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+{"text":"\\section{Introduction}\nThe Minimal Supersymmetric Standard Model (MSSM) is the most attractive framework for the physics beyond the Standard Model.\nIn the MSSM, gauge coupling unification is achieved and the Higgs potential is stabilized. \nDespite these good features, the MSSM has difficulty in the Higgs sector. \nThe MSSM has a $\\mu$ term, $\\mu H_1 H_2$, in the superpotential. To \nmaintain the weak-scale vacuum expectation value (VEV) of a Higgs, \n$|\\mu|$ has to be at the weak scale. However, it is difficult to explain why such a dimensionful parameter\nis much smaller than Plank scale or GUT scale. This problem is the so-called $\\mu$ problem.\n\nA simple way of solving the $\\mu$ problem is to introduce a gauge singlet, and replace the $\\mu$ by the VEV of the gauge singlet field:\n\\begin{eqnarray}\n\\mu H_1 H_2 \\rightarrow \\lambda \\left<S\\right> H_1 \\cdot H_2 \\ .\n\\end{eqnarray}\nThe most famous model to include a gauge singlet is the Next-to-Minimal Supersymmetric Standard Model (NMSSM).\nIn the NMSSM, a new discrete symmetry, $Z_3$, is introduced to forbid the mass term for $S$.\nHowever, $Z_3$ symmetry spontaneously breaks down when electroweak symmetry breaking occurs. \nAt this point, unacceptably large cosmological domain walls appear\\cite{domainwall}.\nIn the Nearly Minimal Supersymmetric Standard \nModel (nMSSM)\\cite{nmssm1}\\cite{nmssm2}\\cite{nmssm3}, the cosmological \ndomain wall problem is solved by tadpoles. The tadpoles are generated by \nsupergravity interactions and explicitly break the discrete symmetry. Therefore, the \ndomain wall problem does not arise.\nThe nMSSM has the same attractive feature of electroweak baryogenesis as \nthe NMSSM has. To achieve successful electroweak baryogenesis,\na strong first-order phase transition is required. Therefore, new sources of CP-violation beyond the CKM matrix have to exist. \nIn the nMSSM, there are additional sources of CP-violation in the singlet sector. Therefore, unlike MSSM, \nnMSSM does not rely on radiative contributions from a light stop for strong first-order phase transition\n\\cite{nmssm_ewbg1}\\cite{nmssm_ewbg2}.\n\nSUSY breaking terms are important in discussing phenomenology, SUSY breaking effects are transmitted to the nMSSM sector from\na hidden sector by one or more mediation schemes.\nOne interesting mediation scheme is anomaly mediation \\cite{am-lisa}\\cite{am-org}\\cite{am-sing}.\nIn anomaly mediation, the supergravity actions of the hidden sector and visible sector are sequestered.\nSUSY breaking effects are transmitted to the visible sector due to the superconformal anomaly. There are studies in which\nsoft breaking terms are derived by anomaly mediation with NMSSM-like models\\cite{am-nmssm}\\cite{am-fat}.\nIn these works, successful electroweak symmetry-breaking is achieved; however, the VEV of $S$ is on the order of a few TeV. \nThis leads to a large higgsino mass: $\\mu_{eff} = \\lambda \\left<S\\right>$.\nTherefore, there are large mass splittings among Higgs (and neutralinos).\n\nTo obtain a moderate value for $\\mu_{eff}$, we consider a deflected anomaly mediation scenario \\cite{dam-org}\\cite{dam-ph}\\cite{dam-pos}, \nwhich introduces an additional messenger sector. The SUSY breaking mass for a messenger is given by a VEV of the gauge singlet field, $X$.\nIn the original deflected anomaly mediation scenario\\cite{dam-org}\\cite{dam-ph},\nthe superpotential of $X$ is extremely flat; therefore, the fermionic \ncomponent of $X$, $\\psi_X$, becomes light and the lightest SUSY particle is $\\psi_X$. \nIn the positively deflected anomaly mediation scenario\\cite{dam-pos}, the superpotential is not flat; therefore, $\\psi_X$ does not have to\nbe light\\cite{dam-recent} and an ordinary SUSY particle can be a candidate for dark matter.\nWe consider the positively deflected anomaly mediation scenario.\nWe also consider SUSY breaking with the Fayet-Iliopoulos D-term. \n\nWe show that when nMSSM and deflected anomaly mediation are combined,\nsuccessful electroweak symmetry breaking occurs for a moderate value of $\\mu_{eff}$.\nWe also show that the lightest neutralino, which is mainly composed of a \nsinglino, is a good candidate for dark matter. We also present \nsparticle mass spectra. \n\nThis paper is organized as follows. In section 2, we introduce the nMSSM Lagrangian and discuss tadpoles. We also discuss the\n direct couplings between nMSSM fields and messenger sector fields.\nIn section 3, we derive the soft SUSY breaking terms of\nthe nMSSM fields in the deflected anomaly mediation scenario.\nSection 4 is devoted to the phenomenology of\nthis scenario. Finally, section 5 presents our conclusions.\n\n\\section{Nearly Minimal Supersymmetric Standard Model}\nIn this section, we discuss tadpoles and direct couplings between \nnMSSM fields and messenger sector fields. First, we introduce the nMSSM\nLagrangian.\n\nThe superpotential and soft breaking terms in the nMSSM are \n\\begin{eqnarray}\nW_{nMSSM} &=& \\lambda \\hat{S} \\hat{H_1} \\cdot \\hat{H_2} + \\frac{m_{12}^2}{\\lambda}\\hat{S} + \ny_u \\hat{Q} \\cdot \\hat{H_2} \\hat{U}^c +\ny_d \\hat{Q} \\cdot \\hat{H_1} \\hat{D}^c \\nonumber \\\\\n&& + y_l \\hat{L} \\cdot \\hat{H_1} \\hat{E}^c \\label{eq:nmssm_sp} \\ ,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n-\\mathcal{L}_{soft} &=& m_S^2 |S|^2 + (a_\\lambda S H_1 \\cdot H_2 + h.c.) + (t_S S + h.c.) \\nonumber \\\\\n&& + \\tilde{m}_{H_1}^2 H_1^\\dagger H_1 + \\tilde{m}_{H_2}^2 H_2^\\dagger H_2 \\nonumber \\\\\n&& + \\tilde{m}_Q^2 \\tilde{Q}^\\dagger \\tilde{Q} + \\tilde{m}_U^2 |\\tilde{u_R}|^2 + \\tilde{m}_D^2 |\\tilde{d_R}|^2 \n+ \\tilde{m}_L^2 \\tilde{L}^\\dagger \\tilde{L} + \\tilde{m}_E^2 |\\tilde{e_R}|^2 \\nonumber \\\\\n&& + (a_u \\tilde{Q} \\cdot H_2 \\tilde{u_R}^* + a_d \\tilde{Q} \\cdot H_1 \n \\tilde{d_R}^* + a_l \\tilde{L} \\cdot H_1 \\tilde{e_R}^* + h.c.) \\ . \\label{eq:nmssm_soft}\n\\end{eqnarray}\n$\\hat{S}$ denotes a gauge singlet chiral superfield, and $S$ is the scalar component of $\\hat{S}$. \nWhen $S$ acquires the VEV, the higgsino mass parameter, $\\mu_{eff} = \\lambda v_s$, is\ngenerated effectively. We take $\\lambda$ to be real positive by suitable redefinitions of $S$, $H_1$ and $H_2$.\nUnlike the NMSSM, there are \nno trilinear terms for the gauge singlet.\n$m_{12}^2 \\hat{S}\/\\lambda$ and $t_S S$ are tadpoles. They are absent at the tree level; \nhowever, they are generated radiatively by supergravity interactions. \nThese terms are on the order of the weak scale, as we describe below.\n\n\\subsection{Tadpoles}\nThe greatest difference between the nMSSM and NMSSM is the existence of \ntadpoles in the former. The tadpoles are generated by supergravity interaction.\nIn the nMSSM, the theory has a global discrete symmetry at tree level.\nThis symmetry guarantees that the generated tadpoles are on the order of the weak scale,\ndespite the fact that supergravity interactions break global symmetries.\nBecause the tadpoles explicitly break the discrete symmetry,\nthe domain wall problem does not appears.\n\nIn nMSSM, the Lagrangian has a discrete R symmetry $Z_{nR'}$ at tree \nlevel. The charge assignment of the fields is shown in \nTable \\ref{table:nmssm_symmetry}.\nThe charge of $Z_{nR'}$, $Q_{nR'}$, is defined as\n\\begin{eqnarray}\nQ_{PQ} + 3 Q_R \\ , \n\\end{eqnarray}\nwhere $Q_{PQ}$ denotes the charge of Peccei-Quinn symmetry, \n$U(1)_{PQ}$, and $Q_R$ denotes the charge of $U(1)_R$.\nUnder $Z_{nR'}$, the nMSSM fields transform as\n\\begin{eqnarray}\n \\Phi_i \\rightarrow \\Phi_i \\exp\\left({i \\frac{Q_{nR'}}{n} \\theta}\\right),\n\\end{eqnarray}\nwhere $\\Phi_i$ denotes nMSSM fields. If the Lagrangian respects $Z_{5R'}$ \nor $Z_{7R'}$ at the tree level, the scale of the generated tadpoles \ncan naturally be the weak scale\\cite{nmssm1}\\cite{nmssm3}.\nWhen the discrete symmetry is $Z_{5R'}$, tadpoles of six-loop order are generated.\n When the discrete symmetry is $Z_{7R'}$, tadpoles of seven-loop order are generated.\nThe tadpoles break the $Z_{5R'}$ or $Z_{7R'}$, and therefore no cosmological domain wall problem exists.\n\n\\TABLE[thbp]{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n& $\\hat{H}_1$ & $\\hat{H}_2$ & $\\hat{S}$ & $\\hat{Q}$ & $\\hat{L}$ & $\\hat{U}^c$ & $\\hat{D}^c$ & $\\hat{E}^c$ & $W$ \\\\\n\\hline\n$U(1)_{PQ}$ & 1 & 1 & -2 & -1 & -1 & 0 & 0 & 0 & 0 \\\\\n\\hline\n$U(1)_R$ & 0 & 0 & 2 & 1 & 1 & 1 & 1 & 1 & 2 \\\\\n\\hline\n$Z_{nR'}$& 1 & 1 & 4 & 2 & 2 & 3 & 3 & 3 & 6 \\\\ \n\\hline\n\\end{tabular}\n\\caption{Charge assignments of fields}\n\\label{table:nmssm_symmetry}\n}\n\nThe generated tadpoles are given by\\cite{nmssm3}:\n\\begin{eqnarray}\nV_{tad} \\sim \\frac{1}{(16\\pi^2)^l} \\left(M_p M_{susy}^2 S +M_{susy} F_S + h.c. \\right),\n\\end{eqnarray}\nwhere $l$ is the number of the loops at which tadpoles first appear. \n$l=6$ in the $Z_{5R'}$ case and $l=7$ in $Z_{7R'}$ case.\nIn a deflected anomaly mediation scenario as well as an anomaly mediation scenario,\n$M_{susy}$ is $\\mathcal{O}(10 {\\rm \nTeV})$; therefore, $Z_{7R'}$ is favorable. \nAs we describe below, $Z_{7R'}$ also forbids direct couplings between \nnMSSM fields and messenger sector fields.\n\n\\subsection{Direct Couplings to the Messenger sector}\nIn a deflected anomaly mediation scenario, the messenger sector is \nintroduced in addition to the hidden sector, which is the origin of SUSY breaking.\nThe messenger sector contains a gauge singlet chiral superfield and messenger superfields.\nThe messengers transmit the SUSY breaking to the nMSSM sector, and this SUSY breaking is \ncomparable to that of anomaly mediation. \nIn this subsection, we show that direct couplings between the messenger sector fields and the nMSSM fields do not exist.\n\nWe consider the following superpotential in the messenger sector.\n\\begin{eqnarray}\nW_{mess} = \\frac{1}{2} m_X \\hat{X}^2 + \\lambda_X \\hat{X} \\bar{\\Psi}_i{\\Psi^i} \\ ,\\label{eq:mess}\n\\end{eqnarray}\nwhere $\\hat{X}$ is a gauge singlet chiral superfield. $\\bar{\\Psi}_i$ and ${\\Psi^i}$ are the messenger fields \nthat transform ${\\bf \\bar{5}}$ and ${\\bf 5}$ for the \n$SU(5)$ GUT gauge group respectively. \nThe $Z_{nR'}$ charge assignment of the fields is shown in Table \\ref{table:nmssm_gm_charge}.\n\n\\TABLE[thbp]{\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n& $X$ & $\\Psi$ & $\\bar{\\Psi}$ & $W_{mess}$ \\\\\n\\hline\n$U(1)_{PQ}$ & 0 & 0 & 0 & 0\\\\\n\\hline\n$U(1)_R$ & 1 & 1\/2 & 1\/2 & 2\\\\\n\\hline\n$Z_{nR'}$& 3 & 3\/2 & 3\/2 & 6\\\\\n\\hline\n\\end{tabular}\n\\caption{Charge assignment of the fields in the messenger sector}\n\\label{table:nmssm_gm_charge}\n}\n\nIn this charge assignment, there are no direct couplings between messenger sector fields and nMSSM fields.\nA direct coupling between the messengers and nMSSM gauge singlet, $S \\bar{\\Psi}_i{\\Psi}^i$, is forbidden by $Z_{nR'}$ symmetry.\n$S X^2$ and $X H_u H_d$ terms are also forbidden. On the other hand, the $S^2 X$ term is forbidden by $Z_{7R'}$ but allowed by $Z_{5R'}$. \nIn the discussion about tadpoles in the previous subsection, we assumed \nthat $Z_{7R'}$ symmetry exists at tree level.\nTherefore, there are no direct couplings between nMSSM fields and the messenger sector fields.\nThe phenomenology of the nMSSM at the weak scale does not depend on the detail of messenger sector.\n\n\\section{SUSY Breaking}\nIn this section, we derive the SUSY breaking terms of the nMSSM in the deflected anomaly \nmediation scenario. We also show the corrections to the soft scalar mass \nwith the Fayet-Iliopoulos D-term.\n\nIn the original deflected anomaly mediation scenario,\n the superpotential of the gauge singlet $\\hat{X}$ is flat; therefore in general,\nthe lightest SUSY particle (LSP) is the fermionic component of $X$, $\\psi_X$.\n Threshold corrections to the sparticle mass squared are negative. \nIn the positively deflected anomaly mediation scenario,\nthe superpotential of $\\hat{X}$ is not flat; \ntherefore, $\\psi_X$ is not necessarily the LSP. Corrections to the sparticle mass squared are positive. \nWe consider the positively deflected anomaly mediation scenario.\nIn Appendix A, we give an \nexplicit example of the positively deflected anomaly mediation scenario \nin which the fermionic partner of $X$ is not the LSP.\n\nWhen $\\hat{X}$ acquires the VEV,\nmessengers obtain SUSY breaking mass, $X + F_X \\theta^2$. This SUSY breaking mass introduces an intermediate threshold \nthat depends on $\\theta^2$.\nIn deflected anomaly mediation, \ncorrections from anomaly mediation to soft breaking terms are generated by the following threshold.\n\\begin{eqnarray}\n\\frac{X+F_X \\theta^2}{\\Lambda \\hat{\\phi}} = \\frac{X}{\\Lambda} \\left[1+\\left(\\frac{F_X}{X}-F_\\phi\\right)\\theta^2 \\right] \\equiv \\frac{X}{\\Lambda}\\left(1+dF_\\phi \\theta^2\\right) , \\label{eq:thre}\n\\end{eqnarray}\nwhere $\\hat{\\phi}$ is the chiral compensator field, $\\hat{\\phi}=1+F_\\phi \n\\theta^2$, and $\\Lambda$ is the ultraviolet cutoff. $d$ is the deflection \nparameter, which denotes the threshold correction to the SUSY breaking. \nIn the positively deflected anomaly mediation scenario, $d$ is positive.\n\n\nMessengers also affect the beta-functions of gauge couplings. The \nbeta-functions of gauge couplings above the scale $|X|$ are written as\n\\begin{eqnarray}\n\\frac{d g_a}{d \\ln \\mu} = -\\frac{g_a^3}{16\\pi^2} (b_a -N_f) ,\n\\end{eqnarray}\nwhere $N_f$ is the number of messengers. \nFor the intermediate threshold and modification of the beta-functions, \nsoft breaking terms in an anomaly mediation are changed to \nthose of a deflected anomaly mediation scenario.\n\nIn a deflected anomaly mediation, the gaugino mass, soft breaking mass and \nscalar trilinear couplings at the scale $\\mu$ are obtained using the following relations\\cite{dam-org}\\cite{dam-pos}.\n\\begin{eqnarray}\n\\frac{m_\\lambda(\\mu)}{g^2(\\mu)} &=& -\\frac{F_\\phi}{2}\\left(\\frac{\\partial}{\\partial \\ln\\mu} -d\\frac{\\partial}{\\partial\\ln |X|}\\right) g^{-2}\\left(\\frac{\\mu}{\\Lambda},\\frac{|X|}{\\Lambda}\\right) \\nonumber \\\\\n\\tilde{m}_i^2(\\mu) &=& -\\frac{|F_\\phi^2|}{4}\\left(\\frac{\\partial}{\\partial\\ln\\mu}-d\\frac{\\partial}{\\partial\\ln|X|}\\right)^2 \\ln Z_i \\left(\\frac{\\mu}{\\Lambda},\\frac{|X|}{\\Lambda}\\right) \\nonumber \\\\\n\\frac{a_{ijk}(\\mu)}{y_{ijk}(\\mu)} &=& - \\frac{F_\\phi}{2}\\left(\\frac{\\partial}{\\partial\\ln\\mu}-d\\frac{\\partial}{\\partial\\ln|X|}\\right) \\sum_{l=i,j,k} \\ln Z_l \\left(\\frac{\\mu}{\\Lambda},\\frac{|X|}{\\Lambda}\\right), \\nonumber \\\\\n\\end{eqnarray}\nwhere $F_\\phi$ is the F-term of the chiral compensator fields and corresponds to the gravitino mass. $|X|$ is the messenger scale and $\\mu < |X|$. $d$ \nis defined in eq. (\\ref{eq:thre}).\n$|X|$ and $d$ can be determined by the superpotential and the soft breaking terms in \nthe messenger sector (see Appendix A). However, we treat them as the parameters of SUSY breaking because we focus on the \nphenomenology at the weak scale.\n\nIn general, the formula for soft breaking terms is complicated. However, by setting the scale as $\\mu = |X|$,\n the soft breaking terms are simplified as\n\\begin{eqnarray}\nm_{\\lambda_a} &=& -\\frac{g_a^2}{(4\\pi)^2}\\left(b_a - dN_f\\right) F_\\phi \n \\ ,\\nonumber \\\\\n\\tilde{m}_i^2 &=& \\frac{|F_\\phi|^2}{2(4\\pi)^4} \\sum_a c_a^i g_a^4 \\left[b_a + d(d+2)N_f\\right] \\nonumber \\\\\n&& - \\frac{|F_\\phi|^2}{4} \\sum_y \\frac{\\partial \\gamma_i(|X|)}{\\partial y} \\beta_y(|X|) \\ ,\\nonumber \\\\\na_{ijk} &=& -\\frac{F_\\phi}{2} \\left[\\gamma_i(|X|) + \\gamma_j(|X|) + \\gamma_k(|X|) \\right] y_{ijk} \\label{eq:formula}.\n\\end{eqnarray}\nHere, $N_f$ is the number of messengers. $b_a$ and $c_a^i$ are the coefficients \nof the gauge coupling beta functions and the anomalous dimensions of the \nfields respectively.\n$b_a = (-33\/5, -1, 3)$, $c_a^L=(3\/5, 3, 0)$, $c_a^{E^c}=(12\/5, 0, 0)$, $c_a^Q=(1\/15, 3, 16\/3)$,\n$c_a^{U^c}=(16\/15, 0,16\/3)$ and $c_a^{D^c}=(4\/15, 0, 16\/3)$.\n\nThe formula for gaugino masses is easily obtained with\n\\begin{eqnarray}\ng_a^{-2}(\\mu) = g_a^{-2}(\\Lambda) + \\frac{b_a}{8\\pi^2} \\ln\\frac{\\mu}{|X|} + \\frac{b_a-N_f}{8\\pi^2}\\ln\\frac{|X|}{\\Lambda}. \\label{eq:gg}\n\\end{eqnarray}\nEquation (\\ref{eq:gg}) can be obtained by integrating the beta-functions explicitly.\nThe derivations of $\\tilde{m}_i^2$ and $a_{ijk}$ are given in Appendix B. \n\nFor the first and second generations of squarks and sleptons, we can\nneglect the contributions from Yukawa couplings. However, for the soft scalar \nmass and the A-term of the third generation of squarks and sleptons,\n the contributions from Yukawa couplings are important. \nFor $\\tilde{m}_S^2$, $\\tilde{m}_{H_1}^2$, $\\tilde{m}_{H_2}^2$ and $a_\\lambda$, contributions from\nYukawa couplings are also important. The anomalous dimensions of $H_1$ and $H_2$ are different from those of the MSSM due to $\\lambda$\nand are given in Appendix C. \nThe anomalous dimensions of the other fields are same as those of the MSSM and are given in \\cite{beta-functions}.\nWhen Yukawa couplings are small,\nwe obtain the results of \\cite{dam-org}\\cite{dam-pos}.\n\nIn a supersymmetric model, there is an additional source of SUSY breaking, the Fayet-Iliopoulos D-term. \nThis term contributes to the square of the scalar mass.\n\nThe Fayet-Iliopoulos D-term is\n\\begin{eqnarray}\n\\mathcal{L} \\ni - \\xi D .\n\\end{eqnarray}\nThe D-term of the Lagrangian is written as\n\\begin{eqnarray}\n\\mathcal{L}_D = \\frac{1}{2} D^2 - g D \\sum_i q_i A_i^\\dagger A_i - \\xi D , \\label{eq:dterm-xi}\n\\end{eqnarray}\nwhere $q_i$ is the U(1) charge of the field $A_i$. After eliminating the \nD-term with the equation of motion,\nthe Lagrangian $\\mathcal{L}_D$ becomes\n\\begin{eqnarray}\n\\mathcal{L}_D = - \\frac{1}{2} \\left( \\sum_i q_i A_i^\\dagger A_i + \\xi \n\t\t\t     \\right)^2 \\ .\n\\end{eqnarray}\nThis leads to additional contributions to the scalar mass terms:\n\\begin{eqnarray}\n\\tilde{m}_{ij}^2 \\rightarrow \\tilde{m}_{ij}^2 + q_i \\xi \\delta_{ij} .\n\\end{eqnarray}\n\nIn the Supersymmetric Standard Model, there is only one $U(1)$ gauge group.\nIn the lepton sector, the hypercharge of the $SU(2)$ doublet is $-1$ and the \nhypercharge of the $SU(2)$ singlet is $+2$. Therefore, we can not solve the tachyonic slepton mass problem \nin anomaly mediation using only the $U(1)_Y$ D-term. \n\nSo far, the additional contributions from the D-term to the soft \nbreaking mass of the nMSSM matter fields\nare\n\\begin{eqnarray}\n&& \\delta \\tilde{m}_L^2 = -D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_E^2 = 2D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_Q^2 = \\frac{1}{3} D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_U^2 = -\\frac{4}{3}D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_D^2 = \\frac{2}{3}D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_{H_1}^2 = -D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)}  \\ ,\\nonumber \\\\\n&& \\delta \\tilde{m}_{H_2}^2 = D_Y \\frac{|F_\\phi|^2}{(4\\pi^4)} \\ , \\label{eq:dtermcont}\n\\end{eqnarray}\nwhere $D_Y$ comes from the common parameter $\\xi$ in eq. (\\ref{eq:dterm-xi}).\n\nWe can now evaluate the soft breaking terms of the nMSSM at the messenger scale using eq. (\\ref{eq:formula}). \nWe solve the renormalization group equations (RGE) using them as the boundary conditions, and then evaluate the soft breaking\nterms at the weak scale. We use RGE codes contained in the NMSSMTools software package\\cite{nmssmtools1}\\cite{nmssmtools2}.\nWe also add the D-term contributions in eq. (\\ref{eq:dterm-xi}) to the soft breaking mass.\nIn the next section, we discuss the phenomenology of the nMSSM with the soft breaking terms obtained by deflected anomaly mediation. \n\n\n\\section{Phenomenology}\nIn this section, we investigate the phenomenological aspects of the nMSSM. \nFirst, we discuss the existence of Landau poles.\nWe demand that $\\lambda$ should not meet the Landau pole up to the scale at\nwhich the tadpoles are generated.\nNext we study the regions of parameter space where successful electroweak breaking occurs, and we evaluate \nthe mass of the lightest Higgs. Subsequently, we discuss the lightest \nneutralino as a dark matter candidate. We evaluate the relic density of the\nlightest neutralino. We also discuss the direct detection of dark matter. Finally, we obtain sparticle mass spectra.\n\n\\subsection{The Landau pole}\nIn the nMSSM, tadpoles generated by supergravity interaction are proportional to powers of $\\lambda$\\cite{nmssm3}.\nTherefore to maintain the tadpoles at the weak scale, $\\lambda$ should not meet \nthe Landau pole up to the scale, at which the tadpoles are generated.\nWe investigate the region of $\\lambda$ and $\\tan\\beta$\n that satisfies the perturbativity condition below the GUT scale.\n\nThe beta-functions of $\\lambda$ and $y_t$ are\n\\begin{eqnarray}\n\\beta_\\lambda = \\frac{1}{16\\pi^2}\\left(4\\lambda^2 + 3 y_t^2 + 3 y_b^2 + y_\\tau^2 -\\frac{3}{5}g_1^2 -3 g_2^2 \\right) \\lambda \\ ,  \\nonumber \\\\\n\\beta_{y_t} = \\frac{1}{16\\pi^2}\\left(\\lambda^2 + 6 y_t^2 + y_b^2 -\\frac{13}{15}g_1^2 -3 g_2^2 -\\frac{16}{3} g_3^2 \\right) y_t \\ .\n\\end{eqnarray}\nThese beta-functions strongly depend on $\\lambda$ and $\\tan \\beta$ \nthrough the top Yukawa coupling.\nFigure \\ref{fig:lpole} shows the allowed region where the perturbativity \nis satisfied up to the GUT scale. \nThe calculation is performed using the RGE code included in the NMSSMTools package. The current experimental value of the top mass is \n$173.1 \\pm 1.3$ GeV\\cite{topmass}. We take the central value for $m_{\\rm top}$ as $173.1$ GeV. \nThe shaded region is consistent with the perturbativity of $\\lambda$. The result depends on \nthe value of $m_{\\rm top}$ and supersymmetric threshold corrections of $\\alpha_s$.\nTherefore there is small difference among the results of \\cite{nmssm4} and \\cite{nmssm_ewbg1} and our results.\nIn our calculation, the region where $\\tan\\beta \\gtrsim 2.0$ and $\\lambda \\lesssim 0.7$ is allowed.\n\n\\FIGURE[htbp]{\n\\epsfig{file=landaupole.eps, width=0.6\\hsize}\n\\caption{\nThe region that is consistent with the perturbativity of $\\lambda$ up to the GUT scale is shown.\nThe gray shaded region below the solid line is allowed. The region above the solid line is excluded owing to the existence of the Landau pole below the GUT scale.\n}\n\\label{fig:lpole}\n}\n\n\\subsection{Electroweak symmetry breaking}\nIn this subsection, we consider the conditions for electroweak \nsymmetry breaking and evaluate $\\mu_{eff} = \\lambda \\left<S\\right>$. We \nalso evaluate the mass of the lightest Higgs. \n\nAfter obtaining the soft breaking terms at the weak scale, we now evaluate the Higgs potential, $V = V_{tree} + \\Delta V$.\nFrom eqs. (\\ref{eq:nmssm_sp}) and (\\ref{eq:nmssm_soft}), the tree-level Higgs \npotential is written as \n\\begin{eqnarray}\nV_{tree} &=& \\tilde{m}_{H_1}^2 H_1^\\dagger H_1 + \\tilde{m}_{H_2}^2 H_2^\\dagger H_2 + m_s^2 |S|^2 + m_{12}^2 (H_1\\cdot H_2 + h.c.) \\nonumber \\\\\n&+& \\lambda^2 |H_1 \\cdot H_2 |^2 + \\lambda^2 |S|^2 (H_1^\\dagger H_1 +H_2^\\dagger H_2 ) + \\frac{g^2}{2}|H_1^\\dagger H_2|^2 \\nonumber \\\\\n&+& \\frac{\\bar{g}^2}{8}(H_2^\\dagger H_2 - H_1^\\dagger H_1)^2 + (t_s S + \n h.c.) +(a_\\lambda S H_1 \\cdot H_2 + h.c.)  \\ , \\label{eq:potential}\n\\end{eqnarray}\nwhere $\\bar{g}^2 = g^2+g'^2$. \n$\\Delta V$ is the one-loop contribution to the effective potential \\cite{cw-potential}:\n\\begin{eqnarray}\n\\Delta V = \\frac{1}{64\\pi^2}\\left(\\sum_b g_b m_b^4 \\left[\\ln\\left(\\frac{m_b^2}{Q^2}\\right)-\\frac{3}{2}\\right]\n- \\sum_f g_f m_f^4 \\left[\\ln\\left(\\frac{m_f^2}{Q^2}\\right)-\\frac{3}{2}\\right]\\right) \\ . \\label{eq:coleman-wein}\n\\end{eqnarray}\n$g_b$ and $g_f$ are the degrees of freedom for bosons and fermions respectively.\nWe determine $\\mu_{eff}\\equiv \\lambda \\left<S\\right>$, $t_s$ and \n$m_{12}^2$ using the stationary conditions of the Higgs potential. \nFrom eqs. (\\ref{eq:potential}) and (\\ref{eq:coleman-wein}), the stationary conditions are\n\\begin{eqnarray}\n \\frac{\\partial V}{\\partial v_1} &=& 2v_1 \\left[\\tilde{m}_{H_1}^2 + (m_{12}^2 + \n\t\t\t\t\t   a_\\lambda \t\t\t\t\t   \nv_s)\\frac{v_2}{v_1}-\\frac{\\bar{g}^2}{4}(v_2^2-v_1^2)+\\lambda^2(v_2^2+v_s\n^2) + \\frac{1}{2v_1}\\frac{\\partial \\Delta V}{\\partial v_1} \\right] = 0 \\ , \\nonumber \\\\\n \\frac{\\partial V}{\\partial v_2} &=& 2v_2 \\left[\\tilde{m}_{H_2}^2 + (m_{12}^2 + \n\t\t\t\t\t   a_\\lambda v_s)\\frac{v_1}{v_2}+\\frac{\\bar{g}^2}{4}(v_2^2-v_1^2)+\\lambda^2(v_1^2+v_s\n^2) + \\frac{1}{2v_2}\\frac{\\partial \\Delta V}{\\partial v_2} \\right] =0  \\ ,\\nonumber \\\\\n \\frac{\\partial V}{\\partial v_s} &=& 2v_s \\left[m_s^2+\\lambda^2(v_1^2+v_2^2)+\\frac{t_s}{v_s}+a_\\lambda \\frac{v_1 v_2}{v_s} + \\frac{1}{2v_s}\\frac{\\partial \\Delta V}{\\partial v_s} \\right]=0 \\label{eq:higgs_kyokuchi} \\ ,\n\\end{eqnarray}\nwhere $v_1=\\left<H_1^0\\right>$, $v_2=\\left<H_2^0\\right>$ and $v_s = \\left<S\\right>$. \nAs we describe later, there is only a small region\nof parameter space where successful electroweak symmetry breaking \noccurs with $v_s > 0$; therefore, we take $v_s < 0$.\nFrom eq. (\\ref{eq:higgs_kyokuchi}), $\\mu_{eff}$ can be determined by,\n\\begin{eqnarray}\n\\mu_{eff}^2 = -\\frac{M_Z^2}{2} + \\frac{\\tilde{m}_{H_1}^2 + \\frac{1}{2v_1}\\frac{\\partial \\Delta V}{\\partial v_1} - \n\\left(\\tilde{m}_{H_2}^2 + \\frac{1}{2v_2}\\frac{\\partial \\Delta V}{\\partial v_2}\\right) \\tan^2\\beta}{\\tan^2\\beta-1} \\label{eq:mueff}.\n\\end{eqnarray}\n$\\mu_{eff}$, $m_{12}^2$ and $t_s$ are determined from eq. (\\ref{eq:higgs_kyokuchi}). We now evaluate the Higgs mass. We \nexpand $H_1^0$, $H_2^0$ and $S$ as\n\\begin{eqnarray}\nH_1^0 &=& v_1 + \\frac{1}{\\sqrt{2}}\\left(h_1^0 + i a_1\\right), \\nonumber \\\\\nH_2^0 &=& v_2 + \\frac{1}{\\sqrt{2}}\\left(h_2^0 + i a_2\\right), \\nonumber \\\\\nS^0 &=& v_s + \\frac{1}{\\sqrt{2}}\\left(s + i a_s\\right). \n\\end{eqnarray}\nUsing these expanded fields, the CP-even Higgs mass matrix is written as\n\\begin{eqnarray}\n\\left(h_1^0 \\ h_2^0 \\ S \\right) M^2 \\left(\n\\begin{array}{c}\nh_1^0 \\\\\nh_2^0 \\\\\nS\n\\end{array}\n\\right) .\n\\end{eqnarray}\nAt tree level, the components of $M^2$ are\n\\begin{eqnarray}\nM_{11}^2 &=& s_\\beta^2 M_a^2 + c_\\beta^2 M_Z^2 \\ , \\nonumber \\\\\nM_{12}^2 &=& -s_\\beta c_\\beta \\left(M_a^2 + M_Z^2 - 2\\lambda^2 v^2 \\right) \\ ,\\nonumber \\\\\nM_{13}^2 &=& v \\left(s_\\beta a_\\lambda + 2 c_\\beta \\lambda^2 v_s^2 \\right) \\ ,\\nonumber \\\\\nM_{22}^2 &=& c_\\beta^2 M_a^2 + s_\\beta^2 M_Z^2 \\ ,\\nonumber \\\\\nM_{23}^2 &=& v \\left(c_\\beta a_\\lambda 2 + s_\\beta \\lambda^2 v_s \\right) \\ ,\\nonumber \\\\\nM_{33}^2 &=& -\\frac{1}{v_s}\\left(t_s + s_\\beta c_\\beta a_\\lambda v_s \\right),\n\\end{eqnarray}\nwhere $c_\\beta = \\cos\\beta$ and $s_\\beta = \\sin\\beta$.\nThe CP-odd Higgs mass matrix at tree-level is\n\\begin{eqnarray}\n\\left(A^0 \\ a_s \\right)\n\\left[\n\\begin{array}{cc}\nM_a^2 & -a_\\lambda v_s \\\\\n-a_\\lambda v_s & -\\frac{1}{v_s}\\left( t_s + s_\\beta c_\\beta a_\\lambda v^2 \\right)\n\\end{array}\n\\right]\n\\left(\n\\begin{array}{c}\nA^0 \\\\\na_s\n\\end{array}\n\\right) ,\n\\end{eqnarray}\nwhere $M_a^2 = -\\left(m_{12}^2 + a_\\lambda v_s \\right)\/c_\\beta s_\\beta$. \n$A^0 = a_d s_\\beta + a_u c_\\beta$, and its orthogonal combination is absorbed by the Z boson.\n\nWe now present the results of numerical calculations.\nFigure \\ref{fig:nmssm_ewsbok} shows the allowed region of successful electroweak symmetry breaking without tachyonic sleptons.\nWe set the messenger scale to $5 F_\\phi \\simeq 150 \\ {\\rm TeV}$.\nSuccessful electroweak symmetry breaking occurs in the region covered by red squares.\nIn the region covered by blue crosses, the mass of the lightest Higgs \nsatisfies the LEP bound with the electroweak symmetry breaking.\nWhen the number of the messengers, $N_f$ increases, the allowed region of the \ndeflection parameter $d$ is shifted downward. Therefore, in the scenario with small \n$d$ ($d < 1$), two or more messengers have to exist. \nFor simplicity, we assume that there is one messenger in the following analysis. \nAlthough there is a region where successful electroweak symmetry \nbreaking occurs with large $\\tan\\beta$ and $v_s >0$, \nthe region is very small. Therefore we take $v_s < 0$.\n\nFigure \\ref{fig:nmssm_mueff} shows the dependence of $\\mu_{eff}$ on SUSY \nbreaking. \nWe see that $|\\mu_{eff}|$ is a decreasing function of $D_Y$, while it is an increasing function of $d$.\nThis can be understood from eqs. (\\ref{eq:formula}), (\\ref{eq:dtermcont}) and (\\ref{eq:mueff}).\nWhen $D_Y$ increases,\n$m_{H_1}^2$ decreases and $m_{H_2}^2$ increases. This implies that $|\\mu_{eff}|$ decreases as $D_Y$ increases.\nWhen $d$ increases, \n$m_{H_1}^2$ and $m_{H_2}^2$ increase at almost the same rate. This implies that $|\\mu_{eff}|$ increases as $d$ increases.\nIn this scenario, moderate values of $\\mu_{eff}$, $100 < |\\mu_{eff}| < \n550$, are obtained without meeting the Landau pole.\n\nFigure \\ref{fig:higgsmass} shows the dependence of the lightest Higgs mass on $d$ and $D_Y$. \nThe calculation is performed with NMSSMTools, including two-loop corrections. We extend the codes to include tadpoles. \nIn this scenario, the mass of the lightest Higgs can be\nheavier than the LEP bound.\n\n\\FIGURE[htbp]{\n\\hspace*{-6mm}\n\\epsfig{file=ewsb1.eps,width=0.45\\hsize}\n\\epsfig{file=ewsb2.eps,width=0.45\\hsize}\n\\epsfig{file=ewsb3.eps,width=0.45\\hsize}\n\\epsfig{file=ewsb4.eps,width=0.45\\hsize}\n\\caption{\nSuccessful electroweak symmetry breaking occurs in the region covered by red squares, and the region covered by blue crosses satisfies\nthe Higgs mass bound of the LEP. In other regions, the sleptons are tachyonic.\nThe calculation is performed with $\\lambda=0.69$ and $m_0 = F_\\phi\/(4\\pi)^4 = 200 \\ {\\rm GeV}$. The messenger scale is taken to be $5 F_\\phi$.\n$\\tan\\beta$ and the number of messengers $N_f$ are $\\tan\\beta=2$ and $N_f=1$ in the top-left figure, \n$\\tan\\beta=3$ and $N_f=1$ in the top-right figure,\n$\\tan\\beta=2$ and $N_f=2$ in the bottom-left figure and\n$\\tan\\beta=20$ and $N_f=1$ in the bottom-right figure. The bottom-right figure is evaluated with $v_s > 0$. \nThe others are evaluated with $v_s < 0$.\n}\n\\label{fig:nmssm_ewsbok}\n}\n\n\\FIGURE[htbp]{\n\\epsfig{file=mueff_dy.eps,width=0.45\\hsize}\n\\epsfig{file=mueff_d.eps,width=0.45\\hsize}\n\\caption{\nThe values of $|\\lambda v_s|$ are shown. The calculations are performed with $\\lambda=0.69$, $\\tan\\beta=2$ and $m_0=200 \\ {\\rm GeV}$.\nModerate values of $|\\lambda v_s|$ are obtained.\n}\n\\label{fig:nmssm_mueff}\n}\n \n\\FIGURE[htbp]{\n \\epsfig{file=mhiggs_dy.eps,width=0.45\\hsize}\n \\epsfig{file=mhiggs_d.eps,width=0.45\\hsize}\n\\caption{\nThe dependence of the Higgs mass on the SUSY breaking parameter is shown. In the left figure we set $d=2.5$, and in the right figure we set $D_Y = 6$. \nOther parameters are chosen as $\\lambda=0.69, \\tan\\beta=2.0$ and $m_0=200 {\\rm GeV}$ in both figures.\n}\n\\label{fig:higgsmass}\n}\n\n\\subsection{Dark matter}\nIn this scenario, the lightest neutralino is the LSP in the wide range of parameter space. \nTherefore, the lightest neutralino is a candidate for dark matter. In this \nsubsection, we evaluate the relic density of the lightest neutralino, \nwhich is mainly composed of a singlino. We also calculate the the neutralino-proton \nscattering cross section, and discuss the direct detection of dark matter.