differs from the one given in \\cite{Ball:2003fq} by an overall sign, which is due to the different convention used in \\cite{Ball:2003fq} (and in \\cite{Ball:2002ps}) for the covariant derivative: $D_\\mu = \\partial_\\mu\n- i e Q_f A_\\mu$ instead of $D_\\mu = \\partial_\\mu + i e Q_f A_\\mu$ as in this analysis.}\nIn QCDF, $F_{A,V}$ are given by $Q_u f_B\/\\lambda_B$ and induce a term $Q_u a_2 b^\\rho$ in  $a_{7L}^{u,{\\rm    ann}}(\\rho^-)$. The long-distance photon contribution to $F_{V,A}$ was found to be \\cite{Ball:2003fq}\n\\begin{equation}\nF^{\\rm soft}_A = -0.07\\pm 0.02 \\equiv Q_u G_A\\,,\\qquad \nF^{\\rm soft}_V = -0.09\\pm\n0.02 \\equiv Q_u G_V\\,.\n\\label{Fsoft}\n\\end{equation}\nwith $G_A+G_V = -0.24\\pm 0.06$ and $G_V-G_A = -0.030\\pm 0.015$.\\footnote{Again, there is a relative sign with respect to the results in \\cite{Ball:2003fq}. This comes from the fact that the product $e F_{A,V}^{\\rm soft}$ is independent of the sign convention for $e$, and as we have changed the overall sign of (\\ref{problem}) with respect to \\cite{Ball:2003fq}, we also have to change the sign of $F_{A,V}^{\\rm soft}$. Stated differently: the relative sign between $F_{A,V}^{\\rm soft}$ and $F_{A,V}^{\\rm hard}$ in \\cite{Ball:2003fq} is wrong because of a mismatch in sign conventions for $e$ in the covariant derivative.}\nIn order to obtain concise expressions for $a_{7L(R)}^{U,{\\rm ann}}$, it proves convenient to define one more hadronic quantity:\n\\begin{equation}\ng^\\rho_{L,R} = \\frac{\\pi^2}{T_1^\\rho}\\,\\frac{m_\\rho f_\\rho}{m_b m_B}\\,\n(G_V\\pm G_A)\n\\end{equation}\nand correspondingly for other mesons. $g_L$ is $\\mathcal{O}(1\/m_b^2)$ as $G_V+G_A$ has the same power scaling in $m_b$ as $T_1$, i.e.\\ $\\sim m_b^{-3\/2}$, as one can read off from the explicit expressions in \\cite{WA}. The difference $G_V-G_A$, on the other hand, is a twist-3 effect due to three-particle light-cone DAs of the photon and is suppressed by one more power of $m_b$, i.e.\\ $g_R\\sim\n1\/m_b^3$. This quantity will enter the CP asymmetry. Our final expressions for $a_{7L(R)}^{U,{\\rm ann}}$ then read:\n\\begin{eqnarray}\na_{7L}^{U,{\\rm ann}}(V) & = & \\left. a_{7L}^{U,{\\rm\n    ann}}(V)\\right|_{\\rm QCDF} (b^V\\to b^V + g^V_L)\\,,\\nonumber\\\\\na_{7R}^{U,{\\rm ann}}(V) & = & \\left. a_{7L}^{U,{\\rm\n    ann}}(V)\\right|_{\\rm QCDF} (b^V\\to g^V_R, d^V\\to 0)\\,.\n    \\label{20A}\n\\end{eqnarray}\nNumerically, one has $g^{\\rho}_L\/b^\\rho = -0.3$, so these corrections, despite being suppressed by one more power in $1\/m_b$, are not small numerically and larger than the known $\\mathcal{O}(\\alpha_s)$\ncorrections to QCDF from $B\\to\\gamma\\ell\\nu$. Based on this, we feel justified in including these long-distance corrections in our analysis, while dropping the radiative ones of Figs.~\\ref{rad_fig2}(a) and (c). For central values of the input parameters we find the following numerical values for the various WA and long-distance photon contributions, including in particular those to which $Q_{1,2}$ contribute (with no Cabibbo suppression):\n\\begin{eqnarray}\na_{7L}^{c,{\\rm ann}}(K^{*0})  &=&  -0.013-0.001\\, {\\rm LD}\\,,\n\\qquad  a_{7L}^{c,{\\rm ann}}(K^{*-}) =  0.004+0.001\\, {\\rm\n  LD}\\,,\\nonumber\\\\\na_{7L}^{u,{\\rm ann}}(\\rho^0) &=&  -0.001-0.004\\, {\\rm LD}\\,,\n \\qquad ~~ a_{7L}^{u,{\\rm ann}}(\\rho^-) =  0.149-0.043\\, {\\rm\n  LD}\\,,\\nonumber\\\\\n a_{7L}^{u,{\\rm ann}}(\\omega)  &=&  -0.024+0.003\\, {\\rm LD}\\,.\n \\label{LDcont}\n\\end{eqnarray}\nThe contribution from the long-distance photon emission is labelled ``LD'' (LD$\\to 1$ at the end). \nThe unexpectedly small $a_{7L}^{u,{\\rm ann}}(\\rho^0)$ is due to a numerical cancellation between the charged-current and  penguin-operator contributions. Comparing these results with those from QCDF,\nEq.~(\\ref{10}), it is evident that WA is, as expected,  largely irrelevant for the branching ratios, except for $B^\\pm\\to \\rho^\\pm\\gamma$.\n\n\n\\subsection{Soft Quark Loops}\n$a_{7L(R)}^{U,{\\rm soft}}$ encodes soft-gluon emission from a (light or heavy quark) loop as shown in Fig.~\\ref{rad_fig1}(b). Soft-gluon emission from a charm loop was first considered in  Ref.~\\cite{KRSW97} as a potentially relevant long-distance contribution to the branching ratio of $B\\to K^*\\gamma$, however, the same diagram also contributes dominantly to the time-dependent CP asymmetry in $B^0\\to K^{*0}\\gamma$ \\cite{grin05}.  As for $a_{7R}^U$, the dominant contributions to $a_{7R}^c(K^*)$ were calculated in Ref.~\\cite{Ball:2006cv} and new to this analysis is their generalisation to the other vector mesons and the inclusion of contributions from light-quark loops. Motivation to include light quark loops stems from the fact that they are doubly CKM-suppressed for $b\\to s\\gamma$ transitions, but not for $b\\to d\\gamma$, for which they are on an equal footing as the heavy quark loops. The quark loop contributions are suppressed by one power of $m_b$ with respect to $a_{7L}^{U,{\\rm QCDF}}$, but they also induce a right-handed photon amplitude which is of the same order in $1\/m_b$ as $a_{7R}^{U,{\\rm QCDF}}$ (\\ref{qcdf_13}), and this amplitude induces the time-dependent CP asymmetry.  The asymmetry is expected to be very small in the SM and $\\propto m_D\/m_b$ due to the chiral suppression of the leading transition (\\ref{qcdf_13}), but could be drastically enhanced by new physics contributions -- thus constituting an excellent ``null test'' of the SM \\cite{Gershon:2006mt,Ball:2006cv}.  It was noticed in Refs.~\\cite{grin04,grin05}  that the chiral suppression is relaxed by emission of a gluon from the quark loop and contributes dominantly to the time-dependent CP asymmetry in $B^0\\to K^{*0}\\gamma$, which motivates the inclusion of these contributions. The task of the present analysis, however, is not so much to calculate these contributions to high accuracy, but to exclude the possibility of  {\\em  large} contributions to the CP asymmetry.  With this in mind, the theoretical uncertainties of the results are very generously estimated  --- which is somewhat unavoidable due to the current uncertainties of the relevant hadronic input parameters.\n\nPotentially the most important contribution to the soft-gluon emission  diagram in Fig.~\\ref{rad_fig1}(b) \ncomes from the charged-current operator $Q_2^U$ with the  large Wilson coefficient $C_2\\sim 1$; it vanishes for $Q_1^U$  by gauge invariance. In addition, the penguin operators $Q_{3,4,6}$ give a non-zero contribution. Details of the derivation of $a_7^{U,{\\rm soft}}$ can be found in Ref.~\\cite{Ball:2006eu} in which the following expression is obtained:\n\\begin{eqnarray}\na_{7L(R)}^{U,{\\rm soft}}(V) & = &  \\frac{\\pi^2}{m_b T_1^{B\\to V}(0)} \\left\\{ Q_U C_2 (l_U\\pm \\tilde l_U)(V) + Q_D C_3 (l_D\\pm \\tilde l_D)(V)\\right.\\nonumber\\\\\n&&\\left. + \\sum_q Q_q (C_4-C_6) (l_q\\pm \\tilde l_q)(V)\\right\\}.\n\\label{24}\n\\end{eqnarray}\nHere the sum over $q$ runs over all five active quarks $u,d,s,c,b$. The contribution from $Q_5$ is proportional to $m_D$ and hence helicity suppressed and neglected.  Assuming $\\rm SU(3)_F$-flavour symmetry for the light quark loops, one has $l_u=l_d=l_s$, and ditto for $\\tilde l_{u,d,s}$, which causes a cancellation of these contributions in the last term of Eq.~(\\ref{24}). ${\\rm SU(3)_F}$-breaking effects are estimated to be around 10\\% \\cite{Ball:2006eu}.  The parameters  $l_c(K^*)$ and $\\tilde{l}_c(K^*)$ were first calculated from three-point sum rules in Ref.~\\cite{KRSW97} and were re-calculated in the more suitable method of LCSR via a local OPE in Ref.~\\cite{Ball:2006cv}. The analysis therein as been updated and extended to $l_b,\\, \\tilde{l}_b$ and the other particles $\\rho,\\,\\omega ,\\,\\bar{K}^*,\\,\\phi$ for the present analysis \\cite{Ball:2006eu}. The results for $l_c$ and $\\tilde l_c$ are given in the upper table of Tab.~\\ref{rad_tab5}. Those for $l_b$ and $\\tilde l_b$ are obtained as\n\\begin{equation}\nl_b = \\frac{m_c^2}{m_b^2}\\, l_c\\,,\\qquad \\tilde l_b = \\frac{m_c^2}{m_b^2}\\, \\tilde l_c\\,.\n\\end{equation}\nFor light-quark loops the photon is almost at threshold and the local OPE does not apply. In Ref.~\\cite{Ball:2006eu} a method was developed for calculating these contributions via LCSRs.  Similar to the method of Ref.~\\cite{Khodjamirian:2000mi}  used for the calculation of soft-gluon contributions to $B\\to\\pi\\pi$, a dispersion relation approach is used to connect the off-shell matrix element to the physical regime $q^2=0$.  The results are presented in the lower table of Tab.~\\ref{rad_tab5}.\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l ||r|r|r|r}\n& \\multicolumn{1}{c|}{l_c} & \\multicolumn{1}{c|}{\\tilde l_c} \n& \\multicolumn{1}{c|}{l_c-\\tilde l_c} & \\multicolumn{1}{c}{l_c +\n    \\tilde l_c} \\\\\n \\hline\n B \\to K^*   & -355 \\pm 280 &  -596 \\pm 520 & 242 \\pm 370 & -952 \\pm 800  \\\\\n B \\to (\\rho,\\omega)   & -382 \\pm 300 & -502\\pm  430 & 120 \\pm 390 & \n-884\\pm 660  \\\\\n B_s \\to \\bar K^*   &  -347\\pm 260 & -342\\pm 400 & -4\\pm 300 & -689\\pm 600 \\\\\n B_s \\to \\phi   & -312 \\pm  240 & -618 \\pm 500 & 306 \\pm 320 & -930 \\pm\n750 \n\\end{array}\n$$\\\\\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l ||r|r|r|r}\n& \\multicolumn{1}{c|}{l_u} & \\multicolumn{1}{c|}{\\tilde l_u} &   \n\\multicolumn{1}{c|}{l_u-\\tilde l_u} & \\multicolumn{1}{c}{l_u + \\tilde l_u} \\\\\n \\hline\n B \\to K^*   & 536 \\pm 70\\% &  635 \\pm 70\\% & -99 \\pm 300 & 1172 \\pm 70 \\% \\\\\n B \\to (\\rho,\\omega)   & 827 \\pm 70\\% & 828\\pm 70\\% & -1\\pm 300&\n 1655\\pm 70\\%  \\\\\n B_s \\to \\bar K^*   &  454\\pm 70\\% & 572\\pm 70\\% & -118\\pm 300 & 1025\\pm\n70\\% \\\\\n B_s \\to \\phi   & 156\\pm 70\\% & 737\\pm 70\\% & -581\\pm 300 & 893\\pm 70\\% \\\\\n\\end{array}\n$$\n\\caption[Soft-gluon contributions from $c$-quark and $u$-quark  loops in units KeV. ]{\\small Soft-gluon contributions from $c$-quark (upper table) and $u$-quark (lower table) loops in units KeV. The quantities $l_{c,u}$ and $\\tilde l_{c,u}$ are defined in Ref.~\\cite{Ball:2006eu}. We assume equal parameters for $\\rho$ and $\\omega$. $l_b$ is obtained as $l_b = l_c m_c^2\/m_b^2$ and correspondingly for $\\tilde l_b$. The uncertainty for $l_u-\\tilde l_u$  is given in absolute numbers because of cancellations. In the $\\rm SU(3)_F$-flavour limit assumed in this calculation one has  $l_u = l_d = l_s  \\equiv l_q$}\n\\label{rad_tab5}\n\\end{table}\n\n\\section{Phenomenological Results}\nIn this section we combine the different contributions to the factorisation coefficients $a_{7L(R)}^U$ and give results for the observables, namely the branching ratios, the isospin asymmetries and the time-dependent CP asymmetries. \n\\subsection{Branching Ratios}\\label{rad_brs}\nThe (non-CP-averaged) branching ratio of the $b\\to D\\gamma$ decay \n$\\bar B\\to V\\gamma$ is given by\n\\begin{eqnarray}\n{\\cal B}(\\bar B\\to V\\gamma) & = & \\frac{\\tau_B}{c_V^2}\\,\n\\frac{G_F^2\\alpha_{\\rm QED} m_B^3  m_b^2}{32 \\pi^4} \\left(1-\\frac{m_V^2}{m_B^2}\\right)^3 \\left[T_1^{B\\to V}(0)\\right]^2\\nonumber\\\\\n&&\\times \\left\\{ \\left| \\sum_{U=u,c} \\lambda_U^{(D)} a_{7L}^U(V)\\right|^2 +  \\left| \\sum_{U=u,c} \\lambda_U^{(D)}\n  a_{7R}^U(V)\\right|^2\\right\\}\n  \\label{BR}\n\\end{eqnarray}\nwith the isospin factors $c_{\\rho^\\pm,K^*,\\phi}=1$  and $c_{\\rho^0,\\omega} = \\sqrt{2}$. The branching ratio for the CP-conjugated channel $B\\to \\bar V\\gamma$ ($\\bar b\\to \\bar D\\gamma$ at parton level) is obtained by replacing $\\lambda_U^{(D)}\\to (\\lambda_U^{(D)})^*$.  With the input parameters from Tab.~\\ref{rad_tab8} and the lifetimes given in Tab.~\\ref{rad_tab9} we find the following CP-averaged branching ratios  for $B\\to K^*\\gamma$, making explicit various sources of uncertainty:\n\\begin{eqnarray}\n\\overline{\\cal B}(B^- \\to K^{*-}\\gamma) & = & (53.3\\pm \\overbrace{13.5}^{T_1}\n\\pm \\overbrace{4.8}^{\\mu}\n\\pm \\overbrace{1.8}^{V_{cb}}\\pm \\overbrace{1.9}^{l_{u,c}} \\pm \\overbrace{1.3}^{\\mbox{other}})\\times\n10^{-6}\\nonumber\\\\\n& =& (53.3\\pm \\underbrace{13.5}_{T_1}\\pm 5.8)\\times 10^{-6}\\,,\n\\nonumber\\\\\n\\overline{\\cal B}(\\bar B^0 \\to K^{*0}\\gamma) & = & (54.2\\pm \\overbrace{13.2}^{T_1}\n\\pm \\overbrace{6.0}^{\\mu}\n\\pm \\overbrace{1.8}^{V_{cb}}\\pm \\overbrace{1.8}^{l_{u,c}} \\pm \\overbrace{1.4}^{\\mbox{other}})\\times\n10^{-6}\\nonumber\\\\\n& = & (54.2\\pm \\underbrace{13.2}_{T_1}\\pm 6.7)\\times 10^{-6}\\,.\n\\label{50}\n\\end{eqnarray}\nWe have added all individual uncertainties in quadrature, except for that induced by the form factor. The uncertainty in $\\mu$ is that induced by the renormalisation-scale dependence, with  $\\mu= m_b(m_b)\\pm 1\\,$GeV.  ``Other'' sources of uncertainty include the dependence on the parameters in Tab.~\\ref{rad_tab7}, on the size of LD WA contributions and the replacement of NLO by LO Wilson coefficients.  The above results agree, within errors, with the experimental ones given in Tab.~\\ref{rad_tab1}, within the large theoretical uncertainty induced by the form factor.\n\nAs the uncertainties of all form factors in Tab.~\\ref{rad_tab8} are of roughly the same size, one might conclude that the predictions for all branching ratios will carry uncertainties similar to those in\n(\\ref{50}). This is, however, not the case: the accuracy of the theoretical predictions can be improved by making use of the fact that the {\\em ratio} of form factors is known much better than the individual form factors themselves. The reason is that the values given in Tab.~\\ref{rad_tab8}, which were calculated using the same method, LCSRs, and with a common set of input parameters,  include common systematic uncertainties (dependence on $f_B$, $m_b$ etc.) which partially cancel in the ratio. In Ref.~\\cite{Ball:2006nr}  the ratio of the $K^*$ and $\\rho$ form factors was found to be\n\\begin{equation}\n\\xi_\\rho \\equiv \\frac{T_1^{B\\to K^*}(0)}{T_1^{B\\to \\rho}(0)} = 1.17\\pm\n0.09\\,.\n\\label{xirho}\n\\end{equation}\nThe uncertainty is by a factor 2 smaller than if we had calculated $\\xi_\\rho$\nfrom the entries in Tab.~\\ref{rad_tab8}; analogously for $\\omega$ one finds\n\\begin{equation}\n\\xi_\\omega \\equiv \\frac{T_1^{B\\to K^*}(0)}{T_1^{B\\to \\omega}(0)} = 1.30\\pm\n0.10\\,.\n\\label{xiomega}\n\\end{equation}\nThe difference between $\\xi_\\rho$ and $\\xi_\\omega$ is mainly due to the difference between $f_\\omega^\\perp$ and $f_\\rho^\\perp$.  For the $B_s$ form factors, we also need the ratio of decay constants $f_{B_s}\/f_{B_d}$. The status of $f_B$ from Lattice QCD was reviewed in  Ref.~\\cite{Onogi}; the present state-of-the-art calculations are unquenched with $N_f=2+1$ active flavours \\cite{unquenchedfB},  whose average is $f_{B_s}\/f_{B_d}=1.23\\pm 0.07$. Again, this ratio is fully consistent with that quoted in Tab.~\\ref{rad_tab8}, but has a smaller uncertainty. One then finds the following ratios for $B_s$ form factors:\n\\begin{equation}\n\\xi_\\phi \\equiv \\frac{T_1^{B\\to K^*}(0)}{T_1^{B_s\\to\\phi}(0)} = 1.01\\pm 0.13\n\\,,\\qquad\n\\xi_{\\bar K^*} \\equiv \\frac{T_1^{B\\to K^*}(0)}{T_1^{B_s\\to\\bar K^*}(0)} \n= 1.09\\pm 0.09\\,.\n\\label{xis}\n\\end{equation}\nThe uncertainty of $\\xi_{\\bar K^*}$ is smaller than that of $\\xi_\\phi$ because the input parameters for $K^*$ and $\\bar K^*$ are the same (except for G-odd parameters like $a_1^\\perp$) and cancel in the ratio; the uncertainty is dominated by that of $f_{B_s}\/f_{B_d}$. To benefit from this reduced theoretical uncertainty in predicting branching ratios, one has to calculate ratios of branching ratios, which mainly depend on $\\xi_V$ and only mildly on  $T_1$  itself: in addition to the  overall normalisation, $T_1$ also enters hard-spectator interactions and power-suppressed corrections, whose size is set by hadronic quantities $\\propto 1\/T_1$. As these corrections are subleading (in  $\\alpha_s$ or $1\/m_b$),  however, a small shift in $T_1$ has only very minor impact on the  branching ratios. The absolute scale for the branching ratios is set by the CP- and isospin-averaged  branching ratio with the smallest experimental uncertainty, i.e.\\ $B\\to K^*\\gamma$; from  Tab.~\\ref{rad_tab1}, one finds:\n\\begin{equation}\n\\overline{\\cal B}(B\\to K^*\\gamma) = \\frac{1}{2}\\left\\{ \\overline{\\cal\n  B}(B^\\pm \\to K^{*\\pm}\\gamma) + \\frac{\\tau_{B^\\pm}}{\\tau_{B^0}}\\, \n  \\overline{\\cal B}(\\bar B^0 \\to K^{*0}\\gamma)\\right\\} = (41.6\\pm\n  1.7)\\times 10^{-6}\\,.\n  \\label{x}\n\\end{equation}\nThat is, we obtain a theoretical prediction for $\\overline{\\cal B}(B\\to\nV\\gamma)$ as \n\\begin{equation}\n\\left.\\overline{\\cal B}(B\\to V\\gamma)\\right|_{\\rm th}= \\left[ \\frac{\\overline{\\cal B}(B\\to V\\gamma)}{\n\\overline{\\cal B}(B\\to K^*\\gamma)}\\right]_{{\\rm th}} \\, \\left.\n\\overline{\\cal B}(B\\to K^*\\gamma)\\right|_{\\rm exp}\\,,\n\\end{equation}\nwhere $\\left[\\dots\\right]_{\\rm th}$ depends mainly on $\\xi_V$ and only in subleading terms on the individual form factors $T_1^{B\\to K^*}$ and $T_1^{B\\to V}$. It is obvious that, except for these subleading terms, this procedure is equivalent to extracting an {\\em effective form factor} $\\left.T_1^{B\\to  K^*}(0)\\right|_{\\rm eff}$ from $B\\to K^*\\gamma$ and using $\\left.T_1^{B\\to V}(0)\\right|_{\\rm eff} = \\left.T_1^{B\\to   K^*}(0)\\right|_{\\rm  eff}\/\\xi_{V}$ for calculating the branching ratios for $B\\to V\\gamma$. From (\\ref{x}) we find\n\\begin{equation}\n\\left. T_1^{B\\to K^*}(0)\\right|_{\\rm eff} = 0.267\\pm \\overbrace{0.017}^{{\\rm th}} \\pm\n\\overbrace{0.006}^{{\\rm exp}} = 0.267\\pm 0.018\\,,\n\\end{equation}\nwhere the theoretical uncertainty  follows from the second uncertainty given in (\\ref{50}). Eqs.~(\\ref{xirho}), (\\ref{xiomega}) and (\\ref{xis}) then yield\n\\begin{eqnarray}\n\\left. T_1^{B\\to \\rho}(0)\\right|_{\\rm eff} &=& 0.228 \\pm 0.023\\,, \\qquad\n\\left. T_1^{B\\to \\omega}(0)\\right|_{\\rm eff} = 0.205 \\pm\n0.021\\,,\\nonumber\\\\\n\\left. T_1^{B_s\\to \\bar K^*}(0)\\right|_{\\rm eff} &=& 0.245 \\pm\n0.024\\,, \n\\qquad\n\\left. T_1^{B_s\\to \\phi}(0)\\right|_{\\rm eff} = 0.260 \\pm\n0.036\\,.\n\\label{56}\n\\end{eqnarray}\nNote that all effective form factors agree, within errors, with the results from LCSRs given in Tab.~\\ref{rad_tab8}, which confirms the results obtained from this method; the crucial point, however, is that the uncertainties are reduced by a factor of 2 (except for $T_1^{B_s\\to \\phi}$). We would like to stress that the motivation for this procedure is to achieve a reduction of the theoretical uncertainty of the\npredicted branching fractions in $B\\to (\\rho,\\omega)\\gamma$ and $B_s$ decays.  The effective form factors do {\\em not} constitute a new and independent theoretical determination, but are derived from the experimental results for $B\\to K^*\\gamma$ under the following assumptions:\n\\begin{itemize}\n\\item there is no new physics in $B\\to K^*\\gamma$;\\footnote{Which is motivated by the\n  results from inclusive $B\\to X_s \\gamma$ decays \\cite{misiak}.}\n\\item QCDF is valid with no systematic uncertainties;\n\\item LCSRs can reliably predict the ratio of form factors at zero\n  momentum transfer.\n\\end{itemize}\n From (\\ref{BR}) and (\\ref{56}), we then predict the following CP-averaged branching ratios:\n\\begin{eqnarray}\n\\overline{\\cal B}(B^-\\to \\rho^-\\gamma) & = & (1.16\\pm \\overbrace{0.22}^{T_1}\\pm \\overbrace{0.13}^{\\rm Other})\\times 10^{-6}\\,, \\nonumber\\\\\n\\overline{\\cal B}(B^0\\to \\rho^0\\gamma) & = & (0.55\\pm 0.11\\pm \n0.07)\\times 10^{-6}\\,, \\nonumber\\\\\n\\overline{\\cal B}(B^0\\to \\omega\\gamma) & = & (0.44\\pm 0.09 \\\n\\pm 0.05)\\times 10^{-6}\\,,\\nonumber\\\\\n\\overline{\\cal B}(B_s\\to \\bar K^*\\gamma) & = & (1.26\\pm 0.25\\pm \n0.18)\\times 10^{-6}\\,, \\nonumber\\\\\n\\overline{\\cal B}(B_s\\to \\phi\\gamma) & = & (39.4\\pm 10.7 \\ \\pm 5.3)\\times 10^{-6}\\,,\n\\label{57}\n\\end{eqnarray}\nwhere the first uncertainty is induced by the effective form factors and  the second includes the variation of all inputs from Tab.~\\ref{rad_tab8}  except for the angle $\\gamma$ of the UT, which is fixed at\n$\\gamma=53^\\circ$.\\footnote{The value of the UT angle\n$\\gamma$ in Tab.~\\ref{rad_tab8} comes from Belle's \nDalitz-plot analysis of the CP asymmetry in $B^-\\to (K_S^0 \\pi^+\\pi^-)\nK^-$, with $K_S^0 \\pi^+\\pi^-$ \\cite{Bellegamma} being a three-body final state common\nto both $D^0$ and $\\bar D^0$. Other determinations all come with theoretical uncertainties and\/or possible contamination by unresolved new physics, so we take this result as a reference point.} The total uncertainty in each channel is $\\sim 20\\%$, except for $B_s\\to \\phi\\gamma$, where it is 30\\%. The results for $\\rho$ and $\\omega$ agree very well with those of {\\sc  BaBar}, Tab.~\\ref{rad_tab1}, but less so with the Belle results, although present experimental and theoretical uncertainties preclude a firm conclusion. Our prediction for $B_s\\to \\phi\\gamma$ is well below the current experimental bound $120\\times 10^{-6}$ \\cite{Yao:2006px}. A branching ratio of the size given in (\\ref{57}) implies that $\\mathcal{O}(10^3)$ $B_s\\to\\phi\\gamma$  events will be seen within the first few years of the LHC.\n\nIn Tab.~\\ref{rad_tab6} we detail the contributions of individual terms to the branching ratios. In all cases ${\\cal B}$ is dominated by the QCDF contribution, with WA most relevant for $B^-\\to \\rho^-\\gamma$. This is expected as WA enters with the large Wilson coefficient $C_2\\sim 1$. The effect is extenuated by long-distance (LD) photon emission, which itself is compensated by soft-gluon emission. The other channels follow a similar pattern, although the size of the effects is smaller.\n\\begin{table}[h]\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{l||c|c|c|c}\n\\mathcal{B}\\times 10^{6}& \\mbox{QCDF} & \\mbox{+ WA (no LD)} & \\mbox{+ WA (inc.\\ LD)}& \n\\mbox{+ soft gluons}\\\\\\hline\nB^-\\to \\rho^-\\gamma & 1.05&1.17 & 1.11&1.16\n\\\\\nB^0\\to \\rho^0\\gamma & 0.49&0.53& 0.53&0.55\n\\\\\nB^0\\to \\omega\\gamma & 0.40& 0.42&0.42 &0.44\n\\\\\nB^-\\to K^{*-}\\gamma &39.7&38.4& 38.3&39.4\n\\\\\nB^0\\to K^{*0}\\gamma & 37.1&39.7 &39.9 &41.0\n\\\\\nB_s^0\\to \\bar K^{*0}\\gamma & 1.12& 1.22& 1.23&1.26\n\\\\\nB_s^0\\to \\phi\\gamma &34.6 & 38.2& 38.3&39.4\n\\end{array}\n$$\n\\caption[Contributions to CP-averaged branching ratios.]{\\small Contributions to CP-averaged branching ratios, using effective form factors and  central values of all other input parameters, Tab.~\\ref{rad_tab8} (in particular $\\gamma=53^\\circ$). LD stands for long-distance photon-emission contributions. Each column  labelled ``+X'' includes the contributions listed in the previous  column plus the contribution induced by X. The entries in the last  column are our total central values.}\n  \\label{rad_tab6}\n\\end{table}\n\\begin{figure}[h]\n$$ \\epsfxsize=0.48\\textwidth\\epsffile{fig7a.eps}\\quad\\epsfxsize=0.48\\textwidth\\epsffile{fig7b.eps}$$\n$$ \\epsfxsize=0.48\\textwidth\\epsffile{fig7c.eps}$$\n  \\caption[CP-averaged branching ratios of $B\\to(\\rho,\\omega)\\gamma$ as function   of angle $\\gamma$.]{\\small CP-averaged branching ratios of $B\\to(\\rho,\\omega)\\gamma$ as function   of UT angle $\\gamma$, using the effective form factors and central values of   other input parameters. (a): $B^\\pm \\to \\rho^\\pm\\gamma$, (b):   $B^0\\to \\rho^0\\gamma$, (c): $B^0\\to\\omega\\gamma$. The boxes indicate the 1$\\sigma$ experimental results from {\\sc BaBar} \\cite{babar_rad}, Tab.~\\ref{rad_tab1}.  Note that the resulting value of $\\gamma$   from the average of all three channels is $\\gamma =  (61.0^{+13.5}_{-16.0}({\\rm exp})^{+8.9}_{-9.2})^\\circ$ -- see Section~\\ref{ckmextract}.} \n  \\label{rad_fig3}\n\\end{figure}\n\nWe would like to close this section by making explicit the  dependence of the three $B\\to (\\rho,\\omega)\\gamma$ branching ratios on $\\gamma$.  In Fig.~\\ref{rad_fig3}  we plot these branching ratios, for central values of the input parameters, as functions of $\\gamma$. We also indicate the present experimental results from {\\sc BaBar} \\cite{babar_rad}, Tab.~\\ref{rad_tab1}, within their 1$\\sigma$ uncertainty. \n\n\\subsection{Isospin Asymmetries}\\label{isosec}\nThe asymmetries $A_I(\\rho)$, $A_I(K^*)$, and   $A(\\rho,\\omega) $ are given by\n\\begin{eqnarray}\nA(\\rho,\\omega) & = &  \\frac{\\overline{\\Gamma}(B^0\\to \\omega   \\gamma)}{\\overline{\\Gamma}(B^0\\to \\rho^0   \\gamma)}-1\\,, \n  \\label{rad_arw} \\\\\nA_{I}(\\rho) & = & \\frac{2\\overline{\\Gamma}(\\bar B^0\\to    \\rho^0 \\gamma)}{\\overline{\\Gamma}(\\bar B^\\pm\\to    \\rho^\\pm \\gamma)} - 1\\,,\n  \\label{rad_air}\\\\\nA_{I}(K^*) & = & \\frac{\\overline{\\Gamma}(\\bar B^0\\to    K^{*0} \\gamma) - \\overline{\\Gamma}(B^\\pm\\to \n  K^{*\\pm} \\gamma)}{\\overline{\\Gamma}(\\bar B^0\\to    K^{*0} \\gamma) + \\overline{\\Gamma}(B^\\pm\\to \n  K^{*\\pm} \\gamma)}\\,,\n  \\label{rad_aik}\n\\end{eqnarray}\nwhere the partial decay rates are CP-averaged. Let us first discuss $A(\\rho,\\omega)$ and $A_{I}(\\rho)$ which are  relevant for the experimental determination of $\\overline{\\cal  B}(B\\to(\\rho,\\omega)\\gamma)$,  which in turn is used for the determination of $|V_{td}\/V_{ts}|$ (or $\\gamma$), see Section~\\ref{ckmextract}. The present experimental statistics for $b\\to d\\gamma$ transitions is rather low, so the experimental value of $\\overline{\\cal B}(B\\to(\\rho,\\omega)\\gamma)$ is obtained under the explicit assumption of perfect symmetry, i.e.\\ $\\overline{\\Gamma}(B^\\pm\\to \\rho^\\pm \\gamma) = 2 \\overline{\\Gamma}(B^0\\to \\rho^0 \\gamma) = 2 \\overline{\\Gamma}(B^0\\to \\omega \\gamma)$. In reality, the symmetry between $\\rho^0$ and $\\omega$ is broken by different values of the form factors, and isospin symmetry between neutral and charged $\\rho$ is broken by photon emission from the spectator quark, the dominant mechanism of which is WA. From the formulas for individual branching ratios, Eq.~(\\ref{BR}),  and the various contributions to the factorisation coefficients $a_{7L(R)}^U$, we find\n\\begin{equation}\nA(\\rho,\\omega) = -0.20\\pm \\overbrace{0.09}^{{\\rm th.}}\\,.\n\\label{eq:AIrw}\n\\end{equation}\nThe uncertainty is dominated by that of the  form factor ratio  $T_1^{B\\to\\omega}(0)\/T_1^{B\\to\\rho}(0)=0.90\\pm 0.05$.\\footnote{Note that this result is dominated by the ratio of  decay constants given in Tab.~\\ref{rad_tab8} and discussed in Ref.~\\cite{Ball:2006eu}. The experimental results entering these averages have a  large spread which may cast a shadow of doubt on the averaged final  branching ratios for $(\\rho^0,\\omega)\\to e^+ e^-$ quoted by PDG  \\cite{Yao:2006px}.} The dependence on all other input parameters is marginal. The result (\\ref{eq:AIrw}) is not compatible with the strict isospin limit $A(\\rho,\\omega) =0$.  $A_{I}(\\rho)$, on the other hand, is very sensitive to $\\gamma$, whereas the form factors drop out. It is driven by the WA contribution and, in the QCDF framework, vanishes if WA is set to zero. In Fig.~\\ref{rad_fig4}(a) we plot $A_{I}(\\rho)$ as  function of $\\gamma$, including the theoretical uncertainties. \n\\begin{table}\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{3pt}\n$$\n\\begin{array}{c||c|c|c|c}\n\\gamma & 40^\\circ & 50^\\circ & 60^\\circ & 70^\\circ\\\\\\hline\nA_I(\\rho) & -(5.3\\pm 6.9)\\% & (0.4\\pm 5.3)\\% & (5.7\\pm 3.9)\\% &\n(10.5\\pm 2.7)\\%\n\\end{array}\n$$\n\\caption[Isospin asymmetry $A_I(\\rho)$ for different values  of $\\gamma$.]{\\small Isospin asymmetry $A_I(\\rho)$  for different values  of $\\gamma$.}\n  \\label{rad_tab7}\n\\end{table}\nAs suggested by the findings of Ref.~\\cite{chamonix}, these results are  not expected to change considerably upon inclusion of  the non-factorisable radiative corrections of  Fig.~\\ref{rad_fig2}(c). In Tab.~\\ref{rad_tab7}, we give the corresponding  results for several values of $\\gamma$, together with the theoretical  uncertainty. Our result agrees very well with that  obtained by the {\\sc BaBar} collaboration: $A_I(\\rho)_{\\rm BaBar} =  0.56\\pm 0.66$ \\cite{babar_rad}.\n\n$A_I(K^*)$ was first discussed in Ref.~\\cite{Kagan:2001zk}, including power-suppressed $\\mathcal{O}(\\alpha_s)$ corrections which unfortunately violate QCDF, i.e.\\ are divergent. It is for this reason that we decide to drop these corrections and include only leading-order terms in $\\alpha_s$. We then find\n\\begin{eqnarray}\nA_I(K^*) &=& (5.4\\pm \\overbrace{1.0}^{\\mu} \\pm \\overbrace{0.6}^{{\\rm NLO}\\leftrightarrow{\\rm\n  LO}} \\pm \\overbrace{0.6}^{f_B} \\pm \\overbrace{0.6}^{{\\rm other}})\\%\\nonumber\\\\\n&=& (5.4\\pm 1.4)\\%\\,,\n\\label{62}\n\\end{eqnarray}\nwhere ${\\rm NLO}\\leftrightarrow{\\rm LO}$ denotes the uncertainty induced by switching from NLO to LO accuracy in the Wilson coefficients and ``other'' summarises all other sources of theoretical uncertainty. \nAs can be inferred from the entries in Tab.~\\ref{rad_tab1}, the  present experimental result is $A_I(K^*)_{\\rm exp}=(3.2\\pm 4.1)\\%$.  In Ref.~\\cite{Kagan:2001zk} it was pointed out that $A_I(K^*)$\nis very sensitive to the values of the Wilson coefficients $C_{5,6}^{\\rm BBL}$ in the combination $a_6\\equiv C_{5}^{\\rm BBL}+C_6^{\\rm   BBL}\/3$. In the SM, varying the renormalisation scale as $\\mu=m_b(m_b)\\pm 1\\,{\\rm GeV}$ and switching between LO and NLO accuracy for the Wilson coefficients, one has $a_6= -0.039\\pm 0.008$, which actually induces the bulk of the uncertainty in Eq.~(\\ref{62}). In Fig.~\\ref{rad_fig4}(b) we plot $A_I(K^*)$ as function of $a_6\/a_6^{\\rm SM}$, with $a_6^{\\rm SM}=-0.039$.\n\\begin{figure}[tb]\n$$\\epsfxsize=0.45\\textwidth\\epsffile{fig8a.eps}\\quad\n  \\epsfxsize=0.45\\textwidth\\epsffile{fig8b.eps}$$\n\\caption[$A_I(\\rho)$  as function of the angle $\\gamma$ and $A_I(K^*)$ as function of $r\\equiv a_6\/a_6^{\\rm     SM}$.]{\\small Left panel: isospin asymmetry $A_I(\\rho)$, Eq.~(\\ref{rad_air}),  as function of the UT angle $\\gamma$. Solid line: central values of input parameters; dashed lines: theoretical uncertainty. Right panel: $A_I(K^*)$, Eq.~(\\ref{rad_aik}), in percent,  as function of the ratio $r\\equiv a_6\/a_6^{\\rm     SM}$ of the combination of penguin Wilson coefficients   $a_6\\equiv C_6+C_5\/3$. Solid line: central value of input parameters,  dashed lines: theoretical uncertainty. The box indicates the  present experimental uncertainty and the straight black lines the  theory uncertainty in $r$.}\n    \\label{rad_fig4}\n\\end{figure}\nThe figure clearly indicates that, although there is presently no\ndiscrepancy between theoretical prediction and experimental result,\na reduction of the experimental uncertainty\nof $A_I(K^*)$ may well reveal some footprints of new physics\nin this observable.\n\n\n\\subsection{CP Asymmetries}\\label{cpsec}\nThe time-dependent CP asymmetry in $\\bar B^0\\to V^0\\gamma$ is given analogously to Eq.~(\\ref{basics_eq13}) as\n\\begin{equation}\nA_{CP}(t)  = S(V\\gamma) \\sin(\\Delta m_D\\, t ) - C(V\\gamma) \n\\cos(\\Delta m_D\\, t)\\,.\n\\label{rad_cpa}\n\\end{equation}\nThe above equation is technically only valid for $\\Delta \\Gamma =0$ and while this is a good assumption for $B^0_d$ decays, it is not so for $B_s^0$ decays. Although Eq.~(\\ref{rad_cpa})  can easily be adapted to non-zero $\\Delta \\Gamma_s$ we refrain from doing so: the whole point in calculating the CP asymmetry is not so much to give precise predictions for $S$ and $C$, but rather to exclude the possibility of large corrections to the naive expectation $S\\sim m_D\/m_b$. With this is mind, small corrections from a non-zero $\\Delta\\Gamma_s$ are irrelevant. The time-dependent CP asymmetries are given in terms of the left- and right-handed photon amplitudes (\\ref{qcdf_8}) by\n\\begin{equation}\nS(V\\gamma) \n =  \\frac{2 \\,{\\rm Im}\\,\\left(\\frac{q}{p}({\\cal A}_L^* \\bar{\\cal A}_L + \n                                       {\\cal A}_R^* \\bar{\\cal A}_R)\\right)}{\n        |{\\cal A}_L|^2 + |{\\cal A}_R|^2 + |\\bar{\\cal A}_L|^2 + |\\bar{\\cal\n                                       A}_R|^2}\\,,\n\\quad\nC(V\\gamma)  =   \\frac{|{\\cal A}_L|^2 + |{\\cal A}_R|^2 - |\\bar{\\cal A}_L|^2 - \n               |\\bar{\\cal A}_R|^2}{\n        |{\\cal A}_L|^2 + |{\\cal A}_R|^2 + |\\bar{\\cal A}_L|^2 + |\\bar{\\cal\n                                       A}_R|^2}\\,.\n\\label{54}\n\\end{equation}\nWith ${\\cal A}_{L,R}$ and $\\bar{\\cal A}_{L,R}$ as given in (\\ref{qcdf_11}). The indirect CP asymmetry $S(V\\gamma)$ relies on the interference of both left- and right-helicity amplitudes and vanishes if one of them is absent; it thus probes indirectly the photon helicity. The direct CP asymmetry\n$C(V\\gamma)$ is less sensitive to $\\bar{\\cal A}_R$, but very sensitive to the    strong phase of $\\bar{\\cal A}_L$ and vanishes if the radiative    corrections to $a_{7L}^{U,{\\rm QCDF}}$, Eq.~(\\ref{10}), are\n     neglected. As the accuracy of the prediction of strong phases in      QCDF is subject to discussion, and in any case $C(V\\gamma)$ is      less sensitive to new physics than $S(V\\gamma)$, we shall\n  not consider direct CP asymmetries in this analysis.\n\nLet us briefly discuss the reason for the expected smallness of $S$. In the process $b\\to D\\gamma$, in the SM, the emitted photon is  predominantly  left-handed in $b$, and right-handed in $\\bar b$ decays. This  is due to the fact that the dominant contribution to the  amplitude comes from the chiral-odd dipole operator $Q_7$. As only left-handed quarks participate in the weak  interaction, an effective operator of this type necessitates, in the SM, a helicity flip on one of the external quark lines, which results in a factor $m_b$ (and a left-handed photon) in $b_R\\to D_L\\gamma_L$ and a factor $m_D$ (and a right-handed photon) in $b_L\\to D_R\\gamma_R$. Hence, the emission of right-handed photons is suppressed by  a factor $m_D\/m_b$, which leads to the QCDF prediction (\\ref{qcdf_13}) for $a_{7R}^U$.  The interesting point is not the smallness of the CP asymmetry {\\em   per se}, but the fact that the helicity suppression can easily be alleviated  in a large number of new physics scenarios where the spin flip occurs on an internal line, resulting in a factor $m_i\/m_b$ instead of $m_D\/m_b$. A prime example is left-right symmetric models \\cite{LRS}, whose impact on the photon polarisation was discussed in\nRefs.~\\cite{alt, grin04,grin05}.  These models also come in a supersymmetric version whose effect on $b\\to s\\gamma$ was investigated in Ref.~\\cite{frank}. Supersymmetry with no left-right symmetry can also provide large  contributions to $b\\to D\\gamma_R$, see Ref.~\\cite{susy} for recent studies. Other\npotential sources of large effects  are warped extra dimensions \\cite{warped} or anomalous right-handed top couplings \\cite{anomalous}.  Unless the amplitude for $b\\to D\\gamma_R$ is of the same order as the SM prediction for $b\\to D \\gamma_L$, or the enhancement of $b\\to D \\gamma_R$ goes along with a suppression of $b\\to D \\gamma_L$, the impact on the branching ratio is small,  as the two helicity amplitudes add incoherently. This implies there can be a substantial contribution of new physics to $b\\to D\\gamma$ escaping detection when only branching ratios are measured.\n\nWe can calculate $S$ directly from (\\ref{54}) and obtain, making explicit the contributions from  different sources:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{array}[b]{rll}\nS(\\rho\\gamma) = \n& \\phantom{-}(\\overbrace{~0.01~}^{m_D\/m_b}+\\overbrace{~0.02~}^{\\rm\n  LD~WA}+\\overbrace{~0.20~}^{\\shortstack{\\footnotesize{soft} \\\\  \\footnotesize{gluons}}}~\\pm~ 1.6)\\%\n& = \\phantom{-}(0.2\\pm 1.6)\\%\\,,\\\\\nS(\\omega\\gamma) =\n& \\phantom{-}(0.01-0.08+0.22\\pm 1.7)\\% \n& = \\phantom{-}(0.1\\pm 1.7)\\%\\,,\\\\\nS(K^*\\gamma) =\n& -(2.9-0+0.6\\pm 1.6)\\%\n& = -(2.3\\pm 1.6)\\%\\,,\\\\\nS(\\bar K^*\\gamma) =\n& \\phantom{-}(0.12+0.03+0.11\\pm 1.3)\\%\n& = \\phantom{-}(0.3\\pm 1.3)\\%\\,,\\\\\nS(\\phi\\gamma) =\n& \\phantom{-}(0+0+5.3\\pm8.2)\\times 10^{-2}\\,\\%\n& =  \\phantom{-}(0.1\\pm 0.1)\\%\\,.\n\\label{eq:SVgamma}\n\\end{array}\n\\end{equation}\nIncluding only the helicity-suppressed contribution, one expects, for $B\\to K^*\\gamma$, neglecting the doubly Cabibbo suppressed amplitude in $\\lambda_u^{(s)}$\n\\begin{equation}\n\\left.S(K^*\\gamma)\\right|_{\\mbox{\\footnotesize no soft gluons}} \n =  -2\\, \\frac{m_s}{m_b}\\,\\sin\\,\\phi_d\\approx -2.7\\%\\,.\\label{75}\n\\end{equation}\nFor $B_s\\to\\phi\\gamma$, one expects the CP asymmetry to vanish if the decay amplitude is proportional to $\\lambda_t^{(s)}$, which, at tree-level, precludes any contributions of type $\\sin(\\phi_s) m_s\/m_b$ and also any contribution from WA. This is     because the mixing angle $\\phi_s$ is given by ${\\rm\n arg}[(\\lambda_t^{(s)})^2]$, Eq.~(\\ref{basics_eq12}),  and the interference of amplitudes in    (\\ref{54}) also yields a factor $(\\lambda_t^{(s)})^2$,  if the individual amplitudes are proportional to  $\\lambda_t^{(s)}$ or \n$(\\lambda_t^{(s)})^*$, respectively; this is indeed the case for the helicity-suppressed term $m_s\/m_b$ induced by the operator $Q_7$ and the WA contributions to $a_{7R}^U(\\phi)$,\nEqs.~(\\ref{15}) and (\\ref{20A}), so that the phases cancel in (\\ref{54}).\n\n\nThe actual results in (\\ref{eq:SVgamma}) disagree with the above\nexpectations because of the contributions from soft-gluon emission,\nwhich enter $a_{7R}^U$. Moreover, for $S(\\phi\\gamma)$ this is because the\nsoft-gluon emission from quark loops is different for $u$ and $c$ loops so that $a_{7R}^c\\neq a_{7R}^u$ and hence  $\\bar {\\cal A}_{R}$ (${\\cal A}_{L}$) is not proportional to   $\\lambda_t^{(s)}$ ($(\\lambda_t^{(s)})^*$). Note that a substantial enhancement of $S(\\phi\\gamma)$ by new physics requires\nnot only an enhancement of $|\\bar{\\cal A}_R|$ (and $|{\\cal\n  A}_L|$), but also the presence of a large phase in (\\ref{54}); \nthis could be  either\na large $B_s$ mixing phase which will also manifest itself in\na sizable CP violation in, for instance, $B_s\\to J\/\\psi \\phi$, see\nRef.~\\cite{BF06,Ball:2006xx}; or it could be a new weak phase in $\\bar{\\cal A}_{R}$\n(and ${\\cal A}_L$); or it could be a non-zero strong phase in\none of the $a_{7R}^{c,u}$ coefficients. Based on the light quark loop results there is not much scope for\na large phase in $a_{7R}^{u}$ (whose contribution is, in addition,\ndoubly Cabibbo suppressed), but the situation could be different for\n$a_{7R}^{c,{\\rm soft}}$, where only the\nleading-order term in a $1\/m_c$ expansion is included, which does not carry a\ncomplex phase \\cite{Ball:2006eu}. It is not excluded that a\nresummation of higher-order terms in this expansion will generate a\nnon-negligible strong phase --- which is not really relevant for our\nresults in Eq.