\n\nIn the nMSSM, the relic density of the lightest neutralino strongly depends \non its mass\\cite{nmssm1}\\cite{nmssm2}. Although the dominant contribution to the\nannihilation cross section is s-channel $Z$ boson exchange, the coupling \nbetween the $Z$ boson and $\\tilde{N}_1$ is significantly small. This is \nbecause the lightest neutralino, $\\tilde{N}_1$ is mainly composed of the \nfermionic component of the nMSSM gauge singlet, $\\hat{S}$. \nThe resonant effect near the $Z$ pole mass is important for the sufficient\nannihilation of the lightest neutralino.\n\nThe neutralino mass matrix is\n\\begin{eqnarray}\n\\left(\\tilde{B} \\ \\tilde{W} \\ \\tilde{H}_1^0 \\ \\tilde{H}_2^0 \\ \\tilde{S}\\right)\n\\left(\n\\begin{array}{ccccc}\nm_{\\lambda_1} & 0 & -c_\\beta s_w M_Z & s_\\beta s_w M_Z & 0 \\\\\n0 & m_{\\lambda_2} & c_\\beta c_w M_Z & -s_\\beta c_w M_Z & 0 \\\\\n-c_\\beta c_w M_Z & c_\\beta c_w M_Z & 0 & \\mu_{eff} & \\lambda v_2 \\\\\ns_\\beta s_w M_Z & -s_\\beta c_w M_Z & \\mu_{eff} & 0 & \\lambda v_1 \\\\\n0 & 0 & \\lambda v_2 & \\lambda v_1 & 0 \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\n\\tilde{B} \\\\\n\\tilde{W} \\\\\n\\tilde{H}_1^0 \\\\\n\\tilde{H}_2^0 \\\\\n\\tilde{S}\n\\end{array}\n\\right)\n,\n\\end{eqnarray}\nwhere $s_\\beta = \\sin\\beta$, $c_\\beta = \\cos\\beta$ and $s_w = \\sin{\\theta_W}$.\n$\\tilde{B}$, $\\tilde{W}$, $\\tilde{H}_{1,2}^0$ and \n$\\tilde{S}$ denote the bino, wino, higgsino and singlino respectively. \nThe mass of the lightest neutralino, $m_{\\chi_1}$, becomes heavier\nas $|\\mu_{eff}|$ decrease. This is because the mixing of the higgsinos becomes small as one can see from eq. (4.10).\nSince $|\\mu_{eff}|$ is a decreasing function of $D_Y$, a larger $D_Y$ leads to a larger $m_{\\chi_1}$. \nThe dependence of the lightest neutralino mass on $D_Y$ is shown in Fig.\\ref{fig:relic}.\nOn the other hand,\na larger $d$ leads to a smaller $m_{\\chi_1}$.\n\nFigure \\ref{fig:relic} shows $m_{\\chi}$ and the relic density of the neutralino, $\\Omega_{\\chi} h^2$. $m_{\\chi}$ and $\\Omega_{\\chi} h^2$ are calculated \nwith NMSSMTools and micrOMEGAs\\cite{microomegas}\\cite{microomegas2}. When $m_{\\chi}$ is large and close to $m_Z$, \n$\\Omega_{\\chi} h^2$ is small. The observed relic density of dark matter is given by\\cite{wmap1}\\cite{wmap2}\n\\begin{eqnarray}\n0.094 < \\Omega_{CDM} h^2 < 0.136 .\n\\end{eqnarray}\nThis condition is satisfied with $m_{\\chi} \\simeq 35$ GeV. With such light dark matter, there are strong limits for the spin-independent\n WIMP-nucleon scattering cross section from CDMS\\cite{cdms} and XENON10\\cite{xenon}. The strongest limit for\n the cross section is for it to be less than $5\\times 10^{-44} {\\rm cm}^2$ for $m_{\\chi_1} \\simeq 30 \\ {\\rm GeV}$.\n\n\\FIGURE[htbp]{\n \\epsfig{file=relic1.eps,width=0.6\\hsize}\n\\caption{\nThe mass and the relic density of the lightest neutralino are shown as functions of the \nSUSY breaking parameter $D_Y$. The other parameters are chosen as $m_0 = 200$ GeV, $d=2.25, \\lambda=0.69$ and $\\tan\\beta=2.0$\n}\n\\label{fig:relic}\n}\n\nThe spin-independent WIMP-nucleon elastic scattering cross section is \nwritten as\n\\begin{eqnarray}\n\\sigma^{\\rm SI} = \\frac{4 m_{\\chi}^2 m_{nucleus}^2}{\\pi (m_{\\chi} + \n m_{nucleus})^2} \\left[Z f_p +(A-Z)f_n\\right]^2\\ .\n\\end{eqnarray}\n$f_{p,n}$ is the coupling between the WIMP and a nucleon given by \\cite{susy_darkmatter}\n\\begin{eqnarray}\nf_{p,n} = \\sum_{q=u,d,s} f_{T_q}^{(p,n)} a_q \\frac{m_{p,n}}{m_q} + \\frac{2}{27} f_{T_G}^{(p,n)} \\sum_{q=c,b,t} a_q \\frac{m_{p,n}}{m_{q}} .\n\\end{eqnarray}\n$a_q$ are the WIMP-quark couplings. We focus on the dark matter-proton scattering cross section in the following discussion. \nThe parameter $f_{T_q}$ is defined by \n\\begin{eqnarray}\nm_p f_{T_q} \\equiv m_q \\left<N|\\bar{q}q|N\\right> \\equiv m_q B_q ,\n\\end{eqnarray}\nand $f_{TG}=1-\\sum_{q=u,d,s} f_{T_q}$. $f_{T_q}$ can be written as \n\\cite{direct_dm_update}\n\\begin{eqnarray}\nf_{T_u} &=& \\frac{m_u B_u}{m_p} = \\frac{2\\sigma_{\\pi N}}{m_p \\left(1+ \\frac{m_d}{m_u}\\right)\\left(1+  \\frac{B_d}{B_u}\\right)} \\ ,\\nonumber \\\\\nf_{T_d} &=& \\frac{m_d B_d}{m_p} = \\frac{2\\sigma_{\\pi N}}{m_p \\left(1+  \n    \\frac{m_u}{m_d}\\right)\\left(1+  \\frac{B_u}{B_d}\\right)} \\ ,\\nonumber \\\\\nf_{T_s} &=& \\frac{m_s B_s}{m_p} = \\frac{ y \\left(\\frac{m_s}{m_d}\\right)  \\sigma_{\\pi N}  }{m_p \\left(1+  \\frac{m_u}{m_d}\\right)} , \\label{eq:ftq}\n\\end{eqnarray}\nwhere $\\sigma_{\\pi N}$ is the $\\pi$-nucleon sigma term:\n\\begin{eqnarray}\n\\sigma_{\\pi N} = \\frac{1}{2}\\left(m_u + m_d\\right)\\left(B_u + B_d\\right) .\n\\end{eqnarray}\nThe phenomenological value of $\\sigma_{\\pi N}$ is $64\\pm 8$ MeV\\cite{direct_dm_update}.\n$y$ denotes the ratio of the strange quark component in the nucleon, defined as\n\\begin{eqnarray}\ny = \\frac{2 B_s}{B_u + B_d}\\ .\n\\end{eqnarray}\n$y$ can be determined by the relation,\n\\begin{eqnarray}\n\\sigma_{0} = \\sigma_{\\pi N} \\left(1-y\\right) = \\frac{1}{2}\\left(m_u + m_d \\right)\\left(B_u + B_d -2B_s \\right) .\n\\end{eqnarray}\n$\\sigma_0$ can be evaluated from baryon mass spectra using chiral perturbation theory. From \\cite{sigma_0}, $\\sigma_0 = 36 \\pm 7$ MeV.\nThere is large ambiguity for $y$. When $(\\sigma_{\\pi N}, \\sigma_0) = (64, 36)$ MeV, $y=0.44$. \nOn the other hand, according to a recent lattice calculation \\cite{ohki},\n $y$ has a small value such as $0.03$.\n\nThe ratios of the quark mass are taken from \\cite{quark_mass_ratio}.\n\\begin{eqnarray}\n\\frac{m_u}{m_d} = 0.553 \\pm 0.043, \\  \\frac{m_d}{m_s} = 18.9 \\pm 0.8 . \\label{eq:mumd}\n\\end{eqnarray}\nThe ratios of the form factors are written as\n\\begin{eqnarray}\n\\frac{B_d}{B_u} = \\frac{2 + (z-1)y}{2z-(z-1)y} , \\label{bdbu}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nz = \\frac{B_u-B_s}{B_d-B_s} .\n\\end{eqnarray}\n$z$ can be calculated from the baryon mass, and its value is \n1.49\\cite{dm_zparameter}. We can now determine $f_{T_q}$ from eqs. \n(\\ref{eq:ftq}), (\\ref{eq:mumd}) and (\\ref{bdbu}). \nWhen $y=0.44$,\n\\begin{eqnarray}\nf_{T_u} \\approx 0.027, \\ \\ f_{T_d} \\approx 0.039, \\ \\ f_{T_s} \\approx 0.365, \\ \\ f_{TG} \\approx 0.569,\n\\end{eqnarray}\nand when $y=0.03$,\n\\begin{eqnarray}\nf_{T_u} \\approx 0.029, \\ \\ f_{T_d} \\approx 0.036, \\ \\ f_{T_s} \\approx 0.025, \\ \\ f_{TG} \\approx 0.91.\n\\end{eqnarray}\nIn these two cases, $f_{T_s}$ and $f_{TG}$ are very different. This \naffects the spin-independent cross section of the WIMP-nucleon scattering significantly.\n\nWIMP-quark couplings, $a_q$, consist of two parts. One part arises from squark \ns-channel exchange and the other arises from the t-channel exchange of \nthe neutral Higgs. \nThe couplings from squark exchange are given by \\cite{aq_squark}\n\\begin{eqnarray}\na_{q_i}^{\\tilde{q}} = -\\frac{1}{2(\\tilde{m}_{1i}^2-m_{\\chi}^2)} {\\rm Re} \\left[X_i Y_i^* \\right] \n- \\frac{1}{2(\\tilde{m}_{2i}^2-m_{\\chi}^2)} {\\rm Re} \\left[W_i \n\t\t\t\t\t\t     V_i^*\\right] \\ ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nX_i &=& \\eta_{11}^* \\frac{g m_{q_i} N_{1,5-i}^*}{2M_w B_i} - \\eta_{12}^* \n e_i g' N_{11}^*  \\ ,\\nonumber \\\\\nY_i &=& \\eta_{11}^* \\left(\\frac{y_i}{2} g' N_{11} + g T_{3i} N_{12}\\right) + \\eta_{12}^* \\frac{g m_{q_i} N_{1,5-i}}{2M_w B_i} \\ ,\\nonumber \\\\\nW_i &=& \\eta_{21}^* \\frac{g m_{q_i} N_{1,5-i}^*}{2M_w B_i} - \\eta_{22}^* e_i g' N_{11}^* \\ ,\\nonumber \\\\\nY_i &=& \\eta_{21}^* \\left(\\frac{y_i}{2} g' N_{11} + g T_{3i} N_{12}\\right) + \\eta_{22}^* \\frac{g m_{q_i} N_{1,5-i}}{2M_w B_i},\n\\end{eqnarray}\nand $i=1$ for an up-type quark and $i=2$ for a down-type quark. $\\tilde{m}_{1i}$ and \n$\\tilde{m}_{2i}$ denote a light squark mass and a heavy squark mass respectively. \n$\\eta$ denotes a squark\nmixing such that\n\\begin{eqnarray}\n\\tilde{q}_l = \\eta_{l1} \\tilde{q}_L + \\eta_{l2} \\tilde{q}_R .\n\\end{eqnarray}\n$y_i$, $T_{3i}$ and $e_i$ denote the hypercharge, isospin and electric \ncharge of the quarks respectively. $B_1 = \\sin\\beta$ and $B_2 = \\cos\\beta$.\n\nThe couplings from neutral Higgs exchange in the nMSSM are given by\\cite{nmssm_aq}\n\\begin{eqnarray}\na_{q_i}^h = \\sum_{a=1}^3 \\frac{1}{m_{h_a^0}^2} {C_Y}_a^i {\\rm Re} \n [C_{H}^a] \\ ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{C_Y}_a^i &=& -\\frac{g m_{q_i}}{4 M_w B_i} S_{a,3-i} \\ ,\\nonumber \\\\\n{C_{H}^a} &=& \\left(-g N_{12}^* + g' N_{11}^* \\right)\\left(S_{a1}N_{13}^* - S_{a2}N_{14}^* \\right) \\nonumber \\\\\n&& - \\sqrt{2} \\lambda \\left[S_{a3}N_{13}^* N_{14}^* + N_{15}^* \n\t\t       \\left(S_{a2}N_{13}^* + S_{a1}N_{14}^* \n\t\t      \\right)\\right] \\ .\\label{eq:a_from_higgs} \n\\end{eqnarray}\n$S_{ij}$ denotes Higgs mixing. One can write the mass eigenstate of the Higgs as\n\\begin{eqnarray}\nh_a^0 = S_{a1} h_d^0 + S_{a2} h_u^0 + S_{a3} h_s .\n\\end{eqnarray}\nWhen $\\lambda=0$, eq. (\\ref{eq:a_from_higgs}) agrees with the couplings \nin the MSSM given in \\cite{aq_squark}.\n\nFigure \\ref{fig:si_cross} shows the spin-independent cross section as a function of $d$ and $D_Y$. When $y=0.44$, $\\sigma^{SI}$ is already excluded\nby the current experiments. On the other hand, when $y=0.03$, \n$\\sigma^{SI}$ is smaller than the upper limit from XENON10 in many regions of the parameter space.\nIn this case, $\\sigma^{SI}$ is large enough to be detected or be \nexcluded by the next-generation experiments.\n\n\\FIGURE[htbp]{\n \\epsfig{file=sigma_d.eps,width=0.49\\hsize}\n \\epsfig{file=sigma_dy.eps,width=0.49\\hsize}\n\\caption{\nThe spin-independent cross sections, $\\sigma^{SI}$ are shown. $\\sigma^{SI}$ are calculated with $m_0 = 200$ GeV, $\\lambda=0.69$ and $\\tan\\beta=2.0$.\nThe upper three lines are calculated with $y=0.44$. $y$ is evaluated with chiral perturbation.\nThe lower three lines are calculated with $y=0.03$, which is the result from a recent lattice calculation.\n}\n\\label{fig:si_cross}\n}\n\n\\subsection{Mass spectrum}\nHere we present the sparticle mass spectrum, the relic density of the \nneutralino and the spin-independent cross section of dark matter-proton scattering.\nThe mass spectra are calculated using NMSSMTools, and the relic \ndensities are calculated using micrOMEGAs.\n\nIn this scenario, the gluino is light in a wide range of the parameter space. This is because the contributions from gauge mediation and anomaly mediation\n cancel. From eq. (\\ref{eq:formula}), the gluino mass at the messenger scale \n is written as\n\\begin{eqnarray}\nm_{\\tilde{g}} = -g_3^2 \\left(3-d\\right)m_0 \\label{eq:mass_gl} \\ ,\n\\end{eqnarray}\nwhere $m_0 = \\displaystyle F_\\phi\/(16\\pi^2)$. Particularly in the region where $d=3$, the gluino mass $m_{\\tilde{g}}$ vanishes.\n\nThe results of the numerical calculation are presented in Table \\ref{tab:masses}. When the deflection parameter $d$ changes, \nthe overall scale of the soft breaking terms changes. However, once we impose the condition of the observed relic density, \n$\\mu_{eff}$ is almost determined by $m_{\\chi}$. Therefore the mass \nspectrum of the Higgs does not change significantly. \nTwo samples in Table \\ref{tab:masses} satisfy the current experimental limits from LEPII and XENON10 and also explain the observed\nrelic abundance of dark matter. The mass of the gluino is $m_{\\tilde{g}} \\sim 200\\  {\\rm GeV}$. \nThe lightness of the gluino is the characteristic feature of this scenario.\n\n\\TABLE[htbp]{\n\\caption{Mass spectra}\n\\begin{tabular}{|c|c c c c c c|}\n\\hline\n\\multicolumn{1}{|l|}{} & $m_0$ & $N_f$ & $d$ & $D_Y$ & $\\lambda$ & $\\tan\\beta$ \\\\ \\hline\ninput p1 & 200 & 1 & 2.2 & 4.84 & 0.69 & 2 \\\\ \\hline\np2 & 200 & 1 & 3.5 & 8.74 & 0.69 & 2 \\\\ \\hline\n\\end{tabular}\n\\begin{tabular}{|c|cccccccc|}\n\\hline\n & $|\\mu_{eff}|$ & $m_{H_1^0}$ & $m_{H_2^0}$ & $m_{H_3^0}$ & $m_{A_1^0}$ & $m_{A_2^0}$ & $m_{H^{\\pm}}$ & $m_{\\chi_1^0}$ \\\\ \\hline\noutput p1 & 264.2 & 127.2 & 328.8 & 460.1 & 285.5 & 487.8 & 433.8 & 34.2 \\\\ \\hline\np2 & 270.6 & 127.5 & 313.5 & 590.0 & 286.6 & 601.6 & 582.7 & 34.3 \\\\ \\hline \\hline\n & $m_{\\chi_2^0}$ & $m_{\\chi_3^0}$ & $m_{\\chi_4^0}$ & $m_{\\chi_5^0}$ & $m_{\\tilde{g}}$ & $m_{\\chi_1^{\\pm}}$ & $m_{\\chi_2^{\\pm}}$ & $m_{\\tilde{\\nu}_L}$ \\\\ \\hline\np1 & 198.3 & 310.5 & 336.4 & 403.8 & 262.1 & 197.2 & 350.2 & 174.5 \\\\ \\hline\np2 & 237.1 & 317.0 & 417.9 & 459.2 & 185.1 & 237.7 & 428.9 & 408.9 \\\\ \\hline \\hline\n & $m_{\\tilde{\\nu}_\\tau}$ & $m_{\\tilde{e}_L}$ & $m_{\\tilde{e}_R}$ & $m_{\\tilde{\\tau}_1}$ & $m_{\\tilde{\\tau}_2}$ & $m_{\\tilde{u}_L}$ & $m_{\\tilde{u}_R}$ & $m_{\\tilde{t}_1}$ \\\\ \\hline\np1 & 174.4 & 184.5 & 651.7 & 184.3 & 651.6 & 1231.7 & 991.9 & 829.5 \\\\ \\hline\np2 & 408.8 & 413.1 & 925.1 & 413.1 & 925.1 & 1682.2 & 1347.1 & 1126.7 \\\\ \\hline \\hline\n & $m_{\\tilde{t}_2}$ & $m_{\\tilde{d}_L}$ & $m_{\\tilde{d}_R}$ & $m_{\\tilde{b}_1}$ & $m_{\\tilde{b}_2}$ & $\\Omega h^2$ & $\\sigma_p^{SI} ({\\rm cm}^2)$ &  \\\\ \\hline\np1 & 1177.4 & 1233.2 & 1169.1 & 1166.9 & 1170.1 & 0.111 & $3.3\\times 10^{-44}$ & \\multicolumn{1}{l|}{} \\\\ \\hline\np2 & 1605.4 & 1683.3 & 1573.3 & 1573.1 & 1598.9 & 0.131 & $3.1\\times 10^{-44}$ & \\multicolumn{1}{l|}{} \\\\ \\hline\n\\end{tabular}\n\\label{tab:masses}\n}\n\n\\section{Conclusions}\nWe investigated the phenomenology of the nMSSM with a Fayet-Iliopoulos D-term in the positively deflected anomaly mediation scenario.\n\nIn the deflected anomaly mediation scenario, the messenger sector is introduced.\nWe showed that the couplings between the nMSSM fields and the messenger sector fields are forbidden by the discrete symmetry,\nand therefore the phenomenology at the weak scale is not affected by the detail of the\nmessenger sector.\nWe evaluated the soft breaking terms at the messenger scale without assuming small Yukawa couplings,\nand showed that the contributions from Yukawa couplings are the same as those of anomaly mediation. \nThe soft breaking parameters are determined by the deflection parameter $d$, \nthe messenger scale and contributions from the Fayet-Iliopoulos D-term.\n\nWe also discussed the phenomenology of the nMSSM at the weak scale.\nWe found that electroweak symmetry breaking is successful, and moderate values of $\\mu_{eff}$ are obtained.\nThe mass of the lightest Higgs is heavier than the LEP bound.\nWe also obtained sparticle mass spectra, and interestingly, the gluino is light.\n\nWe showed that the lightest neutralino is a good candidate for dark matter. \nThe relic density explains the observed abundance of dark matter.\nThe spin-independent dark matter-proton scattering cross section \nsatisfies the upper limit from XENON10 when we consider a small value of the strange quark content of the nucleon as \nindicated by a recent lattice calculation.\nThe cross section is large enough to be detected or excluded by next-generation experiments of direct detection. \n\nWe consider this scenario phenomenologically viable. If the light gluino is discovered, it may imply\nthat SUSY breaking is mediated by supergravity and messengers,\nand these two effects are comparable.\n\n\\section*{Acknowledgements}\nWe thank H. Ohki for useful discussion on the direct detection of dark matter and informing us of the nucleon sigma term.\nWe also thank M. Ibe for discussion on SUSY breaking effects of an intermediate threshold and DM-nucleon scattering cross section. \nWe would like to thank K. Akina and T. Morozumi for careful reading of the manuscript.\nWe thank A. Masiero for discussion on symmetry breaking terms. \nWe acknowledge M. Okawa, K. Ishikawa and T. Inagaki for support and encouragement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\nWeak scale supersymmetry (SUSY) provides an elegant solution to the \nnaturalness problem of the standard model, by invoking a cancellation \nbetween the standard model and its superpartner contributions to the \nHiggs potential.  An interesting consequence of this framework is that \nthe three gauge couplings unify at an extremely high energy of order \n$M_U \\approx 10^{16}~{\\rm GeV}$, if a normalization of the $U(1)_Y$ \ngauge coupling is adopted that allows the embedding of the standard \nmodel gauge group into a larger simple symmetry group: $SU(5) \\supset \nSU(3)_C \\times SU(2)_L \\times U(1)_Y$.  This suggests the existence \nof some unified physics above this energy scale, which in some form \nutilizes $SU(5)$ or a larger group containing it.\n\nThe simplest possibilities for physics above $M_U$ are four dimensional \n(4D) supersymmetric grand unified theories (GUTs)~\\cite{Dimopoulos:1981zb}. \nIn these theories, physics above $M_U$ is described by 4D supersymmetric \ngauge theories in which the standard model gauge group is embedded \ninto a larger (simple) gauge group.  This, however, leads to the \nproblem of doublet-triplet splitting in the Higgs sector, and often \nleads to too rapid proton decay caused by the exchange of colored \ntriplet Higgsinos~\\cite{Murayama:2001ur}.  While several solutions \nto these problems have been proposed within conventional 4D SUSY \nGUTs~[\\ref{Witten:1981kv:X}~--~\\ref{Yanagida:1994vq:X}], their \nexplicit implementations often require the introduction of a larger \nmultiplet(s) and\/or specifically chosen superpotential interactions, \nespecially when one tries to make the models fully realistic.  This \nloses a certain beauty the simplest theory had, especially if one \nadopts the viewpoint that these theories are ``fundamental,'' arising \ndirectly from physics at the gravitational scale, e.g. string theory.\n\nAn alternative possibility for physics above $M_U$ is that the unified \ngauge symmetry is realized in higher dimensional (semi-)classical \nspacetime~[\\ref{Kawamura:2000ev:X}~--~\\ref{Hall:2002ea:X}].  In this \ncase there is no 4D unified gauge symmetry containing the standard \nmodel gauge group as a subgroup --- the unified symmetry in higher \ndimensions is broken locally and explicitly by a symmetry breaking \ndefect.  This structure allows a natural splitting between the doublet \nand triplet components for the Higgs fields, while the successful \nprediction for gauge coupling unification is recovered by diluting \nthe effects from the defect due to a moderately large extra dimension(s). \nDangerous proton decay is suppressed by an $R$ symmetry, arising \nnaturally from the higher dimensional structure of the triplet Higgsino \nmass matrix.  The framework also allows for a simple understanding \nof the observed structure of fermion masses and mixings, in terms \nof wavefunction suppressions of the Yukawa couplings arising for \nbulk quarks and leptons~\\cite{Hall:2001zb,Hebecker:2002re}.\n\nIn this paper we study a framework for physics above $M_U$ in \nwhich the standard model gauge group is unified into a simple gauge \ngroup in precisely the same sense as in conventional 4D SUSY GUTs, \nand yet mechanisms and intuitions developed in higher dimensions \ncan be used to address the various issues of unified theories. \nLet us consider that the standard model gauge group is embedded into \na simple unified gauge group, e.g. $SU(5)$, at energies above $M_U$. \nWe assume that the unified gauge symmetry is broken by strong gauge \ndynamics associated with another gauge group $G$, and that this gauge \ngroup has a large 't~Hooft coupling $\\tilde{g}^2 \\tilde{N}\/16\\pi^2 \n\\gg 1$, where $\\tilde{g}$ and $\\tilde{N}$ are the gauge coupling \nand the number of ``colors'' for the gauge group $G$.  With these \nvalues of the 't~Hooft coupling for $G$, an appropriate (weakly \ncoupled) description of physics is given in higher dimensional \nwarped spacetime (for $\\tilde{N} \\gg 1$), due to the gauge\/gravity \ncorrespondence~\\cite{Maldacena:1997re}. In the simplest setup where \n$\\tilde{g}$ evolves slowly above the dynamical scale, our theories are \nformulated in 5D anti-de~Sitter (AdS) spacetime truncated by two branes, \nwhere the curvature scales on the ultraviolet (UV) and infrared (IR) \nbranes are chosen to be $k \\approx (10^{17}-10^{18})~{\\rm GeV}$ and \n$k' \\approx (10^{16}-10^{17})~{\\rm GeV}$, respectively.  This allows \nus to construct simple ``calculable'' unified theories in which the \nunified gauge symmetry is broken dynamically --- physics above $M_U$ \nis determined simply by specifying parameters in higher dimensional \neffective field theory.\n\nIn this paper we construct realistic unified theories in the \nframework described above.  In general, there are many ways to address \nthe issues of unified theories in our framework.  In one example, \nwhich we discuss in detail, we use the idea that the Higgs doublets \nof the minimal supersymmetric standard model (MSSM) are pseudo-Goldstone \nbosons of a broken global symmetry~\\cite{Inoue:1985cw,Barbieri:1992yy}. \nSpecifically, we assume that the $G$ sector possesses a global $SU(6)$ \nsymmetry, of which an $SU(5)$ ($\\times U(1)$) subgroup is gauged. \nThe gauged $SU(5)$ group contains the standard model gauge group as \na subgroup.  We assume that the dynamics of $G$ breaks the global \n$SU(6)$ symmetry down to $SU(4) \\times SU(2) \\times U(1)$ at the scale \n$M_U$, which leads to the correct gauge symmetry breaking, $SU(5) \n\\rightarrow SU(3)_C \\times SU(2)_L \\times U(1)_Y$, and ensures that \nthe Higgs doublets remain massless after the symmetry breaking, without \nbeing accompanied by their colored triplet partners.  The simplest \nrealization of our theory, corresponding to this symmetry structure, \nis then obtained in 5D truncated warped space in which the bulk $SU(6)$ \ngauge symmetry is broken to $SU(5) \\times U(1)$ and $SU(4) \\times SU(2) \n\\times U(1)$ on the UV and IR branes, respectively.  Realistic unified \ntheories having this symmetry structure were constructed previously \nin flat space in Ref.~\\cite{Burdman:2002se}, where the symmetry \nbreakings on the two branes are both caused by boundary conditions, \nand in Ref.~\\cite{Cheng:1999fw}, where the breakings are by the \nHiggs mechanism.  In our context, we find that the simplest theory \nis obtained if the breakings on the UV and IR branes are caused by \nboundary conditions and the Higgs mechanism, respectively.  Note that, \nin the ``4D description'' of the theory, the Higgs breaking on the IR \nbrane corresponds to dynamical GUT breaking, and the low-energy Higgs \ndoublets are interpreted as composite particles of the dynamical \nGUT-breaking sector.  This theory thus provides a simple explicit \nrealization of the composite pseudo-Goldstone Higgs doublets, in which \nthe origin of the global $SU(6)$ symmetry can be understood as the \n``flavor'' symmetry of the dynamical GUT-breaking sector. \n\nBelow the GUT-breaking scale $M_U$, our theory is reduced to the \nMSSM (supplemented by small seesaw neutrino masses).  The successful \nunification prediction for the low-energy gauge couplings is \npreserved as long as the threshold corrections from the dynamical \nGUT-breaking sector are sufficiently small.  Our higher dimensional \ndescription of the theory allows us to estimate the size of these \ncorrections, and we find that this can be the case.  Dimension five \nproton decay does not exclude the theory, because of the existence \nof these threshold corrections.  Realistic quark and lepton mass \nmatrices can also be reproduced, where the observed hierarchies \nin masses and mixings are understood in terms of the wavefunction \nprofiles of the quark and lepton fields.  In the 4D description of \nthe theory, these hierarchies arise through mixings between elementary \nstates and composite states of $G$, which are given by powers of \n$M_U\/M_*$, where $M_*$ is the fundamental scale of the theory, close \nto the 4D Planck scale.  Unwanted unified mass relations for the \nfirst two generation fermions do not arise, because of GUT breaking \neffects in the $G$ sector.\n\nWe also discuss other possible theories in our framework.  We show \nthat it allows for the construction of large classes of models, \nincluding missing partner type and product group type models. \nIn most of them, the Higgs doublets arise as states localized to \nthe IR brane, corresponding to composite states of the strong $G$ \ndynamics. A 4D scenario related to these theories was discussed \npreviously in Ref.~\\cite{Kitano:2005ez}, based on a supersymmetric \nconformal field theory (CFT), where a possible AdS interpretation \nwas also noted.  In all of these theories, our higher dimensional \nframework allows a straightforward implementation of the mechanism \ngenerating the hierarchical fermion masses and mixings, in terms of \nthe wavefunction profiles of matter fields in the extra dimension.\n\nThe organization of the paper is as follows.  In the next section we \ndescribe the basic structure of our theory using the 4D description. \nWe describe how the MSSM arises naturally at low energies in this \ntheory.  In section~\\ref{sec:model} we construct an explicit model \nin truncated 5D warped space.  We show that the model does not suffer \nfrom problems of conventional 4D SUSY GUTs, e.g. the doublet-triplet \nsplitting and dimension five proton decay problems, and also that \nthe observed hierarchies in the quark and lepton mass matrices can \nbe understood in terms of the wavefunction profiles of these fields \nin the extra dimension.  In section~\\ref{sec:other}, we discuss \nother possible theories in our framework, including missing partner \ntype and product group type models.  Discussion and conclusions \nare given in section~\\ref{sec:concl}, which include a comment \non the possibility of having a theory with $\\tilde{g}^2 \n\\tilde{N}\/16\\pi^2 \\simlt 1$. \n\n\n\\section{Basic Picture}\n\\label{sec:picture}\n\nIn this section we describe our theory using the 4D description. \nHere we focus on the case where the light Higgs doublets of the MSSM \narise as pseudo-Goldstone supermultiplets of the GUT scale dynamics. \nThis has the virtue that the success of the theory is essentially \nguaranteed by its symmetry structure, without relying on specifically \nchosen matter content or interactions.  Other possibilities will be \ndiscussed in section~\\ref{sec:other}.\n\nWe consider that the standard model gauge group is embedded into \na simple gauge group $SU(5)$, which is spontaneously broken at \nthe scale $M_U \\approx 10^{16}~{\\rm GeV}$.  What is the underlying \ndynamics of this symmetry breaking?  A hint will come from considering \nhow the MSSM arises below the symmetry breaking scale $M_U$.  In \nparticular, considering how the MSSM matter content naturally appears \nat energies below $M_U$ and why interactions among these particles \n-- the gauge and Yukawa interactions -- take the observed form and \nvalues will provide a guide to the physics of this symmetry breaking. \nThe suppression of certain operators allowed by standard model gauge \ninvariance, e.g. the ones leading to dangerous dimension five proton \ndecay, may also give hints regarding the structure of this physics.\n\nWe focus on the possibility that the unified gauge group, $SU(5)$, is \nspontaneously broken by dynamics associated with another gauge group \n$G$.  In this setup, the $G$ sector is charged under $SU(5)$, as it \nbreaks $SU(5)$ dynamically.  The setup also allows the existence of other \nfields -- elementary fields -- that are singlet under $G$ and charged \nunder $SU(5)$.  Suppose now that the theory has a matter content that \nsatisfies $n_{{\\bf 5}^*} - n_{\\bf 5} = n_{\\bf 10} - n_{{\\bf 10}^*} = 3$ \nand $n_{\\bf r} - n_{{\\bf r}^*} = 0$ (${\\bf r} \\neq {\\bf 5}, {\\bf 10}$), \nwhere $n_{\\bf r}$ represents the number of $SU(5)$ multiplets in \na complex representation ${\\bf r}$.  The matter content is arbitrary \notherwise.  (Note that this is not a very strong requirement on the \nspectrum --- with $n_{\\bf r} - n_{{\\bf r}^*} = 0$ for ${\\bf r} \\neq \n{\\bf 5}, {\\bf 10}$, the condition $n_{{\\bf 5}^*} - n_{\\bf 5} = n_{\\bf 10} \n- n_{{\\bf 10}^*}$ arises automatically as a consequence of anomaly \ncancellation.)  With this assumption, the low energy matter content \nis expected to be just the three generations of quarks and leptons, \nno matter what happens associated with the dynamics of the GUT-breaking \nsector $G$.  In general, the gauge dynamics of $G$ will produce an \narbitrary number of split GUT multiplets as composite states, by picking \nup the effect of GUT breaking.  These states can then mix with the \nelementary states, so that the low energy states are in general mixtures \nof elementary and composite states and thus a collection of various \nincomplete $SU(5)$ multiplets.  Nevertheless, conservation of chirality \nguarantees that we always have three generations of quarks and leptons \nat low energies, although they may not arise simply from three copies \nof $({\\bf 5}^* + {\\bf 10})$.  Assuming that all the fields vector-like \nunder the standard model gauge group obtain masses of order $M_U$ \nthrough nonperturbative effects of $G$, the matter content below \n$M_U$ is exactly the three generations of quarks and leptons.\n\nThe above argument shows that we can naturally obtain a low-energy \nchiral matter content that fills complete $SU(5)$ multiplets for chirality \nreasons (although each component in a multiplet may come from several \ndifferent $SU(5)$ multiplets at high energies).  It also implies that \nany multiplets that do not fill out a complete $SU(5)$ multiplet must \nbe vector-like.  It is interesting that the MSSM has exactly this \nstructure.  Unless there is some special reason, however, the vector-like \nstates are all expected to have masses of order $M_U$ from nonperturbative \neffects of $G$.  What could the special reason be for the Higgs doublets? \n\nThe lightness of the Higgs doublets can be understood group theoretically \nif we identify these states as pseudo-Goldstone bosons of a broken global \nsymmetry~\\cite{Inoue:1985cw}.  Suppose that the $G$ sector possesses \na global $SU(6)$ symmetry, of which an $SU(5)$ ($\\times U(1)$) subgroup \nis gauged and identified as the unified gauge symmetry.  We assume that \nthe dynamics of $G$ breaks the global $SU(6)$ symmetry down to $SU(4) \n\\times SU(2) \\times U(1)$ at the dynamical scale $\\approx M_U$ in such \na way that the gauged $SU(5)$ subgroup is broken to the standard model \ngauge group $SU(3)_C \\times SU(2)_L \\times U(1)_Y$ (321).  This leads \nto Goldstone chiral supermultiplets, whose quantum numbers under 321 \nare given by $({\\bf 3}, {\\bf 2})_{-5\/6} + ({\\bf 3}^*, {\\bf 2})_{5\/6} \n+ ({\\bf 1}, {\\bf 2})_{1\/2} + ({\\bf 1}, {\\bf 2})_{-1\/2}$.  While the \nfirst two of these are absorbed by the broken $SU(5)$ gauge multiplets \n(the massive XY gauge supermultiplets), the last two are left in the \nlow energy spectrum.  Although the global $SU(6)$ symmetry of the $G$ \nsector is explicitly broken by the gauging of the $SU(5)$ ($\\times U(1)$) \nsubgroup, the supersymmetric nonrenormalization theorem guarantees that \nthe mass term for $({\\bf 1}, {\\bf 2})_{1\/2} + ({\\bf 1}, {\\bf 2})_{-1\/2}$ \nis not generated without picking up the effect of supersymmetry breaking, \nallowing us to identify these states as the two Higgs doublets of the \nMSSM: $H_u({\\bf 1}, {\\bf 2})_{1\/2}$ and $H_d({\\bf 1}, {\\bf 2})_{-1\/2}$. \nThis provides a complete understanding of the MSSM field content in \nour framework.  The MSSM states -- the gauge, matter and Higgs fields \n-- are the only states that could not get a mass of order $M_U$ from \n$G$, because they are protected by gauge invariance, chirality, and \nthe (pseudo-)Goldstone mechanism.\n\nSince the two Higgs doublets arise from the dynamical breaking of \n$SU(6)$, they are composite states of $G$.  Suppose now that the \ndynamics of $G$ also produces composite states that have the same 321 \nquantum numbers as the MSSM quarks and leptons, ${\\cal Q}, {\\cal U}, \n{\\cal D}, {\\cal L}$ and ${\\cal E}$.  These composite states will then \nhave ``Yukawa couplings'' with the Higgs fields at $M_U$, $W \\approx \n{\\cal Q} {\\cal U} H_u + {\\cal Q} {\\cal D} H_d + {\\cal L} {\\cal E} H_d$, \nwhere the sizes of the couplings are naturally of order $4\\pi$.  These \ncouplings, however, disappear at low energies after integrating out all \nthe heavy modes, because the strong $G$ dynamics respects $SU(6)$ and \nthe Higgs doublets are the Goldstone bosons associated with the dynamical \nbreaking of $SU(6)$.  Now, suppose that the theory also has several \nelementary fields that transform as ${\\bf 5}^*$ and ${\\bf 10}$ under \n$SU(5)$.  In this case the low-energy quarks and leptons, \n$Q, U, D, L$ and $E$, are in general linear combinations of the \nelementary and composite states.  The Yukawa couplings for these \nlow-energy fields, $W \\approx Q U H_u + Q D H_d + L E H_d$, can then \nbe nonzero because the elementary states do not respect the full \n$SU(6)$ symmetry.  The sizes of the Yukawa couplings are determined by \nthe strengths of the mixings between the elementary and composite states, \nwhich are in turn determined by the dimensions of the $G$-invariant \noperators that interpolate the composite states.  This situation is \nanalogous to the case where the standard model Higgs boson is identified \nas a pseudo-Goldstone boson of strong gauge dynamics at the TeV \nscale~\\cite{Contino:2003ve}.  By choosing operator dimensions to be \nlarger for lighter generations, we can naturally understand the origin \nof the hierarchical structure for the quark and lepton masses and \nmixings.  The unwanted mass relations for the quarks and leptons \ncan be avoided because low-energy quarks and leptons feel the \nGUT-breaking effects in the $G$ sector.\n\nDangerous dimension four and five proton decay can be suppressed if \nthe theory possesses a continuous or discrete $R$ symmetry, under which \nthe low-energy MSSM fields carry the charges $Q(1)$, $U(1)$, $D(1)$, \n$L(1)$, $E(1)$, $H_u(0)$ and $H_d(0)$ (and $N(1)$ if we introduce \nright-handed neutrino superfields $N$).  This $R$ symmetry is most \nlikely spontaneously broken by the dynamics of the $G$ sector (unless \nthere is a low-energy singlet field that transforms nonlinearly \nunder this symmetry; see discussion in section~\\ref{sec:other}). \nThe $R$ symmetry should also be broken to the $Z_2$ subgroup, \nthe $R$ parity of the MSSM, in order to give weak scale masses to \nthe gauginos.  Supersymmetry breaking produces supersymmetric and \nsupersymmetry-breaking masses for the Higgs doublets, as well as \nmasses for the gauginos, squarks and sleptons, ensuring the stability \nof the desired vacuum.  Successful supersymmetric gauge coupling \nunification is preserved if the threshold corrections associated \nwith the $G$ sector are sufficiently small. \n\nWe have depicted the basic picture of the theory in Fig.~\\ref{fig:basic}. \n\\begin{figure}[t]\n\\begin{center} \n\\begin{picture}(310,170)(0,0)\n  \\CArc(10,50)(10,180,270)  \\CArc(240,50)(10,270,360)\n  \\CArc(10,160)(10,90,180)  \\CArc(240,160)(10,0,90)\n  \\Line(10,40)(240,40)  \\Line(10,170)(240,170)\n  \\Line(0,50)(0,160)  \\Line(250,50)(250,160)\n  \\Text(15,152)[l]{gauged $SU(5)$}  \\Text(125,152)[]{$\\subset$}\n  \\Text(32,130)[l]{${\\bf 5}^*_1, {\\bf 5}^*_2, \\cdots$}\n  \\Text(32,114)[l]{${\\bf 10}_1, {\\bf 10}_2, \\cdots$}\n  \\Text(32,98)[l]{$\\cdots \\cdots$}\n  \\CArc(192,95)(43,0,360)  \\Text(228,152)[r]{global $SU(6)$}\n  \\Text(192,109)[]{\\Large $G$}\n  \\Text(156,89)[l]{\\tiny $SU(6) \\rightarrow$}\n  \\Text(160,82)[l]{\\tiny $SU(4) \\!\\times\\! SU(2) \\!\\times\\! U(1)$}\n  \\Text(264,161)[l]{$n_{{\\bf 5}^*} - n_{\\bf 5} = 3$}\n  \\Text(264,146)[l]{$n_{{\\bf 10}} - n_{{\\bf 10}^*} = 3$}\n  \\Text(264,131)[l]{$n_{{\\bf r}} - n_{{\\bf r}^*} = 0$}\n  \\Text(285,117)[l]{\\small $({\\bf r} \\neq {\\bf 5}, {\\bf 10})$}\n  \\Line(22,140)(22,25) \\Line(22,25)(19,31) \\Line(22,25)(25,31)\n  \\Text(22,20)[t]{$V_{SU(3)_C}, V_{SU(2)_L}, V_{U(1)_Y}$}\n  \\Line(140,50)(172,72)  \\Line(140,50)(76,94)\n  \\Line(140,50)(140,25) \\Line(140,25)(137,31) \\Line(140,25)(143,31)\n  \\Text(140,20)[t]{$3 \\times (Q,U,D,L,E)$}\n  \\Line(220,72)(220,25) \\Line(220,25)(217,31) \\Line(220,25)(223,31)\n  \\Text(220,20)[t]{$H_u, H_d$}\n  \\Text(253,15)[l]{$\\cdots \\cdots$}\n  \\Text(294,16)[l]{MSSM}\n\\end{picture}\n\\caption{The basic picture of the theory in the 4D description.}\n\\label{fig:basic}\n\\end{center}\n\\end{figure}\nHow can we realize this picture in explicit models?  It is not so \nstraightforward to construct such models in the conventional 4D framework. \nIn particular, it is not easy to find explicit gauge group and matter \ncontent for the $G$ sector having all the features described above.  (The \ndifficulty increases if some of the relevant composite states are excited \nstates of the $G$ sector.  We then cannot use beautiful exact results for \n${\\cal N} = 1$ supersymmetric gauge theories~\\cite{Intriligator:1995au}, \nwhich are applicable to lowest-lying modes.)  In our framework, however, \nthis problem is in some sense ``bypassed.''  Suppose that the $G$ sector \npossesses a large 't~Hooft coupling, $\\tilde{g}^2 \\tilde{N} \/16\\pi^2 \n\\gg 1$.  In this case, the theory is so strongly coupled that the gauge \ntheory description in terms of ``gluons'' and ``quarks'' does not make \nmuch sense.  Instead, in this parameter region, the theory is better \nspecified by composite ``hadron'' states, which have a tower structure. \nFor $\\tilde{N} \\gg 1$, these ``hadronic'' tower states are weakly \ncoupled~\\cite{'tHooft:1973jz}, and under certain circumstances \nthey can be identified as the Kaluza-Klein (KK) states of a weakly \ncoupled higher dimensional theory.  In particular, if the $G$ \nsector is quasi-conformal ($\\tilde{g}$ evolves very slowly) \nabove its dynamical scale, the corresponding higher dimensional \ntheory is formulated in warped AdS spacetime truncated by \nbranes~\\cite{Maldacena:1997re,Arkani-Hamed:2000ds}.  In the next section \nwe construct an explicit unified model in truncated 5D warped spacetime, \nwhich has all the features described in this section.  In practice, \nonce we have a theory in higher dimensions, we can forget about the \n``original'' 4D picture for most purposes --- our higher dimensional \ntheory is an effective field theory with which we can consistently \ncalculate various physical quantities.  The theory does not require \nany more information than the gauge group, matter content, boundary \nconditions, and values of various parameters, to describe physics \nat energies below the cutoff scale $M_*$ ($\\gg M_U$). \n\n\n\\section{Model}\n\\label{sec:model}\n\n\\subsection{Basic symmetry structure}\n\\label{subsec:symm}\n\nFollowing the general picture presented in the previous section, we \nconsider 5D warped spacetime truncated by two branes: the UV and IR \nbranes.  The spacetime metric is given by\n\\begin{equation}\n  ds^2 = e^{-2ky} \\eta_{\\mu\\nu} dx^\\mu dx^\\nu + dy^2,\n\\label{eq:metric}\n\\end{equation}\nwhere $y$ is the coordinate for the extra dimension and $k$ denotes the \ninverse curvature radius of the warped AdS spacetime.  The two branes \nare located at $y=0$ (the UV brane) and $y=\\pi R$ (the IR brane).  This \nis the spacetime considered in Ref.~\\cite{Randall:1999ee}, in which the \nAdS warp factor is used to generate the large hierarchy between the weak \nand the Planck scales by choosing the scales on the UV and IR branes \nto be the Planck and TeV scales, respectively ($kR \\sim 10$).  Here \nwe choose instead the UV-brane and IR-brane scales to be $k \\approx \n(10^{17}-10^{18})~{\\rm GeV}$ and $k' \\equiv k\\, e^{-\\pi kR} \\approx \n(10^{16}-10^{17})~{\\rm GeV}$, respectively, so that the IR brane serves \nthe role of breaking the unified symmetry.  (A more detailed discussion \non the determination of the scales is provided in later subsections.) \nIn this sense, we may loosely call the UV and IR branes the Planck \nand GUT branes, respectively. \n\nWe consider supersymmetric unified gauge theory on this gravitational \nbackground.  We choose the gauge symmetry in the bulk to be $SU(6)$, \ncorresponding to the global symmetry that the dynamical GUT-breaking \nsector possesses in the 4D description of the model.  The bulk \n$SU(6)$ gauge symmetry is broken to $SU(5) \\times U(1)$ and $SU(4) \n\\times SU(2) \\times U(1)$ on the UV and IR branes, respectively, \nleaving an unbroken $SU(3) \\times SU(2) \\times U(1) \\times U(1)$ gauge \nsymmetry at low energies.  There are two ways to break a gauge symmetry \non a brane: by boundary conditions and by the Higgs mechanism.  Let \nus first consider $SU(6) \\rightarrow SU(5) \\times U(1)$ on the UV \nbrane.  If this breaking is caused by the Higgs mechanism, then in \nthe corresponding 4D description the fundamental gauge symmetry \nof the theory is $SU(6)$, which is spontaneously broken to $SU(5) \n\\times U(1)$ at a very high energy $E \\gg M_U$.  In this case, we \nmust introduce matter fields in representations of $SU(6)$, so that \nthe standard $SU(5)$ embedding of matter fields~\\cite{Georgi:1974sy} \nshould be modified\/extended.  On the other hand, if $SU(6) \\rightarrow \nSU(5) \\times U(1)$ on the UV brane is caused by boundary conditions, \nthen in the corresponding 4D description only the $SU(5) \\times \nU(1)$ subgroup of the global $SU(6)$ symmetry is explicitly gauged \n(see Fig.~\\ref{fig:basic}), so that we can employ the standard $SU(5)$ \nembedding for matter fields.  We thus adopt the latter option to \nconstruct our minimal model here, although models based on the former \noption can also be accommodated in our framework.\n\n\\begin{figure}[t]\n\\begin{center}\n  \\input{figure.tex}\n\\caption{A schematic picture of the model in 5D.}\n\\label{fig:5D}\n\\end{center}\n\\end{figure}\nWhat about the symmetry breaking $SU(6) \\rightarrow SU(4) \\times SU(2) \n\\times U(1)$ on the IR brane?  If we break $SU(6)$ to $SU(4) \\times SU(2) \n\\times U(1)$ by boundary conditions on the IR brane, the two massless \nHiggs doublets, whose existence is guaranteed by the general symmetry \nargument presented in the previous section, arise from extra-dimensional \ncomponents of the bulk $SU(6)$ gauge fields.  This setup, however, leads \nto extra states lighter than $k' \\approx (10^{16}-10^{17})~{\\rm GeV}$ \nonce matter fields are introduced in the bulk with the zero modes \nlocalized towards the UV brane (such matter fields are used to naturally \nexplain the observed hierarchies in the fermion masses and mixings; \nsee subsection~\\ref{subsec:matter}).  These extra states generically \ndo not fill complete $SU(5)$ representations and thus induce large \nthreshold corrections for the standard model gauge couplings.  Large \nthreshold corrections can be avoided if we judiciously choose boundary \nconditions for matter fields, but bulk $SU(6)$ gauge invariance then \nstill requires complicated structure for the matter sector to reproduce \nthe observed fermion masses and mixings.  These issues do not arise \nif the breaking $SU(6) \\rightarrow SU(4) \\times SU(2) \\times U(1)$ \nis caused by the Higgs mechanism on the IR brane, as we will see later. \nWe therefore adopt the Higgs breaking of $SU(6)$ on the IR brane. \nThis completely determines the basic symmetry structure of our model, \nwhich is depicted in Fig.~\\ref{fig:5D}.\n\n\n\\subsection{Gauge-Higgs sector and scales of the system}\n\\label{subsec:Higgs}\n\nLet us start by describing the gauge-Higgs sector of the model.  Using \n4D $N=1$ superfield language, in which the gauge degrees of freedom \nare contained in $V(A_\\mu, \\lambda)$ and $\\Phi(\\phi+iA_5, \\lambda')$, \nthe boundary conditions for the 5D $SU(6)$ gauge supermultiplet are \ngiven by\n\\begin{eqnarray}\n  && V:\\: \\left( \\begin{array}{ccccc|c}\n    (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \n    (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \n    (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \n    (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \n    (+,+) & (+,+) & (+,+) & (+,+) & (+,+) & (-,+) \\\\ \\hline\n    (-,+) & (-,+) & (-,+) & (-,+) & (-,+) & (+,+) \n  \\end{array} \\right),\n\\label{eq:bc-gauge-1} \\\\\n  && \\Phi:\\: \\left( \\begin{array}{ccccc|c}\n    (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \n    (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \n    (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \n    (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \n    (-,-) & (-,-) & (-,-) & (-,-) & (-,-) & (+,-) \\\\ \\hline\n    (+,-) & (+,-) & (+,-) & (+,-) & (+,-) & (-,-) \n  \\end{array} \\right),\n\\label{eq:bc-gauge-2}\n\\end{eqnarray}\nwhere $+$ and $-$ represent Neumann and Dirichlet boundary conditions, \nrespectively, and the first and second signs in parentheses represent \nboundary conditions at $y=0$ and $y=\\pi R$, respectively.  These boundary \nconditions lead to $SU(6) \\rightarrow SU(5) \\times U(1)$ on the UV \nbrane.  Since only $(+,+)$ components have zero modes, we obtain 4D \n$N=1$ $SU(5) \\times U(1)$ gauge supermultiplets as massless fields at \nthis point (coming from the upper left $5 \\times 5$ block and the lower \nright element in $V$).  All the other KK modes have masses of order \n$\\pi k'$ or larger.\n\nThe symmetry breaking $SU(6) \\rightarrow SU(4) \\times SU(2) \\times \nU(1)$ on the IR brane is caused by the vacuum expectation value (VEV) \nof a field $\\Sigma({\\bf 35})$ localized to the IR brane, where the \nnumber in the parenthesis represents the transformation property under \n$SU(6)$.  We here consider that the $\\Sigma$ field is strictly localized \non the IR brane and has the following superpotential:\n\\begin{equation}\n  {\\cal L}_\\Sigma = \\delta(y-\\pi R) \n    \\biggl[ \\int\\! d^2\\theta \\Bigl( \\frac{M}{2}\\, {\\rm Tr}(\\Sigma^2) \n      + \\frac{\\lambda}{3}\\, {\\rm Tr}(\\Sigma^3) \\Bigr) \n    + {\\rm h.c.} \\biggr],\n\\label{eq:IR-Higgs}\n\\end{equation}\nwhere the metric factor is absorbed into the normalization of \nthe $\\Sigma$ field.  (We will always absorb the metric factor into \nthe normalizations of fields in similar expressions below, denoted \nby ${\\cal L}$.)  The field $\\Sigma$ is canonically normalized in 4D, \nso that natural values of the parameters $M$ and $\\lambda$ are of \norder $M'_* = M_* e^{-\\pi kR}$ and $4\\pi$, respectively.  Here, \n$M_*$ is the cutoff scale of the 5D theory.  In general, the IR-brane \npotential for $\\Sigma$ also has higher dimension terms suppressed \nby $M'_*$, in addition to Eq.~(\\ref{eq:IR-Higgs}).  The presence \nof these terms, however, does not affect the qualitative conclusions \nof our paper.  Below, we assume that the parameter $M$ is a factor \nof a few smaller than its naive size, e.g. $M \\sim k'$, to make \nour analysis better controlled.  In this case, the effect of higher \ndimension terms are expected to be suppressed even quantitatively.\n\nThe superpotential of Eq.~(\\ref{eq:IR-Higgs}) has the following vacuum:\n\\begin{eqnarray}\n  \\langle \\Sigma \\rangle \n    &=& {\\rm diag}\\biggl(-\\frac{2M}{\\lambda}, -\\frac{2M}{\\lambda}, \n      \\frac{M}{\\lambda}, \\frac{M}{\\lambda}, \\frac{M}{\\lambda}, \n      \\frac{M}{\\lambda} \\biggr),\n\\label{eq:Sigma-VEV}\n\\end{eqnarray}\nwhere we have chosen $\\lambda, M > 0$ without loss of generality. \nThe VEV of Eq.~(\\ref{eq:Sigma-VEV}) leads to $SU(6) \\rightarrow \nSU(4) \\times SU(2) \\times U(1)$ on the IR brane, making a part of \nthe $SU(5) \\times U(1)$ gauge multiplet $V$ massive.  The remaining \nmassless 4D $N=1$ gauge multiplet is that of $SU(3) \\times SU(2) \n\\times U(1) \\times U(1)$, which we identify as the standard model \ngauge group with an extra $U(1)_X$: $SU(3)_C \\times SU(2)_L \\times \nU(1)_Y \\times U(1)_X$.\n\nAn important aspect of the model is that the vacuum of \nEq.~(\\ref{eq:Sigma-VEV}) is a part of a continuum of vacua, \nwhich can easily be seen by studying the excitations.  Under \nthe unbroken $SU(3)_C \\times SU(2)_L \\times U(1)_Y \\times \nU(1)_X$ gauge symmetry, the $\\Sigma$ field decomposes as\n\\begin{eqnarray}\n  \\Sigma &=& \\Sigma_G({\\bf 8}, {\\bf 1})_{(0,0)}\n    + \\Sigma_W({\\bf 1}, {\\bf 3})_{(0,0)}\n    + \\Sigma_B({\\bf 1}, {\\bf 1})_{(0,0)}\n\\nonumber\\\\\n  && {} + \\Sigma_D({\\bf 3}^*, {\\bf 1})_{(1\/3,2)}\n    + \\Sigma_{\\bar{D}}({\\bf 3}, {\\bf 1})_{(-1\/3,-2)}\n    + \\Sigma_L({\\bf 1}, {\\bf 2})_{(-1\/2,2)}\n    + \\Sigma_{\\bar{L}}({\\bf 1}, {\\bf 2})_{(1\/2,-2)}\n\\nonumber\\\\\n  && {} + \\Sigma_X({\\bf 3}, {\\bf 2})_{(-5\/6,0)}\n    + \\Sigma_{\\bar{X}}({\\bf 3}^*, {\\bf 2})_{(5\/6,0)}\n    + \\Sigma_{S}({\\bf 1}, {\\bf 1})_{(0,0)},\n\\label{eq:Sigma-comp}\n\\end{eqnarray}\nwhere the numbers in parentheses represent the quantum numbers \nunder $SU(3)_C \\times SU(2)_L \\times U(1)_Y \\times U(1)_X$. \nThe normalization of $U(1)_Y$ is chosen to match the conventional \ndefinition of hypercharge, while that of $U(1)_X$ is chosen, when \nmatter fields are introduced, to match the conventional definition \nfor the ``$U(1)_\\chi$'' symmetry arising from $SO(10)\/SU(5)$. \nExpanding the superpotential of Eq.~(\\ref{eq:IR-Higgs}) around the \nvacuum, we find that all the components of $\\Sigma$ obtain masses \nof order $M$ except for $\\Sigma_X$, $\\Sigma_{\\bar{X}}$, $\\Sigma_L$ \nand $\\Sigma_{\\bar{L}}$.  Among these four, the first two are absorbed \ninto the massive $SU(5)\/321$ gauge fields, but the last two remain \nas massless chiral superfields, which parameterize the continuous \ndegeneracy of vacua.  This degeneracy is a consequence of the \nspontaneously broken $SU(6)$ symmetry, and the massless fields have \nthe quantum numbers of a pair of Higgs doublets.  We thus identify \nthese fields as the two Higgs doublets of the MSSM: $H_u$ and $H_d$. \n\nWe have found that the gauge-Higgs sector of our model gives only \nthe 4D $N=1$ gauge supermultiplet for $SU(3)_C \\times SU(2)_L \\times \nU(1)_Y \\times U(1)_X$ and the two Higgs doublets $H_u$ and $H_d$ \nbelow the scale of order $M \\sim k'$.  The $U(1)_X$ gauge symmetry \ncan be broken at a scale somewhat below $k'$ by the Higgs mechanism. \nFor example, we can introduce the superpotential on the UV brane \n\\begin{equation}\n  {\\cal L}_X = \\delta(y) \n    \\biggl[ \\int\\! d^2\\theta\\, Y (X \\bar{X} - \\Lambda^2) \n    + {\\rm h.c.} \\biggr],\n\\label{eq:UV-X}\n\\end{equation}\nwhere $Y$, $X$ and $\\bar{X}$ are UV-brane localized chiral superfields \nthat are singlet under $SU(5)$ and have charges of $0$, $10$ and \n$-10$ under $U(1)_X$, respectively.  The scale $\\Lambda$ is that \nfor $U(1)_X$ breaking, which may be generated by some other dynamics. \nThe superpotential of Eq.~(\\ref{eq:UV-X}) produces the VEVs $\\langle \nX \\rangle = \\langle \\bar{X} \\rangle = \\Lambda$, leading to $U(1)_X$ \nbreaking at the scale $\\Lambda$.  The nonvanishing VEV for $\\bar{X}$ \ncan also be used to generate small neutrino masses through the \nconventional seesaw mechanism, as we will see later.  This motivates\nthe values of the $X$ and $\\bar{X}$ charges.\n\nVarious scales of our system -- the AdS inverse curvature radius \n$k$, the size of the extra dimension $R$, and the cutoff scale \nof the effective 5D theory $M_*$ -- are constrained by the scale \nof gauge coupling unification, the size of the unified gauge coupling \n$g_U \\simeq 0.7$, and the value of the 4D (reduced) Planck scale \n$M_{\\rm Pl} \\simeq 2.4 \\times 10^{18}~{\\rm GeV}$.  In our warped \n5D theory, it is natural to consider that parameters in the bulk and \non the IR brane obey naive dimensional analysis (at least roughly) \nwhile those on the UV brane do not, because the former represent \nstrongly coupled $G$ dynamics while the latter represent the weakly \ncoupled elementary sector.  Using naive dimensional analysis in \nhigher dimensions~\\cite{Chacko:1999hg}, we obtain the following \nLagrangian for the graviton and the gauge fields:\n\\begin{equation}\n  {\\cal L} \\approx \\delta(y) \n    \\Biggl[ \\frac{\\tilde{M}^2}{2} {\\cal R}^{(4)}\n      - \\frac{1}{4 \\tilde{g}^2} F^{\\mu\\nu} F_{\\mu\\nu} \\Biggr]\n    + \\Biggl[ \\frac{1}{2} \\frac{M_*^3}{16\\pi^3} {\\cal R}^{(5)} \n      - \\frac{1}{4} \\frac{C M_*}{16\\pi^3} F^{MN} F_{MN} \\Biggr]\n\\label{eq:gravity-gauge}\n\\end{equation}\nwhere ${\\cal R}^{(4)}$ and ${\\cal R}^{(5)}$ are the 4D and 5D Ricci \ncurvatures, respectively, $M, N = 0,1,2,3,5$, and $C$ is a group \ntheoretical factor, $C \\simeq 6$.  This leads to the following relations:\n\\begin{eqnarray}\n  \\frac{1}{g_U^2} &\\simeq& \\frac{1}{\\tilde{g}^2} \n    + \\frac{C}{16\\pi^2} \\biggl(\\frac{M_*}{\\pi k} \\biggr) \\pi kR,\n\\label{eq:g_U}\\\\\n  M_{\\rm Pl}^2 &\\simeq& \\tilde{M}^2 \n    + \\frac{k^2}{16} \\biggl(\\frac{M_*}{\\pi k} \\biggr)^3 .\n\\label{eq:M_Pl}\n\\end{eqnarray}\nNow, gauge coupling unification at $M_U \\approx 10^{16}~{\\rm GeV}$ \nimplies that we should choose $M$ to be around this scale, and thus \n$k' = k\\, e^{-\\pi kR} \\approx (10^{16}-10^{17})~{\\rm GeV}$.  Then, \nchoosing $M_*\/\\pi k$ to be a factor of a few, e.g. $M_*\/\\pi k \\simeq \n(2\\!\\sim\\!3)$, to make the higher dimensional description trustable, \nwe obtain $k \\simlt 10^{18}~{\\rm GeV}$ from Eq.~(\\ref{eq:M_Pl}) \n(and $\\tilde{M}^2 > 0$).  We thus find that the scales of our 5D \ntheory should be chosen as $k \\approx (10^{17}-10^{18})~{\\rm GeV}$ \nand $k' \\approx (10^{16}-10^{17})~{\\rm GeV}$, which implies \n$kR \\sim 1$, with the cutoff scale $M_*$ a factor of a few larger \nthan $\\pi k$.  The UV-brane gauge coupling $\\tilde{g}$ is then likely \nto be nonzero, implying that the elementary $SU(5)$ gauge field has \nnonvanishing tree-level kinetic terms in the 4D description.  In \nparticular, this implies that elementary $SU(5)$ gauge interactions \nare likely to be weakly coupled at energies $E \\gg M_U$.\n\n\n\\subsection{Matter sector and quark and lepton masses and mixings}\n\\label{subsec:matter}\n\nLet us now include matter fields in the model.  In the 4D description \nof the theory, low-energy quark and lepton fields arise from mixtures \nof elementary states, which transform as ${\\bf 10}$'s and ${\\bf 5}^*$'s \nunder the gauged $SU(5)$, and composite states of $G$, which form \nmultiplets of the global $SU(6)$.  In the 5D theory, this situation is \nrealized by introducing matter hypermultiplets in the bulk, which are \nrepresentations of $SU(6)$, and by imposing $SU(6)$-violating boundary \nconditions on the UV brane.  We here present an explicit realization \nof this picture, leading to realistic phenomenology at low energies.\n\nWe begin by considering the structure of the matter sector for \na single generation.  For quarks and leptons that are incorporated \ninto the ${\\bf 10}$ representation of $SU(5)$, $\\{ Q, U, E \\}$, \nwe introduce a bulk hypermultiplet $\\{ {\\cal T}, {\\cal T}^c \\}$ \ntransforming as ${\\bf 20}$ under $SU(6)$:\n\\begin{eqnarray}\n  {\\cal T}({\\bf 20})\n    &=& {\\bf 10}^{(+,+)}_{1}\n      \\oplus {\\bf 10}^{*(-,+)}_{-1},\n\\label{eq:T} \\\\\n  {\\cal T}^c({\\bf 20})\n    &=& {\\bf 10}^{*(-,-)}_{-1}\n      \\oplus {\\bf 10}^{(+,-)}_{1},\n\\label{eq:Tc}\n\\end{eqnarray}\nwhere ${\\cal T}$ and ${\\cal T}^c$ represent 4D $N=1$ chiral superfields \nthat form a hypermultiplet in 5D.  (Our notation is such that \n``non-conjugated'' and ``conjugated'' chiral superfields have the \nopposite gauge quantum numbers; see e.g.~\\cite{Arkani-Hamed:2001tb}. \nThey have the same quantum numbers for ${\\bf 20}$ of $SU(6)$ \nbecause ${\\bf 20}$ is a (pseudo-)real representation.)  The \nright-hand-side of Eqs.~(\\ref{eq:T},~\\ref{eq:Tc}) shows the \ndecomposition of ${\\cal T}$ and ${\\cal T}^c$ into representations \nof $SU(5) \\times U(1)_X$ (in an obvious notation), as well as the \nboundary conditions imposed on each component (in the same notation \nas that in Eqs.~(\\ref{eq:bc-gauge-1},~\\ref{eq:bc-gauge-2})).  With \nthese boundary conditions, the only massless state arising from \n$\\{ {\\cal T}, {\\cal T}^c \\}$ is ${\\bf 10}_1$ of $SU(5) \\times U(1)_X$ \nfrom ${\\cal T}$, which we identify as the low-energy quarks and \nleptons $Q, U$ and $E$.\n\nA bulk hypermultiplet $\\{ {\\cal H}, {\\cal H}^c \\}$ can generically \nhave a mass term in the bulk, which is written as \n\\begin{equation}\n  S = \\int\\!d^4x \\int_0^{\\pi R}\\!\\!dy \\, \n    \\biggl[ e^{-3k|y|}\\! \\int\\!d^2\\theta\\, \n    c_{\\cal H}\\, k\\, {\\cal H} {\\cal H}^c + {\\rm h.c.} \\biggr],\n\\label{eq:bulk-mass}\n\\end{equation}\nin the basis where the kinetic term is given by $S_{\\rm kin} = \\int\\!d^4x \n\\int\\!dy\\, [e^{-2k|y|} \\int\\!d^4\\theta\\, ({\\cal H}^\\dagger {\\cal H} + \n{\\cal H}^c {\\cal H}^{c\\dagger}) + \\{ e^{-3k|y|} \\int\\!d^2\\theta\\, \n({\\cal H}^c \\partial_y {\\cal H} - {\\cal H} \\partial_y {\\cal H}^c)\/2 + \n{\\rm h.c.} \\}]$~\\cite{Marti:2001iw}.  The parameter $c_{\\cal H}$ controls \nthe wavefunction profile of the zero mode.  For $c_{\\cal H} > 1\/2$ \n($< 1\/2$) the wavefunction of a zero mode arising from ${\\cal H}$ is \nlocalized to the UV (IR) brane; for $c_{\\cal H} = 1\/2$ it is conformally \nflat.  (If a zero mode arises from ${\\cal H}^c$, its wavefunction \nis localized to the IR (UV) brane for $c_{\\cal H} > -1\/2$ ($< -1\/2$) \nand conformally flat for $c_{\\cal H} = -1\/2$.)  We choose these $c$ \nparameters to take values larger than about $1\/2$ for matter fields. \nFor these values of $c$ parameters, all the KK excited states of \n$\\{ {\\cal T}, {\\cal T}^c \\}$ have masses of order $\\pi k'$ or larger, \nso that the $\\{ {\\cal T}, {\\cal T}^c \\}$ multiplet gives only the \nmassless ${\\bf 10}_1$ state below the energy scale of $k'$.\n\nFor quarks and leptons incorporated into the ${\\bf 5}^*$ representation \nof $SU(5)$, $\\{ D, L \\}$, we introduce a bulk hypermultiplet $\\{ {\\cal F}, \n{\\cal F}^c \\}$ transforming as ${\\bf 70}^*$ under $SU(6)$:\n\\begin{eqnarray}\n  {\\cal F}({\\bf 70}^*)\n    &=& {\\bf 5}^{*(+,+)}_{-3}\n      \\oplus {\\bf 10}^{*(-,+)}_{-1}\n      \\oplus {\\bf 15}^{*(-,+)}_{-1}\n      \\oplus {\\bf 40}^{*(-,+)}_{1},\n\\label{eq:F} \\\\\n  {\\cal F}^c({\\bf 70})\n    &=& {\\bf 5}^{(-,-)}_{3}\n      \\oplus {\\bf 10}^{(+,-)}_{1}\n      \\oplus {\\bf 15}^{(+,-)}_{1}\n      \\oplus {\\bf 40}^{(+,-)}_{-1},\n\\label{eq:Fc}\n\\end{eqnarray}\nwhere the right-hand-side again shows the decomposition into \nrepresentations of $SU(5) \\times U(1)_X$, together with the boundary \nconditions imposed on each component.%\n\\footnote{Note that the signs $\\pm$ for the boundary conditions in \nEqs.~(\\ref{eq:F},~\\ref{eq:Fc}) represent the Neumann\/Dirichlet boundary \nconditions in the interval $y: [0, \\pi R]$.  In the orbifold picture, \nthe boundary conditions of Eqs.~(\\ref{eq:F},~\\ref{eq:Fc}) can be \nobtained effectively as follows.  We prepare a hypermultiplet obeying \nthe boundary conditions ${\\cal F}({\\bf 70}^*) = {\\bf 5}^{*(+,+)}_{-3} \n\\oplus {\\bf 10}^{*(-,+)}_{-1} \\oplus {\\bf 15}^{*(-,+)}_{-1} \\oplus \n{\\bf 40}^{*(+,+)}_{1}$ and ${\\cal F}^c({\\bf 70}) = {\\bf 5}^{(-,-)}_{3} \n\\oplus {\\bf 10}^{(+,-)}_{1} \\oplus {\\bf 15}^{(+,-)}_{1} \\oplus \n{\\bf 40}^{(-,-)}_{-1}$, where the first and second signs in the \nparentheses represent transformation properties under the reflection \n$y \\leftrightarrow -y$ and $(y-\\pi R) \\leftrightarrow -(y - \\pi R)$, \nrespectively.  We then introduce a UV-brane localized chiral superfield \ntransforming as ${\\bf 40}_{-1}$ under $SU(5) \\times U(1)_X$, and couple \nit to the ${\\bf 40}^{*(+,+)}_{1}$ state from ${\\cal F}({\\bf 70}^*)$. \nThis reproduces the boundary conditions of Eqs.~(\\ref{eq:F},~\\ref{eq:Fc}) \nin the limit that this coupling (brane mass term) becomes large.  (For \nthe relation between a large brane mass term and the Dirichlet boundary \ncondition, see e.g.~\\cite{Nomura:2001mf}.)  The fact that the boundary \nconditions of Eqs.~(\\ref{eq:F},~\\ref{eq:Fc}) can be reproduced in \nthe orbifold picture by taking a consistent limit guarantees their \nconsistency.  In the 4D description, this corresponds to introducing \nonly a ${\\bf 5}^*_{-3}$ elementary state, which couples to \na component of a $G$-invariant operator transforming as ${\\bf 70}$ \nunder the global $SU(6)$.  Similar remarks also apply to other \nfields, e.g. the $\\{ {\\cal N}, {\\cal N}^c \\}$ hypermultiplet \nin Eqs.~(\\ref{eq:N},~\\ref{eq:Nc}).}\nWith these boundary conditions, the only massless state arising from \n$\\{ {\\cal F}, {\\cal F}^c \\}$ is ${\\bf 5}^*_{-3}$ of $SU(5) \\times \nU(1)_X$ from ${\\cal F}$, which we identify as the low-energy quarks \nand leptons $D$ and $L$.  All the KK excited states have masses of \norder $\\pi k'$ or larger for $c_{\\cal F} \\simgt 1\/2$.\n\nThe right-handed neutrino $N$ arises from a bulk hypermultiplet \n$\\{ {\\cal N}, {\\cal N}^c \\}$ transforming as ${\\bf 56}$ of $SU(6)$:\n\\begin{eqnarray}\n  {\\cal N}({\\bf 56})\n    &=& {\\bf 1}^{(+,+)}_{5}\n      \\oplus {\\bf 5}^{(-,+)}_{3}\n      \\oplus {\\bf 15}^{(-,+)}_{1}\n      \\oplus {\\bf 35}^{(-,+)}_{-1},\n\\label{eq:N} \\\\\n  {\\cal N}^c({\\bf 56}^*)\n    &=& {\\bf 1}^{(-,-)}_{-5}\n      \\oplus {\\bf 5}^{*(+,-)}_{-3}\n      \\oplus {\\bf 15}^{*(+,-)}_{-1}\n      \\oplus {\\bf 35}^{*(+,-)}_{1}.\n\\label{eq:Nc}\n\\end{eqnarray}\nThe zero mode arises only from ${\\bf 1}_5$ in ${\\cal N}$, which is \nidentified as the right-handed neutrino supermultiplet $N$. The other \nKK states are all heavier than of order $\\pi k'$ for $c_{\\cal N} \n\\simgt 1\/2$.\n\nThe Yukawa couplings for the quarks and leptons arise from IR-brane \nlocalized terms\n\\begin{equation}\n  {\\cal L}_{\\rm Yukawa} = \\delta(y-\\pi R) \n    \\biggl[ \\int\\! d^2\\theta \\Bigl( \n      y_{\\cal T} {\\cal T} {\\cal T} \\Sigma\n      + y_{\\cal F} {\\cal T} {\\cal F} \\Sigma \n      + y_{\\cal N} {\\cal F} {\\cal N} \\Sigma \\Bigr) \n    + {\\rm h.c.} \\biggr].\n\\label{eq:IR-Yukawa}\n\\end{equation}\n(The Yukawa couplings also receive contributions from higher dimension \nterms as will be seen later in this subsection.)  Note that these \ninteractions, as well as those in Eq.~(\\ref{eq:IR-Higgs}), respect \nthe usual $R$ parity of the MSSM, with $\\Sigma$ even.\n\nThe interactions of Eq.~(\\ref{eq:IR-Yukawa}) give the Yukawa \ncouplings of the quark and lepton chiral superfields, $Q, U, D, L, E$ \nand $N$, with the Higgs doublets, $H_u$ and $H_d$, at low energies \n($W = QUH_u$, $QDH_d + LEH_d$ and $LNH_u$ from the first, second \nand third terms, respectively).  Recall that the two Higgs doublets \nof the MSSM, $H_u$ and $H_d$, arise from $\\Sigma$ as pseudo-Goldstone \nchiral superfields of the broken $SU(6)$ symmetry.  For matter fields \nwith $|c| > 1\/2$, the Yukawa couplings receive suppressions due \nto the fact that the fields effectively feel only the IR brane \n(strong dynamics) or the UV brane (explicit breaking of $SU(6)$), \nboth of which are needed to generate nonvanishing Yukawa couplings \nat low energies~\\cite{Contino:2003ve}.  