~(\\ref{eq:SVgamma}), but could be relevant for the\ninterpretation of any new physics to be found in that observable. For\n$S(K^*\\gamma)$, on the other hand, no new phases are required, and\nany enhancement of $ |\\bar{\\cal A}_R|$ (and $|{\\cal A}_L|$) by new physics will\nresult in a larger value of $S(K^*\\gamma)$.\n\nFor all $S$ except $S(K^*\\gamma)$,\nthe uncertainty is entirely dominated by that of the soft-gluon emission\nterms $l_{u,c}-\\tilde l_{u,c}$, whose uncertainties have been doubled with\nrespect to those given in Tab.~\\ref{rad_tab5}. The smallness of\n$S((\\rho,\\omega)\\gamma)$ is due to the fact that the helicity\nfactor is given by $m_d\/m_b$ (we use $m_{u,d}\/m_s = 1\/24.4$ from\nChPT). For $\\bar K^*$,\nthe suppression from the small mixing\nangle is relieved by the fact that both weak amplitudes in\n$\\lambda_U^{(d)}$ contribute\nso that the CP asymmetry is comparable\nwith that of $\\rho$ and $\\omega$. Despite the generous uncertainties, it is\nobvious that none of these CP symmetries is larger than\n4\\% in the SM, which makes these observables very interesting\nfor new physics searches. The present experimental result from the $B$\nfactories, $S(K^*\\gamma)=-0.28\\pm 0.26$ \\cite{Barberio:2007cr}, certainly encourages\nthe hope that new physics may manifest itself in that\nobservable. While a measurement of the $b\\to d$ CP asymmetries is\nprobably very difficult even at a super-flavour factory,\n$S(K^*\\gamma)$ is a promising observable for $B$ factories \\cite{superB}, but\nnot for the LHC.\\footnote{$K^*$ has to be traced via its decay into a CP\neigenstate, i.e.\\ $K_S\\pi^0$. Neutrals in the final state are not\nreally LHC's favourites.} $B_s\\to \\phi(\\to K^+K^-)\\gamma$, on the\nother hand, will be studied in detail at the LHC, and in particular at\nLHCb, and any largely enhanced value of $S(\\phi\\gamma)$ \nwill be measured within the first years of running. \n\n\\section{Extraction Of CKM Parameters}\\label{ckmextract}\n\nLet us now turn to the determination of CKM parameters from the\nbranching ratios determined in Section~\\ref{rad_brs}. In this context, two particularly interesting\nobservables are \n\\begin{equation}\\label{58}\nR_{\\rho\/\\omega}\\equiv\\frac{\\overline{\\cal B}(B\\to (\\rho,\\omega)\\gamma)}{\n\\overline{\\cal B}(B\\to K^*\\gamma)}\\,,\\qquad\nR_{\\rho}\\equiv\\frac{\\overline{\\cal B}(B\\to \\rho\\gamma)}{\n\\overline{\\cal B}(B\\to K^*\\gamma)}\\,,\\qquad\n\\end{equation}\ngiven in terms of the CP- and isospin-averaged branching ratios of\n$B\\to(\\rho,\\omega)\\gamma$ and $B\\to \\rho\\gamma$, respectively, \nand $B\\to K^*\\gamma$ decays. $R_{\\rho\/\\omega}$\nhas been measured by both {\\sc BaBar} and Belle \\cite{babar_rad,belle_rad}, a first\nvalue of $R_\\rho$ has been given by {\\sc BaBar} \\cite{babar_rad}. \nThe experimental determinations\nactually assume exact isospin symmetry, i.e.\\\n$\\overline{\\Gamma}(B^\\pm\\to \\rho^\\pm\\gamma) \\equiv\n2 \\overline{\\Gamma}(B^0\\to \\rho^0\\gamma)$, and also\n$\\overline{\\Gamma}(B^0\\to \\rho^0\\gamma) \\equiv \n\\overline{\\Gamma}(B^0\\to \\omega\\gamma)$; and as we have seen in Section~\\ref{isosec}, these relations are not in fact exact. Hence, the present\nexperimental results for $R_{\\rho\/\\omega}$ are theory-contaminated. \nAs the isospin asymmetry between the charged and neutral $\\rho$ decay \nrates turns out to be smaller than the asymmetry\n between $\\rho^0$ and $\\omega$, \nit would actually be preferable, from an experimental point of view,\nto drop the $\\omega$ channel and\nmeasure $R_\\rho$ instead of $R_{\\rho\/\\omega}$, as done in the most\nrecent {\\sc BaBar} analysis on that topic \\cite{babar_rad}. \nWe will give numerical results and\ntheory uncertainties for both $R_{\\rho\/\\omega}$ and $R_{\\rho}$.\n\nOne parametrisation of $R_{\\rho\/\\omega}$ often quoted, \nin particular in experimental papers, is\n\\begin{equation}\\label{Brat}\nR_{\\rho\/\\omega} = \\left|\\frac{V_{td}}{V_{ts}}\\right|^2\n\\left(\\frac{1-m_{\\rho}^2\/m_B^2}{1-m_{K^*}^2\/m_B^2}\\right)^3\n\\frac{1}{\\xi^2_{\\rho}} \\left [ 1 +  \\Delta R\\right],\n\\end{equation}\nwith $\\Delta R=0.1\\pm 0.1$ \\cite{BVga1} and again assuming isospin\nsymmetry for $\\rho$ and $\\omega$. This parametrisation creates the impression\nthat $\\Delta R$ is a quantity completely unrelated to\nand with a fixed value independent of\n$|V_{td}\/V_{ts}|$. We would like to point out here that this \nimpression is {\\em wrong}: $\\Delta R$ contains both QCD\n(factorisable and non-factorisable) effects and such from weak\ninteractions. In Ref.~\\cite{Ball:2006nr}  $\\Delta R$ is expressed in\nterms of the factorisation coefficients $a_{7L}^U$, assuming isospin\nsymmetry for $\\rho^0$ and $\\omega$, as\n\\begin{eqnarray}\n1+\\Delta R & = & \\left|\n  \\frac{a_{7L}^c(\\rho)}{a_{7L}^c(K^*)}\\right|^2 \\left( 1 +\n  {\\rm Re}\\,(\\delta a_\\pm + \\delta a_0) \\left[\\frac{R_b^2 - R_b\n  \\cos\\gamma}{1-2 R_b \\cos\\gamma + R_b^2}\\right]\\right.\\nonumber\\\\\n& & \\left. + \\frac{1}{2}\\left( |\\delta a_\\pm|^2 + |\\delta a_0|^2\\right)\n  \\left\\{ \\frac{R_b^2}{1-2 R_b \\cos\\gamma + R_b^2}\\right\\} \\right)\n\\label{delR}\n\\end{eqnarray}\nwith $\\delta a_{0,\\pm}=\na_{7L}^u(\\rho^{0,\\pm})\/a_{7L}^c(\\rho^{0,\\pm})-1$. Eq.~(\\ref{delR}) shows explicitly that $\\Delta R$ depends both on QCD\n ($\\delta a_{\\pm,0}$) and CKM parameters ($R_b,\\gamma$).\nThe point we would like to make is that the calculation of $\\Delta R$\nrequires input values for $R_b$ and $\\gamma$. Once these parameters\n(and the Wolfenstein parameter $\\lambda$)\nare fixed, however, $|V_{td}\/V_{ts}|$ is also fixed and given by\n\\begin{equation}\\label{61}\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = \\lambda \\sqrt{1-2 R_b \\cos\\gamma\n  + R_b^2} \\left[ 1 + \\frac{1}{2}\\,( 1 - 2 R_b \\cos\\gamma) \\lambda^2 +\n  \\mathcal{O}(\\lambda^4)\\right]\\,.\n\\end{equation} \nHence, as $|V_{td}\/V_{ts}|$ and $(R_b,\\gamma)$ are not independent\nof each other, \nit is {\\em impossible} to extract $|V_{td}\/V_{ts}|$ from (\\ref{Brat})\nwith a fixed value of $\\Delta R$. Of course $R_{\\rho\/\\omega}$ and $R_\\rho$ of (\\ref{58}) \n{\\em can} be used to extract information\nabout CKM parameters, but in order to do so one has to settle for a set\nof truly independent parameters. Based on (\\ref{61}), one can\nexchange, say, $\\gamma$ for $|V_{td}\/V_{ts}|$.\\footnote{Strictly speaking, (\\ref{61}) only\n  fixes $\\cos\\gamma$ as function of $|V_{td}\/V_{ts}|$, leaving\n  a twofold degeneracy of $\\gamma$. Eq.~(\\ref{delR}), however, only\n  depends on $\\cos\\gamma$, so that indeed one can unambiguously \nreplace $\\gamma$ by $|V_{td}\/V_{ts}|$.}  So we can either consider $R_V$ as a\nfunction of the CKM parameters $R_b$ and $\\gamma$ (let us call this\nthe $\\gamma$ set of parameters) or as a function of $R_b$ and\n$|V_{td}\/V_{ts}|$ (to be called the $|V_{tx}|$ set). Using the\n$\\gamma$ set, a measurement of $R_V(\\gamma,R_b)$ allows a\ndetermination of $\\gamma$, whereas \n$R_V(|V_{td}\/V_{ts}|,R_b)$ allows the\ndetermination of $|V_{td}\/V_{ts}|$. \nIn either case, the simple quadratic relation\n(\\ref{Brat}) between $R_V$ and $|V_{td}\/V_{ts}|$ becomes more\ncomplicated.\n\n\n In Figs.~\\ref{rad_fig5} and \\ref{rad_fig6} we plot the resulting values of $|V_{td}\/V_{ts}|^2$ and $\\gamma$, respectively,  as a  function of $R_V$. Although the curve in Fig.~\\ref{rad_fig5}(a) looks like a straight line, as naively expected from (\\ref{Brat}), this is not exactly the case, because of the dependence of $\\Delta R$ on $|V_{td}\/V_{ts}|$. In  Fig.~\\ref{rad_fig5}(b) we plot $\\Delta R$ for the $|V_{tx}|$ set of parameters. The dependence of $\\Delta R$ on $|V_{td}\/V_{ts}|$ is rather strong. Apparently indeed $\\Delta R=0.1\\pm 0.1$ in the expected range $0.16<|V_{td}\/V_{ts}|<0.24$, but this estimate does not reflect  the true theoretical uncertainty which is indicated by the dashed lines in the figure. \n\\begin{figure}\n$$\n\\epsfxsize=0.45\\textwidth\\epsffile{rad_fig4a.eps}\\qquad\\epsfxsize=0.45\\textwidth\\epsffile{rad_fig4b.eps}\n$$\n\\caption[$|V_{td}\/V_{ts}|^2$ as function of   $R_{\\rho\/\\omega}$ and $\\Delta R$ as function of   $|V_{td}\/V_{ts}|$.]{\\small Left panel: $|V_{td}\/V_{ts}|^2$ as function of   $R_{\\rho\/\\omega}$, Eq.~(\\ref{58}),   in the $|V_{tx}|$ basis -- see   text.  Solid line: central values. Dash-dotted lines: theoretical   uncertainty induced by $\\xi_\\rho = 1.17\\pm 0.09$, (\\ref{xirho}).  Dashed lines: other   theoretical uncertainties, including those induced   by $|V_{ub}|$, $|V_{cb}|$ and the hadronic parameters of   Tab.~\\ref{rad_tab8}. Right panel: $\\Delta R$ from   Eq.~(\\ref{delR}) as function of\n  $|V_{td}\/V_{ts}|$   in the $|V_{tx}|$ basis. Solid   line: central values. Dashed   lines: theoretical uncertainty.}\\label{rad_fig5} \n  $$\\epsfxsize=0.45\\textwidth\\epsffile{rad_fig5.eps}$$\n\\caption[The UT angle $\\gamma$ as function of   $R_{\\rho\/\\omega}$.]{\\small The UTangle $\\gamma$ as function of   $R_{\\rho\/\\omega}$ in the $\\gamma$ set of CKM parameters. Solid   lines: central values of input parameters. Dash-dotted lines: theoretical   uncertainty induced by $\\xi_\\rho = 1.17\\pm 0.09$. Dashed lines: other   theoretical uncertainties.}\\label{rad_fig6}\n$$\\epsfxsize=0.45\\textwidth\\epsffile{rad_fig6.eps}$$\n\\caption[Central values of  $R_{\\rho\/\\omega}$ and $R_{\\rho}$ as functions of $|V_{td}\/V_{ts}|$]{\\small Central values of  $R_{\\rho\/\\omega}$ (solid line) and $R_{\\rho}$   (dash-dotted line) as functions of $|V_{td}\/V_{ts}|$.}\\label{rad_fig7}\n\\end{figure}\n\nIt is now basically a matter of choice whether to use\n$R_{\\rho\/\\omega}$ to determine $|V_{td}\/V_{ts}|$ or $\\gamma$. Once one of\nthese parameters is known, the other one follows from Eq.~(\\ref{61}). In\nFig.~\\ref{rad_fig6} we plot $\\gamma$ as a function of\n$R_{\\rho\/\\omega}$, together with the theoretical uncertainties. In\nFig.~\\ref{rad_fig7} we also compare the central values of\n$R_{\\rho\/\\omega}$ with those of $R_{\\rho}$, as a function of\n$|V_{td}\/V_{ts}|$. Although the difference is small, $R_{\\rho}$ is\nexpected to be larger than $R_{\\rho\/\\omega}$.  $R_{\\rho\/\\omega}$ and $R_{\\rho}$ are dominated by the  uncertainties of $\\xi_{\\rho}$ and as discussed in Ref.~\\cite{Ball:2006nr}, a reduction of this uncertainty  would require a reduction of the uncertainty of the transverse decay constants $f_V^\\perp$ of $\\rho$ and $K^*$. With the most recent results from {\\sc BaBar}, $R_{\\rho\/\\omega} = 0.030\\pm 0.006$ \\cite{babar_rad}, and from Belle, $R_{\\rho\/\\omega} = 0.032\\pm 0.008$ \\cite{belle_rad}, we then find\n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{array}[b]{l@{\\quad}l@{\\quad\\leftrightarrow\\quad}l}\n\\mbox{{\\sc BaBar}:} & \\displaystyle\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = 0.199\\overbrace{^{+0.022}_{-0.025}}^{{\\rm exp}}\\pm\n\\overbrace{0.014}^{{\\rm th}} &\\displaystyle\n\\gamma = (61.0\\overbrace{^{+13.5}_{-16.0}}^{{\\rm exp}}\\overbrace{^{+8.9}_{-9.3}}^{\n{\\rm th}})^\\circ\\,,\\\\[10pt]\n\\mbox{Belle:} & \\displaystyle\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = 0.207\\,^{+0.028}_{-0.033}\\,\n^{+0.014}_{-0.015} &\\displaystyle\n\\gamma = (65.7\\,^{+17.3}_{-20.7}\\,^{+8.9}_{-9.2})^\\circ\\,.\n\\end{array}\n\\label{63}\n\\end{equation}\nThese numbers  compare well with the Belle result  \\cite{Bellegamma} \nfrom tree-level processes, $\\gamma=(53\\pm 20)^\\circ$, quoted in\nTab.~\\ref{rad_tab8}, and results from  global fits\n\\cite{global}. We also would like to point out that the above\ndetermination of $\\gamma$ is actually a determination of\n$\\cos\\gamma$, via Eq.~(\\ref{61}), and implies, in principle, a twofold\ndegeneracy $\\gamma\\leftrightarrow 2\\pi-\\gamma$. This is in contrast to the\ndetermination from $B\\to D^{(*)} K^{(*)}$ in \\cite{Bellegamma}, which\ncarries a twofold degeneracy\n$\\gamma \\leftrightarrow \\pi+\\gamma$. Obviously these two\ndeterminations taken together remove the degeneracy and \nselect $\\gamma\\approx 55^\\circ<180^\\circ$. If \n$\\gamma\\approx 55^\\circ+180^\\circ$ instead, one would have \n$|V_{td}\/V_{ts}|\\approx 0.29$ from\n(\\ref{61}), which is definitely ruled out by data. Hence, the result\n(\\ref{63}) confirms the SM interpretation of $\\gamma$ from \nthe tree-level CP asymmetries in $B\\to D^{(*)} K^{(*)}$.\n\n\\begin{table}[ht]\n$$\n\\begin{array}{c|c|c}\n\\tau_{B^0} & \\tau_{B^\\pm}\/\\tau_{B^0} & \\tau_{B_s^0}\/\\tau_{B^0}\n\\\\\\hline\n1.530(9)\\,{\\rm ps} & 1.071(9) & 0.958(39)\n\\end{array}\n$$\n\\caption[$B$ lifetimes from HFAG.]{\\small $B$ lifetimes from HFAG \\cite{Barberio:2007cr}.}\n\\label{rad_tab9}\n\\addtolength{\\arraycolsep}{3pt}\n\\renewcommand{\\arraystretch}{1.3}\n$$\n\\begin{array}{c|c|c|c|c|c}\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{CKM parameters and couplings}}\\\\\\hline\n\\lambda \\mbox{~\\cite{Yao:2006px}} & |V_{cb}| \\mbox{~\\cite{inclmoments}} & \n|V_{ub}| & \\gamma \\mbox{~\\cite{Bellegamma}} & \\alpha_s(m_Z)\n\\mbox{~\\cite{Yao:2006px}} & \\alpha_{\\rm QED}\\\\\\hline\n0.227(1) & 42.0(7)\\times 10^{-3} & 4.0(7)\\times\n10^{-3} & (53\\pm 20)^\\circ & 0.1176(20) & 1\/137\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{B parameters}}\\\\\\hline\nf_{B_q}\\mbox{~\\cite{Onogi}} & f_{B_s}\\mbox{~\\cite{Onogi}} &\n    \\lambda_{B_q}(\\mu_h) \\mbox{~\\cite{Ball:2006nr}} & \\lambda_{B_s}(\\mu_h)\n& \\mu_h     \\\\\\hline\n200(25)\\,{\\rm MeV} & 240(30)\\,{\\rm MeV} & 0.51(12)\\,{\\rm GeV} &\n0.6(2)\\,{\\rm GeV} & 2.2\\,{\\rm GeV}\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{$\\rho$ parameters}}\\\\\\hline\nf_{\\rho} & f_{\\rho}^\\perp & a_1^\\perp({\\rho}) &\na_2^\\perp({\\rho}) & T_1^{B\\to\\rho}(0)\\\\\\hline\n216(3)\\,{\\rm MeV} & 165(9)\\,{\\rm MeV} & 0 & 0.14(6) & 0.27(4)\n\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{$\\omega$  parameters}}\\\\\\hline\nf_{\\omega} & f_{\\omega}^\\perp & a_1^\\perp({\\omega}) &\na_2^\\perp({\\omega}) & T_1^{B\\to\\omega}(0)\\\\\\hline\n187(5)\\,{\\rm MeV} & 151(9)\\,{\\rm MeV} & 0 & 0.15(7) & 0.25(4)\n\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{$K^*$ parameters}}\\\\\\hline\nf_{K^*} & f_{K^*}^\\perp & a_1^\\perp({K^*})\\mbox{~\\cite{Ball:2005vx}} & \na_2^\\perp({K^*}) & T_1^{B_q\\to K^*}(0) & \nT_1^{B_s\\to \\bar  K^*}(0)\\\\\\hline\n220(5)\\,{\\rm MeV} & 185(10)\\,{\\rm MeV} & 0.04(3) & 0.15(10) &\n0.31(4) & 0.29(4)\\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{$\\phi$ parameters}}\\\\\\hline\nf_{\\phi} & f_{\\phi}^\\perp & a_1^\\perp({\\phi}) &\na_2^\\perp({\\phi}) & T_1^{B_s\\to\\phi}(0)\\\\\\hline\n215(5)\\,{\\rm MeV} & 186(9)\\,{\\rm MeV} & 0 & 0.2(2) & 0.31(4) & \\\\\\hline\\hline\n\\multicolumn{6}{c}{\\mbox{quark masses}}\\\\\\hline\n\\multicolumn{2}{c|}{m_s(2\\,{\\rm GeV})\\mbox{~\\cite{ms}}} \n& m_b(m_b)\\mbox{~\\cite{inclmoments}} & \nm_c(m_c)\\mbox{~\\cite{czakon}} & \\multicolumn{2}{c}{\nm_t(m_t)\\mbox{~\\cite{mt}}}\\\\\\hline\n\\multicolumn{2}{c|}{100(20)\\,{\\rm MeV}} \n& 4.20(4)\\,{\\rm GeV} & 1.30(2)\\,{\\rm GeV} & \n\\multicolumn{2}{c}{163.6(2.0)\\,{\\rm GeV}} \\\\\\hline\\hline\n\\end{array}\n$$\n\\caption[Summary of input parameters.]{\\small Summary of input parameters. The  value of $|V_{ub}|$ is an average over inclusive and exclusive  determinations and the result from UTangles Refs.~\\cite{Barberio:2007cr,global,Vub}. None of our results is very sensitive  to $|V_{ub}|$. For an explanation of our choice of the value of the UT angle  $\\gamma$, see text. $\\lambda_{B_s}$ is obtained from   $\\lambda_{B_q}$, see Eq.~(\\ref{rad_bs}).  The vector meson  decay constants $f_V$, $f_V^\\perp$ are discussed in Ref.~\\cite{Ball:2006eu}; the  values of the Gegenbauer moments $a_i^\\perp$ are compiled from  various sources   \\cite{Ball:2006nr,Ball:1996tb,Ball:1998sk,Ball:2003sc} and include only small ${\\rm SU(3)_F}$-breaking,  in line with the findings for pseudoscalar mesons \\cite{Ball:2006wn}. The form factors $T_1$  are \nupdates of previous LCSR results  \\cite{Ball:2004rg}, including the updated values of the decay constants\n $f_{\\rho,\\omega,\\phi}$ and of $a_1^\\perp({K^*})$ \\cite{Ball:2005vx,Ball:2006fz}. \n All scale-dependent quantities are given at the scale $\\mu=1\\,$GeV unless stated otherwise.}\n\\label{rad_tab8}\n\\end{table}\n\\chapter{Summary and Conclusions}\\label{chapter8_conc}\nThis thesis has consisted of three main analyses centred on the investigations and determinations of meson light-cone distribution amplitudes.  We have seen how the determinations of decay observables in $B$ decays are reliant on the sound understanding of both theoretical and experimental uncertainties with the work presented in this thesis striving towards the former. To summarise:\n\nWe began, in Chapter~\\ref{chapter1_basics}, with a brief introduction defining the QCD Lagrangian, discussing CP violation and the $\\Delta B =1$ effective Hamiltonian.\n\nIn Chapter~\\ref{chapter2_DAs} we investigated the structure of vector mesons distribution amplitudes to twist-3 accuracy. We included all $\\rm SU(3)_F$-breaking and G-parity violating effects. The QCD equations of motion were implemented to unpick the interwoven relations between the distribution amplitudes ultimately expressing the two-particle twist-3 distribution amplitudes in terms of the three-particle twist-3 and two-particle twist-2 distribution amplitudes. The equations of motion result in integral equations which are readily solved order-by-order in conformal spin and to the order considered all the distribution amplitudes are then expressed by a small number of non-perturbative parameters. Finite quark mass effects appear in the equation of motion and therefore impact the two-particle twist-3 distribution amplitudes (\\ref{das_eq31}-\\ref{das_eq33}). Such effects also cause mixing between the twist-3 hadronic parameters under renormalisation scale evolution, see Eq.~(\\ref{das_eq37}).\n\nIn Chapter~\\ref{chapter3_SR} we discussed the methods of QCD sum rules (the SVZ method) and QCD sum rules on the light-cone. We outlined the procedures with example correlation functions and ended the chapter with an example calculation of the $\\alpha_s$ corrections to the gluon condensate contribution to a $K$ meson sum rule. The calculation made use of the background field technique and served to illustrate the calculation of radiative corrections to -- and extraction of -- vacuum condensates in the SVZ method. The result of the calculation is in conflict with that in the literature, see Eqs.~(\\ref{last}) and (\\ref{least}).\n\n\nIn Chapter~\\ref{chapter4_det}  we determined the leading hadronic parameters defined in Chapter~\\ref{chapter2_DAs} via SVZ sum rules. We calculated the three-particle twist-3 parameters to NLO in conformal spin, also including all G-parity violating terms and finite strange quark mass effects. The determination of the twist-3 parameters is new for $K^*$ and $\\phi$. The results for the $\\rho$ agree within uncertainties  with previous determinations and are presented in Tabs.~\\ref{det_tab1} and \\ref{det_tab2}. We also calculate $\\mathcal{O}(\\alpha_s)$ and $\\mathcal{O}(m_s^2)$ corrections to the quark condensate for the sum rules for $a_n^{\\parallel,\\perp}(V)$, which for $n=2$ is the first non-trivial Gegenbauer coefficient of the G-even particles $\\rho$ and $\\phi$. We add this contribution to the existing sum rules taken from the literature and update the value of $a_2^{\\parallel,\\perp}(\\phi)$ which we find to be consistent with that found for $K^*$ and $\\rho$;  $a^\\perp_{1,2}(V)=a^\\parallel_{1,2}(V)$ within uncertainties. The results find direct application in QCD factorisation descriptions of $B\\to V$ decays, and the light-cone sum rule analyses of $B\\to V$ transition form factors.\n\nIn Chapter~\\ref{chapter5_eta} we calculated the form factors of $B\\to \\eta^{\\prime}$\nsemileptonic transitions from light-cone sum rules,\nincluding the gluonic singlet contributions. We built upon the previous light-cone sum rule determination of the $B\\to \\eta$ form factor by casting the calculation consistently within the phenomenologically motivated $\\eta$-$\\eta^{\\prime}$ mixing scheme of Refs.~\\cite{Feldmann:1998vh,Feldmann:1998sh}. We found that, as\nexpected, these contributions are more relevant for $f_+^{\\eta^{\\prime}}$ than\nfor $f_+^\\eta$ and can amount up to 20\\% in the former, \ndepending on the only poorly\nconstrained leading Gegenbauer moment $B^g_2$ of the gluonic twist-2\ndistribution amplitude of $\\eta^{\\prime}$.  The numerical results, with each contribution listed separately, are given by Eqs.~(\\ref{fp1}) and (\\ref{fp0}). Consequently, it seems unlikely that the large exclusive $B\\to \\eta^{\\prime} K$ and inclusive $B\\to \\eta^{\\prime} X$ branching ratios can be explained by a large $B^g_2$, as it would have to assume a very extreme value. We also found that the form factors\nare sensitive to the values of the twist-2 two-quark Gegenbauer\nmoments $a_2^{\\eta,\\eta^{\\prime}}$ which, given the uncertainty of independent\ndeterminations, we have set equal to $a_2^\\pi$, see Fig.\\ref{eta_diags3}.\n\nThe ratio\nof branching ratios ${\\cal B}(B\\to\\eta^{\\prime} e\\nu)\/{\\cal B}(B\\to\\eta e\\nu)$\nis sensitive to both $a_2$ and $B^g_2$ and may be used to constrain\nthese parameters, once it is measured with sufficient accuracy, see Fig.~\\ref{eta_fig8}. The\nextraction of $|V_{ub}|$ from these semileptonic decays, in particular\n$B\\to\\eta e\\nu$, with negligible singlet contribution, although\npossible in principle, at the moment is obscured by the lack of\nknowledge of $a_2$. We\nwould also like to stress that, in the framework of the quark-flavour\nmixing scheme for the $\\eta$-$\\eta^{\\prime}$ system as used in this analysis,\n$B\\to \\eta^{\\prime}$ transitions probe only the $\\eta_q$ component of these\nparticles. The $\\eta_s$ component could be probed directly for\ninstance in the $b\\to s$ penguin transition $B_s\\to \\eta^{\\prime}\n\\ell^+\\ell^-$, although such a measurement would also be sensitive\nto new physics in the penguin diagrams.\n\nIn Chapter~\\ref{chapter6_QCDF} we discussed the QCD factorisation (QCDF) approach of Refs.~\\cite{Beneke:1999br, Beneke:2000ry} and its application to the radiative $B$ decays $B \\to V \\gamma$ of Refs.~\\cite{Bosch:2001gv,Bosch:2002bw}.   We discussed the appearance of distribution amplitudes in the factorisation formulas and focused on the leading contributions to the $B\\to V \\gamma$ decays.\n\n\nIn Chapter~\\ref{chapter7_rad} we performed a phenomenological analysis of the radiative $B$ decays to vector mesons $B\\to V \\gamma$, using the framework discussed in Chapter~\\ref{chapter6_QCDF}. We investigated the most relevant power-suppressed corrections to the QCDF predictions for the radiative decays $B_{u,d} \\to (\\rho, \\omega, K^*)\\gamma$ and  $B_{s} \\to (\\phi, \\bar{K}^*)\\gamma$. We use the QCDF framework presented in Refs.~\\cite{Bosch:2001gv,Bosch:2002bw} in which we find use for the twist-2 DA parameters determined in Chapter~\\ref{chapter4_det}. Besides the leading QCDF contributions we included long-distance photon emission and soft-gluon mission from quark loops. These effects, although formally $\\sim 1\/m_b$ with respect to the leading contributions, augment the QCDF predictions for the branching ratios, CP and isospin asymmetries. \n\nThe impact of the power-suppressed corrections on the branching ratios is found to be very small, with the exception of the weak annihilation contributions to $B^\\pm\\to \\rho^\\pm \\gamma$ which are large due to a large combination of Wilson coefficients $C_2+C_1\/3=1.02$ and no CKM-suppression. Moreover, long-distance photon emission also impacts most here, see Eq.~(\\ref{LDcont}). An explicit break down  of the results are given in Tab.~\\ref{rad_tab6}. \n\nThe isospin asymmetries $A(\\rho,\\omega)$, $A_I(\\rho)$ and $A_I(K^*)$  are driven by weak annihilation and long-distance photon emission contributions. We found a non-zero asymmetry $A(\\rho,\\omega)=-0.20\\pm0.09$ which suggests the  explicit assumption of perfect symmetry, i.e.\\ $\\overline{\\Gamma}(B^\\pm\\to \\rho^\\pm \\gamma) = 2 \\overline{\\Gamma}(B^0\\to \\rho^0 \\gamma) = 2 \\overline{\\Gamma}(B^0\\to \\omega \\gamma)$ used to obtain the experimental value of $\\overline{\\cal B}(B\\to(\\rho,\\omega)\\gamma)$ is not so well justified. We found $A_I(\\rho)$ to depend strongly on the UT angle $\\gamma$, as shown in Tab.~\\ref{rad_tab7}. With our central value of $\\gamma=53^\\circ$ (see Tab~\\ref{rad_tab8}) our result agrees very well with the {\\sc BaBar} result $A_I(\\rho)_{\\rm BaBar} =  0.56\\pm 0.66$ \\cite{babar_rad}. For $A_I(K^*)$ we found a result consistent with the experimental result $A_I(K^*)_{\\rm exp}=(3.2\\pm4.1)\\%$ and, via its sensitivity to the Wilson coefficient combination $C_5+C_6\/3$ conclude that a reduction in the experimental uncertainty may uncover signs of new physics contributing to these Wilson coefficients, see Fig.~\\ref{rad_fig4}.\n\nThe  indirect CP asymmetries $S(V\\gamma)$ are caused by the interference between the amplitudes describing the production of left and right-handed photons, see Eqs.~(\\ref{qcdf_8}) and (\\ref{54}). The right-handed amplitude is suppressed by $m_D\/m_b$ with respect to the left-handed one for $\\bar B =b \\bar q$ decays (and vice versa for $B$ decays). Due to this natural suppression in  the SM we expect the CP asymmetries to be small, and this suppression can be relieved by many new physics senarios. We investigated the soft-gluon effects arising from soft heavy and soft quark loops. The calculation of these contributions makes use of the three-particle twist-3 DA parameters determined in Chapter~\\ref{chapter4_det}. They contribute to both the left and right-handed amplitudes, and so may also relieve to SM suppression. We found that although they do divert the results from the values naively expected, there is no scope for a large enhancement due to these power-suppressed contributions. The results are given in Eq.~(\\ref{eq:SVgamma}).\n\nFinally, using the most recent results from {\\sc BaBar} and Belle,  we extracted the CKM parameter ratio $|V_{td}\/V_{ts}|$ and equivalently the UT angle $\\gamma$ from the  ratio of branching ratios $R_{\\rho\/\\omega}$. The results are \n\\begin{equation}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{array}[b]{l@{\\quad}l@{\\quad\\leftrightarrow\\quad}l}\n\\mbox{{\\sc BaBar}:} & \\displaystyle\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = 0.199\\overbrace{^{+0.022}_{-0.025}}^{{\\rm exp}}\\pm\n\\overbrace{0.014}^{{\\rm th}} &\\displaystyle\n\\gamma = (61.0\\overbrace{^{+13.5}_{-16.0}}^{{\\rm exp}}\\overbrace{^{+8.9}_{-9.3}}^{\n{\\rm th}})^\\circ\\,,\\\\[10pt]\n\\mbox{Belle:} & \\displaystyle\n\\left|\\frac{V_{td}}{V_{ts}}\\right| = 0.207\\,^{+0.028}_{-0.033}\\,\n^{+0.014}_{-0.015} &\\displaystyle\n\\gamma = (65.7\\,^{+17.3}_{-20.7}\\,^{+8.9}_{-9.2})^\\circ\\,.\n\\end{array}\n\\end{equation}\nand agree well with the Belle result $\\gamma=(53\\pm20)^\\circ$ obtained from tree-level processes, and results from global fits \\cite{global}. The result confirms the SM interpretation of $\\gamma$ from \nthe tree-level CP asymmetries in $B\\to D^{(*)} K^{(*)}$.\n\n\\chapter*{Acknowledgements}\n\nFirst and foremost, I would like to thank my supervisor Patricia Ball for all her help and guidance over the last three years. It has been a great opportunity to work with her, and a fantastic learning experience. I must also thank Roman Zwicky for always finding the time to quell my confusions, and with whom it was a pleasure to collaborate. Also, I thank Angelique Talbot for all her friendly discussions, and I wish Aoife Bharucha all the best with her future projects.\n\nI also thank my office mates Ciaran Williams, Karina Williams, Kemal Ozeren, Martyn Gigg and Stefan Hoeche, and the many other friends who have made my time in Durham and the IPPP so enjoyable.\n\nTo those whose support cannot be appreciated enough; I must thank my parents. I thank my brother too for all the discussions and debates we had over coffee, and finally, I must also thank my grandparents. \n\nThis work was supported by a PPARC studentship which is gratefully acknowledged.\n\n\\chapter*{Declaration}\nI declare that no material presented in this thesis has previously been submitted\nfor a degree at this or any other university. The research described in this thesis has been carried out in collaboration with Prof.~Patricia Ball and Dr.~Roman Zwicky and has been published as follows:\n\\begin{itemize}\n  \\item{``$B \\to V \\gamma$ beyond QCD factorisation,''\\newline\n  P.~Ball, G.~W.~Jones and R.~Zwicky, Phys.\\ Rev.\\  D {\\bf 75} (2007) 054004,\\newline [arXiv:hep-ph\/0612081].}\n\n    \\item{``Twist-3 distribution amplitudes of $K^*$ and $\\phi$ mesons,''\\newline\nP.~Ball and G.~W.~Jones, JHEP {\\bf 03} (2007) 069, \\newline [arXiv:hep-ph\/0702100].}\n   \n    \\item{``$B \\to \\eta^{(\\prime)}$ Form Factors in QCD,''\\newline\nP.~Ball and G.~W.~Jones, JHEP {\\bf 08} (2007) 025, \\newline  arXiv:0706.3628 [hep-ph].}\n   \n\\end{itemize}\nThe copyright of this thesis rests with the author.  No quotation from it should be published without their prior written consent and information derived from it should be acknowledged. \n\\begin{flushright}\ngarethwarrenjones@gmail.com\n\\end{flushright}\n\\tableofcontents\n\\listoffigures\n\\listoftables\n\\input{chapter0_intro.tex}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $K$ be a complete discretely valued field of mixed characteristic with perfect residue field $k$. Fix a separable closure of $\\overline{K}$ of $K$ and let $G_K$ be the absolute Galois group of $K$. The study of stable lattices in crystalline representations of $G_K$ plays an important role in number theory. For example, in many modularity lifting results, one wants to understand liftings of mod $p$ representations of the Galois group of a number field $F$ to Galois representations over $\\mathbb Z_p$-lattices with nice properties when restricted to the Galois groups of $F_v$ for all places $v$ of $F$. And a reasonable property at places over $p$ is that the representation of the Galois group of the local field is crystalline. There are various theories about characterizing $G_K$-stable lattices in crystalline representations, for example, theory of strongly divisible lattices of Breuil(cf. \\cite{BreuilIntegral}), Wach modules(cf. \\cite{Wach96} and \\cite{Berger}), Kisin modules(cf. \\cite{KisinFcrystal}), Kisin-Ren's theory(cf. \\cite{KisinRen}) and the theory of $(\\varphi, \\widehat{G})$-modules(cf. \\cite{liu-notelattice}). The theories above state that one can describe lattices in crystalline representations using certain linear algebraic data over certain commutative rings $A$. \n\nIn a recent work of Bhatt-Scholze\\cite{BS2021Fcrystals}, they give a different characterization of the category of lattices in crystalline representations. To explain their result, let $\\O_K$ be the ring of integers in $K$, and they consider the absolute prismatic site $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, which is defined as the opposite category of all bounded prisms over $\\O_K$ and equipped with the faithfully flat topology. Let $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$ be the structure sheaf over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, and $\\mathcal{I}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\subset \\O_{\\mathlarger{\\mathbbl{\\Delta}}}$ be the ideal sheaf of the Hodge-Tate divisor, then $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$ carries a $\\varphi$-action coming from the $\\delta$-structures. A prismatic $F$-crystal in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is defined as a crystal $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$ over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules together with an isomorphism $(\\varphi^\\ast \\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}})[1\/\\mathcal{I}_{{\\mathlarger{\\mathbbl{\\Delta}}}}] \\simeq \\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/\\mathcal{I}_{{\\mathlarger{\\mathbbl{\\Delta}}}}]$. The main result in \\cite{BS2021Fcrystals} is the following:\n\n\\begin{theorem}\\label{thm-intro-main-1}(\\cite[Theorem 1.2]{BS2021Fcrystals} and Theorem~\\ref{Thm-main-1})\nThere is an equivalence of the category of prismatic $F$-crystals in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ and the category of Galois stable lattices in crystalline representations of $G_K$.\n\\end{theorem}\n\nTo relate the result of Bhatt-Scholze with previous works of characterizing lattices in crystalline representations using linear algebraic data, one should first realize the base rings $A$ used in those theories as certain prisms $(A,I)$ over $\\O_K$. Then one should expect that evaluating the prismatic $F$-crystals on $(A,I)$ should recover the corresponding theory. For example, in the theory of Kisin \\cite{KisinFcrystal}, he uses the base ring $A=\\mathfrak{S}:=W(k)[\\![u]\\!]$ with $\\delta(u)=0$, and if one fixes a uniformizer $\\varpi$ of $\\O_K$ which is a zero of an Eisenstein polynomial $E \\in W(k)[u]$, then it is well-known that $(A,(E))$ is the so-called Breuil-Kisin prism which is inside $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$. And Kisin was able to attach any lattice $T$ in a crystalline representation of $G_K$ a finite free $A$-module together with an isomorphism $(\\varphi^\\ast \\mathfrak{M})[1\/E] \\simeq \\mathfrak{M}[1\/E]$. Now, if $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is the prismatic $F$-crystal attaching to $T$ under Theorem~\\ref{thm-intro-main-1}, then Bhatt-Scholze show that the evaluation of $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$ on $(A,(E))$ recovers Kisin's theory (cf. Theorem 1.3 of $loc.cit.$). \n\nThe first question answered in this paper is whether and how one can recover the theory of $(\\varphi,\\hat{G})$-modules from the prismatic $F$-crystals characterization of Bhatt-Scholze. The category of $(\\varphi,\\hat{G})$-modules, roughly speaking, consisting of pairs $((\\mathfrak{M},\\varphi_{\\mathfrak{M}}),\\hat{G})$, where $(\\mathfrak{M},\\varphi_{\\mathfrak{M}})$ is a Kisin module, and $\\hat{G}$ is a $G_K$-action on $\\mathfrak{M}\\otimes_{\\mathfrak{S},\\varphi} \\widehat {\\mathcal R}$ that commutes with $\\varphi_{\\mathfrak{M}}$ and satisfying some additional properties. Here $\\widehat {\\mathcal R}$ is a subring of $\\Ainf$ that is stable under $\\varphi$ and $G_K$, where $\\Ainf=W(\\O_{\\overline{K}}^\\flat)$ introduced by Fontaine, and there is a surjection $\\theta: \\Ainf:=W(\\O_{\\overline{K}}^\\flat) \\to \\widehat{\\O_{\\overline{K}}}$. However, the period ring $\\widehat {\\mathcal R}$ introduced by Liu is not known to be $p$-adically complete or not, and it is even harder to determine whether it can be shown up as a prism. So in order to relate the theory of $(\\varphi,\\hat{G})$-modules with the category of prismatic $F$-crystals of Bhatt-Scholze, we develop a theory of prismatic $(\\varphi,\\hat{G})$-modules, in which theory the ring $\\widehat {\\mathcal R}$ is replaced by $A^{(2)}_{\\st}$, a subring of $\\Ainf$ constructed as certain prismatic envelope in \\S \\ref{subsec-Ast}. \n\nThe first result of this paper is about the theory of prismatic $(\\varphi,\\hat{G})$-modules. We can show similar to the classical $(\\varphi,\\hat{G})$-module theory, there is an equivalence between the category of prismatic $(\\varphi,\\hat{G})$-modules and lattices in semi-stable representations of $G_K$. Moreover, $(A^{(2)}_{\\st},(E))$ is indeed a prism in $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, it admits a map $(A,(E)) \\to (A^{(2)}_{\\st},(E))$ of prisms, and carries an action of $G_K$. For a $G_K$-stable lattice $T$ in a crystalline representation, if $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is the prismatic $F$-crystal attaches to $T$, then evaluating $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ on the morphism $(A,(E)) \\to (A^{(2)}_{\\st},(E))$ recovers the prismatic $(\\varphi,\\hat{G})$-module attaches to $T$. We can also show the map $A^{(2)}_{\\st} \\to \\Ainf \\xrightarrow{\\varphi} \\Ainf$ factor through $\\widehat {\\mathcal R}$, so the theory of prismatic $(\\varphi,\\hat{G})$-modules recovers the classical theory. The ring $A^{(2)}_{\\st}$ is simpler than $\\widehat {\\mathcal R}$ in many ways, although it is still very complicated and non-noetherian, it is more explicitly described and is $p$-adic complete. In particular, our new theory can be used to fix the gap \\cite{liu-Fontaine} indicated by \\cite[Appendix B]{gao2021breuilkisin}.\n\nThe second attempt made in this paper is to provide a new approach to the equivalence between the category of prismatic $F$-crystals and the category of lattices in crystalline representation established by Bhatt and Scholze as in Theorem~\\ref{thm-intro-main-1}. That is, using the known equivalence between lattices in semi-stable representations and prismatic $(\\varphi, \\hat G)$-modules, we will establish a functor from the category of prismatic $(\\varphi, \\hat G)$-modules that correspond to crystalline representations to prismatic $F$-crystals, and show this functor is an equivalence. \n\nTo be more precise, let $T$ be a $G_K$-stable lattice in a crystalline representation with positive Hodge-Tate weights, let $(A, E)$ be the Breuil-Kisin prism, and let $(A^{(2)},(E))$ (resp. $(A^{(3)},(E))$) be the self-product (self-triple-product) of $(A, (E))$ in $(\\O_K)_{\\mathlarger{\\mathbbl{\\Delta}}}$. Then evaluating prismatic $F$-crystals on the diagram $(A,(E)) \\xrightarrow{i_1} (A^{(2)},(E)) \\xleftarrow{i_2} (A,(E))$ induces an equivalence of the category of prismatic $F$-crystals and Kisin modules with descent data, that is pairs $((\\mathfrak{M},\\varphi_{\\mathfrak{M}}),f)$ where $(\\mathfrak{M},\\varphi_{\\mathfrak{M}})$ is a Kisin module and \n$$\nf: \\mathfrak{M}\\otimes_{\\mathfrak{S},i_1} A^{(2)} \\simeq \\mathfrak{M}\\otimes_{\\mathfrak{S},i_2} A^{(2)}\n$$\nis an isomorphism of $A^{(2)}$-modules that is compatible with $\\varphi$ and satisfies cocycle condition over $A^{(3)}$. Using this, to establish an equivalence between prismatic $(\\varphi, \\hat G)$-modules that correspond to crystalline representations and prismatic $F$-crystals, it remains to find certain correspondence between the $\\hat{G}$-action and the descent isomorphism $f$. We will show the descent isomorphism can be obtained by taking the $G_K$-action of the $(\\varphi,\\widehat{G})$-module at a specific element. To be more precise, fix a Kummer tower $K _\\infty = \\bigcup_{n = 1}^\\infty K (\\varpi _n )$ used in the theory of Kisin, where $\\{\\varpi _n\\}_{n}$ is a compatible system of $p^n$-th roots of $\\varpi_0=\\varpi$, and let $L$ be the normalization of $K _\\infty$ inside $\\overline{K}$. Choose $\\tau\\in \\hat{G}:=\\Gal(L\/K)$ satisfying $\\tau(\\varpi_n)=\\zeta_{p^n}\\varpi_n$ such that $\\{\\zeta_{p^n}\\}$ is a compatible system of primitive $p^n$-th roots of $1$, then our slogan is that the descent isomorphism corresponds to the $\\tilde{\\tau}$-action on the Kisin module $\\mathfrak{M}$ inside $T^{\\vee}\\otimes \\Ainf$ where $\\tilde{\\tau}\\in G_K$ is any lifting of $\\tau$ under the quotient map $G_K \\to \\widehat{G}$. \n\nTo sketch our idea, first we have the maps $u\\mapsto [{\\varpi}^\\flat]$ and $u\\mapsto [{\\tau}({\\varpi}^\\flat)]$ defines two morphisms of $(A,(E))$ to $(\\Ainf,\\Ker\\theta)$. By the universal property of $(A^{(2)},(E))$, these two maps induce a morphism $(A^{(2)},(E)) \\to (\\Ainf,\\Ker\\theta)$. We can show this map is injective, and the embedding factors through $A^{(2)}_{\\st}$, which is the base ring used in our prismatic $(\\varphi, \\hat G)$-module theory. That is,  we have a chain of subrings $A\\subset A^{(2)} \\subset A^{(2)}_{\\st}$ of $\\Ainf$, such that $\\tilde{\\tau}(A)$ is also contained in $A^{(2)}$. We can show a prismatic $(\\varphi, \\hat G)$-module corresponds to a crystalline representation if and only if the coefficients of the $\\tilde{\\tau}$-action on $\\mathfrak{M}$ in $T^{\\vee}\\otimes \\Ainf$ lie inside $A^{(2)}$. And once this is proved, the $\\tilde{\\tau}$-action will induce an isomorphism:\n$$\nf_{\\tau}: \\mathfrak{M}\\otimes_{\\mathfrak{S},\\tau} A^{(2)} \\simeq \\mathfrak{M}\\otimes_{\\mathfrak{S}} A^{(2)}.