Then, considering that \n$y_{\\cal T} \\sim y_{\\cal F} \\sim y_{\\cal N} = O(4\\pi^2\/M'_*)$ from \nnaive dimensional analysis, we find that the low-energy Yukawa \ncoupling $y$ arising from the IR-brane term $\\int\\!d^2\\theta\\, \n{\\cal M}_1 {\\cal M}_2 \\Sigma$ (${\\cal M}_1, {\\cal M}_2 \n= {\\cal T}, {\\cal F}, {\\cal N}$) takes a value \n\\begin{equation}\n  y \\approx 4\\pi f_1 f_2\\,\n    \\biggl( \\frac{\\pi k}{M_*} \\biggr),\n\\label{eq:Yukawa-value}\n\\end{equation}\nwhere $f_i \\simeq (k'\/k)^{|c_{{\\cal M}_i}|-1\/2}$ for $|c_{{\\cal M}_i}| \n> 1\/2$ and $f_i \\simeq 1$ for $|c_{{\\cal M}_i}| < 1\/2$ ($i=1,2$); for \n$|c_{{\\cal M}_i}| \\simeq 1\/2$, $f_i$ receives a logarithmic suppression, \n$f_i \\simeq 1\/(\\ln(k\/k'))^{1\/2}$.  This allows us to explain the observed \nhierarchies of fermion masses and mixings by powers of $k'\/k = e^{-\\pi kR} \n= O(0.1)$, by choosing different values of $c_{\\cal T}, c_{\\cal F}$ and \n$c_{\\cal N}$ for different generations.  This is similar to the situation \nwhere the hierarchies are explained by overlaps of matter and Higgs \nwavefunctions~\\cite{Gherghetta:2000qt,Hebecker:2002re}, although \nin the present setup the low-energy Yukawa couplings are also suppressed \nfor $c_{{\\cal M}_i} < -1\/2$, where apparent overlaps between matter \nand Higgs fields are large, due to the pseudo-Goldstone boson nature \nof the Higgs doublets.  This opens the possibility of localizing the \nfirst two generations to the IR brane, rather than to the UV brane \nas we will do shortly, to generate the observed hierarchies of fermion \nmasses and mixings.\n\nThe right-handed neutrino superfield $N$ can obtain a large mass \nterm through the UV-brane operator\n\\begin{equation}\n  {\\cal L}_N = \\delta(y) \n    \\biggl[ \\int\\! d^2\\theta\\, \\frac{\\eta}{2} \\bar{X} N^2 \n    + {\\rm h.c.} \\biggr],\n\\label{eq:UV-N}\n\\end{equation}\nwhere $\\bar{X}$ is a $U(1)_X$-breaking field, having the VEV \n$\\langle \\bar{X} \\rangle = \\Lambda$ (see Eq.~(\\ref{eq:UV-X})). \nThis gives a small mass for the observed left-handed neutrino \nthrough the conventional seesaw mechanism~\\cite{Seesaw}.%\n\\footnote{An alternative possibility to generate a small neutrino \nmass is to strongly localize the $N$ field to the IR brane by taking \n$c_{\\cal N} \\ll -1\/2$, in which case the neutrino Yukawa coupling \nis strongly suppressed and we can obtain a small Dirac neutrino \nmass.  The scale of the neutrino mass, however, is unexplained \nin this case.}\n\nIt is rather straightforward to generalize the analysis so far \nto the case of three generations.  We simply introduce a set of \nbulk hypermultiplets $\\{ {\\cal T}, {\\cal T}^c \\}$, $\\{ {\\cal F}, \n{\\cal F}^c \\}$ and $\\{ {\\cal N}, {\\cal N}^c \\}$ for each generation. \nThe couplings $y_{\\cal T}$, $y_{\\cal F}$ and $y_{\\cal N}$ in \nEq.~(\\ref{eq:IR-Yukawa}) and $\\eta$ in Eq.~(\\ref{eq:UV-N}) then \nbecome $3 \\times 3$ matrices.  We assume that there is no special \nstructure in these matrices, so that all the elements in $y_{\\cal T}$, \n$y_{\\cal F}$ and $y_{\\cal N}$ are of order $4\\pi^2\/M'_*$, suggested \nby naive dimensional analysis.  The observed fermion masses and \nmixings, however, can still be reproduced through the dependence of \nthe low-energy Yukawa couplings on the values of bulk hypermultiplet \nmasses $c_{\\cal T}, c_{\\cal F}$ and $c_{\\cal N}$.  Let us take, for \nexample, the bulk masses to be\n\\begin{equation}\n  c_{{\\cal T}_1} \\simeq \\frac{5}{2}, \\quad \n  c_{{\\cal T}_2} \\simeq \\frac{3}{2}, \\quad \n  c_{{\\cal T}_3} \\simeq \\frac{1}{2}, \\quad\n  c_{{\\cal F}_1} \\simeq c_{{\\cal F}_2} \\simeq \n  c_{{\\cal F}_3} \\simeq \\frac{3}{2}, \\quad\n  c_{{\\cal N}_1} \\simeq c_{{\\cal N}_2} \\simeq \n  c_{{\\cal N}_3} \\simeq \\frac{1}{2}.\n\\label{eq:c-flavor}\n\\end{equation}\nThen, taking $M_*\/\\pi k$ to be a factor of a few, e.g. $2\\!\\sim\\!3$, \nwe obtain the following low-energy Yukawa matrices from \nEq.~(\\ref{eq:Yukawa-value}):\n\\begin{equation}\n  y_u \\approx \n  \\pmatrix{\n     \\epsilon^4 & \\epsilon^3 & \\epsilon^2 \\cr\n     \\epsilon^3 & \\epsilon^2 & \\epsilon   \\cr\n     \\epsilon^2 & \\epsilon   & 1          \\cr\n  },\n\\quad\n  y_d \\approx y_e^T \\approx\n  \\epsilon\n  \\pmatrix{\n     \\epsilon^2 & \\epsilon^2 & \\epsilon^2 \\cr\n     \\epsilon   & \\epsilon   & \\epsilon   \\cr\n     1          & 1          & 1          \\cr\n  },\n\\quad\n  y_\\nu \\approx\n  \\epsilon\n  \\pmatrix{\n     1 & 1 & 1 \\cr\n     1 & 1 & 1 \\cr\n     1 & 1 & 1 \\cr\n  },\n\\label{eq:y-values}\n\\end{equation}\nwhere $y_u$, $y_d$, $y_e$ and $y_\\nu$ are defined in the low-energy \nsuperpotential by\n\\begin{equation}\n  W = (y_u)_{ij} Q_i U_j H_u + (y_d)_{ij} Q_i D_j H_d\n    + (y_e)_{ij} L_i E_j H_d + (y_\\nu)_{ij} L_i N_j H_u,\n\\label{eq:y-def}\n\\end{equation}\nwith $i,j, = 1,2,3$, and\n\\begin{equation}\n  \\epsilon \\equiv \\frac{k'}{k} \\simeq \\frac{1}{20}\n  \\quad {\\rm for} \\,\\,\\, kR \\simeq 1.\n\\label{eq:epsilon}\n\\end{equation}\nTogether with a structureless Majorana mass matrix for the \nright-handed neutrinos, $M_N = \\eta \\langle \\bar{X} \\rangle$, \nthe Yukawa matrices of Eq.~(\\ref{eq:y-values}) well reproduces \ngross features of the observed quark and lepton masses and \nmixings~\\cite{Hall:1999sn}.  It is straightforward to make \nfurther refinements on this basic picture; for example, we can \nmake $c_{{\\cal F}_1}$ somewhat larger than $3\/2$ to better reproduce \ndown-type quark and charged lepton masses, as well as the neutrino \nmixing angles $\\theta_{12}$ and $\\theta_{13}$.  A schematic picture \nfor the zero-mode wavefunctions (for the $\\{ {\\cal T}, {\\cal T}^c \\}$ \nmultiplets) is depicted in Fig.~\\ref{fig:5D}.\n\nUnwanted $SU(5)$ mass relations for the first two generation \nfermions can be avoided by using higher dimension operators, e.g. \nof the form ${\\cal L} \\sim \\delta(y-\\pi R) \\int\\!d^2\\theta\\, {\\cal T} \n{\\cal F} \\Sigma^2$.  (Violation of $SU(5)$ relations may also come \nfrom $SU(5)$-violating mixings between the matter zero modes and \nthe corresponding KK excited states, arising from the IR-brane \nterms of Eq.~(\\ref{eq:IR-Yukawa}) through the $\\Sigma$ VEV.)  Since \nthe effects are higher order in $\\langle \\Sigma \\rangle\/M'_*$, \nwhich we assume somewhat small, $O(1)$ violation in the Yukawa \ncoupling requires a somewhat suppressed coefficient for the leading \n$SU(5)$-invariant piece coming from Eq.~(\\ref{eq:IR-Yukawa}). \nA realistic pattern for the fermion masses and mixings can be \nobtained if (only) the 22 element of the $y_{\\cal F}$ matrix \nis somewhat suppressed~\\cite{Georgi:1979df}.\n\nThe three generation model allows IR-brane operators of the form \n${\\cal L} \\sim \\delta(y-\\pi R) \\int\\!d^2\\theta\\, \\epsilon^{ij} \n{\\cal T}_i {\\cal T}_j$, where the antisymmetry in the generation \nindices $i,j$ arises from the pseudo-real nature of the ${\\bf 20}$ \nrepresentation.  The existence of these operators, however, does \nnot significantly affect predictions of the model.\n\nTo summarize, we have obtained an $SU(3)_C \\times SU(2)_L \\times \nU(1)_Y \\times U(1)_X$ gauge theory below the scale of $M \\sim k'$, \nwith three generations of matter, $Q, U, D, L, E$ and $N$, and \ntwo Higgs doublets, $H_u$ and $H_d$.  The Yukawa couplings of \nEq.~(\\ref{eq:y-def}) are obtained with realistic patterns for quark \nand lepton masses and mixings.  The $U(1)_X$ gauge symmetry is \nspontaneously broken at the scale $\\Lambda$, somewhat below $k'$, \ngiving masses to the right-handed neutrino superfields of order \n$\\Lambda$.  We thus have the complete MSSM, supplemented by seesaw \nneutrino masses, below the unification scale $\\sim k'$.  We emphasize \nthat the successes of our model depend only on its basic features, \nsuch as the symmetry structure and locations of fields.  They are \nthus quite robust.  For example, the existence of higher dimension \noperators in the IR-brane potential, e.g. terms of the form ${\\rm \nTr}(\\Sigma^n)$ ($n$: integers $>3$) added to Eq.~(\\ref{eq:IR-Higgs}), \ndoes not destroy these successes.\n\n\n\\subsection{Gauge coupling unification and proton decay}\n\\label{subsec:analysis}\n\nIn this subsection, we present a study on proton decay and gauge \ncoupling unification in our model, to demonstrate that it can \naccommodate realistic phenomenology at low energies.  In this \nsubsection we consider matter configurations such that lighter \ngenerations are localized more towards the UV brane, as in the \nexample of Eq.~(\\ref{eq:c-flavor}).\n\nWe first note that the terms in Eqs.~(\\ref{eq:IR-Yukawa}) introduce, \nthrough the VEV of $\\Sigma$, $SU(5)$-violating mass splittings \ninto the KK towers for the matter fields: $\\{ {\\cal T}, {\\cal T}^c \\}$, \n$\\{ {\\cal F}, {\\cal F}^c \\}$ and $\\{ {\\cal N}, {\\cal N}^c \\}$. \nThese splittings, in turn, give threshold corrections to gauge \ncoupling unification.  Similar corrections also arise from the \ngauge KK towers.  We expect, however, that these corrections are \nnot large.  Using the AdS\/CFT correspondence, we estimate the \nsize of the corrections to be of order $(C\/16\\pi^2)(M_*\/\\pi k)$ \nfor $1\/g_a^2$, where $g_a$ are the 4D gauge couplings.  Moreover, \nif the value of $\\langle \\Sigma \\rangle$ (and thus $M$) is somewhat \nsuppressed compared with its naive size of $M'_*\/4\\pi$, as we assume \nhere, the threshold corrections receive additional suppressions \nof $O(4\\pi \\langle \\Sigma \\rangle\/M'_*)$ because the spectrum \nof the KK towers becomes $SU(5)$ symmetric for $4\\pi \\langle \\Sigma \n\\rangle\/M'_* \\ll 1$.  The contributions from tree-level IR-brane \noperators, such as $\\int\\!d^2\\theta\\, \\Sigma {\\cal W}^\\alpha \n{\\cal W}_\\alpha$, are also sufficiently small, of order $C\/16\\pi^2$ \nfor $1\/g_a^2$ with an additional suppression of $O(4\\pi \\langle \n\\Sigma \\rangle\/M'_*)$ for small $\\langle \\Sigma \\rangle$.\n\nAnother important issue in supersymmetric unified theories is \ndimension five proton decay caused by low-energy operators of the \nform $W \\sim QQQL$, $UUDE$.  There are two independent sources for \nthese operators: tree-level operators existing at the gravitational \nscale and operators generated by the GUT (breaking) dynamics.  In \nour theory, the former correspond to tree-level operators ${\\bf 10}_1 \n{\\bf 10}_1 {\\bf 10}_1 {\\bf 5}^*_{-3} \\supset QQQL$, $UUDE$ located \non the UV brane, where the subscripts on ${\\bf 10}_1 \\subset {\\cal T}$ \nand ${\\bf 5}^*_{-3} \\subset {\\cal F}$ denote the $U(1)_X$ charges. \nWhile the coefficients of these operators are suppressed by the \nfundamental scale $M_*$, which is larger than the unification scale, \nit is still problematic, especially because we do not have any \nYukawa suppressions in the coefficients.  Therefore, to suppress \nthese contributions, we impose a discrete $Z_{4,R}$ symmetry on the \ntheory, whose charge assignment is given in Table~\\ref{table:Z4R} \n(in the normalization that the $R$ charge of the superpotential \nis $2$).\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|cc|c|cccccc|ccc|}  \\hline \n  & $V$ & $\\Phi$ & $\\Sigma$ & \n    ${\\cal T}$ & ${\\cal T}^c$ & ${\\cal F}$ & \n    ${\\cal F}^c$ & ${\\cal N}$ & ${\\cal N}^c$ & \n    $Y$ & $X$ & $\\bar{X}$ \\\\ \\hline\n  $Z_{4,R}$ & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & \n    2 & 0 & 0 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{$Z_{4,R}$ charges for fields.}\n\\label{table:Z4R}\n\\end{table}\nAs is clear from the terms in Eq.~(\\ref{eq:IR-Higgs}), this symmetry \nshould be broken on the IR brane, i.e. broken by the dynamics of \nthe GUT breaking sector $G$.  This can be incorporated by introducing \na spurion chiral superfield $\\phi$ with $\\langle \\phi \\rangle \\sim \nM'_*\/4\\pi$ on the IR brane, whose $Z_{4,R}$ charge is $+2$, or \nequivalently introducing fields $\\phi$ and $\\bar{\\phi}$ of $Z_{4,R}$ \ncharges $+2$ and $-2$ with the superpotential giving the VEVs for \nthese fields.  This introduces ``$O(1)$'' breaking of $Z_{4,R}$ on \nthe IR brane, keeping $Z_{4,R}$ invariance for the UV-brane terms.%\n\\footnote{The $Z_{4,R}$ symmetry can be gauged in 5D if we cancel \nthe discrete $Z_{4,R}$-$SU(5)^2$ anomaly by the Green-Schwarz \nmechanism~\\cite{Green:1984sg}, by introducing a singlet field $S$ \nthat transforms nonlinearly under $Z_{4,R}$ and couples to the \n$SU(5)$ gauge kinetic term on the UV brane.  We consider that the \n$S$ field appears only in front of the kinetic term of the $SU(5)$ \ngauge superfields, and not in UV-brane superpotential terms.  Such \nterms would potentially induce dimension five proton decay, although \nthey are suppressed in a certain (broad) region for the $S$ VEV.}\n\nAfter killing the UV-brane operators, the low-energy dimension five \nproton decay operators can still be generated through strong $G$ \ndynamics, since the $Z_{4,R}$ symmetry is spontaneously broken by \nthis dynamics.  One source is tree-level dimension five operators \non the IR brane.  These operators, however, receive suppressions \nof order the Yukawa couplings in 4D, because the wavefunctions for \nlight generation matter are suppressed on the IR brane due to the \nbulk hypermultiplet masses, and so are not particularly dangerous. \n(In the 4D picture, these suppressions arise from small mixings \nbetween the elementary and composite matter states for light \ngenerations.)  The only potentially dangerous contribution to \ndimension five proton decay in our model then comes from the exchange \nof the colored triplet Higgsinos -- composite states of $G$ arising \nas components of $\\Sigma$ -- because the mass of these states can \nbe smaller than $M'_*$.  To suppress this contribution, we can simply \nraise the mass of the colored triplet Higgsino states compared with \nthe unification scale; in fact, the mass is expected to be larger \nthan the GUT breaking VEV because the coupling $\\lambda$ in \nEq.~(\\ref{eq:IR-Higgs}) is naturally of order $4\\pi$.  Note that \nbecause of the existence of threshold corrections from KK towers \nto gauge coupling unification, there is no tight relation between \nthe mass of the triplet Higgsinos and the low-energy values of \nthe gauge couplings, which excluded the minimal SUSY $SU(5)$ GUT \nin 4D~\\cite{Murayama:2001ur}.\n\n\n\\subsection{Supersymmetry breaking}\n\\label{subsec:SUSY-breaking}\n\nOur model can be combined with almost any supersymmetry breaking \nscenario.  If the mediation scale of supersymmetry breaking is \nlower than the unification scale, there are essentially no particular \nimplications from our theory on the pattern of supersymmetry breaking. \nOn the other hand, if the mediation scale is higher, there can be \ninteresting implications, e.g., on the flavor structure of supersymmetry \nbreaking masses.  For example, if the supersymmetry breaking sector \nis localized on the IR brane, i.e. arises as a result of the dynamics \nof $G$, the third generation superparticles (presumably only the \nones coming from the ${\\bf 10}$ representation of $SU(5)$) can \nhave different masses than the lighter generation superparticles, \nwhich receive universal masses from the gauginos through loop \ncorrections~\\cite{Kitano:2006}.%\n\\footnote{We thank R.~Kitano for discussions on this issue.}\nThese are consequences of our way of generating hierarchies \nin fermion masses and mixings.\n\nThe supersymmetric mass (the $\\mu$ term) and supersymmetry-breaking \nmasses (the $\\mu B$ term and non-holomorphic scalar squared masses) \nfor the Higgs doublets are both generated through supersymmetry \nbreaking.  In the case that the supersymmetry breaking sector is \nlocalized on or directly communicates with the IR brane, these masses \nare generated through IR-brane operators of the form, ${\\cal L} \n\\sim \\delta(y-\\pi R) \\int\\!d^2\\theta\\, \\{ Z M \\Sigma^2\/M'_* + \nZ \\Sigma^3\/M'_*\\} + {\\rm h.c.}$ and $\\delta(y-\\pi R) \\int\\!d^4\\theta\\, \n\\{ (Z+Z^\\dagger)\\Sigma^\\dagger \\Sigma\/M'_* + Z^\\dagger Z (\\Sigma^2 \n+ \\Sigma^{\\dagger 2})\/M_*^{\\prime 2} + Z^\\dagger Z \\Sigma^\\dagger \n\\Sigma\/M_*^{\\prime 2} \\}$, where $Z$ is a chiral superfield responsible \nfor supersymmetry breaking, $\\langle Z \\rangle = \\theta^2 F_Z$, \nand we have omitted $O(1)$ coefficients.  These operators produce \nsupersymmetry breaking terms in the $\\Sigma$ potential, which lead \nto a slight shift of the vacuum from Eq.~(\\ref{eq:Sigma-VEV}) and \nconsequently generate weak scale masses for components of $H_u$ \nand $H_d$.  The generated masses respect the relation\n\\begin{equation}\n  \\mu B = \\left||\\mu|^2 + m_{H_u}^2 \\right|,\n\\qquad\n  m_{H_u}^2 = m_{H_d}^2,\n\\label{eq:Higgs-1}\n\\end{equation}\nreflecting the fact that the scalar potential for $\\Sigma$ still \nhas a global $SU(6)$ symmetry, where $m_{H_u}^2$ and $m_{H_d}^2$ are \nnon-holomorphic supersymmetry breaking squared masses for $H_u$ and \n$H_d$, and we have taken the phase convention that $\\mu B > 0$.  Note \nthat, unlike the case where the Higgs fields are non pseudo-Goldstone \nfields~\\cite{Giudice:1988yz}, the K\\\"ahler potential terms of the \nform $\\delta(y-\\pi R) \\int\\!d^4\\theta\\, \\Sigma^2$, $\\delta(y-\\pi R) \n\\int\\!d^4\\theta\\, \\{ Z^\\dagger \\Sigma^2\/M'_* + {\\rm h.c.} \\}$ and \n$\\delta(y-\\pi R) \\int\\!d^4\\theta\\, \\{ Z^\\dagger Z \\Sigma^2\/M_*^{\\prime 2} \n+ {\\rm h.c.} \\}$ do not produce a weak scale $\\mu$ term; we need \nsupersymmetry breaking interactions for $\\Sigma$, generated by \nsuperpotential terms or $\\delta(y-\\pi R) \\int\\!d^4\\theta\\, \n(Z+Z^\\dagger)\\Sigma^\\dagger \\Sigma\/M'_*$. \n\nAn interesting case arises if supersymmetry is broken in a hidden \nsector that does not have direct interactions with the GUT breaking \nsector.  In this case, the Higgs sector supersymmetry breaking \nparameters arise through gravitational effects and obey the \ntighter relation\n\\begin{equation}\n  \\mu B = |\\mu|^2, \\qquad m_{H_u}^2 = m_{H_d}^2 = 0.\n\\label{eq:Higgs-2}\n\\end{equation}\nIn the language of the compensator formalism (see \ne.g.~\\cite{Randall:1998uk}), these terms arise from ${\\cal L} \\sim \n\\delta(y-\\pi R) \\int\\!d^2\\theta\\, \\phi (M\/2) \\Sigma^2 + {\\rm h.c.}$, \nwhere $\\phi = 1 + \\theta^2 m_{3\/2}$ is the compensator field \nwith $m_{3\/2}$ the gravitino mass.  (Here, we have assumed that \nsupersymmetry breaking in the compensator field is not canceled \nby the conformal dynamics of the GUT breaking sector.)  The relations \nof Eqs.~(\\ref{eq:Higgs-1},~\\ref{eq:Higgs-2}) hold at the unification \nscale of $O(k')$, so that their connections to low energy parameters \nmust involve renormalization group effects between the unification \nand the weak scales.  It is also possible that there are \nadditional contributions to supersymmetry breaking parameters, \ne.g. $m_{H_u}^2$ and $m_{H_d}^2$, in addition to the ones in \nEqs.~(\\ref{eq:Higgs-1},~\\ref{eq:Higgs-2})\n\nAlternatively, the $\\mu$ and $\\mu B$ terms may be generated below \nthe unification scale.  For example, they may be generated associated \nwith the dynamics of $U(1)_X$ breaking~\\cite{Hall:2002up}.  In this \ncase there is no trace in the Higgs sector parameters that the Higgs \nfields are pseudo-Goldstone fields of the GUT breaking dynamics.\n\n\n\\section{Other Theories: GUT Engineering on the IR Brane}\n\\label{sec:other}\n\nSo far, we have considered a theory in which the lightness of the \ntwo Higgs doublets is understood by the pseudo-Goldstone mechanism \nassociated with the dynamics of GUT breaking.  As we have seen, this \ncan be elegantly implemented in our framework by considering the \nbulk $SU(6)$ gauge symmetry, broken to $SU(5) \\times U(1)$ and \n$SU(4) \\times SU(2) \\times U(1)$ on the UV and IR branes, respectively. \nThe mass of the light Higgs doublets is protected from the existence \nof explicit breaking by localizing these fields ($\\subset \\Sigma$) \non the IR brane, which is geometrically separated from the UV brane \nwhere explicit breaking of $SU(6)$ resides.  In the 4D description \nof the model, the global $SU(6)$ symmetry of the GUT breaking sector \nis understood as a ``flavor'' symmetry of this sector, and the extreme \nsuppression of explicit symmetry breaking effects in the $\\Sigma$ \npotential comes from the fact that $\\Sigma$ is a composite field, \nwith the corresponding operator having a (very) large canonical mass \ndimension.  An interesting thing about our construction is that it \nallows us to implement these mechanisms in simple and controllable \nways in effective field theory, giving a simple and calculable unified \ntheory above $M_U$ in which the lightness of the Higgs doublets is \nunderstood by a symmetry principle.\n\nLet us now consider if we can construct simpler theories in our warped \nspace framework.  Suppose we consider a supersymmetric $SU(5)$ gauge \ntheory in the 5D warped spacetime of Eq.~(\\ref{eq:metric}), and suppose \nthat the bulk $SU(5)$ gauge symmetry is broken to the $SU(3)_C \\times \nSU(2)_L \\times U(1)_Y$ subgroup on the IR brane by boundary conditions. \nIn this case, the doublet Higgs fields may be light without being \naccompanied by their triplet partners if they propagate in the \nbulk with appropriate boundary conditions imposed at the GUT breaking \nbrane~\\cite{Kawamura:2000ev,Hall:2001pg}, or if they are simply \nlocated on that brane~\\cite{Hebecker:2001wq}.  In the 4D description, \nhowever, this seems to be simply ``postulating'' particular dynamics \nof the GUT breaking sector that splits the mass of the doublet components \nfrom that of the triplet partners, and it is not clear if this can be \nregarded as a ``solution'' to the doublet-triplet splitting problem. \nFor example, we have a continuous parameter, the tree-level mass of the \nHiggs doublets on the IR brane, that has to be chosen to be very small \nto achieve the splitting.  The situation may be better if this parameter \nis forbidden by a symmetry, e.g. an $R$ symmetry~\\cite{Hall:2001pg}. \nThis symmetry may be imposed as a global symmetry in 5D, but in that \ncase it is not entirely clear if such a symmetry is preserved by strong \n$G$ dynamics (5D quantum gravity effects).  To avoid this and to give \na non-trivial meaning to the symmetry in the context of gauge\/gravity \nduality, we can gauge the symmetry in higher dimensions (although it \ncan still be broken on the UV brane, eliminating the existence of the \ncorresponding gauge field in 4D).  In this case, anomaly cancellation \nconditions become an issue, and we find that for a continuous $U(1)_R$ \nor a discrete $Z_{4,R}$ symmetry (with the charge assignment given \nby $V_{SU(5)}(0)$, $T_{\\bf 10}(1)$, $F_{{\\bf 5}^*}(1)$, $N_{\\bf 1}(1)$, \n$H_{\\bf 5}(0)$, $\\bar{H}_{{\\bf 5}^*}(0)$, assuming the MSSM matter \ncontent at low energies) we need to cancel the low energy anomalies \nvia the Green-Schwarz mechanism~\\cite{Green:1984sg}.  This requires the \nintroduction of a singlet field $S$ on the IR brane which transforms \nnonlinearly under the $R$ symmetry and couples to the $SU(3)_C$, $SU(2)_L$ \nand $U(1)_Y$ gauge kinetic terms with appropriate coefficients.  (Anomaly \ntransmission across the bulk~\\cite{Callan:1984sa} may also be necessary \nto make the full 5D theory anomaly free, depending on the symmetry \nand matter content.)  We assume that the $S$ field appears only in \nfront of the gauge kinetic terms, and not in IR-brane superpotential \nterms, so that a large mass term for the Higgs doublets is not \nregenerated.  (This may naturally occur in a UV theory in the absence \nof other gauge groups.)  Note that, in the 4D description, this \nsetup corresponds to the situation where the $R$-$SU(5)^2$ anomaly \nis canceled between the elementary-field and $G$-sector contributions.%\n\\footnote{We can show that this construction is not available in \na 4D SUSY GUT theory where the GUT-breaking (Higgs) sector does not \ngive tree-level contributions to the low energy anomalies.  Assuming \nthe MSSM matter content below the unification scale, with the $U(1)_R$ \ncharges given by $V_{321}(0)$, $Q(1)$, $U(1)$, $D(1)$, $L(1)$, $E(1)$, \n$H_u(0)$ and $H_d(0)$, we find the low-energy $U(1)_R$-$SU(3)_C^2$, \n$U(1)_R$-$SU(2)_L^2$ and $U(1)_R$-$U(1)_Y^2$ anomalies to be $3$, \n$1$ and $-3\/5$, respectively, which cannot be matched to high \nenergy theories, where these anomalies arise as a $U(1)_R$-$SU(5)^2$ \nanomaly and are thus universal.  (Here, the $SU(5)$ normalization \nis employed for the $U(1)_Y$ charges.)  This implies that $U(1)_R$ \nshould either be spontaneously broken, or there is explicit \n$SU(5)$-violating physics in the effective field theory.  In our case, \nthis conclusion can be avoided because the (dynamical) GUT-breaking \nsector carries the $U(1)_R$-$SU(5)^2$ anomaly, a part of which can \nbe manifested as Green-Schwarz terms at low energies.} \nIn this setup, doublet-triplet splitting seems ``natural,'' at least \nin the higher dimensional picture.  Thus, while the theory with the \n$R$ symmetry still seems to correspond to a particular choice of GUT \nbreaking dynamics in the 4D description, we may say that the theory \ndoes not have the problem of doublet-triplet splitting.%\n\\footnote{An interesting feature of this class of theories is \nthat the low energy theory contains an axion field $S$ that \ncouples to the QCD gauge fields with the decay constant of order \nthe unification scale.  This can be used to solve the strong $CP$ \nproblem~\\cite{Peccei:1977hh}, although the initial amplitude of \nthis field in the early universe must be (accidentally) small to \navoid the cosmological difficulty of overclosing the universe.}\nAfter all, the ``formulation'' of the doublet-triplet splitting problem \nmay have to be changed in the large 't~Hooft coupling regime, where \nphysics is specified by the ``hadronic'' quantities, i.e. matter \ncontent, location, and boundary conditions in higher dimensions.%\n\\footnote{If we break the bulk $SU(5)$ gauge symmetry by boundary \nconditions at the UV brane, it leads to a theory which is interpreted \nas an $SU(3)_C \\times SU(2)_L \\times U(1)_Y$ gauge theory in the 4D \ndescription.  The doublet-triplet splitting problem does not arise \nas the theory is not unified, and yet the successful unification of \ngauge couplings arises at the leading-log level in the limit that \nthe tree-level gauge kinetic terms on the UV brane are small.  This \ncorresponds in the 4D description that the 321 gauge couplings \nat the unification scale are dominated by the asymptotically non-free \ncontribution from a strong sector that has a global $SU(5)$ symmetry, \nof which the $SU(3) \\times SU(2) \\times U(1)$ subgroup is gauged \nand identified as the low-energy 321 gauge group.  While this theory \nis somewhat outside the framework described in this paper, it is \ninteresting on its own.}\n\nIn these respects, our framework offers many possible ways to \naddress the problems of conventional 4D SUSY GUTs.  For example, we \ncan again consider a 5D $SU(5)$ gauge theory in the warped spacetime \nof Eq.~(\\ref{eq:metric}), but then break the bulk $SU(5)$ by the \nVEV of a chiral superfield located on the IR brane, generated by \nan appropriate IR-brane superpotential.  Then, if this superpotential \ndoes not have the problem of doublet-triplet splitting, e.g. by having \nthe form of missing partner type models~\\cite{Masiero:1982fe}, then \nwe may say that the problem has been solved.  (As discussed before, \nit is better if the IR-brane superpotential is protected by a (discrete) \nsymmetry; otherwise, it would correspond to ``artificially'' choosing \nthe dynamics of the GUT breaking sector.  In practice, this may be \ndifficult, since we cannot use the non-universal Green-Schwarz terms \non the IR brane because the GUT breaking there is due to the Higgs \nmechanism.  We do not pursue this issue further here.)  An advantage \nof this approach over conventional 4D model building is that we need \nnot care about physics above the unification scale when engineering \nGUT breaking physics, i.e. the GUT-breaking Higgs content and \nsuperpotential.  In the conventional 4D SUSY GUT framework, theories \nsolving the doublet-triplet and\/or dimension five proton decay \nproblems often have too large matter content, leading to the problem \nof the unified gauge coupling hitting a Landau pole (well) below \nthe gravitational\/Planck scale.  In our case, all (possibly large) \nmultiplets located on the IR brane correspond to composite fields \nof the GUT breaking dynamics in the 4D description, and do not \ncontribute to the running of the unified gauge coupling above the \nunification scale, $M_U \\sim k'$ (see e.g.~\\cite{Goldberger:2002cz}). \nPotential complication of this sector may also not bother us, because \nit is the result of ``dynamics'' of the GUT breaking sector.  We note \nthat this makes the extension to $SO(10)$ unified theories trivial \n--- we can break $SO(10)$ on the IR brane by arbitrary combinations \nof boundary condition and Higgs breakings with an arbitrary field \ncontent.\n\nThere are many applications of the ideas described above.  For instance, \nwe can apply it to product-group theories~\\cite{Yanagida:1994vq,%\nIzawa:1997he,Weiner:2001pv}.  Let us once again consider \na supersymmetric $SU(5)$ gauge theory in the 5D warped spacetime \nof Eq.~(\\ref{eq:metric}).  We then introduce an additional gauge \ngroup $SU(3) \\times SU(2) \\times U(1)$ on the IR brane, with the Higgs \ndoublets charged under this IR-brane gauge group (and thus without \nbeing accompanied by any partner).  Now, we can consider that our \nlow-energy $SU(3)_C \\times SU(2)_L \\times U(1)_Y$ gauge group is \na diagonal subgroup of the bulk $SU(5)$ and the IR-brane $SU(3) \\times \nSU(2) \\times U(1)$.  (This breaking can be caused by the VEV of an \nappropriate IR-brane localized field).  Then, if the gauge couplings, \n$\\tilde{g}_a$ of the original $SU(3)$, $SU(2)$ and $U(1)$ are large, \n$\\tilde{g}_a \\approx 4\\pi$ ($a = 1,2,3$), the low-energy MSSM gauge \ncouplings are effectively unified at the scale where $SU(5) \\times \nSU(3) \\times SU(2) \\times U(1) \\rightarrow SU(3)_C \\times SU(2)_L \n\\times U(1)_Y$ occurs $\\sim k'$, since the gauge couplings of $SU(3)_C$, \n$SU(2)_L$ and $U(1)_Y$, $g_a$, are given by $1\/g_a^2 = 1\/g_5^2 + \n1\/\\tilde{g}_a^2 \\approx 1\/g_5^2$ at that scale.  Here, $g_5$ ($= O(1)$) \nis the coupling of the zero mode of the bulk $SU(5)$ gauge field. \nA problem of the corresponding scenario in 4D~\\cite{Weiner:2001pv} \nis that, since the original $U(1)$ gauge coupling is strong at the \nunification scale, it hits the Landau pole immediately above that scale. \nThere, it is also not clear why the three independent gauge couplings \nof $SU(3)$, $SU(2)$ and $U(1)$ become strong at a single scale, which \nmust also coincide with the scale of diagonal breaking to avoid large \nthreshold corrections.  Our theory addresses all of these issues \nnaturally --- since the $SU(3)$, $SU(2)$ and $U(1)$ gauge fields are \ncomposite states of the GUT breaking dynamics, they all have strong \ncouplings, $\\tilde{g}_a \\approx 4\\pi$, at the scale where breaking \nto the diagonal subgroup occurs, and there is no issue of a Landau \npole above this scale.  A potentially large mass for the Higgs \ndoublets can be avoided by introducing a (discrete) gauge symmetry \nwith the anomalies canceled by the Green-Schwarz terms on the IR brane.%\n\\footnote{Another possibility for canceling anomalies is to add extra \nmatter fields (in complete $SU(5)$ multiplets) that obtain masses from \nsupersymmetry breaking~\\cite{Kurosawa:2001iq}.  We thank N.~Maru for \npointing out this work to us.}\n(To do all of these completely within the regime of effective field \ntheory, the scale of diagonal breaking should be somewhat below the \nIR-brane cutoff, and the $SU(3)$, $SU(2)$ and $U(1)$ gauge couplings \nshould be asymptotically non-free.  These can be arranged with an \nappropriate introduction of massive fields on the IR brane.  In the \nlimit that the scale of diagonal breaking approaches the IR-brane \ncutoff, this theory is reduced to one of the theories discussed in \nthe second paragraph of this subsection, where the Higgs doublets \nare located on the GUT-breaking IR brane.)  Note that since the quarks \nand leptons are introduced in the bulk in representations of the \nbulk $SU(5)$, we are still considering a unified theory of quarks \nand leptons (although it is possible to introduce them on the IR brane \nin representations of $SU(3) \\times SU(2) \\times U(1)$).  In particular, \nproton decay from unified gauge boson exchange still exists.  The Yukawa \ncouplings of matter to the Higgs fields arise through the IR-brane \nVEV, breaking $SU(5) \\times SU(3) \\times SU(2) \\times U(1)$ down to \n$SU(3)_C \\times SU(2)_L \\times U(1)_Y$. \n\nIn most of the theories described above, the Higgs fields are localized \nto the IR brane, so that they are composite fields of the dynamical \nGUT breaking sector in the 4D description.  (This need not be the case. \nOne of the theories described in the second paragraph of this subsection \ncontains Higgs fields that propagate in the bulk, with appropriate \nboundary conditions imposed at the IR brane.  We can, however, always \nlocalize them to the IR brane by introducing appropriate hypermultiplet \nmasses.)  The observed hierarchies of quark and lepton masses and mixings \ncan then always be explained by wavefunction overlaps between the matter \nand Higgs fields, by appropriately choosing the bulk hypermultiplet \nmasses for the matter fields such that lighter generations are localized \nmore towards the UV brane, as e.g. in Eq.~(\\ref{eq:c-flavor}).  (The \noption of localizing lighter generations towards the IR brane is not \navailable unless the Higgs fields are pseudo-Goldstone boson multiplets.)\nWe find it very interesting that our framework of ``holographic grand \nunification'' accommodates many different ideas of solving the problems \nof conventional SUSY GUTs, developed mainly in the 4D context, with \nthe automatic bonus of explaining the observed hierarchies of fermion \nmasses and mixings through the wavefunction profiles of matter fields \nin the extra dimension.\n\n\n\\section{Discussion and Conclusions}\n\\label{sec:concl}\n\nIn this paper we have studied a framework in which grand unification \nis realized in truncated warped higher dimensional spacetime, \nwhere the UV and IR branes set the Planck and unification scales, \nrespectively.  In the 4D description, this corresponds to theories \nin which the the grand unified gauge symmetry is spontaneously \nbroken by strong gauge dynamics having a large 't~Hooft coupling, \n$\\tilde{g}^2 \\tilde{N}\/16\\pi^2 \\gg 1$ (and a large number of ``colors'', \n$\\tilde{N} \\gg 1$).  In this parameter region, an appropriate (weakly \ncoupled) description of physics is obtained in higher dimensions, and \nphysics above the unification scale is determined by higher dimensional \nfield theories, e.g. by specifying the spacetime metric, gauge group, \nmatter content, boundary conditions, and Lagrangian parameters.  This \nallows us to control certain dynamical properties of the GUT breaking \nsector in the regime where effective field theory applies.  For example, \nwe can make the size of threshold corrections small by making the \nsymmetry breaking VEV on the IR brane (slightly) smaller than its naive \nvalue.  Moreover, the framework allows us to straightforwardly adopt \nintuitions and mechanisms arising from the higher dimensional picture. \nIn particular, we can explain the observed hierarchies in quark and \nlepton masses and mixings in terms of the wavefunction profiles of \nmatter fields in higher dimensions.  The generated hierarchies are \nnaturally of the right size, of order $M_U\/M_* \\simeq 1\/20$.\n\nWe have presented several realistic models within this framework.  In \none model, on which we have focused the most, the lightness of the Higgs \ndoublets is explained by the pseudo-Goldstone mechanism.  The strong \ngauge dynamics sector possesses a global $SU(6)$ symmetry as a ``flavor'' \nsymmetry, of which the $SU(5)$ ($\\times U(1)$) subgroup is gauged and \nidentified as the unified gauge group.  When the global symmetry is \nbroken dynamically to $SU(4) \\times SU(2) \\times U(1)$, the unified \ngauge symmetry is broken to the standard model gauge group, and the two \nMSSM Higgs doublets arise as massless pseudo-Goldstone supermultiplets. \nIn our framework, this is realized by postulating a bulk $SU(6)$ gauge \nsymmetry, broken to $SU(5) \\times U(1)$ and $SU(4) \\times SU(2) \\times \nU(1)$ on the UV and IR branes, respectively.  One of the difficulties \nin implementing this mechanism in the conventional 4D framework is to \nfind a way to suppress effects of explicit breaking in the potential \ngenerating the spontaneous $SU(6)$ breaking, since such effects would \nreintroduce an unacceptably large mass for the Higgs doublets.  In our \ncase, these effects are (exponentially) suppressed by a large mass \ndimension for the operator generating the spontaneous $SU(6)$ breaking. \nSuch an assumption is easy to implement in higher dimensions -- simply \nassume that the Higgs field breaking $SU(6)$ is localized to the IR \nbrane.  This provides another example of the ``controllability'' of \nstrong gauge dynamics in the large 't~Hooft coupling regime.\n\nWe have also demonstrated that many ideas for solving the problems of \nconventional 4D SUSY GUTs can be naturally implemented on the IR brane. \nWe have presented several realistic models of this kind, for example, \nones based on missing partner type or product group type scenarios. \nThese models have the interesting feature that the GUT scale physics \non the IR brane does not affect physics at higher energies, since the \nrelevant physics arises as a result of the strong GUT breaking dynamics \n(as composite states) in the 4D description.  For example, large \nGUT multiplets, often needed to solve the problems of SUSY GUTs, \ndo not contribute to the evolution of the unified gauge coupling \nat higher energies, and gauge fields having very large gauge couplings \ncan naturally arise at the GUT scale without having the problem of \na Landau pole.  These features open up new possibilities for GUT model \nbuilding.\n\nOne can view the ``success'' of the present framework in several \ndifferent ways.  For one who is interested in addressing the \nphenomenology of unified theories, such as gauge coupling unification \nand proton decay, models in our framework can be used to give predictions \nof observable quantities.  For example, we can explore relations \nbetween the branching ratios of proton decay and matter configurations \nin the extra dimension, as in the case of unified theories in flat \nspace~\\cite{Nomura:2001tn}.  Models of fermion masses and mixings, \nas well as models of supersymmetry breaking, can also be developed \nwithin the framework.%\n\\footnote{For example, we can take one of the supersymmetry breaking \nmodels in~\\cite{Goldberger:2002pc}, with the boundary conditions at \nthe UV brane changed to be trivial, and glue that spacetime (the 5D \nwarped spacetime with the scales at the UV and IR branes taken to be the \nPlanck and TeV scales, respectively) to one of our holographic warped \nGUT spacetimes discussed in section~\\ref{sec:other}, at the UV branes \nof both spacetimes.  (The consistency of such constructions in effective \nfield theory has been discussed recently in~\\cite{Cacciapaglia:2006tg}.) \nIn the 4D description, this corresponds to the situation where both \nthe unified gauge symmetry and supersymmetry are broken by strong \ngauge dynamics, at the unification scale and the TeV scale, respectively. \nIn practice, this system is analyzed most efficiently by first \nintegrating out the GUT scale physics.  Then the low energy effective \ntheory is simply reduced to one of the models in~\\cite{Goldberger:2002pc}, \nbut now we have an understanding of the hierarchical structure of the \nYukawa couplings, located on the UV brane of the effective theory. \nWhile this effective field theory may be at the border of the weak \nand strong coupling regimes in 5D, it may still reproduce gross \nfeatures of physical quantities, e.g. the superparticle spectrum, \nas is the case in higher dimensional formulations of QCD.}\nOn the other hand, one may be interested in exploring possible \n``UV completions'' of models formulated in warped spacetime.  It is \npossible, after all, that there may be some nontrivial consistency \nconditions in higher dimensional field theories, which are difficult \n(though not impossible) to catch in effective theory, and one way \nof ensuring the consistency of such theories is to ``derive'' them \nfrom complete UV theories.  Such ``UV completions'' may be achieved, \nfor example, by embedding models into string theory, identifying \na ``dual'' 4D theory, or by finding a 4D theory whose infrared \nfixed point has similar features as the original models in warped \nspace~\\cite{Arkani-Hamed:2001ca}.  From this perspective, our framework \noffers a guide on which models ``UV theorists'' should aim to reproduce; \nfor example, string theorists may want to reproduce unified theories \nin 5D warped spacetime, with the unified gauge symmetry broken at \nan IR throat, rather than 4D unified theories directly from \ncompactification.\n\nWe finally comment on the possibility that the unified gauge \nsymmetry is broken by strong gauge dynamics whose 't~Hooft coupling \nis large but not extremely large, i.e. $\\tilde{g}^2 \\tilde{N}\/16\\pi^2 \n\\sim 1$.  In this case, the picture based on higher dimensional \nspacetime is not fully justified, but even then some properties of \ntheories, especially properties associated with the IR brane physics \n(GUT breaking dynamics), may be effectively described by our higher \ndimensional warped unified theories.  In fact, such an approach had \na certain level of successes in describing physics of lowest-lying \nexcitations in QCD~\\cite{Erlich:2005qh}.  In this sense, our framework \nmay have a larger applicability than what is naively expected.\n\nIn summary, we have presented a framework in which dynamical GUT \nbreaking models are realized in a regime that has a weakly coupled \n``dual'' picture.  Grand unified theories are realized in warped \nhigher dimensional spacetime, with the UV and IR spacetime cutoffs \nproviding the Planck and the unification scales, respectively. \nSeveral types of realistic models are discussed, with interesting \nimplications for quark and lepton masses and mixings.  It would \nbe interesting to study further implications of these models, such \nas those on proton decay, precise gauge coupling unification, \nsupersymmetry breaking, and flavor physics.\n\n\n\\section*{Acknowledgments}\n\nThis work was supported in part by the Director, Office of Science, Office \nof High Energy and Nuclear Physics, of the US Department of Energy under \nContract DE-AC02-05CH11231.  The work of Y.N. was also supported by the \nNational Science Foundation under grant PHY-0403380, by a DOE Outstanding \nJunior Investigator award, and by an Alfred P. Sloan Research Fellowship.\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Statement of Conjectures}\n\nLet $E$ be an elliptic curve defined over the field $\\mathbf{Q}$ of rational numbers. Assume that $E$ does not have complex multiplication. Denote by $N$ the conductor of $E$, so that $E$ has good reduction at primes $p$ not dividing $N$. Let $\\{p_k\\}_{k \\geq 1}$ be the set of primes in ascending order. For $p_k$ not dividing $N$, define as usual the quantity $a_{p_k}:= p_k+1-\\#E(\\mathbf{F}_{p_k})$ (here $\\mathbf{F}_{p_k}$ is the finite field of cardinality $p_k$, and $E(\\mathbf{F}_{p_k})$ is the group of points of $E$ over $\\mathbf{F}_{p_k}$). By the Hasse bound one has $|a_{p_k}| \\leq 2 p_k^{1\/2}$. We define $x_k \\in [0,1]$ for $k \\geq 1$, referred to as the (normalized) Frobenius angle of $E$ at the prime $p_k$, by the condition:\n\\[\na_{p_k} = 2p_k^{1\/2} \\cos(\\pi x_k)\n\\]\nif $p_k$ does not divide $N$, and we simply define $x_k=1\/2$ if $p_k$ divides $N$. The Sato-Tate conjecture, as established by Taylor {\\it et. al.} \\cite{CHT,T,HSBT,BLGHT}, states that the sequence $\\{x_k\\}_{k \\geq 1}$ is uniformly distributed with respect to the Sato-Tate measure on $[0,1]$. Specifically define the Sato-Tate measure $\\mu_{ST}$ on $[0,1]$ (which is a probability measure) by:\n\\[\nd \\mu_{ST} = 2 \\sin^2(\\pi u) \\, du\n\\] \n(where $du$ is the Lebesgue measure on $[0,1]$) and for $x_k$ as above, denote by $\\delta_{x_k}$ the Dirac point mass distribution on $[0,1]$ supported at $x_k$. For any integer $K \\geq 1$ consider the probability distribution on $[0,1]$ given by:\n\\begin{eqnarray}\n\\frac{1}{K} \\sum_{k=1}^K \\delta_{x_k}\n\\end{eqnarray}\nthen the Sato-Tate conjecture states that the probability distribution (1.1) converges weakly to the Sato-Tate measure $\\mu_{ST}$ as $K$ tends to infinity.\n\n\\bigskip\nIn this paper we propose the following refinement of the original Sato-Tate conjecture, that in addition to uniform distribution, we conjecture that the sequence $\\{x_k\\}_{k \\geq 1}$ is pseudorandom, in other words the $x_k$'s are in fact {\\it statistically independently} distributed with respect to the Sato-Tate measure. Specifically for any integer $s \\geq 1$, denote by $\\mu_{ST}^{[s]}$ the probability measure on $[0,1]^s$ given by the product of $s$ (independent) copies of the original one dimensional Sato-Tate measure $\\mu_{ST}$. We consider the joint distributions for $s$ successive terms from the sequence $\\{x_k\\}_{k \\geq 1}$. Thus define $X_k \\in [0,1]^s$ for $k \\geq 1$ to be the $s$-dimensional vector given by:\n\\[\nX_k = (x_k,x_{k+1}, \\cdots,x_{k+s-1})\n\\]\nand denote by $\\delta_{X_k}$ the Dirac point mass distribution on $[0,1]^s$ that is supported at $X_k$.\n\n\\bigskip\nFor integer $K \\geq 1$ consider similarly the probability distribution on $[0,1]^s$ given by:\n\\begin{eqnarray}\n\\frac{1}{K} \\sum_{k=1}^K \\delta_{X_k}\n\\end{eqnarray}\n\n\\bigskip\nWe propose the following:\n\\begin{conjecture}\nFor any integer $s \\geq 1$, the sequence $\\{X_k\\}_{k \\geq 1}$ is uniformly distributed with respect to $\\mu_{ST}^{[s]}$. In other words the probability distribution on $[0,1]^s$ as given by (1.2), converges weakly to $\\mu_{ST}^{[s]}$, as $K$ tends to infinity. \n\\end{conjecture}\n\n\\bigskip\nFor $s=1$ this is the original Sato-Tate conjecture (which is already proved). For $s \\geq 2$ this asserts the statistical independence of the distribution of $s$ successive terms from the sequence $\\{x_k\\}_{k \\geq 1}$ with respect to Sato-Tate measure. Thus in terms of statistical distribution, Conjecture 1.1 asserts that, with respect to the Sato-Tate measure, the sequence $\\{x_k\\}_{k \\geq 1}$ is $\\infty$-distributed in the sense of Knuth, {\\it c.f.} Definition C on page 151 of \\cite{K}. It is in this sense that we say that the sequence $\\{x_k\\}_{k \\geq 1}$ is pseudorandom (with respect to the Sato-Tate measure). We present numerical evidences for Conjecture 1.1 in section 2 below.\n\n\\bigskip\n\\begin{remark}\n\\end{remark}\n\\noindent Let $E$ and $E^{\\prime}$ be elliptic curves over $\\mathbf{Q}$ without complex multiplication, and assume that $E$ and $E^{\\prime}$ are non-isogenous. Let $\\{x_k\\}_{k \\geq 1}$ and $\\{x_k^{\\prime} \\}_{k \\geq 1}$ be the sequences of Frobenius angles associated to $E$ and $E^{\\prime}$ respectively. Consider the two dimensional vectors $(x_k,x_k^{\\prime}) \\in [0,1]^2 $ for $k \\geq 1$. Harris \\cite{H} established that the sequence $\\{ (x_k,x_k^{\\prime})\\}_{k \\geq 1}$ is uniformly distributed with respect to $\\mu_{ST}^{[2]}$. By contrast, the setting of our Conjecture 1.1 concerns the statistical independence of the distribution of Frobenius angles for a single elliptic curve.\n\\bigskip\n\\bigskip\n\nEven more optimistically, we propose the following quantitive refinement of Conjecture 1.1. Firstly define the extreme discrepancy $D^{[s]}_K$ (with respect to the measure $\\mu_{ST}^{[s]}$) for integer $K \\geq 1$ as follows. For any rectangular region $\\mathcal{R} \\subset [0,1]^s$ of the form:\n\\[\n\\mathcal{R} = [a_1,b_1) \\times \\cdots \\times [a_s,b_s)\n\\]\ndefine\n\\[\nA(\\mathcal{R};K) = \\#\\{ 1\\leq k\\leq K \\,\\ | \\,\\ X_k \\in \\mathcal{R}       \\}\n\\]\n\\bigskip\nThen define\n\\[\nD^{[s]}_K = \\sup_{\\mathcal{R} } \\Big|    \\frac{A(\\mathcal{R};K)}{K}        -  \\mu_{ST}^{[s]}  (\\mathcal{R})   \\Big|\n\\]\nwhere $\\mathcal{R}$ ranges over all rectangular regions in $[0,1]^s$ as above.\n\n\\bigskip\n\n\nIn general for a point $W =(w^{(1)},\\cdots,w^{(s)}) \\in [0,1]^s$, we denote by $\\mathcal{R}_W \\subset [0,1]^s$ the rectangular region:\n\\[\n\\mathcal{R}_W = [0,w^{(1)} ) \\times \\cdots \\times  [0,w^{(s)} ) \n\\]\n\n\\bigskip\n\nWe define the star discrepancy $D_K^{*,[s]}$ (with respect to the measure $\\mu_{ST}^{[s]}$) as:\n\\[\nD^{*,[s]}_K = \\sup_{W \\in [0,1]^s } \\Big|    \\frac{A(\\mathcal{R}_W;K )}{K}        -  \\mu_{ST}^{[s]}  (\\mathcal{R}_W)   \\Big|\n\\]\n\n\\bigskip\nWe have $0 \\leq D_K^{[s]} ,D_K^{*,[s]}\\leq 1$, and the inequalities ({\\it c.f.} p. 93 of \\cite{KN}):\n\\begin{eqnarray}\nD_K^{*,[s]} \\leq D_K^{[s]} \\leq 2^s \\cdot D_K^{*,[s]}\n\\end{eqnarray}\n(remark that in {\\it loc. cit.} the notion of discrepancy with respect to the Lebesgue measure on $[0,1]^s$ is considered, but the same considerations apply verbatim with respect to the measure $\\mu_{ST}^{[s]}$ as well).\n\n\\bigskip\nThe discrepancies $D^{[s]}_K $ and $D_K^{*,[s]}$ quantify the uniformity of distribution of the finite set $\\{ X_k \\}_{k=1}^K$ with respect to the measure $\\mu_{ST}^{[s]}$ on $[0,1]^s$.\n\n\\bigskip\n\\begin{conjecture}\nFor any integer $s \\geq 1$ and $\\epsilon >0$, there exists a constant $C=C(E,s,\\epsilon)$ (depending only on the elliptic curve $E$, $s$ and $\\epsilon$), such that:\n\\[\nD^{[s]}_K \\leq C K^{\\epsilon- \\frac{1}{2}}\n\\]\nfor any integer $K \\geq 1$.\n\\end{conjecture}\n\n\\bigskip\nOf course, Conjecture 1.3 is interesting only when $ \\epsilon < \\frac{1}{2} $; in addition it can also be stated equivalently in terms of the star discrepancy $D_K^{*,[s]}$ instead of $D_K^{[s]}$, by virtue of the inequalities (1.3).\n\n\\bigskip\n\n\nWhen $s=1$, Conjecture 1.3 was originally formulated by Akiyama-Tanigawa (Conjecture 1 of \\cite{AT}), which refines the original Sato-Tate conjecture. In general Conjecture 1.3 is a refinement of Conjecture 1.1; namely that by standard results on uniform distribution ({\\it c.f.} p. 93 of \\cite{KN}), Conjecture 1.1 is equivalent to the assertion:\n\\[\n\\lim_{K \\rightarrow \\infty} D^{[s]}_K =0.\n\\]\n\n\\bigskip\n\n Conjecture 1.3 is a very strong statement. Indeed Akiyama-Tanigawa showed that their conjecture (i.e. Conjecture 1.3 in the case $s=1$) implies the validity of the Riemann Hypothesis for the $L$-function associated to $E$ (that the $L$-function associated to $E$ has analytic continuation is of course the consequence of the modularity of $E$); more generally their conjecture implies the validity of the Riemann Hypothesis for all the higher symmetric power $L$-functions associated to $E$, {\\it c.f.} Poposition 3.5 of \\cite{M} for the precise statement (in {\\it loc. cit.} suitable analytic hypotheses on the higher symmetric power $L$-functions are assumed, which are in any case consequences of the Langlands Functoriality Conjecture with respect to the symmetric power functorial liftings of the modular form associated to $E$. The existence of the symmetric power functorial liftings of the modular form associated to $E$ is established by Newton-Thorne \\cite{NT} under quite general conditions, including all semistable $E$ for instance). \n \n \\bigskip\n Conversely Nagoshi showed (see Theorem 2 of \\cite{N}) that the conjecture of Akiyama-Tanigawa holds (at least) for $ \\epsilon > 1\/4$, if one supposes the validity of the Riemann Hypothesis for all the higher symmetric power $L$-functions associated to $E$ (again assuming suitable analytic hypotheses on the higher symmetric power $L$-functions); see also \\cite{RT} for the explicit version. At this moment we do not know whether our Conjecture 1.3 for $s \\geq 2$ can be approached using the theory of $L$-functions. Nevertheless we present numerical evidences for Conjecture 1.3 in section 3 below.\n\n\\bigskip\n\nIn this paper the computations of the orders of the group of points of elliptic curves over finite fields were performed using the {\\it GP\/PARI} program. The rest of the computations were then performed using {\\it Mathematica 9.0}.\n\n\n\n\\section{Numerical Evidences for Conjecture 1.1}\n\nWith the setting as in Conjecture 1.1, for continuous function $f$ defined on $[0,1]^s$, we would like to test whether:\n\\[\n \\frac{1}{K} \\sum_{k=1}^K f(X_k)  \\stackrel{?}{\\rightarrow} \\int_{[0,1]^s} f \\, d \\mu_{ST}^{[s]}\n\\]\n as $K$ tends to infinity. \n\n\\bigskip\nWe consider the following six elliptic curves over $\\mathbf{Q}$ without complex multiplication, taken from the {\\it $L$-functions and modular forms database}, whose affine Weierstrass equations are given as follows, with conductor $N$ and Mordell-Weil rank $r$ as indicated:\n\n\\[\nE_1: y^2+y=x^3-x^2, \\,\\ N=11,\\,\\ r=0\n\\]\n\n\\[\nE_2: y^2+y =x^3-x, \\,\\ N=37, \\,\\ r=1\n\\]\n\n\\[\nE_3: y^2 +xy=x^3+1, \\,\\ N=433, \\,\\ r=2\n\\]\n\n\\[\nE_4:y^2+y=x^3-7x+6, \\,\\ N=5077, \\,\\ r=3\n\\]\n\n\\[\nE_5: y^2+y=x^3-7x+36, \\,\\ N=545723, \\,\\ r=4\n\\]\n\n\\[\nE_6: y^2+y=x^3-79x+342, \\,\\ N=19047851, \\,\\ r=5\n\\]\n\n\\bigskip\n\n\n\n\\bigskip\nWe denote the coordinates on $[0,1]^s$ as $u^{(1)},\\cdots,u^{(s)}$. For testing statistical independence, it is good enough to choose test functions $f$ of the form:\n\\[\nf(u^{(1)},\\cdots,u^{(s)}) = \\prod_{i=1}^s f_i(u^{(i)})\n\\]\nfor continuous functions $f_i$ on $[0,1]$, in which case we have\n\\begin{eqnarray}\n\\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]} = \\prod_{i=1}^s \\int_{[0,1]}  f_i (u^{(i)})  \\cdot 2 \\sin^2(\\pi u^{(i)}) \\, d u^{(i)}\n\\end{eqnarray}\n\n\\bigskip\nWe first consider the case $s=10$. Define the function $f^{[10]}$ on $[0,1]^{10}$:\n\n\\begin{eqnarray*}\n& & f^{[10]}(u^{(1)},\\cdots,u^{(10)}) \\\\ \n&= & \\ln(2 + u^{(1)}) \\cdot \\ln (3 + u^{(2)}) \\cdot \\exp(-u^{(3)}) \\cdot (1+ u^{(4)})^2 \\cdot   (2+ u^{(5)})  \\cdot \\\\\n& &  \\sqrt{2 + u^{(6)}} \\cdot \\sqrt{ 3 + u^{(7)}} \\cdot (4+ u^{(8)})^{\\frac{1}{3}} \\cdot  (8+ u^{(9)})^{\\frac{1}{4}}  \\cdot  \\exp( \\sqrt{1+u^{(10)}}   )\n\\end{eqnarray*}\n\n\\bigskip\nUsing the command NIntegrate of {\\it Mathematica}, the numerical value of the integral $\\int_{[0,1]^{10}}  f^{[10]} \\, d\\mu_{ST}^{[10]}$ is computed as in (2.1) to be:\n\\[ \n\\int_{[0,1]^{10}}  f^{[10]} \\, d\\mu_{ST}^{[10]} \\stackrel{.}{=}114.076\n\\]\n\nThe numerical results for $\\frac{1}{K} \\sum_{k=1}^K f^{[10]}(X_k)$ for $K=5000$, $K=10000$, $K=20000$, $K=50000$, and $K=100000$ are tabulated in Figure 1 below.\n\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\n\t\t\\toprule \n\t\t& $K=5000$  &  $   K=10000$   &  $   K=20000$   &   $K=50000 $ & $K=100000$   \\\\\n\t\t\n\t\t\\midrule \n        $E_1$    &    113.87      &  113.753        &    113.903      &  114.009     &  114.032\\\\    \n        $E_2$   & 114.196       &  114.08        &  114.074            &   114.128     & 114.181\\\\\n\n        $E_3$ &    114.534     &  114.576        &       114.493          &     114.154  & 114.237 \\\\\n\n        $E_4$ &   115.375      &     115.011  &     114.683        &   114.441      &  114.248\\\\\n\n        $E_5$ & 116.127         &    115.137   &      114.499       &    114.62        & 114.474 \\\\\n        $E_6$ &  116.559      & 115.371      &         115.312     &    114.471       &114.519\\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K f^{[10]}(X_k)$}  \n    \\label{Numres}\n\\end{figure}\n\n\\bigskip\nWe next consider examples with larger values of $s$. For $s \\geq 1$ define the function $g^{[s]}$ on $[0,1]^s$ given by:\n\\begin{eqnarray*}\n g^{[s]}(u^{(1)},\\cdots,u^{(s)}) =  100 \\cdot \\prod_{i=1}^s \\exp ( - u^{(i)} \/i)\n\\end{eqnarray*}\n\n\\bigskip\nWe have:\n\\begin{eqnarray}\n& & \\int_{[0,1]^s} g^{[s]} \\, d \\mu_{ST}^{[s]}  \\\\ &=& 100 \\cdot \\prod_{i=1}^s \\int_{[0,1]} \\exp (-u^{(i)}\/i) \\cdot 2 \\sin^2(\\pi u^{(i)})\\, d u^{(i)}  \\nonumber \\\\\n  &=& 100 \\cdot  \\prod_{i=1}^s    \\left( \\big(   1 - \\exp (-1\/i)  \\big)\n         \\cdot \\frac{4 \\pi^2 i^3}{1+ 4\\pi^2 i^2} \\right) \\nonumber\n\\end{eqnarray}\n\n\\bigskip\nThe numerical values of $\\int_{[0,1]^s} g^{[s]} \\, d \\mu_{ST}^{[s]}$ for $s=500$, $s=1000$, and $s=2000$, are computed as in (2.2) to be:\n\n\n\\begin{eqnarray*}\n\\int_{[0,1]^{500}} g^{[500]} \\, d \\mu_{ST}^{[500]} \\stackrel{.}{=}\n3.44034\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{1000}} g^{[1000]} \\, d \\mu_{ST}^{[1000]} \\stackrel{.}{=}\n2.43333\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{2000}} g^{[2000]} \\, d \\mu_{ST}^{[2000]} \\stackrel{.}{=}\n1.72086\n\\end{eqnarray*}\n\n\\bigskip\nWhile the numerical results for $\\frac{1}{K} \\sum_{k=1}^K g^{[s]}(X_k)$ for $K=5000$, $K=10000$, $K=20000$, $K=50000$, and $K=100000$, in the cases $s=500$, $s=1000$, and $s=2000$ respectively, are tabulated in Figures 2, 3, 4 below.\n\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$  &  $   K=10000$   &  $   K=20000$   &   $K=50000 $ & $K=100000$   \\\\\n\t\t\n\t\t\\midrule \n$E_1$&      3.4513 & 3.4541 & 3.45074 & 3.4417 & 3.44034 \\\\\n $E_2$&3.4208 & 3.43017 & 3.43423 & 3.43228 & 3.42744 \\\\\n $E_3$&3.4013 & 3.38951 & 3.40058 & 3.43055 & 3.42734 \\\\\n $E_4$&3.33751 & 3.36335 & 3.38431 & 3.40763 & 3.42225 \\\\\n $E_5$&3.27776 & 3.35011 & 3.40524 & 3.3932 & 3.40618 \\\\\n $E_6$&3.24535 & 3.33898 & 3.33186 & 3.4024 & 3.39838 \\\\\n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K g^{[500]}(X_k)$}  \n    \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$  &  $   K=10000$   &  $   K=20000$   &   $K=50000 $ & $K=100000$   \\\\\n\t\t\n\t\t\\midrule \n      \n $E_1$&2.4422 & 2.44414 & 2.44149 & 2.43447 & 2.43337 \\\\\n $E_2$&2.41925 & 2.42568 & 2.42922 & 2.42732 & 2.42337 \\\\\n $E_3$&2.40409 & 2.39574 & 2.40298 & 2.426 & 2.42349 \\\\\n $E_4$&2.35616 & 2.37465 & 2.39071 & 2.40813 & 2.4195 \\\\\n $E_5$&2.31073 & 2.3628 & 2.40653 & 2.39708 & 2.40692 \\\\\n $E_6$&2.28672 & 2.35609 & 2.3498 & 2.40406 & 2.40116 \\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K g^{[1000]}(X_k)$}  \n    \\label{Numres}\n\\end{figure}\n\n\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$  &  $   K=10000$   &  $   K=20000$   &   $K=50000 $ & $K=100000$   \\\\\n\t\t\n\t\t\\midrule \n \n $E_1$&1.72737 & 1.72919 & 1.72717 & 1.72179 & 1.72091 \\\\\n $E_2$&1.71064 & 1.71501 & 1.71777 & 1.71642 & 1.71329 \\\\\n $E_3$&1.69752 & 1.6939 & 1.69775 & 1.71531 & 1.71333 \\\\\n $E_4$&1.66527 & 1.6769 & 1.68915 & 1.70186 & 1.71047 \\\\\n $E_5$&1.63056 & 1.66685 & 1.70142 & 1.6933 & 1.70075 \\\\\n $E_6$&1.61208 & 1.66286 & 1.65744 & 1.69837 & 1.69624 \\\\       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K g^{[2000]}(X_k)$}  \n    \\label{Numres}\n\\end{figure}\n\n\n\n\n\n\n\n\\bigskip\n\n\n\n\nIn a similar way for $s \\geq 1$ define the function $h^{[s]}$ on $[0,1]^s$ given by:\n\\begin{eqnarray*}\n& & h^{[s]}(u^{(1)},\\cdots,u^{(s)}) \\\\ & =& 100 \\cdot  \\prod_{i=1}^s \\cos\\left(\\frac{ \\pi u^{(i) }}{2 i^{1\/2}} \\right)\n\\end{eqnarray*}\n\n\\bigskip\n\nWe have:\n\\begin{eqnarray}\n& & \\int_{[0,1]^s} h^{[s]} \\, d \\mu_{ST}^{[s]}  \\\\ &=& 100 \\cdot \\prod_{i=1}^s \\int_{[0,1]} \\cos\\left(\\frac{\\pi u^{(i)}}{2 i^{1\/2}}\\right) \\cdot 2 \\sin^2(\\pi u^{(i)})\\, d u^{(i)}  \\nonumber \\\\\n  &=& 100 \\cdot \\prod_{i=1}^s  \\left(           \\frac{2 i^{1\/2}}{\\pi} \\cdot \\sin\\left( \\frac{\\pi}{2 i^{1\/2}}\\right)  \\cdot \\frac{16 i}{16i-1}  \\right)  \\nonumber\n\\end{eqnarray}\n\n\\bigskip\nThe numerical values of $\\int_{[0,1]^s} h^{[s]} \\, d \\mu_{ST}^{[s]}$ for $s=500$, $s=1000$, $s=1500$, and $s=2000$, are computed as in (2.3) to be:\n\n\\begin{eqnarray*}\n\\int_{[0,1]^{500}} h^{[500]} \\, d \\mu_{ST}^{[500]}  \\stackrel{.}{=}  8.814\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{1000}} h^{[1000]} \\, d \\mu_{ST}^{[1000]}  \\stackrel{.}{=} 6.92239\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{1500}} h^{[1500]} \\, d \\mu_{ST}^{[1500]}   \\stackrel{.}{=}6.0099   \n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\int_{[0,1]^{2000}} h^{[2000]} \\, d \\mu_{ST}^{[2000]}   \\stackrel{.}{=}    5.43635\n\\end{eqnarray*}\n\n\\bigskip\n\nWhile the numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[s]}(X_k)$ for $K=5000$, $K=10000$, $K=20000$, $K=50000$, and $K=100000$, in the cases $s=500$, $s=1000$, $s=1500$, and $s=2000$ respectively, are tabulated in Figures 5, 6, 7, 8 below.\n\n\n\\bigskip\n\\bigskip\n\n\nSumming up: the numerical results of this section supply evidences for the validity of Conjecture 1.1; this conjecture can be described as saying that, the sequence $\\{x_k\\}_{k \\geq 1}$ is a pseudorandom sequence in $[0,1]$ with distribution law given by the Sato-Tate measure. \n\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$  &  $   K=10000$   &  $   K=20000$   &   $K=50000 $ & $K=100000$   \\\\\n\t\t\n\t\t\\midrule \n       $E_1$&$8.84564$&$ 8.8739$&$ 8.85475$&$ 8.82749$& $8.81591$\\\\\n $E_2$&$8.74331$&$ 8.7891$&$ 8.80022$&$ 8.78814$&$ 8.7736$\\\\\n $E_3$&$8.70232$&$ 8.63423$&$ 8.67626$&$ 8.79667$&$ 8.77114$\\\\\n $E_4$&$8.49599$&$ 8.54784$&$ 8.62823$&$ 8.69612$&$ 8.76489$\\\\\n $E_5$&$8.27357$&$ 8.53379$&$ 8.69698$&$ 8.67127$&$ 8.70012$\\\\\n $E_6$&$8.18894$&$ 8.50111$&$ 8.45486$&$ 8.70109$&$ 8.69085$\\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[500]}(X_k)$}  \n    \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$  &  $   K=10000$   &  $   K=20000$   &   $K=50000 $ & $K=100000$   \\\\\n\t\t\n\t\t\\midrule \n$E_1$&$6.9513$&$ 6.97341$&$ 6.95763$&$ 6.93425$&$ 6.9241$\\\\\n $E_2$&$6.86542$&$ 6.90182$&$ 6.91247$&$ 6.90096$&$ 6.8879$\\\\\n $E_3$&$6.82891$&$ 6.77459$&$ 6.80619$&$ 6.90821$&$ 6.8863$\\\\\n $E_4$&$6.65655$&$ 6.69763$&$ 6.76587$&$ 6.822$&$ 6.88$\\\\\n $E_5$&$6.47181$&$ 6.67989$&$ 6.82385$&$ 6.80146$&$ 6.8251$\\\\\n $E_6$&$6.40304$&$ 6.65958$&$ 6.61765$&$ 6.82616$&$ 6.8182$\\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[1000]}(X_k)$}  \n    \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$  &  $   K=10000$   &  $   K=20000$   &   $K=50000 $ & $K=100000$   \\\\\n\t\t\n\t\t\\midrule \n         $E_1$&$ 6.03675      $&$   6.0559         $&$     6.04206        $&$    6.02086         $&$   6.01151        $\\\\\n $E_2$&$   5.96016         $&$    5.99199         $&$   6.00129          $&$    5.99081         $&$   5.97872        $\\\\\n $E_3$&$     5.9214       $&$       5.87986      $&$    5.90473         $&$     5.9971        $&$     5.97718      $\\\\\n $E_4$&$     5.77546       $&$     5.80653        $&$   5.86934          $&$   5.91933          $&$   5.97276        $\\\\\n $E_5$&$     5.60953       $&$     5.7892        $&$     5.92279        $&$    5.90058         $&$    5.92183       $\\\\\n $E_6$&$     5.54555       $&$   5.77334          $&$    5.73411         $&$   5.92231          $&$   5.91537        $\\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[1500]}(X_k)$}  \n    \\label{Numres}\n\\end{figure}\n\n\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5000$  &  $   K=10000$   &  $   K=20000$   &   $K=50000 $ & $K=100000$   \\\\\n\t\t\n\t\t\\midrule \n       $E_1$&$5.46061$&$ 5.47935$&$ 5.46676$&$ 5.4467$&$ 5.4379$\\\\\n $E_2$&$5.39035$&$ 5.41982$&$ 5.42836$&$ 5.41881$&$ 5.4073$\\\\\n $E_3$&$5.3513$&$ 5.31787$&$ 5.33875$&$ 5.42458$&$ 5.4058$\\\\\n $E_4$&$5.2241$&$ 5.24905$&$ 5.30693$&$ 5.35259$&$ 5.4022$\\\\\n $E_5$&$5.06843$&$ 5.23096$&$ 5.35741$&$ 5.33487$&$ 5.3545$\\\\\n $E_6$&$5.01111$&$ 5.21909$&$ 5.18121$&$ 5.35458$&$ 5.3483$\\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $\\frac{1}{K} \\sum_{k=1}^K h^{[2000]}(X_k)$}  \n    \\label{Numres}\n\\end{figure}\n\n\n\n\n\n\\section{Numerical Evidences for Conjecture 1.3, part I}\n\nFor numerics related to Conjecture 1.3 in the case $s=1$ (i.e. the original conjecture of Akiyama-Tanigawa), we refer to \\cite{AT} and \\cite{St}. To test Conjecture 1.3 directly one would need to evaluate the values the $D_K^{[s]}$ or $D_K^{*,[s]}$ for $K$ large. But in higher dimensions $s$ there are serious combinatorial difficulties in computing (even just numerically) the values of $D_K^{[s]}$ or $D_K^{*,[s]}$; this is the well known phenomenon known as the {\\it Curse of Dimensionality}. \n\n\\bigskip\n\nWe first note that, Conjecture 1.3 is obviously equivalent to the statement:\n\\[\n\\liminf_{K \\rightarrow \\infty} - \\frac{\\ln  D^{[s]}_K}{\\ln K} \\geq \\frac{1}{2}\n\\]\n(and similarly with $D_K^{[s]}$ being replaced by $D_K^{*,[s]}$).