\n$$\nWe will see $f_{\\tau}$ gives the descent isomorphism. As a result, we give a new proof for Theorem~\\ref{thm-intro-main-1}.\n\nAn advantage of our approach is that our new method can be easily generalized to the semi-stable representations cases. It turns out that the prism $(A^{(2)}_{\\st},(E))$ is isomorphic to the self-coproduct of $(A,(E))$ in the category of logarithmic prisms over $\\O_K$ defined by Koshikawa\\cite{Koshikawa2021log-prism}. Using the equivalence between prismatic $(\\varphi, \\hat G)$-modules and lattices in semi-stable representations of $G_K$. we will show in \\S\\ref{sec-logprismandsemistablereps} the following generalization of Theorem~\\ref{thm-intro-main-1} for semi-stable representations. \n\n\\begin{theorem}\\label{thm-intro-log-main}(Theorem~\\ref{thm-log-main-1})\nThere is an equivalence of the category of prismatic $F$-crystals in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules over $(\\O_K)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ and the category of Galois stable lattices in semi-stable representations of $G_K$.\n\\end{theorem}\n\nAnother interesting and natural question one can ask is whether Theorem~\\ref{thm-intro-main-1} and Theorem~\\ref{thm-intro-log-main} can accommodate more general base rings. Motivated by our strategy, it seems to us that the answer should be affirmative if a suitable theory of $(\\varphi, \\hat G)$-module can accommodate more general base rings, for example, if the base ring $R$ is a complete DVR with \\emph{imperfect} residue field that admits a finite $p$-basis. We are working on such direction and hopefully will report our progress in the future. So part of our paper, for example, \\S~\\ref{sec-ring-strcuture} do allow specific general base rings.  \n\n\\subsection*{Acknowledgments} It is our pleasure to thank Hui Gao,  Wansu Kim, Teruhisa Koshikawa, Zeyu Liu,  Yong Suk Moon, Peter Scholze, Koji Shimizu, Yupeng Wang,  Zhiyou Wu and Min Yu for comments and conversations during the preparation of this paper. \n\n\\section{Ring Structures on certain prismatic envelope} \\label{sec-ring-strcuture}\n\nRecall that $K$ is a completed discrete valuation field in mix characteristic $(0 , p)$ with ring of integers of $\\O_K$ and prefect residue field $k$.   Write $W= W(k)$. Let $\\varpi\\in \\O_K$ be a uniformizer and $E= E(u)\\in W[u]$ be the Eisenstein polynomial of $\\varpi$. \nLet $\\mathbb C_p$ be the $p$-adic completion of $\\overline{K}$, and $\\O_{\\mathbb C_p}$ be the ring of integers. Let $R_0$ be a $W(k)$-algebra which admits Frobenius lift $\\varphi : R_0 \\to R_0 $. Set $R: = R_0 \\otimes_{W(k)}\\O_K$. We make the following assumptions for $R_0$ and $R$: \n\\begin{enumerate}\n\\item Both $R_0$ and $R$ are $p$-adically complete integral domains, and $R_0 \/ p R_0= R\/ \\varpi R$ is an integral domain;  \n    \\item Let $\\Breve{R}_0 =W\\langle t _1,  \\dots , t _m \\rangle $. $R_0$ is a $\\Breve{R}_0 $-\\emph{formally \\'etale} algebra with $p$-adic topology;   \n\\item $\\breve{R}_0$ admits a Frobenius lift such that $\\breve{R_0} \\to R_0$ defined in (2) is $\\varphi$-equivalent. \n\n\\item The $k$-algebra $R_0 \/ p R_0$ has finite $p$-basis in the sense of \\cite[Definition 1.1.1]{deJong}.\n\\end{enumerate}\nOur main example is $R_0= \\breve R_0 =  W(k). $ We will not use the finite $p$-basis assumption until \\S4. The following are other examples of  $R_0$:  \n\\begin{example}\\label{Eg-1} \n\\begin{enumerate}\n    \\item $R_0 = W(k) \\langle t _1^{\\pm 1} ,  \\dots , t _m ^{\\pm 1}\\rangle$ with $\\varphi (t_j) = t ^p_j$\n\\item $ R_ 0 = W(k) [\\![t]\\!]$ with $\\varphi (t) = t^p$ or $(1+t)^p -1 $.\n\\item $ R_0$ is an unramified complete DVR with imperfect field $\\kappa$ with finite $p$-basis. See  \\S\\ref{subsec-baserings} for more discussions. \n\\end{enumerate}\n\\end{example} \n\nWe reserve $\\gamma_i(\\cdot)$ to denote $i$-th divided power. \n\\subsection{Construction of \\texorpdfstring{$A^{(2)}$}{A(2)}} \\label{subsrc-construct-A2}\nLet $A=\\mathfrak{S}=R_0[\\![u]\\!]$ and extend $\\varphi : A \\to A$ by $\\varphi (u)= u^p$. It is well-known that $(A, E)$ is a prism and we can define a surjection $\\theta: A \\to R$ via $u\\mapsto \\varpi$. We have $\\Ker\\theta = (E(u))$.  Let $\\breve A := \\breve R_0 [\\![u]\\!]$ and define $\\varphi$ and $\\breve \\theta:  \\breve A \\to \\breve R : = \\O_K \\otimes _W \\breve R_0$ similarly. \nWe set \\[ A ^{\\ho 2}: = A [\\![y -x, s_1 - t_1, \\dots , s_m - t_m]\\!], \\ A^{\\ho 3}: = A[\\![ y -x, w-x , \\{ s_i - t _i , r_i - t_i\\}_{j= 1, \\dots , m}]\\!].\\] \nNote that $A ^{\\ho 2}$  (resp. $ A^{\\ho 3}$) is $\\breve A \\otimes_{\\mathbb Z_p} \\breve A $(resp. $\\breve A \\otimes_{\\mathbb Z_p} \\breve A \\otimes_{\\mathbb Z_p} \\breve A$)-algebra by $ u \\otimes 1 \\mapsto x$, \n$1\\otimes u \\mapsto y$ and $1 \\otimes t_i \\mapsto s_i$ (resp. $1\\otimes 1 \\otimes u \\mapsto w$ and $1 \\otimes 1 \\otimes  t_i \\mapsto r_i$). So in this way, we can extend Frobenius $\\varphi$ of $A$, which is compatible with that on $\\breve A$ to $A ^{\\ho 2}$ and $A^{\\ho 3}$. \nSet $J ^{(2)}= (E, y -x , \\{s_i- t _i \\}_{i = 1, \\dots,  m} )\\subset A^{\\ho 2}$ and $J ^{(3)} = (E, y-x , w-x  , \\{s_i-t_i , r_i - t_i\\}_{i = 1, \\dots , m}) \\subset A ^{\\ho 3}.$ Clearly, we have $A^{\\ho i}\/ J ^{(i)}\\simeq R$ for $i = 2, 3$. And we have $A^{\\ho 2}\/(p,E)$ (resp. $A^{\\ho 3}\/(p,E)$) is a formal power series ring over the variables $\\bar{y}-\\bar{x}, \\{\\bar{s}_i-\\bar{t}_i\\}_{i = 1, \\dots,  m}$ (resp. $\\bar{y}-\\bar{x} , \\bar{w}-\\bar{x}  , \\{\\bar{s}_i-\\bar{t}_i , \\bar{r}_i - \\bar{t}_i\\}_{i = 1, \\dots , m}$), so $(A,(E)) \\to (A^{\\ho i}, J ^{(i)})$ satisfies the requirements of  in \\cite[Prop. 3.13]{BS19}, and we can construct the prismatic envelope with respect to this map, which will be denoted by $A^{(i)}$. More precisely,  $A^{(i)}\\simeq A^{\\ho i}\\left \\{\\frac{J ^{(i)}}{E}\\right\\}_\\delta^{\\wedge}$, here $\\{\\cdot\\}_\\delta^{\\wedge}$ means freely adjoining elements in the category of $(p, E(u))$-completed $\\delta$-$A$-algebras. We will see  $A^{(i)}$, $i = 2 ,3$ are the self product and triple product of $A$ in category $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ in \\S  \\ref{subsec-pris-crystal}. \n\n\\subsection{The ring \\texorpdfstring{$A^{(2)}_{\\max}$}{A2max}} Now we set $t_0 = x$, $s_0 = y$ and \n\\[ z_j = \\frac{s_i - t _i}{E} \\textnormal{ and } z_0 = z=  \\frac{y -x }{E}= \\frac{s_0 - t_0}{E}. \\]\nNote that $A^{(i)}$ are  $A$-algebras via $u \\mapsto x$.\n\\begin{definition}\nLet $\\Omax$ be the $p$-adic completion of the $A$-subalgebra of $A[\\frac{1}{p}]$ generated by $p^{-1}E$. And let $A_{\\max}^{(2)}$ be the $p$-adic completion of the $A$-subalgebra of $ A [z_j ,  \\frac{1}{p}; j = 0 , \\dots , m ]$ generated by $p^{-1}E$ and $\\{\\gamma_i(z_j)\\}_{i\\geq 1, j = 0 , \\dots , m}$. \n\\end{definition}\nWe first note that $A^{(2)}_{\\max}$ is an $A^{\\widehat \\otimes 2}$-algebra via $ (s_j - t_j ) = E z_j, j =0 ,\\dots,  m$. Write $\\iota: A^{\\ho 2} \\to A^{(2)}_{\\max}$ for the structure map. \nBy construction, it is easy to see that $A^{(2)} _{\\max}\\subset R_0[\\frac 1 p] [\\![ E, z_j, j = 0, \\dots , m]\\!]$. In particular, $A^{(2)}_{\\max}$ is a domain and    \nany element $b\\in A^{(2)}_{\\max}$ can be \\emph{uniquely} written as \n$\\sum\\limits_{i_0= 0}^\\infty\\cdots \\sum\\limits_{i_m= 0 }^\\infty  b_{i_1 , \\dots, i _m} \\prod\\limits_{j= 0}^m\\gamma_{i_j} (z_{j})$ with $b _{i_0 , \\dots , i_m} \\in \\Omax$ and $b_{i_0, \\dots , i_m}\\to 0$ $p$-adically when $i_0 + \\cdots+  i_m \\to \\infty$.  \nOur next aim is to define $\\varphi$ on $A^{(2)}_{\\max}$. For this, we need a little preparation. \n\\begin{lemma}\\label{lem-Omax}\n$c: = \\frac{\\varphi(E)}{p}\\in \\Omax$ and $ c^{-1} \\in \\Omax$. \n\\end{lemma}\n\\begin{proof}\nWe have $A$ is a $\\delta$-ring, and $E$ is a distinguished element, so in particular\n$$\n\\varphi(E)\/p=c_0+E^p\/p\n$$\nwhere $c_0=\\delta(E)\\in A^\\times$. So $c = \\varphi(E)\/p\\in \\Omax$, and \n$c ^{-1} = c_0 ^{-1} \\sum\\limits_{i = 0}^\\infty \\frac {(- c_0^{-1} E^p ) ^i }{p ^i }\\in \\Omax. $\n\\end{proof}\n\nNow we define $\\varphi(z)=\\varphi(z_0)= \\frac{y^p-x^p}{\\varphi(E)}$ and $\\varphi (z_j) = \\frac{\\varphi(s_j)  - \\varphi(t_j) }{\\varphi (E)}$. Since \n\\begin{IEEEeqnarray*}{+rCl+x*}\n\\varphi(z)=\\frac{y^p-x ^p}{\\varphi(E)}=c^{-1}\\frac{y^p-x^p}{p}=c^{-1}\\frac{(x+ Ez)^p-x^p}{p} &= & c^{-1}\\sum_{i=1}^px^{p-i}(Ez)^i\\binom{p}{i}\/p\\\\\n&=& c^{-1}\\sum_{i=1}^{p}a_iz^i,\n\\end{IEEEeqnarray*}\nwhere $a_i\\in W(k)[\\![x]\\!][\\frac{E^p}{p}]\\subset \\Omax\\subset A^{(2)}_{\\max}$ and $c$ is a unit in $\\Omax$, we have $ \\varphi (z) \\in A^{(2)}_{\\max}$.\nThen \n$$\n \\gamma_n(\\varphi(z))=\\frac{\\varphi(z)^n}{n!}=\\frac{z^n}{n!}(c^{-1}\\sum_{i=1}^{p}a_iz^{i-1})^n\n$$\nis in $A^{(2)}_{\\max}.$  The argument for $\\varphi (z_j)$ for $j >1$ need a little more details. Note that $\\varphi (t_j) = t_j ^p + p \\delta (t_j)$ with $\\delta(t_j) \\in \\breve R_0$ by our assumptions. It is clear that $\\delta(s_j)-\\delta (t_j) = (s_j - t_j) \\lambda_j$ with $\\lambda_j \\in A ^{\\ho  2}$. Using that $(s_j -t_j) = E z_j$, so\n\\begin{equation}\\label{eqn-special-shape}\n\\varphi (z_j) = c^{-1} (\\frac{s^p _j - t^p_j}{p} + E z_j \\lambda_j)    \n\\end{equation}\nThe same argument as that for $\\varphi (z_0)$ also shows that $\\gamma_n (z_j)\\in A^{(2)}_{\\max}$, for $j =1 , \\dots , m$. \n\nSince any element $b\\in A^{(2)}_{\\max}$ can be uniquely written as $\\sum\\limits_{i_0= 0}^\\infty\\cdots \\sum\\limits_{i_m= 0 }^\\infty  b_{i_1 , \\dots, i _m} \\prod\\limits_{j= 0}^m\\gamma_{i_j} (z_{j})$ with $b _{i_0 , \\dots , i_m} \\in \\Omax$ and $b_{i_0, \\dots , i_m}\\to 0$ $p$-adically when $i_0 + \\cdots+  i_m \\to \\infty$, this allows to extend Frobenius map  $\\varphi $ on $ A$ to a \\emph{ring} map $\\varphi: A^{(2)}_{\\max} \\to A^{(2)}_{\\max}$ by sending $u \\mapsto u ^p$,  $z \\mapsto \\frac{y ^p -x^p}{\\varphi(E)}$,  $\\varphi (z_j) = \\frac{\\varphi(s_j)  - \\varphi(t_j)}{\\varphi (E)}$,  and $\\gamma_i (z_j) \\mapsto \\gamma_i (\\varphi (z_j))$ as the above.\n\n\\begin{remark}\\label{rem-not-Frob-lift} The ring map $\\varphi: A^{(2)}_{\\max}\\to A^{(2)}_{\\max}$ is \\emph{not} a Frobenius lift of $A^{(2)} _{\\max}\/ p$ because $\\varphi (E\/p)- (E\/p) ^p \\not \\in p A^{(2)}_{\\max}$. In particular, $A^{(2)}_{\\max}$ is not a $\\delta$-ring. \n\\end{remark}\n\nRecall that $A^{(2)}_{\\max}$ is an $A^{\\ho 2}$-algebra via map  $\\iota : A^{\\ho 2}\\to A^{(2)}_{\\max}$. The above construction of Frobenius $\\varphi$ on $A^{(2)} _{\\max}$ is obviously compatible with  $\\iota$. \n\nOur next goal is to show that $\\iota$ induces a map $A^{(2)}\\to A^{(2)}_{\\max}$ so that $A^{(2)} $ is a subring of $A^{(2)}_{\\max}$ which is compatible with $\\varphi$-structures and filtration. We need a little preparation. \nWrite $\\mathfrak z_{n}= \\delta^n (z)$ with $\\delta_0(z)= z= \\mathfrak z_0$, and $A_0 = W(k) [\\![u]\\!]$. \n\\begin{lemma}\\label{lem-delta-n} $$\\delta^n (Ez) = b_n \\mathfrak z_{n} + \\sum_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i.  $$\nwhere $a^{(n)}_i \\in A_0 [\\mathfrak z_0, \\dots, \\mathfrak z_{n -2}]$ so that $a^{(n)}_p\\in A_0 ^\\times$ and for $0 \\leq i \\leq p-1$ each monomials of  $a^{(n)}_i $ contains a  factor $\\mathfrak z_{j}^p$  for some  $ 0 \\leq j\\leq n -2$. Furthermore,  $b_{n+1 } =p\\delta (b_n ) + b^p_n $ and $b_1 = p \\delta (E) + E^p$. \n\\end{lemma}\n\\begin{proof} Given  $f \\in A_0[x_1 , \\dots , x_m]$, if each monomials of $f$  contains $x_j^l$ for some $j$ and $l \\geq p$ then we call $f$  \\emph{good}.  For example, $f= x_1^p x_2 + 2 x_ 1x_2^{p +3}.$ So we need to show that $a^{(n)}_i\\in A_0[\\mathfrak z_0, \\dots, \\mathfrak z_{n -2}]$ is good.  Before making induction on $n$, we discuss some properties of good polynomial. It is clear that the set of good polynomials is closed under addition and multiplications. Note that \n\\begin{equation}\\label{eqn-delta}\n\\delta(\\mathfrak z_{l}^i)= \\frac 1 p (\\varphi (\\mathfrak z_{l}^i) - \\mathfrak z_{l} ^{p i})= \\frac 1 p \\big( (p\\mathfrak z_{l +1} + \\mathfrak z_{l } ^p)^i - \\mathfrak z_{l}^{p i}\\big) = \\sum \\limits_{j  = 1 }^{i} \\binom{i}{j}(p^{j -1} \\mathfrak z_{l }^{p(i-j)} ) \\mathfrak z_{l+1} ^{j }. \n\\end{equation}\nIn particular, given an $f\\in A_0[\\mathfrak z_0 , \\dots , \\mathfrak z_{m}]$,  $\\delta(\\mathfrak z_{m}^p f)= f^p \\delta (\\mathfrak z_{m} ^p) +  \\mathfrak z_{m}^{p ^2}\\delta (f) + p \\delta (\\mathfrak z_{m} ^p)\\delta (f)$  is  a good polynomial in $A[\\mathfrak z_0 , \\dots, \\mathfrak z_{m+1}]$. Using the fact that \n$\\delta (a+b)=\\delta(a) + \\delta (b) + F(a, b)$ where  $F(X, Y) = \\frac 1 p ( X^p + Y^p - (X+Y)^p) = - \\sum\\limits_{i = 1}^{p-1} \\binom{p}{i}\/p X^i Y^{p-i}$, together with the above argument of $\\delta(\\mathfrak z_l^p f)$, it is not hard to show that if $g\\in A_0[\\mathfrak z_0 , \\dots , \\mathfrak z_{m}]$ is good  then $\\delta (g) \\in A_0[\\mathfrak z_0 , \\dots , \\mathfrak z_{m}, \\mathfrak z_{m +1}]$ is also good. \n\nNow we make induction on $n$. When $n =1$, we have \n$$\\delta (Ez)= E^p \\mathfrak z_{1} + z^p \\delta(E) + p \\delta (E) \\mathfrak z_{1}= (p \\delta (E) + E^p) \\mathfrak z_{1} + \\delta(E) z^p.$$ \nThen $b_1 = p \\delta (E) + E^p $,  $a^{(1)}_p= \\delta(E)\\in A_0 ^\\times$ and $a^{(1)}_i = 0$ for $1 \\leq i \\leq p-1$ are required. Now assume the formula is correct for $n$, then \n$$\\delta ^{n +1} (Ez) = \\delta (b_n \\mathfrak z_{n} + \\sum_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i) = \\delta (b_n \\mathfrak z_{n}) + \\delta (\\sum_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i)) + F(b_n \\mathfrak z_{n}, \\sum_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i)), $$\nClearly, $ F(b_n \\mathfrak z_{n}, \\sum\\limits_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i))  = \\sum\\limits_{j = 1} ^{p-1} \\tilde a^{(n)}_j \\mathfrak z_{n} ^j$ with $\\tilde a^{(n)}_j$ being good. An easy induction shows that \n$\\delta (\\sum\\limits_{i = 0} ^p a^{(n)}_i \\mathfrak z_{n -1} ^i) = \\sum\\limits_{i = 0} ^p \\delta (a^{(n)}_i \\mathfrak z_{n -1} ^i) + f$ with $f \\in A_0[\\mathfrak z_0, \\dots, \\mathfrak z_{n -1}]$ being good. Since \n$\\delta (a^{(n)}_i \\mathfrak z_{n -1} ^i)= (a^{(n)}_i)^p \\delta (\\mathfrak z_{n -1}^i) + (\\mathfrak z_{n-1}^{pi})\\delta (a_i ^{(n)}) + p \\delta (\\mathfrak z_{n -1}^i  ) \\delta (a_i^{(n)})$, by using formula of $\\delta(\\mathfrak z_{n-1}^i)$ in \\eqref{eqn-delta} and that $a^{(n)}_i$ is good implies that \n$\\delta (a_i^{(n)}) $ is also good, we conclude that for $0 \\leq i \\leq p-1$,  $$\\sum\\limits_{i = 0} ^{p-1} \\delta (a^{(n)}_i \\mathfrak z_{n -1} ^i) = \\sum_{i =0}^{p-1} \\alpha_i \\mathfrak z_{n} ^i$$ with $\\alpha_i \\in A_0 [\\mathfrak z_0, \\dots, \\mathfrak z_{n-1}]$ being good polynomials. Using that $a _p^{(n)} \\in A_0 ^\\times$, we compute that  $\\delta (a_p^{(n)}\\mathfrak z^p_{n-1}) = \\sum \\limits_{i = 0}^p \\beta_i \\mathfrak z_{n}^i$ with $\\beta_p \\in p A_0$ and $\\beta_j\\in A_0 [\\mathfrak z_0 , \\dots , \\mathfrak z_{n-1}]$ being good for $1\\leq j \\leq p-1$. \n Now we only need to analyze $\\delta (b_n \\mathfrak z_{n})$, which is \n $\\delta (b_n ) \\mathfrak z_{n}^p + b_n^p \\mathfrak z_{n+1} + p \\delta (b_n)\\mathfrak z_{n+1}$. So $b_{n+1} = p \\delta(b_n) + b_n ^p$ and $a_p^{(n+1)} = \\delta (b_n) + \\beta_p$. Since $\\delta(b_n) \\in A_0^\\times$, we see that $a_p^{(n+1)}= \\delta(b_n) + \\beta_p \\in A_0^\\times$ as required. \n\\end{proof}\n\nLet $\\widetilde A^{(2)} := A^{\\ho 2} [z_j]_\\delta= A^{\\ho2} [\\delta ^n (z_j), n \\geq 0, j =0 , \\dots , m]$ and natural map $\\alpha : \\widetilde A^{(2)} \\to \\widetilde A^{(2)} [\\frac 1 p]$ (we do not know $\\alpha$ is injective at this moment).  \n\\begin{lemma}\\label{lem:gamma(z)-polynomial-in-E\/p}\nFor $i\\geq 0$ and $j=0,1,\\ldots,d$, there exists $f_{ij}(X) \\in \\widetilde A^{(2)} [X]$ such that, as elements of $\\widetilde A^{(2)}[\\frac 1 p] $ via $\\alpha: \\widetilde A^{(2)} \\to \\widetilde A^{(2)} [\\frac 1 p]$, \n\\[\n\\gamma_i(z_j) = f_{ij}\\Bigl(\\frac{E}{p}\\Bigr).\n\\]\n\\end{lemma}\n\\begin{proof}\nWrite $z = z_{j}$ for simplicity, and let $\\tilde{\\gamma}(z)= \\frac{z^p}{p}$ and $\\tilde{\\gamma}^n = \\underbrace{\\tilde{\\gamma} \\circ \\tilde{\\gamma} \\cdots \\circ \\tilde{\\gamma}}_n$. It suffices to show that for each $n \\geq 1$, we have $\\tilde{\\gamma}^n(z) = f_n(\\frac{E}{p})$ inside $\\widetilde A^{(2)}[\\frac 1 p]$ for some $f_n(X) \\in \\widetilde A^{(2)} [X]$. For an element $x \\in A[\\delta^i(z)]_{i \\geq 0}$, we say that $x$ has \\emph{$\\delta$-order $\\leq n$}  if $x\\in \\sum_{0\\leq j\\leq n}A[\\{\\delta^i(z)\\}_{0 \\leq i \\leq n }] \\delta^j(z)$, namely, if $x$ can be written as a sum of monomials such that each term is divisible by $\\delta^j(z)$ for some $0 \\leq j \\leq n$. \n\nWe claim that the following two equations hold for each $n \\geq 1$:\n\\begin{enumerate}\n\\item We have\n\\begin{equation} \\label{eq:delta(z)}\n    \\delta^n(z) = \\nu_n \\tilde{\\gamma}^n(z)+P_n\\Bigl(\\frac{E}{p}\\Bigr)+\\frac{E^p}{p}d_n\\delta^n(z)\n\\end{equation}\nfor some $\\nu_n \\in A^{\\times}$, $d_n \\in A$, and $P_n(X) \\in (A[\\delta^i(z)]_{i \\geq 0})[X]$ such that each coefficient of $P_n(X)$ has $\\delta$-order $\\leq n-1$. \n\n\\item We have\n\\begin{equation} \\label{eq:gamma(delta(z))}\n    \\tilde{\\gamma}(\\delta^{n-1}(z)) = \\mu_{n-1}\\tilde{\\gamma}^n(z)+Q_{n-1}\\Bigl(\\frac{E}{p}\\Bigr)\n\\end{equation}\nfor some $\\mu_{n-1} \\in A^{\\times}$ and $Q_{n-1}(X) \\in (A[\\delta^i(z)]_{i \\geq 0})[X]$ such that each coefficient of $Q_{n-1}(X)$ has $\\delta$-order $\\leq n-1$.\n\\end{enumerate}\n\nWe prove claims (1) and (2) by induction. For $n = 1$, since\n\\[\n\\delta(Ez) = z^p\\delta(E)+(p\\delta(E)+E^p)\\delta(z)\n\\]\nand $\\delta(E) \\in \\mathfrak{S}^{\\times}$, we have\n\\[\n\\delta(z) = -\\tilde{\\gamma}(z)+\\delta(E)^{-1}\\frac{\\delta(Ez)}{p}-\\delta(E)^{-1}\\frac{E^p}{p}\\delta(z). \n\\]\nBy easy induction, we also have $\\delta^i(Ez) \\in (Ez)A$ for each $i \\geq 1$. So claim (1) holds. Claim (2) holds for $n = 1$ trivially with $Q_0(X) = 0$.\n\nSuppose that claims (1) and (2) hold for $1 \\leq n \\leq m$. We will verify claims (1) and (2) for $n = m+1$. We first consider claim (2). Since each coefficient of $P_m(X)$ has $\\delta$-order $\\leq m-1$, $\\frac{E^p}{p}=p^{p-1}\\bigl(\\frac{E}{p}\\bigr)^p$, and Equations \\eqref{eq:delta(z)} and \\eqref{eq:gamma(delta(z))} hold for $1\\leq n \\leq m$, applying $\\tilde{\\gamma}(\\cdot)$ to Equation \\eqref{eq:delta(z)} for $n = m$ yields\n\\[\n\\tilde{\\gamma}(\\delta^m(z)) = \\nu_m^p \\tilde{\\gamma}^{m+1}(z)+Q_m\\Bigl(\\frac{E}{p}\\Bigr)\n\\]\nfor some $Q_m(X) \\in (\\mathfrak{S}[\\delta^i(z)]_{i \\geq 0})[X]$ such that each coefficient of $Q_m(X)$ has $\\delta$-order $\\leq m$. This proves the claim (2) for $n = m+1$.\n\n\nWe now consider claim (1) for $n = m+1$. By the above Lemma for $n = m+1$ and that $b_n= p\\alpha_n +\\beta_n  E^p$\nfor some $\\alpha_n \\in A^{\\times}$ and $\\beta_n \\in  A$ (via an easy induction on $n$), we have \n\\[\n\\alpha_{m+1}\\delta^{m+1}(z) = \\frac{\\delta^{m+1}(Ez)}{p}-\\beta_{m+1}\\frac{E^p}{p}\\delta^{m+1}(z)-a_p^{(m+1)}\\tilde{\\gamma}(\\delta^m(z))-\\frac{1}{p}\\sum_{j=0}^{p-1} a_j^{(m+1)}(\\delta^{m}(z))^j.\n\\]\nAs noted above, we have $\\delta^{m+1}(Ez) \\in (Ez)A$. Furthermore, by the condition on $a_j^{(m+1)}$, the last term $\\frac{1}{p}\\sum_{j=0}^{p-1} a_j^{(m+1)}(\\delta^{m}(z))^j$ is a linear combination of terms involving $\\tilde{\\gamma}(\\delta^l(z))=\\frac{1}{p}(\\delta^l(z))^p$ for some $0\\leq l\\leq m-1$.\nThus, by applying Equations~\\eqref{eq:delta(z)} and \\eqref{eq:gamma(delta(z))} for $1 \\leq n \\leq m$, we see that claim (1) also holds for $n = m+1$ with $\\nu_{m+1} = -\\alpha_{m+1}^{-1}a_p^{(m+1)}\\mu_m$ and $d_{m+1} = -\\alpha_{m+1}^{-1}\\beta_{m+1}$.\nThis completes the induction and prove the lemma .\n\\end{proof}\n\n\\begin{remark}\nIn the above proof, by equation~\\eqref{eq:gamma(delta(z))}, we even have for each $i,j\\geq 0$, $\\gamma_i(\\delta^j(z))=f(\\frac{E}{p})$ for some $f\\in \\widetilde A^{(2)} [X]$. \n\\end{remark}\n\nAn easy induction by \\eqref{eq:delta(z)} implies that $\\alpha (\\delta^n (z) ) \\in A^{\\ho 2}[\\{\\gamma_i (z_j )\\}_{i \\geq 0 , j =1 , \\dots , m}, \\frac E p]\\subsetA^{(2)}_{\\max}$, which satisfies equations in Lemma \\ref{lem-delta-n} by replacing \n$\\mathfrak z_n$ by $\\alpha (\\delta^n (z))$ inside $A^{(2)}_{\\max}$. It is clear that $\\iota$ is still Frobenius compatible (because both $A ^{\\ho 2}$ and $A^{(2)}_{\\max}$ are domains). Since $E = p \\frac E p$, $\\iota$ is a continuous for $(p , E)$-topology on $\\widetilde A^{(2)} $ and $p$-topology on $A^{(2)}_{\\max}$. Finally, we construct a ring map $\\iota: A^{(2)} \\to A^{(2)}_{\\max}$ so that $\\iota$ is compatible with Frobenius.  \n\nOur next goal is to show that $\\iota$ is injective. Define ${\\textnormal{Fil}} ^i A^{(2)}_{\\max}[\\frac 1 p]:  = E^i A^{(2)}_{\\max}[\\frac 1 p]$. For any subring $B \\subset A^{(2)}_{\\max}[\\frac 1 p]$, set \n$${\\textnormal{Fil}} ^i B : = B \\cap {\\textnormal{Fil}} ^i A^{(2)}_{\\max}[\\frac 1 p]= B \\cap E^i A^{(2)}_{\\max}[\\frac 1 p].$$\nLet $D_z$ be the $p$-adic completion of $R [\\gamma_i (z_j), i \\geq 0;  j = 0, \\dots , m]$. \n\\begin{proposition} \\label{prop-key-property}\n\\begin{enumerate}\n    \\item $\\widetilde A^{(2)}\/ E = R [\\gamma_i (z_j ), i \\geq 0; j = 0, \\dots , m]$.\n    \\item $ A^{(2)}\/ E \\simeq D_z$.\n    \\item $\\iota$ is injective.\n    \\item ${\\textnormal{Fil}} ^1 A^{(2)}= E A ^{(2)}$.\n    \\item $A^{(i)}$ are flat over $A$ for $i = 2, 3$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n(1)  By definition, $\\widetilde A^{(2)}  = A ^{\\ho  2}[ z^{(n)}_j , n\\geq 0; j = 0 , \\dots, m ]\/ J $ where $\\mod J  $ is equivalent the following relations (note that $z_0 = z$):  $E z= (x-y), E z_j = s_j - t_j, \\delta  (z^{(n)}_j)= z^{(n+1)}_{j}, \\delta ^n (Ez) = \\delta^n (y -x) , \\delta ^n (Ez_j)= \\delta ^n (s_j-t _j). $ Since $\\delta (x-y)= \\frac{(x^p - y ^p)- (x-y)^p}{p}$ and $\\delta(s_j - t_j) = \\frac{\\varphi (s_j - t_j)- (s_j - t_j)^p}{p}$, \nit is easy to prove by induction that $\\delta^n (x-y)$  and $\\delta^n (s_j - t_j)$ always contains a factor $(x-y)$, $s_j - t_j$ and hence $\\delta^n (x-y), \\delta(s_j - t_j)\\equiv 0 \\mod E$. Therefore $\\delta ^n (Ez_j) \\equiv 0 \\mod E$. By Lemma \\ref{lem-delta-n}, we see that \n$$p \\mu_n z^{(n )}_{j} = -\\sum_{i = 0} ^p  \\overline{a^{(n)}_i} (z^{(n -1)}_j) ^i  \\mod E \\text{ and } pz^{(1)}_j = z_j^p \\mod E $$\nwhere $\\overline {a^{(n)}_i} = a^{(n)}_i \\mod E$ and $\\mu_n = \\frac{\\delta (b_n)}{p}\\mod E \\in \\O_K^\\times$. Using that $a_p^{(n)} \\in A_0^\\times$, and $a_i^{(n)}, 1 \\leq i \\leq p-1$ are good in the sense that they contains factor of $(z^{(l)}_j) ^p$ for some $l = 0 , \\dots, n-2$, we easily see by induction that \n$\\widetilde A^{(2)} \/E = R [\\widetilde \\gamma^n (z_j ), n \\geq 0; j = 0 , \\dots , m  ]$. But it is well-known that $R [\\widetilde \\gamma^n (z_j), n \\geq 0; j = 0 , \\dots , m ] = R [\\gamma_n (z_j), n \\geq 0; j = 0 , \\dots , m]. $\n\nNow we show that the natural map $\\iota:  \\widetilde A^{(2)}\\to A^{(2)}_{\\max}[\\frac  1 p]$ induced by $\\alpha (\\delta ^n(z_j))$ is injective. Note that $\\widetilde A^{(2)} $ is the direct limit of $\\widetilde A^{(2)}_n : =  A ^{\\hat \\otimes 2}[\\{ \\delta^i (z_j)\\}_{i = 1 , \\dots , n, j = 0 , \\dots , m}]$. A similar argument similar as above show that $\\widetilde  A^{(2)}_n \/ E $ injects to $A^{(2)}_{\\max}[\\frac 1 p]\/E = D_z [\\frac 1 p]$. \nSince $\\widetilde A^{(2)}_n$ is $E$-separate and  $A^{(2)}_{\\max}$ is a domain, this implies that $\\widetilde A^{(2)}_n$ injects to $A^{(2)}_{\\max}[\\frac 1 p]$. So \n$\\widetilde A^{(2)}$ injects to $A^{(2)}_{\\max}$ via $\\iota$. \n\n(2) Since $A^{(2)}$ is $(p, E)$-completion of $\\widetilde A^{(2)}$ \\footnote{Indeed, $A^{(2)}$ is \\emph{derived} $(p, E)$-completion. Since $\\widetilde A^{(2)}\/ E$  is $\\mathbb Z_p$-flat, then derived completion coincides with the classical completion, which is used here.}, we have a natural map from $\\bar \\iota: A^{(2)}\/E \\to D_z$. The surjectivity of $\\bar \\iota$ is straightforward as $A^{(2)}$ is also $p$-complete. To see injectivity, given an sequence $f_n $ so that $f_{n +1}- f _n \\in  (p , E)^n \\widetilde A^{(2)} $ and $ f_n = E g_n$ for all $n$, we have to show that $g_n$ is a convergent sequence in $A^{(2)}$.  Since $E (g_{n +1} - g_n) = \\sum_{i = 0} ^n p ^i E ^{n - i} h_i$ with $h _i \\in \\widetilde A^{(2)} $. Then $ E|p ^n h_n $. Since $\\widetilde A^{(2)} \/E$ has no $p$-torsion,\nwe have $E | h_n $ and write $h_n  = E h'_n $. Since $\\widetilde A^{(2)} $ is a  domain as it is inside the fraction field of $A ^{\\ho 2}$, we see that \n$ g_{n +1}- g_n = p ^n h'_n + \\sum\\limits_{i = 0} ^{n -1} p ^i E^{n - i - 1} h_i $. Hence $g_n$ converges in $ A^{(2)}$ as required. \n \n(3) It is clear that $A^{(2)}_{\\max}[\\frac 1 p]\/ E \\simeq D_z [\\frac 1 p]$. So the map $\\iota \\mod E(u)$ induces an injection $D_z \\hookrightarrow D_z [\\frac 1 p]$. So for any $x\\in \\Ker (\\iota)$, we see that $x= Ea$ for some $a \\in A^{(2)}$. As $A^{(2)}_{\\max}$ is $E$-torsion free and $A^{(2)}$ is $E$-complete, we see that $x= 0$ as required.  \n \n(4) By the definition of ${\\textnormal{Fil}} ^1 A^{(2)}  $, we see that $E A ^{(2)} \\subset {\\textnormal{Fil}} ^1 A ^{(2)}$ and $ A^{(2)} \/ {\\textnormal{Fil}} ^1 A^{(2)} $ injects to $A^{(2)}_{\\max}[\\frac 1 p ]\/E= D_z[\\frac 1 p]$. But we have seen that $A^{(2)}\/E = D_z$ injects to $D_z$. Then ${\\textnormal{Fil}} ^1 A^{(2)} = E A^{(2)}$. \n\n(5) Both $A^{(2)}$ and $A^{(3)}$ are obtained by the construction of \\cite[Proposition 3.13]{BS19}, which implies that they are $(p,E)$-complete flat over $A$. Since $A$ is Noetherian, by \\cite[Tag 0912]{stacks-project}, we have both $A^{(2)}$ and $A^{(3)}$ are $A$-flat.\n\\end{proof}\n\\begin{corollary}\\label{cor-filtration-shape}\n\\begin{enumerate}\n    \\item ${\\textnormal{Fil}} ^i A ^{(2)} = E^i A^{(2)}. $\n    \\item $A^{(i)}$ are bounded prisms for $i = 2, 3$. \n\\end{enumerate}\n\\end{corollary} \n\\begin{proof}These follow that $A^{(2)} \/ E A^{(2)} \\simeq D_z$ which is $\\mathbb Z_p$-flat. For (2), we have $A^{(2)}$ and $A^{(3)}$ are $(p,E)$-complete flat over $A$, so boundedness follows from (2) in \\cite[Lemma 3.7]{BS19}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem-subring} $A^{(2)}$ is a closed subset inside $A^{(2)}_{\\max}$.  \n\\end{lemma}\n\\begin{proof} We need to show the following statement:  Given $ x \\in \\widetilde A^{(2)} $, if $x= p ^n y$ with $y \\in A^{(2)}_{\\max}$ then $x = \\sum\\limits_{i = 0}^n p ^{n-i} E^i x_i$ with $x_i \\in \\widetilde A^{(2)} . $ Indeed, since $A^{(2)}\/E \\simeq A^{(2)}_{\\max}\/ {{\\textnormal{Fil}} ^1}$, there exists $x_0, w_1 \\in \\widetilde A^{(2)} $ so that $x= p ^n x_0 + E w_1$. Then $Ew_1 \\in p ^n  A^{(2)}_{\\max}$. Write $ E w_1= p ^n \\sum \\limits_{i =0} ^\\infty\\sum\\limits_{j =0}^m f_{ij} \\gamma_i (z_j)$, we see that $f_{ij}= \\sum_{l \\geq 1} a_{ijl}  \\frac{E^l}{p^l}\\in {\\textnormal{Fil}} ^1 \\Omax$. So it is easy to see that $p ^n E^{-1}f_{ij} \\in p ^{n -1} \\Omax$ and then \n$w_1 = p ^{n -1} x_1 $ with $x_1 \\in A^{(2)}_{\\max}$. Then we may repeat the above argument to $w_1$, and finally $x= \\sum\\limits_{i = 0}^n p ^{n-i} E^i x_i$ with $x_i \\in \\widetilde A^{(2)}$ as required. \n\\end{proof}\n\nNow we realize $A^{(2)}$ as a subring of $A^{(2)}_{\\max}$ via $\\iota$. We need to introduce some auxiliary rings. By the description of elements in $A^{(2)}_{\\max}$, we define ${\\widetilde S}_0$ be the subring of $A^{(2)}_{\\max}$ as follow\n$$\n\\widetilde{S} :=  A^{(2)} [\\![ \\frac {E ^p}{p} ]\\!] := \\{ \\sum_{i \\geq 0} a_i (\\frac{E^p}{p})^i \\mid a_i\\in A^{(2)} \\}.\n$$\nAnd when $p=2$, we define \n$\n\\widehat{S} := A^{(2)} [\\![\\frac{E^4}{2}]\\!]\n$ simiarly. \nWe will have $\\widehat{S} \\subset \\widetilde{S} \\subset A^{(2)}_{\\max}$. Viewing $\\widetilde S$ and $\\widehat{S}$ as subrings of $A^{(2)}_{\\max}$, we give them the filtration induced from $A^{(2)}_{\\max}$. The following lemma is crucial for later applications and we thank Yong Suk Moon for many useful comments to improve many details in the proof.  \n\n\\begin{lemma}\\label{lem-auxiliaryrings}\nFix $h \\in \\mathbb N$, then we have\n\\begin{enumerate}\n    \\item We have $\\varphi(A^{(2)}_{\\max}) \\subset \\widetilde{S} \\subset A^{(2)}_{\\max}$, and when $p=2$, we have $\\varphi(\\widetilde{S}) \\subset \\widehat{S} \\subset \\widetilde{S}$;\n    \\item $x \\in {\\textnormal{Fil}} ^h \\widetilde S $ if and only if $x$ can be written as\n    $$\n    x = \\sum\\limits_{i \\geq h } a_i \\frac {E ^i}{p ^{\\lfloor \\frac i p\\rfloor}} \n    $$\n    with $a_i\\in A^{(2)}$. \n    \\item when $p>2$, there is a $h_0>h$ such that $\\varphi ({\\textnormal{Fil}} ^m \\widetilde S ) \\subset A ^{(2)} + E^h{\\textnormal{Fil}}^{m +1} \\widetilde S$ for all $m > h_0$;\n    \\item when $p=2$, then $x \\in {\\textnormal{Fil}} ^h \\widehat S $ if and only if $x$ can be written as\n    $$\n    x = \\sum\\limits_{i \\geq h } a_i \\frac {E ^i}{2 ^{\\lfloor \\frac i 4\\rfloor}} \n    $$\n    with $a_i\\in A^{(2)}$;\n    \\item when $p=2$, there is a $h_0>h$ such that $\\varphi ({\\textnormal{Fil}} ^m \\widehat{S}  ) \\subset A ^{(2)} + E^h{\\textnormal{Fil}}^{m +1} \\widehat{S} $ for all $m > h_0$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nFor $(1)$, any $a\\in A^{(2)}_{\\max}$, we can write \n$$\na = \\sum _{i_0 = 0}^\\infty \\cdots \\sum_{i _m = 0}^\\infty \\sum_{l = 0}^\\infty  a_{i_0 , \\dots , i_m, l } \\left (\\frac E p\\right ) ^l \\prod_{j = 0}^m \\gamma_{i_j} (z_j)\n$$\nwhere $a_{i_0 , \\dots , i_m, l}\\in A$ and $a_{i_0 , \\dots , i_m, l}\\to 0$ $p$-adically when $\\sum_j i _j + l \\to \\infty$. Thanks for Lemma \\ref{lem:gamma(z)-polynomial-in-E\/p}, we see that $b_{i_0 , \\dots , i_m , l}: = \\varphi \\left  (\\left (\\frac E p\\right ) ^l \\prod_{j = 0}^m \\gamma_{i_j} (z_j) \\right)\\in \\widetilde S$. So $\\varphi (a) = \\sum a_{i_0 , \\dots , i_m , l} b_{i_0 , \\dots , i_m , l}$ converges in $\\widetilde S$. \n\nFor the claim in $(1)$ for $p=2$, we have $\\varphi(\\frac{E^2}{2})=(E^2+2b')^2\/2= \\frac{E^4}{2} + 2b$ for some $b,b'\\in A$. And for $a = \\sum_{i \\geq 0} a_i (\\frac{E^p}{p})^i \\in \\widetilde{S}$, we have \n$$\n\\varphi(a) = \\sum_{i \\geq 0} \\varphi(a_i) (\\frac{\\varphi(E^2)}{2})^i= \\sum_{i \\geq 0} \\varphi(a_i) \\sum_{j=0}^{i} c_{ij}(2b)^{i -j} (\\frac{E^4}{2})^j = \\sum_{j \\geq 0} \\left (\\sum_{i=j}^{\\infty} \\varphi(a_i)c_{ij} (2b)^{i-j} \\right ) (\\frac{E^4}{2})^j\n$$\nfor some $c_{ij} \\in \\mathbb Z$. So we have $\\varphi(a) \\in \\widehat{S}$.\n\nFor $(2)$, the if part is trivial. For the other direction, any $x \\in {\\textnormal{Fil}} ^h \\widetilde S $, we have \n$$\nx = \\sum\\limits_{i \\geq 0 } a_i \\frac {E^{i}}{p^{\\lfloor \\frac i p  \\rfloor}} \n$$\nas element in $\\widetilde S$. And if we also have $x \\in {\\textnormal{Fil}} ^h A^{(2)}_{\\max}[\\frac  1 p ] = E^h A^{(2)}_{\\max}[\\frac 1 p]$, this implies for $\\tilde a_0=\\sum\\limits_{0\\leq i \\leq h} a_i \\frac {E^{i}}{p^{\\lfloor \\frac i p \\rfloor}}$ is in ${\\textnormal{Fil}} ^h A^{(2)} [\\frac 1 p]$. This implies $p^{\\lfloor \\frac h p\\rfloor}\\tilde a_0 \\in {\\textnormal{Fil}} ^h A^{(2)}= E^h A^{(2)}$. That is $\\tilde a_0={p^{-\\lfloor \\frac h p\\rfloor}}{E^h} b$ for some $b \\in A^{(2)}$. So $x$ is of the given form. The proof for $(4)$ is similar.\n\nFor $(3)$, we have by $(2)$, $x \\in {\\textnormal{Fil}} ^m \\widetilde S$, $x$ can be written as\n$$\nx = \\sum\\limits_{i \\geq m } a_i \\frac {E ^i}{p ^{\\lfloor \\frac i p\\rfloor}}.\n$$\nAnd use the fact $\\varphi(E)=E^p+pb$ for some $b \\in A^{(2)}$, we have \n$$\n\\varphi(x) = \\sum\\limits_{i \\geq m } \\varphi(a_i) \\sum_{j=0}^{i} \\frac {c_{ij}E^{p(i-j)} p^j}{p ^{\\lfloor \\frac i p\\rfloor}} = \\sum_{i \\geq m} \\sum_{ j \\geq \\lfloor \\frac i p\\rfloor}^{i} \\frac {b_{ij}E^{p(i-j)} p^j}{p ^{\\lfloor \\frac i p\\rfloor}} + \\sum_{i \\geq m} \\sum_{0 \\leq j < \\lfloor \\frac i p\\rfloor} E^h\\frac {b_{ij}E^{p(i-j)-h} p^j}{p ^{\\lfloor \\frac i p\\rfloor}}\n$$    \nwith  $b_{ij} \\in A^{(2)}$.\n\nIn particular, we have $\\sum_{i \\geq m} \\sum_{ j \\geq \\lfloor \\frac i p\\rfloor}^{i} \\frac {b_{ij}E^{p(i-j)} p^j}{p ^{\\lfloor \\frac i p\\rfloor}}$ is inside $A^{(2)}$. To prove $(3)$, it is amount to find $h_0$ such that whenever $m>h_0$, $i \\geq m$ and $0 \\leq j < \\lfloor \\frac i p\\rfloor$, we have \n$$\n\\sum_{i \\geq m} \\sum_{0 \\leq j < \\lfloor \\frac i p\\rfloor} \\frac {b_{ij}E^{p(i-j)-h} p^j}{p ^{\\lfloor \\frac i p\\rfloor}} \\in {\\textnormal{Fil}}^{m +1} \\widetilde S.\n$$\nThe claim follows if we can find $h_0 > h$ such that $\\frac {E^{p(i-j)-h} p^j}{p ^{\\lfloor \\frac i p\\rfloor}} \\in \\widetilde S$ and $p (i-j)- h \\geq m +1$ for all $m>h_0$, $i \\geq m$ and $0 \\leq j < \\lfloor \\frac i p\\rfloor$. That is $\\lfloor\\frac{p (i -j)-h}{p} \\rfloor +j \\geq \\lfloor \\frac i p \\rfloor $ and $p(i-j)-h\\geq m+1$ for all $i,j,m$ in this range. And solve this we have it is enough to choose $h_0 > \\max \\{h, \\frac{p(h+1)+1}{p(p-2)}\\}$, which is valid for $p>2$.\n\nStatement in $(5)$ is similar to $(3)$. Any $x \\in {\\textnormal{Fil}} ^m \\widehat S$, $x$ can be written as\n$$\nx = \\sum\\limits_{i \\geq m } a_i \\frac {E ^i}{2^{\\lfloor \\frac i 4\\rfloor}}.\n$$\nWe have $\\varphi(E)=E^2+2b$ for some $b \\in A^{(2)}$, so \n$$\n\\varphi(x) = \\sum\\limits_{i \\geq m } \\varphi(a_i) \\sum_{j=0}^{i} \\frac {c_{ij}E^{2(i-j)} 2^j}{2 ^{\\lfloor \\frac i 4\\rfloor}} = \\sum_{i \\geq m} \\sum_{ j \\geq \\lfloor \\frac i 4\\rfloor}^{i} \\frac {b_{ij}E^{2(i-j)} 2^j}{2 ^{\\lfloor \\frac i 4\\rfloor}} + \\sum_{i \\geq m} \\sum_{0 \\leq j < \\lfloor \\frac i 4\\rfloor} E^h\\frac {b_{ij}E^{2(i-j)-h} 2^j}{2 ^{\\lfloor \\frac i 4\\rfloor}}.\n$$ \nSimilar to the argument in $(3)$, it is amount to find $h_0$ such that whenever $m>h_0$, $i \\geq m$ and $0 \\leq j < \\lfloor \\frac i 4\\rfloor$, we have $\\lfloor (i -j)-\\frac h 2 \\rfloor + j  \\geq \\lfloor \\frac i  4 \\rfloor $ and $2(i-j)-h \\geq m +1$. It is enough to choose $h_0 > 2(h+2)$. \n\\end{proof}\n\n\nIf  $A$ is a ring then we denote by  ${\\rm M} _d (A)$ the set of $d\\times d$-matrices with entries in $A$.\n\n\\begin{proposition}\\label{prop-desecnt} Let $Y \\in {\\rm M}_d (A^{(2)}_{\\max})$ so that $E^ h Y = B  \\varphi (Y) C$ with $B$ and $C$ in ${\\rm M}_d (A ^{(2)})$ then   $Y$ is in ${\\rm M}_d (A ^{(2)}[\\frac 1 p])$. \n\\end{proposition}\n\\begin{proof} First, we claim that there is a constant $s$ only depends on $h$, such that the entries of $p^s Y$ is in $\\widetilde S$. By $(1)$ of Lemma~\\ref{lem-auxiliaryrings}, entries of  $E ^ h Y$ are in $\\widetilde S$. So for each entry $a$ of $Y$, we can write $E^h a = \\sum \\limits_{i = 0}^\\infty a_i \\frac{ E^{pi}}{p^i}$ with $a_i \\in A ^{(2)}$. It is clear that $E^h p ^h a = a' + E^h \\sum\\limits_{i \\geq h} a_j \\frac {E^{pi-h }}{p ^i} $ so that $a' \\in A^{(2)}$. Therefore, $a' \\in {\\textnormal{Fil}}^h A^{(2)} = E^h A ^{(2)}$ by Corollary \\ref{cor-filtration-shape}. So write $a' =  E^h b$, we have $ p ^h a = b' + \\sum\\limits_{i \\geq h} a_j \\frac {E^{pi-h }}{p ^i}$. In particular, we see that $p ^{2h} a \\in \\widetilde S$, this proves our claim. When $p=2$, then we may repeat the above argument and we can assume $p^s Y$ is in ${\\rm M} _d (\\widehat{S})$.\n\nLet $R=\\widetilde S$ when $p>2$ and $R=\\widehat{S}$ when $p=2$, then we may assume $Y$ is inside ${\\rm M} _d (R)$. Then we claim there is another constant $r$ only depends on $h$, such that for each entry $a$ of $Y$, there is a sequence $\\{b_i\\}_{i\\geq 1}$ in $A^{(2)}$ such that we have $a - \\sum\\limits_{i = 0} ^m b _i E^ i \\in {\\textnormal{Fil}} ^{m +1} R$. Note that once this is known, we will have $\\sum\\limits_{i = 0} ^m b _i E^ i$ converges to an element $b$ in $A^{(2)}$, and $a-b=0$ since it is in ${\\textnormal{Fil}} ^{m} R$ for all $m\\in \\mathbb N$. \n\nSo it remains to show our claim. When $p>2$, let $h_0$ be the integer in $(3)$ of Lemma~\\ref{lem-auxiliaryrings}, then it is easy to show there is a constant $r$ only depends on $h_0$ (so only on $h$) and sequence $\\{b_i\\}_{i=1}^{h_0}$ such that for each entry $a$ of $Y':=p^rY$, we have\n$$\na - \\sum\\limits_{i = 0} ^{h_0} b _i E^ i \\in {\\textnormal{Fil}} ^{h_0 +1} R.\n$$\nNow we show our claim by induction, assume for each entry $a$ in $Y'$, there is a sequence $\\{b_i\\}_{i=1}^{m}$ such that,\n$$\na - \\sum\\limits_{i = 0} ^{m} b _i E^ i \\in {\\textnormal{Fil}} ^{m +1} R.\n$$\nfor some $m \\geq h_0$.  So we can write $Y'$ as\n$$\n\\sum_{i = 0}^m Y_i E^i + Z_{m+1},   \n$$\nwith $Y_i \\in {\\rm M}_d (A^{(2)})$ and $Z_{m+1} \\in {\\rm M}_d ({\\textnormal{Fil}} ^{m +1} R)$. Writing $X_{m}= \\sum_{i = 0}^m Y_i E^i$, then $E^h Y' = B \\varphi (Y') C$ implies \n$$\nE^hZ_{m+1} = B\\varphi(X_m)C -E^hX_m + B\\varphi(Z_{m+1})C.\n$$\nBy $(3)$ in Lemma~\\ref{lem-auxiliaryrings}, we have $B\\varphi(Z_{m+1})C = A_{m+1} + E^h B_{m+1}$, with $A_{m+1} \\in {\\rm M}_d (A^{(2)})$ and $B_{m+1} \\in {\\rm M}_d ({\\textnormal{Fil}} ^{m +2} R)$. One can check $B\\varphi(X_m)C -E^hX_m + A_{m+1} \\in {\\rm M}_d ({\\textnormal{Fil}}^{h +m +1 } A^{(2)} )$, so $B\\varphi(X_m)C -E^hX_m + A_{m+1}=E^{h+m+1} Y_{m+1}$ with $Y_{m+1} \\in {\\rm M}_d (A^{(2)})$. And we have $Y - \\sum_{i = 0}^{m+1} Y_i E^i = B_{m+1} \\in {\\rm M}_d ({\\textnormal{Fil}} ^{m +2}R)$ as required.