\n\n\\bigskip\n\\begin{proposition}\nLet $f$ be a function defined on $[0,1]^s$, which is of bounded variation in the sense of Hardy and Krause. Then Conjecture 1.3 implies:\n\\[\n\\liminf_{K \\rightarrow \\infty} - \\frac{\\ln \\big|       \\frac{1}{K}\\sum_{k=1}^K f(X_k)   - \\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]}  \\big|}{\\ln K} \\geq \\frac{1}{2}\n\\]\n\\end{proposition}\n\\begin{proof}\nThis is an immediate consequence of the Koksma-Hlawka inequality ({\\it c.f.} p. 151 of \\cite{KN} and p. 967 of \\cite{Ni2}):\n\\[\n\\Big|       \\frac{1}{K}\\sum_{k=1}^K f(X_k)   - \\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]}  \\Big|   \\leq V(f) \\cdot D_K^{*,[s]}\n\\]\nwhere $V(f)$ is the total variation of $f$ in the sense of Hardy and Krause (the version of the Koksma-Hlawka inequality stated in {\\it loc. cit.} is with respect to the Lebesgue measure on $[0,1]^s$, but the same proof works verbatim with respect to the measure $\\mu_{ST}^{[s]}$ on $[0,1]^s$).\n\\end{proof}\n\n\\bigskip\nIn view of Proposition 3.1. we may then test Conjecture 1.3 indirectly as follows. With $f$ defined on $[0,1]^s$ as above (of bounded variation in the sense of Hardy and Krause), denote the relative error:\n\n\\[\n\\OP{RelErr}(f,K) =    \\frac{\\frac{1}{K}\\sum_{k=1}^K f(X_k)   - \\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]} }{ \\int_{[0,1]^s} f \\, d\\mu_{ST}^{[s]} }\n\\]\n\n\\bigskip\n\\noindent (assuming that the integral is nonzero). \n\n\\bigskip\n\nThen we evaluate:\n\n\\[\n-\\frac{\\ln |\\OP{RelErr} (f,K)|}{\\ln K}\n\\]\nwith $K$ being large. By virtue of Proposition 3.1, Conjecture 1.3 implies that\n\n\\[\n\\liminf_{K \\rightarrow \\infty}   \\frac{-\\ln |\\OP{RelErr}(f,K) |}{\\ln K} \\geq \\frac{1}{2}\n\\]\n\n\\bigskip\n\\bigskip \n\nIn the following numerical examples the dimensions $s$ and the test functions $f$ on $[0,1]^s$ are chosen as in section 2, namely: \n\n\\[\nf^{[10]},g^{[500]},g^{[1000]},g^{[2000]},h^{[500]},h^{[1000]},h^{[1500]}, h^{[2000]}\n\\]\n\n\\bigskip\nThe results are tabulated in Figures 9 -16 below.\n\n\\bigskip\n\\bigskip\n\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$  &  $   K=   10^6$   &  $   K= 2 \\times 10^6$   &   $K=5 \\times 10^6 $ & $K=10^7$   \\\\\n\t\t\n\t\t\\midrule \n       $E_1$    &     0.69949&0.679386&0.630927&0.66101&0.691336 \\\\    \n        $E_2$   &     0.633969&0.581946&0.687151&0.641671&0.670621\\\\\n\n        $E_3$ &     0.573905&0.57178&0.585748&0.576866&0.592804\\\\\n\n        $E_4$ &      0.523103&0.542605&0.531523&0.566703&0.525969\\\\\n\n        $E_5$ &  0.514324&0.516659&0.513771&0.50841&0.504189\\\\\n        $E_6$ &   0.494667&0.524282&0.505647&0.490742&0.491219\\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (f^{[10]},K)|}{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$  &  $   K=   10^6$   &  $   K= 2 \\times 10^6$   &   $K=5 \\times 10^6 $ & $K=10^7$   \\\\\n\t\t\n\t\t\\midrule \n        $E_1$    & 0.638723& 0.648979& 0.631947& 0.862372& 0.808676          \\\\    \n        $E_2$   &  0.575948&0.505878&0.633176&0.552754&0.543251     \\\\\n\n        $E_3$ &  0.487867&0.496663&0.514921&0.501386&0.498532  \\\\\n\n        $E_4$ & 0.444677&0.462&0.462372&0.493716&0.460063  \\\\\n\n        $E_5$ &  0.431359&0.436573&0.438725&0.445434&0.44006      \\\\\n        $E_6$ &   0.404953&0.436986&0.430817&0.421285&0.426719   \\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (g^{[500]},K)|}{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\n\n\n\\newpage\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$  &  $   K=   10^6$   &  $   K= 2 \\times 10^6$   &   $K=5 \\times 10^6 $ & $K=10^7$   \\\\\n\t\t\n\t\t\\midrule \n        $E_1$    &  0.631169&0.645835&0.630361&0.712336&0.706233       \\\\    \n        $E_2$   &   0.567706&0.497695&0.617072&0.544564&0.534925  \\\\\n\n        $E_3$ &   0.480746&0.489362&0.507584&0.494107&0.492241   \\\\\n\n        $E_4$ &   0.437471&0.455814&0.455769&0.487224&0.453975  \\\\\n\n        $E_5$ &    0.424289&0.429748&0.432216&0.439139&0.434175  \\\\\n        $E_6$ &    0.39818&0.430663&0.424525&0.414925&0.420545          \\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (g^{[1000]},K)|}{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$  &  $   K=   10^6$   &  $   K= 2 \\times 10^6$   &   $K=5 \\times 10^6 $ & $K=10^7$   \\\\\n\t\t\n\t\t\\midrule \n            $E_1$    &   0.627366& 0.638355&0.622921&0.675519&0.66696        \\\\    \n        $E_2$   &    0.559385&0.490252&0.605272&0.53634&0.526713  \\\\\n\n        $E_3$ &      0.473656&0.482631&0.500549&0.487117&0.485837  \\\\\n\n        $E_4$ & 0.430963&0.449776&0.449301&0.480237&0.447863\\\\\n\n        $E_5$ &      0.418162&0.423543&0.426154&0.4331&0.428402  \\\\\n        $E_6$ &      0.391767&0.424534&0.418375&0.408966&0.414847        \\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (g^{[2000]},K)|}{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$  &  $   K=   10^6$   &  $   K= 2 \\times 10^6$   &   $K=5 \\times 10^6 $ & $K=10^7$   \\\\\n\t\t\n\t\t\\midrule \n        $E_1$    &   0.606447& 0.634717&0.596622&0.769728&0.72325 \\\\    \n        $E_2$   &    0.592215&0.483297&0.643853&0.573784&0.538485    \\\\\n\n        $E_3$ &      0.469397&0.46504&0.491604&0.482104&0.506944   \\\\\n\n        $E_4$ &    0.428098&0.451091&0.44635&0.467237&0.44823 \\\\\n\n        $E_5$ & 0.417505&0.421578&0.424573&0.429395&0.423905\\\\\n        $E_6$ &  0.398343&0.426324&0.412744&0.399427&0.40276  \\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (h^{[500]},K)|}{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\n\n\\newpage\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$  &  $   K=   10^6$   &  $   K= 2 \\times 10^6$   &   $K=5 \\times 10^6 $ & $K=10^7$   \\\\\n\t\t\n\t\t\\midrule \n        $E_1$    &   0.599239&0.632049&0.594434&0.699701&0.677617        \\\\   \n        $E_2$   &    0.58362&0.4755&0.618896&0.561171&0.528059       \\\\\n\n        $E_3$ &       0.462718&0.458196&0.484627&0.474634&0.499336 \\\\\n\n        $E_4$ &    0.421188&0.445555&0.440249&0.46107&0.442045\\\\\n\n        $E_5$ &   0.410704&0.414996&0.418179&0.423116&0.41825\\\\\n        $E_6$ &  0.391969&0.420306&0.406692&0.393337&0.396837       \\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (h^{[1000]},K)|}{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$  &  $   K=   10^6$   &  $   K= 2 \\times 10^6$   &   $K=5 \\times 10^6 $ & $K=10^7$   \\\\\n\t\t\n\t\t\\midrule \n        $E_1$  &  0.596406     & 0.624722         &      0.588404              &            0.675777              &    0.659381     \\\\    \n        $E_2$   &   0.578855         &    0.471502                 &     0.612237                      &  0.555763       &   0.523419 \\\\\n\n        $E_3$ &   0.458443            &   0.454467                &       0.480825                   &   0.470964      &    0.495892            \\\\\n\n        $E_4$ &     0.41748           &    0.442316              &       0.436737                    &  0.457416        &      0.438645       \\\\\n\n        $E_5$ &       0.40721        &    0.411428                &         0.414829               &     0.419795        &    0.415218      \\\\\n        $E_6$ &    0.388235          &   0.416779                  &     0.403099                &     0.390014       &  0.393701         \\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (h^{[1500]},K)|}{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\n\n\n\\bigskip\n\\bigskip\n\n\n\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^5$  &  $   K=   10^6$   &  $   K= 2 \\times 10^6$   &   $K=5 \\times 10^6 $ & $K=10^7$   \\\\\n\t\t\n\t\t\\midrule \n             $E_1$    &     0.594196&0.620986&0.584018&0.670973&0.654911       \\\\    \n        $E_2$   &    0.574501&0.468545&0.607293&0.55214&0.520405   \\\\\n\n        $E_3$ &     0.456099&0.452267&0.478601&0.468548&0.493642\\\\\n\n        $E_4$ &    0.415082&0.440149&0.434365&0.45504&0.436376\\\\\n\n        $E_5$ &   0.404872&0.409085&0.412494&0.417499&0.413082\\\\\n        $E_6$ &  0.385872&0.414499&0.40085&0.387842&0.391689     \\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln |\\OP{RelErr} (h^{[2000]},K)|}{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\n\n\n\\section{Numerical Evidences for Conjecture 1.3, part II}\n\n\n\n\nFinally we test Conjecture 1.3 directly for the dimensions $s=2$ and $s=3$. We first recall the following general result of Niederreiter \\cite{Ni1} in order to compute the star discrepancy $D_K^{*,[s]}$ of $\\{X_k\\}_{k=1}^K \\subset [0,1]^s$ (with respect to the measure $\\mu_{ST}^{[s]}$).\n\n\\bigskip\n\nFor $1 \\leq i \\leq s$, denote by $0= \\beta^{(i)}_1 < \\cdots  <\\beta^{(i)}_{n_i}=1$ the set of distinct values of the set of $i$-th coordinates of the points $\\{X_k\\}_{k=1}^K$, with the values $0$ and $1$ being included. Denote by $\\mathfrak{q}$ the collection of rectangular regions $Q \\subset [0,1]^s$ of the form:\n\\[\nQ =  \\prod_{i=1}^s (  \\beta^{(i)}_{j_i}  , \\beta^{(i)}_{j_i +1}    ],  \\,\\  1 \\leq j_i < n_i \\mbox{ for } 1 \\leq i \\leq s\n\\]\nwhich thus forming a partition of $(0,1]^s$. For $Q$ as above, denote:\n\\[\nY(Q) =   (\\beta^{(1)}_{j_1 +1},\\cdots, \\beta^{(s)}_{j_s +1}  )  \\in [0,1]^s\n\\]\nthe upper end point of $Q$, and\n\\[\nZ(Q) = (\\beta^{(1)}_{j_1},\\cdots, \\beta^{(s)}_{j_s}  ) \\in [0,1]^s  \n\\]\nthe lower end point of $Q$. \n\n\n\n\n\\bigskip\nWe then have:\n\\begin{proposition} The star discrepancy $D_K^{*,[s]}$ is equal to:\n\\begin{eqnarray*}\n\\max_{Q \\in \\mathfrak{q}} \\Big(\\max \\Big(    \\Big|   \\frac{A(  \\mathcal{R}_{Y(Q)};K)  }{K}     - \\mu_{ST}^{[s]}(     \\mathcal{R}_{Y(Q)}  )        \\Big|,       \\Big|          \\frac{A(\\mathcal{R}_{Z(Q)} ;K)}{K}     - \\mu_{ST}^{[s]}(     \\mathcal{R}_{Z(Q)}  )    \\Big|         \\Big)  \\Big) \n\\end{eqnarray*}\n\\end{proposition}\n\\begin{proof}\nThis is Theorem 2 of \\cite{Ni1}. In {\\it loc. cit.} it is stated with respect to the Lebesgue measure on $[0,1]^s$, but the argument works verbatim with respect to the measure $\\mu_{ST}^{[s]}$.\n\\end{proof}\n\n\\bigskip\n\\begin{remark}\n\\end{remark}\n\\noindent To compute $D_K^{*,[s]}$ using Proposition 4.1 ({\\it i.e.} Theorem 2 of \\cite{Ni1}) requires the evaluation of  $O(K^s)$ terms. Although there are algorithms that improve upon that of \\cite{Ni1} for computing the star discrepancy ({\\it c.f.} for example \\cite{DE}), the time complexity of the known algorithms is still exponential in terms of the dimension $s$; this is an instance of the {\\it Curse of Dimensionality}. In fact it is known that the computation of star discrepancy belongs to the class of NP-hard problems \\cite{GSW}.\n\n\\bigskip\n\\bigskip\n\nBelow we use Proposition 4.1 to compute the numerical values of $D_K^{*,[s]}$ and hence test Conjecture 1.3 (with respect to $D_K^{*,[s]}$), in the cases $s=2$ and $s=3$. The results are tabulated in Figures 17, 18 below.\n\n\\bigskip\nSumming up: the numerical results of section 3 and section 4 supply evidences for the validity of Conjecture 1.3. It is a refinement of Conjecture 1.1, and can be regarded as a qualitative form, of the Law of Iterated Logarithm for random numbers ({\\it c.f.} Chapter 7 of \\cite{Ni3}). In particular, Conjecture 1.3 implies that, with respect to the Sato-Tate measure, the sequence of Frobenius angles of an elliptic curve over $\\mathbf{Q}$ without complex multiplication forms a pseudorandom sequence in $[0,1]$ with strong randomness property, (at least) as far as statistical distribution is concerned.\n\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c|c|c|c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^3$  &  $   K=   10^4$   &  $   K= 2 \\times 10^4$   &   $K=5 \\times 10^4 $ & $K=10^5$   \\\\\n\t\t\n\t\t\\midrule \n        $E_1$    &   0.513743         &     0.481735               &     0.493825                     &    0.508233   &  0.506597\\\\    \n        $E_2$   &   0.506241         &     0.511887                &       0.468917                    &   0.494157      & 0.492688\\\\\n\n        $E_3$ &    0.484667           &     0.483204              &       0.484577                   &   0.526097      & 0.487613\\\\\n\n        $E_4$ &   0.442152             &    0.418237              &        0.434046                   &   0.440515       &0.467903\\\\\n\n        $E_5$ &    0.423393           &    0.443996                &    0.46427                    &  0.441762           & 0.42277\\\\\n        $E_6$ &    0.413569          &    0.421813                 &      0.419844               &  0.458051          &0.426849\\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln  D_K^{*,[2]} }{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\\bigskip\n\\bigskip\n\n\\begin{figure}[hp] \n\t\n\t\\centering  \n\t\n\t\\begin{tabular}{c|c} \n\t\t\n\t\t\\toprule \n\t\t& $K=5 \\times 10^3$   \\\\\n\t\t\n\t\t\\midrule \n        $E_1$   &      0.495306           \\\\    \n        $E_2$   &     0.479892             \\\\\n\n        $E_3$ &  0.472156                   \\\\\n\n        $E_4$ &     0.413948               \\\\\n\n        $E_5$ &          0.405477           \\\\\n        $E_6$ &      0.39623                \\\\\n       \n\t\t\\bottomrule \n\t\t\n\t\\end{tabular}\n    \\caption{Numerical results for $-\\frac{\\ln  D_K^{*,[3]} }{\\ln K}$}  \n    \\label{Numres}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion and Final Remarks}\n\nIn this paper we propose conjectures that refine the Sato-Tate conjecture, specifically we conjecture that the Frobenius angles of a given elliptic curve over $\\mathbf{Q}$ without complex multiplication, are statistically independently distributed with respect to the Sato-Tate measure, including the more quantitative version involving the discrepancy of joint distributions. Numerical evidences are presented to support the conjectures. \n\n\\bigskip\n\nTaylor {\\it et. al.} \\cite{CHT,T,HSBT,BLGHT} had established the Sato-Tate conjecture for elliptic curves over totally real fields without complex multiplication, and more generally \\cite{BLGG} established the Sato-Tate conjecture for Hilbert modular forms over totally real fields ({\\it c.f.} \\cite{ACC+} for the latest results on the Sato-Tate conjecture for automorphic forms over number fields). Thus it is natural to expect that our Conjectures 1.1 and 1.3 extend to the more general setting as well. \n\n\\bigskip\nIt would be intriguing to find possible connections between Conjecture 1.3 in the case $s \\geq 2$ and properties of $L$-functions.\n\n\\bigskip\n\nFinally and most interestingly, as observed experimentally from the numerics, the rate of convergence to the measure $\\mu_{ST}^{[s]}$, is slower in the case of curves with higher Mordell-Weil ranks (in the one dimensional case $s=1$ this was already observed in \\cite{St}). Heuristically, in accordance with the original form of the Birch and Swinnerton-Dyer conjecture, this can be seen as due to the fact that, for curves of high Mordell-Weil rank, there is a Chebyshev bias for the quantities $a_{p_k}$ towards being negative, {\\it c.f.} \\cite{M}, \\cite{S}, \\cite{KM}. It would be important to understand the rate of convergence to the measure $\\mu_{ST}^{[s]}$ in a more precise form (for example along the lines suggested in \\cite{St} in the case $s=1$).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{State Counting in Fermi Liquids}\n\nIn a non-interacting system, the number of low-energy addition states\nper electron per spin is equal to one. Should the number of low-energy\naddition states per electron per spin exceed unity, Fermi liquid\ntheory fails and new electronic states emerge at low energy that\ncannot be constructed from the non-interacting system. To show that\nthis state of affairs obtains in a doped Mott insulator, we compare\nthe number of electrons per site ($n_h$) that can be added to the\nholes created by the dopants with the number of single-particle\n addition states per site at low energy,\n\\begin{eqnarray}\\label{dos}\nL=\\int_\\mu^\\Lambda N(\\omega)d\\omega,\n\\end{eqnarray}\ndefined as the integral of the single-particle density of states\n($N(\\omega)$) from the chemical potential, $\\mu$, to a cutoff energy\nscale, $\\Lambda$, demarcating the IR and UV scales.  \nConsider first\nthe case of a Fermi liquid or non-interacting system.  As illustrated\nin Fig. (\\ref{fig1}), the total weight of the valence band is 2, that\nis, there are 2 states per site.  The integrated weight of the valence\nband up to the chemical potential determines the filling.\nConsequently, the unnocupied part of the spectrum, which determines $L$,\nis given by $L=2-n$. The number of electrons that can be added to the\nempty sites is also $n_h=2-n$ (see Fig. (\\ref{fig2})).  Consequently, the number\nof low-energy states per electron per spin is identically unity. The\nkey fact on which this result hinges is that the total weight of the\nvalence band is a constant independent of the electron density. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9.0cm]{mhtransfer.eps}\n\\caption{Evolution of the single-particle density of states from\n  half-filling to the one-hole limit in a doped Mott insulator\n  described by the Hubbard model.  Removal of an electron results\n  in two empty states at low energy as opposed to one in the\n  band-insulator limit. The key difference with the Fermi liquid is\n  that the total weight spectral weight carried by the lower Hubbard\n  band (analogue of the valence band in a Fermi liquid) is not a\n  constant but a function of the filling. }\n\\label{fig2}\n\\end{figure}\n\n\\section{Doped Mott Insulators: Not just electrons}\n\nFor a doped Mott insulator, the situation is quite different as a\ncharge gap splits the spectrum into lower and upper Hubbard bands (LHB\nand UHB, hereafter)\ndepicted in Fig. (\\ref{fig2}).  At half-filling the chemical potential\nlies in the gap.  The sum rule that 2 states exist per site \napplies only to the combined weight of both bands.  At any finite\ndoping,\nthe weight in the LHB and UHB is determined by the density.    For\nexample, at half-filling each carries half the spectral weight. Even\nin the atomic limit, the spectral weights in the LHB and UHB are\ndensity dependent as shown in Fig. (\\ref{fig2}).\nNonetheless, for a doped Mott insulator in the atomic limit, i.e. one\nelectron per site with infinite on-site repulsion $U$, it is still true that $L\/n_h=1$, because creating a hole leaves behind an empty site which\ncan be occupied by either a spin-up or a spin-down electron. Hence,\nwhen $x$ electrons (see Fig. (\\ref{dos})) are removed, $L=n_h=2x=2-2n$\nin the atomic limit. Recall for a Mott system, $x=1-n$ as the hole\ndoping occurs relative to half-filling.  Hence, for a Mott system in\nthe atomic limit, the total weight in the LHB increases from 1\/2 at\nhalf-filling to $1-x+2x=2-n$ as the system is doped.  This result\nillustrates that the total weight in the LHB goes over smoothly to the\nnon-interacting limit when $n=0$.  That is, 2 states exist per site at\nlow energy entirely in the LHB.  The change from half the spectral\nweight at $n=1$ to all the spectral weight residing in the LHB at\n$n=0$ is a consequence of spectral weight transfer.  The atomic limit,\nhowever, only captures the static (state counting) part of the\nspectral weight.  In fact the $2x$ sum rule,in which $L\/n_h=1$, is\ncaptured by the widely used\\cite{lee,lee1,lee2} $t-J$ model of a doped\nMott insulator in which no doubly occupied sites are allowed.   However, real Mott systems are not in the\natomic limit. Finite hopping with matrix element $t$ creates double\noccupancy, and as a result empty sites with weight $t\/U$. Such empty sites with fractional weight contribute to $L$. Consequently,\nwhen $0<t\/U\\ll 1$, $L$ is strictly larger than $2x$. The actual behaviour,\nred curve in Fig. (\\ref{dos}), has been confirmed\nboth experimentally by oxygen 1s x-ray-absorption spectroscopy\\cite{xray} and\ntheoretically by exact diagonalization\\cite{diag} and\nperturbation theory\\cite{harris}. The number of {\\it electrons}\\cite{frac} that can be\nadded at {\\it  low energy} remains twice the number of dopants, $n_h=2x$, even\nwhen the hopping\nis finite. That is, the quantum fluctuations mediated by the hopping\ndo not change the number of electrons that can be added at low energy.\nAs a result, for a doped Mott insulator,\n$L\/n_h>1$. Consequently, in contrast to a Fermi liquid, simply counting\nthe number of electrons that can be added does {\\it not} exhaust the\navailable phase space to add an electron at low energy. Thus,\nadditional degrees of freedom at low-energy, not made out of the elemental\nexcitations, must exist. They arise from the hybridization with the doubly occupied sector\nand hence must emerge at low energy from a collective charge 2e\nexcitation. The new charge $e$ state that emerges at low energy must correspond to a\nbound state of the collective charge $2e$ excitation and the hole that\nis left behind. It is the physics of this new charge $e$ state that\nmediates the non-Fermi liquid behaviour in a doped Mott insulator.\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.0cm,angle=-90]{dos.eps}\n\\caption{a) Integrated low-energy spectral\nweight, $L$, defined in Eq. (\\ref{dos}), as a function of the\nelectron filling, n: 1) the dashed line is the non-interacting limit, vanishing on-site\ninteraction ($U=0$), in which $L=2-n$, 2) atomic limit\n(blue line) of a doped\nMott insulator, $U=\\infty$, in which $L=2(1-n)=2x$, $x$ the doping\nlevel and 3) a real Mott insulator in which $0<t\/U\\ll 1$, red curve. For\n$0<t\/U\\ll 1$, $L$ must lie strictly above the $U=\\infty$ limit and hence $L>2x$ away from the\natomic limit. (b) Hopping processes mediated by the $t\/U$ terms in\nthe expansion of the projected transformed operators in terms of the bare electron operators (see Eq. (\\ref{trans})). As a\nresult of the $t\/U$ terms in Eq. (\\ref{eq:sc1}), the low-energy\ntheory in terms of the bare fermions does not preserve double\noccupancy. The process\nshown here illustrates that mixing between the high and low-energy\nscales obtains only if double occupancy neighbours a hole. In\nthe exact low-energy theory, such processes are mediated by the new\ndegree of freedom, $\\varphi_i$, the charge $2e$ bosonic field which\nbinds a hole and produces a new charge $e$ excitation, the collective\nexcitation in a doped Mott insulator. }\n\\label{dos}\n\\end{figure}\n\nIn a series\\cite{lowen1} of recent papers, we found the collective charge mode in a doped Mott insulator by integrating out exactly\nthe high-energy scale in the Hubbard model, thereby obtaining an exact description of the IR physics. Consistent with the physical argument presented above, the collective mode is a\ncharge $2e$ bosonic field which mediates new electron dynamics at low\nenergy and is not made out of the elemental excitations in the UV. In\nan attempt to clarify this theory, we establish the relationship\nbetween the standard perturbative approach and the physics mediated by the charge $2e$ boson. By comparing how the operators\ntransform in both theories, we are able to unambiguously associate\nthe charge $2e$ boson with dynamical (hopping-dependent) spectral weight\ntransfer across the Mott gap\\cite{diag,eskes,slavery}. Further, the\nspectral weight transfer is mediated by a bound excitation of the\ncharge $2e$ boson and a hole in accord with the physical argument\npresented above.\n\n\\section{Standard Approach}\n\nOf course the standard approach for treating the fact that $L\/n_h>1$\nin a doped Mott insulator is through perturbation theory, not by explicitly constructing the missing degree of freedom. The goal of the perturbative approach is to bring the Hubbard model\n\\begin{eqnarray}\nH&=&-t\\sum_{i,j,\\sigma} g_{ij}\na^\\dagger_{i,\\sigma}a_{j,\\sigma}+\nU\\sum_i a^\\dagger_{i,\\uparrow}a^\\dagger_{i,\\downarrow}a_{i,\\downarrow}a_{i,\\uparrow}\n\\end{eqnarray}\ninto block diagonal form in which each block has a fixed number of `fictive' doubly occupied sites. Here $i,j$ label lattice sites,\n$g_{ij}$ is equal to one iff $i,j$ are nearest neighbours and $a_{i,\\sigma}$\nannihilates an electron with spin $\\sigma$ on lattice site $i$. We say `fictive' because the operators which make\ndouble occupancy a conserved quantity are not the physical electrons but\nrather a transformed (dressed) fermion we call $c_{i\\sigma}$\ndefined below.\nFollowing Eskes et al.\\cite{eskes},\nfor any operator $O$, we define $\\tilde O$ such that\n$ O\\equiv {\\bf O}(a)$ and $\\tilde{O}\\equiv {\\bf O}(c)$,\nsimply by replacing the Fermi operators $a_{i\\sigma}$ with the\ntransformed fermions $c_{i\\sigma}$. Note that $O$ and $\\tilde O$ are only\nequivalent in the $U=\\infty$ limit. The procedure which makes the Hubbard model\nblock diagonal is now well known\\cite{eskes,spalek,sasha,girvin,anderson}.\nOne constructs a similarity\ntransformation $S$ which connects sectors that differ by at most one\n`fictive' doubly occupied site such that\n\\begin{eqnarray}\nH=e^S\\tilde H e^{-S}\n\\end{eqnarray}\nbecomes block diagonal, where $\\tilde H$ is expressed in terms of the transformed fermions. In the new basis,\n$[H,\\tilde V]=0$,\nimplying that double occupation of the transformed fermions\nis a good quantum number, and all of the eigenstates\ncan be indexed as such. This does not mean that $[H,V]=0$. If it\nwere, there would have been no reason to do the similarity transformation in\nthe first place. $\\tilde V$, and\nnot $V$, is conserved. Assuming that $V$ is the conserved\nquantity results in a spurious local SU(2)\\cite{lsu21,lsu22}\nsymmetry in the strong-coupling limit at\nhalf-filling.\n\nOur focus is on the relationship between the physical and `fictive'\nfermions. To leading order\\cite{eskes} in $t\/U$, the bare fermions,\n\\begin{eqnarray}\na_{i\\sigma}&=&e^Sc_{i\\sigma}e^{-S}\n\\simeq c_{i\\sigma}-\\frac{t}{U}\\sum_{\\langle j, i\\rangle}\n\\left[(\\tilde n_{j\\bar\\sigma}-\\tilde n_{i\\bar\\sigma})c_{j\\sigma}\\right.\\nonumber\\\\\n&-&\\left.c^\\dagger_{j\\bar\\sigma}c_{i\\sigma}c_{i\\bar\\sigma}+c^\\dagger_{i\\bar\\sigma}c_{i\\sigma}c_{j\\bar\\sigma}\\right],\n\\end{eqnarray}\nare linear combinations of\nmultiparticle states in the transformed basis as is expected in\ndegenerate perturbation theory\nBy inverting this relationship, we find that to leading order, the transformed operator is simply,\n\\begin{eqnarray}\nc_{i\\sigma}\\simeq a_{i\\sigma}+\\frac{t}{U}\\sum_{j} g_{ij}X_{ij\\sigma}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nX_{ij\\sigma}=\\left[(n_{j\\bar\\sigma}-n_{i\\bar\\sigma})a_{j\\sigma}-a^\\dagger_{j\\bar\\sigma}a_{i\\sigma}a_{i\\bar\\sigma}+a^\\dagger_{i\\bar\\sigma}a_{i\\sigma}a_{j\\bar\\sigma}\\right].\n\\end{eqnarray}\nWhat we would like to know is what do the transformed fermions look\nlike in the lowest energy sector. We accomplish this by computing\nthe projected operator\n\\begin{eqnarray}\n(1-\\tilde n_{i\\bar\\sigma})c_{i\\sigma}&\\simeq &(1-n_{i\\bar\\sigma})a_{i\\sigma}+\\frac{t}{U}\\sum_{j}g_{ij}\\left[\n(1-n_{i\\bar\\sigma})X_{ij\\sigma}\\right.\\nonumber\\\\\n&&\\left.-X^\\dagger_{ij\\bar\\sigma}a_{i\\bar\\sigma}a_{i\\sigma}-a^\\dagger_{i\\bar\\sigma}X_{ij\\bar\\sigma}a_{i\\sigma}\\right].\n\\end{eqnarray}\nSimplifying, we find that\n\\begin{eqnarray}\\label{trans}\n(1-\\tilde n_{i\\bar\\sigma})c_{i\\sigma}&\\simeq &(1-n_{i\\bar\\sigma})a_{i\\sigma}+\\frac{t}{U}V_\\sigma\na_{i\\bar\\sigma}^\\dagger b_i\\nonumber\\\\\n&+&\\frac{t}{U}\\sum_{j}g_{ij}\\left[\nn_{j\\bar\\sigma}a_{j\\sigma}+n_{i\\bar\\sigma}(1-n_{j\\bar\\sigma})a_{j\\sigma}\\right.\\nonumber\\\\\n&&\\left.+(1-n_{j\\bar\\sigma})\\left(a_{j\\sigma}^\\dagger\na_{i\\sigma}-a_{j\\sigma}a^\\dagger_{i\\sigma}\\right)a_{i\\bar\\sigma}\\right].\n\\end{eqnarray}\nHere $V_\\sigma=-V_{\\bar\\sigma}=1$ and\n$b_i= \\sum_{j\\sigma} V_\\sigma c_{i\\sigma}c_{j\\bar\\sigma}$ where $j$ is summed over the nearest neighbors of $i$.\nAs is evident, the projected `fictive' fermions involve the projected\nbare fermion, $(1-n_{i\\bar\\sigma})a_{i\\sigma}$, which yields the $2x$ sum\nrule plus admixture with the doubly occupied sector\nmediated by the $t\/U$ corrections.  These $t\/U$ terms, which are\nentirely local and hence cannot be treated at the mean-field level, generate the $>2x$ or\nthe dynamical part of the spectral weight transfer.  This physics (which has been shown to play a significant role even at half-filling\\cite{trem}) is absent from projected models such as the standard implementation\\cite{lee,lee1,lee2} of the $t-J$ model\nin which double occupancy is prohibited. As we have pointed out in the introduction, the physics left out by projecting out double occupancy is important because it tells us immediately that $L\/n_h>1$; that is, new degrees of freedom must be present at low energy. Put another way, the operator in Eq. (\\ref{trans}) is not a free excitation and but rather describes a non-Fermi liquid ($L\/n_h>1$).   A process mediated by the\nlast term in Eq. (\\ref{trans}), depicted in Fig. (\\ref{fig1}), obtains only\nif a doubly occupied and empty site are neighbors. This underscores\nthe fact that in Mott systems, holes can be heavily dressed by the upper\nHubbard band. It is this dressing that generates dynamical\nspectral weight transfer.\n\nBefore we demonstrate how a single collective degree of freedom describes\nsuch dressing, we focus on the low-energy Hamiltonian in the bare\nelectron basis. The answer in the transformed basis\nis well-known\\cite{eskes} and involves\nthe spin-exchange term as well as the three-site hopping term. Our\ninterest\nis in what this model corresponds\nto in terms of the bare electron operators which do not preserve\ndouble occupancy. To accomplish this, we simply undo the\nsimilarity transformation after we have projected the transformed\ntheory onto the lowest energy sector. Hence, the quantity of interest\nis $H_{sc}=e^{-S}P_0e^SHe^{-S}P_0e^S$. Of course, without projection, the\nanswer in the original basis at each order of perturbation theory\nwould simply be the Hubbard\nmodel. However, the question at hand is what does the low-energy\ntheory look like in the original electron basis. Answering this\nquestion is independent of the high energy sectors in the transformed\nbasis because all such subspaces lie at least $U$ above the $m=0$\nsector. Hence, it is sufficient to focus on $P_0e^SHe^{-S}P_0$. To\nexpress $P_0e^SHe^{-S}P_0$ in the bare electron operators, we substitute\nEq. (\\ref{trans}) into the first of Eqs. (14) of Eskes, et\nal.\\cite{eskes} to obtain\n\\begin{widetext}\n\\begin{eqnarray} H_{sc}&=&e^{-S}P_0e^SHe^{-S}P_0e^S\\nonumber\\\\\n &=& -t\\sum_{\\langle i,j\\rangle}\\xi_{i\\sigma}^{\\dagger}\\xi_{j\\sigma}-\\frac{t^{2}}{U}\\sum_{i}b_{i}^{(\\xi)\\dagger}b_{i}^{(\\xi)}\n-\\frac{t^{2}}{U}\\sum_{\\langle i,j\\rangle,\\langle\ni,k\\rangle,\\sigma}\\left\\{\n\\xi_{k\\sigma}^{\\dagger}\\left[(1-n_{i\\bar{\\sigma}})\\eta_{j\\sigma}+\\xi_{j\\bar{\\sigma}}^{\\dagger}\\xi_{i\\bar{\\sigma}}\\eta_{i\\sigma}+\\xi_{i\\bar{\\sigma}}^{\\dagger}\\xi_{i\\sigma}\\eta_{j\\bar{\\sigma}}\\right]+h.c.\\right\\}\n\\label{eq:sc1}\n\\end{eqnarray}\n\\end{widetext}\nas the low-energy theory in terms of the original electron\noperators. Here, $\\xi_{i\\sigma}=a_{i\\sigma}(1-n_{i\\bar\\sigma})$ and\n$\\eta_{i\\sigma}=a_{i\\sigma}n_{i\\bar\\sigma}$. The first two terms correspond to the $t-J$ model in\nthe bare electron basis plus 3-site hopping. However, terms in the bare-electron basis\nwhich do not preserve the number of doubly\noccupied sites explicitly appear. As\nexpected, the\nmatrix elements which connect sectors which differ by a single doubly\noccupied site are reduced from the bare hopping $t$ to $t^2\/U$. All\nsuch terms arise from the fact that the transformed and bare electron\noperators differ at finite $U$. Hence, Eq. (\\ref{eq:sc1}) makes\ntransparent that the standard\nimplementation\\cite{lee,lee1,lee2} of the $t-J$ model in which the transformed and bare\nelectron operators are assumed equal is inconsistent because the terms\nwhich are dropped are precisely of the same order, namely $O(t^2\/U)$,\nas is the spin-exchange.\n\n\\section{New Approach}\n\nExplicitly integrating out\\cite{lowen1} the high-energy scale in the\nHubbard model uncloaks the collective degree of freedom that accounts\nfor the key difference between the bare and\ntransformed electrons and the ultimate origin of the breakdown of\nFermi liquid theory.\nThe central element of this theory is an elemental field, $D_i$, which we associated with the\ncreation of double occupation via a constraint. Our approach is in the spirit of Bohm and Pines\\cite{bohm} who also extended the Hilbert space with a constrained field to decipher the collective behaviour of the interacting electron gas. A Lagrange multiplier\n$\\varphi_i$, the charge $2e$ bosonic field, enters the action much the way the\nconstraint $\\sigma$ does in the non-linear $\\sigma$ model. The\ncorresponding Euclidean Lagrangian is\n\\begin{eqnarray}\\label{LE}\n{\\cal L}&&=\\int d^2\\theta\\left[\\bar{\\theta}\\theta\\sum_{i,\\sigma}(1- n_{i,-\\sigma}) a^\\dagger_{i,\\sigma}\\dot a_{i,\\sigma} +\\sum_i D_i^\\dagger\\dot D_i\\right.\\nonumber\\\\\n&&+U\\sum_j D^\\dagger_jD_j- t\\sum_{i,j,\\sigma}g_{ij}\\left[ C_{ij,\\sigma}a^\\dagger_{i,\\sigma}a_{j,\\sigma}\n+D_i^\\dagger a^\\dagger_{j,\\sigma}a_{i,\\sigma}D_j\\right.\\nonumber\\\\\n&&+\\left.\\left.(D_j^\\dagger \\theta a_{i,\\sigma}V_\\sigma a_{j,-\\sigma}+h.c.)\\right]+H_{\\rm con}\\right]\n\\end{eqnarray}\nwhere\n$C_{ij,\\sigma}\\equiv\\bar\\theta\\theta\\alpha_{ij,\\sigma}\\equiv\\bar\\theta\\theta(1-n_{i,-\\sigma})(1-n_{j,-\\sigma})$\nand $d^2\\theta$ represents a complex Grassman integration. The constraint Hamiltonian $H_{\\rm con}$ is taken to be\n\\begin{eqnarray}\\label{con}\nH_{\\rm con} = s\\bar{\\theta}\\sum_j\\varphi_j^\\dagger (D_j-\\theta a_{j,\\uparrow}a_{j,\\downarrow})+h.c.\n\\end{eqnarray}\nThe Grassman variable $\\theta$ is needed to\nfermionize double occupancy so that it can properly be associated with the\nhigh energy Fermi field, $D_i$. The constant $s$ has been inserted to\ncarry the units of energy. The precise value of $s$ will be determined\nby comparing the low-energy transformed electron with that in\nEq. (\\ref{trans}). This Lagrangian was constructed so that if we solve\nthe constraint, that is, integrate over $\\varphi$ and then $D_i$, we\nobtain exactly $\\int d^2\\theta \\bar\\theta\\theta L_{\\rm Hubb}=L_{\\rm Hubb}$, the\nLagrangian of the Hubbard model.\n\nThe advantage of this construction,\nhowever, is that we have been able to coarse-grain cleanly over the\nphysics on the scale $U$. That is, all the physics on the scale $U$ appears as the mass term of the new fermionic degree of freedom, $D_i$. It makes sense to integrate out $D_i$ as it\nis a massive field in the new theory. The low-energy theory to\n$O(t^2\/U)$,\n\\begin{eqnarray}\n\\label{HIR-simp}\nH_{\\rm eff}&=&-t\\sum_{i,j,\\sigma}g_{ij} \\alpha_{ij\\sigma}a^\\dagger_{i,\\sigma}a_{j,\\sigma}\\nonumber\\\\\n&&-\\frac{t^2}U \\sum_{j} b^\\dagger_{j} b_{j}-\\frac{s^2}U\\sum_{i}\\varphi_i^\\dagger \\varphi_i\\nonumber\\\\\n&&-s\\sum_j\\varphi_j^\\dagger a_{j,\\uparrow}a_{j,\\downarrow}-\\frac{ts}U \\sum_{i}\\varphi^\\dagger_i\nb_{i}+h.c.\\;\\;.\n\\end{eqnarray}\ncontains explicitly the charge $2e$ boson, $\\varphi_i$. Here $b^{(a)}_i=\\sum_{\\sigma j}V_{\\sigma}a_{i\\sigma}a_{j\\bar\\sigma}$ where $j$ is the nearest-neighbour of $i$. To fix the\nenergy scale $s$, we determine how the electron operator transforms in\nthe exact theory. As is standard, we add a source term to the\nstarting Lagrangian which generates the canonical electron operator\nwhen the constraint is solved. For hole-doping, the appropriate\ntransformation that yields the canonical electron operator in the UV\nis\n\\begin{eqnarray}\n{\\cal L}\\rightarrow {\\cal L}+\\sum_{i,\\sigma} J_{i,\\sigma}\\left[\\bar\\theta\\theta(1-n_{i,-\\sigma} ) a_{i,\\sigma}^\\dagger + V_\\sigma D_i^\\dagger \\theta a_{i,-\\sigma}\\right] +\nh.c.\\nonumber\n\\end{eqnarray}\nHowever, in the IR in which we only integrate over the heavy degree of\nfreedom, $D_i$, the electron creation operator becomes\n\\begin{eqnarray}\\label{cop}\na^\\dagger_{i,\\sigma}&\\rightarrow&(1-n_{i,-\\sigma})a_{i,\\sigma}^\\dagger\n+ V_\\sigma \\frac{t}{U} b_i a_{i,-\\sigma}\\nonumber\\\\\n&+& V_\\sigma \\frac{s}{U}\\varphi_i^\\dagger a_{i,-\\sigma}\n\\end{eqnarray}\nto linear order in $t\/U$. This equation bares close resemblance to the\ntransformed electron operator in Eq. (\\ref{trans}), as it should. In\nfact, the first two terms are identical. The last term in\nEq. (\\ref{trans}) is associated with double occupation. In\nEq. (\\ref{cop}), this role is played by $\\varphi_i$. Demanding that Eqs. (\\ref{trans}) and (\\ref{cop}) agree requires that $s= t$, thereby eliminating\nany ambiguity associated with the constraint\nfield. Consequently, the complicated interactions appearing in\nEq. (\\ref{eq:sc1}) as a result of the inequivalence between\n$c_{i\\sigma}$ and $a_{i\\sigma}$ are replaced by a single charge $2e$ bosonic field\n$\\varphi_i$ which generates dynamical spectral weight transfer across the\nMott gap. The interaction in Fig. (\\ref{fig1}), corresponding to the\nsecond-order process in the term $\\varphi_i^\\dagger b_i$, is the key physical process that enters the dynamics at low-energy. That the dynamical spectral weight transfer can be captured\nby a charge $2e$ bosonic degree of freedom is the key outcome of the\nexact integration of the high-energy scale. This bosonic field represents a\ncollective excitation of the upper and lower Hubbard bands. However, we should not immediately conclude that $\\varphi_i$ gives rise to a propagating charge $2e$ bosonic mode, as it does not have canonical kinetics; at the earliest, this could be generated at order $O(t^3\/U^2)$ in perturbation theory. Alternatively, we believe that $\\varphi$ appears as a bound degree of freedom. Since the\ndominant process mediated by $\\varphi_i$ requires a hole and a doubly\noccupied site to be neighbours (see Fig. (\\ref{fig1})) we identify $\\varphi_i^\\dagger a_{i\\bar\\sigma}$ as a new\ncharge $e$ excitation responsible for dynamical spectral weight\ntransfer. It is the appearance of this state at low-energy that\naccounts for the breakdown of Fermi liquid theory in a doped Mott\ninsulator. Physically, $\\varphi_i$ is the dressing of a hole by the\nhigh-energy scale. We have\npreviously shown\\cite{lowen1} that the formation of this bound state can produce the experimentally observed\nbifurcation\nof the electron dispersion below the chemical potential seen in\nPbBi2212\\cite{graf}, the mid-infrared band in the optical\nconductivity and the pseudogap\\cite{lowen2}.  Further, the breakup of\nthe bound state beyond a critical doping leads to $T-$ linear \nresistivity\\cite{lowen2}.\n\nThe essential problem of Mottness is that in a hole-doped Mott\ninsulator, empty sites can arise from doping or from hopping processes\nwhich mix the upper and lower Hubbard bands. Both contribute to $L$.\nHowever, the spectral weight on the empty sites resulting from mixing\nwith the high-energy scale is proportional to $t\/U$. Hence, such\nempty sites effectively represent holes with fractional charge $-e(t\/U)$ not \n$-e$ as is the the case with the holes resulting from doping. Consequently,\nthey make no contribution to $n_h$, thereby giving rise to $L\/n_h>1$\nfor a doped Mott insulator and a general breakdown of the standard\nFermi liquid theory of metals.  At half-filling, such fractionally charged\nsites still persist. Adding an electron to such a system at low\nenergies would require adding it coherently to a number of sites equal to $U\/t\\gg 1$. Such coherent addition of an electron at low energies has vanishing probability. The result is a gap for charge $e$ but not for\n$e(t\/U)$\nexcitations.  \nAt finite doping, holes in a Mott insulator are linear superpositions\nof both kinds of empty sites. As a result, holes in the\nhard-projected\\cite{lee,lee1,lee2} $t-J$ model, in which $L=2x$, are not equivalent to holes in the Hubbard model. Approximations which prohibit explicit\ndouble occupancy, such as the standard treatment\\cite{lee,lee1,lee2} of the $t-J$ model in\nwhich the operators are not transformed, miss completely\\cite{lh1,prelovsek,haule} the localizing\\cite{lh2,kotliar,choy} physics\nresulting from the orthogonality between charge $e$ excitations and\nthe sites with spectral weight $t\/U$. In the exact theory, the\nphysics associated with a finite length scale for double occupancy is\ncontained straightforwardly in a charge $2e$ bosonic field, instead of being buried\nin complicated interaction terms in\nEq. (\\ref{eq:sc1}).\n\nAs Polchinski\\cite{polchinski}  (as well as others\\cite{shankar}) have emphasized that from the point of\nview of the renormalization group,  $T-$linear resistivity in the\ncuprates makes a Fermi liquid description untenable.  We believe that\nour low energy theory containing the charge $2e$ bosonic field is in\nthis sense a suitable replacement for Fermi liquid theory as it can\nexplain\\cite{lowen2} $T-$linear resistivity.  \nWe have shown above that the bosonic field accounts for what would be\na consequence of complicated non-linear dependences on electron\noperators in projected models. What is clear from\nPolchinski's\\cite{polchinski} arguments is that projected models do\nnot give a good basis upon which to build a theory -- they mask the\nubiquitous physics of strong coupling, namely that new degrees of freedom emerge at low energy.\n\n\\acknowledgements This work was initiated at the Kavli Institute for\nTheoretical Physics and funded partially through PHY05-51164. We thank George Sawatzky for several conversations\nthat initiated this work, R. Bhatt, M. Hastings, R. Shankar, M. Sobol,\nA. Chernyshev, A. -M. Tremblay, and O. Tchernyshyov for helpful discussions and the NSF, Grant Nos. DMR-0605769.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAg$_3$Co(CN)$_6$ has attracted a lot of attention due to its colossal positive and negative thermal expansion~\\cite{GoodwinAgCoCN2008,Conterio2008}, and also because of its giant negative linear compressibility~\\cite{GoodwinAgCoCNnlc2008}. The negative thermal expansion (NTE) along the $c$-axis and the positive thermal expansion (PTE) along the $a$($b$) axes are an order of magnitude larger than that observed in many other crystalline solids. The material also shows negative linear compressibility (NLC), namely  along the $c$-axis, that is several times greater than the typical value found in crystals. As shown in Fig.~\\ref{fig:primitive}, the ambient-pressure phase of Ag$_3$Co(CN)$_6$ has a trigonal structure with space group $P\\bar{3}1m$. The structure consists of layers of Kagome sheets of Ag atoms in the $(001)$ crystal plane at height $z=1\/2$, with Co--CN--Ag--NC--Co chains along the $\\langle011\\rangle$ lattice directions linking  [Co(CN)$_6$]$^{3-}$ octahedra. These chains are hinged together in a way that gives the structure a high degree of flexibility; expansion in the trigonal $(001)$ plane is accompanied by a shrinkage in the orthogonal direction in a way that does not change the relevant bond lengths.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{PrimitiveKagome2.pdf}\n\\end{center}\n\\caption{\\label{fig:primitive} Ambient phase $P\\bar{3}1m$ of Ag$_3$Co(CN)$_6$: (a) unit cell with silver in red, cobalt in blue, carbon in black and nitrogen in white grey; (b) looking down the $\\left[0,\\,0,\\,1\\right]$ direction with Ag atoms (red) in a Kagome sheet connected to the octahedra [Co(CN)$_6$]$^{3-}$ anions (blue) above and below.}\n\\end{figure}\n\n\nPrevious \\emph{ab initio} density functional theory (DFT) calculations were unable to reproduce the correct ground-state structure and the high-pressure phase of the material~\\cite{Calleja2008,Hermet2013,Mittal2012}. Whilst these studies were able to reproduce the lengths of the Co--C, C--N and N--Ag bonds which characterise the structure, the predicted lattice parameters differ considerably from the experimental values. The key interatomic distance that changes as the structure flexes is the Ag\\ldots Ag distance, which is equal to half the value of the $a$ lattice parameter. The first of the DFT studies~\\cite{Calleja2008} showed that a \\textit{post hoc} correction for dispersive interactions between the Ag cations was sufficient to shift the equilibrium DFT structure into good agreement with the experimental crystal structure. The same study also showed that there is no significant covalent bonding between neighbour Ag atoms; it was this factor, combined with the fact that DFT calculations on the structural analogue in which hydrogen or deuterium atoms replace the Ag atoms are in excellent agreement with experiment, that suggested an important role for dispersive Ag$\\ldots$Ag interactions.\n\nOn this basis, it would be useful to see if a DFT calculation that explicitly includes a correction for the long-range dispersive forces will reproduce the ground state and the high-pressure phase of Ag$_3$Co(CN)$_6$ correctly. If so, it should then be possible to obtain reliable phonons via such calculation in order to better understand the exotic behaviour of this material.\n\nModern implementations of DFT now include a correction for the long-range dispersive interactions~\\cite{Grimme2004,Grimme2006,Dobson2006,Dion2004,Thonhauser2007,Tkatchenko2009,Roman-Perez2009}. One widely used method is called `DFT+D2'~\\cite{Grimme2006} where a dispersive interaction that is dampened at short range to avoid double counting of energy is added to the DFT energy from the generalised-gradient approximation (GGA) calculation. Semi-empirical parameters in such a dispersive interaction are provided in Ref.~\\onlinecite{Grimme2006} for most elements in the periodic table. The method has been successfully applied to various materials in which the dispersive interactions are important. One good example is the recent work on cesium halides by Zhang et al.~\\cite{Zhang2013}, where the DFT+D2 formalism gives both an improved agreement between the optimised and experimental crystal structures and a correct prediction of the ground-state phases.\n\nIn this work, we have carried out DFT+D2 calculations for Ag$_3$Co(CN)$_6$. This has confirmed that the inclusion of dispersive forces give the correct ground state structure, as anticipated in the first DFT study of this material~\\cite{Calleja2008}. It is also shown that the DFT+D2 model correctly gives the structure of the high-pressure phase; without the dispersive interaction DFT gives a structure without the interdigitation found experimentally~\\cite{GoodwinAgCoCNnlc2008}. On the basis of these successes it is now reasonable to investigate the lattice dynamics of Ag$_3$Co(CN)$_6$, from which we have been able to study a number of physical and thermodynamic properties. These form the focus of this paper.\n\n\n\\section{Methods}\n\n\\subsection{DFT calculations}\n\nThe DFT calculations were performed using the CASTEP code~\\cite{Segafll2002}. For comparison, we used both local-density approximation (LDA) and GGA of Perdew-Burke-Ernzerhof (PBE)~\\cite{Perdew1996} for the exchange-correlation functional. Optimized norm conserving pseudopotentials generated using the RRKJ method~\\cite{Rapper1990} as implemented in the OPIUM package and with parameters from the Rappe and Bennett library~\\cite{link} were used in various calculations. A plane-wave basis set was used with the cut-off energy of 1800~eV. Sampling of the Brillouin zone was performed on a $6\\times6\\times6$ Monkhorst-Pack (MP)~\\cite{Monkhorst1976} grid.\n\nThe geometries of all structures were optimised using the BFGS method to achieve a convergence of less than $10^{-6}$ eV per atom change in energy per cycle and a force residual of $5\\times10^{-4}$~eV\/{\\AA}. At different pressures, tolerance for accepting convergence of the maximum stress component during unit cell optimization is $5\\times10^{-3}$ GPa.\n\n\n\n\\begin{table*}[t]\n\\caption{\\label{tab:groundstate} Calculated ground-state structures (from GGA+D, GGA and LDA), including the unit-cell edges ($a=b$ and $c$), fractional coordinates of C and N, and the nearest-neighbouring ion distances. $V$ is the volume of one formula unit (note that there is one formulate unit per unit cell). The Ag--Ag distance is equal to $a\/2$. $\\Delta_\\mathrm{GGA+D}$, $\\Delta_\\mathrm{GGA}$ and $\\Delta_\\mathrm{LDA}$ represent the deviations of the different calculations compared to experiment at a temperature of 10~K from reference \\onlinecite{Conterio2008}.}\n\\begin{tabular}{@{\\extracolsep{8pt}}c|ccc|c|ccc}\n\\hline  & LDA & GGA & GGA+D & Experiment & $\\Delta_\\mathrm{GGA+D}$  & $\\Delta_\\mathrm{GGA}$& $\\Delta_\\mathrm{LDA}$ \\\\\n\\hline\n$a\\left(=b\\right)$ (\\AA) & $6.118$ & $7.629$ & $6.664$ & $6.754$ & $-1.3$\\% & $+13$\\% & $-9$\\% \\\\\n$c$ (\\AA)                & $7.626$ & $6.621$ & $7.416$ & $7.381$ & $+0.5$\\% & $-10$\\% & $+3$\\%\\\\\n$V$ (\\AA$^3$)           & $247.2$ & $333.7$ & $285.2$ & $291.6$  & $-2$\\% & $+14$\\% & $-15$\\% \\\\\nC$_x$                      & $0.238$ & $0.202$ & $0.225$ & $0.220$ & $+0.003$ & $-0.020$  & $+0.016$ \\\\\nC$_z$                      & $0.154$ & $0.171$ & $0.158$ & $0.153$ & $+0.002$ & $+0.015$  & $-0.002$ \\\\\nN$_x$                      & $0.364$ & $0.321$ & $0.347$ & $0.342$ & $+0.008$ & $-0.018$  & $+0.025$ \\\\\nN$_z$                      & $0.269$ & $0.282$ & $0.270$ & $0.266$ & $+0.006$ & $+0.018$  & $+0.005$ \\\\\nC--N (\\AA)               & $1.170$ & $1.164$ & $1.164$ & $1.170$ & $-0.5$\\% & $-0.5$\\%  & $0$\\% \\\\\nAg--N (\\AA)              & $1.948$ & $1.988$ & $1.983$ & $2.034$ & $-2.5$\\% & $-2.3$\\%  & $-4$\\%\\\\\nCo--C (\\AA)              & $1.868$ & $1.914$ & $1.906$ & $1.865$ & $+2.2$\\% & $+2.6$\\%  & $+0.2$\\%\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{DFT+D2 calculations}\n\nThe dispersive contribution was directly added to the DFT GGA energy using a semi-empirical form introduced by Grimme~\\cite{Grimme2006},\n\\begin{eqnarray}\\label{g06}\nE_\\mathrm{disp}  =  - s_6 \\sum\\limits_{i = 1}^{N - 1} {\\sum\\limits_{j = i + 1}^N {\\frac{{C_6^{ij} }}{{R_{ij}^6 }}} } f_\\mathrm{damp} \\left( {R_{ij} } \\right)\n\\end{eqnarray}\n\n\\noindent where $N$ the number of atoms in the system. $C_6^{ij}$ is the dispersion coefficient of atomic pair $\\left(i,j\\right)$ that can be computed from the dispersion coefficient of the individual atoms as\n\\begin{eqnarray}\\label{dispcoefficient}\nC_6^{ij}  = \\sqrt {C_6^i C_6^j }\n\\end{eqnarray}\n\n\\noindent where $R_{ij}$ is the distance between the two atoms, and $R_r$ is the sum of the atomic van der Waals radii of the pair. The dampening factor $f_\\mathrm{damp}$ is defined as\n\\begin{eqnarray}\\label{dampenfactor}\nf_\\mathrm{damp} \\left( {R_{ij} } \\right) = \\frac{1}{{1 + \\exp \\left[ { - d\\left( {R_{ij} \/R_r  - 1} \\right)} \\right]}}\n\\end{eqnarray}\n\n\\noindent with $d=20$. $s_6$ is a scaling factor dependent on the functional used in the calculation; for PBE, $s_6=0.75$. This method has been implemented in CASTEP for geometry optimisation. In what follows we will refer to this method as `GGA+D'; calculations without the dispersion correction will simply be labelled as `LDA' or `GGA' as appropriate.\n\n\n\n\\subsection{Lattice dynamics with DFPT+D2}\\label{section:DFTPD2methods}\n\n\nDensity functional perturbation theory (DFPT)~\\cite{Baroni2001,Refson2006} was used to calculate phonons on a $5\\times5\\times5$ grid of wave vectors, and frequencies for phonons of other wave vectors were then obtained using interpolation~\\cite{Baroni2001}. Phonon density of states (DoS) were calculated using a $25\\times25\\times25$ MP grid~\\cite{Monkhorst1976} corresponding to a total of 1470 independent wave vectors.\n\nAt the present time CASTEP can only support a DFT+D2 calculation for phonons using the supercell method of finite displacement~\\cite{Karki1997}, which turns out to be too expensive to be feasible for Ag$_3$Co(CN)$_6$. Therefore, we first carried out a regular DFPT phonon calculation using CASTEP to get the corresponding dynamical matrices of different wave vectors. We then used the dispersive interaction of Eq.~\\eqref{g06} implemented in the lattice simulation program GULP~\\cite{Gale1997} to calculate its contribution to the dynamical matrices separately, all based on the same optimised structure from GGA+D. The dynamical matrices from the two codes are added together using a combination of Python scripts and the use of MATLAB, and the combined dynamical matrix was diagonalised to give the  phonon frequencies with effects of the dispersive interaction included. For future convenience, we call this the `DFPT+D' method.\n\nTo check the accuracy of our scripts for the DFPT+D method, we performed a benchmark phonon calculation for NaI, chosen because it has a large refractive index (the largest among alkali halides~\\cite{Li1976}) and hence likely to have a significant dispersive energy term. This material has a simple structure with only 2 atoms in the primitive cell, so that it was feasible to carry out a DFT+D2 phonon calculation using the supercell method in CASTEP (here called the `supercell+D' method). By comparing the calculated phonon frequencies from DFPT+D and supercell+D, we found the two agree with each other extremely well, with a mean relative discrepancy less than $2\\%$ (see phonon dispersion curves in the Supplemental Material~\\cite{supplemental}).\n\nWith the calculated phonon frequencies, the linear Gr\\\"{u}neisen parameter $\\gamma_{ab}$ is calculated by varying the $a$ and $b$ dimensions of the unit cell by $0.005\\%$ with fixed $c$ dimension,\n\\begin{eqnarray}\\label{gammaa}\n\\gamma_{ab}=\\left(-\\partial \\ln\\omega\/\\partial \\ln a\\right)_c\n\\end{eqnarray}\nand the linear Gr\\\"{u}neisen parameter $\\gamma_{c}$ is calculated by varying the $c$ dimension of the unit cell by $0.005\\%$ with fixed $a$ and $b$ dimensions,\n\\begin{eqnarray}\\label{gammac}\n\\gamma_{c}=\\left(-\\partial \\ln\\omega\/\\partial \\ln c\\right)_{ab}.\n\\end{eqnarray}\nWe will show later how these two quantities determine the coefficients of linear thermal expansion $\\alpha_a = \\partial \\ln a\/\\partial T$ and $\\alpha_c =\\partial \\ln c\/\\partial T$.\n\n\\section{Ground-state properties of Ag$_3$Co(CN)$_6$}\\label{groundstate}\n\n\\subsection{Crystal structure}\n\nThe details ground-state structures of Ag$_3$Co(CN)$_6$ optimised using GGA, with and without the dispersive interaction, and using LDA are reported in Table~\\ref{tab:groundstate}, where they are compared to the experimental values~\\cite{Conterio2008}. It is clear that, without the dispersive interaction, the calculated ground-state structure is wrong. Inclusion of the dispersive interaction results in the correct structure with small deviations from experiment.\n\nIt is worth remarking on the role the Ag$\\ldots$Ag dispersive interaction has on the structure. The dispersive interaction is a weak attractive interaction, which opposing the repulsive Coulomb interaction, Thus the effect of the dispersive interaction is to reduce the overall Ag$\\ldots$Ag interaction. On this basis, addition of the dispersive interaction to the GGA model enables the structure to relax with a shorter Ag$\\ldots$Ag distance and hence a smaller value of the $a$ lattice parameter, as see in the results in Table \\ref{tab:groundstate}. On the other hand, the well-known tendency of LDA to overbind already results in a shorter Ag$\\ldots$Ag distance.\n\nWe can quantify this point. The DFT calculations give an approximate value for the charge of the Ag cation of $+0.65|e|$\\cite{Segall1996}, where $e$ is the electronic charge. Calculation of the Ag$\\ldots$Ag forces due to the Coulomb and dispersive interactions (taking $f_\\mathrm{damp}=1$ in Eq.~\\ref{g06}) over the range of distances 3.3--3.5~\\AA\\ shows that the dispersive interaction reduces the net force between neighbouring Ag ions by nearly a factor of 2.\n\n\\subsection{Elasticity}\n\nThe GGA+D computed elastic compliances are given in Table~\\ref{tab:compliance}. The linear compressibilities along the $a$($b$) and $c$ crystal axes were calculated using the elastic compliances as\n\\begin{eqnarray}\\label{linearcompressibility}\n\\beta _{ab}  = -\\partial \\ln a\/\\partial p  =s_{11}  + s_{12}  + s_{13}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{linearcompressibility2}\n\\beta _c  =  -\\partial \\ln c\/\\partial p = 2s_{13}  + s_{33},\n\\end{eqnarray}\nrespectively. The volume compressibility was calculated as the sum\n\\begin{eqnarray}\\label{linearcompressibility3}\n\\beta = -\\partial \\ln V\/\\partial p =2\\beta_{ab}  + \\beta_c\n\\end{eqnarray}\nThe linear elastic moduli $B_{ab}$ along $a$ and $b$ axes as well as $B_c$ along $c$ axis are the inverse of the $\\beta_{ab}$ and $\\beta_c$, respectively. Their relations with the elastic constants are given in the supplemental material~\\cite{supplemental}.\n\nAs shown in Table~\\ref{tab:compliance}, the GGA+D calculated $s_{33}$ and $s_{13}$ have almost the same magnitude but with opposite sign, showing that the $c$ dimension would response equivalently to a stress acting on the $a$ or $b$ dimension and a tension directly acting on the $c$ dimension. This shows the  effectiveness of the hinging mechanism in the material. In comparison, the small value of $s_{12}$ shows that the change in dimension $a$ (or $b$) is barely correlated to the change in $b$ (or $a$) dimension.\n\nNegative values of $\\beta_c$ and $B_c$ correspond to the NLC of the material, namely the material will \\textit{elongate} in the $c$ dimension under hydrostatic compression. The bulk modulus and its first derivative were calculated as $B=15.8(8)$~GPa and $B^\\prime=-4.9(8)$, respectively~\\cite{supplemental}. Using the 3rd-order Birch-Murnaghan (BM) equation of state (EoS)~\\cite{Birch1947} to fit to the calculated isotherm data from 0 to 0.6~GPa also results in a negative value of $B^\\prime$ of $-3(2)$. These results predict that the material will have pressure-induced softening~\\cite{Fangzeolite2013,Fangexp2013,Fangexpression2014} at low pressures.\n\n\n\\begin{table}[t]\n\\setlength{\\tabcolsep}{4pt}\n\\caption{\\label{tab:compliance} Calculated compliances at different pressures for the ambient phase of Ag$_3$Co(CN)$_6$ obtained from calculating the change of energy corresponding to a set of given strains $\\varepsilon _{ij}$ generated according to the trigonal symmetry. Results were obtained using GGA+D, and the LDA results are from Ref.~\\onlinecite{Hermet2013}.The compliances of a trigonal phase have the symmetry~\\cite{Nye1985} $s_{ij}=s_{ji}, s_{22}=s_{11}, s_{55}=s_{44}, s_{23}=s_{13}, s_{24}=-s_{14}, s_{66}=2(s_{11}-s_{12})$. The corresponding elastic constants and elastic moduli are given in the Supplemental Material~\\cite{supplemental}. The linear compressibility $\\beta_{ab}$ and $\\beta_c$ as well as the volume compressibility $\\beta$ are calculated from the compliances using Eqs~\\eqref{linearcompressibility} to~\\eqref{linearcompressibility3}.}\n\\centering\n\\begin{tabular}{ccccc}\n\\hline Compliance (TPa$^{-1}$) & 0.0 GPa & 0.04 GPa & 0.1 GPa & LDA \\\\\n\\hline\n$s_{11}$ & 61(3)  & 62(3)  & 64(4)  & 85 \\\\\n$s_{33}$ & 22(1)  & 21.4(9)   & 23(2)  & 16\\\\\n$s_{44}$ & 38.5(9)   & 37.7(7)   & 44(3)  & 73 \\\\\n$s_{12}$ & 2(1)   & 1(1)   & 3(2)   & $-22$ \\\\\n$s_{13}$ & $-21(1)$   & $-21(1)$   & $-23(2)$   & $-17$ \\\\\n$s_{14}$ & $15(1)$ & $15(1)$ & $17(2)$ & $-41$ \\\\\n$\\beta_{c}$    & $-21(2)$ & $-21(2)$ & $-23(4)$ & $-19$ \\\\\n$\\beta_{ab}$   & $42(4)$  & 42(4)    & 44(5)    & 45    \\\\\n$\\beta$        & $63(6)$  & 63(6)   & 65(8)    & 72    \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nThe calculated bulk modulus at 0 K, as the inverse of $\\beta$ in Eq.~\\ref{linearcompressibility3}, is 15.8(8)~GPa which is significantly larger than the experimental value of $B=6.5(3)$~GPa at 300~K~\\cite{GoodwinAgCoCNnlc2008}. This apparent overestimation of the calculation may actually be due to a considerable softening of the material on heating, as will be discussed later in Section~\\ref{softening}. The same idea can be used to explain the apparent large underestimation of the compressibilities: the calculated values $\\beta_{ab}=42(4)$~TPa$^{-1}$ and $\\beta_c=-21(2)$~TPa$^{-1}$ are much lower than the experimental values of $\\beta_{ab}=115(8)$~TPa$^{-1}$ and $\\beta_c=-79(9)$~TPa$^{-1}$ at 300~K~\\cite{GoodwinAgCoCNnlc2008}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{phaseIInews.pdf}\n\\end{center}\n\\caption{\\label{fig:phaseII} Structures of the high-pressure phase of Ag$_3$Co(CN)$_6$ (space group $C2\/m$) optimised using (a) GGA+D , and (b) either GGA or LDA without a correction for the dispersion energy. The experimentally observed interdigitated structure, characterised by the indented Ag atoms, can be seen only when dispersion corrections are used.}\n\\end{figure}\n\n\n\\section{High-pressure phase of Ag$_3$Co(CN)$_6$}\n\n\\subsection{Crystal structure of the high-pressure phase}\nAg$_3$Co(CN)$_6$ undergoes a structural phase transition at $0.19$ GPa to a monoclinic phase~\\cite{GoodwinAgCoCNnlc2008} and denoted as Phase-II. The phase transition involves displacements of Ag atoms in alternative rows, which cause the high-pressure phase to possess an interdigitated structure as seen by viewing down the $\\left[0,\\,0,\\,1\\right]$ direction. This is indicated in Fig.