\n\nAt last when $p=2$. We know we can assume $Y$ is inside ${\\rm M}_d (\\widehat{S})$. Then repeat the above arguments by replacing $(3)$ in Lemma~\\ref{lem-auxiliaryrings} with $(5)$, we can also prove our claim.\n\\end{proof}\n\n\\subsection{The ring \\texorpdfstring{$A^{(2)}_{\\st}$}{A(2)st}}\\label{subsec-Ast} We assume that $R = \\O _K$ in the following two subsections. \nFor our later use for semi-stable representations, we construct $A^{(2)}_{\\st}$ as the following: Define $\\varphi$ on $W(k)[\\![x, \\mathfrak y]\\!]$ by $\\varphi(x)= x^p$ and $\\varphi (\\mathfrak y ) = (1+\\mathfrak y)^p -1$ and set $w = \\frac{\\mathfrak y}{ E}$. Set $ A^{(2)}_{\\st}: = W(k)[\\![x,\\mathfrak y ]\\!]\\{w\\}_\\delta^\\wedge$ where $\\wedge$ means $(p, E)$-completion. Similarly, we define $A^{(3)}_{\\st}=W(k)[\\![x,\\mathfrak y, \\mathfrak z ]\\!]\\{\\frac{\\mathfrak y}{E},\\frac{\\mathfrak z}{E}\\}^\\wedge_{\\delta}$, with the $\\delta$-structure on $W(k)[\\![x,\\mathfrak y, \\mathfrak z ]\\!]$ given by $\\delta(x)=0$, $\\varphi(\\mathfrak y)=(\\mathfrak y+1)^p-1$ and $\\varphi(\\mathfrak z)=(\\mathfrak z+1)^p-1$. Define $A^{(2)}_{\\st,\\max}$ to be the $p$-adic completion of $W(k)[\\![x, \\mathfrak y]\\!][w, \\frac E p , \\gamma_i (w), i \\geq 0].$ It is clear that for any $f \\in A^{(2)}_{\\st ,\\max}$ can be written uniquely $a = \\sum\\limits_{i= 0}^\\infty f_i \\gamma_i (w) $ with $f_i \\in \\Omax$ and $f_i \\to 0$ $p$-adically. For any subring $B\\subset A^{(2)}_{\\st , \\max}[\\frac 1 p]$, we set ${\\textnormal{Fil}} ^i B : = B \\cap E^i A^{(2)}_{\\st , \\max}[\\frac 1 p]$ and $D_w $ the $p$-adic completion of $\\O_K [\\gamma_i(w), i \\geq 0]$. \n\nIt turns out that $A^{(2)}$ and $A^{(2)}_{\\st}$ share almost the same properties by replacing $z$ with $w$. \nSo we summarize all these properties in the following: \n\\begin{proposition}\\label{prop-Ast-properties}\n\\begin{enumerate}\n \\item One can extend Froebnius from $A$ to $A^{(2)}_{\\st, \\max}$.  \n    \\item There exists an embedding $\\iota : A^{(2)}_{\\st} \\hookrightarrow A^{(2)}_{\\st , \\max}$ so that $\\iota$ commutes with Frobenius. \n    \\item $A^{(2)}_{\\st} \\cap E ^ i A^{(2)}_{\\st , \\max}[\\frac 1 p] = E A^{(2)} _{\\st}$. \n    \\item $A ^{(2)}_{\\st}\/E \\simeq D_w = A ^{(2)}_{\\st , \\max}\/ {\\textnormal{Fil}} ^1 A^{(2)}_{\\st , \\max}.$\n    \\item $A^{(2)}_{\\st}$ is closed in $A^{(2)}_{\\st, \\max}$. \n    \\item $A^{(2)}_{\\st}$ and $A^{(3)}_{\\st}$ are flat over $A$, and in particular they are bounded. \n    \\item Proposition \\ref{prop-desecnt} holds by replacing $A^{(2)}_{\\max}$ and $A^{(2)}$ by \n    $A^{(2)}_{\\st} $ and $A^{(2)}_{\\st, \\max}$ respectively. \n\\end{enumerate}\n\n\\end{proposition}\n\\begin{proof}\nAll previous proof applies by noting the following difference \n$$\\varphi(w)= \\varphi ( \\frac {\\mathfrak y}{E}) = c ^{-1} \\frac {1}{p} \\sum_{i =1}^p \\binom{p}{i} \\mathfrak y ^i= c ^{-1 }\\sum _{i =1}^{p-1} \\mathfrak y ^i\\binom{p}{i}\/ p  + c^{-1}\\frac{E^p w^p}{p}.   $$\nAlso $\\delta (\\mathfrak y) = \\sum\\limits _{i =1}^{p-1} \\mathfrak y ^i\\binom{p}{i}\/ p$ always contains $\\mathfrak y$-factor and this is a key input for the analogy of Lemma \\ref{lem:gamma(z)-polynomial-in-E\/p}. \n\nFor the boundedness of $A^{(3)}_{\\st}$, we have \n$$ \nW(k)[\\![x,\\mathfrak y, \\mathfrak z]\\!]\/(p,E)\\simeq (\\O_K\/p)[\\![\\bar{\\mathfrak y}, \\bar{\\mathfrak z}]\\!]\n$$ so $\\{\\mathfrak y, \\mathfrak z\\}$ form a $(p,E)$-complete regular sequence, and by \\cite[Proposition 3.13]{BS19}, $A^{(3)}_{\\st}$ is also $A$-flat, and this implies $A^{(3)}_{\\st}$ is bounded by (2) in Lemma 3.7 of $loc.cit.$. \n\\end{proof}\n\nNote that $A^{\\ho 2}= W(k) [\\![x, y]\\!]\\subset W(k)[\\![x,\n \\mathfrak y]\\!]$ via $y = x(\\mathfrak y +1)$ or equivalently $\\mathfrak y = \\frac y x -1 $. It is clear that this inclusion is a map of $\\delta$-rings. By the universal property of prismatic envelope to construct $A^{(2)}$, the inclusion induces a map of prisms $\\alpha: A^{(2)}\\to A^{(2)}_{\\st}$. Since $z= xw$, we easily see that $A^{(2)}_{\\max} \\subset A^{(2)}_{\\st, \\max}$. So $A^{(2)}\\subset A^{(2)} _{\\st}$ via $\\alpha$. We will see that $A^{(2)}$ (resp. $A^{(2)}_{\\st}$) is the self product of $A$ in category $X_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ (resp. $(X, M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\text{log}}}$) in \n \\S \\ref{subsec-pris-crystal} and \\S \\ref{sec-logprismandsemistablereps}. Then the existence of $\\alpha: A^{(2)} \\to A^{(2)}_{\\st}$ can be explained by the universal property of self product. See \\S\\ref{sec-logprismandsemistablereps} for details. \n\nTo simplify our notation, let $B^{(2)}_{\\st}$ (resp. $B^{(3)}_{\\st}$, $B^{(2)}$, $B ^{(3)}$) be the $p$-adic completion of $ {A^{(2)}_{\\st}} [\\frac 1 E]$ (resp. $A^{(3)}_{\\st}[\\frac 1 E]$, $A^{(2)} [\\frac 1 E]$, $A^{(3)}[\\frac 1 E]$). \n\\begin{lemma}\\label{lem-intersection} \\begin{enumerate}\n\\item $ A^{(i)}_{\\st}  \\subset B^{(i)}_{\\st}\\subset B^{(i)}_{\\st}[\\frac 1 p]$ and $ A^{(i)} \\subset B ^{(i)} \\subset B ^{(i)}[\\frac 1 p]$ for $i = 2, 3$. \n    \\item $B^{(2)}_{\\st} \\cap {A^{(2)}_{\\st}} [\\frac 1 p] = A^{(2)}_{\\st}$ and $B ^{(2)} \\cap {A^{(2)}} [\\frac 1 p] = A^{(2)}$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} Here we only prove the case $A^{(2)}$ while the proofs for $A^{(2)}_{\\st}$, $A^{(3)}$ and $A^{(3)}_{\\st}$ are almost the same. \n\nBy Proposition \\ref{prop-key-property}, $A^{(2)}$ is a subring of $A^{(2)} _{\\max}\\subset K_0  [\\![x, z]\\!]$. So $A^{(2)}$ and hence $A^{(2)} [\\frac 1 E]$\nis an integral domain. Then $B^{(2)} $ has no $p$-torsion: Assume that $x \\in B^{(2)}$ so that $p x = 0 $. Suppose that $x_n \\in A^{(2)} [\\frac 1 E]$ so that $x \\equiv x _n \\mod p ^n$. Then $p x_n \\equiv 0 \\mod p ^n A^{(2)} [\\frac 1 E]$. Since $A^{(2)} [\\frac 1 E]$ is domain,  $x_{n}\\equiv 0 \\mod p ^{n -1}$. Hence  $x = 0 $. As $B^{(2)}$ has no $p$-torsion, we see that $B^{(2)}\\subset B^{(2)}[\\frac 1 p]$. \nTo see the natural map $A^{(2)} \\to B^{(2)}$ is injective, it suffices to show that $A^{(2)} \/ p A^{(2)} $ injects to $ A^{(2)} \/ pA^{(2)} [\\frac 1 u]= B ^{(2)}\/ pB^{(2)}$. Clearly, this is equivalent to that $A^{(2)} \/ pA^{(2)}$ has no $u$-torsion. Note that $A^{(2)}$ is obtained by taking prismatic envelope of $A^{\\ho 2}= W(k)[\\![x, z]\\!]$ for the ideal $ I = (z)$. As mentioned before, we can apply \\cite[Prop. 3.13]{BS19} to our situation. So $A^{(2)}$ is flat over $A$ and hence $A^{(2)} \/ p A^{(2)}$ has no $u$-torsion as desired. \n\n\nNow we can regard $B^{(2)}$ and $A^{(2)} [\\frac 1 p]$ as  subrings of $B^{(2)}[\\frac 1 p]$. In particular, $ B^{(2)} \\cap A^{(2)} [\\frac 1 p ]$ makes sense and contains $A^{(2)}$. For any $x\\in B^{(2)} \\cap A^{(2)} [\\frac 1 p ]$, if $x\\not \\in A^{(2)}$ but $p x \\in A^{(2)}$. Then the image of $y = px  $ inside $A^{(2)}\/ p A^{(2)}$ is nonzero but the image of $y$ in $B ^{(2)}\/ p B ^{(2)}$ is zero. This contradicts to that $A^{(2)} \/ p A^{(2)} $ injects to $B^{(2)}\/ p B^{(2)}$. So such $x$ can not exist and we have $B ^{(2)} \\cap {A^{(2)}} [\\frac 1 p] = A^{(2)}$ as required. \\end{proof}\n\n\nBy \\cite[Lem. 3.9]{BS19}, any prism $(B, J)$ admits its perfection $(B,J)_{\\perf}=(B_{\\perf}, JB _{\\perf})$. \n\\begin{remark}\nIn \\cite{BS19}, the underlying $\\delta$-ring of $(B,J)_{\\perf}$ is denoted by $(B_{\\infty},JB_{\\infty})$, and $B_{\\perf}$ is defined as the direct perfection of $B$ in the category of $\\delta$-rings. In this paper, we write $B_{\\perf}$ as the $(p,J)$-adic completion of $\\mathrm{colim}_{\\varphi} B$, which also coincides with the derived $(p, I)$-completion of $\\mathrm{colim}_{\\varphi} B$ (cf. Lemma 3.9 of $loc.cit.$).\n\\end{remark}\n\n\\begin{lemma}\\label{lem-perfisflat}\nWe have $(A^{(2)})_{\\perf}$ and $(A^{(2)}_{\\st})_{\\perf}$ are $A$-flat.\n\\end{lemma}\n\\begin{proof} We have seen that $A^{(2)}$ is $A$-flat via $i_1$. And it is easy to see $\\varphi$ on $A$ is flat. Since $i_1$ is a $\\delta$-map, so we have $\\varphi^n\\circ i_1 =i_1 \\circ \\varphi^n$ which is flat. So $\\mathrm{colim}_\\varphi A^{(2)}$ is flat over $A$. In particular, we will have $A_{\\perf}$ is $(p , E)$-complete flat over $A$. Now since $A$ is Noetherian, by \\cite[Tag 0912]{stacks-project}, we have $(A^{(2)})_{\\perf}$ is $A$-flat. The proof for $(A^{(2)}_{\\st})_{\\perf}$ is the same. \n\\end{proof}\n\n\\subsection{Embedding \\texorpdfstring{$A^{(2)}$}{A(2)} and \\texorpdfstring{$A^{(2)}_{\\st}$}{A(2)st} to \\texorpdfstring{$\\Ainf$}{Ainf}}\\label{subsec-embedding}\nLet $\\Ainf=W(\\O_{\\mathbb C_p}^\\flat)$, then there is a surjection $\\theta: \\Ainf \\to \\O_{\\mathbb C_p}$ and $\\Ker\\theta=(E)$. And let $\\BdR^+$ be the $\\ker\\theta$-adic completion of $\\Ainf[\\frac{1}{p}]$.\n\n\\begin{definition}\nLet $\\A_{\\max}$ be the $p$-adic completion of the $\\Ainf$-subalgebra of $\\BdR^+$ generated by $E\/p$. \n\\end{definition}\nIt can be easily seen that $\\varphi(E\/p):=\\varphi(E)\/p\\in A_{\\mathrm{cris}}\\subset \\A_{\\max}$ is well-defined and it extends the Frobenius structure on $\\Ainf$ to an endomorphism on $\\Amax$.\n\nLet $\\{\\varpi_n\\}_{n\\geq 0}$ be a compatible system of $p^n$-th roots of $\\varpi_0=\\varpi$ and $\\{\\zeta_n\\}_{n\\geq 0}$ be a compatible system of $p^n$-th roots of 1. Write $\\varpi^\\flat  : = \\{\\varpi_n\\}_{n\\geq 0}, \\zeta^\\flat : = \\{\\zeta_n\\}_{n\\geq 0}\\in \\O_{\\mathbb C_p}^\\flat$ and  let $u=[\\varpi^\\flat ]$, $\\epsilon=[\\zeta^\\flat]$, $v=\\epsilon u$ and $\\mu=\\epsilon-1$ be elements inside $\\Ainf$. We can regard $W(k)[\\![x,y]\\!]$ as a subring of  $\\Ainf$ via $x \\mapsto u$ and $y\\mapsto v$. Consider $z' = \\frac{u-v}{E}\\in \\Ainf [\\frac 1 E]$. Since $u -v = u (\\epsilon -1)$ is clearly inside $\\Ker (\\theta )$ and $ \\Ker (\\theta) = E \\Ainf$, we conclude that $ z' \\in \\Ainf$. Hence we have a natural map (of $\\delta$-rings) $\\iota_A : \\widetildeA^{(2)} \\to \\Ainf$ via $z\\mapsto z'$,  which naturally extends to $\\iota _A : A^{(2)}\\to \\Ainf$ because $(p, E)$-topology of $A^{(2)}$ matches with the weak topology of $\\Ainf$. Similarly, we have map of $\\delta$-rings $\\iota_{\\st}: A^{(2)}_{\\st} \\to A_{\\inf}$ via $ x \\mapsto u$ and $\\mathfrak y \\mapsto \\epsilon-1$ and $w\\mapsto \\frac{\\epsilon-1}{E}$. \n\\begin{remark}\\label{rem-embedding-depend} Once we know that $A^{(2)}$ is self-product of $A$ inside $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ with $X= \\Spf (\\O_K)$ as explained in \\S \\ref{subsec-pris-crystal}. The map $\\iota_A$ can be constructed as the following: First we fix an embedding $A\\to \\Ainf$ by sending $x \\mapsto u =  [\\varpi^\\flat]$. Then $A\\to \\Ainf$ by $x \\to v= \\epsilon u$ is another map of prisms. By universal property of $A^{(2)}$, these two maps extends to a map $\\iota_A : A^{(2)} \\to \\Ainf$. Clearly, the map $\\iota_A : A^{(2)} \\to \\Ainf$ depends on choice of ${\\varpi}^\\flat = (\\varpi_n)_{n \\geq 0}$ and ${\\zeta}^\\flat = (\\zeta_n)_{n \\geq 0}$. Also $\\iota_A$ is a special case of $\\iota^{(2)}_\\gamma$ defined by \\eqref{equ-diagram-prisms} in \\S\\ref{subsec-pris-crystal-proof}. Indeed if $\\gamma ([w^\\flat])= [\\zeta^\\flat][w ^\\flat]$ then $\\iota_A = \\iota^{(2)}_{\\gamma}$. Similarly comment also applies to $\\iota_{\\st}$. \n\\end{remark}\n\n\n\\begin{proposition}\nThere is a unique embedding \n$\n\\begin{tikzcd}\nA_{\\max}^{(2)} \\arrow[r, hookrightarrow] & \\A_{\\max}\n\\end{tikzcd}\n$ such that\n$$\n\\begin{tikzcd}\nW(k)[\\![x,y]\\!] \\arrow[r, hookrightarrow]\\arrow[d, hookrightarrow] & \\Ainf \\arrow[d, hookrightarrow]  & \\\\\nA_{\\max}^{(2)} \\arrow[r, hookrightarrow] & \\A_{\\max}  \\arrow[r, hookrightarrow] & \\BdR^+\n\\end{tikzcd}\n$$\ncommutes. Furthermore, ${\\textnormal{Fil}} ^ i \\BdR^+ \\cap A^{(2)}_{\\max}= {\\textnormal{Fil}} ^i A^{(2)}_{\\max}$. The same result holds when $A^{(2)}_{\\max}$ is replaced by $A^{(2)}_{\\st , \\max}$.  \n\\end{proposition}\n\\begin{proof} In the following, we only treat the case of $A^{(2)}_{\\st , \\max}$ while the proof of $A^{(2)}_{ \\max}$ is the same by noting that $z= u w$ in $A_{\\inf}$. \n\nThe uniqueness is clear. To show the existence of the embedding, it is enough to show $\\gamma_i(w)\\in\\Amax$ for all $i\\geq 1$. \n\nIt is a well-known fact that $\\Amax$ is isomorphic to the $p$-adic completion of $\\Ainf[\\frac{u^e}{p}]$, and $\\Amax[1\/p]$ is a Banach $\\mathbb Q_p$-algebra, which is the completion of $\\Ainf[1\/p]$ under the norm $\\lvert \\cdot \\rvert_{p^{-1}}$ such that\n$$\n\\lvert x \\rvert_{p^{-1}} = \\sup_{n} \\{p^{-n}\\lvert x_n \\rvert_{\\O_{C}^\\flat}\\}\n$$\nwhere $x=\\sum_{n\\gg 0} [x_n]p^n \\in \\Ainf[1\/p]$. And we have for $x\\in \\Amax[1\/p]$, $x\\in \\Amax$ if and only if $\\lvert x \\rvert_{p^{-1}} \\leq 1$. Moreover $\\lvert \\cdot \\rvert_{p^{-1}}$ is multiplicative. So now it is enough to show for $x=\\gamma_i(w)$ considered as an element inside $ \\Amax[1\/p]$, we have $\\lvert x^{p-1} \\rvert_{p^{-1}}\\leq 1$. To show this, we have by \\cite[Proposition 3.17]{BMS1}, $\\xi:=\\mu\/\\varphi^{-1}(\\mu)$ is a generator of $\\Ker\\theta$ with $\\mu = \\epsilon-1$. In particular, $w=\\mu\/E = a  \\varphi^{-1} (\\mu) \\in \\Ainf$ with $a\\in \\Ainf^\\times$. \nAnd we can check $\\overline{w}^{p-1} = c \\overline{u}^e$ inside $\\O_C^\\flat=\\Ainf\/p\\Ainf$, with $c$ a unit. So $w^{p-1}=au^e+bp$ with $a,b\\in \\Ainf$, and\n$$\nx^{p-1}=\\frac{(au^e+bp)^i}{(i!)^{p-1}}.\n$$\nUsing the fact $v_p(i!)< \\frac{i}{p-1}$, one can show each term in the binomial expansion on the right hand side of the equation has $\\lvert \\cdot \\rvert_{p^{-1}}$-norm less or equal to $1$, so in particular, $\\lvert x^{p-1} \\rvert_{p^{-1}}\\leq 1$.\n\nTo prove that ${\\textnormal{Fil}} ^i \\BdR^+ \\cap  A ^{(2)}_{\\st, \\max} = {\\textnormal{Fil}} ^i A ^{(2)}_{\\st , \\max}$, it suffices to show that $E \\BdR ^+ \\cap A ^{(2)}_{\\st, \\max}[\\frac 1 p] = E A ^{(2)}_{\\st , \\max}[\\frac 1 p]$. By  Proposition  \\ref{prop-key-property},  we reduces to prove that the map $$\\theta : D_w= A ^{(2)}_{\\st , \\max}[\\frac 1 p ]\/E \\to \\BdR^+ \/ E= \\mathbb C_p$$ is injective. Let $f(w) = \\sum_{i \\geq 0} a_i \\gamma_i (w)\\in \\Ker \\theta$ with $a_i \\in \\O_K$ limits to $0$ $p$-adically. Then $f(w_0) = 0$ with $w_0: = \\theta (w)= \\theta (\\frac{\\epsilon-1}{E}) \\in \\mathbb C_p$. Note  $v_p (w_0)\\geq \\frac{1}{p-1}$ because it is well-known  $ \\frac{\\epsilon-1}{\\varphi^{-1}(\\epsilon) -1}$ is another generator of kernel $\\theta: A_{\\inf}  \\to \\O_{\\mathbb C_p}$ and then  \n$v_p (w_0) = v_p (\\theta (\\varphi^{-1} (\\epsilon)-1))= \\frac{1}{p-1}$. \nSince we are aiming to show that $f= 0$, without loss of generality, we can assume that $K$ contains $p_1= \\sqrt[p-1]{p}$. Note that \n$v_p (i !)\\leq \\frac{1}{p-1}$, we conclude that $\\frac{w_0}{p_1}$ is a root of $f(p_1 w)$ which is in $\\O_K\\langle w\\rangle$. By Weierstrass preparation theorem, $w_0$ is algebraic over $K$ unless $f=0$.  By Lemma below,  $w_0: = \\theta (w) \\in \\mathbb C_p$ is transcendental over $K$ and hence $f= 0$. \n\\end{proof}\n\\begin{lemma}\\label{lem-transcendental}\n$w_0 = \\theta (\\frac{\\epsilon-1}{E})$ is transcendental over $K$. \n\\end{lemma}\n\\begin{proof}\nIf $w_0$ is contained in an algebraic extension $L$ over $K$, we define $L_{0,\\infty}=\\bigcup_n L(\\varpi_n)$. For $g\\in G_{L_{0,\\infty}}$, we will have \n$$\n\\theta(g(\\frac{\\epsilon-1}{E}))=g(w_0)=w_0=\\theta(\\frac{\\epsilon-1}{E}).\n$$\nSince $G_{L_{0,\\infty}}$ fix $E$,  $\\theta(\\frac{g(\\epsilon-1)-(\\epsilon-1)}{E})=0$. This implies $g(\\epsilon-1)-(\\epsilon-1)\\in {\\textnormal{Fil}}^2\\BdR^+$. Recall for $t=\\log \\epsilon$, $t-(\\epsilon-1)\\in {\\textnormal{Fil}}^2\\BdR^+$, so we have $g(t)-t \\in {\\textnormal{Fil}}^2\\BdR^+$. But this can't be true. Since $L_{0,\\infty}$ can only contain finitely many $p^n$-th roots of $1$, for $g\\in G_{L_{0,\\infty}}$, $g(t)=c(g)t$ satisfying $c(g) \\in \\mathbb Q_p$ and $c(g)\\neq 1$. This implies $g(t)-t = (c(g)-1)t \\in {\\textnormal{Fil}}^1\\BdR^+ \\setminus {\\textnormal{Fil}}^2\\BdR^+$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor-inj}\nThe natural maps $\\iota _ A : A ^{(2)} \\to A_{\\inf}$ and $\\iota_{\\st} : A^{(2)}_{\\st} \\to A_{\\inf}$ are injective. \n\\end{corollary}\n\nTo summarize, we have the following commutative diagram of rings inside $\\BdR^+$:\n$$\n\\begin{tikzcd}\nA^{(2)}  \\arrow[d, hookrightarrow] \\arrow[r, hookrightarrow] & A^{(2)}_{\\st}\\arrow[r, hookrightarrow] \\arrow[d, hookrightarrow]  & \\Ainf \\arrow[d, hookrightarrow] \\\\\nA^{(2)}_{\\max} \\arrow[r, hookrightarrow] & A^{(2)}_{\\st, \\max} \\arrow[r, hookrightarrow] & \\Amax.  \n\\end{tikzcd}\n$$\n\n \\section{Application to semi-stable Galois representations}\nIn this section, we assume that $R= \\O_K$. We explain how to use the period ring $A^{(2)}$ and $A^{(2)}_{\\st}$ to understand lattices in crystalline and semi-stable representations. Roughly speaking, we are going to use $A^{(2)}$ and $A^{(2)}_{\\st}$ to replace $\\widehat{\\mathcal R}$ in the theory of $(\\varphi , \\hat G)$-modules developed in \\cite{liu-notelattice}. \n\nLet $K _\\infty =\\bigcup_{n = 1}^\\infty K (\\varpi _n )$, $G_\\infty: = {\\rm Gal } (\\overline  K \/ K_\\infty)$ and $G_K: = {\\rm Gal } (\\overline K  \/ K)$.  Recall that $A= \\mathfrak S = W(k)[\\![u]\\!]$. Let $S $  be the $p$-adic completion of $ W(k) [\\![u , \\frac{E^i}{i !}, i \\geq 1]\\!]$, which is the  PD envelope of $W(k)[u]$ for the ideal $(E)$. It is clear that $S\\subset \\Omax$. We define $\\varphi$ and ${\\textnormal{Fil}} ^i$ on $S$ induced that from those on $\\Omax$, in particular, ${\\textnormal{Fil}} ^i S = S \\cap E ^i \\Omax[\\frac 1 p]$. Note that $A$ embeds to $\\Ainf$ via $u \\mapsto [\\varpi^\\flat ]$ is not stable under $G_K$-action but only on $G_\\infty$-action. For any $g \\in G_K$, define ${\\underline{\\varepsilon}} (g)= \\frac{g(u)}{u}$. It is clear that ${\\underline{\\varepsilon}} (g) = \\epsilon ^{a(g)}$ with $a(g) \\in \\mathbb Z_p$. We define \\emph{two}  differential operators $N_S$ and $\\nabla_S$ on $S$ by $N_S(f) = \\frac{d f}{du}u$ and $\\nabla_S (f) = \\frac{ df }{du}$. We need $\\nabla_S$ to treat crystalline representations.  \n\n\\subsection{Kisin module attached to semi-stable representation}\\label{subsec-Kisin-st} Fix $h \\geq 0$, \na \\emph{Kisin module of height $h$} is a finite free $A$-module $\\mathfrak{M} $ with a semi-linear endomorphism $\\varphi_{\\mathfrak{M}}: \\mathfrak{M} \\to \\mathfrak{M}$ so that $\\coker (1 \\otimes \\varphi_\\mathfrak{M})$ is killed by $E^h$, where $1 \\otimes \\varphi_{\\mathfrak{M}} : \\mathfrak{M} ^* : = A \\otimes _{\\varphi, A}\\mathfrak{M} \\to \\mathfrak{M} $ is linearization of  $\\varphi_\\mathfrak{M}$. Note here we are using classical setting of Kisin modules used in \\cite{liu-notelattice} but it is good enough for this paper. The following summarizes the results on Kisin modules attached to $G_K$-stable $\\mathbb Z_p$-lattices in semi-stable representations. The details and proofs of these facts can be found in \\cite{liu-notelattice}. \n\nLet $T$ be a $G_K$-stable $\\mathbb Z_p$-lattice inside a semi-stable representation $V$ of $G_K$ with Hodge-Tate weights in $\\{0, \\dots , h\\}$. Let $D: = D^*_{\\st} (V)= \\mathrm{Hom}_{\\mathbb Q_p, G_K} (V , B_{\\st})$ be the filtered $(\\varphi , N)$-module attached to $V$ and $D_K : = K \\otimes_{K_0} D$. Then there exists a unique Kisin module $\\mathfrak{M} : = \\mathfrak{M} (T) $ of height $h$ attached to $T$ so that \n\\begin{enumerate}\n    \\item $\\mathrm{Hom}_{\\varphi , A} (\\mathfrak{M} , \\Ainf)\\simeq T|_{G_\\infty}$. \n    \\item There exists an $S$-linear isomorphism  \n    $$\\iota_S : S [\\frac 1 p] \\otimes _{\\varphi, A}\\mathfrak{M} \\simeq D \\otimes _{W(k)} S $$ so that $\\iota_S$ is compatible with $\\varphi$ on the both sides. \n\\item $\\iota_S$ also induces an isomorphism ${\\textnormal{Fil}}^h (S [\\frac 1 p] \\otimes _{\\varphi, A}\\mathfrak{M}) \\simeq {\\textnormal{Fil}} ^h (D\\otimes_{W(k)} S) $. The filtration on the both sides are defined as following: \n\\[{\\textnormal{Fil}}^h (S [\\frac 1 p] \\otimes _{\\varphi, A}\\mathfrak{M}): =\\left  \\{ x \\in S [\\frac 1 p] \\otimes _{\\varphi, A}\\mathfrak{M}| (1\\otimes \\varphi_\\mathfrak{M} (x)) \\in {\\textnormal{Fil}} ^h S[\\frac 1 p ] \\otimes_A \\mathfrak{M} \\right \\}.  \\] \nTo define \nfiltration on ${\\mathcal D} : = S \\otimes _ {W(k)} D$, we first extend the monodromy operator $N_{\\mathcal D}$ (resp. $\\nabla_{\\mathcal D}$) on $D$ to ${\\mathcal D}$ by $N_{{\\mathcal D}}= 1 \\otimes N_D + N_S \\otimes 1$ (resp. $\\nabla_{\\mathcal D} = 1 \\otimes N_D + \\nabla_S \\otimes 1$). Then we define ${\\textnormal{Fil}} ^i {\\mathcal D}$ by induction: set ${\\textnormal{Fil}}^0 {\\mathcal D} = {\\mathcal D}$ and \n\\[ {\\textnormal{Fil}} ^i{\\mathcal D}: = \\{x \\in {\\mathcal D}|  N_{{\\mathcal D}}(x) \\in {\\textnormal{Fil}}^{i-1}{\\mathcal D}, f_\\varpi (x) \\in {\\textnormal{Fil}} ^i D_K\\}\\]\nwhere $f_\\varpi : {\\mathcal D} \\to D_K$ is induced by $S\\to \\O_K$ via $u \\mapsto \\varpi. $\n\\end{enumerate}\n\\begin{remark}[Griffith transversality]\\label{rem-GT} From the construction of ${\\textnormal{Fil}} ^i {\\mathcal D}$, we see that $N_{{\\mathcal D}} ({\\textnormal{Fil}}  ^i{\\mathcal D}) \\subset {\\textnormal{Fil}}^{i -1}{\\mathcal D}$. This property is called Griffith transversality. \n\nWe only use $\\nabla_{\\mathcal D}$ when $N_D = 0$, that is, when $V$ is crystalline. In this case, it is clear that $N_{\\mathcal D} = u \\nabla_{{\\mathcal D}}$. So it is clear that $\\nabla_{\\mathcal D} ({\\textnormal{Fil}} ^i{\\mathcal D}) \\subset {\\textnormal{Fil}} ^{i-1}{\\mathcal D}$. \n\\end{remark}\nFor ease of notations, we will write $N = N_{\\mathcal D}$ and $\\nabla = \\nabla_{\\mathcal D}$ in the following. \nLet $T^\\vee: = \\mathrm{Hom}_{\\mathbb Z_p} (T , \\mathbb Z_p)$ and $V ^\\vee : = T^\\vee \\otimes_{\\mathbb Z_p}\\mathbb Q_p$ denote the dual representations.  \nThen there exists an $A_{\\inf}$-linear injection \n\\begin{equation}\\label{eqn-iota-A}\n\\iota_\\mathfrak{M}: A_{\\inf} \\otimes _A \\mathfrak{M} \\to  T ^\\vee \\otimes_{\\mathbb Z_p} \\Ainf, \n\\end{equation}\nwhich is compatible with $G_\\infty$-actions ($G_\\infty$ acts on $\\mathfrak{M}$ trivially) and $\\varphi$ on both sides. Applying $S \\otimes_{\\varphi, A}$ and using $\\iota _S: =S \\otimes_{\\varphi, A} \\iota_\\mathfrak{M} $, we obtain the following commutative diagram \n$$\n\\xymatrix@C=5em{ \\Acris[\\frac 1 p] \\otimes_{\\varphi , A}\\mathfrak{M} \\ar[d]_\\wr  ^{\\Acris \\otimes_S \\iota _S} \\ar[r] ^{S \\otimes_{\\varphi, A} \\iota _\\mathfrak{M}} & V^\\vee \\otimes_{\\mathbb Z_p} \\Acris \\ar@{=}[d]\\\\   \\Acris \\otimes_{W(k)} D \\ar[r]^{\\alpha} & V^\\vee \\otimes _{\\mathbb Z_p} \\Acris}\n$$\nwhere the second row  $\\alpha$ is built by the classical comparison $$B_{\\st} \\otimes_{K_0} D^*_{\\st}(V) \\simeq   V^\\vee \\otimes_{\\mathbb Q_p} B_{\\st},   $$\nand  $\\alpha$ is $G_K$-stable on the both sides. The left side of $\\alpha$ is defined by \n\\[ \\forall x \\in D, \\forall g \\in G_K, g (x) = \\sum_{i = 0}^\\infty N^i (x) \\gamma_i (\\log ({\\underline{\\varepsilon}}(g))) \\]\nTherefore, if we regard $\\mathfrak{M}^* : = A \\otimes_{\\varphi, A} \\mathfrak{M}$ as an  $A$-submodule of  \n$ V^\\vee \\otimes_{\\mathbb Z_p} \\Acris$ via injection $\\iota^* : = S \\otimes_{\\varphi, A} \\iota_A$, one can show that: \n\\begin{equation}\\label{eqn-G-action}\n\\forall g \\in G_K, x \\in \\mathfrak{M}^*, g(x) = \\sum_{i = 0}^\\infty N_{\\mathcal D}^i (x) \\gamma_i (\\log ({\\underline{\\varepsilon}}(g))).  \n\\end{equation}\n When $V$ is crystalline, or equivalently, $N_D = 0$, we have (\\cite[\\S8.1]{LL2021comparison})\n \\begin{equation}\\label{eqn-G-action-2}\n\\forall g \\in G_K, x \\in \\mathfrak{M}^*, g(x) = \\sum_{i = 0}^\\infty \\nabla_{\\mathcal D}^i (x) \\gamma_i (u {\\underline{\\varepsilon}}(g)).  \n\\end{equation}\n\n\\subsection{Descent of the \\texorpdfstring{$G_K$}{GK}-action}\\label{subsec-G-image}\n\nLet us first discuss the  $G_K$-action on $\\mathfrak{M} \\subset T ^\\vee \\otimes_{\\mathbb Z_p}\\Ainf$ via $\\iota_\\mathfrak{M}$ in \\eqref{eqn-iota-A} in more details.  \nWe select an $A$-basis $e_1, \\dots , e_d$ of $\\mathfrak{M}$ so that $\\varphi (e_1, \\dots, e_d)= (e_1, \\dots , e_d )\\mathfrak A$ with $\\mathfrak A \\in {\\rm M}_d (A)$.  Then there exists a matrix $B\\in {\\rm M}_d (A)$ so that $\\mathfrak A B = B \\mathfrak A = E^h I_d$. For any $g \\in G_K, $ let $X_g$ be the matrix so that \n\\[ g (e_1, \\dots , e_d) = (e_1, \\dots , e_d) X_g. \\]\nIn this section, we are interested in where the entries of $X_g$ locates. \n\\begin{theorem}\\label{Thm-1}The entries of $X_g$ are in $A^{(2)}_{\\st}$. If $V$ is crystalline and $g(u)- u = Ez$ then $X_g \\in {\\rm M}_{d} (A^{(2)}). $\n\\end{theorem}\n\nFirst,  it is well-known that $ W (\\mathbb C_p^\\flat) \\otimes_{\\Ainf} \\iota_\\mathfrak{M} $ is an isomorphism. So $X_g\\in {\\rm M}_d (W (\\mathbb C_p^\\flat))$. Since $G_K$-actions and $\\varphi$-commutes, we have $$ \\mathfrak A \\varphi(X_g)= X_g g (\\mathfrak A)  .$$  \n Define $${\\textnormal{Fil}} ^h \\mathfrak{M} ^* : = \\{ x \\in \\mathfrak{M}^* | (1 \\otimes \\varphi_{\\mathfrak{M}}) (x) \\in E^h \\mathfrak{M}\\}. $$\nSince $\\mathfrak{M}$ has $E$-height $h$, it is easy to show that ${\\textnormal{Fil}} ^h \\mathfrak{M}^*$ is a finite free $A$-module and ${\\textnormal{Fil}}^h {\\mathcal D}$ is generated by ${\\textnormal{Fil}} ^h \\mathfrak{M}^*$. \n\nTo be more precise, let $\\{e^*_i : =1 \\otimes e_i, i =1 , \\dots , d\\}$ be an $A$-basis of $\\mathfrak{M}^*$. It is easy to check that $(\\alpha_1, \\dots , \\alpha_d)= (e_1^* , \\dots , e_d^*) B$ is an $A$-basis of ${\\textnormal{Fil}} ^h \\mathfrak{M}^*$,  and it is  also an $S[\\frac 1 p]$-basis of ${\\textnormal{Fil}} ^h {\\mathcal D}$. \nSo for any $g \\in G_K$, we have $g (\\alpha_j) = \\sum\\limits_{i = 0}^\\infty N ^i (\\alpha_j) \\gamma _i (\\log ({\\underline{\\varepsilon}} (g))) $. By Griffith transversality in Remark \\ref{rem-GT}: \n$N ({\\textnormal{Fil}} ^i {\\mathcal D})\\subset {\\textnormal{Fil}}^{i-1}{\\mathcal D},   $\n we have, \n \\begin{equation}\\label{eqn-action-g}\n g (\\alpha_j)= \\sum_{i = 0}^h N ^i (\\alpha_j) E^i \\gamma _i (\\frac {\\log ({\\underline{\\varepsilon}} (g))}{E}) + \\sum_{i > h}^\\infty N^i (\\alpha_j) \\gamma_i(E) (\\frac{\\log ({\\underline{\\varepsilon}} (g))}{E})^i.      \n \\end{equation}\nSince $N^i (\\alpha_j) E^i \\in {\\textnormal{Fil}} ^h {\\mathcal D}$,  $\\gamma_i (E)$ in $\\Omax$ and $w ^n \\to 0$ inside $A^{(2)}_{\\st, \\max}$,  we see that $g (\\alpha_1, \\dots , \\alpha_d) = (\\alpha_1 , \\dots , \\alpha_d) Y_g $ with $Y_g \\in A^{(2)}_{\\st , \\max}[\\frac 1 p ]. $\n\nIn the case that $V$ is crystalline, using \\eqref{eqn-G-action-2}, we have \n$$g (\\alpha_j)= \\sum_{i = 0}^h \\nabla ^i (\\alpha_j) E^i \\gamma _i (\\frac {u{\\underline{\\varepsilon}} (g)}{E}) + \\sum_{i >  h}^\\infty \\nabla^i (\\alpha_j) \\gamma_i(E) (\\frac{u{\\underline{\\varepsilon}} (g)}{E})^i $$\n\\emph{If $g $ is chosen so that  $ g (u)-u = Ez$} then, a similar argument can shows that $g (\\alpha_1, \\dots , \\alpha_d) = (\\alpha_1 , \\dots , \\alpha_d) Y^\\nabla_g $ with $Y^\\nabla_g \\in A^{(2)}_{\\max}[\\frac 1 p]. $\n\nNow $g (e_1^*, \\dots , e_d^*) = (e_1^*, \\dots, e_d ^*) \\varphi (X_g)$.  Using similar arguments, we see that $\\varphi(X_g)$'s entry are in $A^{(2)}_{\\st , \\max}[\\frac 1 p ]$ and $A^{(2)}_{\\max}[\\frac 1 p]$ respectively. Since $(\\alpha_1 , \\dots , \\alpha_d) = (e_1^*, \\dots, e_d^*)B$, we conclude that $$ \\varphi(X_g) g (B) = B Y_g.$$ \nUsing the formula that $\\mathfrak A \\varphi (X_g) = X_g g (\\mathfrak A)$ and $\\mathfrak A B = B \\mathfrak A = E^h I_d$, we conclude that \n$Y_g = (\\frac{g (E)}{E}) ^h X_g$. Write  $r= \\frac{g (E)}{E}$. We claim that $r$ is a unit in $A^{(2)}_{\\st}$. Indeed, \n$\\frac{g(E)}{E}= \\frac{E (u \\epsilon ^{a(g)})}{E(u)}= \\sum\\limits_{i =0}^e E^{(i)} (u) \\frac{ u ^i (\\epsilon^{a(g)}-1)^i}{E i !}$ is again inside $A_{\\st}^{(2)}$, where $E^{(i)}$ means the $i$-th derivative of $E$. And it is easy to show $g(E)$ is also a distinguished element $A_{\\st}^{(2)}$, so by \\cite[Lemma 2.24]{BS19}, $r$ is a unit. Similarly, when $g(u)-u=Ez$, we will have $r=\\frac{g (E)}{E}\\in (A^{(2)})^\\times$. Hence  \n\\begin{equation}\\label{eqn-key-eqn}\nE^h X_g  = r ^{-h} \\mathfrak A  \\varphi (X_g) g (B).\n\\end{equation}\n\n\n\nNow we can apply Proposition \\ref{prop-desecnt} and Proposition \\ref{prop-Ast-properties} (5) to the above formula, we conclude that for $g\\in G_K$ (resp. $g\\in G_K$ such that $g(u)-u=Ez$ and $V$ is crystalline), we have $X_g$ has entries in $A^{(2)}_{\\st}[\\frac 1 p]$ (resp. $A^{(2)}[\\frac 1 p]$). \n\nTo complete the proof of Theorem \\ref{Thm-1}, it suffices to show that  entries $X_g$ are in $A^{(2)}_{\\st}$ (resp. $A^{(2)}$). Unfortunately, the proof to remove ``$\\frac 1 p$\" is much harder, which needs \\S \\ref{subsec-phi-tau} and \\S \\ref{subsec-pris-crystal-proof}. For the remaining of this subsection, we only show that the proof of Theorem \\ref{Thm-1} can be reduced to the case that $g = \\tilde \\tau$ for a special selected $\\tilde \\tau \\in G_K$.  \n\nLet $L = \\bigcup\\limits_{n =1}^\\infty K_{\\infty} (\\zeta_{p ^n})$, $K_{1^\\infty}: = \\bigcup_{n =1}^\\infty K (\\zeta_{p ^n})$,  $\\hat G : = \\Gal(L \/K) $ and $H_K : = \\Gal (L \/ K _\\infty)$. \nIf $p > 2$ then it is known that $\\hat G \\simeq \\Gal (L\/ K_{1 ^\\infty}) \\rtimes H_K $ with \n$\\Gal (L\/ K_{1 ^\\infty}) \\simeq \\mathbb Z_p$. Let $\\tau$ be a topological generator of $\\Gal (L\/ K_{1 ^\\infty}) $. We have $\\tau (u) = \\epsilon^a u$ with $a\\in \\mathbb Z_p ^\\times$. Without loss of generality, we may assume that $\\tau (u) = \\epsilon u$. If $p=2$ then we can still select $\\tau \\in \\hat G $ so that $\\tau (u)= \\epsilon u$ and $\\tau, H_K$ topologically generate $\\hat G$. Pick $\\tilde \\tau \\in G_K$ a lift of $\\tau$. Clearly, we have $\\tilde \\tau (u ) - u = E z$. \n\n\\begin{proposition}\\label{thm-1prime} \nFor $g = \\tilde \\tau, $ the entries of $X_g$ are in $A^{(2)}_{\\st}$, and if further  $V$ is crystalline, then $X_g \\in {\\rm M}_{d} (A^{(2)}).$\n\\end{proposition}\n\\begin{lemma}\\label{lem-equivalenceofthm}\nProposition ~\\ref{thm-1prime} is equivalent to Theorem~\\ref{Thm-1}.\n\\end{lemma}\n\\begin{proof}\nSince  $\\hat{G}$ is topologically generated by $\\tau$ and $H_K$. So $G_K$ is topologically generated by $G_\\infty$ and $\\tilde{\\tau}$. And we have $\\tau(u)-u=(\\epsilon-1)u=Ez$. Now if $X_{\\tilde{\\tau}}$ has coefficient in $A^{(2)}_{\\st}$ and $X_g=I_d$ for all $g\\in G_\\infty$  then to show that $X_g \\in A^{(2)}_{\\st}$ for all $g \\in G_K$, it suffices to show that $X_{\\tilde \\tau^{p ^n }}$ converges to $I_d$ inside ${\\rm M}_d (A^{(2)}_{\\st})$. Since $A^{(2)}_{\\st}$ is closed in $A^{(2)}_{\\st, \\max}$ by Proposition \\ref{prop-Ast-properties} (5), it suffices to show that $X_{\\tilde \\tau^{p ^n}}$ converges inside $A^{(2)}_{\\st, \\max}$. Since $X_g= (\\frac{E}{g (E)})^r Y_g$ and $Y_g$ is defined by \\eqref{eqn-action-g}, we easily check that $X_{\\tilde \\tau^{p ^n}}$ converges to $I_d$ in $A^{(2)}_{\\st,\\max}$ by using that ${\\underline{\\varepsilon}}(\\tilde \\tau^{p ^n})$ converges to $0$ in $(p , \\epsilon-1)$-topology. The proof for the crystalline case is similar by replacing $A^{(2)} _{\\st}$ with $A^{(2)}$.\n\\end{proof}\nSo it remains to prove Proposition ~\\ref{thm-1prime} to complete the proof of  Theorem~\\ref{Thm-1}. We will prove Proposition~\\ref{thm-1prime} in \\S\\ref{subsec-pris-crystal-proof}. Briefly speaking, for $g = \\tilde \\tau$, we have shown that the linearization of the $g$-action defines a $\\varphi$-equivariant isomorphism:\n$$\nf_g: \\mathfrak{M}\\otimes_{A,\\iota_g} A_{\\st}^{(2)}[\\frac{1}{p}] \\simeq \\mathfrak{M}\\otimes_{A} A_{\\st}^{(2)}[\\frac{1}{p}]\n$$\nof $A_{\\st}^{(2)}[\\frac{1}{p}]$-modules, and since $g(u)- u = Ez$ and $V$ is crystalline, $f_g$ defines a $\\varphi$-equivariant isomorphism:\n$$\nf_g: \\mathfrak{M}\\otimes_{A,\\iota_g} A^{(2)}[\\frac{1}{p}] \\simeq \\mathfrak{M}\\otimes_{A} A^{(2)}[\\frac{1}{p}]\n$$\nof $A^{(2)}[\\frac{1}{p}]$-modules. Here $\\iota_g: A \\to A^{(2)}_{\\st}$ (resp. $\\iota_g: A \\to A^{(2)})$) is defined by $u \\to g(u)$. On the other hand, by \\cite[Theorem 5.6]{wu2021galois}, we will see the $g$-action on $T^\\vee \\otimes W(\\mathbb C_p^\\flat)$ also descent to a $\\varphi$-equivariant morphism $c$ of $B^{(2)}$-modules, and recall that $B^{(2)}$ the is $p$-adic completion of $A^{(2)} [\\frac 1 E]$. Then by comparing $c$ and $f_g$ using the technique developed in \\S\\ref{subsec-phi-tau}, we will deduce Proposition~\\ref{thm-1prime} from Lemma~\\ref{lem-intersection}.\n \n\\begin{remark}\\label{rem-inputofWu} Our original strategy to prove Theorem \\ref{Thm-1} is to show $A^{(2)}_{\\st} [\\frac 1 p] \\cap W ({\\mathbb C^\\flat_p}) = A^{(2)}_{\\st}$ (resp. $A^{(2)} [\\frac 1 p] \\cap W (\\O^\\flat_{\\mathbb C_p}) = A^{(2)}$).  This is equivalent to that $ A^{(2)}\/ p , A^{(2)}_{\\st}\/ p$ injects in $\\mathbb C_p^\\flat$. Unfortunately, it does not work out though we can show  $ \\widetilde A^{(2)}\/ p , \\widetilde {A^{(2)}_{\\st}}\/ p$ injects in $\\mathbb C_p^\\flat$.\n\n\\end{remark}\n\n\n\\subsection{Relation to \\texorpdfstring{$(\\varphi, \\hat G)$}{(phi,Ghat)}-modules}\\label{subsec-phiGhatmodules} In this subsection, we show that the base ring $\\widehat{{\\mathcal R}}$ for the theory of $(\\varphi, \\hat G)$-modules can be replaced by $A^{(2)}_{\\st}$. To this end, this builds a new theory of $(\\varphi, \\hat G)$-modules  with new base ring $A^{(2)}_{\\st}$. Since the idea of this new theory is almost the same as that of the old one, We will use \\emph{classical} to indicate we are using the theory over $\\widehat {\\mathcal R}$. For example, when we say classical $(\\varphi, \\hat G)$-module, it means a $(\\varphi , \\hat G)$-module over $\\widehat {\\mathcal R}$. \nRecall $L = \\bigcup\\limits_{n =1}^\\infty K_{\\infty} (\\zeta_{p ^n})$, $\\hat G : = \\Gal(L \/K) $ and $H_K : = \\Gal (L \/ K _\\infty)$. Let $\\mathfrak m $  be the maximal ideal of $\\O_{\\mathbb C_p}^\\flat$ and set $I_+ = W(\\mathfrak m)$ so that $\\Ainf\/ I _+ = W(\\bar k)$. For any subring $B\\subset \\Ainf$ set $ I_+ B = B \\cap I_+$. Let $t = \\log \\epsilon$,  $t ^{(i)} = t ^{r(i)} \\gamma_{\\tilde q(i)}(\\frac{t ^{p-1}}{p})$ where $ i = (p-1) \\tilde q (i) + r(i)$ with $ 0 \\leq r(i)<p-1. $\n Recall that $\\widehat {\\mathcal R} : = \\Ainf \\cap {\\mathcal R}_{K_0 }$ where \n$${\\mathcal R}_{K_0}:=\\left  \\{ \\sum_{i = 0}^\\infty f_i t ^{(i)}, f_i\\in S[\\frac 1 p], f_i \\to 0\\  p{\\text{-adically}}  \\right \\}.$$\n\n\n\\begin{lemma} \\label{lem-properties-Ast}\n\\begin{enumerate}\n    \\item As a subring of $\\Ainf$, $A^{(2)}_{\\st}$ is stable under $G_K$-action and the $G_K$-action factors through $\\hat G$. \n\\item $A^{(2)}_{\\st}\/ I_+A^{(2)}_{\\st} = W(k)$. \n\\item $I_+ A^{(2)} \\subset u A^{(2)}_{\\st}$. \n\\item $\\varphi (A^{(2)}_{\\st} ) \\subset \\widehat {\\mathcal R}$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}(1) It is clear that the $G_K$-action is stable on $W(k)[\\![u , \\epsilon -1]\\!]$. Since $A^{(2)}_{\\st}$ is $(p , E)$-completion of $W(k)[\\![u , \\epsilon -1]\\!] [\\delta^i (w), i \\geq 0]$, to show that $A^{(2)}_{\\st}$ is $G_K$-stable, it suffices to show that $g (w)\\in A^{(2)}_{\\st}$ (because $g$ and $\\delta$ commutes, if $g(x)\\in A^{(2)} _{\\st}$ then so is $g (\\delta (x))$). Now $E w = \\epsilon-1$, we have $g (E) g (w) = g (\\epsilon -1)= \\epsilon ^{a(g)} -1 $. Then $g (w) = \\frac{E}{g(E)} \\frac {\\epsilon^{a(g)} -1}{E}$. By \\cite[Lemma 2.24]{BS19}, $E\/g(E)$ is a unit in $A^{(2)}_{\\st}$, then $g (w)\\in A^{(2)}_{\\st}$. \n\n(2) It is clear that both $u , \\epsilon-1 $ are in $I_+$. Hence $w \\in I_+$ because $E w = \\epsilon-1 \\in I_+$ and $E \\equiv p \\mod I_+$. For any $x=\\sum\\limits_{i = 0}^\\infty  p ^i[x_i] \\in \\Ainf$, $x \\in I_+$ if and only of $x_i \\in \\mathfrak m$. Then it is easy to check that $\\delta (I_+)\\subset I_+$, and consequently  all $\\delta^i (w) \\in I_+$. So $I_+ A^{(2)}_{\\st}$ is topologically generated by $u, y=\\epsilon-1 , \\delta ^i (w), i \\geq 0$ and hence $A^{(2)}_{\\st}\/ I_+A^{(2)}_{\\st} = W(k) $ as required. \n\n(3) $I_+ A^{(2)}$ is topologically generated by $u, v=\\epsilon u, \\{\\delta ^i (z)\\}, i \\geq 0$. And (3) follows from $z= u w$ and $\\delta^n (z)= u^{p ^n} \\delta^n (w)$.  \n\n(4) Since $A^{(2)}_{\\st} \\subset A^{(2)}_{\\st , \\max}$, it suffices to show that \n$\\varphi (A^{(2)}_{\\st , \\max}) \\subset {\\mathcal R}_{K_0}$. Since $\\varphi(\\Omax) \\subset A[\\![\\frac{E^p }{p}]\\!]\\subset S$, it suffices to show that $\\varphi (\\gamma_n (w) ) \\in {\\mathcal R}_{K_0}$. Note that $\\varphi (E) = p \\nu$ with $\\nu \\in A[\\![\\frac{E^p }{p}]\\!] ^\\times$ and $\\gamma _i (\\epsilon -1) \\in {\\mathcal R}_{K_0}$. And we have\n$$\n\\varphi(w)=\\varphi (\\frac{(\\epsilon -1)}{E})= \\nu^{-1} (\\epsilon-1) \\sum_{i = 1}^p \\left ( \\binom{p}{i} \/ p \\right ) (\\epsilon -1)^{i -1} \n$$\nwhich is a polynomial with coefficients in $\\mathbb Z$ and in variables $\\nu^{-1}$ and $\\gamma_i(\\epsilon -1)$'s. In particular $\\varphi (\\gamma_n (w) ) \\in {\\mathcal R}_{K_0}$ by basic properties of divided powers. \n\\end{proof}\n\n\\begin{definition} A (finite free) $(\\varphi, \\hat G)$-module of height $h$ is a (finite free) Kisin module $(\\mathfrak{M} , \\varphi_{\\mathfrak{M}})$ of height $h$ together with an $A^{(2)}_{\\st}$-semi-linear $\\hat G$-action on $\\widehat \\mathfrak{M} : = A^{(2)}_{\\st} \\otimes _A \\mathfrak{M}$ so that \n\\begin{enumerate}\n    \\item The actions of $\\varphi$ and $\\hat G$ on $\\widehat \\mathfrak{M}$ commutes; \n    \\item $\\mathfrak{M} \\subset \\widehat \\mathfrak{M} ^{H_K}$; \n    \\item $\\hat G$-acts on $\\widehat \\mathfrak{M} \/ I^+ A^{(2)}_{\\st}$ trivially. \n\\end{enumerate}\n\\end{definition}\n\nThe category of $(\\varphi, \\hat G)$-modules consists of the above objects and morphism of two $(\\varphi, \\hat G)$-modules is morphism of Kisn modules that commutes with actions of $\\hat G$. Given a $(\\varphi, \\hat G)$-modules $\\widehat \\mathfrak{M} : = (\\mathfrak{M} , \\varphi, \\hat G)$, we define a $\\mathbb Z_p$-representation of $G_K$, \n$$\\widehat T (\\widehat \\mathfrak{M}) : = \\mathrm{Hom}_{A^{(2)}_{\\st}, \\varphi} (A^{(2)}_{\\st} \\otimes _A \\mathfrak{M}, \\Ainf).$$\n\n\nSince $\\varphi (A^{(2)}_{\\st})\\subset \\widehat {\\mathcal R}$, given a $(\\varphi, \\hat G)$-module $\\widehat \\mathfrak{M} : = (\\mathfrak{M}, \\varphi, \\hat G)$-defined as the above, $(\\mathfrak{M}, \\varphi)$ together $\\hat G$-action on $\\widehat{\\mathcal R} \\otimes_{\\varphi, A}\\mathfrak{M}$ is a \\emph{classical} $(\\varphi, \\hat G)$-modules $\\widehat \\mathfrak{M}_c$. It is easy to check that $\\widehat T (\\widehat \\mathfrak{M} ) = \\widehat T(\\widehat \\mathfrak{M}_c): =\\mathrm{Hom}_{\\widehat {\\mathcal R}, \\varphi} (\\widehat {\\mathcal R} \\otimes _{\\varphi , A} \\mathfrak{M}, \\Ainf). $ \n\n\n\\begin{theorem}\\label{thm-2}\nThe functor $\\widehat T$ induces an anti-equivalence between the category of $(\\varphi, \\hat G)$-modules of height $h$ and the category of $G_K$-stable $\\mathbb Z_p$-lattices in semi-stable representations with Hodge-Tate weights in $[0 , \\dots , h]$. \n\\end{theorem}\n\\begin{proof}\nGiven an $(\\varphi, \\hat G)$-module $\\widehat \\mathfrak{M}= (\\mathfrak{M}, \\varphi, \\hat G)$, $\\widehat\\mathfrak{M}_c $ is a classical $(\\varphi, \\hat G)$-module. So $\\widehat T (\\widehat \\mathfrak{M})= \\widehat T (\\widehat \\mathfrak{M}_c)$ is a lattice inside semi-stable representation with Hodge-Tate weights in $[0, \\dots , h]$. Conversely, given a lattice in semi-stable representation $T$ with Hodge-Tate weights in $[0, \\dots , h]$, following the proof for the existence of classical $(\\varphi, \\hat G)$-module $\\widehat \\mathfrak{M} $ so that $\\widehat T (\\widehat \\mathfrak{M} ) = T$, it suffices to show that for any $g \\in G_K$, \n$g (\\mathfrak{M}) \\subset A^{(2)}_{\\st} \\otimes _A \\mathfrak{M}$, here $\\mathfrak{M} $ and $ A^{(2)}_{\\st} \\otimes _A \\mathfrak{M}$ are regarded as submodules of $T^\\vee \\otimes _{\\mathbb Z_p} \\Ainf$ via $\\iota_\\mathfrak{M}$ in \\eqref{eqn-iota-A} and uses the $G_K$-action on $T^\\vee \\otimes _{\\mathbb Z_p} \\Ainf$. This follows Theorem \\ref{Thm-1}. \n\\end{proof}\n\nNow let us discuss when $\\widehat T (\\widehat \\mathfrak{M})$ becomes a crystalline representation. Recall that $\\tau$ is a selected topological generator of $\\Gal (L\/ K_{1 ^\\infty}) $, and we have $\\tau (u) = \\epsilon u$ and $\\tau, H_K$ topologically generate $\\hat G$. \n\n\n\n\\begin{corollary}\\label{cor-crystalline} Select $\\tau \\in \\hat G$ as the above. Then $\\widehat T (\\widehat \\mathfrak{M} )$ is crystalline if and only if $\\tau (\\mathfrak{M}) \\subset A^{(2)} \\otimes _{A} \\mathfrak{M}$. \n\\end{corollary}\n\\begin{proof} Clearly for the selected $\\tau$, we have $\\tau (u) - u = Ez$. If $T: = \\widehat T (\\widehat \\mathfrak{M})$ is crystalline then Theorem \\ref{Thm-1} proves that $\\tau (\\mathfrak{M}) \\subset A^{(2)} \\otimes _{A} \\mathfrak{M}$. Conversely,  Suppose $\\tau (\\mathfrak{M}) \\subset A^{(2)} \\otimes _{A} \\mathfrak{M}$. \nThen we see that $(\\tau-1)\\mathfrak{M} \\subset u \\Ainf \\otimes_A \\mathfrak{M}$ by Lemma \\ref{lem-properties-Ast} (3). And we have this is enough to show that $\\widehat T (\\widehat \\mathfrak{M})$ is crystalline. For example, We will have $\\mathfrak{M}\\otimes_{A}(\\Ainf[\\frac{1}{p}]\/\\mathfrak{p})$ has a $G_K$-fixed basis given by a basis of $\\mathfrak{M}$, where the ideal $\\mathfrak{p}$ is defined as $\\mathfrak{p}:=\\cup_{n\\in \\mathbb N} \\varphi^{-n}(u)\\Ainf[\\frac{1}{p}] \\subset \\Ainf[\\frac{1}{p}]$. Then one can prove by the same method in \\cite[Thm. 3.8]{Ozeki} or directly use \\cite[Theorem 4.2.1]{Duthesis} that $T$ is crystalline.  \n\\end{proof}\n\\begin{remark}\\label{rem-Ast-is-good} Though $A^{(2)}_{\\st}$ is still complicated, for example, it is not noetherian, $A^{(2)}_{\\st}$ is still better than old $\\widehat {\\mathcal R}$: at least it has explicit topological generators. Furthermore, $A^{(2)}_{\\st}$ is $p$-adic complete. This can help to close the gap in \\cite{liu-Fontaine} mentioned in \\cite[Appendix B]{gao2021breuilkisin}. Indeed, as indicated by Remark B.0.5 \\emph{loc.cit.}, if $\\widehat {\\mathcal R}$ can be shown to be $p$-adic complete then the gap in \\cite{liu-Fontaine} can be closed. So by replacing $\\widehat {\\mathcal R}$ by $A^{(2)}_{\\st}$, we close the gap of \\cite{liu-Fontaine} (\\cite{gao2021breuilkisin} provides another similar way to close the gap). \n\\end{remark}\n\n\\section{Crystalline representations and prismatic \\texorpdfstring{$F$}{F}-crystals}\nIn this section, we reprove the theorem of Bhatt and Scholze on the equivalence of prismatic $F$-crystal and lattices in crystalline representations of $G_K$ and complete the proof of Theorem \\ref{Thm-1}. We start to discuss some  general facts on the absolute prismatic site (which allows general base rings). \n\n\\subsection{Prismatic \\texorpdfstring{$F$}{F}-crystals in finite projective modules}\\label{subsec-pris-crystal}\nLet $R=R_0\\otimes_W \\O_K=R_0[u]\/E$ as in the beginning of \\S2 and $X=\\Spf(R)$ with the $p$-adic topology.\n\n\\begin{definition}\nThe (absolute) prismatic site $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ of $X$ is the opposite of the category of bounded prisms $(A, I)$ that are $(p,I)$-completed together with a map $R \\to A\/I$, and a morphism of prisms $(A, I) \\to (B, J)$ is a covering map if and only if $A \\to B$ is $(p, I)$-completely faithfully flat. \\\\\n\nDefine the following functors: \n$$\n\\O_{\\mathlarger{\\mathbbl{\\Delta}}}: (A,I)\\mapsto A,\n$$ \nand for all $h\\in \\mathbb N$, let\n$$\n\\mathcal{I}_{\\mathlarger{\\mathbbl{\\Delta}}}^h: (A,I)\\mapsto I^h.\n$$\nIt is known in \\cite{BS19} that these are sheaves on $X_{\\mathlarger{\\mathbbl{\\Delta}}}$. We will also use $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/\\mathcal{I}_{{\\mathlarger{\\mathbbl{\\Delta}}}}]^\\wedge_p$ to denote functor assign $(A,I)$ to the $p$-adic completion of $A$ with $I$ inverted.\n\\end{definition} \n\nNow we verify $A^{(2)} $(resp.  $A^{(3)})$  constructed in \\S\\ref{subsrc-construct-A2} is indeed self (resp. triple) product of $A$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$. We mainly discuss the situation of $A^{(2)}$ while the proof of $A^{(3)}$ is almost the same. Recall that $\\breve A = \\breve R_0 [\\![u]\\!]= W\\langle t_1, \\dots , t_m \\rangle[\\![u]\\!]$. \n\nFirst we want to make a remark on the existence of nonempty self-coproduct in the category of prisms over $R$. We thank Peter Scholze for answering our question on Mathoverflow. And we just repeat his answer here. Let $(A_i,I_i)$ for $i=1,2$ that are prisms over $R$, let $A_0=A_1\\hat{\\otimes}_{\\mathbb Z_p} A_2$ where the completion is taken for the $(p,I_1,I_2)$-adic topology. Let $J$ be the kernel of the map:\n$$\nA_0 \\to A_1\/I_1\\otimes_{R} A_2\/I_2.\n$$\nLet $(A,I)$ be the prismatic envelope of $(A_1,I_1)\\to (A_0,J)$, one can  check this is the initial object in the category of prisms over $R$ that admits maps from $(A_i,I_i)$ such that the two $R \\to A_i\/I_i \\to A\/I$ agree. Also we want to note that in general, we don't know if the boundedness of $(A_1,I_1)$ and $(A_2,I_2)$ will imply the boundedness of their coproduct. But we have seen $A^{(2)}$ and $A^{(3)}$ are indeed bounded by Corollary \\ref{cor-filtration-shape}.\n \nTo start, note that there exists a $W$-linear map $\\breve i_2: \\breve A \\to A^{\\ho 2}$ induced by $u \\mapsto y$ and $t_i \\mapsto s_i$. We claim that $\\breve i_2$ uniquely extends to $ i_2 : A \\to A^{\\ho 2}$ which is compatible with $\\delta$-structures. Indeed, consider the following commutative diagram\n$$\\xymatrix@C=45pt{ A \\ar[r] ^-{\\overline i_2}\\ar@{-->}[rd]^{i_{2, n}} & A^{\\ho 2} \/ (p , J^{(2)} )  \\\\ {\\breve A}  \\ar[u]\\ar[r]^-{\\breve i_{2, n} } & A^{\\ho 2} \/ (p , J ^{(2)})^n\\ar[u]}  $$\nHere $\\breve i_{2, n}= \\breve i_2 \\mod (p , J ^{(2)})^n$ and $\\overline{i}_2$ is induced by $A \\to A\/(p, E)  \\simeq A^{\\ho 2}\/ (p , J ^{(2)}) $. Since $\\breve i_2 (u) = y = x+ (y -x)$ and $\\breve i_2 (t_i) = s_i = t _i + (s_i - t_i)$, we see that the above (outer) diagram commutes. Since $A$ is formally \\'etale over $\\breve A $ by $(p, u)$-adic topology, we conclude that there exists a unique map $i_{2, n} : A \\to A^{\\ho 2} \/ (p , J^{(2)})^n$ so that the above diagram commutes. Since $A ^{\\ho 2}$ is $(p, J ^{(2)}  )$-complete,  there uniquely exists  $i _2 : A \\to A^{\\ho 2}$ which extends $\\breve i_2$. To see $i_2$ is compatible with $\\delta$-structures. it suffices to show that $\\varphi \\circ i _2 = i_2 \\circ \\varphi$. But both of $\\varphi \\circ i _2$  and $ i_2 \\circ \\varphi$ extend $ \\breve A \\overset \\varphi \\to \\breve A \\to A^{\\ho 2}$.  Again by formally \\'etaleness of $A$ over $\\breve A$, we see that $\\varphi \\circ i _2 = i_2 \\circ \\varphi$. Hence we obtain a map $ 1 \\otimes i_2: A \\otimes _{\\mathbb Z_p }A \\to A^{\\ho 2}$. Define $\\theta^{\\otimes 2}: A\\otimes_{\\mathbb Z_p} A \\to R$ via $\\theta^{\\otimes 2} (a \\otimes b)= \\theta (a) \\theta (b)$. By the construction of $i _2$, we have the following commutative diagram\n\\[ \\xymatrix{ A \\otimes _{\\mathbb Z_p} A \\ar[r] ^{1 \\otimes i _2} \\ar[d]^{\\theta ^{\\otimes 2 }} & A ^{\\ho 2}\\ar[d]  \\\\ R \\ar[r]^- \\sim & A^{\\ho 2}\/ J ^{(2)}  }\\]\nLet $\\widehat {A ^{\\otimes 2}} $ be the $(p , \\ker (\\theta ^{\\otimes 2}))$-completion of $A^{\\otimes 2}: = A \\otimes_{\\mathbb Z_p} A$. \nHence $1 \\otimes i_2$ induces a map $\\hat i_2$ from the $\\widehat {A ^{\\otimes 2}}$ to $ A ^{\\ho 2}$ because $A ^{\\ho 2}$ is clearly $(p , J ^{(2)})$-complete. To treat $A^{\\ho 3}$, we construct $ i _3: A \\to A ^{\\ho 3}$ by extending $\\breve i _3: A \\to A^{\\ho 3}$ by sending $u \\mapsto w $ and $t _j \\mapsto r_j$. The same method shows that $i_3$ is compatible with $\\delta$-structure and we obtain a map $1 \\otimes i _2 \\otimes i_3 : A^{\\otimes 3} \\to A^{\\ho 3}$ with $A ^{\\otimes 3}: A \\otimes_{\\mathbb Z_p} A \\otimes_{\\mathbb Z_p}A$. Similarly, we obtain a natural map $\\hat i _3 : \\widehat {A ^{\\otimes 3}} \\to A ^{\\ho 3} $. \n\\begin{lemma} For $s= 2, 3$, $\\hat i_s : \\widehat {A ^{\\otimes s}} \\to A ^{\\ho s}$ are isomorphisms. \n\\end{lemma}\n\\begin{proof}\nWe need to construct an inverse of $\\hat i_s$. We only show for $\\hat i _2$ and the proof for $\\hat i _3$ is the same. \nLet $g: A^{\\ho 2} \\to \\widehat {A ^{\\otimes 2}}$ be the $A$-linear map by sending $y -x \\mapsto 1 \\otimes u - u \\otimes 1 $ and $s_j - t _j  \\mapsto  1 \\otimes t_j  - t_j \\otimes 1$.  Clearly $g$ is well-defined because $ 1 \\otimes u - u \\otimes 1$ and $1 \\otimes t_j  - t_j \\otimes 1$ are in $\\Ker (\\theta ^{\\otimes 2})$. Since $i_2 (u) = y$ and $i_2 (t_j) = s_j$,  $\\hat i _2 \\circ g $ is identity on $A ^{\\ho 2}$. Now it suffices to show that $h : = g \\circ \\hat i_2 $ is identity. Write $K = (p , \\Ker (\\theta ^{\\otimes 2}))$. Note that we have map $ A \\otimes_{\\mathbb Z_p} \\breve A \\to  \\widehat {A ^{\\otimes 2}} \\overset h \\to   \\widehat {A ^{\\otimes 2}}$ induced by $h  $ which we still call it $\\breve h $. \nNow  we  have the following commutative diagram  \n$$\\xymatrix@C=55pt{ A\\otimes_{\\mathbb Z_p} A  \\ar[r] ^-{\\mod K }\\ar@{-->}[rd]^{\\mod K^n}_{h_{ n} } & (A\\otimes_{\\mathbb Z_p} A ) \/ K  \\\\  A \\otimes_{\\mathbb Z_p}{\\breve A}  \\ar[u]\\ar[r]^-{\\breve h \\mod K^n  } & (A \\otimes_{\\mathbb Z_p} A )\/ K ^n\\ar[u]},   $$\nwhere $h_n$ is induced by $h \\mod K^n$. \nWe see that both $ h_n $ and $\\mod K^n$ on the dashed arrow can make the diagram commute. Then by the formal \\'etaleness of $A$ over $\\breve A$, we conclude that $h_n  = \\mod K^n$ and $h$ is the identity map. \n\\end{proof}\n\\begin{proposition}\\label{prop-selfproduct} $A^{(2)}$ and $A^{(3)}$ is self-product  and triple product of  $A$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$. \n\\end{proposition}\n\\begin{proof} \nIn the following, we only treat the case of $A^{(2)}$ while the proof for $A^{(3)}$ is the same. \nWe need to prove that  for any $B = (B,J)$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$, \n$$\n\\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A^{(2)},B)=\\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A, B ) \\times \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A,B).\n$$\nBy the above lemma, we have natural maps $A \\otimes_{\\mathbb Z_p} A \\to \\widehat{A^{\\otimes 2}} \\simeq A^{\\ho 2}$. Combined with natural map $A^{\\ho 2}\\to A^{(2)}$ as $A^{(2)}$ is the prismatic envelope of $ A^{\\ho 2}$ for the ideal $J^{(2)}$, we have map $\\alpha : A \\otimes_{\\mathbb Z_p} A \\to A^{(2)}$ which is compatible with $\\delta$-structures. Then $\\alpha$ induces map $$\n\\beta: \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A^{(2)},B)\\to \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A, B ) \\times \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}^{\\mathrm{opp}}}(A,B).\n$$\nTo prove the surjectivity of $\\beta$, given $f_i \\in  \\mathrm{Hom}_{X_{\\mathlarger{\\mathbbl{\\Delta}}}}(A, B )$ for $i = 1,2$, we obtain a map  $f _1 \\otimes f _2: A \\otimes_{\\mathbb Z_p} A \\to B$. It is clear that $(f_1 \\otimes f_2) (\\Ker (\\theta ^{\\otimes 2})) \\subset J$. Since $B$ is $(p, J)$-derived complete, $f \\otimes f_2$ extends to a map $ f _1 \\ho f_2 : \\widehat{A ^{\\otimes 2}}\\simeq A ^{\\ho 2} \\to B$ which is compatible with $\\delta$-structures, Hence $f_1 \\ho f _2$ is a morphism of $\\delta$-algebra. Finally, by the universal properties of prismatic envelope, $f _1 \\ho f_2 $ extends to   a map of prisms $ f_1 \\ho_{{\\mathlarger{\\mathbbl{\\Delta}}}} f_2: A^{(2)} \\to B$ as required. \n\nFinally, we need to show that $\\beta$ is injective. It suffices to show that $A$-algebra structure map $i _1 : A\\to A^{(2)} $ and $i'_2: A \\overset{i_2}{\\to } A^{\\ho 2} \\to A^{(2)}$ both are  injective. \nSince all rings here are $(p, E)$-complete integral domains, it suffices to check that $i_1 , i_2' \\mod (p, E)$ are injective. By Proposition \\ref{prop-key-property}, we see that $i_1 \\mod (p, E)$ is $R\/ pR \\to R\/pR [\\{\\gamma _i (z_j)\\}] $, so it is injective. By the construction $i'_2$ and $i_2$, we see that $i'_2 \\mod (p, E)$ is the same as $A\/(p, E) \\to A ^{\\ho 2}\/ (p , J ^{(2)}) \\to A^{(2)} \/(p, E)$, which is same as $R\/ pR \\to R\/pR [\\{\\gamma _i (z_j)\\}] $.  So it is injective. \n\\end{proof}\n\n\\begin{remark}\\label{rem-Astprelog}\n    When $R=\\O_K$ is a complete DVR with perfect residue field $k$, we know a priori, the self-product $A^{(2)}$ of $(A,(E))$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ can be constructed as the prismatic envelope of $(A,(E))\\to (B,I)$, where $B$ is the $(p,E(u),E(v))$-adic completion of $W(k)[\\![u]\\!] \\otimes_{\\mathbb Z_p} W(k)[\\![v]\\!]$ and $I$ is the kernel of the map:\n    $$\n    B \\to A\/(E)\\otimes_{R} A\/(E)=R.\n    $$\n    On the other hand,  $W(k)$ is formally \\'etale over $\\mathbb Z_p$ for the $p$-adic topology, so for all $(C,J)\\in X_{\\mathlarger{\\mathbbl{\\Delta}}}$, the map $W(k)\\to R \\to C\/J$ lifts uniquely to a map $W(k) \\to C$. In particular, for all $(C,J)\\in X_{\\mathlarger{\\mathbbl{\\Delta}}}$, $C$ has a natural $W(k)$-algebra structure. So when we construct the self-product, we can also consider $A^{(2)}$ as the prismatic envelope of $(A,(E))\\to (C,J)$, where $C$ is the $(p,E(u),E(v))$-adic completion of $A\\otimes_{W(k)} A$ and $J$ is the kernel of the map:\n    $$\n    C \\to A\/(E)\\otimes_{R} A\/(E)=R.\n    $$\n    We  have $C\\simeq W(k)[\\![u,v]\\!]$, $J=(E(u),u-v)$ and $A^{(2)}=W(k)[\\![u,v]\\!]\\{\\frac{u-v}{E}\\}^\\wedge_\\delta$.\n\\end{remark}\n\n\\begin{definition}\\label{def-Fcrystal}\n\\begin{enumerate}\n    \\item \nA prismatic crystal over $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules (resp. $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/I]^\\wedge_p$-modules) is a finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-module (resp. $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/I]^\\wedge_p$-module) $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ such that for all morphisms $f: (A, I) \\to (B, J)$ of prisms, it induces an isomorphism:\n$$\nf^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},A} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,I))\\otimes_A B \\simeq \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},B} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B,J))\n$$\n$$\n(resp.\\quad f^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},A} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,I))\\otimes_{A[1\/I]^\\wedge_p} B[1\/I]^\\wedge_p \\simeq \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},B} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B,J))).\n$$\n\n\\item A prismatic $F$-crystal over $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ of height $h$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules is a prismatic crystal $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules together with a $\\varphi_{\\O_{\\mathlarger{\\mathbbl{\\Delta}}}}$-semilinear endomorphism $\\varphi_{\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}}$ of the $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-module $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}: \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\to \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ such that the cokernel of the linearization $\\varphi^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\to \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is killed by $\\mathcal{I}^h$.\n\\end{enumerate}\n\n\\end{definition}\n\n\\begin{proposition}\\label{prop-cover-final-object}\nIf the sheaf represented by $(B,I)$ in $\\Shv(X_{\\mathlarger{\\mathbbl{\\Delta}}})$ covers the final object $\\ast$ in $\\Shv(X_{\\mathlarger{\\mathbbl{\\Delta}}})$, i.e., for any $(C,J)$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$, there is a $(P, J)$ lies over $(B,I)$ and covers $(C,J)$. Also assume that the self-coproduct $B^{(2)}$ and self-triple-coproduct $B ^{(3)}$ of $(B,I)$ are inside $X_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, i.e., they are bounded. Then a prismatic crystal $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ over $X$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules is the same as a finite projective module $\\mathfrak{M}$ over $B$ together with a descent data $\\psi: \\mathfrak{M}\\otimes_{i_1,B} B^{(2)}\\simeq \\mathfrak{M}\\otimes_{i_2,B} B^{(2)}$ satisfies the cocycle condition. Here  $i _j : B \\to B^{(2)}$ $(j=1,2)$ are the two natural maps.\n\\end{proposition}\n\n\\begin{proof}\nFirst let $\\mathfrak{M}$ be a prismatic crystal in finite projective modules. Define $\\mathfrak{M}= \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B,I))$, and the descent data comes from the crystal property:\n$$\n\\psi:\\mathfrak{M}\\otimes_{i_1,B} B^{(2)}\\simeq  \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B^{(2)},I)) \\simeq \\mathfrak{M}\\otimes_{i_2,B} B^{(2)}.\n$$\nNow  given $(\\mathfrak{M}, \\psi)$, then for any $(C,J)$ in $X_{\\mathlarger{\\mathbbl{\\Delta}}}$, we need to construct a finite projective module over $C$. We choose the $(P, J)$ as in the assumption, let $\\mathfrak{M}_P=\\mathfrak{M}  \\otimes_B P$, and consider the following diagram:\n$$\n\\begin{tikzcd}\nC \\arrow[r] & P \\arrow[rr,\"f_1\"] & & P^{(2)}_C \\\\\n & B \\arrow[u] \\arrow[r,\"i_1\"] & B^{(2)} \\arrow[ur,\"f\"] & \\\\\n & & B \\arrow[u,\"i_2\"] \\arrow[r] & P \\arrow[uu,\"f_2\"] \\\\\n & & & C \\arrow[u]\n\\end{tikzcd}\n$$\nHere $(P^{(2)}_C,J)$ is the self-coproduct of $(P,J)$ in the category of prisms over $(C,J)$, and the existence of $(P^{(2)}_C,J)$ is from \\cite[Corollary 3.12]{BS19}, where they also show that $P^{(2)}_C$ is the derived $(p, J)$-completion of $P\\otimes^\\mathbb L_C P$ and $(P^{(2)}_C,J)$ is bounded. As a bounded prism over $(C,J)$, $(P^{(2)}_C,J)$ is naturally inside $X_{\\mathlarger{\\mathbbl{\\Delta}}}$, so $f$ exists by the universal property of $B^{(2)}$. So if we take the base change of $\\psi$ along $f$, we get \n$$\nf^\\ast\\psi: (\\mathfrak{M}\\otimes_{i_1,B} B^{(2)})\\otimes_{B^{(2)},f}P^{(2)}_C \\simeq (\\mathfrak{M}\\otimes_{i_2,B} B^{(2)})\\otimes_{B^{(2)},f}P^{(2)}_C\n$$\nwhich is the same as an isomorphism:\n$$\n\\psi_C: \\mathfrak{M}_{P}\\otimes_{P,f_1}P^{(2)}_C \\simeq \\mathfrak{M}_{P}\\otimes_{P,f_2}P^{(2)}_C.\n$$\nSimilar arguments will show $\\psi_C$ satisfies the cocycle condition. And $\\mathfrak{M}_{P}$ descents to a finite projective module over $C$ by \\cite[Proposition A.12]{ALB}.\n\\end{proof}\n\n\\begin{remark}\nWe want to note that the structures of finite nonempty coproducts in the category of bounded prisms over a prism $(A,I)$ is much simpler compared with the structure of finite nonempty products in the category $(R\/A)_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ (cf. \\cite[Lecture V, Corollary 5.2]{BhaNotes18}).\n\\end{remark}\n\n\\begin{lemma}\\label{lem-AEcoversfinal}\nThe prism $(A,(E))$ defined in \\S\\ref{subsrc-construct-A2} covers the final object $\\ast$ in $\\Shv(X_{\\mathlarger{\\mathbbl{\\Delta}}})$ in the sense of Proposition~\\ref{prop-cover-final-object}. And $A^{(2)}$ and $A^{(3)}$ are bounded.\n\\end{lemma}\n\\begin{proof}\nThe proof is similar to \\cite[Lemma 5.2.8]{ALB}, we need to show for $R$ defined as in \\S\\ref{subsrc-construct-A2}, there exists a quasi-syntomic perfectoid cover of $R$. We will construct this perfectoid cover similar to \\cite[\\S 7.1]{Kim12}.\n\nFirst recall we have $R=\\O_K\\otimes_{W}R_0$, and we fix a compatible system $\\{\\varpi_n\\}_{n\\geq 0}$ of $p^n$-th roots of a uniformizer $\\varpi_0$ of $\\O_K$ inside $E$. Let $\\widehat K_\\infty$ be the $p$-adic completion of $\\cup_n K(\\varpi_n)$, we know $\\widehat K_\\infty$ is perfectoid. Use $\\overline{R}_0[\\![u]\\!]$ to denote $A\/(p)=R\/(\\varpi)=R_0\/(p)[\\![u]\\!]$, and let $\\overline{R}_0[\\![u]\\!]_{\\rm perf}^\\wedge$ to be the $u$-adic completion of the direct perfection of $\\overline{R}_0[\\![u]\\!]$, it can be checked directly that $(\\overline{R}_0[\\![u]\\!]_{\\rm perf}^\\wedge[1\/u],\\overline{R}_0[\\![u]\\!]_{\\rm perf}^\\wedge)$ is a perfectoid affinoid $\\widehat K_\\infty^\\flat$-algebra, by tilt equivalence, there is a corresponded perfectoid affinoid $\\widehat K_\\infty$-algebra. More explicitly, let $\\tilde{R}_\\infty = W(\\overline{R}_0[\\![u]\\!]_{\\rm perf}^\\wedge)\\otimes_{W(\\O_{\\widehat K_\\infty}^\\flat),\\theta} \\O_{\\widehat K_\\infty}$. Then  $\\tilde{R}_\\infty$ is naturally an  $R$-algebra, and we claim it is a quasi-syntomic cover of $R$.\n\nTo show this, by \\cite[\\S 7.1.2]{Kim12}, we have\n$$\n\\tilde{R}_\\infty = (R_0\\widehat {\\otimes}_{W}\\O_{\\widehat K_\\infty})\\widehat {\\otimes}_{\\mathbb Z_p} \\mathbb Z_p\\langle T_i^{-p^\\infty}\\rangle\n$$\nwhere $T_i \\in R_0$ is any lift of a $p$-basis of $R_0\/(p)$. We have $\\O_K\\to \\O_{\\widehat K_\\infty}$ is a quasi-syntomic cover so by (2) of \\cite[Lemma 4.16]{BMS2}, $R \\to R_0\\widehat {\\otimes}_{W}\\O_{\\widehat K_\\infty}$ is also a quasi-syntomic cover. And we have $S=\\mathbb Z_p\\langle T_i^{-p^\\infty}\\rangle$ is a quasi-syntomic ring, this can be seen by constructing  a perfectoid quasi-syntomic covering of it, so by Lemma 4.34 of $loc.cit.$, we have the complex $\\mathbb{L}_{S\/\\mathbb Z_p} \\in D(S)$ has $p$-complete Tor amplitude in $[-1,0]$. In particular, $\\mathbb Z_p \\to \\mathbb Z_p\\langle T_i^{-p^\\infty}\\rangle$ is also a quasi-syntomic cover, so applying (1) in Lemma 4.16 of $loc. cit.$,  $R \\to \\tilde{R}_\\infty$ is a quasi-syntomic perfectoid cover.\n\nThe  boundedness of $A^{(2)}$ and $A ^{(3)}$ is from (2) in Corollary~\\ref{cor-filtration-shape}.\n\\end{proof}\n\n\\begin{corollary}\\label{cor-crystal-descentdata}\nAssume the the base $X=\\Spf(R)$ satisfies the condition in \\S\\ref{sec-ring-strcuture}, and let $A$, $A^{(2)}$ and $A^{(3)}$ be  defined as in \\S\\ref{subsrc-construct-A2}, then a prismatic $F$-crystal $(\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}, \\varphi_{\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}})$ in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules of height $h$ over $X$ is the same as a Kisin module $(\\mathfrak{M},\\varphi_\\mathfrak{M})$ of height $h$ over $A$ with a descent datum\n$$\nf: \\mathfrak{M} \\otimes_{A,i_1} A^{(2)} \\simeq \\mathfrak{M} \\otimes_{A,i_2} A^{(2)}\n$$\nthat compatible with the $\\varphi$-structure and satisfies the cocycle condition over $A^{(3)}$.\n\\end{corollary}\n\n\n\\begin{theorem}(\\cite[Theorem 1.2]{BS2021Fcrystals})\\label{Thm-main-1}  \nLet $T$ be a crystalline representation of $G_K$ over a $\\mathbb Z_p$-lattice of Hodge-Tate weights in $[0,h]$, then there is a prismatic $F$-crystal $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}(T)$ over $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ of height $h$ over $X$ such that $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,E))$ is the Kisin module associated to $T$. Moreover, the association of $T\\mapsto \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}(T)$ induces an equivalence of the above two categories. \n\\end{theorem}\nWe will prove this theorem in \\S \\ref{subsec-pris-crystal-proof}. \n\n\\begin{remark}\nTheorem~\\ref{Thm-main-1} was first established by Bhatt-Scholze in \\cite[Theorem 1.2]{BS2021Fcrystals}. The harder direction of \\cite[Theorem 1.2]{BS2021Fcrystals} is to show for all $\\mathbb Z_p$-lattices inside crystalline representations of $G_K$, one can attach a prismatic $F$-crystal. Using the theory of $(\\varphi,\\hat{G})$-modules, we have shown in \\S\\ref{subsec-G-image}, given a crystalline representation of $G_K$ over a $\\mathbb Z_p$-lattices $T$, we can attach a Kisin module $\\mathfrak{M}$ and a descent data\\footnote{Strictly speaking, \\S\\ref{subsec-G-image} only constructs an isomorphism but have not checked that  it satisfies cocycle condition, which will be proved in \\S \\ref{subsec-pris-crystal-proof}.} \n$$\nf_{\\tilde{\\tau}}: \\mathfrak{M}\\otimes_{A,i_1} A^{(2)}[\\frac{1}{p}] \\simeq \\mathfrak{M}\\otimes_{A,i_2} A^{(2)}[\\frac{1}{p}]\n$$\ncomes from the $\\tau$-action. We just show this is a $\\varphi$-equivariant isomorphism, and we need to show it gives rise to a descent data over $A^{(2)}$. As we have mentioned in Remark~\\ref{rem-inputofWu}, we can not find a direct ring theoretic proof of this. Our idea is to use result of \\cite{wu2021galois} or \\cite[Corollary 3.7]{BS2021Fcrystals}: the underlying Galois representation $T$ gives a descent data over $A^{(2)}[\\frac{1}{E}]^\\wedge_p$. To finish our proof, we need to compare this descent data with $f_{\\tilde{\\tau}}$ over $A^{(2)}[\\frac{1}{E}]^\\wedge_p[\\frac{1}{p}]$. This lead us to develop a ``prismatic\" $(\\varphi,\\tau)$-module theory in the next subsection, where we will have Lemma~\\ref{lem-evaluation-1} and Lemma~\\ref{lem-evaluation-2} to help us compare descent data over $A^{(2)}[\\frac{1}{E}]^\\wedge_p$ and $A^{(2)}[\\frac{1}{E}]^\\wedge_p[\\frac{1}{p}]$ via an evaluation map to $W(\\O_{\\hat{L}}^\\flat)$.\n\\end{remark}\n\n\n\\subsection{\\texorpdfstring{$(\\varphi,\\tau)$}{(phi,tau)}-modules and prismatic \\texorpdfstring{$F$}{F}-crystals}\\label{subsec-phi-tau} In this subsection, we make some preparations to prove Proposition \\ref{thm-1prime} and Theorem \\ref{Thm-main-1}. So we restrict to the case that $R=\\O_K$ is a complete DVR with perfect residue field. \n\\begin{definition}\nAn \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ is a pair $(\\mathcal M, \\varphi_\\mathcal M)$ such that $\\mathcal M$ is a finite free module over $A[1\/E]^\\wedge_p$, and $\\varphi_\\mathcal M$ is an isomorphism\n$$\n\\varphi_\\mathcal M: \\varphi^\\ast \\mathcal M: = A[1\/E]^\\wedge_p\\otimes_{\\varphi , A[1\/E]^\\wedge_p} \\mathcal M  \\simeq \\mathcal M\n$$\nof $A[1\/E]^\\wedge_p$-modules. And we define an \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p[1\/p]$ to be a $\\varphi$-module over $A[1\/E]^\\wedge_p[1\/p]$ such that it is obtained from an \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ by base change.\n\nAn \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ (resp. $A[1\/E]^\\wedge_p[1\/p]$) with descent data is a triple $(\\mathcal M, \\varphi_\\mathcal M, c)$, such that $(\\mathcal M, \\varphi_\\mathcal M)$ is an \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ (resp. $A[1\/E]^\\wedge_p[1\/p]$), and $c$ is an isomorphism\n$$\nc: \\mathcal M \\otimes_{A[1\/E]^\\wedge_p,i_1} B^{(2)} \\simeq \\mathcal M \\otimes_{A[1\/E]^\\wedge_p,i_2} B^{(2)}\n$$\n$$\n(\\text{resp. }c: \\mathcal M \\otimes_{A[1\/E]^\\wedge_p[1\/p],i_1} B^{(2)}[1\/p] \\simeq \\mathcal M \\otimes_{A[1\/E]^\\wedge_p[1\/p],i_2} B^{(2)}[1\/p])\n$$\nthat compatible with the $\\varphi$-structure and satisfies the cocycle condition over $B^{(3)}$ (resp. $B^{(3)}[\\frac 1 p]$). Here for $j=1,2$, $i_j: A[1\/E]^\\wedge_p \\to B^{(2)}$ is the map induced from $i_j: (A,(E)) \\to (A^{(2)},(E))$. \n\\end{definition}\n\n\\begin{remark}\\label{rmk-Wuandevaluation}\nIt is the main result in \\cite{wu2021galois} and \\cite[\\S2]{BS2021Fcrystals} that there is an equivalence of the category of lattices in representations of $G_K$ and the category of prismatic $F$-crystals in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/I]^\\wedge_p$-modules over $\\O_K$. Also by \\cite[Proposition 2.7]{BS2021Fcrystals}, one can show prismatic $F$-crystals in finite locally free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}[1\/I]^\\wedge_p$-modules is the same as \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ with descent data. \n\\end{remark}\n\n\nThe aim of this subsection is to use the ideas in \\cite{wu2021galois} and \\cite[\\S 5.5]{KedlayaLiu-relativeII} show that \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ (resp. $A[1\/E]^\\wedge_p[1\/p]$) with descent data are equivalence to $\\mathrm{Rep}_{\\Z_p}(G_K)$ (resp. $\\mathrm{Rep}_{\\Q_p}(G_K)$). More importantly, for all $\\gamma \\in \\hat{G}$, we will construct an evaluation at $\\gamma$ map\n$$\ne_\\gamma: B^{(2)} \\to W(\\hat{L}^\\flat)\n$$\nand use it to study $\\varphi$-equivariant morphisms between finite free $B^{(2)}$ and $B^{(2)}[1\/p]$-modules. We will see the evaluation at $\\tau$ map will play a crucial role in our proof of Proposition~\\ref{thm-1prime} and the Theorem~\\ref{Thm-main-1} below.\n\n\nRecall in \\S\\ref{subsec-phiGhatmodules}, we define $L = \\bigcup\\limits_{n =1}^\\infty K_{\\infty} (\\zeta_{p ^n})$, $\\hat G : = \\Gal(L \/K) $ and $H_K : = \\Gal (L \/ K _\\infty)$. Moreover, we define $\\widehat K_{1^\\infty}$ to be the $p$-adic completion of $\\cup_{n\\geq 0} K(\\zeta_{p^n})$, and we let $\\hat{L}$ to be the $p$-adic completion of $L$. It is clear that $A[1\/E]_p^\\wedge\\subset W(\\hat L ^\\flat)^{H_K}$.\nRecall the following definition and theorem in \\cite{Caruso-phitau}:\n\n\\begin{theorem}\\label{thm-caruso}\nAn \\'etale $(\\varphi,\\tau)$-module is a triple $(\\mathcal M, \\varphi_{\\mathcal M}, \\hat{G})$ where\n\\begin{itemize}\n    \\item $(\\mathcal M, \\varphi_{\\mathcal M})$ is an \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$;\n    \\item $\\hat{G}$ is a continuous $W(\\hat{L}^\\flat)$-semi-linear $\\hat{G}$-action on $\\hat{\\mathcal M}:=W(\\hat{L}^\\flat)\\otimes_{A[1\/E]^\\wedge_p}\\mathcal M$, and $\\hat{G}$ commutes with $\\varphi_{\\mathcal M}$;\n    \\item regarding $\\mathcal M$ as an $A[1\/E]^\\wedge_p$-submodule of $\\hat{\\mathcal M}$, we have $\\mathcal M\\subset \\hat{\\mathcal M}^{H_K}$.\n\\end{itemize} \nThen there is an anti-equivalence of the category of \\'etale $(\\varphi,\\tau)$-modules and $\\mathrm{Rep}_{\\Z_p}(G_K)$, such that if $T$ corresponds to $(\\mathcal M, \\varphi_{\\mathcal M}, \\hat{G})$, then\n$$\nT^\\vee = (\\hat{\\mathcal M}\\otimes_{W(\\hat{L}^\\flat)}W(\\mathbb C_p^\\flat))^{\\varphi=1}.\n$$\n\\end{theorem}\n\nOne of the basic facts used in the theory of \\'etale $(\\varphi,\\tau)$-modules developed in \\cite{Caruso-phitau} is that $\\Gal(\\hat{L}\/\\widehat K_{1^\\infty})\\simeq \\mathbb Z_p$, and we write $\\tau$ to be a topological generator of $\\Gal(\\hat{L}\/K_{1^\\infty})$ determined by $\\tau(\\varpi_n)=\\zeta_{p^n}\\varpi_n$ as the discussion before Corollary \\ref{cor-crystalline}. Also  $\\hat{G}$ is topologically generated by $\\tau$ and $H_K$, so in particular,  the $\\hat{G}$-action on $\\hat{\\mathcal M}$ is determined by the action of $\\tau$ on $\\mathcal M$ inside $\\hat{\\mathcal M}$. As discussed before, we will provides a direct correspondence of the category of \\'etale $(\\varphi,\\tau)$-modules and the category of \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ with descent data. Moreover, we will construct an evaluation at $\\tau$ map:\n$$\ne_\\tau: B^{(2)} \\to W(\\hat{L}^\\flat),\n$$\nand show that the $\\tau$-action on $\\mathcal M$ inside $\\hat{\\mathcal M}$ is given by the base change of the descent data along $e_\\tau$. \n\n\\begin{remark}\nIn \\cite[Theorem 5.2]{wu2021galois}, they prove a similar equivalence but for \\'etale $(\\varphi,\\Gamma)$-modules. The theory of \\'etale $(\\varphi,\\Gamma)$-module is defined for the cyclotomic tower $K_{1^\\infty}$ over $K$ while the theory of \\'etale $(\\varphi,\\tau)$-modules is defined using the Kummer tower $K_{\\infty}$. We will use a lot of ideas and results developed in \\cite{wu2021galois} when proving our claims in this subsection. The main difficulty in our situation is that the Kummer tower $K_\\infty$ is not a Galois tower over $K$. To deal with this, we have to use the idea in \\cite[\\S 5.5]{KedlayaLiu-relativeII}. Roughly speaking, we will take the Galois closure $L$ of $K_\\infty$,  then prove results over $\\hat{L}$, then descent back to $K_\\infty$ using the fact $\\widehat K_\\infty=\\hat{L}^{H_K}$. \n\nOne should be able to construct the evaluation map in the content of \\cite{wu2021galois} the same way as we define in this subsection. This map will give a more direct correspondence of the descent data and the $\\Gamma$-actions on \\'etale $(\\varphi,\\Gamma)$-modules.\n\\end{remark}\n\nBy \\cite[Lem 3.9]{BS19}, any prism $(B , J)$ admits a map into its perfection $(B_{\\perf}, JB_{\\perf})$. The following theorem (\\cite[Thm 3.10]{BS19}) is the key to understand perfect prisms.  \n\\begin{theorem}\\label{thm-perfectprismandperfectoidring}\n$(A,I)\\to A\/I$ induces an equivalence of the category of perfect prisms over $\\O_K$ with the category of integral perfectoid rings over $\\O_K$.\n\\end{theorem}\n \nLet $(A,(E))$ be the Breuil-Kisin prism defined in \\S\\ref{subsrc-construct-A2}, we have\n\\begin{lemma}\\label{lem-perfectionofA} $A_{\\perf}\\simeq W(\\O_{\\widehat K_\\infty}^\\flat)$.\n\\end{lemma}\n\\begin{proof}\nExactly the same as the proof of \\cite[Lemma 2.17]{wu2021galois}\n\\end{proof}\n\n\\begin{lemma}\nLet $\\Perfd_K$ be the category of perfectoid $K$-algebras, then $\\Perfd_{K}$ admits finite non-empty coproducts. \n\\end{lemma}\n\\begin{proof}\nLet $R$ and $S$ be two perfectoid $K$-algebras, it follows from \\cite[Corollary 3.6.18]{KedlayaLiu-relativeI} that the uniform completion $(R\\otimes_K S)^u$ of the tensor product $(R\\otimes_K S)$ is again a perfectoid $K$-algebra, and it is easy to show this is the coproduct of $R$ and $S$ in the category of perfectoid $K$-algebras.\n\\end{proof}\n\nFor $i\\in \\mathbb N_{>0}$, let $(A^{(i)},(E))$ (resp. $(\\Ainf(\\O_{\\hat{L}})^{(i)},(E))$) denote the $i$-th self-coproduct of $(A,(E))$ (resp. $(\\Ainf(\\O_{\\hat{L}}),(E))$) in the category of prisms over $\\O_K$, where $\\Ainf(\\O_{\\hat{L}}):=W(\\O_{\\hat{L}}^\\flat)$. The following is  a description of $(A^{(i)})_{\\perf}[1\/E]^\\wedge_p$ and $(\\Ainf(\\O_{\\hat{L}})^{(i)})_{\\perf}[1\/E]^\\wedge_p$. \n\n\\begin{lemma}\\label{lem-Aiperf}\nLet $\\widehat K_\\infty^{(i)}$ (resp. $\\hat{L}^{(i)}$) be the $i$-th self-coproduct of $\\widehat K_\\infty$ (resp. $\\hat{L}$) in $\\Perfd_K$, then $(A^{(i)})_{\\perf}[1\/E]^\\wedge_p \\simeq W((\\widehat K_\\infty^{(i)})^\\flat)$ (resp. $(\\Ainf(\\O_{\\hat{L}})^{(i)})_{\\perf}[1\/E]^\\wedge_p \\simeq W((\\hat{L}^{(i)})^\\flat)$).\n\\end{lemma}\n\\begin{proof}\nWe will only prove the lemma for $(A^{(i)})_{\\perf}[1\/E]^\\wedge_p$, and the case for $(\\Ainf(\\O_{\\hat{L}})^{(i)})_{\\perf}[1\/E]^\\wedge_p $ is similar.\n\nWe use similar arguments as in \\cite[Lemma 5.3]{wu2021galois}. Fix $i$, first we can show $(A^{(i)})_{\\perf}$ is the $i$-th self-coproduct of $(A_{\\perf}, (E))$ in the category of perfect prisms over $\\O_K$, i.e. $(A^{(i)})_{\\perf}=(A_{\\perf})^{(i)}_{\\perf}$. By Theorem~\\ref{thm-perfectprismandperfectoidring}, Lemma~\\ref{lem-perfectionofA} and \\cite[Proposition 2.15]{wu2021galois}, if we let $S=(A^{(i)})_{\\perf}\/E$, then $S[1\/p]$ is the $i$-th self-coproduct of $\\widehat K_\\infty$ in the category of perfectoid $K$-algebras. Now we have \n$$\n(A^{(i)})_{\\perf}[1\/E]^\\wedge_p\\simeq W(S^\\flat)[1\/[\\varpi^\\flat ]]^\\wedge_p=W(S^\\flat[1\/\\varpi^\\flat ])=W((S[1\/p])^\\flat)\\simeq W((\\widehat K_\\infty^{(i)})^\\flat).\n$$\n\\end{proof}\n\n\\begin{remark}\\label{rem-diamonds}\nThere is another way to view $\\widehat K_\\infty^{(i)}$ in terms of diamonds over $\\mathrm{Spd}(K,\\O_K)$ which is used in the proof of \\cite[Lemma 5.3]{wu2021galois}, that there exist a ring of integral elements $\\widehat K_\\infty^{(i),+}$ in $\\widehat K_\\infty^{(i)}$, such that we have \n\\begin{equation}\\label{eq-diamondKi}\n    \\Spa(\\widehat K_\\infty^{(i)},\\widehat K_\\infty^{(i),+})^\\diamond \\simeq \\underbrace{\\Spa(\\widehat K_\\infty,\\widehat K_\\infty^{+})^\\diamond \\times_{\\mathrm{Spd}(K,\\O_K)} \\ldots \\times_{\\mathrm{Spd}(K,\\O_K)}\\Spa(\\widehat K_\\infty,\\widehat K_\\infty^{+})^\\diamond}_{i\\text{-copies of } \\Spa(K_\\infty,K_\\infty^{+})^\\diamond}.\n\\end{equation}\nAnd similar results hold for $\\hat{L}$. Using this description and the fact that functor from perfectoid spaces over $\\Spa(K, \\O_K)$ to diamonds over $\\mathrm{Spd}(K, \\O_K)$ is an equivalence, we have $\\hat{L}^{(i)}$ has a natural action of $\\hat{G}^i$ coming from the action on the diamond spectrum. Since $\\hat{L}^{H_K}=\\widehat K_\\infty$, we have\n$$\n\\Spa(\\widehat K_\\infty^{(i)},\\widehat K_\\infty^{(i),+})^\\diamond \\simeq \\left (\\Spa(\\hat{L},\\O_{\\hat{L}})^\\diamond \\times \\ldots \\times_{\\mathrm{Spd}(K,\\O_K)}\\Spa(\\hat{L},\\O_{\\hat{L}})^\\diamond \\right )^{H_K^i} \\simeq (\\Spa(\\hat{L}^{(i)},\\hat{L}^{(i),+})^\\diamond  )^{H_K^i}.\n$$\nThat is, $(\\hat{L}^{(i)})^{H_K^i}=\\widehat K_\\infty^{(i)}$.\n\\end{remark}\n\nNow we use ideas in \\cite{wu2021galois} and \\cite[\\S 5.5]{KedlayaLiu-relativeII} to study \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ with descent data. We will show this category is the same as generalized $(\\varphi,\\Gamma)$-modules in the work of Kedlaya-Liu. The following is a quick review of Example 5.5.6 and 5.5.7 in \\cite{KedlayaLiu-relativeII}.\n\nFirstly, one has $\\hat{L}^{(i)}\\simeq \\Cont(\\hat{G}^{i-1}, \\hat{L})$, here $\\Cont$ means the set of continuous functions. One can see this fact from the proof of \\cite[Theorem 5.6]{wu2021galois}. When $i=2$, we choose the two canonical maps $i_1,i_2:\\hat{L} \\to \\hat{L}^{(2)}$, corresponds to $j_1,j_2: \\hat{L} \\to \\Cont(\\hat{G}, \\hat{L})$ given by \n\\begin{equation}\\label{eq-j1j2}\n   j_1(x): \\gamma \\mapsto \\gamma (x) \\quad \\text{ and } \\quad j_2(x): \\gamma \\mapsto x.