~\\ref{fig:phaseII}(a) by the indented Ag atoms.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{phasetransition2.pdf}\n\\end{center}\n\\caption{\\label{fig:phasetransition2} The calculated enthalpy of the high-pressure phase relative to that of the ambient phase using GGA+D. An overestimated phase-transition pressure of $2.5$~GPa is predicted.}\n\\end{figure}\n\nOur calculations show that neither LDA or GGA without the dispersive interaction can give the correct optimised high-pressure phase with the interdigitated structure~\\cite{GoodwinAgCoCNnlc2008}, as shown by Fig.~\\ref{fig:phaseII}(b). It is only by including the dispersive interaction in the GGA+D calculation that the interdigitated structure of the high-pressure phase can be reproduced, as shown in Fig.~\\ref{fig:phaseII}(a).\n\nFig.~\\ref{fig:phasetransition2} shows the difference in enthalpy between the two phases as calculated using the GGA+D method. The predicted phase-transition pressure of about 2.5 GPa overestimates the experimental value of 0.19 GPa~\\cite{GoodwinAgCoCNnlc2008}. Although this appears to be a large discrepancy, it is magnified by the fact that the experimental transition pressure is so low. Phase transition pressures are hard to calculate; we attribute the discrepancy to an accumulation of small errors associated with a number of approximations in the DFT method and the dispersion correction. The calculated relative change of the cell volume at the phase transition is $11\\%$, smaller than the experimental value of $16\\%$~\\cite{GoodwinAgCoCNnlc2008}. Table~\\ref{tab:highpphase} compares the optimised structure with the $C2\/m$ space group in GGA+D with the experiment values at 0.23 GPa.\n\n\n\\begin{table}[t]\n\\caption{\\label{tab:highpphase} Comparison of optimised (GGA+D2) and experimental \\cite{GoodwinAgCoCNnlc2008} crystal structures of the high-pressure phase (space group $C2\/m$) at a pressure of 0.23 GPa. $\\Delta_\\mathrm{GGA+D}$ represents the differences between the two. $V$ is the volume of one formula unit (note that there are 2 formula units in the unit cell). The fractional coordinates of Ag1 are $(1\/2, 0, 1\/2)$. }\n\\begin{tabular}{@{\\extracolsep{10pt}}c|ccc}\n\\hline  & GGA+D & Experiment & $\\Delta_\\mathrm{GGA+D}$ \\\\\n\\hline\n$a$ (\\AA) & $6.485$  & 6.693  & $-3.1$\\%\\\\\n$b$ (\\AA) & $11.144$ & 11.539 & $-3.4$\\% \\\\\n$c$ (\\AA) & $6.658$  & 6.566  & $+1.4$\\%\\\\\n$\\beta$ ($^\\circ$)     & $101.84$ & $101.48$ & $+0.36$ \\\\\n$V$ (\\AA$^3$)& 235.6    & 248.5   & $+5.2$\\% \\\\\nC1$_x$       & 0.790& 0.825  & $-0.035$  \\\\\nC1$_z$       & 0.163& 0.182  & $-0.019$   \\\\\nN1$_x$       & 0.664& 0.715  & $-0.051$   \\\\\nN1$_z$       & 0.264& 0.302  & $-0.038$   \\\\\nC2$_x$       & 0.145& 0.163  & $-0.019$   \\\\\nC2$_y$       & 0.123& 0.119  & $+0.004$   \\\\\nC2$_z$       & 0.177& 0.157  & $+0.0209$   \\\\\nN2$_x$       & 0.241& 0.258  & $-0.017$   \\\\\nN2$_y$       & 0.197& 0.185  & $+0.012$   \\\\\nN2$_z$       & 0.280& 0.259  & $+0.021$   \\\\\nAg2$_y$      & 0.243& 0.240  & $+0.002$   \\\\\nC1--N1 (\\AA) & 1.161 & 1.183 & $-1.8$\\%\\\\\nC2--N2 (\\AA) & 1.170 & 1.126 & $+3.9$\\%\\\\\nAg1--N1 (\\AA)&2.069 & 2.123 & $-2.5$\\%\\\\\nAg2--N2 (\\AA)&2.097 & 2.199 & $-4.6$\\%\\\\\nCo--C1 (\\AA) &1.907 & 1.830 & $+4.2$\\%\\\\\nCo--C2 (\\AA) &1.922 & 1.924 & $-0.1$\\%\\\\\nAg--Ag(1) (\\AA)&2.868&2.996 & $-4.3$\\%\\\\\nAg--Ag(2) (\\AA)&5.407&5.548 & $-2.5$\\%\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nOriginally, it was found~\\cite{GoodwinAgCoCNnlc2008} that the high-pressure phase of the material has a space group of $C2\/m$. However, recently, it was proposed~\\cite{Hermet2013} that the high-pressure phase should have the lower symmetry of space group $Cm$, because a structure with this symmetry can be obtained as a subgroup of the space group of the ambient-pressure phase, $P\\bar{3}1m$, whereas a structure with space group $C2\/m$ cannot. Our calculations indicate that the optimised structures starting from both space groups $C2\/m$ and $Cm$ have exactly the same enthalpy up to a pressure of 7~GPa (the highest we examined), with relaxed structures that differ only by a small origin offset. We conclude that the structure of the high-pressure phase has the originally-proposed $C2\/m$ structure.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{celldistance2.pdf}\n\\end{center}\n\\caption{\\label{fig:phaseIIparameters} Upper panel: various lattice parameters of the monoclinic phase of Ag$_3$Co(CN)$_6$ at different pressures using GGA+D. The calculated (solid lines) and the experimental (symbol) values of each parameter are in the same colour. Lower panel: the calculated nearest Ag$\\ldots$Ag distance in two phases. In the high-pressure phase, the GGA+D result (symbol line) shows the correct trend of two types of Ag$\\ldots$Ag distances changing with compression, while the GGA\/LDA (dashed line) result does not.}\n\\end{figure}\n\n\\subsection{Elasticity}\n\nA fit of the 3rd-order BM EoS to the calculated isotherm of the high-pressure phase yields $B=17(6)$~GPa and $B^\\prime=17(7)$; experimental values are $B=11.8(7)$~GPa and $B^\\prime=13(1)$, respectively~\\cite{GoodwinAgCoCNnlc2008}. Thus, unlike the ambient-pressure phase, which has pressure-induced softening at low pressures, the high-pressure phase of the material quickly becomes harder under compression.\n\nThe calculated change of lattice parameters of the high-pressure monoclinic phase-II are presented in Fig.~\\ref{fig:phaseIIparameters}, and compared to the experimental values. The agreement between the two are good with the largest relative deviation below $10\\%$. By fitting to a 3rd-order polynomial of pressure $\\left(p-p_\\mathrm{c}\\right)$ with the phase-transition pressure $p_\\mathrm{c}=2.5$~GPa, the linear compressibilities of $a_\\mathrm{II}$, $b_\\mathrm{II}$ and $c_\\mathrm{II}$ were obtained at different pressures. Their averaged values over $2.5$--$8.0$~GPa are $19(1)$, $6.9(4)$ and $-4.1(3)$~TPa$^{-1}$, respectively. These values are in good agreement with experimental values~\\cite{GoodwinAgCoCNnlc2008} of $15.9(9)$, $9.6(5)$ and $-5.3(3)$~TPa$^{-1}$.\n\nAs pointed out in Ref.~\\onlinecite{GoodwinAgCoCNnlc2008}, the relatively small compressibility along $b_\\mathrm{II}$ is due to the interdigitation in the high-pressure phase. Upon compression, the structure becomes more indented (Fig.~\\ref{fig:phaseII}(b)), resulting in the Ag$\\ldots$Ag(1) distance between the indented Ag atom and its nearest neighbour increases with pressure, while the Ag$\\ldots$Ag(2) distance between the two indented Ag atoms at the opposite sites decreases. This behaviour of the Ag$\\ldots$Ag distances under pressure is seen in the GGA+D calculated results shown in the lower panel of Fig.~\\ref{fig:phaseIIparameters}.\n\n\\begin{table}[t]\n\\setlength{\\tabcolsep}{3pt}\n\\caption{\\label{tab:raman} The calculated Raman and infrared spectrums (in THz) of Ag$_3$Co(CN)$_6$ using DFPT+D compared to the experimental values at 80 K (Raman)~\\cite{Rao2011} and 295 K (Infrared)~\\cite{Hermet2013}. $\\Delta_\\mathrm{DFPT+D}$ is the deviation of the DFPT+D calculated frequencies compared to the experiment (in THz). The first derivative of the frequency with respect to pressure is in unit of THz\/GPa.}\n\\centering\n\\begin{tabular}{ccccc}\n\\hline Raman & $\\omega_\\mathrm{DFPT+D}$ & $\\Delta_\\mathrm{DFPT+D}$ & $\\left(\\partial \\omega \/\\partial p\\right)_\\mathrm{Exp.}$~\\cite{Rao2011} & $\\left(\\partial \\omega \/\\partial p\\right)_\\mathrm{DFPT+D}$ \\\\\n\\hline\n$2.6$  & 2.9  & 0.3 & 0.3   & 0.4  \\\\\n$4.2$  & 4.3  & 0.1 & 0.6   & 0.4  \\\\\n$4.9$  & 5.0  & 0.1 & 0.3   & 0.6  \\\\\n$9.7$  & 9.8  & 0.1 & $-0.3\\footnotemark[1]$&$-0.04$ \\\\\n$14.2$ & 13.8 & $-0.4$ & 0.1  &$-0.006$ \\\\\n$14.2$ & 13.9 & $-0.3$ & 0.1  &$-0.04$ \\\\\n$15.6$ & 16.1 & 0.5    & 0.7   & 0.06  \\\\\n$15.6$ & 16.1 & 0.5    & 0.7   & 0.2  \\\\\n$65.5$ & 65.2 & $-0.3$ & 0.2  & 0.1 \\\\\n$66.1$ & 66.0 & $-0.1$ & 0.3   & 0.1 \\\\\n\\hline Infrared & $\\omega_\\mathrm{DFPT+D}$ & $\\Delta_\\mathrm{DFPT+D}$ & $\\left(\\partial \\omega \/\\partial p\\right)_\\mathrm{Exp.}$~\\cite{Hermet2013} & $\\left(\\partial \\omega \/\\partial p\\right)_\\mathrm{DFPT+D}$  \\\\\n\\hline\n1.2  & 1.4   & 0.2 & --     & 0.2       \\\\\n1.4  & 1.5   & 0.1  & --     & $0.1$   \\\\\n1.6  & 2.2   & 0.6 & --     & $-0.3$   \\\\\n4.0  & 4.2   & 0.2  & $-0.2$ & $0.01$   \\\\\n5.3  & 5.6   & 0.3  & --     & $-0.4$   \\\\\n5.5  & 5.7   & 0.2  & --     & $-0.2$  \\\\\n8.0  & 8.6   & 0.6  & --     & 0.3    \\\\\n8.0  & 8.8   & 0.8 & --     & 0.2       \\\\\n13.0 & 12.8  & $-0.2$  & --     & $-0.03$   \\\\\n14.5 & 14.8  & 0.3  & $-0.02$ & $0.01$  \\\\\n14.8 & 14.9  & 0.1  & 0.03    & 0.01    \\\\\n17.6 & 17.8  & 0.2  & --     & 0.3     \\\\\n--   & 18.0  & --   & --     & 0.2     \\\\\n--   & 65.1  & --   & --     & $0.1$ \\\\\n--   & 65.2  & --   & --     & 0.1    \\\\\n\\hline\n\\end{tabular}\n\\footnotetext[1]{From non-hydrostatic experiment~\\cite{Rao2011}.}\n\\end{table}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.97\\textwidth]{phonons2.pdf}\n\\end{center}\n\\caption{\\label{fig:phonon} (a) DFPT+D calculated phonon dispersion curves along the high-symmetry directions in the Brillouin zone. (b) and (c) are dispersion curves coloured according to the values of linear Gr\\\"{u}neisen parameters along the $a$($b$) axes ($\\gamma_{ab}$) and $c$ axis ($\\gamma_c$), respectively, with values $\\leq-20$ in red gradually passing to values $\\geq+20$ in blue.}\n\\end{figure*}\n\n\n\\section{Lattice dynamics calculations}\n\nThe phonon calculations were performed using the DFPT+D method as discussed in Section~\\ref{section:DFTPD2methods}. Table~\\ref{tab:raman} shows that the calculated Raman and infrared spectra are in good agreement with the experiment~\\cite{Rao2011,Hermet2013}. The phonon dispersion curves along the high-symmetry directions in the Brillouin zone for frequencies up to 18~THz are presented in Fig.~\\ref{fig:phonon}(a).\n\nWe have studied the eigenvectors of different vibrational modes as shown by the animations in the Supplemental Materials~\\cite{supplemental}. We found that the infrared-active modes at $1.4$--$1.5$~THz showing negative linear Gr\\\"{u}neisen parameters $\\gamma_{ab}$ and positive linear Gr\\\"{u}neisen parameters $\\gamma_c$ correspond to the rotation of Ag-triangle pairs against each other in the Kagome sheet about their shared apex. The Raman-active mode at $2.9$~THz, having positive $\\gamma_{ab}$ and negative $\\gamma_c$, corresponds to the rotations of  CoC$_6$ octahedra that pulls the connected layers of Ag atoms along the $c$ axis closer together. The Raman-active modes at $4.3$ and $5.0$~THz correspond to similar type of vibrations but with CoC$_6$ octahedra deforming, and these also show positive $\\gamma_{ab}$ and negative $\\gamma_c$.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{Amode1s.pdf}\n\\end{center}\n\\caption{\\label{fig:Amode1} Vibration corresponds to the first mode at point A $\\left(0,\\,0,\\,1\/2\\right)$ from its eigenvector looking down the $\\left[1,\\,0,\\,0\\right]$ direction. Each Ag atom (red) is connected to two [Co(CN)$_6$] octahedra (blue) in the upper and lower layers via the Co--CN--Ag--NC--Co linkages. Arrows show the transverse motion of the nearly-rigid bridging group CN--Ag--NC resulted from the concerted rotation of the octahedra. The dashed square shows the unit cell.}\n\\end{figure}\n\n\nThe dispersion curves are also shown in Fig.~\\ref{fig:phonon}(b) and (c) with colours that reflect the calculated values of $\\gamma_{ab}$ and $\\gamma_c$ as given by Eqs~\\ref{gammaa} and~\\ref{gammac}, respectively. One can see that it is almost the same set of low-frequency modes that contribute to the PTE along the $a$($b$) axes and NTE along the $c$ axis, i.e. their values of $\\gamma_{ab}$ and $\\gamma_c$ show similar magnitudes but are opposite in sign. This is directly related to the hinging structure in the material where any level of expansion in the $a$($b$) axes would transfer into a similar level of contraction in the $c$ axis via the Co--CN--Ag--NC--Co linkage. Modes around the wave vector A $\\left(0,\\,0,\\,1\/2\\right)$ and around the middle point along the H$\\left(-1\/3,\\,2\/3,\\,1\/2\\right)$$\\rightarrow$K$\\left(-1\/3,\\,2\/3,\\,0\\right)$ direction have the lowest frequencies ($< 1.0$ THz) and hence have the most extreme values of Gr\\\"{u}neisen parameters, The first two degenerate modes at A correspond to concerted rotations of rigid Co(CN)$_6$ octahedra together with the nearly-rigid CN--Ag--NC linkages moving sideways~\\cite{supplemental}, as shown by its eigenvector in Fig.~\\ref{fig:Amode1}. The first mode at the middle point $\\left(-1\/3,\\,2\/3,\\,1\/4\\right)$ along the H$\\rightarrow$K direction corresponds to the Ag atoms vibrating along the $c$ axis, producing a transverse wave passing through each Kagome sheet~\\cite{supplemental}.\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.97\\textwidth]{dosgp.pdf}\n\\end{center}\n\\caption{\\label{fig:dosgp} Calculated DoS of modes with frequencies $\\leq 9$ THz. At pressures (a) 0.0 GPa, (b) 0.04 GPa and (c) 0.1 GPa, the DoS in the upper panel is coloured according to the averaged value of $\\gamma_{ab}$ and the DoS in the lower panel is coloured according to the averaged value of $\\gamma_{c}$ around each energy. Values $\\leq -10$ are in red and $\\geq 10$ are in blue. $\\gamma_{ab}$ of the low-frequency modes, especially the modes below 1.0 THz, decrease largely upon compression and even change their signs at 0.1 GPa as indicated by the change of colour from blue to red. (d) Coloured DoS according to the average value of relative frequency change with pressure (in GPa$^{-1}$) around each energy bin. The upper panel shows the frequency change from 0.0 to 0.04 GPa and the lower panel shows that from 0.04 to 0.1 GPa. Stiffened phonons ($\\partial \\ln \\omega\/\\partial p \\geq +0.1$) are in blue and softened phonons ($\\partial \\ln \\omega\/\\partial p \\leq -0.1$) are in red.}\n\\end{figure*}\n\nThe picture shown in Fig.~\\ref{fig:phonon} is reflected in plots of the vibrational densities of states (DoS), which are shown in Fig.~\\ref{fig:dosgp}. These were calculated from the full set of DFPT+D vibrations computed on a $25\\times25\\times25$ grid (corresponding to a total of 1470 wave vectors in the Brillouin zone). Plots of the DoS are plotted for three pressures and coloured according to the averaged value of $\\gamma_{ab}$ and $\\gamma_c$ of the modes around each energy. The plots for vibrations at ambient pressure (Fig.~\\ref{fig:dosgp}(a)) show that the same low-frequency modes contribute positively to $\\gamma_{ab}$ and negatively to $\\gamma_c$. This situation changes under pressure, as we will now discuss.\n\n\n\\section{Effect of compression on thermal expansion}\n\n\\subsection{Increase of linear thermal expansion on compression}\n\nFrom the calculated Gr\\\"{u}neisen parameters and the compliances given in Table~\\ref{tab:compliance}, the linear coefficients of thermal expansion of Ag$_3$Co(CN)$_6$ along the $a$($b$) and the $c$ axes were calculated within the quasi-harmonic approximation as~\\cite{Barron1980}\n\\begin{eqnarray}\\label{linearcteab}\n\\alpha _{ab}  &=& \\frac{1}{\\Omega }\\sum\\limits_{s,\\textbf{k}} {\\left\\{ {c_{s,\\textbf{k}} \\left[ {\\frac{{\\left( {s_{11}  + s_{12} } \\right)\\gamma _{ab}(s,\\textbf{k}) }}{2} + s_{13} \\gamma _c(s,\\textbf{k}) } \\right]} \\right\\}} \\nonumber \\\\\n&=& \\frac{1}{\\Omega }\\left[ {\\frac{{\\left( {s_{11}  + s_{12} } \\right)\\overline \\gamma _{ab} }}{2} + s_{13} \\overline \\gamma _c } \\right]\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{linearctec}\n\\alpha _c  &=& \\frac{1}{\\Omega }\\sum\\limits_{s,\\textbf{k}} {\\left\\{ {c_{s,\\textbf{k}} \\left[ {s_{13} \\gamma _{ab}(s,\\textbf{k})  + s_{33} \\gamma _c(s,\\textbf{k}) } \\right]} \\right\\}} \\nonumber \\\\\n&=& \\frac{1}{\\Omega }\\left[ {s_{13} \\overline \\gamma _{ab}  + s_{33} \\overline \\gamma _c } \\right],\n\\end{eqnarray}\nrespectively, where\n\\begin{eqnarray}\\label{specificheat}\nc_{s,\\textbf{k}}  = \\hbar \\omega _{s,\\textbf{k}} \\frac{{\\partial n_{s,\\textbf{k}} }}{{\\partial T}}\n\\end{eqnarray}\nis the contribution of the normal-mode $\\left\\{s,\\textbf{k}\\right\\}$ to the specific heat with $n_{s,\\textbf{k}}  = \\left[ {\\exp \\left( {\\hbar \\omega _{s,\\textbf{k}} \/k_BT } \\right) - 1} \\right]^{ - 1}$, and $\\Omega$ is the volume of the unit cell. The overall Gr\\\"{u}neisen parameters are defined as\n\\begin{eqnarray}\\label{overallgamma}\n\\overline \\gamma_{ab}&=&\\sum\\limits_{s,\\textbf{k}} {c_{s,\\textbf{k}} \\gamma_{ab}(s,\\textbf{k})} \\nonumber \\\\\n\\overline \\gamma_{c}&=&\\sum\\limits_{s,\\textbf{k}}  {c_{s,\\textbf{k}} \\gamma_{c}(s,\\textbf{k})}.\n\\end{eqnarray}\nThe volume CTE is calculated as\n\\begin{eqnarray}\\label{linearctev}\n\\alpha _V  = 2\\alpha _{ab}  + \\alpha _c.\n\\end{eqnarray}\n\nThe calculated values of $\\alpha_{ab}$ and $\\alpha_c$ at different temperatures and pressures are shown in Fig.~\\ref{fig:cote}. The averaged values of $\\alpha_{ab}$ and $\\alpha_c$ over 50--500 K are $+127$~MK$^{-1}$ and $-101$~MK$^{-1}$, respectively. These exceptionally large values are in reasonable agreement with the experimental values~\\cite{GoodwinAgCoCN2008} of $\\alpha_{ab}=+135$~MK$^{-1}$ and $\\alpha_c=-131$~MK$^{-1}$. The hinging mechanism of the material as discussed previously results in similar magnitude of the PTE along the $a$($b$) axes and the NTE in the $c$ axis.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{cotenew.pdf}\n\\end{center}\n\\caption{\\label{fig:cote} DFPT+D calculated coefficients of thermal expansion at different temperatures for pressures of 0.0 (solid line), 0.04 (dashed line) and 0.1 GPa (dotted line) using quasi-harmonic approximation, compared to the experiment at ambient pressure (in open circle)~\\cite{GoodwinAgCoCN2008}. A plot from GGA calculated phonons based on the correct structure optimised using the GGA+D method is given in the Supplemental Material~\\cite{supplemental} for comparison.}\n\\end{figure}\n\nIn addition to reproducing the experimentally-observed~\\cite{GoodwinAgCoCN2008} colossal PTE and NTE of Ag$_3$Co(CN)$_6$, an interesting finding from Fig.~\\ref{fig:cote} is that $\\partial \\alpha_c \/ \\partial p > 0$, that is $\\alpha_{c}$, which has a negative value, becomes less negative on compression. This is opposite to the usual behaviour that $\\partial \\alpha\/ \\partial p<0$ as found in most PTE materials such as metals, metal oxides and alkali halides~\\cite{Fangmetal2010,Zhang2007,Song2012,Sun2013} and also in many isotropic NTE materials~\\cite{Chapman2005,Fangmd2013,Fangzeolite2013,Fangmodel2014}.\n\nAccording to the standard thermodynamic relation~\\cite{Fangexp2013}\n\\begin{eqnarray}\\label{linearwarmhardening}\n\\left( {\\frac{{\\partial B_c }}{{\\partial T}}} \\right)_p  = B_c^2 \\left( {\\frac{{\\partial \\alpha _c }}{{\\partial p}}} \\right)_T,\n\\end{eqnarray}\na positive value of $\\partial \\alpha_c \/ \\partial p$ means a corresponding positive value of $\\partial B_c \/ \\partial T$. If $B_c$ were positive as would usually be the case, this would give the unusual property of the material becoming harder at higher temperature~\\cite{Fangmodel2014}, but in this case $B_c$, as the inverse of $\\beta_c$ is negative (see Table~\\ref{tab:compliance}), and thus $B_c$ becomes less negative on heating with $\\beta_c$ becoming more negative. Hence higher temperatures enhance NLC.\n\nTo understand this, we note that the values of $\\alpha_{ab}$ and $\\alpha_c$ depend on $\\overline \\gamma_{ab}$ and $\\overline \\gamma_{c}$ weighted by the compliances, as given in Eqs~\\ref{linearcteab} and~\\ref{linearctec}. Since the compliances listed in Table~\\ref{tab:compliance} change little with pressure, any significant change of the CTE with pressure must be due to the change of the overall Gr\\\"{u}neisen parameters.\n\nIn the temperature range of 0--500 K, only contributions from the low-frequency modes ($\\leq 9$ THz) (Fig.~\\ref{fig:dosgp}(a)) are important. At zero pressure, contributions from the low-frequency modes result in positive $\\overline \\gamma_{ab}$ and negative $\\overline \\gamma_c$ as shown in Fig.~\\ref{fig:overallgamma}. Since $s_{11}$ and $s_{12}$ are positive and $s_{13}$ is negative, both $\\overline \\gamma_{ab}$ and $\\overline \\gamma_{c}$ would contribute constructively to the positive value of $\\alpha_{ab}$ according to Eq.~\\ref{linearcteab}. Similarly, since $s_{13}$ is negative and $s_{33}$ is positive, $\\overline \\gamma_{ab}$ and $\\overline \\gamma_{c}$ would also contribute constructively to the negative value of $\\alpha_{c}$. Thus, the increase of $\\overline \\gamma_{ab}$ (becoming more positive) and decrease of $\\overline \\gamma_{c}$ (becoming more negative) would enhance the linear PTE and NTE, while the decrease of $\\overline \\gamma_{ab}$ and increase of $\\overline \\gamma_{c}$ would reduce the linear PTE and NTE of the material.\n\nFig.~\\ref{fig:overallgamma} shows that there is a significant decrease of $\\overline \\gamma_{ab}$ and a smaller decrease of $\\overline \\gamma_{c}$ on compression. According to Eqs~\\ref{linearcteab} and~\\ref{linearctec}, the first effect is more dominant and results in the large decrease in the magnitude of both $\\alpha_{ab}$ and $\\alpha_c$ with pressure, corresponding to the conventional decrease of elastic moduli on heating ($\\partial B_{ab}\/\\partial T \\propto \\partial \\alpha_{ab}\/\\partial p < 0$) and the heat enhancement of NLC ($\\partial B_{c}\/\\partial T \\propto \\partial \\alpha_{c}\/\\partial p > 0$), respectively.\n\nIt is interesting to note this enhancement of NLC on heating could not happen without the hinging mechanism in the structure working efficiently, because it is this mechanism that gives almost the same magnitudes to $s_{13}$ and $s_{33}$ (as discussed in Section~\\ref{groundstate}), which in turn provide the same weighting of $\\overline \\gamma_{ab}$ and $\\overline \\gamma_{c}$ in their contributions to $\\alpha_c$. If we had the case where the hinging is not effective, a much smaller value of $s_{13}$ compared to $s_{33}$ would make the decrease of $\\overline \\gamma_c$ dominate, resulting in a decrease of $\\alpha_c$ on compression (corresponding to $\\partial B_{c}\/\\partial T \\propto \\partial \\alpha_{c}\/\\partial p < 0$); in this case the enhancement of NLC on heating would not be observed.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{averagegammanew.pdf}\n\\end{center}\n\\caption{\\label{fig:overallgamma} Calculated overall Gr\\\"{u}neisen parameters $\\overline \\gamma_{ab}$ (in blue) and $\\overline \\gamma_{c}$ (in red) by Eq.~\\ref{overallgamma}. $\\overline \\gamma_{ab}$ at 0.0, 0.04 and 0.1~GPa correspond to solid, dashed and dotted lines, respectively. $\\overline \\gamma_{c}$ at the same pressures corresponds to solid, dashed and dotted lines, respectively. From 0.0 to 0.1~GPa, $\\overline \\gamma_{ab}$ decreases significantly and even becomes negative at 0.1~GPa, while $\\overline \\gamma_c$ decreases much less.}\n\\end{figure}\n\n\n\n\n\\subsection{Exceptionally large $\\partial \\alpha_V\/\\partial p$}\\label{softening}\n\n\nAnother interesting finding in Fig.~\\ref{fig:cote} is the exceptionally large reduction in $\\alpha_V$ on compression. The magnitude of $\\partial \\alpha_V \/ \\partial p$ is found to be about 1125~MK$^{-1}$\/GPa from 0.0 to 0.04~GPa and 2083~MK$^{-1}$\/GPa from 0.04 to 0.1~GPa, values that are more than 10 times larger than what is normally considered as a large value~\\cite{Cetinkol2008} (\\textit{ca} 100~MK$^{-1}$\/GPa) and more than 10$^4$ times larger than that of a hard metal~\\cite{Fangmetal2010}.\n\nFrom 0.0 to 0.1~GPa, the linear CTE of the material is reduced from its colossal value to a more moderate value of about $\\pm25$~MK$^{-1}$ which is similar to the values found in the NTE metal cyanides~\\cite{Goodwin2005,Fangmd2013}. As discussed in the previous section, such significant reduction in the magnitudes of $\\alpha_{ab}$ and $\\alpha_c$ is due to the large decrease of $\\overline \\gamma_{ab}$. In particular, when $\\overline \\gamma_{ab}$ becomes negative at 0.1 GPa, it begins to contribute to $\\alpha_{ab}$ and $\\alpha_c$ (Eqs~\\ref{linearcteab} and~\\ref{linearctec}) with opposite sign to that of $\\overline \\gamma_{c}$.\n\nThe significant decrease on compression of $\\overline \\gamma_{ab}$ is attributed to the large decrease in $\\gamma_{ab}$ of most low-frequency modes ($\\leq9$~THz), especially the modes with frequencies $<1.0$~THz (as those at wave vector A in Fig.~\\ref{fig:phonon}). On one hand, such a decrease is related to the increase of mode frequencies (see Eq.~\\ref{gammaa}) upon hydrostatic compression as shown in Fig.~\\ref{fig:dosgp}(d). On the other hand, the sign change of $\\gamma_{ab}$ at 0.1 GPa is indicated in Figs~\\ref{fig:dosgp}(a) to (c) by the coloured DoS according to the values of $\\gamma_{ab}$ at different pressures.\n\nThe sign change of $\\gamma_{ab}$ of the low-frequency modes under pressure can be explained with the help of Fig.~\\ref{fig:Amode1}. As discussed previously, the transverse vibration of the CN--Ag--NC bridge of such modes can pull the connected Co closer hence contract the $c$ dimension of the crystal. With relaxed Co--CN--Ag--NC--Co linkage at zero pressure, reducing the $a$ and $b$ dimensions of the unit cell tends to extend the $c$ dimension due to the hinging mechanism. This would make the transverse vibration that contracts the dimension more difficult and result in positive $\\gamma_{ab}$ in Eq.~\\ref{gammaa}. However, at high hydrostatic pressures, large elongation in the $c$ dimension (due to the giant NLC of the material) would largely extend the Co--CN--Ag--NC--Co linkage. This time, reducing the $a$ and $b$ dimensions with fixed $c$ of the unit cell can accommodate part of the extension in the linkage and make the linkage less taut. This would in turn make it easier for the CN--Ag--NC linkage to vibrate transversely, which would result in negative $\\gamma_{ab}$ in Eq.~\\ref{gammaa}.\n\nThe scissor-like behaviour of the change of linear CTE seen in the upper panel of Fig.~\\ref{fig:cote}, namely the decrease of $\\alpha_{ab}$ accompanied by the increase of $\\alpha_c$ upon compression, makes the combined $\\alpha_V$ in Eq.~\\ref{linearctev} close to zero at high pressure. The large value of $s_{11}$ due to the weak interaction between Ag atoms in the $a$--$b$ plane makes sure that the contribution from $\\overline \\gamma_{ab}$ to $\\alpha_{ab}$ in Eq.~\\ref{linearcteab} dominates, so that $\\alpha_{ab}$ would decrease largely according to the decrease of $\\overline \\gamma_{ab}$. On the other hand, as discussed in the previous section, the effective hinging mechanism guarantees the similarly large increase of $\\alpha_c$. Thus, it is the dispersive interaction together with the hinging mechanism that make $\\alpha_{ab}$ and $\\alpha_c$ change with pressure like a scissor.\n\nAccording to the relation~\\cite{Fangexp2013}\n\\begin{eqnarray}\\label{bsoftening}\n\\left( {\\frac{{\\partial B_V }}{{\\partial T}}} \\right)_p  = B_V^2 \\left( {\\frac{{\\partial \\alpha _V }}{{\\partial p}}} \\right)_T,\n\\end{eqnarray}\nthe giant reduction of $\\alpha_V$ with pressure implies a giant decrease of $B$ on heating. From Eq.~\\ref{bsoftening}, $B(T)$ can be calculated as\n\\begin{eqnarray}\\label{bulkmodulustemperature}\nB(T) = \\left( {\\frac{1}{{B_{T = 0} }} - \\int_0^T {\\frac{{\\partial \\alpha }}{{\\partial p}}} dT} \\right)^{ - 1},\n\\end{eqnarray}\nand is shown in Fig.~\\ref{fig:bt}. From 0.0 to 300 K, $B$ is reduced by $75\\%$ which is much larger than the observed giant softening ($\\sim 45\\%$) of the isotropic NTE material ZrW$_2$O$_8$~\\cite{Pantea2006} on heating. Such softening results in a value of $B$ in much better agreement with the experimental value of 6.5(3) GPa at room temperature~\\cite{GoodwinAgCoCN2008}, as shown in Fig.~\\ref{fig:bt}.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{BT3.pdf}\n\\end{center}\n\\caption{\\label{fig:bt} Calculated temperature dependence of the bulk modulus $B$ of Ag$_3$Co(CN)$_6$ at zero pressures using Eq.~\\eqref{bulkmodulustemperature}. The great softening of $B$ on heating brings the calculated value in better agreement to the experimental one at room temperature.}\n\\end{figure}\n\n\n\\section{Conclusions}\n\n\nBy including the dispersive correction in the DFT GGA calculation, we are now able to correctly reproduce the ground state of Ag$_3$Co(CN)$_6$ as well as the the high-pressure phase of the material having the interdigitated structure.\n\nWe found that, by using the DFPT+D calculated phonons, it is almost the same set of low-frequency modes that contribute to both linear PTE and NTE of the material with their linear Gr\\\"{u}neisen parameters showing similar magnitudes but with opposite sign. Such modes, as those around the wave vector A and the middle point along the H$\\rightarrow$K, correspond to the transverse vibrations of the CN--Ag--NC bridge within the Co--CN--Ag--NC--Co linkage that can transfer the expansion in the $a$($b$) dimension to the contraction in the $c$ dimension.\n\nFrom the DFPT+D results, we have predicted that the value of $\\alpha_c$ of Ag$_3$Co(CN)$_6$ increases on compression, contrary to what is normally seen in PTE and NTE materials. In turn this suggests that the NLC of Ag$_3$Co(CN)$_6$ will be enhanced on heating. We also predicted an exceptionally large reduction in volume CTE on compression, which corresponds to the change of sign of the linear Gr\\\"{u}neisen parameters under pressure together with the right elasticity of the material. The latter is based on the weak interactions between Ag atoms in the $a$--$b$ plane and the effective hinging mechanism in the structure. This property also suggests a giant softening of the material on heating with a reduction in the bulk modulus of about $75\\%$ from 0--300 K.\n\nThe method and results presented in this work would be able to apply to other framework materials, such as KMn[Ag(CN)$_2$]$_3$ and Zn[Au(CN)$_2$]$_2$, that have atoms (e.g. Ag and Au) with large dispersive interactions and show large anisotropic properties of PTE\/NTE as well as NLC~\\cite{Cairns2012,Kamali2013,Cairns2013,Gatt2013}. It would be interesting in a future study to see if the phenomena of heat enhancement of NLC and giant reduction of volume CTE on compression predicted for Ag$_3$Co(CN)$_6$ can also be found in these other materials. It would be also interesting to use other schemes to include the van der Waals dispersion correction (such as the use of non-local Langreth-Lundqvist functional~\\cite{Dion2004} in the DFT) in calculating properties of these materials and compare the results.\n\n\n\\begin{acknowledgements}\nWe gratefully acknowledge financial support from the Cambridge International Scholarship Scheme (CISS) of the Cambridge Overseas Trust and Fitzwilliam College of Cambridge University (HF). We thank the CamGrid high-throughput environment of the University of Cambridge. We thank the UK HPC Materials Chemistry Consortium, funding by EPSRC (EP\/F067496), to allow us to use the HECToR\/ARCHER national high-performance computing service provided by UoE HPCx Ltd at the University of Edinburgh, Cray Inc and NAG Ltd, and funded by the Office of Science and Technology through EPSRC's High End Computing programme.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}