\n\\end{equation}\n\nFrom Remark~\\ref{rem-diamonds}, there is a natural action of $\\hat{G}^2$ on $\\hat{L}^{(2)}$. One can check this corresponds to the $\\hat{G}^2$-action on $\\Cont(\\hat{G},\\hat{L})$ given by:\n$$\n(\\sigma_1,\\sigma_2)(f)(\\gamma)=\\sigma_2 f(\\sigma_2^{-1}\\gamma\\sigma_1).\n$$\n\n\\begin{remark}\nWe interchange the roles of $j_1$ and $j_2$ comparing with the isomorphism defined in \\cite[Example 5.5.6]{KedlayaLiu-relativeII}, so the $\\hat{G}^2$-action is different from that in Example 5.5.7 of $loc. cit.$, we will see this definition is more convenient when relating the descent data with the semilinear group actions. \n\\end{remark}\n\nOne can show $\\Cont(\\hat{G},-)$ commutes with tilting and the Witt vector functor, as been discussed in \\cite[Lemma 5.3]{wu2021galois}, so in particular, we have \n$$\nW((\\hat{L}^{(i)})^\\flat) \\simeq \\Cont(\\hat{G}^{i-1}, W(\\hat{L}^\\flat)).\n$$\nFor $i=2$, we still use $j_1$ and $j_2$ to represent the two canonical maps from $W(\\hat{L}^\\flat)$ to $\\Cont(\\hat{G}, W(\\hat{L}^\\flat))$ that comes from \\eqref{eq-j1j2}. The above isomorphism also is compatible with the action of $\\hat{G}^2$, so we have\n\\begin{equation}\\label{eq-K(2)andhatGaction}\nW((\\widehat K_\\infty^{(2)})^\\flat) \\simeq \\Cont(\\hat{G}, W(\\hat{L}^\\flat))^{H_K^2}\n\\end{equation}\nWe prove the following lemma for our later use.\n\nNow let $\\mathcal M$ be an \\'etale $\\varphi$-module over $W(\\widehat K_{\\infty}^\\flat)$ with a descent data: \n$$\n\\psi: \\mathcal M \\otimes_{W(\\widehat K_\\infty^\\flat),j_1} W((\\widehat K_\\infty^{(2)})^\\flat) \\simeq \\mathcal M \\otimes_{W(\\widehat K_\\infty^\\flat),j_2} W((\\widehat K_\\infty^{(2)})^\\flat)\n$$\nas \\'etale $\\varphi$-modules over $W((\\widehat K_\\infty^{(2)})^\\flat)$ and satisfies cocycle condition over $W((\\widehat K_\\infty^{(3)})^\\flat)$. Using \\eqref{eq-K(2)andhatGaction}, we have $\\psi$ is the same as a descent data:\n\\begin{equation}\\label{eq-descentdata-1}\n\\hat{\\psi}: {\\mathcal M} \\otimes_{W(\\widehat K_\\infty^\\flat),j_1} \\Cont(\\hat{G}, W(\\hat{L}^\\flat))^{H_K^2} \\simeq {\\mathcal M} \\otimes_{W(\\widehat K_\\infty^\\flat),j_2} \\Cont(\\hat{G}, W(\\hat{L}^\\flat))^{H_K^2}.\n\\end{equation}\n\nFor each $\\gamma \\in \\hat{G}$, we have an evaluation map $\\tilde{e}_\\gamma: \\Cont(\\hat{G}, W(\\hat{L}^\\flat)) \\to W(\\hat{L}^\\flat)$ given by evaluating at $\\gamma$. Using \\eqref{eq-j1j2}, one can check $\\tilde{e}_\\gamma \\circ j_2: W(\\widehat K_\\infty^\\flat) \\to W(\\hat{L}^\\flat)$ is given by the natural embedding and $\\tilde{e}_\\gamma \\circ j_1: W(\\widehat K_\\infty^\\flat) \\to W(\\hat{L}^\\flat)$ is given by $x\\mapsto \\gamma(x)$. So for each $\\gamma \\in \\hat{G}$, if we tensor \\eqref{eq-descentdata-1} against the evaluation map $\\tilde{e}_\\gamma$, we get an isomorphism:\n$$\n\\psi_\\gamma: {\\mathcal M}\\otimes_{W(\\widehat K_\\infty^\\flat),\\gamma} W(\\hat{L}^\\flat) \\simeq {\\mathcal M}\\otimes_{W(\\widehat K_\\infty^\\flat)} W(\\hat{L}^\\flat). \n$$\nAnd similar to the classical Galois descent theory, the cocycle condition for $\\psi$ implies $\\{\\psi_\\gamma\\}_\\gamma$ satisfies \n$$\n\\psi_{\\sigma \\gamma} = \\psi_\\sigma \\circ \\sigma^\\ast \\psi_\\gamma.\n$$\nHence $\\{\\psi_\\gamma\\}_\\gamma$ defines a continuous semilinear action of $\\hat{G}$ on $\\hat{\\mathcal M}:=\\mathcal M\\otimes_{W(\\widehat K_\\infty^\\flat)} W(\\hat{L}^\\flat)$. One can check for $\\gamma \\in H_K$, we have the composition\n$$\nW(\\widehat K_\\infty^\\flat) \\xrightarrow{j_k} W((\\widehat K_\\infty^{(2)})^\\flat) \\to \\Cont(\\hat{G}, W(\\hat{L}^\\flat)) \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat)\n$$\nis the natural embedding $W(\\widehat K_\\infty^\\flat) \\hookrightarrow W(\\hat{L}^\\flat)$ for $k=1,2$. And using the cocycle condition, one can show $\\psi_\\gamma=\\id$ for $\\gamma \\in H_K$, so in particular, $\\mathcal M \\subset \\hat{\\mathcal M}^{H_K}$. Conversely, given a semilinear action of $\\hat{G}$ on $\\hat{\\mathcal M}$ such that $\\mathcal M \\subset \\hat{\\mathcal M}^{H_K}$, $\\{\\psi_\\gamma\\}_\\gamma$ defines a descent data $\\psi$ over $\\Cont(\\hat{G}, W(\\hat{L}^\\flat))^{H_K^{2}}$ if and only if the semilinear action is continuous. In summary, we have\n\n\\begin{theorem}\\label{thm-evaluation-1}\n\\begin{enumerate}\n    \\item The category of \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ with descent data over $A^{(2)}[1\/E]^\\wedge_p$ is equivalent to the category of \\'etale $(\\varphi,\\tau)$-modules over $A[1\/E]^\\wedge_p$;\n    \\item Given a descent data $f$ of an \\'etale $\\varphi$-module $\\mathcal M$ over $A[1\/E]^\\wedge_p$, and $\\gamma\\in \\hat{G}$, we can define the evaluation $f_\\gamma$ of $f$ at $\\gamma$, defined by the base change of $f$ along \n$$\ne_{\\gamma}: A^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat),\n$$\nwhich defines an isomorphism:\n$$\nf_\\gamma: \\mathcal M \\otimes_{A[1\/E]^\\wedge_p,\\tilde{\\iota}_\\gamma} W(\\hat{L}^\\flat) \\simeq \\mathcal M \\otimes_{A[1\/E]^\\wedge_p} W(\\hat{L}^\\flat)\n$$\nwhere $\\tilde{\\iota}_\\gamma: A[1\/E]^\\wedge_p \\to W(\\hat{L}^\\flat) \\xrightarrow{\\gamma} W(\\hat{L}^\\flat)$. Suppose that  $(\\mathcal M,f)$ corresponds to a $\\mathbb Z_p$-representation $T$ of $G_K$, then $f_\\gamma$ corresponds to the semilinear action of $\\gamma$ on $\\mathcal M$ inside $\\mathcal M \\otimes_{A[1\/E]^\\wedge_p} W(\\mathbb{C}_p^\\flat)\\simeq T^\\vee \\otimes W(\\mathbb{C}_p^\\flat)$. Moreover, two descent data $f , g$ are equal if and only if $f_{\\tau} = g_{\\tau}$.\n\\end{enumerate} \n\\end{theorem}\n\\begin{proof} The discussion above the theorem establishes the equivalence between the category of \\'etale $\\varphi$-modules over $A_{\\perf}[1\/E]^\\wedge_p$ with descent data over $(A^{(2)})_{\\perf}[1\/E]^\\wedge_p$ is equivalent to the category of \\'etale $(\\varphi,\\tau)$-modules over $A[1\/E]^\\wedge_p$. Now  (1) follows \\cite[Theorem 4.6]{wu2021galois} which shows that the category of \\'etale $\\varphi$-modules over $B[\\frac 1 I]^\\wedge_p$ is equivalent to the category of \\'etale $\\varphi$-modules over $B_{\\perf}[\\frac 1 I]^\\wedge_{p}$ for bounded prism $(B, I)$ satisfying $\\varphi (I) \\mod p$ is generated by a non-zero\ndivisor in $B\/p$. Then it just remains to prove the last statement in (2). Actually one can check (2) by chasing all the functors used in (1), and use the fact that for any \\'etale $(\\varphi,\\tau)$-module, the $\\hat{G}$-action on $\\hat{\\mathcal M}$ is determined by the $\\tau$-action on $\\mathcal M$. However, this can also been seen directly from the following lemma.\n\\end{proof}\n\n\\begin{lemma}\\label{lem-evaluation-1}\nGiven two finite free \\'etale $\\varphi$-modules $\\mathcal M,\\mathcal{N}$ over $A^{(2)}[1\/E]^\\wedge_p$ and two morphisms $f, g: \\mathcal M \\to \\mathcal{N}$ of \\'etale $\\varphi$-modules over $A^{(2)}[1\/E]^\\wedge_p$. Let $f_\\tau,g_\\tau$ be the base changes of $f,g$ along the map \n$$\ne_{\\tau}: A^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\tau} W(\\hat{L}^\\flat).\n$$\nThen $f=g$ if and only if $f_\\tau=g_\\tau$.\n\\end{lemma}\n\n\\begin{proof}\nWe take the natural base change of $f$ and $g$ along $A^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p$, we get two morphisms $\\psi$ and $\\psi'$ between \\'etale $\\varphi$-modules over $(A^{(2)})_{\\perf}[1\/E]^\\wedge_p$. Since the base change functor between \\'etale $\\varphi$-modules over $A^{(2)}[1\/E]^\\wedge_p$ and $(A^{(2)})_{\\perf}[1\/E]^\\wedge_p$ is an equivalence of categories, it reduces to show that $\\psi=\\psi'$ if and only if their base change along \n$$\n\\tilde{e}_{\\tau}: (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{} W(\\hat{L}^\\flat)\n$$\nis equal. Since $\\mathcal M$ and $\\mathcal{N}$ are finite free, it is enough to show the evaluation map:\n$$\n\\tilde{e}_{\\tau}: \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\to W\\big((\\hat{L}^{(2)})^\\flat\\big)\n$$\nis injective. Suppose $h\\in \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}}$ satisfies $h(\\tau)=0$, then \n$$\n(\\sigma_1,\\sigma_2)(h)(\\tau)=\\sigma_2 h(\\sigma_2^{-1}\\tau\\sigma_1)=0\n$$\nfor $(\\sigma_1,\\sigma_2)\\in H_K^{2}$. Since $\\hat{G}$ is topologically generated by $H_K$ and $\\tau$, we get $h\\equiv 0$.\n\\end{proof}\n\nNow we give the $\\mathbb Q$-isogeny versions of Theorem \\ref{thm-evaluation-1} and Lemma \\ref{lem-evaluation-1}. \nRecall that  the \\'etale $(\\varphi,\\tau)$-modules over $A[1\/E]^\\wedge_p[\\frac{1}{p}]$ is equivalent to the category of $\\mathbb Q_p$-representations of $G_K$, and recall the following definition of \\'etale $(\\varphi,\\tau)$-modules over $B[1\/J]^\\wedge_p[\\frac{1}{p}]$ for a prism $(B,J)\\in X_{\\mathlarger{\\mathbbl{\\Delta}}}$.\n\n\\begin{definition}\\label{def-etalephimodule-2}\nAn (globally) \\'etale $\\varphi$-module $\\mathcal M$ over $B[1\/J]^\\wedge_p[\\frac{1}{p}]$ is a (finite projective) $\\varphi$-module over $B[1\/J]^\\wedge_p[\\frac{1}{p}]$ that arises by base extension from an \\'etale $\\varphi$-module $B[1\/J]^\\wedge_p$.\n\\end{definition}\n\nFrom this definition, we immediately deduce the following result from \\cite[Theorem 4.6]{wu2021galois}\n\\begin{proposition}\nFor any prism $(B,J)\\in X_{\\mathlarger{\\mathbbl{\\Delta}}}$ satisfying $\\varphi (J) \\mod p$ is generated by a non-zero\ndivisor in $B\/p$, the base change functor defined by $B[1\/J]^\\wedge_p[\\frac{1}{p}]\\to B_{\\perf} [1\/J]^\\wedge_p[\\frac{1}{p}]$ induces an equivalence between the category of \\'etale $\\varphi$-modules over $B[1\/J]^\\wedge_p[\\frac{1}{p}]$ and the category of \\'etale $\\varphi$-modules over $B_{\\perf}[1\/J]^\\wedge_p[\\frac{1}{p}]$.\n\\end{proposition}\n\nAnd similar to Theorem~\\ref{thm-evaluation-1} and Lemma~\\ref{lem-evaluation-1}, we have\n\\begin{theorem}\\label{thm-evaluation-2}\nThe category of \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p[\\frac{1}{p}]$ with descent data over $A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}]$ is equivalent to the category of \\'etale $(\\varphi,\\tau)$-modules over $A[1\/E]^\\wedge_p[\\frac{1}{p}]$. Moreover, \n$$\n\\Cont\\big(\\hat{G}, W(\\hat{L}^\\flat)[\\frac{1}{p}]\\big)^{H_K^{2}} \\simeq W(\\widehat K_\\infty^{(2)})^\\flat[\\frac{1}{p}]. \n$$\nFor  $\\gamma\\in \\hat{G}$, we can define the evaluation map\n$$\n\\tilde{e}_\\gamma: \\Cont\\big(\\hat{G}, W(\\hat{L}^\\flat)[\\frac{1}{p}]\\big) \\to W(\\hat{L}^\\flat)[\\frac{1}{p}].\n$$\nAnd given a descent data $f$ of an \\'etale $\\varphi$-module $\\mathcal M$ over $A[1\/E]^\\wedge_p[\\frac{1}{p}]$, and $\\gamma\\in \\hat{G}$, we can define the evaluation $f_\\gamma$ of $f$ at $\\gamma$, defined by the base change of $f$ along \n$$\ne_\\gamma: A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}] \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p[\\frac{1}{p}] \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat)[\\frac{1}{p}],\n$$\nwhich defines an isomorphism:\n$$\n\\mathcal M \\otimes_{A[1\/E]^\\wedge_p[1\/p],\\tilde{\\iota}_\\gamma} W(\\hat{L}^\\flat)[\\frac{1}{p}] \\simeq \\mathcal M \\otimes_{A[1\/E]^\\wedge_p[1\/p]} W(\\hat{L}^\\flat)[\\frac{1}{p}]\n$$\nwhere $\\tilde{\\iota}_\\gamma: A[1\/E]^\\wedge_p[\\frac{1}{p}] \\to W(\\hat{L}^\\flat)[\\frac{1}{p}] \\xrightarrow{\\gamma} W(\\hat{L}^\\flat)[\\frac{1}{p}]$. If $(\\mathcal M,f)$ corresponds to a $\\mathbb Q_p$-representation $V$ of $G_K$, then $f_\\gamma$ corresponds to the semilinear action of $\\gamma$ on $\\mathcal M $ inside $ V^\\vee \\otimes W(\\mathbb{C}_p^\\flat)[1\/p]$. Moreover, two descent data $f , g$ are equal if and only if $f_{\\tau} = g_{\\tau}$.\n\\end{theorem}\n\n\\begin{lemma}\\label{lem-evaluation-2}\nGiven two finite free \\'etale $\\varphi$-modules $\\mathcal M,\\mathcal{N}$ over $A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}]$ and two morphisms $f, g: \\mathcal M \\to \\mathcal{N}$ of \\'etale $\\varphi$-modules over $A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}]$. Let $f_\\tau,g_\\tau$ be the base changes of $f,g$ along the map \n$$\ne_\\tau:A^{(2)}[1\/E]^\\wedge_p[\\frac{1}{p}] \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p[\\frac{1}{p}] \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat[\\frac{1}{p}]\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\tau} W(\\hat{L}^\\flat)[\\frac{1}{p}].\n$$\nThen $f=g$ if and only if $f_\\tau=g_\\tau$.\n\\end{lemma}\n\n\\begin{proof}\nThe proofs are exactly the same as the proof of Theorem~\\ref{thm-evaluation-1} and Lemma~\\ref{lem-evaluation-1}, plus the following fact that\n$$\n\\Cont\\big(\\hat{G}, W(\\hat{L}^\\flat)[\\frac{1}{p}]\\big) = \\Cont\\big(\\hat{G}, W(\\hat{L}^\\flat)\\big)[\\frac{1}{p}],\n$$\nwhich can be shown by the compactness of $\\hat{G}$. \n\\end{proof}\n\n\\subsection{Proofs of Proposition~\\ref{thm-1prime} and Theorem \\ref{Thm-main-1}}\\label{subsec-pris-crystal-proof} We keep the assumption that  $R=\\O_K$ is a mixed characteristic complete DVR with perfect residue field in this subsection, and keep our notations in \\S 2.1.\n\nLet us first prove Proposition~\\ref{thm-1prime} using Lemma~\\ref{lem-intersection} and results in \\S\\ref{subsec-phi-tau}. First, we give a different interpretation of the ``evaluation map\":\n$$\ne_\\gamma: A^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat)\n$$\nin Theorem~\\ref{thm-evaluation-1} when restricted on $A^{(2)}$ . Recall that we fix a compatible system $\\{\\varpi_n\\}_n$ of $p^n$-th roots of a uniformizer $\\varpi \\in \\O_K$, this defines a map of prisms $\\iota: (A,(E)) \\to (\\Ainf,(E))$ maps $u$ to $[{\\varpi}^\\flat ]$, and given a $\\gamma \\in G_K$, we define $\\iota_{\\gamma}$ to be the composition of $\\iota$ with $\\gamma: (\\Ainf,(E)) \\to (\\Ainf,(E))$ where the second map is defined as $a \\mapsto \\gamma(a)$. Since $(E)\\subset \\Ainf$ is equal to $\\Ker(\\theta)$ and $\\theta$ is  $G_K$-equivariant,  $\\gamma$ is a well-defined map of $\\delta$-pairs. By the universal property of $A^{(2)}$, we can define a map of prisms $\\iota_{\\gamma}^{(2)} : (A^{(2)},(E)) \\to (\\Ainf,(E))$ so that the following diagram commutes:  \n\\begin{equation}\\label{equ-diagram-prisms}\n\\begin{tikzcd}\n(A,(E)) \\arrow[rr,\"i_1\"] \\arrow[ddrr,\"\\iota_\\gamma\",swap] & & (A^{(2)}, (E) )\\arrow[dd,\"\\iota^{(2)}_\\gamma\"] & & (A,(E)) \\arrow[ll,\"i_2\",swap] \\arrow[ddll,\"\\iota\"]\\\\\n& & & &\\\\\n& & (\\Ainf, (E)) & &\n\\end{tikzcd}  \n\\end{equation}\nWe have $\\iota^{(2)}_{\\gamma}$ induces a morphism $\\tilde{\\iota}^{(2)}_{\\gamma}: A^{(2)}[1\/E]^\\wedge_p \\to W(\\mathbb C_p^\\flat)$. We claim for all $\\gamma \\in G_K$, $\\tilde{\\iota}^{(2)}_{\\gamma}$ is the same as the \n$$\nA^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\gamma} W(\\hat{L}^\\flat) \\hookrightarrow W(\\mathbb C_p^\\flat).\n$$\nTo see this, by the universal property of direct perfection, we have \\eqref{equ-diagram-prisms} factorizes as:\n$$\n\\begin{tikzcd}\n(A,(E)) \\arrow[d]\\arrow[rr,\"i_1\"]  & & (A^{(2)}, (E) )\\arrow[d] & & (A,(E)) \\arrow[d] \\arrow[ll,\"i_2\",swap]\\\\\n(A_{\\perf},(E)) \\arrow[rr,\"i'_1\"] \\arrow[ddrr,\"\\iota'_\\gamma\",swap] & & ((A^{(2)})_{\\perf}, (E) )\\arrow[dd,\"\\iota'^{(2)}_\\gamma\"] & & (A_{\\perf},(E)) \\arrow[ll,\"i'_2\",swap] \\arrow[ddll,\"\\iota'\"]\\\\\n& & & &\\\\\n& & (\\Ainf, (E)) & &\n\\end{tikzcd}  \n$$\nSo $\\tilde{\\iota}^{(2)}_\\gamma$ has a factorization\n$$\nA^{(2)}[1\/E]^\\wedge_p \\to (A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\to W(\\mathbb C_p^\\flat).\n$$\nWe just need to check $\\iota'^{(2)}_\\tau$ induces the evaluation map \n$$\n(A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\tau} W(\\hat{L}^\\flat) \\xhookrightarrow{} W(\\mathbb{C}_p^\\flat).\n$$\nAnd this follows from the isomorphism of $(A^{(2)})_{\\perf}[1\/E]^\\wedge_p\\simeq W((K^{(2)}_\\infty)^\\flat)$, then one check directly for $j_1,j_2$ defined in \\eqref{eq-j1j2}, $\\tilde{e}_\\gamma\\circ j_1: A_{\\perf}[1\/E]^\\wedge_p \\to W(\\hat{L}^\\flat)$ is equal to the map induced from $\\iota'_\\gamma$ and $\\tilde{e}_\\gamma\\circ j_2: A_{\\perf}[1\/E]^\\wedge_p \\to W(\\hat{L}^\\flat)$ is equal to the map induced from $\\iota'$. In particular, we have a commutative diagram:\n\\begin{equation}\\label{eq-iotaandevaluation}\n\\begin{tikzcd}\nA^{(2)} \\arrow[d, hook] \\arrow[rrr, \"\\iota^{(2)}_\\gamma\"] &&& \\Ainf \\arrow[d, hook]\\\\\nA^{(2)}[1\/E]^\\wedge_p \\arrow[r] & (A^{(2)})_{\\perf}[1\/E]^\\wedge_p  \\arrow[r,\"\\tilde{e}_\\gamma\",hook] & W(\\hat{L}^\\flat) \\arrow[r,hook] & W(\\mathbb C_p^\\flat).\n\\end{tikzcd}  \n\\end{equation}\nNow we can prove Proposition~\\ref{thm-1prime}.\n\n\\begin{proof}[Proof of Proposition~\\ref{thm-1prime}]\nFirst we pick $\\gamma=\\tilde{\\tau}$ that is a preimage of $\\tau$ under the map $G_K \\to \\hat{G}$, we have $\\gamma(u)-u=Ez$ and $\\iota^{(2)}_{\\gamma}$ defined as above is the embedding defined in \\S\\ref{subsec-embedding} by Remark~\\ref{rem-embedding-depend}. In particular, composing the embedding $A^{(2)} \\hookrightarrow \\Ainf$ defined in \\S\\ref{subsec-embedding} with $\\Ainf \\hookrightarrow W(\\mathbb C_p^\\flat)$, one get the evaluation map \n$$\n(A^{(2)})_{\\perf}[1\/E]^\\wedge_p \\simeq \\Cont\\Big(\\hat{G}, W\\big((\\hat{L}^{(2)})^\\flat\\big)\\Big)^{H_K^{2}} \\xrightarrow{\\tilde{e}_\\tau} W(\\hat{L}^\\flat) \\xhookrightarrow{} W(\\mathbb{C}_p^\\flat).\n$$\nrestricted on $A^{(2)}$.\n\nKeep the notations as in \\S\\ref{subsec-G-image}, and let $\\mathcal M_{\\Ainf}=W(\\mathbb C_p^\\flat)\\otimes_A \\mathfrak{M}$ and $\\mathcal M_A \\simeq \\mathfrak{M}\\otimes_A A[1\/E]^\\wedge_p$. By  Theorem~\\ref{thm-evaluation-1} and Theorem~\\ref{thm-caruso}, recall we use $B^{(2)}= A^{(2)} [\\frac 1 E]^\\wedge_p$ and $B^{(2)}_{\\st}= A^{(2)}_{\\st} [\\frac 1 E]^\\wedge_p$ to simplify our notations, we have there is a descent data \n$$\nc: \\mathcal M_A \\otimes _{A[1\/E]^\\wedge_p, \\tilde{i}_1} B^{(2)} \\to \\mathcal M_A \\otimes_{A[1\/E]^\\wedge_p, \\tilde{i}_2} B^{(2)} \n$$\nof $\\mathcal M_A$ over $B^{(2)}$ that corresponds to the representation $T$. And the semilinear action of $\\gamma=\\tilde{\\tau}$ on $\\mathcal M_{\\Ainf}$ is given by the evaluation $c_\\tau$, that is,  we have the linearization of the $\\tilde{\\tau}$-action is defined by\n$$\nc_\\tau: W(\\mathbb C_p^\\flat) \\otimes_{\\tilde{\\iota}_\\gamma,A[1\/E]^\\wedge_p} \\mathcal M_A \\simeq W(\\mathbb C_p^\\flat) \\otimes_{\\tilde{\\iota} , A[1\/E]^\\wedge_p} \\mathcal M_A.\n$$\nBy  base change $c$ along $B^{(2)} \\to B^{(2)}[\\frac{1}{p}]$, we get a $B^{(2)}[\\frac{1}{p}]$-linear $\\varphi$-equivariant morphism:\n$$\nc': \\mathcal M_A \\otimes _{A[1\/E]^\\wedge_p, \\tilde{i}_1} B^{(2)}[\\frac{1}{p}] \\to \\mathcal M_A \\otimes_{A[1\/E]^\\wedge_p, \\tilde{i}_2} B^{(2)}[\\frac{1}{p}]. \n$$\nOn the other hand, from the discussions after Proposition~\\ref{thm-1prime}, $\\tilde{\\tau}$-action also defines a $\\varphi$-equivariant morphism \n$$\nf_{\\tilde{\\tau}}: \\mathfrak{M}\\otimes_{A,\\iota_{\\tilde{\\tau}}} A_{\\st}^{(2)}[\\frac{1}{p}] \\simeq \\mathfrak{M}\\otimes_{A} A_{\\st}^{(2)}[\\frac{1}{p}].\n$$\nWe will see in Proposition~\\ref{prop-descentBsttoB2} below that $f_{\\tilde{\\tau}}$ actually descents to a $B^{(2)}[1\/p]$-linear morphism. Assuming this fact, then if we base change $f_{\\tilde{\\tau}}$ along $A^{(2)}[\\frac{1}{p}] \\to W(\\mathbb C_p^\\flat)[\\frac{1}{p}]$, we will have $f_{\\tilde{\\tau}}\\otimes W(\\mathbb C_p^\\flat)[\\frac{1}{p}]=c_\\tau$ since the way we define $f_{\\tilde{\\tau}}$ is by taking the ${\\tilde{\\tau}}$-action. From the discussion at the beginning of the proof and Lemma~\\ref{lem-evaluation-2}, we have $f_{\\tilde{\\tau}}=c'$ as a $B^{(2)}[\\frac{1}{p}]$-linear isomorphism between $\\mathcal M_A \\otimes _{A[1\/E]^\\wedge_p, \\tilde{i}_1} B^{(2)}[\\frac{1}{p}] $ and $ \\mathcal M_A \\otimes_{A[1\/E]^\\wedge_p, \\tilde{i}_2} B^{(2)}[\\frac{1}{p}]$.\n\nWe fix a basis $\\{e_i\\}$ of $\\mathfrak{M}$, for $j=1,2$ let $\\{e^j_i\\}$ be the basis of $\\mathcal M_A \\otimes _{A, \\tilde{i'}_j} B^{(2)}[\\frac{1}{p}] $ defined by $e^j_i=e_i\\otimes 1$ and the tensor is via $A \\to A[1\/E]^\\wedge_p \\xrightarrow{\\tilde{i}_j} B^{(2)}[1\/p]$. So we can interpret $f_{\\tilde{\\tau}}=c'$ as matrix using this two basis, this matrix is $X_{\\tilde{\\tau}}$ from this definition, so it has coefficients inside $A_{\\st}^{(2)}[\\frac{1}{p}]$ by the discussion before \nProposition~\\ref{thm-1prime}. On the other hand, $X_{\\tilde{\\tau}}$ has coefficients in $B^{(2)}\\subset B_{\\st}^{(2)}$ since $c'$ is defined by the $B^{(2)}$-linear map $c$. So by Lemma~\\ref{lem-intersection}, we have $X_{\\tilde{\\tau}}$ has coefficients inside $A_{\\st}^{(2)}$. The same argument shows when $T$ is crystalline, then $X_{\\tilde{\\tau}}$ has coefficients inside $A^{(2)}$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop-descentBsttoB2}\nBase change along $B^{(2)} \\to A^{(2)}_{\\st}[1\/E]^\\wedge_p$ defines an equivalence of categories of \\'etale $\\varphi$-modules over $B^{(2)}$ and $A^{(2)}_{\\st}[1\/E]^\\wedge_p$ and an equivalence of categories of \\'etale $\\varphi$-modules over $B^{(2)}[1\/p]$ and $A^{(2)}_{\\st}[1\/E]^\\wedge_p[1\/p]$.\n\\end{proposition}\n\\begin{proof}\nBy \\cite[Theorem 4.6]{wu2021galois}, we just need to show the same result after perfections, we will show $(A^{(2)})_{\\perf}= (A^{(2)}_{\\st})_{\\perf}$ in Lemma~\\ref{lem-perfectionofA2andAst2} using the logarithmic prismatic site.\n\\end{proof}\n\nNow, let us prove Theorem~\\ref{Thm-main-1} by first producing a functor $\\mathcal T$ from prismatic $F$-crystals in finite $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-modules to lattices inside a crystalline representation. For prism $A$, we use $i_k: A \\to A^{(2)}$ or $A^{(3)}$ for natural map from $A$ to $k$-th factor of $A^{(2)}$ or $A^{(3)}$. The notation $i_{kl}: A ^{(2)} \\to A ^{(3)}$ has the similar meaning. \n\nBy Corollary \\ref{cor-crystal-descentdata}, given a prismatic $F$-crystal ${\\mathfrak{M}}_{\\mathlarger{\\mathbbl{\\Delta}}}$, we obtain a Kisin module $(\\mathfrak{M} , \\varphi _{\\mathfrak{M}})$ of height $h$ together with descent data\n$f: \\mathfrak{M} \\otimes _{A, i_1} A^{(2)} \\to \\mathfrak{M} \\otimes_{A, i_2}A^{(2)} $ so that $f$ satisfies the following cocycle condition $ i _{13} \\otimes f = (i _{23} \\otimes f) \\circ (i _{12} \\otimes f ) $, where $i_{kl} \\otimes f$ is the base change of $f$ along $i_{kl}$, and $f$ also compatible with the $\\varphi$-structure on the both sides of $f$. Note that the existence of $f$ follows from the crystal property of   $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$: \n\\begin{equation}\\label{eqn-cocyclefromcrystal}\nf: \\mathfrak{M} \\otimes _{A, i_1} A^{(2)} \\simeq \\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}((A^{(2)},(E))) \\simeq \\mathfrak{M} \\otimes_{A, i_2}A^{(2)} \n\\end{equation}\n\nWe let $\\mathcal M=\\mathfrak{M}\\otimes_A A[1\/E]^\\wedge_p$ and  $c=f\\otimes_{A^{(2)}} B^{(2)}$, then  $(\\mathcal M,c)$ is an \\'etale $\\varphi$-module with descent data, which corresponds to a $\\mathbb Z_p$-representation of $G_K$. Moreover the semilinear action of $G_K$ on $\\mathfrak{M}\\otimes_A W(\\mathbb{C}_p^\\flat)$ comes from $\\{c_\\gamma\\}_{\\gamma\\in G_K}$ using the evaluation maps. If we define \n$$\nf_\\gamma: \\Ainf \\otimes_{\\iota_\\gamma,A} \\mathfrak{M} \\to \\Ainf \\otimes_{\\iota , A} \\mathfrak{M}\n$$\nas the base change of $f$ along $\\iota_{\\gamma}^{(2)}$, then by \\eqref{eq-iotaandevaluation}, we have $c_\\gamma=f_\\gamma$. The $G_K$-semilinear action commutes with $\\varphi$ as $f$ does. For any $\\gamma \\in G_K$, we have $\\gamma (A) \\subset W(k)[\\![u , \\epsilon-1]\\!] \\subset A^{(2)}_{\\st} \\subset \\Ainf$. Therefore, the $G_K$-action on the $\\Ainf \\otimes_A \\mathfrak{M}$ defined the above factors through $A^{(2)}_{\\st} \\otimes_A \\mathfrak{M}$. We claim that $G_K$-action on $\\widehat \\mathfrak{M} : = A^{(2)}_{\\st}\\otimes_A \\mathfrak{M}$ defines a $(\\varphi, \\hat G)$-module which corresponds to a crystalline representation.\n\nFirst, for $\\gamma \\in G_\\infty$, $\\gamma(A) = A$ in $\\Ainf$, we conclude $\\iota^{(2)}_\\gamma : A^{(2)} \\to \\Ainf$ satisfies $\\iota^{(2)}_{\\gamma}\\circ i_1=\\iota^{(2)}_{\\gamma}\\circ i_2$. In particular, for any $\\gamma \\in G_\\infty$ and $j=1,2$, using \\eqref{eqn-cocyclefromcrystal} and the crystal property of $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$,  $f_\\gamma$ comes from the base change of \\eqref{eqn-cocyclefromcrystal} along $\\iota^{(2)}_\\gamma : A^{(2)} \\to \\Ainf$, in particular, we have \n$$\nf_\\gamma: \\mathfrak{M} \\otimes _{A, \\iota^{(2)}_{\\gamma}\\circ i_1} \\Ainf \\simeq \\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}((\\Ainf,\\Ker\\theta)) \\simeq \\mathfrak{M} \\otimes_{A, \\iota^{(2)}_{\\gamma}\\circ i_2}\\Ainf. \n$$\nSince $\\iota^{(2)}_{\\gamma}\\circ i_1=\\iota^{(2)}_{\\gamma}\\circ i_2$, we have $f_\\gamma={\\rm id}$ which means $\\mathfrak{M} \\subset (\\widehat \\mathfrak{M}) ^{G_\\infty}$. Similarly, $G_K$ acts on $\\widehat \\mathfrak{M}\/ I_+$ corresponds the base change of $f$ along \n$$A^{(2)} \\xrightarrow{\\iota^{(2)}_\\gamma} \\Ainf \\to W(\\bar{k}) $$\nwhere the last arrow is the reduction modulo $W(\\mathfrak m)$ ($\\mathfrak m$ is the maximal ideal of $\\O_{\\mathbb C_p}^\\flat$). One can check for all $\\gamma\\in G_K$ and $j=1,2$, we have \n$$\nA \\xrightarrow{i_j} A^{(2)} \\xrightarrow{\\iota^{(2)}_\\gamma} \\Ainf \\to W(\\bar{k})\n$$\nare all equal to $A \\to W(k) \\hookrightarrow W(\\overline{k})$ with the first arrow given by $u \\mapsto 0$. The above map induces a morphism of prisms $(A,(E)) \\to (W(k),(p))$, then using \\eqref{eqn-cocyclefromcrystal} and the crystal condition of $\\mathfrak{M}_{\\mathlarger{\\mathbbl{\\Delta}}}$, we can similarly prove that $G_K$ acts on $\\widehat \\mathfrak{M}\/ I_+$-trivially, so $(\\mathfrak{M}, \\varphi_{\\mathfrak{M}}, G_K)$ is a $(\\varphi, \\hat G)$-module. Furthermore, $\\widehat T (\\widehat \\mathfrak{M})$ is crystalline by Corollary \\ref{cor-crystalline} and Theorem~\\ref{Thm-1}. \n\n\\begin{remark}\nIn \\S\\ref{sec-logprismandsemistablereps}, we will consider a category consisting of modules with descent data, and similar arguments about the triviality of the Galois actions can be shown directly using the cocycle condition of the descent data. We summarize this fact in the following easy fact.\n\\end{remark}\n\\begin{lemma}\nLet $q:(A^{(2)},(E)) \\to (B,J)$ be a map of prisms satisfying $q\\circ i_1 =q\\circ i_2$, then for any descent data $f$ over $A^{(2)}$, the base change of $f$ along $q$ is the identity map.\n\\end{lemma}\n\nTo show the fully faithfulness of this functor, first let $(\\mathfrak{M}, f)$, $(\\mathfrak{M}', f')$ be two Kisin modules with descent data $f , f'$ respectively. Suppose that there exists a map $ \\alpha: \\mathcal T ((\\mathfrak{M} , f)) \\to \\mathcal T ((\\mathfrak{M}' , f'))$ as lattices of crystalline representations, then from our construction of $\\mathcal{T}$ and Theorem~\\ref{thm-2}, $\\alpha$ is induced from a map $\\hat{\\alpha}: (\\mathfrak{M} , \\varphi_{\\mathfrak{M}}, \\hat G_{\\mathfrak{M}}) \\to (\\mathfrak{M}' , \\varphi _{\\mathfrak{M}'}, \\hat G_{\\mathfrak{M}'})$ between $(\\varphi, \\hat G)$-modules. The faithfulness of $\\mathcal{T}$ follows the fact that $A \\to A[1\/E]^\\wedge_p$ induces a fully faithful functor between Kisin modules over $A$ and \\'etale $\\varphi$-modules over $A[1\/E]^\\wedge_p$ from \\cite[Proposition 2.1.12]{KisinFcrystal}. On the other hand,  $\\hat{\\alpha}$ gives morphisms $\\hat{\\alpha}_1: \\mathfrak{M} \\otimes_{A,i_1} A^{(2)} \\to \\mathfrak{M}' \\otimes_{A,i_1}A^{(2)}$ and $\\hat{\\alpha}_2: \\mathfrak{M} \\otimes_{A,i_2} A^{(2)} \\to \\mathfrak{M}' \\otimes_{A,i_2}A^{(2)}$. If we view $A$ and $A^{(2)}$ as subrings of $\\Ainf$ using diagram \\eqref{equ-diagram-prisms}, then the following diagram commutes by the fact that $\\hat{\\alpha}: \\widehat{\\mathfrak{M}} \\to \\widehat{\\mathfrak{M}'}$ is compatible with $\\tau$-action.\n$$\n\\begin{tikzcd}\n\\mathfrak{M} \\otimes_{A,i_1} A^{(2)} \\arrow[r,\"f\"] \\arrow[d,\"\\hat{\\alpha}_1\"] & \\mathfrak{M} \\otimes_{A,i_2} A^{(2)} \\arrow[d,\"\\hat{\\alpha}_2\"] \\\\\n\\mathfrak{M}' \\otimes_{A,i_1} A^{(2)} \\arrow[r,\"f'\"] & \\mathfrak{M}' \\otimes_{A,i_2} A^{(2)}  \\\\\n\\end{tikzcd}\n$$\nThus we produces a morphism between $(\\mathfrak{M}, f)$ and $(\\mathfrak{M}', f')$, i.e. $\\mathcal{T}$ is also full. \n\nIt remains to show the functor $\\mathcal{T}$ is essential surjective. Given a lattice $T$ in a crystalline representation of $G_K$, let $\\mathfrak{M}$ be the corresponded Kisin module, it suffices to construct a descent data of $\\mathfrak{M}$ over $A^{(2)}$. We have shown in our proof of Proposition~\\ref{thm-1prime} that if we view $A^{(2)}$ as a subring of $\\Ainf$ via $\\iota^{(2)}_{\\tilde{\\tau}}$, then $X_{\\tilde{\\tau}}$ defines a $\\varphi$-equivariant isomorphism $f: \\mathfrak{M}\\otimes_{A,i_1} A^{(2)} \\simeq \\mathfrak{M}\\otimes_{A,i_2} A^{(2)}$ of $A^{(2)}$-modules. We also show the base change of $f$ along $A^{(2)} \\to B^{(2)}$ is equal to the descent data  $c$ of the \\'etale $\\varphi$-module $\\mathcal M_A=\\mathfrak{M}\\otimes_A A[1\/E]^\\wedge_p$ that corresponds to $G_K$-action on $T$. In particular, $c: \\mathfrak{M}\\otimes_{A,i_1} B^{(2)} \\simeq \\mathfrak{M}\\otimes_{A,i_2} B^{(2)}$ satisfies the cocycle condition. By Lemma \\ref{lem-intersection}, $A^{(2)}$ (resp. $A^{(3)}$) injects into $B^{(2)}$ (resp. $B^{(3)}$), so we have $f$ also satisfies the cocycle condition. In particular, $(\\mathfrak{M},f)$ together produce a primatic $F$-crystals in finite free $\\O_{\\mathlarger{\\mathbbl{\\Delta}}}$-module by Corollary~\\ref{cor-crystal-descentdata}.\n\n\\begin{remark}\nGiven an \\'etale $\\varphi$-module $(\\mathcal M_A,\\varphi_{\\mathcal M_A}, c)$ over $A[1\/E]^\\wedge_p$ with descent datum $c$, we call $(\\mathcal M_A,\\varphi_{\\mathcal M_A}, c)$ is \\emph{of finite $E$-height} if $\\mathcal M_A$ is of finite $E$-height, i.e., if there is a finite free Kisin module $(\\mathfrak{M},\\varphi_{\\mathfrak{M}})$ of finite height and defined over $A$ such that $\\mathfrak{M}\\otimes_A A[1\/E]^\\wedge_p \\simeq \\mathcal M_A$ as $\\varphi$-modules. Since $(\\mathcal M_A, \\varphi_{\\mathcal M_A})$ is the \\'etale $\\varphi$-module for $T|_{G_\\infty}$, our definition of finite $E$-height is compatible with the one given by Kisin under the equivalence in (1) of Theorem~\\ref{thm-evaluation-1}. \n\nWe expect same arguments in the proof of Proposition~\\ref{thm-1prime} will be used to study representations of finite $E$-height. Similar result has been studied using the theory of $(\\varphi,\\tau)$-modules by Caruso. For example, in the proof of \\cite[Lemma 2.23]{Caruso-phitau}, Caruso shows for representations of finite $E$-height, the $\\tau$-actions descents to $\\mathfrak{S}_{u\\text{-np},\\tau}$, which is a subring of $\\Ainf$ closely related to $\\tilde{\\iota}^{(2)}_{\\tilde\\tau}(B^{(2)})\\cap \\Ainf$, where $\\tilde{\\tau}$ is a preimage of $\\tau$ in $G_K$. \n\\end{remark}\n\n\\begin{remark}\nWe can also establish the compatibility of our Theorem~\\ref{Thm-main-1}, the theory of Kisin and \\cite[Theorem 1.2]{BS2021Fcrystals}. Given a lattice $T$ in a crystalline representation of $G_K$ with non-negative Hodge-Tate weight, and let $\\mathfrak{M}$ be the Kisin module corresponds to $T$ in \\cite{KisinFcrystal}, and let $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ (reso. $\\mathfrak{M}'_{{\\mathlarger{\\mathbbl{\\Delta}}}}$) be the prismatic $F$-crystal corresponds to $T^\\vee$ under \\cite[Theorem 1.2]{BS2021Fcrystals} (resp. $T$ under Theorem~\\ref{Thm-main-1}). Note that we need to take $T^\\vee$ since in the work of Bhatt-Scholze, the equivalence is covariant. By our construction of $\\mathfrak{M}'_{{\\mathlarger{\\mathbbl{\\Delta}}}}$, we have $\\mathfrak{M}'_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,(E)))\\simeq \\mathfrak{M}$. By \\cite[Remark 7.11]{BS2021Fcrystals}, $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,(E)))\\simeq \\mathfrak{M}$. Next we need to show the descent data over $A^{(2)}$ constructed respectively are the same. By Corollary~\\ref{cor-inj}, we just need to show they are the same as descent data of \\'etale $\\varphi$-modules over $A^{(2)}[1\/E]^\\wedge_p$, but they are the same by our $\\tau$-evaluation criteria in Lemma~\\ref{lem-evaluation-1}.  \n\\end{remark}\n\n\n\\section{Logarithmic prismatic \\texorpdfstring{$F$}{F}-crystals and semi-stable representations}\\label{sec-logprismandsemistablereps}\nIn this section, we will propose a possible generalization of Theorem~\\ref{Thm-main-1} to semi-stable representations using the absolute logarithmic prismatic site. The main reference of this subsection is \\cite{Koshikawa2021log-prism}. We will restrict ourselves to the base ring $R=\\O_K$, a complete DVR with perfect residue field. And we give $R$ the log structure associated to the prelog structure $\\alpha: \\mathbb N \\to R$ such that $\\alpha(1)=\\varpi$ is a uniformizer in $R$, i.e., let $D=\\{\\varpi=0\\}$, then the log structure on $X=\\Spf(R)$ is defined by \n$$\nM_X=M_D \\hookrightarrow \\O_X \\text{ where } M_D(U):=\\{f\\in \\O_X(U) \\,|\\, f|_{U\\backslash D}\\in \\O^\\times(U\\backslash D) \\}.\n$$\nLet us introduce the absolute logarithmic site over $(X,M_X)$.\n\\begin{definition}\\cite[Definition 2.2 and Definition 3.3]{Koshikawa2021log-prism}\n\\begin{enumerate}\n    \\item A $\\delta_{\\log}$-ring is a tuple $(A,\\delta, \\alpha:M\\to A, \\delta_{\\log}:M\\to A)$, where $(A,\\delta)$ is a $\\delta$-pair and $\\alpha$ is a prelog-structure on $A$. And $\\delta_{\\log}$ satisfies:\n    \\begin{itemize}\n        \\item $\\delta_{\\log}(e)=0$,\n        \\item $\\delta(\\alpha(m))=\\alpha(m)^p\\deltalog(m)$,\n        \\item $\\deltalog(mn)=\\deltalog(m)+\\deltalog(n)+p\\deltalog(m)\\deltalog(n)$\n        for all $m,n\\in M$. And we will simply denote it by $(A,M)$ if this is no confusion. Morphisms are morphisms of $\\delta$-pairs that compatible with the perlog structure and $\\deltalog$-stucture.\n    \\end{itemize}\n    \\item A $\\delta_{\\log}$-triple is $(A,I,M)$ such that $(A,I)$ is a $\\delta$-pair and $(A,M)$ is a $\\delta_{\\log}$-ring.\n    \\item A $\\delta_{\\log}$-triple $(A,I,M)$ is a prelog prism if $(A,I)$ is a prism, and it is bounded if $(A,I)$ is bounded.\n    \\item A bounded prelog prism is a log prism if it is $(p, I )$-adically log-affine (cf. \\cite[Definition 3.3]{Koshikawa2021log-prism}). \n    \\item A bounded (pre)log prism is integral if $M$ is an integral monoid.\n    \\item A $\\delta_{\\log}$-triple $(A,I,M)$ is said to be over $(R,\\mathbb N)$ if $A\/I$ is an $R$-algebra and there is a map $M\\to \\mathbb N$ of monoids such that the following diagram commutes.\n    $$\n    \\begin{tikzcd}\n    M \\arrow[rr] \\arrow[d] & & A \\arrow[d] \\\\\n    \\mathbb N \\arrow[r] & R \\arrow[r] & A\/I\n    \\end{tikzcd}\n    $$\n    All $\\delta_{\\log}$-triples over $(R,\\mathbb N)$ form a category. Similarly, we can define the category of prelog prisms over $(R,\\mathbb N)$ and the category of bounded log prisms over $(R,\\mathbb N)^a$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\nIf $A$ is an integral domain, or more general if $\\alpha(M)$ consists of non-zero divisors, then $\\deltalog$ is uniquely determined by $\\delta$ if exists. In particular, morphisms between such $\\delta_{\\log}$-rings are just morphisms of $\\delta$-rings.\n\\end{remark}\n\n\\begin{remark}\nNote that in this paper, for a $\\delta$-pair $(A,I)$, we always assume $A$ is $(p,I)$-adic complete, but in \\cite{Koshikawa2021log-prism}, non-$(p,I)$-adic completed $\\delta_{\\log}$-triples are also been studied. By Lemma 2.10 of loc.cit., we can always take the $(p,I)$-adic completions of the $\\delta$-pair $(A,I)$ and the $\\delta_{\\log}$-structure will be inherited. \n\\end{remark}\n\n\\begin{proposition}\\cite[Corollary 2.15]{Koshikawa2021log-prism}\nGiven a bounded prelog prism $(A,I,M)$, one can associate it with a log prism\n$$\n(A,I,M)^a=(A,I,M^a)\n$$\n\\end{proposition}\n\\begin{remark}\nWhen we deal with log prisms in this paper, we will always take it as the log prism associated with some prelog prism. And by the above proposition, we know taking the associated log prism does not change the underlying $\\delta$-pair. Moreover, it is a general fact that $(A,I,M)^a$ is integral if $(A,I,M)$ is a integral.\n\\end{remark}\n\n\\begin{definition}\\label{def-logprism}\nThe absolute logarithmic prismatic site $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ is the opposite of the category whose objects are \n\\begin{enumerate}\n    \\item bounded log prisms $(A,I,M_A)$ with \\textit{integral} log structure,\n    \\item maps of formal schemes $f_A: \\Spf(A\/IA) \\to X$,\n    \\item the map $f_A$ satisfies\n$$\n(\\Spf(A\/IA),f_A^\\ast M_X) \\to (\\Spf(A),M_A)^a\n$$\ndefines an exact closed immersion of log formal schemes.\n\\end{enumerate}\nA morphism  $(A,I,M_A) \\to (B,I,M_B)$ is a cover if and only if $A \\to B$ is $(p,I)$-complete faithfully flat and the pullback induces an isomorphism on log structure. We define the structure sheaf $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ on $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ by $(A,I,M_A) \\mapsto A$.\n\\end{definition}\n\nThere is a variant of the about definition that we will also use in this subsection, we define $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$ be the full subcategory of $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ whose  objects are $(A,I,M_A)$ with $A$ perfect.\n\n\\begin{remark}\nOur definition of the absolute logarithmic prismatic site is different from \\cite[Definition 4.1]{Koshikawa2021log-prism}. First, we need to consider the absolute prismatic site, not the relative one. Furthermore, we use the $(p,I)$-complete faithfully flat topology compared with the $(p,I)$-complete \\'etale topology. Also we require the log-structures to be integral. \n\\end{remark}\n\n\\begin{proposition}\\label{prop-logprismisasite}\n$(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ forms a site.\n\\end{proposition}\n\\begin{proof}\nSimilar to \\cite[Corollary 3.12]{BS19}, we need to show for a given diagram \n$$\n\\begin{tikzcd}\n(C,I,M_C) & (A,I,M_A) \\arrow[l,\"c\",swap] \\arrow[r,\"b\"] & (B,I,M_B)\n\\end{tikzcd}\n$$\nin $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ such that $b$ is a cover, then the pushout of $b$ along $c$ is a covering. From the argument in $loc. cit.$, we known for the underlying prisms, the pushout of $b$ along $c$ is the $(p,I)$-completed tensor product $D=C\\widehat {\\otimes}_A B$, and $(D,I)$ is a bounded prism covers $(C,I)$ in the $(p,I)$-complete faithful flat topology. And we give $D$ the log structure $M_D$ defined by viewing $\\Spf(D)$ as the fiber product via \\cite[Proposition 2.1.2]{Ogus_logbook}, then $(C,M_C)\\to (D,M_D)$ is strict morphism by Remark 2.1.3 of $loc.cit.$,  so in particular, $M_D$ is integral since $M_C$ is. For the same reason, \n$$\n(\\Spf(D\/ID),f_D^\\ast M_X) \\to (\\Spf(D),M_D)^a\n$$ is strict since it is the base change of a strict morphism. It is an exact closed immersion since pushout of a surjective map of monoids is again surjective.\n\\end{proof}\n\n\\begin{example}\\cite[Example 3.4]{Koshikawa2021log-prism}\\label{exa-logprism}\n\\begin{enumerate}\n    \\item Let $(A,(E))$ be the Breuil-Kisin prism, then we can define a perlog structure to $(A,(E))$ given by $\\mathbb N \\to A; n\\mapsto u^n$, one have $(A,(E),\\mathbb N)^a$ is in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$, where (3) in Definition~\\ref{def-logprism} follows from the prelog structures $\\mathbb N \\to R \\to A\/(E)$ and $\\mathbb N \\to A \\to A\/(E)$ induce the same log structure.\n    \\item For any prism $(B,J)$ over $(A,(E))$, it has a natural prelog structure $\\mathbb N \\to A \\to B$, and similar to $(1)$, $(B,J,\\mathbb N)^a$ is in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$.\n    \\item A special case of (2) is that $(B,J)=(A_{\\perf},(E))$, the perfection of $(A,(E))$. One has the prelog structure in (2) can be directly defined as $1\\mapsto [\\varpi^\\flat]$. And $(A,(E),\\mathbb N)^a \\to(B,J,\\mathbb N)^a$ is a covering of log prisms in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$.\n    \\end{enumerate}\n\\end{example}\n\nActually, all logarithmic structures of log prisms in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ is the log structure associated to a prelog structure defined by $\\mathbb N$. We thank Teruhisa Koshikawa for letting us know the following lemma.\n\n\\begin{lemma}\\label{lem-Nchart}\nFor any log prism $(B,J,M_B)$ inside $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$, $(B,M_B)^a$ admits a chart $\\mathbb N \\to B$ defined by $n \\mapsto u_B^n$ for some $u_B\\in B$ satisfying $u_B \\equiv \\varpi \\mod J$. \n\\end{lemma}\n\\begin{proof}\nFor any log prism $(B,J,M_B)$ inside $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$, we have \n$$\n(\\Spf(B\/J),f_B^\\ast M_X) \\to (\\Spf(B),M_B)^a\n$$\ndefines an exact closed immersion of log formal schemes. So by the proof of \\cite[Proposition 3.7]{Koshikawa2021log-prism}, if we let $N^a_{B\/J}:=\\Gamma(\\Spf(B\/J),\\underline{\\mathbb N}^a)$ for the prelog structure $\\mathbb N \\to \\O_K \\to B\/J$ induced from the given prelog structure on $\\O_K$, then the fiber product $M_B \\times_{N^a_{B\/J}} \\mathbb N$ is a chart for $(B,M_B)^a$. Moreover, since we assume $M_B$ to be integral, we have $(\\Spf(B\/J),f_B^\\ast M_X) \\to (\\Spf(B),M_B)^a$ is a log thickening with ideal $J$ in the sense of \\cite[Definition 2.1.1.]{Ogus_logbook}, and one can show $M_B \\times_{N^a_{B\/J}} \\mathbb N \\simeq \\mathbb N \\times (1+J)$. Now $(1+J)^\\times =(1+J)$, so \n$$\n\\mathbb N \\to \\mathbb N \\times (1+J) \\simeq M_B \\times_{N^a_{B\/J}} \\mathbb N \\to B\n$$\nis also a chart for $(B,M_B)^a$. And the prelog structure given by $n \\mapsto u_B^n$ for some $u_B\\in B$ satisfying the image of $u_B$ in $B\/J$ coincides with the image of $\\varpi$ under $\\O_K \\to B\/J$.\n\\end{proof}\n\nIn the rest of this subsection, we will try to generalize results we proved in \\S\\ref{subsec-pris-crystal}-\\S\\ref{subsec-pris-crystal-proof} for the logarithmic prismatic site. \n\n\\begin{lemma}\\label{lem-log-nonemptyproduct}\n\\begin{enumerate}\n    \\item For $(A,I_A,M_A)^a, (B,I_B,M_B)^a\\in (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ such that $M_A,M_B$ are integral and $(A,M_A)\\to (A\/I_A,\\mathbb N)$ and $(B,M_B)\\to (B\/I_B,\\mathbb N)$ are exact surjective, there is a prelog prism $(C,I_C,M_C)$ with integral log structure that is universal in the sense that the diagram \n    $$\n    \\begin{tikzcd}\n    (A,I_A,M_A) \\arrow[r] & (C,I_C,M_C) & (B,I_B,M_B) \\arrow[l] \n    \\end{tikzcd}\n    $$\n    is initial in the category of diagrams \n    $$\n    \\begin{tikzcd}\n    (A,I_A,M_A) \\arrow[r] & (D,I_D,M_D) & (B,I_B,M_B) \\arrow[l] \n    \\end{tikzcd}\n    $$\n    of prelog prisms over $(R,\\mathbb N)$, and $(D,M_D)\\to (D\/I_D,\\mathbb N)$ is an exact surjective.\n    \\item If $(C,I_C)$ in (1) is bounded, then $(C,I_C,M_C)^a$ is the product of $(A,I_A,M_A)^a$ and $(B,I_B,M_B)^a$ inside $ (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$.\n    \\item If $(A,I_A,M_A)^a, (B,I_B,M_B)^a$ in (1) are in $ (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$, and let $(C_{\\perf},I_C)$ be the perfection of $(C,I_C)$ defined in (1). Let $(C_{\\perf},I_C,M_C)$ be the prelog prism with prelog structure induced from $C$. Then $(C_{\\perf},I_C,M_C)^a$ is the product of $(A,I_A,M_A)^a$ and $(B,I_B,M_B)^a$ in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet $(A,I_A,M_A),(B, I_B, M_B)\\in (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$, define $C_0$ to be the $(p,I_A,I_B)$-adic completion of $A\\otimes_{W(k)}B$ and let $J$ be the kernel of\n$$\nC_0 \\to A\/I_A \\widehat {\\otimes}_R B\/I_B.\n$$\nThen $(C_0,J,M_A\\times M_B)$ is a $\\deltalog$-triple over $(A,I_A,M_A)$. And we have $(C_0,J,M_A\\times M_B) \\to (C_0\/J,\\mathbb N)$ is surjective. Then we can apply \\cite[Proposition 3.6]{Koshikawa2021log-prism} to get a universal prelog prism $(C,I_C,M_C)$ over $(A,I_A,M_A)$ and $(B, I_B, M_B)$ and satisfies $(C,M_C)\\to (C\/J,\\mathbb N)$ is exact surjective. Just recall in the proof of \\cite[Proposition 3.6]{Koshikawa2021log-prism}, we first construct a $\\deltalog$-triple $(C',J',M_C')$ which is universal in the sense that it is a $\\deltalog$-triple over both $(A,I_A,M_A)$ and $(B, I_B, M_B)$ satisfying $C'\/J'$ is over $A\/I_A$ and $B\/I_B$ as $R$-algebra and $(C',M_C')\\to (C'\/J',\\mathbb N)$ is exact surjective. Then we take the prismatic envelope with respect to $(A,I_A) \\to (C',J')$ to get $(C,I_C)$. Then we can check such $(C,I_C,M_C)$ satisfies the universal property. For (2), when $(C,I_C)$ is bounded, the fact that $(C,I_C,M_C)^a$ is the product of $(A,I_A,M_A)^a$ and $(B,I_B,M_B)^a$ inside $ (X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ follows from Proposition 3.7 of $loc.cit.$. For (3), we have $(C_{\\perf},I_C)$ is automatic bounded, and one can check $(C_{\\perf},I_C)$ is universal using exactly the same proof of Proposition 3.7 of $loc.cit.$.\n\\end{proof}\n\nWe thank Koji Shimizu for the following lemma on $A^{(2)}_{\\st}$. \n\n\\begin{lemma}\\label{lem-Ast2islogprism}\nLet $(A,I,\\mathbb N)^a$ be the Breuil-Kisin prism defined in $(1)$ of Example~\\ref{exa-logprism}, then the self-product (resp. self-triple product) of $(A,I,\\mathbb N)^a$ in $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ exist. Moreover, if we let $(A^{\\langle 2 \\rangle},I,M^2)^a$ (resp. $(A^{\\langle 3 \\rangle},I,M^3)^a$) be self-product (resp. self-triple product) of $(A,I,\\mathbb N)^a$, then $A^{\\langle i \\rangle}\\simeq A^{(i)}_{\\st}$ for $i=2,3$.\n\\end{lemma}\n\\begin{proof}\nBy our construction in Lemma~\\ref{lem-log-nonemptyproduct}, $(A^{\\langle 2 \\rangle},I,M)$ is the prelog prismatic envelope $(C,I_C,M_C)$ with respect to\n    $$\n    (A,(E),\\mathbb N) \\to (C_0,J,\\mathbb N^2) \\text{ and } (C_0\/J,\\mathbb N^2)\\to (R,\\mathbb N)\n    $$\n    where $C_0=W[\\![u,v]\\!]$, $J=(E(u),u-v)$ with the prelog structure given by $\\beta: (1,0)\\mapsto u, (0,1)\\mapsto v$. The prelog prismatic envelope is constructed using the technique of exactification: consider $\\pi: (C_0,\\mathbb N^2)\\to (R=C\/J,\\mathbb N)$ where the map between log structures is given by $\\pi_{\\log}: \\mathbb N\\times \\mathbb N \\to \\mathbb N;(m,n)\\mapsto m+n$, here $\\pi_{\\log}$ is surjective but not exact, so to constructsthe exactification of $\\pi: (C,\\mathbb N^2)\\to (R,\\mathbb N)$ (cf. \\cite[Construction 2.18]{Koshikawa2021log-prism}), first we have the exactification of $\\pi_{\\log}$ is \n    $$\n    \\alpha: M^2 \\to \\mathbb N \\quad \\text{ given by } \\quad (m,n) \\mapsto m+n,\n    $$\n    where $M^2=\\{(m,n)\\in \\mathbb Z\\times \\mathbb Z \\,|\\, m+n\\in \\mathbb N \\}$. Since $M^2$ is generated by $(-1,1)$, $(1,-1)$, $(0,1)$ and $(1,0)$, one has the exactification of $\\pi$ is \n    $$\n    \\Big( W(k)[\\![u,v]\\!]\\big[\\frac{v}{u},\\frac{u}{v}\\big]^\\wedge_{(p,J')}, J',M^2; \\alpha: (1,0)\\mapsto {u}, (0,1)\\mapsto v, (1,-1)\\mapsto \\frac{u}{v}, (-1,1)\\mapsto \\frac{v}{u} \\Big)\n    $$\n    where $J':=\\ker(W(k)[\\![u,v]\\!]\\big[\\frac{v}{u},\\frac{u}{v}\\big] \\to R)$. \n    \n    We have the $(p,J')$-adic completion of $W(k)[\\![u,v]\\!]\\big[\\frac{v}{u},\\frac{u}{v}\\big]$ is $W(k)[\\![u,\\frac{v}{u}-1]\\!]$. Then take prismatic envelope of \n    $\n    (A,(E))\\to (W(k)[\\![u,\\frac{v}{u}-1]\\!], (E,\\frac{v}{u}-1)).\n    $ One can check \n    $$ W(k)[\\![u,\\frac{v}{u}-1]\\!]\\big\\{\\frac{v\/u-1}{E(u)}\\big\\}^\\wedge_\\delta \\simeq A_{\\st}^{(2)}$$ \n    directly from the definition of $A_{\\st}^{(2)}$.\n    \n    Similarly, we can show $A^{\\langle 3 \\rangle}\\simeq A^{(3)}_{\\st}$ which is also bounded. \n\\end{proof}\n\nThe following is one of our key observations.\n\\begin{lemma}\\label{lem-perfectionofA2andAst2}\nWe have $(A^{\\langle 2 \\rangle})_{\\perf} \\simeq (A^{(2)})_{\\perf}$.\n\\end{lemma}\n\\begin{proof}\nLet $u_1,u_2$ be the image of $u$ under the two natural maps $i_{j}: A_{\\perf} \\to (A^{(2)})_{\\perf}$ for $j=1,2$. We claim that $u_2\/u_1$ is inside $(A^{(2)})_{\\perf}$.\n\nFirstly, we have already shown $A_{\\perf}\\simeq W(\\widehat{\\O}_{K_\\infty}^\\flat)$ and $u=[\\varpi^\\flat]$, here $\\varpi^\\flat=(\\varpi_n)$ with $\\{\\varpi_n\\}_{n\\geq 0}$ being a compatible system of $p^n$-th roots of $\\varpi$ inside $\\O_{\\widehat{K}_\\infty}$, and $(\\varpi_n) \\in \\O_{\\widehat K_\\infty}^\\flat$ via the identification $\\O_{\\widehat K_\\infty}^\\flat \\simeq \\lim_{x \\mapsto x^p} \\O_{\\widehat K_\\infty}$. Let $S=(A^{(2)})_{\\perf}\/(E)$, this is an integral perfectoid ring over $\\O_K$ in the sense of \\cite{BMS1}. We have $S^\\flat\\simeq (A^{(2)})_{\\perf}\/(p)$. For $j=1,2$, define $\\varpi_j^\\flat=u_j \\mod (p) \\in S^\\flat$, then we have $u_j=[\\varpi_j^\\flat]$ for $j=1,2$.\n\nRecall in \\S~\\ref{subsrc-construct-A2}, we have $z = \\frac{y -x}{E(x)}$ in $A^{(2)}$. Since  $ E(x) \\equiv  x^e \\mod p$, we have $ x  (1 + x^{e-1} z) \\equiv  y \\mod p$. If we denote $\\iota : A^{(2)} \\to (A^{(2)})_{\\perf} $ the natural map, then $\\iota(x)=u_1$ and $\\iota(y)=u_2$ in our definition, and $u _1 (1 + u_1^{e-1} \\iota(z)) \\equiv u _2 \\mod p$ inside $S^\\flat=A^{(2)}_{\\perf}\/(p)$. This is the same as $\\varpi_1^\\flat \\mu =  \\varpi_2^\\flat$ with $\\mu = (1 + u_1^{e-1} \\iota(z)) \\mod p$ in $S ^\\flat$. So we have $ [\\mu] u _1 = [\\mu] [\\varpi_1^\\flat] = [\\varpi_2^\\flat]=  u_2$, which proves our claim.\n\n\nNow by symmetry, $u_1\/u_2$ is also inside $(A^{(2)})_{\\perf}$, so $u_1\/u_2$ is a unit in $(A^{(2)})_{\\perf}$. So we can give $(A^{(2)})_{\\perf}$ a prelog structure\n$$\n\\alpha: M^2 \\to (A^{(2)})_{\\perf} \\text{ with } (1,-1)\\mapsto \\frac{u_1}{u_2}, (-1,1)\\mapsto \\frac{u_2}{u_1}, (1,0)\\mapsto {u_1}, (0,1)\\mapsto {u_2}\n$$\nwith the monoid $M^2$ defined as in the proof of Lemma~\\ref{lem-Ast2islogprism}, then  $((A^{(2)})_{\\perf},(E),M^2)^a$ is in $X_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$. \n\nOne can check the maps $i_1,i_2: (A,(E)) \\to (A^{(2)},(E)) \\to ((A^{(2)})_{\\perf},(E))$ induce $i_1,i_2: (A_{\\perf},(E),\\mathbb N) \\to ((A^{(2)})_{\\perf},(E),M^2)$ of prelog prisms. So by Lemma~\\ref{lem-Ast2islogprism}, there is a unique map $(A^{\\langle 2 \\rangle},I,M^2)\\to ((A^{(2)})_{\\perf},(E),M^2)$, which factors through $((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)$.  So it induces a map $((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2) \\to ((A^{(2)})_{\\perf},(E),M^2)$ inside $X_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$. On the other hand, by the universal property of $A^{(2)}$, we know there is a map $(A^{(2)})_{\\perf} \\to (A^{\\langle 2 \\rangle})_{\\perf}$ fits into the coproduct diagram in $X_{{\\mathlarger{\\mathbbl{\\Delta}}}}^{\\perf}$, which is the full subcategory of $X_{\\mathlarger{\\mathbbl{\\Delta}}}$ containing perfect prisms.\n\nOne can check the composition $\\eta: ((A^{(2)})_{\\perf},(E)) \\to ((A^{\\langle 2 \\rangle})_{\\perf},(E)) \\to ((A^{(2)})_{\\perf},(E))$ satisfies $\\eta\\circ i_j= i_j \\circ \\eta$ for $i_1,i_2:(A_{\\perf},(E))\\to ((A^{(2)})_{\\perf},(E))$. Such a map is unique inside $X_{{\\mathlarger{\\mathbbl{\\Delta}}}}^{\\perf}$, so $\\eta=\\id_{((A^{(2)})_{\\perf},(E))}$. \n\nOn the other hand, the composition \n$$\n\\eta': ((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)^a \\to ((A^{(2)})_{\\perf},(E),M^2)^a \\to ((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)^a\n$$ satisfies $\\eta\\circ i'_j= i'_j \\circ \\eta$ for $i'_1,i'_2:(A_{\\perf},(E),\\mathbb N)^a\\to ((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)^a$ induced from $i'_1,i'_2:(A,(E),\\mathbb N)\\to (A^{\\langle 2\\rangle},(E),M^2)$. Such map is also unique inside $X_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}^{\\perf}$, so $\\eta'=\\id_{((A^{\\langle 2 \\rangle})_{\\perf},(E),M^2)^a}$. So in particular we have $(A^{\\langle 2 \\rangle})_{\\perf}\\simeq (A^{(2)})_{\\perf}$.\n\\end{proof}\n\n\n\\begin{theorem}\\label{thm-log-phitau}\nThe category of \\'etale $\\varphi$-module over $A[1\/E]^\\wedge_p$ with a descent data over $A_{\\st}^{(2)}[1\/E]^\\wedge_p$ is equivalent to the category of lattice in representations of $G_K$. Moreover, for all $\\gamma\\in\\hat{G}$, we can define the evaluation map\n$$\ne_\\gamma: A_{\\st}^{(2)}[1\/E]^\\wedge_p \\to W(\\hat{L}^\\flat)\n$$\nsuch that Lemma~\\ref{lem-evaluation-1} is still valid. Moreover, the $\\mathbb Q$-isogeney version of this theorem also holds.\n\\end{theorem}\n\n\\begin{remark}\nThe above theorem should be related to the \\'etale comparison theorem in the log prismatic settings, which has not been studied in \\cite{Koshikawa2021log-prism} yet.\n\\end{remark}\n\nMoreover, we have a log version of Lemma~\\ref{lem-AEcoversfinal} also holds. We thank Teruhisa Koshikawa for hints of the following result.\n\n\\begin{proposition}\\label{prop-logcoverfinalobject}\nThe sheaf represented by $(A,(E),\\mathbb N)^a$ covers the final object $\\ast$ in in $\\Shv((X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}})$.\n\\end{proposition}\n\\begin{proof}\nFor any log prism $(B,J,M_B)$, by Lemma~\\ref{lem-Nchart}, we can assume $(B,J,M_B)^a=(B,J,\\mathbb N)^a$, with prelog structure defined by $n \\mapsto u_B^n$ with $u_B \\equiv \\varpi \\mod J$.\n\nUsing deformation theory, we have there is a unique $W(k)$-algebra structure for $B$, and we define $C=B[\\![u]\\!][\\frac{u_B}{u},\\frac{u}{u_B}]\\{\\frac{u_B\/u-1}{J}\\}^\\wedge_\\delta$, where the completion is taken for the $(p,J)$-adic topology. Similar to the proof of Lemma~\\ref{lem-Ast2islogprism}, we have $(C,JC,\\mathbb N)^a$ is the product of $(A,(E),\\mathbb N)^a$ and $(B,J,\\mathbb N)^a$ inside $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$. Moreover, we have $B \\to C$ is $(p, J)$-complete flat by \\cite[Proposition 3.13]{BS19}. It remains to show that $(B,J) \\to (C,J)$ is a covering, i.e., $B \\to C$ is $(p, J)$-complete faithfully flat. Let \n$$\nC^{nc}:=B[\\![u]\\!][\\frac{u_B}{u},\\frac{u}{u_B}]\\{\\frac{u_B\/u-1}{J}\\}_{\\delta}\n$$\nbe the non-complete version of $C$ that we have the $(p,J)$-adic completion of $C^{nc}$ is $C$. Now we just need to show the flat ring map $B\/(p,J) \\to C\/(p,J)=C^{nc}\/(p,J)$ is also faithful. \n\nWe claim that $C\/ (p , J)$ is free over $B \/ (p , J)$. One has $JC=E(u)C$, and $(p , J)= (p , E) = (p , J , E)$ in $C$. So $C\/ (p , J)=C^{nc}\/(p,J)$ is equal to \n\\[ B [\\![u ]\\!][\\frac{u_B}{u}, \\frac{u}{u _B}][\\delta^i(z), i \\geq 0 ]\/ \\left (p , J , E , Ez = \\frac{u_B}{u }-1 , \\delta^i (\\frac{u_B}{u }-1))= \\delta^i (Ez), i \\geq 1 \\right ).\\]\nAfter modulo $(p,J)$, the above is the direct limit of\n\\[B \/ (p , J)[\\delta ^i (z)]\/ \\left (\\delta^i (\\frac{u_B}{u }-1))= \\delta^i (Ez) \\mod (p, E , J) \\right )\\]\nfor $i\\geq 0$.\n\nNow we use Lemma \\ref{lem-delta-n} to compute $\\delta^i (\\frac{u_B}{u }-1)= \\delta^i (Ez) \\mod (p, E , J)$. We keep the notations in Lemma \\ref{lem-delta-n}, by induction, we have $b_n = 0 \\mod (p , E)$. Using that $a_p^{(j)}\\in A_0^\\times$, $\\delta^i (\\frac{u_B}{u }-1)= \\delta^i (Ez) \\mod (p, E , J)$ gives a relation $ (z_{i -1}) ^ p =   \\sum\\limits_{j= 0}^{p -1} \\tilde a_j^{(i)} (z_{i-1}) ^j$ where $z_i = \\mathfrak z_i \\mod (p , J , E)$ and $\\tilde a_j^{(i)} \\in B \/ (p , J)[z_0, z_1, \\dots, z_{i-2}]$. In summary, we have \n$$C\/ (p , J) = B\/(p, J)[z_i, i \\geq 0]\\Bigg\/ \\left ((z_{i}) ^ p -   \\sum\\limits_{j= 0}^{p -1} \\tilde a_j^{(i)} (z_{i}) ^j, i\\geq 1 \\right )$$\nwhich is free over $B \/(p, J)$. \n\\end{proof}\n\n\\begin{definition}\\label{def-logFcrystal}\n\\begin{enumerate}\n    \\item \nA prismatic crystal over $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ in finite locally free $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-modules is a finite locally free $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-module $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ such that for all morphisms $f: (A, I,M_A) \\to (B, J,M_B)$ of log prisms, it induces an isomorphism:\n$$\nf^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},A} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((A,I,M_A))\\otimes_A B \\simeq \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}},B} := \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}((B,J,M_B))\n$$\n\n\\item A prismatic $F$-crystal over $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ of height $h$ (in finite locally free $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-modules) is a prismatic crystal $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ in finite locally free $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-modules together with a $\\varphi_{\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}}$-semilinear endomorphism $\\varphi_{\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}}$ of the $\\O_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$-module $\\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}: \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\to \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ such that the cokernel of the linearization $\\varphi^\\ast \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}} \\to \\mathfrak{M}_{{\\mathlarger{\\mathbbl{\\Delta}}}}$ is killed by $\\mathcal{I}_{\\mathlarger{\\mathbbl{\\Delta}}}^h$.\n\n\\end{enumerate}\n\\end{definition}\n\nIn particular, with help of Theorem~\\ref{thm-log-phitau} and Proposition~\\ref{prop-logcoverfinalobject}, a direct translation of proofs in \\S\\ref{subsec-pris-crystal-proof} with $A^{(2)}$ replaced by $A^{(2)}_{\\st}$ shows the following theorem.\n\\begin{theorem}\\label{thm-log-main-1}\nThe category of prismatic $F$-crystals over $(X,M_X)_{{\\mathlarger{\\mathbbl{\\Delta}}}_{\\log}}$ of height $h$ is equivalent to the category of lattices in semi-stable representations of $G_K$ with Hodge-Tate weights between $0$ and $h$.\n\\end{theorem}\n\n\\section{Some discussions on base rings}\\label{subsec-baserings} In this section, we show that our base ring assumed at the beginning of \\S \\ref{sec-ring-strcuture} covers many situations of base rings used in \\cite{Kim12} and \\cite{Brinon}. \n\nLet $K$ be complete DVR with perfect residue field $k$, and let $K_0=W[\\frac{1}{p}]$ with $W=W(k)$, fix a uniformizer $\\varpi\\in \\O_K$ and $E(u)\\in W[u]$ a minimal polynomial of $\\varpi$ over $K_0$. Let $R$ be a normal domain and satisfies that $R$ is a $p$-complete flat $\\O_K$-algebra that is complete with respect to $J$-adic topology, for an ideal  $J=(\\varpi, {t_1},\\ldots,{t_d})$ of $R$ containing $\\varpi$. We also assume $\\overline{R}=R\/(\\varpi)$ is a finite generated $k$-algebra with \\emph{finite $p$-basis} discussed in \\cite[\\S 1.1]{deJong}.\n\n\\begin{lemma}[\\cite{Kim12} Lemma 2.3.1 and lemma 2.3.4]\\label{Kim-lemma}\n\\begin{enumerate}\n    \\item In the above setting, there is a $p$-adic formally smooth flat $W$-algebra $R_0$ equipped with a Frobenius lift $\\varphi_0$ such that $\\overline{R}: = R_0\/(p)$. Moreover let $J_0$ be the preimage of $\\overline{J}$ inside $R_0$, then $R_0$ is $J_0$-adically complete, and under this topology, $R_0$ is formally smooth. \n    \\item $R_0\/(p)\\xrightarrow{\\sim}R\/(\\varpi)$ lifts to a $W$-algebra morphism $R_0 \\to R$ and the induced $\\O_K$-algebra morphism $\\O_K\\otimes_W R_0 \\to R$ is an isomorphism. Moreover this isomorphism is continuous with respect to the $J_0$-adic topology.\n\\end{enumerate}\n\\end{lemma}\n\nLet $(R_0, \\varphi_{R_0})$ denote a flat $W$-lift of $R\/(\\varpi)$ obtained from the above lemma. And we will have $J_0=(p, t_1, \\ldots, t_d)\\in R_0$, and we write $\\overline{J}=(\\overline{t_1},\\ldots, \\overline{t_d})\\subset \\overline{R}$. \n\n\\begin{definition}\\label{RAE}\nLet $R_0$ be a $p$-complete $\\mathbb Z_p$-algebra, we say $R_0$ satisfies the ``refined almost \u00e9talenes\" assumption, or simply RAE assumption, if $\\hat{\\Omega}_{R_0}=\\oplus_{i=1}^m R_0 dT_i$ with $T_i\\in R_0^\\times$. Where $\\hat{\\Omega}_{R_0}$ is the module of of $p$-adically continuous K\\\"ahler differentials.\n\\end{definition}\nThe following are examples of $R_0$ and $R$ which satisfy assumptions of Lemma \\ref{Kim-lemma} and RAE assumption. \n\\begin{example}\n\\begin{enumerate}\n    \\item If $R\/(\\varpi)$ is a completed noetherian regular local ring with residue field $k$, then Cohen structure theorem implies\n     $R\/(\\varpi)=k[\\![\\overline{x_1},\\ldots,\\overline{x_d}]\\!]$. In this case, $R_0=W[\\![x_1,\\ldots, x_d]\\!]$ and $J_0=(p,x_1,\\ldots, x_d)$. Then $R=W[\\![x_1,\\ldots, x_d]\\!][u]\/E$, with $E\\in W[u]$ is a Eisenstein polynomial.\n    \\item Let $R_0 = W(k) \\langle t _1^{\\pm 1} ,  \\dots , t _m ^{\\pm 1}\\rangle$ and $J_0=(p)$, in this example, $\\overline{R}=k[\\overline{t}_1^{\\pm 1} ,  \\dots , \\overline{t}_m ^{\\pm 1}]$ is not local.\n    \\item An unramified complete DVR $(R_0 , p)$ with residue field $k$ so that $[k : k ^p]<\\infty$. \n    \\item Note the the Frobenius liftings in Lemma ~\\ref{Kim-lemma} is not unique. In (2) we can choose $\\varphi_{R_0}(t_i)=t_i^p$. In (1), we can choose the $\\varphi_{R_0}(x_i)=x_i^p$ or $\\varphi_{R_0}(x_i)=(x_i+1)^p-1$.\n\\end{enumerate}\n\\end{example}\nLet $R_0$ be $p$-complete algebra which satisfies the RAE assumption, Set $\\breve R_0 = W\\langle t_1 , \\dots , t_m \\rangle$ and $f : \\breve R_0 \\to R_0$ by sending $t_i$ to $T_i$. \n\\begin{proposition}\\label{prop-fetale} Assume that $R_0$ is a $p$-complete integral domain which admits finite $p$-basis and satisfies RAE assumption.  \nThen $f$ is formally \\'etale $p$-adically. \n\\end{proposition}\n\\begin{proof} We thanks for Wansu Kim providing the following proof. By standard technique using \\cite[Ch.III, Corollaire 2.1.3.3]{Illusie1} (e.g., see the proof in \\cite[Lem. 2.3.1]{Kim12}), it suffices to show that the cotangent complex \n$\\mathbb L_{R_0 \/ \\breve R_0}$ is acyclic. Since both $R_0$ and $\\breve R_0$ are $\\mathbb Z_p$-flat, it suffice to show that $\\mathbb L_{R_1 \/ \\breve R_1}$ is acyclic where $R_1 = R_ 0 \/ p R_0$ and $\\breve R_1 = \\breve R_0 \/ p \\breve R_0$. Since $R_0$ has finite $p$-basis, by \\cite[Lem. 1.1.2]{deJong}, $\\mathbb L_{R_1 \/k}\\simeq \\Omega_{R_1\/k}$. Note that maps $k \\to \\breve R_1 \\to R_1$ induces a fiber sequence \n\\[ \\mathbb L_{\\breve R_1 \/k}\\otimes^{\\mathbb L}_{\\breve R_1} R_1 \\to \\mathbb L _{R_1 \/ k} \\to \\mathbb L_{R_1 \/ \\breve R_1}\\]\nSince that $ \\mathbb L_{\\breve R_1 \/k} \\simeq \\Omega_{\\breve R_1\/k}$ and $\\Omega_{\\breve R_1\/k}\\simeq \\Omega_{R_1\/k}$ by RAE condition, we conclude that $\\mathbb L_{R_1\/ \\breve R_1}= 0$ as required. \n\\end{proof}\nLet us end with a discussion about our base rings and the base rings used in \\cite{Brinon}. As explained in the beginning of \\cite[Chap. 2]{Brinon}, his base ring $R_0$ in \\cite{Brinon} is  obtained from $W\\langle t_1^{\\pm 1}, \\ldots, t_m^{\\pm 1}\\rangle$ by a finite number of iterations of certain  operations and is also assumed to satisfy certain properties.  By Prop. 2.0.2 \\emph{loc. cit.}, we see that $R_0$ has finite $p$-basis and satisfies RAE assumption. So the base ring $R_0$ in \\cite{Brinon} also satisfies the requirement that $f:W\\langle t_1, \\ldots, t_m\\rangle \\to R_0$ is formally \\'etale by Proposition \\ref{prop-fetale}. \n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\newbox\\ncinttbox\n\t\\setbox0=\\hbox{$-$}\n\t\\setbox2=\\hbox{$\\displaystyle\\int$}\n\t\\setbox\\ncintdbox=\\hbox{\\rlap{\\hbox\n\t\tto \\wd2{\\hskip-.125em \\box2\\relax\\hfil}}\\box0\\kern.1em}\n\t\\setbox0=\\hbox{$\\vcenter{\\hrule width 4pt}$}\n\t\\setbox2=\\hbox{$\\textstyle\\int$}\n\t\\setbox\\ncinttbox=\\hbox{\\rlap{\\hbox\n\t\tto \\wd2{\\hskip-.175em \\box2\\relax\\hfil}}\\box0\\kern.1em}\n\\newcommand{\\ncint}{\\mathop{\\mathchoice{\\copy\\ncintdbox}%\n\t\t\t\t\t{\\copy\\ncinttbox}{\\copy\\ncinttbox}%\n\t\t\t\t\t{\\copy\\ncinttbox}}\\nolimits}\n\n\\newcommand{\\commentA}[1]{\\textcolor{red}{\\textsf{A: #1}}}\n\\newcommand{\\commentJ}[1]{\\textcolor{blue}{\\textsf{J: #1}}}\n\\newcommand{\\commentS}[1]{\\textcolor{green}{\\textsf{S: #1}}}\n\n\\newcommand{\\raisebox{-2pt}{$\\overset{\\cdotp\\,\\,\\cdotp}{\\frown}$}}{\\raisebox{-2pt}{$\\overset{\\cdotp\\,\\,\\cdotp}{\\frown}$}}\n\\newcommand{\\raisebox{-2pt}{$\\overset{\\cdotp\\,\\,\\cdotp}{\\smile}$}}{\\raisebox{-2pt}{$\\overset{\\cdotp\\,\\,\\cdotp}{\\smile}$}}\n\n\n\\hyphenation{geo-me-try ma-ni-fold ma-ni-folds pro-duct pro-ducts}\n\n\n\n\\begin{document}\n\n\\maketitle\n\n\\vspace{-2pc}\n\n\n\n\\begin{abstract}\nLet $B$ be a $C^{*}$-algebra, $X$ a Hilbert $C^{*}$-module over $B$ and $M,N\\subset X$ a pair of complemented submodules. We prove the $C^{*}$-module version of von Neumann's alternating projections theorem: the sequence $(P_{N}P_{M})^{n}$ is Cauchy in the $*$-strong module topology if and only if $M\\cap N$ is the complement of $\\overline{M^{\\perp}+N^{\\perp}}$. In this case, the $*$-strong limit of $(P_{M}P_{N})^{n}$ is the orthogonal projection onto $M\\cap N$. We use this result and the local-global principle to show that the cosine of the Friedrichs angle $c(M,N)$ between any pair of complemented submodules $M,N\\subset X$ is well-defined and that $c(M,N)<1$ if and only if $M\\cap N$ is complemented and $M+N$ is closed. \n\\end{abstract}\n\n{\\bf Keywords:}\n{\\small two projections, von Neumann's alternating projection theorem, Friedrichs angle, Hilbert $C^{*}$-module, local-global principle.}\n\n{\\bf MSC2020:} {\\small 46L08, 47A46}\n\n\\parskip=6pt\n\\parindent=0pt\n\\allowdisplaybreaks\n\\section*{Introduction}\n\nIn this note we offer a new and general approach to the two projection problem in Hilbert $C^{*}$-modules.\nAs an application we extend and improve upon several of the main results in the recent work of \\cite{Luo} by giving new proofs that allow for the removal of a key hypothesis. \n\nBriefly, we begin by proving the Hilbert $C^{*}$-module version of von Neumann's alternating projections theorem, which computes the projection onto $M\\cap N$ for a concordant pair of complemented submodules $M,N$ (see below). We then proceed to use this result to define the Friedrichs angle between an arbitrary pair of complemented submodules. The angle is realised as a function on the space of representations of the coefficient algebra of the module. The properties of the Friedrichs angle give necessary and sufficient conditions for the sum and intersection of two complemented submodules to again be complemented.\nWe now give a little more detail on these results. \n\nGiven two closed subspaces $M,N$ of a Hilbert space $H$ there is an orthogonal direct sum decomposition\n\\begin{equation}\n\\label{directsum}\nH=(M\\cap N)\\oplus \\overline{(M^{\\perp}+N^{\\perp})}.\n\\end{equation}\nA fundamental result of von Neumann, the \\emph{method of alternating projections}, states that the projection $P_{M\\cap N}$ onto $M\\cap N$ can be obtained as the $*$-strong limit\n\\[P_{M\\cap N}=s-\\lim_{n\\to\\infty} (P_{M}P_{N})^{n}=s-\\lim_{n\\to\\infty} (P_{N}P_{M})^{n}.\\]\nThe (cosine) of the \\emph{Friedrichs angle between $M$ and $N$} is the quantity \n$$\nc(M,N):=\\|P_{M}P_{N}-P_{M\\cap N}\\|,\n$$ \nand the subspace $M^{\\perp}+N^{\\perp}$ is closed if and only if $c(M,N)<1$.\n\nIn this paper we consider a pair $(M,N)$ of complemented submodules of a Hilbert $C^{*}$-module $X$ over a $C^{*}$-algebra $B$. It is well-known that closed submodules of Hilbert $C^*$-modules need not be orthogonally complemented. This one technical constraint necessitates the discussion of adjointable endomorphisms and regular (unbounded) operators for these modules, \\cite{FL,Lance}. \n\nThe complementability issue does not arise for finite dimensional vector spaces of course, but already in the case of finite rank, locally trivial vector bundles on compact Hausdorff base spaces we see examples \nof non-complementability of intersections. Classically the issue gives rise to the notion of a strict homomorphism of vector bundles \\cite[Section 1.3]{Atiyah}, and we relate the vector bundle situation to the complementability problem in Remarks \\ref{eg:VB-unstrict}, \\ref{discontangle} and \\ref{Atiyah} below.\n\nIn Theorem \\ref{vN} we show that the pair $(M,N)$ induces a direct sum decomposition like \\eqref{directsum} of the Hilbert $C^*$-module $X$ if and only if von Neumann's theorem on alternating projections is valid for this pair of submodules. We call such pairs \\emph{concordant} and characterise them in terms of their Hilbert space localisations in Theorem \\ref{locharm}. Our results have implications for the Hilbert module version of the two projection problem, \\cite{Luo}. The Hilbert space version first gained prominence in the work of Halmos \\cite{H}, and has since had numerous incarnations and applications: for a recent survey see \\cite{BS}.\n\nIn \\cite{Luo}, the Friedrichs angle between complemented submodules has been defined under the constraint that $M\\cap N$ is complemented. In Section \\ref{sec:angle} of this note we remove this hypothesis and extend the definition of the Friedrichs angle to arbitrary pairs of complemented submodules via the local-global principle of \\cite{Pierrot}. We interpret the Friedrichs angle as a function on the space $\\widehat{B}$ of irreducible representations of $B$ and prove that $c(M,N)=c(M^{\\perp},N^{\\perp})$. We deduce that $c(M,N)<1$ if and only if the sequence $(P_{N}P_{M})^{n}$ is Cauchy for the operator norm if and only if $M\\cap N$ is complemented and $M^{\\perp}+N^{\\perp}$ is closed.\n\n{\\bf Notation.} For a Hilbert $C^{*}$-module $X$ over a $C^{*}$-algebra $B$ we denote by $\\End^{*}_{B}(X)$ the unital $C^{*}$-algebra of adjointable operators on $X$ and by $\\mathbb{K}(X)\\subset \\End^{*}_{B}(X)$ the ideal of compact operators. The symbols $\\otimes^{\\textnormal{alg}}_{B}, \\widehat{\\otimes}_{B}$ and $\\otimes_{B}$ denote the balanced algebraic, projective and $C^{*}$-module tensor products, respectively.\n\n{\\bf Acknowledgements.} We thank Marcel de Jeu for helpful conversations and Michael Frank for valuable correspondence.\n\\section{Concordant submodules}\nLet $X$ be a Hilbert $C^{*}$-module over the $C^{*}$-algebra $B$.\nGiven two complemented submodules $M$ and $N$ of $X$, we write $P_{M},P_{N}$ respectively for the corresponding projections in $\\End^{*}_{B}(X)$. The intersection $M\\cap N$ is a closed submodule of $X$, and there is an inclusion\n\\[M^{\\perp}+N^{\\perp}\\subset (M\\cap N)^{\\perp}.\\]\nThe submodule $M^{\\perp}+N^{\\perp}$ need not be closed, but since $(M\\cap N)^{\\perp}$ is closed,\n\\[\\overline{M^{\\perp}+N^{\\perp}}\\subset (M\\cap N)^{\\perp},\\]\nas well. In case $X$ is a Hilbert space there is an equality (see (\\cite[Theorem 4.6.4]{Dbook})\n\\begin{equation}\n\\label{concordant}\n \\overline{M^{\\perp}+N^{\\perp}}= (M\\cap N)^{\\perp},\n\\end{equation} \nand thus the projections $P_{M\\cap N}$ and $P_{\\overline{M^{\\perp}+N^{\\perp}}}$ exist and satisfy $1-P_{M\\cap N}=P_{\\overline{M^{\\perp}+N^{\\perp}}}$. \n\nIn general, the projections do not exist unless the submodules are complemented.  To our knowledge, it is an open question whether the intersection of complemented submodules is again complemented. In \\cite[Section 3]{Luo} it was shown that even in case all the projections exist, \\eqref{concordant} need not not hold (see Remark \\ref{LMXcounter} below).\n\\begin{defn}\nLet $M$ and $N$ be complemented submodules of a Hilbert $C^{*}$-module $X$. The pair $(M,N)$ is \\emph{concordant} if $X=(M\\cap N)\\oplus\\overline{(M^{\\perp}+N^{\\perp})}$. If the pair $(M,N)$ is not concordant, we say it is \\emph{discordant}. \n\\end{defn}\nThe pair $(M,N)$ is concordant if their intersection $M\\cap N$ is complemented and its complement is $\\overline{M^{\\perp}+N^{\\perp}}$.\n\\begin{rmk}\\label{LMXcounter} The pair $(M,N)$ being concordant is strictly stronger than the requirement that $M\\cap N$ be complemented. In \\cite[Section 3]{Luo} it is shown that for $X=B=C([0,\\frac{\\pi}{2}], M_{2}(\\mathbb{C}))$, the submodules\n\\[M=\\textnormal{Ran } \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix},\\quad N= \\textnormal{Ran } \\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix},\\]\nsatisfy $M\\cap N =0$, which is complemented, whereas $\\overline{M^{\\perp}+N^{\\perp}}\\neq X$ so $(M,N)$ is not concordant.\n\\end{rmk}\n\n\\begin{rmk} \nNote that $(M,N)$ is \\emph{harmonious} in the sense of \\cite[Definition 4.1]{Luo} if each of the submodules\n\\[\n\\overline{M+N},\\,\\, \\overline{M+N^{\\perp}},\\,\\,\\overline{M^{\\perp}+N},\\,\\, \\overline{M^{\\perp}+N^{\\perp}} \n\\]\nis complemented. In this case the respective complements are\n\\[\nM^{\\perp}\\cap N^{\\perp},\\,\\,M^{\\perp}\\cap N,\\,\\, M+N^{\\perp},\\,\\, M\\cap N,\n\\]\nas explained in the discussion after \\cite[Definition 4.1]{Luo}. Thus $(M,N)$ is harmonious if and only if each of the pairs $(M,N)$, $(M,N^{\\perp})$, $(M^{\\perp},N)$ and $(M^{\\perp},N^{\\perp})$ is concordant.\n \\end{rmk}\n \\begin{rmk} If $M+N$ is closed, then by \\cite[Proposition 4.6]{LSX} $M^{\\perp}+N^{\\perp}$ is closed and $X=(M\\cap N)\\oplus (M^{\\perp}+N^{\\perp})$. In particular, $M+N$ is closed if and only if $M^{\\perp}+N^{\\perp}$ is closed and in this case both $(M,N)$ and $(M^{\\perp},N^{\\perp})$ are concordant (see Proposition 3.10 below).\n \\end{rmk}\n \\begin{rmk}\n \\label{rmkuniversal} \n In \\cite{RS} it was shown that the universal $C^{*}$-algebra $C^{*}(p,q)$ generated by two projections $p$ and $q$ admits the following concrete model\n\\[\nC^{*}(p,q)\\simeq\\left\\{A(t)\\in C([0,\\pi\/2], M_2(\\mathbb{C})): A(0)\\ \\mbox{and}\\ A(\\pi\/2)\\ \\mbox{diagonal}\\right\\},\n\\]\nwith the isomorphism is determined by\n\\[\np\\mapsto P:=\\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix}, \\quad q\\mapsto Q\n:=\\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix}.\n\\]\nFrom this point of view, the counterexample of \\cite[Section 3]{Luo} discussed in Remark \\ref{LMXcounter} above arises from the universal example. This shows that specific properties such as being concordant or harmonious hold in some representations of $C^{*}(p,q)$, but not in all of them.\n\\end{rmk}\n\nWe will now characterise concordant pairs by looking at their Hilbert space localisations.\n\n\nLet $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ be a representation of $B$ on the Hilbert space $H_{\\pi}$ and write $X_{\\pi}:=X\\otimes_{B}H_{\\pi}$. There is a representation\n\\begin{equation}\n\\widehat{\\pi}:\\End^{*}_{B}(X)\\to \\mathbb{B}(X_{\\pi}),\\quad T\\mapsto T\\otimes 1.\n\\label{eq:bob}\n\\end{equation}\nWrite $M_{\\pi}:=M\\otimes_{B}H_{\\pi}\\subset X\\otimes_{B}H_{\\pi}$, and similarly for $N$. Then $M_{\\pi}$ and $N_{\\pi}$ are closed subspaces of the Hilbert space $X_{\\pi}$ and we have $P_{M_{\\pi}}:=\\widehat{\\pi}(P_{M})=P_{M}\\otimes 1$, as well as  $P_{N_{\\pi}}:=\\widehat{\\pi}(P_{N})=P_{N}\\otimes 1$. Since the subspace $M_{\\pi}\\cap N_{\\pi}$ is closed, there is a projection $P_{M_{\\pi}\\cap N_{\\pi}}\\in\\mathbb{B}(X_{\\pi})$ that projects onto $M_{\\pi}\\cap N_{\\pi}$. In general, the equality $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$ need not hold, even if $M\\cap N$ is complemented. \nWe recall the following fact.\n\\begin{prop}[Local-global principle for complemented submodules \\cite{Pierrot}]\n\\label{locglob} \nLet $\\Omega\\subset X$ be a closed submodule. Then $\\Omega$ is complemented if and only if for every irreducible representation $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ there is an equality $(\\Omega_{\\pi})^{\\perp}=(\\Omega^{\\perp})_{\\pi}$.\n\\end{prop}\n\\begin{proof}\nBy \\cite[Corollaire 1.17]{Pierrot}, we have that $X=\\Omega\\oplus \\Omega^{\\perp}$ if and only if for every irreducible representation $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ there is an equality\n\\[\nX_{\\pi}=X\\otimes_{B}H_{\\pi}=(\\Omega\\oplus \\Omega^{\\perp})\\otimes_{B}H_{\\pi}=\\Omega\\otimes_{B}H_{\\pi}\\oplus \\Omega^{\\perp}\\otimes_{B}H_{\\pi}=\\Omega_{\\pi}\\oplus (\\Omega^{\\perp})_{\\pi}.\n\\]\nSince  $(\\Omega^{\\perp})_{\\pi}\\subset (\\Omega_{\\pi})^{\\perp}$, this holds if and only if $(\\Omega^{\\perp})_{\\pi}=(\\Omega_{\\pi})^{\\perp}$.\n\\end{proof}\nA weaker form of this result was proved independently, though several years later, in \\cite{KL12}. There, the local side of the equivalence involved \\emph{all} representations of the $C^{*}$-algebra $B$. The two results are equivalent because the proof of the implication $\\Rightarrow$ in Proposition \\ref{locglob} holds verbatim for an arbitrary representation of the $C^{*}$-algebra $B$, see \\cite{KL17}. We will use both instances of the result.\n\\begin{lemma}\n\\label{inclusion} \nLet $X$ be a Hilbert $C^{*}$-module over $B$, $M,N$ complemented submodules and $\\pi:B\\to\\mathbb{B}(H_{\\pi})$ a representation. Then there is an equality of closed subspaces\n\\[\n\\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}=\\big(\\,\\overline{M^{\\perp}+N^{\\perp}}\\,\\big)_{\\pi}.\n\\]\n\\end{lemma}\n\\begin{proof}\nThe inclusion of subspaces\n\\begin{align*}\n(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}\\subset \\Big(\\overline{M^{\\perp}+N^{\\perp}}\\Big)_{\\pi}\n\\end{align*}\nshows that we have an inclusion of closed linear subspaces\n\\begin{equation*}\n\\overline{(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}}\\subset \\left(\\overline{M^{\\perp}+N^{\\perp}}\\right)_{\\pi}.\\end{equation*} \n The subspace $(M^{\\perp}+N^{\\perp})\\otimes^{\\textnormal{alg}}_{B}H_{\\pi}$ is dense in $(\\,\\overline{M^{\\perp}+N^{\\perp}}\\,)_{\\pi}$ and since\n \\[\n (M^{\\perp}+N^{\\perp})\\otimes^{\\textnormal{alg}}_{B}H_{\\pi}\\subset(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}\\subset \\overline{(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}}\\subset \\left(\\overline{M^{\\perp}+N^{\\perp}}\\right)_{\\pi},\\]\n it follows that $\\overline{(M^{\\perp})_{\\pi}+(N^{\\perp})_{\\pi}}=\\left(\\overline{M^{\\perp}+N^{\\perp}}\\right)_{\\pi}$. Since $M$ and $N$ are complemented we have $(M_{\\pi})^{\\perp}=(M^{\\perp})_{\\pi}$ and $(N_{\\pi})^{\\perp}=(N^{\\perp})_{\\pi}$ and thus $\\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}=\\left(\\overline{M^{\\perp}+N^{\\perp}}\\right)_{\\pi}$.\n\\end{proof}\n\\begin{thm}\n\\label{locharm}\nLet $X$ be a Hilbert $C^{*}$-module over $B$ and $M$ and $N$ complemented submodules. Then \nthe pair $(M,N)$ is concordant if and only if for every irreducible \nrepresentation $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ there is an equality of closed subspaces $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$.\n\\end{thm}\n\\begin{proof} \nSuppose that $M$ and $N$ are concordant so that\n\\[\nX=(M\\cap N)\\oplus \\left(\\overline{M^{\\perp}+N^{\\perp}}\\right).\n\\]\nTherefore Proposition \\ref{locglob} and Lemma \\ref{inclusion} give\n \\[\n ((M\\cap N)_{\\pi})^{\\perp}=((M\\cap N)^{\\perp})_{\\pi}=(\\overline{M^{\\perp}+N^{\\perp}})_{\\pi}=\\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}.\n \\]  \n\nTaking orthogonal complements we find \n$(M\\cap N)_{\\pi}=\\Big(\\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}\\Big)^{\\perp}=M_{\\pi}\\cap N_{\\pi}$.\n\nConversely, suppose that $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$ for all irreducible \nrepresentations $\\pi$. \nBy Lemma \\ref{inclusion} and Equation \\eqref{concordant} we have \n\\[\n(\\overline{M^{\\perp}+N^{\\perp}})_{\\pi}= \\overline{(M_{\\pi})^{\\perp}+(N_{\\pi})^{\\perp}}=(M_{\\pi}\\cap N_{\\pi})^{\\perp},\n\\] \n\nand we deduce that\n\\begin{align*}\n(M\\cap N)_{\\pi}\\oplus (\\overline{M^{\\perp}+N^{\\perp}})_{\\pi}&= (M_{\\pi}\\cap N_{\\pi})\\oplus (M_{\\pi}\\cap N_{\\pi})^{\\perp}=X_{\\pi}.\n\\end{align*}\n\nBy Proposition \\ref{locglob} we conclude that $X=(M\\cap N)\\oplus \\overline{M^{\\perp}+N^{\\perp}}.$\n\\end{proof}\nIn line with the local-global principle, Proposition \\ref{locglob}, we obtain the same result when we consider all representations of the base algebra $B$.\n\\begin{corl}\n\\label{locharmcor}\nLet $X$ Hilbert $C^{*}$-module over $B$ and $M$ and $N$ complemented submodules. Then $(M,N)$ is concordant if and only if for every representation $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ there is an equality of closed subspaces $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$.\n\\end{corl}\n\\begin{proof} \nThe proof of $\\Rightarrow$ in Theorem \\ref{locharm} shows that $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$ for every representation whenever $(M,N)$ is concordant.\n\\end{proof}\n\\begin{rmk}\n\\label{eg:VB-unstrict}\nConsider $B=C([0,\\frac{\\pi}{2}])$, $X=C([0,\\frac{\\pi}{2}], \\mathbb{C}^{2})$ and consider the submodules\n\\[\nM=\\textnormal{Ran }\\!\\!\\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix},\\quad N\n= \\textnormal{Ran }\\!\\!\\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix}.\n\\] We have $M\\cap N=0$ and for the irreducible representations given by $t\\in[0,\\pi\/2]$ we have\n\\[\nM_t\\cap N_t=\\begin{cases} 0 &  t=0\\\\ \\C & t\\neq 0,\\end{cases}\n\\]\nso $(M,N)$ is discordant by Theorem \\ref{locharm}.\n\\end{rmk}\n\n\n\\section{Von Neumann's theorem of alternating projections}\nLet $P,Q\\in \\End^{*}_{B}(X)$ be projections\n\\[\nP^{*}=P^{2}=P,\\quad Q^{*}=Q^{2}=Q.\n\\]\nThe submodules $\\textnormal{Ran } P$ and $\\textnormal{Ran } Q$ are complemented in $X$, and every complemented submodule is the range of an adjointable projection. As noted before, it is an open question whether the intersection $\\Omega:=\\textnormal{Ran } P\\cap \\textnormal{Ran } Q$, which is a closed submodule, is complemented. In case $B=\\mathbb{C}$ and $X$ is a Hilbert space this is true and thus there is a projection $P_{\\Omega}$ with $\\textnormal{Ran } P_{\\Omega}=\\Omega$. For $n\\geq 0$, write\n\\[\n(P,Q)_{n}:=\\cdots PQPQ,\\quad \\textnormal{the product of exactly } n \\,\\,\\textnormal{alternating factors ending in } Q.\n\\]\nVon Neumann proved the following well-known theorem.\n\\begin{thm}[{\\cite[Lemma 22]{vN}}]\n\\label{vN} \nLet $H$ be a Hilbert space, $M,N\\subset H$ closed subspaces and $\\Omega:=M\\cap N$. Let $P=P_{M}$ and $Q=P_{N}$ be the orthogonal projections onto $M$ and $N$ respectively. The orthogonal projection $P_{\\Omega}$ onto $\\Omega$ can be obtained as the strong limit of any of the sequences\n\\begin{equation}\n\\label{alternatingseq}\n(PQ)^{n},\\quad (QP)^{n}, \\quad (P,Q)_{n},\\quad (Q,P)_{n},\n\\end{equation}\nor any of their subsequences. \n\\end{thm}\nIn a Hilbert $C^{*}$-module $X$, the analogue of the $*$-strong topology is  defined by the seminorms\n\\[\n\\|T\\|_{x}:=\\max\\{\\|Tx\\|,\\|T^{*}x\\|\\},\\quad x\\in X,\n\\]\nand we refer to this topology as the $*$-\\emph{strong module topology}.\nOn bounded sets the $*$-strong module topology coincides with the \n\\emph{strict topology} on $\\End_{B}^{*}(X)$ relative to the ideal $\\mathbb{K}(X)$, \\cite[Proposition 5.5.9]{Troitsky}.\nThe following fact is well-known.\n\\begin{lemma}\\label{complete}\nThe $*$-strong module topology is complete on bounded sets.\n\\end{lemma}\n\\begin{proof} Let $T_{n}\\in\\End^{*}_{B}(X)$ be a sequence that is Cauchy for the seminorms $\\|\\cdot \\|_{x}$, $x\\in X$. By the Uniform Boundedness Principle, the operators \\[Tx:=\\lim_{n\\to\\infty} T_{n}x, \\quad \\textnormal{and} \\quad T^{*}x:=\\lim_{n\\to\\infty} T^{*}_{n}x,\\] are well-defined, bounded and mutually adjoint.\n\\end{proof}\n\\begin{lemma}\\label{reductionlemma} Let $P,Q\\in\\End^{*}_{B}(X)$ be projections. Then $(PQ)^{n}$ and $(QP)^{n}$ are $*$-strongly Cauchy if and only if $(PQP)^{n}$ and $(QPQ)^{n}$ are $*$-strongly Cauchy  if and only if $(P,Q)_{n}$ and $(Q,P)_{n}$ (as defined in \\eqref{alternatingseq}) are $*$-strongly Cauchy. The same statement holds for the norm topology.\n\\end{lemma}\n\\begin{proof} \nSince\n\\[\n(P,Q)_{n}=\\left\\{\\begin{matrix}  (PQ)^{\\frac{n}{2}} & n\\quad \\textnormal{even}\\\\ (QPQ)^{\\frac{n-1}{2}} & n\\quad  \\textnormal{odd},\\end{matrix}\\right.\\quad (Q,P)_{n}=\\left\\{\\begin{matrix}  (QP)^{\\frac{n}{2}} & n\\quad \\textnormal{even}\\\\ (PQP)^{\\frac{n-1}{2}} & n\\quad  \\textnormal{odd},\\end{matrix}\\right. \n\\]\nit suffices to prove that $(PQ)^{n}$ and $(QP)^{n}$ are $*$-strongly Cauchy if and only $(PQP)^{n}$ and $(QPQ)^{n}$ are $*$-strongly Cauchy. The same holds for the norm topology.\n\nAny projection $P$ satisfies $\\langle Px,Px\\rangle\\leq \\langle x,x\\rangle$ and $Q(PQ)^{n}=(QPQ)^{n}$ so that\n\\begin{align}\n\\label{sandwich}\n\\langle (PQ)^{n}x,(PQ)^{n}x\\rangle&=\\langle (Q(PQ)^{n}+(1-Q)(PQ)^{n})x,(Q(PQ)^{n}+(1-Q)(PQ)^{n})x\\rangle\\nonumber\\\\\n&\\geq \\langle (QPQ)^{n}x,(QPQ)^{n}x\\rangle\\nonumber\\\\\n&=\\langle (P(QPQ)^{n}+(1-P)(QPQ)^{n})x,(P(QPQ)^{n}+(1-P)(QPQ)^{n})x\\rangle\\nonumber\\\\\n&\\geq \\langle (PQ)^{n+1}x,(PQ)^{n+1}x\\rangle.\n\\end{align}\nNow for $m>n$ we have \n\\[\n(PQ)^{n}-(PQ)^{m}=(PQ)^{n}(1-(PQ)^{m-n}),\\quad  ((QPQ)^{n}-(QPQ)^{m})=(QPQ)^{n}(1-(QPQ)^{m-n}),\n\\]\nwhich, together with \\eqref{sandwich} gives\n\\begin{align*}\n\\langle ((PQ)^{n}-(PQ)^{m})x ,((PQ)^{n}&-(PQ)^{m})x\\rangle =\n\\langle (PQ)^{n}(1-(PQ)^{m-n})x,(PQ)^{n}(1-(PQ)^{m-n})x\\rangle \\\\ &\\geq \\langle (QPQ)^{n}(1-(PQ)^{m-n})x,(QPQ)^{n}(1-(PQ)^{m-n})x\\rangle\\\\\n&= \\langle ((QPQ)^{n}-(QPQ)^{m})x,((QPQ)^{n}-(QPQ)^{m})x\\rangle\\\\\n&= \\langle (QPQ)^{n}(1-(QPQ)^{m-n})x,(QPQ)^{n}(1-(QPQ)^{m-n})x\\rangle\\\\\n&\\geq \\langle (PQ)^{n+1}(1-(QPQ)^{m-n})x,(PQ)^{n+1}(1-(QPQ)^{m-n})x\\rangle\\\\\n&\\geq \\langle( (PQ)^{n+1}-(PQ)^{m+1})x,((PQ)^{n+1}-(PQ)^{m+1})x\\rangle.\n\\end{align*}\nThis proves that $(PQ)^{n}$ is pointwise Cauchy if and only if $(QPQ)^{n}$ is pointwise Cauchy. Thus $(PQ)^{n}$ and $(QP)^{n}$ are both $*$-strongly Cauchy if and only if $(PQP)^{n}$ and $(QPQ)^{n}$ are both $*$-strongly Cauchy. The statements for the norm topology follow from the same inequalities. This completes the proof.\n\\end{proof}\n\\begin{prop} \n\\label{Cauchy} \nSuppose that $(PQ)^{n}$ is $*$-strongly Cauchy. Then so are $(QP)^{n}$, $(PQP)^{n}$, $(QPQ)^{n}$, $(Q,P)_{n}$ and $(P,Q)_{n}$. The $*$-strong limits of each of these sequences is a projection $P_{\\Omega}$ with range $\\Omega:=\\textnormal{Ran } P \\cap \\textnormal{Ran } Q$. In particular $\\Omega$ is complemented.\n\\end{prop}\n\\begin{proof} \nSince $((PQ)^{n})^{*}=(QP)^{n}$, the first statement follows from Lemma \\ref{reductionlemma}. \nWe will prove that $s-\\lim_{n\\to\\infty} (PQP)^n=s-\\lim_{n\\to \\infty} (QPQ)^{n}$ and that this operator is a projection $P_{\\Omega}$ with range $\\Omega$. It then follows that $\\Omega$ is complemented and that \n\\[\nP_{\\Omega}=s-\\lim_{n\\to\\infty}(P,Q)_{n}=s-\\lim_{n\\to\\infty}(Q,P)_{n},\n\\]\nsince $(PQP)^{n}$ is a subsequence of $(Q,P)_{n}$ and $(QPQ)^{n}$ is a subsequence of $(P,Q)_{n}$. Then $(PQ)^{n}$ and $(QP)^{n}$ are subsequences of $(P,Q)_{n}$ and $(Q,P)_{n}$, respectively it follows that\n\\[\nP_{\\Omega}=s-\\lim_{n\\to\\infty}(PQ)^{n}=s-\\lim_{n\\to\\infty}(QP)^{n},\n\\]\nas well.\n\nBy Lemma \\ref{complete} the $*$-strong limit $\\tilde{P}:=\\lim (PQP)^{n}$ exists, is self-adjoint and $\\|\\tilde{P}\\|\\leq 1$. To prove that $\\tilde{P}$ is a projection let $x\\in X$ and $\\varepsilon >0$. Choose $N$ such that for all $k\\geq N$ we have\n\\[\n\\|\\tilde{P}x-(PQP)^{k}x\\|<\\varepsilon.\n\\]\nNow consider\n\\begin{align*}\n\\|\\tilde{P}^{2}x-\\tilde{P}x\\|\n&=\\|\\tilde{P}(PQP)^{k}x-\\tilde{P}x\\|+ \\|\\tilde{P}(\\tilde{P}-(PQP)^{k})x\\|\\\\\n&\\leq \\|\\tilde{P}(PQP)^{k}x-\\tilde{P}x\\|+ \\|(\\tilde{P}-(PQP)^{k})x\\|\\\\\n&< \\|\\tilde{P}(PQP)^{k}x-\\tilde{P}x\\|+ \\varepsilon\\\\\n&=\\lim_{n\\to\\infty}\\|(PQP)^{n+k}x-\\tilde{P}x\\|+\\varepsilon=\\varepsilon,\n\\end{align*}\nand as $\\varepsilon$ was arbitrary, it follows that $\\tilde{P}^{2}x=\\tilde{P}x$.\n\nTo prove that $\\textnormal{Ran }\\tilde{P}=\\Omega$, first observe that if $x\\in \\Omega$ then $$x=Px=Qx=PQPx,$$ so $\\tilde{P}x=x$ and $\\Omega\\subset \\textnormal{Ran } \\tilde{P}$. \n\nFor the reverse inclusion we will show that $\\tilde{P}=P\\tilde{P}=Q\\tilde{P}$. The equalities \n\\[\nP\\tilde{P}x=\\tilde{P}x,\\quad \\textnormal{and}\\quad PQ\\tilde{P}x=\\tilde{P}x,\n\\] \nhold by construction. Now for any $x\\in X$ we have\n\\[\n\\langle Px, Px\\rangle\\leq \\langle x,x\\rangle,\\quad\\langle Qx, Qx\\rangle\\leq \\langle x,x\\rangle,\n\\]\n from which we deduce that\n \\[\n \\langle\\tilde{P}x,\\tilde{P}x\\rangle=\\langle PQ\\tilde{P}x,PQ\\tilde{P}x\\rangle\\leq \\langle Q\\tilde{P}x,Q\\tilde{P}x\\rangle\\leq  \\langle\\tilde{P}x,\\tilde{P}x\\rangle.\n \\]\nTherefore $\\langle Q\\tilde{P}x,Q\\tilde{P}x\\rangle= \\langle\\tilde{P}x,\\tilde{P}x\\rangle$ and $\\langle (1-Q)\\tilde{P}x,(1-Q)\\tilde{P}x\\rangle=0$. It follows that $(1-Q)\\tilde{P}x=0$ so $Q\\tilde{P}x=\\tilde{P}x$. This shows that $Q\\tilde{P}=\\tilde{P}$ and thus $\\textnormal{Ran } \\tilde{P}\\subset\\Omega$. Therefore $\\Omega$ is complemented and $P_{\\Omega}=\\tilde{P}=s-\\lim (PQP)^{n}$ in the $*$-strong module topology. By exhanging the r\\^oles of $P$ and $Q$, we find that $P_{\\Omega}=s-\\lim (QPQ)^{n}$ as well.\n\\end{proof}\nIn order to address the appropriate converse to Proposition \\ref{Cauchy}, we need a description of the Banach space dual $X^{*}:=\\mathbb{B}(X,\\mathbb{C})$ of bounded linear functionals on a Hilbert $C^{*}$-module $X$. To this end we first recall the \\emph{dual or conjugate $C^{*}$-module}. \n\nThe space of compact operators $\\mathbb{K}(X,B)$ from $X$ to $B$ is a left $B$-module via $(b\\cdot K)(x):=bK(x)$ and carries a natural left $B\\simeq \\mathbb{K}(B,B)$ valued inner product $\\langle K, L\\rangle:=KL^{*}$. The \\emph{conjugate module} $\\overline{X}$ is defined to be the set $X$ with the conjugate $\\mathbb{C}$-vector space structure, and we write elements of $\\overline{X}$ as $\\overline{x}$ with $x\\in X$. The left $B$-module structure and inner product\n\\[b\\cdot \\overline{x}:=\\overline{xb^{*}},\\quad \\langle \\overline{x},\\overline{y}\\rangle:=\\langle x,y\\rangle.\\]\nThese left Hilbert $C^{*}$-modules over $B$ are isomorphic, by the following well-known theorem \\cite[page 13]{Lance}.\n\\begin{prop}[Riesz-Fr\\'echet theorem for Hilbert $C^{*}$-modules]\nThe map \n\\[\nT:\\overline{X}\\rightarrow \\mathbb{K}(X,B),\\quad \\overline{x}\\mapsto T_{x},\\quad T_{x}(y):=\\langle x,y\\rangle, \\quad x,y\\in X, \n\\]\nis a unitary isomorphism of left Hilbert $C^{*}$-modules over $B$.\n\\end{prop}\nThe dual Banach space of the $C^{*}$-algebra $B$, $B^{*}:=\\mathbb{B}(B,\\mathbb{C})$, is a right Banach $B$-module via\n\\[\n(\\varphi\\cdot b) (a):=\\varphi(ba),\\quad a,b\\in B.\n\\]\nLastly, for a right Banach $B$-module $V$ and a left Banach $B$-module $W$, we denote by $V\\widehat{\\otimes}_{B}W$ the balanced Banach space projective tensor product of $V$ and $W$. We are now ready to recall a result of Schweizer, \\cite[Proposition 3.1]{Schweizer},  giving a complete description of the dual Banach space $X^{*}$ of the module $X$.\n\\begin{prop}\\label{Schweizer} \nLet $X$ be a Hilbert $C^{*}$-module, $\\overline{X}:=\\mathbb{K}(X,B)$ the conjugate module and $X^{*}=\\mathbb{B}(X,\\mathbb{C})$ the dual Banach space of $X$. The  map $\\psi:B^{*}{\\otimes}_{B}^{\\textnormal{alg}}\\overline{X}\\to X^{*}$ given by\n\\[\n\\psi(\\phi\\otimes\\overline{y})(x):=\\phi(\\langle y,x\\rangle), \\quad \\phi\\in B^{*},\\quad x,y\\in X,\n\\]\nextends to an isometric isomorphism $B^{*}\\widehat{\\otimes}_{B}\\overline{X}\\to X^{*}$ of Banach spaces. \n\\end{prop}\nFor a Banach space $W$, the \\emph{weak topology} on $W$ is the locally convex topology defined by the seminorms $\\|w\\|_{\\varphi}:=\\|\\varphi(w)\\|$. In general the weak topology is \\emph{not} complete, that is, weak Cauchy sequences need not have a weak limit in $X$. However, we do have the following fundamental result for weakly convergent sequences. \n\\begin{thm}[{\\cite[Chap II, Section 38]{RSz}}]\n\\label{Mazur}\nLet $W$ be a Banach space and $C\\subset W$ a convex set. Then the weak closure of $C$ coincides with the norm closure of $C$. In particular, if $w_j\\to w$ in the weak topology, then there exists a sequence of convex combinations $y_j:=\\sum_{k=j}^{n_{j}} t_j w_j$ such that $\\|y_j-w\\|\\to 0$.\n\\end{thm}\nIn the sequel we will freely use the following computational tool.\n\\begin{lemma}\n\\label{powers} \nLet $P,Q\\in\\End^{*}_{B}(X)$ be projections such that $\\Omega:=\\textnormal{Ran } P\\cap \\textnormal{Ran } Q$ is complemented. Then for all $k\\geq 1$ we have\n\\begin{align*}\n(PQ-P_{\\Omega})^{k}&=(PQP)^{k}-P_{\\Omega},\\quad (QP-P_{\\Omega})^{k}=(QPQ)^{k}-P_{\\Omega},\\\\\n(PQP-P_{\\Omega})^{k}&=(PQP)^{k}-P_{\\Omega},\\quad (QPQ-P_{\\Omega})^{k}=(QPQ)^{k}-P_{\\Omega}.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nThe statement holds for $k=1$. Since $P_{\\Omega}=P_{\\Omega}P=PP_{\\Omega}=P_{\\Omega} Q=QP_{\\Omega}$ we have \n\\[\n(PQ)^{k+1}-P_{\\Omega}=(PQ-P_{\\Omega})((PQ)^{k}-P_{\\Omega}),\n\\quad (QP)^{k+1}-P_{\\Omega}=(QP-P_{\\Omega})((QP)^{k}-P_{\\Omega}),\n\\]\nand\n\\[(PQP)^{k+1}-P_{\\Omega}=P((QP)^{k+1}-P_{\\Omega}),\\quad (QPQ)^{k+1}-P_{\\Omega}=Q((PQ)^{k+1}-P_{\\Omega}),\\]\nso the result follows by induction on $k$.\n\\end{proof}\nWe are now ready to prove our main theorem.\n\n\\begin{thm}\n\\label{complementedalternating} \nLet $M,N$ be complemented submodules of a Hilbert $C^{*}$-module $X$. Then $(M,N)$ is a concordant pair if and only if the sequence $(P_{N}P_{M})^{n}$ is Cauchy in the $*$-strong module topology on $\\End^{*}_{B}(X)$.\n\\end{thm}\n\\begin{proof} \nWe write $P=P_{M}$, $Q=P_{N}$ and $\\Omega:=M\\cap N$.\n\n\n$\\Leftarrow$ In Proposition \\ref{Cauchy} it was proved that $\\Omega$ is complemented and $\\lim (PQ)^{n}x=P_{\\Omega}x$. Now if $\\pi:B\\to\\mathbb{B}(H_{\\pi})$ is an irreducible representation then\n\\begin{align*}\nP_{M_{\\pi}\\cap N_{\\pi}}(x\\otimes h)&=\\lim_{n\\to\\infty}(P_{M_{\\pi}}P_{N_{\\pi}})^{n}(x\\otimes h)=\\lim_{n\\to\\infty}\\widehat{\\pi}(P_{M}P_{N})^{n}(x\\otimes h)\\\\\n&=\\lim_{n\\to\\infty}((PQ)^{n}x)\\otimes h=P_{\\Omega}x\\otimes h=\\widehat{\\pi}(P_{\\Omega})(x\\otimes h),\n\\end{align*}\nso $P_{M_{\\pi}\\cap N_{\\pi}}=\\widehat{\\pi}(P_{\\Omega})$ and thus $\\Omega_{\\pi}=M_{\\pi}\\cap N_{\\pi}$, so $(M,N)$ is concordant by Theorem \\ref{locharm}.\n\nFor the converse, assume that $(M,N)$ is concordant and write $P_{\\Omega}$ for the projection onto $\\Omega$. By Lemma \\ref{reductionlemma} it suffices to prove that $(PQP)^{n}x\\to P_{\\Omega} x$ and $(QPQ)^{n}x\\to P_{\\Omega} x$ for all $x\\in X$. \n\nWe first prove that $(PQP)^{n}x$ converges to $P_{\\Omega}x$ in the weak topology on $X$. To this end observe that since $\\|(PQP)^{n}\\|\\leq \\|PQP\\|^{n}\\leq 1$ the sequence $(PQP)^{n}x$ is bounded in norm. Therefore, by Proposition \\ref{Schweizer}  it suffices to show that $(\\phi\\otimes \\overline{y}) ( (PQP)^{n}x)\\to (\\phi\\otimes \\overline{y})(P_{\\Omega}x)$ for all $\\phi\\in B^{*}$ and $y\\in X$, as such functionals generate the weak topology. Since every functional on the $C^{*}$-algebra $B$ is a linear combination of four states (see \\cite{T}), we may restrict ourselves to states $\\sigma\\in B^{*}$. In the universal representation $H_{u}$ of $B$, every state $\\sigma$ arises as a vector state associated to a unit vector $h_{\\sigma}\\in H_{u}$. Denote by $\\pi_{\\sigma}$ the GNS-representation associated to the state $\\sigma$. Then by Theorem \\ref{vN} we find\n\\begin{align*}\n(\\sigma\\otimes \\overline{y})((PQP)^{n}x)&=\\sigma(\\langle y, (PQP^{n})x\\rangle )=\\langle h_{\\sigma}, \\langle y, (PQP)^{n}x\\rangle h_{\\sigma}\\rangle\\\\\n&=\\langle y\\otimes h_{\\sigma}, (PQP)^{n}x\\otimes h_{\\sigma}\\rangle \\to \\langle y\\otimes h_{\\sigma}, P_{\\Omega_{\\sigma}} (x\\otimes h_{\\sigma})\\rangle.\n\\end{align*}\nBy Corollary \\ref{locharmcor}, $M_{\\pi_\\sigma}\\cap N_{\\pi_\\sigma}=\\Omega_{\\pi_\\sigma}$ so $P_{\\Omega_{\\sigma}}=P_{\\Omega}\\otimes 1=\\widehat{\\pi}_{\\sigma}(P_{\\Omega})$,\nand\n$(PQP)^{n}x\\otimes h_{\\sigma}\\to P_{\\Omega} x\\otimes h_{\\sigma}$ in the Hilbert space $X\\otimes_{B} H_{u}$. Therefore $(PQP)^{n}x\\to P_{\\Omega}x$ weakly in $X$. \n\nBy Theorem \\ref{Mazur}, there is a sequence of convex combinations $y_{k}=\\sum_{i=k}^{n_{k}} t_{i}(PQP)^{i}x$ such that $y_k\\to P_{\\Omega}x$ in norm in $X$. Since for all $n$ we have\n\\[\nP_{\\Omega}(PQP)^{n}=(PQP)^{n}P_{\\Omega}=P_{\\Omega},\\quad (PQP)^{m}\\leq (PQP)^{n},\\quad m\\geq n,\n\\]\nwe can estimate\n\\begin{align*}\n\\langle (y_{k}-P_{\\Omega})x,  (y_{k}-P_{\\Omega})x\\rangle&=\n\\left\\langle \\left(\\sum_{i=k}^{n_{k}} t_{i}(PQP)^{i}-P_{\\Omega}\\right)x , \\left(\\sum_{i=k}^{n_{k}} t_{i}(PQP)^{i}-P_{\\Omega}\\right)x\\right\\rangle\\\\ \n&=\\left\\langle \\left(\\sum_{i=k}^{n_{k}} t_{i}(PQP)^{i}-P_{\\Omega}\\right)^{2}x , x\\right\\rangle\\\\\n&=\\left\\langle \\left(\\sum_{i,j=k}^{n_{k}}t_{i}t_{j}(PQP)^{i+j}-P_{\\Omega}\\right)x, x\\right\\rangle \\\\\n&\\geq \\left\\langle \\left(\\sum_{i,j=k}^{n_k}t_{i}t_{j}(PQP)^{2n_{k}}-P_{\\Omega}\\right)x, x\\right\\rangle \\\\\n&=\\langle ((PQP)^{2n_k}-P_{\\Omega})x, x\\rangle \\\\%-\\langle P_{\\Omega}x, P_{\\Omega}\\rangle\n&=\\langle ((PQP)^{n_k}-P_{\\Omega})x, ((PQP)^{n_k}-P_{\\Omega})x\\rangle,\n\\end{align*}\nwhere the last step follows using Lemma \\ref{powers}. \nTherefore it follows that the subsequence $(PQP)^{n_{k}}$ is such that for all $x\\in X$  we have norm convergence $(PQP)^{n_{k}}x\\to P_{\\Omega}x$ as $k\\to \\infty$. Since for any $m\\geq n$ we have \n\\[\n\\langle ((PQP)^{n}-P_{\\Omega})x, ((PQP)^{n}-P_{\\Omega})x\\rangle \\geq\\langle( (PQP)^{m}-P_{\\Omega})x,( (PQP)^{m}-P_{\\Omega})x\\rangle,\n\\]\nwe find that \n\\begin{align*}\n\\|((PQP)^{n}-P_{\\Omega})x\\|\\geq \\|((PQP)^{m}-P_{\\Omega})x\\|.\n\\end{align*}\nThus it follows that $\\lim_{n\\to\\infty} \\|((PQP)^{n}-P_{\\Omega})x\\|\\to 0$. By swapping the r\\^oles of $P$ and $Q$ we find that $\\lim_{n\\to\\infty} \\|((QPQ)^{n}-P_{\\Omega})x\\|\\to 0$ as well. This completes the proof.\n\\end{proof}\n\n\\section{Angle, sum and intersection}\n\\label{sec:angle}\nWe now consider the applications of our main result to various problems concerning pairs of complemented submodules of Hilbert $C^*$-modules.\n\\subsection{The Friedrichs angle between complemented submodules}\nIn \\cite{Luo}, the following definition for the Friedrichs angle between complemented submodules was given, which we now recall. Let $M,N\\subset X$  be complemented submodules such that $M\\cap N$ is complemented and write $P_{M},P_{N}$ and $P_{M\\cap N}$ respectively for the corresponding projections. The quantity\n\\begin{equation}\n\\label{moduleangle}\nc(M,N):=\\|P_{M}P_{N}(1-P_{M\\cap N})\\|=\\|P_{M}P_{N}-P_{M\\cap N}\\|,\n\\end{equation}\nis called the (cosine of the) \\emph{Friedrichs angle between $M$ and $N$}. \n\nFor the above definition, the existence of the projection $P_{\\Omega}$ seems necessary. This is undesirable and ideally the angle should be an invariant associated to any pair $(M,N)$ of complemented submodules. We propose the following generalisation, based on Hilbert space localisation.  \n\\begin{defn} \nLet $M,N\\subset X$  be complemented submodules. Let $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ be a representation of $B$ on $H_{\\pi}$. The quantity\n\\begin{equation}\\label{localangle}\nc_{\\pi}(M,N):=c(M_{\\pi},N_{\\pi})=\\|P_{M_{\\pi}}P_{N_{\\pi}}(1-P_{M_{\\pi}\\cap N_{\\pi}})\\|=\\|P_{M_{\\pi}}P_{N_{\\pi}}-P_{M_{\\pi}\\cap N_{\\pi}}\\|,\n\\end{equation}\nis called the (cosine of the) \\emph{local Friedrichs angle between $M$ and $N$ at $\\pi$}. \n\\end{defn}\n\n\\begin{prop}\n\\label{independent} Suppose that $\\pi:B\\to \\mathbb{B}(H_{\\pi})$ is faithful. Then\n\\begin{enumerate}\n\\item If $(M,N)$ is concordant, then $c_{\\pi}(M,N)=c(M,N)$;\n\\item If $(M,N)$ is discordant, then $c_{\\pi}(M,N)=1$.\n\\end{enumerate}\nIn particular the (cosine of the) local Friedrich angle $c_{\\pi}(M,N)$ is independent of the choice of faithful representation $\\pi$.\n\\end{prop}\n\\begin{proof} Suppose $(M,N)$ is concordant, so that by Corollary \\ref{locharmcor}, $M_{\\pi}\\cap N_{\\pi}=(M\\cap N)_{\\pi}$ and $P_{M_{\\pi}\\cap N_{\\pi}}=\\widehat{\\pi}(P_{M\\cap N})$. \n Since $\\pi$ is faithful, the representation $\\End^{*}_{B}(X)\\to \\mathbb{B}(X_{\\pi})$ is faithful and hence isometric. Therefore \n\\[c_{\\pi}(M,N)=\\|\\widehat{\\pi}(P_{N}P_{M}-P_{M\\cap N})\\|=\\|P_{N}P_{M}-P_{M\\cap N}\\|=c(M,N),\\]\nwhich proves {\\em1}.\n\nClearly $0\\leq c_{\\pi}(M,N)\\leq 1$, so suppose that $c_{\\pi}(M,N)<1$ and write $P=P_{M}$ and $Q=P_{N}$. We will show that the sequence $(PQ)^{n}$ is Cauchy for the norm topology. Then by Theorem \\ref{complementedalternating}, $(M,N)$ is concordant, which proves {\\em2}. So for $m\\geq n$ recall the representation $\\widehat{\\pi}$ from Equation \\eqref{eq:bob} and consider\n\\begin{align*}\n\\|(PQ)^{n}-(PQ)^{m}\\|&=\\|\\widehat{\\pi}((PQ)^{n}-(PQ)^{m})\\|\\\\\n&\\leq \\|\\widehat{\\pi}(PQ)^{n}-P_{M_{\\pi}\\cap N_{\\pi}}\\|+\\|\\widehat{\\pi}(PQ)^{m}-P_{M_{\\pi}\\cap N_{\\pi}}\\|\\\\\n&=\\|(P_{M_\\pi}P_{N_\\pi})^{n}-P_{M_{\\pi}\\cap N_{\\pi}}\\|+\\|(P_{M_\\pi}P_{N_\\pi})^{m}-P_{M_{\\pi}\\cap N_{\\pi}}\\|\\\\\n&=\\|(P_{M_\\pi}P_{N_\\pi}-P_{M_{\\pi}\\cap N_{\\pi}})^{n}\\|+\\|(P_{M_\\pi}P_{N_\\pi}-P_{M_{\\pi}\\cap N_{\\pi}})^{m}\\|\\quad(\\textnormal{by Lemma \\ref{powers}})\\\\\n&\\leq \\|P_{M_\\pi}P_{N_\\pi}-P_{M_{\\pi}\\cap N_{\\pi}}\\|^{n}+\\|P_{M_\\pi}P_{N_\\pi}-P_{M_{\\pi}\\cap N_{\\pi}}\\|^{m}\\\\\n&=c_{\\pi}(M,N)^{n}+c_{\\pi}(M,N)^{m}\\to 0,\n\\end{align*}\nsince $c_{\\pi}(M,N)<1$. This completes the proof.\n\\end{proof}\nWe denote by $\\widehat{B}$ the space of unitary equivalence classes of irreducible representations of the $C^{*}$-algebra $B$, by $\\mathcal{P}(B)$ the pure state space of $B$ and by $\\pi_{\\sigma}$ the $GNS$-representation associated to the state $\\sigma$. We can view the local Friedrichs angles as a function $\\widehat{B}\\to [0,1]$ and via the composition $\\mathcal{P}(B)\\to \\widehat{B}$, also as a function on $\\mathcal{P}(B)$.\n\\begin{corl} \\label{localglobalangle} The Friedrichs angle \\eqref{moduleangle} and the local Friedrichs angles \\eqref{localangle} are related by $c(M,N)=\\sup_{\\pi\\in\\widehat{B}}c_{\\pi}(M,N)=\\sup_{\\sigma\\in\\mathcal{P}(B)}c_{\\pi_{\\sigma}}(M,N)$.\n\\end{corl}\n\\begin{proof}\nThe representations $\\widehat{H}=\\bigoplus_{\\pi\\in\\widehat{B}}H_{\\pi}$ and $H_{\\mathcal{P}}:=\\bigoplus_{\\sigma\\in\\mathcal{P}(B)}H_{\\pi_{\\sigma}}$ are faithful.\n\\end{proof}\nIn view of Proposition \\ref{independent} and Corollary \\ref{localglobalangle}, we \\emph{define} the Friedrichs angle between an arbitrary pair of complemented submodules to be  $c(M,N):=c_{\\pi}(M,N)$, with $\\pi$ faithful. \nIt was shown in \\cite{Luo} that \n\\begin{equation}\n\\label{Deutschangle}\nc(M,N)=c(M^{\\perp},N^{\\perp}),\n\\end{equation}\nprovided that $M\\cap N$ and $M^{\\perp}\\cap N^{\\perp}$ are complemented. In particular, the equality holds for any pair of subspaces of a Hilbert space, \\cite[Theorem 2.16]{D}. \nWe will now show that the equality \\eqref{Deutschangle} holds for an arbitrary pair of complemented submodules. This gives an extension, and a different proof, of \\cite[Theorem 5.12]{Luo}.\n\\begin{thm}\n\\label{anglesymmetry}\nLet $X$ be a Hilbert $C^{*}$-module and $M,N\\subset X$ complemented submodules. Then $c(M,N)$ is well-defined and  $c(M,N)=c(M^{\\perp},N^{\\perp})$.\n\\end{thm}\n\\begin{proof}\nFor any representation $\\pi:B\\to\\mathbb{B}(H_{\\pi})$ there is an equality of submodules $(M_{\\pi})^{\\perp}=(M^{\\perp})_{\\pi}$ whenever $M$ is complemented. Moreover Equation \\eqref{Deutschangle} holds for the subspaces $M_{\\pi},N_{\\pi}$ of the Hilbert space $X_{\\pi}$. Thus by Proposition \\ref{independent} we have\n\\begin{align*}\nc(M,N)&=c_{\\pi}(M,N)=c(M_{\\pi},N_{\\pi})\\\\ &=c((M_{\\pi})^{\\perp},(N_{\\pi})^{\\perp})=c((M^{\\perp})_{\\pi},(N^{\\perp})_{\\pi})\\\\&=c_{\\pi}(M^{\\perp},N^{\\perp})=c(M^{\\perp},N^{\\perp}),\\end{align*}\nas claimed.\n\\end{proof}\nNow we further analyse the properties of the local Friedrichs angles as a function on $\\widehat{B}$.\n\\begin{prop}\n\\label{continuity}\nSuppose $(M,N)$ is concordant. Then the map\n\\[\\widehat{B} \\to [0,1],\\quad\\pi\\mapsto c_{\\pi}(M,N),\\]\nis lower semi-continuous. If $X$ is full and $\\widehat{B}$ is Hausdorff, $\\pi\\mapsto c_{\\pi}(M,N)$ is continuous.\n\\end{prop}\n\\begin{proof} Let $J:=\\left\\langle B,B\\right\\rangle$ and $\\widehat{B}\\to\\widehat{J}$ the restriction map, which is continuous. The $C^{*}$-algebras $J$ and $\\mathbb{K}(X)$ are Morita equivalent, so by the Rieffel correspondence \\cite{RieffelInd} the map $\\pi\\mapsto\\widehat{\\pi}$ is a homeomorphism $\\widehat{J} \\to \\widehat{\\mathbb{K}(X)}$. Since $\\mathbb{K}(X)\\subset \\End^{*}_{B}(X)$ is an essential ideal, there is a continuous inclusion $\\widehat{\\mathbb{K}(X)}\\to \\widehat{\\End^{*}_{B}(X)}$, see \\cite[Section 2]{Dauns}. When $(M,N)$ is concordant the map $\\pi\\mapsto c_{\\pi}(M,N)$ can be written as a composition\n\\[\\pi\\mapsto\\widehat{\\pi}\\mapsto \\|\\widehat{\\pi}(P_{M}P_{N}-P_{M\\cap N})\\|,\\]\nand is thus lower semicontinuous by \\cite[Lemma A.30]{RW}. For $X$ full and $\\widehat{B}$ Hausdorff, continuity follows by \\cite[Lemma 5.2]{RW}.\n\\end{proof}\n\\begin{corl} \nSuppose $X$ is full, $B$ is unital, $\\widehat{B}$ is Hausdorff and $(M,N)$ is concordant. Then $c(M,N)<1$ if and only if $c_{\\pi}(M,N)<1$ for every irreducible representation $\\pi$.\n\\end{corl}\n\\begin{proof} \nSince $\\widehat{B}$ is compact Hausdorff and the Friedrichs angle is continuous, \nthe pointwise estimate  $c_{\\pi}(M,N)<1$ implies that $c(M,N)=\\sup_{\\pi\\in\\widehat{B}} c_{\\pi}(M,N) <1$.\n\\end{proof}\n\\begin{rmk}\n\\label{discontangle}\nIn Proposition \\ref{continuity}, the condition that $(M,N)$ be concordant cannot be relaxed. Consider $B=C([0,\\frac{\\pi}{2}])$, $X=C([0,\\frac{\\pi}{2}], \\mathbb{C}^{2})$ and consider the submodules\n\\[\nM=\\textnormal{Ran } \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix},\\quad N\n= \\textnormal{Ran } \\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix}.\n\\]\nFor $t\\in [0,\\pi\/2]$ we write $c_{t}(M,N)$ for the Friedrichs angle at $t$. For $0<t\\leq \\frac{\\pi}{2}$ we have $M_{t}\\cap N_{t}=\\{0\\}$ whereas at $t=0$ we have $M_0=N_0$. We thus have\n\\[\nc_{t}(M,N)^{2}=\\|P_{M_{t}}P_{N_{t}}\\|^{2}=\\left\\|\\begin{pmatrix}\\cos^{2}t & \\sin t \\cos t \\\\ 0 & 0\\end{pmatrix}\\right\\|^{2}\n=\\cos^{2}t <1,\\quad 0<t\\leq \\frac{\\pi}{2},\n\\]\nand $c_{t}(M,N)=|\\cos t|$, whereas $c_0(M,N)=0$. Thus the angle is discontinuous at $0$ and in particular we conclude once more that $(M,N)$ is discordant. This is another instance where the universal example (see Remark \\ref{rmkuniversal}) provides a counterexample to a specific property. \n\n\\end{rmk}\n\n\n\\subsection{Sum and intersection}\n\\label{sec: sumint}\nWith our extended definition of the Friedrichs angle we now examine the case $c(M,N)<1$ in more detail. Our results \nare all derived from results in \\cite{Luo}, where complementability of $M\\cap N$ is an assumption. We first recall the following well-known fundamental result and a relevant corollary.\n\\begin{thm}\\label{range} Let $T\\in \\End^{*}_{B}(X)$, then\n\\[\\overline{\\textnormal{Ran}(T)}=\\overline{\\textnormal{Ran}(TT^{*})},\\]\nand $T$ has closed range if and only if $T^{*}$ has closed range. If $T$ has closed range then\n\\[\\textnormal{Ran}(T)=\\textnormal{Ran}(TT^{*}),\\]\nand $\\textnormal{Ran}(T)$ is a complemented submodule of $X$ with $\\textnormal{Ran}(T)^{\\perp}=\\ker T^{*}$.\n\\end{thm}\n\\begin{proof}\nSee \\cite[Theorem 3.2, Proposition 3.7]{Lance}. \n\\end{proof}\n\\begin{corl}[\\cite{Luo}, Remark 5.8]\\label{sumrange}\nLet $P,Q$ be adjointable projections on a Hilbert $C^{*}$-module $X$, then \n\\[\\overline{\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)}=\\overline{\\textnormal{Ran}(P+Q)}.\\]\nIn particular\n$\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)$ is closed if and only if $\\textnormal{Ran}(P+Q)$ is closed and in that case\n\\[\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)=\\textnormal{Ran}(P+Q),\\]\nwhich is a complemented submodule of $X$.\n\\end{corl}\n\\begin{proof}\nBy the following well-known observation\n\\[\n\\begin{pmatrix} 0 & 0 \\\\ 0 & P+Q\\end{pmatrix}=\\begin{pmatrix}0 & 0 \\\\ P & Q\\end{pmatrix}\\cdot \\begin{pmatrix}0 & P \\\\ 0 & Q\\end{pmatrix}=\\begin{pmatrix}0 & 0 \\\\ P & Q\\end{pmatrix}\\cdot \\begin{pmatrix}0 & 0 \\\\ P & Q\\end{pmatrix}^{*},\n\\]\nas operators on $X\\oplus X$, it follows that $\\overline{\\textnormal{Ran}(P+Q)}=\\overline{\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)}$ by Theorem \\ref{range}. Therefore,\n $\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)$ is closed if and only if $\\textnormal{Ran}(P)+\\textnormal{Ran}(Q)=\\textnormal{Ran}(P+Q)$ is closed.\n\\end{proof}\n\\begin{lemma}\n\\label{kernel}\nLet $P,Q$ be adjointable projections on a Hilbert $C^{*}$-module $X$, then \n\\[\\ker(1-PQ)=\\ker(1-QP)=\\textnormal{Ran } P\\cap \\textnormal{Ran } Q.\\]\n\\end{lemma}\n\\begin{proof} It is clear that $\\textnormal{Ran } P\\cap \\textnormal{Ran } Q\\subset \\ker (1-PQ)\\cap \\ker (1-QP)$. We prove the reverse inclusion.\nSuppose that $x\\in \\ker(1-PQ)$, so that $x=PQx$. Then clearly $x=Px$ and \n\\[\\langle Qx, Qx\\rangle=\\langle PQx, PQx\\rangle+\\langle (1-P)Qx,(1-P)Qx\\rangle\\geq \\langle PQx,PQx\\rangle=\\langle x,x\\rangle,\\]\nand thus $(1-Q)x=0$ so $x=Qx$. By reversing the roles of $P$ and $Q$ this shows that\n\\[\\ker(1-PQ)=\\ker(1-QP)=\\{x\\in X: x=Px=Qx\\}=\\textnormal{Ran } P\\cap \\textnormal{Ran } Q,\\]\nas claimed.\\end{proof}\nLastly we recall the following remarkable result from \\cite{LSX}, concerning the case where the internal sum $M+N$ is closed.\n\\begin{prop}[{\\cite[Proposition 4.6]{LSX}}]\\label{closedsum} Let $M$ and $N$ be complemented submodules of $X$, such that $M + N$ is closed in $X$. Then both $M + N$ and\n$M \\cap N$ are  complemented and $X=(M\\cap N)\\oplus (M^{\\perp}+N^{\\perp})$.\n\\end{prop}\nIn the above result, complementability of $M\\cap N$ is concluded as opposed to assumed, and we now characterise closedness of $M+N$ in terms of our extended definition of $c(M,N)$, as well as in terms of properties of the operators $1-P_{M}P_{N}$ and $1-P_{N}P_{M}$.\n\\begin{prop}\n\\label{prop: closed range}\nLet $X$ be a Hilbert $C^{*}$-module over a $C^{*}$-algebra $B$, and $M$ and $N$ complemented submodules. \nThen the following are equivalent: \n\\begin{enumerate}\n\\item $c(M,N)<1$;\n\\item The sequence $(P_{M}P_{N})^{n}$ is Cauchy for the operator norm;\n\\item $M\\cap N$ is complemented and $ \\| P_{M}P_{N}-P_{M\\cap N} \\| = \\|P_{N}P_{M}-P_{M\\cap N}\\|<1$;\n\\item $M\\cap N$ is complemented and the operators $1-P_{M}P_{N}:(M\\cap N)^{\\perp}\\to (M\\cap N)^{\\perp}$ and $1-P_{M}P_{N}:(M\\cap N)^{\\perp}\\to (M\\cap N)^{\\perp}$ are bijective; \n\\item $\\textnormal{Ran}(1-P_{M}P_{N})$ and $\\textnormal{Ran}(1-P_{N}P_{M})$ are closed;\n\\item $X=(M\\cap N)\\oplus (M^{\\perp}+N^{\\perp})$;\n\\item $M^{\\perp}+N^{\\perp}$ is closed;\n\\item $X=(M^{\\perp}\\cap N^{\\perp})\\oplus (M+N)$;\n\\item $M+N$ is closed;\n\\item $M\\cap N$ is complemented and $(M\\cap N)^{\\perp}\\cap M+(M\\cap N)^{\\perp}\\cap N$ is closed.\n\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nWe write $P=P_{M}$, $Q=P_{N}$ and $\\Omega:=M\\cap N$.\n\n\n$1.\\Rightarrow 2.$ This was shown in the proof of Proposition \\ref{independent}.\n\n$2.\\Rightarrow 3.$ By Theorem \\ref{complementedalternating}, $\\Omega$ is complemented and $(PQ)^{n}\\to P_{\\Omega}$ in norm. By Lemma \\ref{reductionlemma}, $(QPQ)^{n}\\to P_{\\Omega}$ in norm as well. Thus for $n$ sufficiently large $\\|(QPQ)^{2^n}-P_{\\Omega}\\|<1$. Then \napplying Lemma \\ref{powers} and the $C^{*}$-identity we find\n\\begin{align*}\n\\|(QPQ)^{2^n}-P_{\\Omega}\\|=\\|(QPQ-P_{\\Omega})^{2^{n}}\\|=\\|QPQ-P_{\\Omega}\\|^{2^{n}},\n\\end{align*}\nand it follows that $\\|QPQ-P_{\\Omega}\\|<1$. Now, again by the $C^{*}$-identity, \\[\\|PQ-P_{\\Omega}\\|^{2}=\\|(QP-P_{\\Omega})(PQ-P_{\\Omega})\\|=\\|QPQ-P_{\\Omega}\\|,\\] so we find that $\\|PQ-P_{\\Omega}\\|<1$.\n\n$3.\\Rightarrow 4.$ Since $\\| PQ-P_{\\Omega}\\|<1$. We find that the series\n\\[\n\\sum_{n=0}^{\\infty}(PQ-P_{\\Omega})^{n}=P_{\\Omega}+\\sum_{n=0}^{\\infty}((PQ)^{n}-P_{\\Omega})\\quad\\textnormal{(by Lemma \\ref{powers})}, \n\\]\nis norm convergent to $(1+P_{\\Omega}-PQ)^{-1}$. Since $\\Omega=\\ker (1-PQ)$ by Lemma \\ref{kernel}, $1+P_{\\Omega}-PQ$ is bijective and commutes with $P_{\\Omega}$, it follows that it maps $\\Omega^{\\perp}$ bijectively onto itself. The same argument applies to $QP$.\n\n\n$4.\\Rightarrow 5.$ Since $X=\\Omega\\oplus \\Omega^{\\perp}$ and, by Lemma \\ref{kernel}, $\\Omega=\\ker (1-PQ)=\\ker (1-QP)$ it follows that \n\\[\\textnormal{Ran}(1-PQ)=\\textnormal{Ran}(1-QP)=\\Omega^{\\perp},\\] is closed.\n\n$5.\\Rightarrow 6.$\nSince $\\textnormal{Ran}(1-PQ)$ and $\\textnormal{Ran}(1-QP)$ are closed, they are complemented by Theorem \\ref{range}. Since $(1-PQ)^{*}=1-QP$, we have\n\\[\\textnormal{Ran}(1-QP) =\\ker(1-PQ)^{\\perp}=\\Omega^{\\perp}=\\ker(1-QP)^{\\perp}=\\textnormal{Ran}(1-PQ),\\]\nand by Lemma \\ref{kernel} $\\Omega=\\ker(1-PQ)=\\ker(1-QP)$.\nSince\n\\[1-PQ=(1-P)Q+1-Q,\\quad 1-QP=(1-Q)P+1-P,\\]\nit follows that \n\\[\n\\Omega^{\\perp}\n=\\textnormal{Ran}(1-PQ)\\subset \\textnormal{Ran}(1-P)+\\textnormal{Ran}(1-Q)\n\\subset \\Omega^{\\perp}.\n\\]\nTherefore\n\\[\n\\Omega^{\\perp}=\\textnormal{Ran}(1-P)+\\textnormal{Ran}(1-Q)=M^{\\perp}+N^{\\perp}.\n\\]\n\n$6.\\Rightarrow 7.$ Since $\\Omega^{\\perp}$ is closed, $M^{\\perp}+N^{\\perp}$ is closed.\n\n$7.\\Rightarrow 8.$ By Proposition \\ref{closedsum} $M^{\\perp}\\cap N^{\\perp}$ is complemented with complement $(M+N)^{\\perp}$\n\n$8.\\Rightarrow 9.$ Since $(M^{\\perp}\\cap N^{\\perp})^{\\perp}$ is closed $M+N$ is closed. \n\n$9.\\Rightarrow 10.$ Since $\\Omega$ is complemented, this follows from \\cite[Lemma 5.11]{Luo}, and Corollary \\ref{sumrange}  and Proposition \\ref{closedsum} above. \n\n$10.\\Rightarrow 1.$ By \\cite[Lemma 5.11]{Luo} we have $c(M^{\\perp}, N^{\\perp})<1$, so by Theorem \\ref{anglesymmetry} we have  $c(M,N)<1$.\n\\end{proof}\n\\begin{corl} Let $M,N\\subset X$ be complemented submodules. Then $c(M,N)<1$  if and only if $M\\cap N$ is complemented and $M+N$ is closed if and only if $M^{\\perp}\\cap N^{\\perp}$ is complemented and $M^{\\perp}+N^{\\perp}$ is closed.\n\\end{corl}\n\\begin{rmk}\n\\label{Atiyah}\nLet $E$ and $F$ be vector bundles over a compact Haursdorff space $Y$. In \\cite[Definition 1.3.2]{Atiyah} a vector bundle morphism $\\varphi:E\\to F$ is called \\emph{strict} if the map $y\\mapsto \\dim\\ker\\varphi_{y}$ is continuous, hence locally constant, on $Y$. By \\cite[Proposition 1.32]{Atiyah}, if $\\varphi$ is strict then $\\bigcup_{y\\in Y}\\ker \\varphi_{y}$ is a subbundle of $E$, and thus its module of sections is complemented.\n\nAs in Remarks \\ref{eg:VB-unstrict} and \\ref{discontangle} consider $B=C([0,\\frac{\\pi}{2}])$, $X=C([0,\\frac{\\pi}{2}], \\mathbb{C}^{2})$ and consider the submodules\n\\[\nM=\\textnormal{Ran } \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix},\\quad N\n= \\textnormal{Ran } \\begin{pmatrix} \\cos^{2}t & \\sin t \\cos t \\\\ \\sin t \\cos t &\\sin^{2}t\\end{pmatrix}.\n\\]\nWe obtain two globally trivial rank one vector bundles over $[0,\\pi\/2]$ whose modules of sections are $M$ and $N$ respectively. \nThe vector bundle homomorphism $\\varphi:X\\to X$ defined by $\\varphi:=(1-P_{M}P_{N})$\nis not strict since\n\\[\n\\ker\\varphi_t=\\ker (1-P_{M_t}P_{N_t})=M_{t}\\cap N_{t},\n\\]\nso $t\\mapsto\\dim\\ker\\varphi_{t}$ is discontinuous at $0$ by Remark \\ref{eg:VB-unstrict}. \n\\end{rmk}\nFor a commutative unital $C^{*}$-algebra $B$ the following corollary to Proposition \\ref{prop: closed range} corresponds to the situation where cosine of the angle between the corresponding sub-vector bundles is $<1$ in which case the bundle endomorphism $1-P_{M}P_{N}$ is strict.\n\\begin{corl}If the $C^{*}$-algebra $B$ is unital and $X$ is finitely generated and projective over $B$, then for any pair $M,N\\subset X$ of complemented submodules with $c(M,N)<1$ both $M\\cap N$ and $M+N$ are finitely generated projective $B$-modules.\n\\end{corl}\n\\begin{proof}\nIf $X$ is finitely generated and projective over the unital algebra $B$, then any complemented submodule is finitely generated and projective, so in particular $M\\cap N$ is. Since $M+N$ is finitely generated and closed in $X$, it is a Hilbert $C^{*}$-module and hence is projective by \\cite[Theorem 1.4.6]{Troitsky}.\n\\end{proof}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}