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+{"text":"\\section{INTRODUCTION}\n\\label{secz:intro}\nThe pion--nucleus interaction on and below the $\\Delta_{33}$ resonance\nhas been extensively studied, and microscopic models of elastic scattering\ndo a reasonable job \\cite{john92} of describing the data.\nThe pion--nucleus interaction at energies above the $\\Delta_{33}$ resonance\nhas received \\cite{rokn88,will89,marl84,hipi} much less attention. In this\nenergy region (300 MeV $\\le T_\\pi \\le $ 1 GeV), the pion has a much shorter\nwavelength. For example, the wavelength at resonance is about 4 fm (about\nthe size of the nucleus)\nwhile at 1 GeV the wavelength is 1 fm (about the size of a single nucleon).\nThe shorter wavelength implies that elastic and inelastic data at the\nhigher energies will provide sensitivity to the details of the spatial\ndependence of the ground--state and transition densities. The data may also\nprove sensitive to modifications \\cite{sieg84,brow88} of the properties of\nthe nucleon in the nuclear medium, or reveal significant contributions from\nexchange currents \\cite{jiang92}.\n\nMoreover, the pion--nucleon two--body interaction becomes much weaker as\none goes to energies above the $\\Delta_{33}$ resonance.\nThe two--body total cross section becomes less than 30 mb, which is about 15\n\\% of that on the $\\Delta_{33}$ resonance. The first implication of this\nweaker amplitude is that the pion is able to penetrate deeper into the\nnucleus. A simple estimate \\cite{erns79} from total reaction cross\nsection studies shows that a projectile can penetrate into a target to\na radius that is equal to the impact parameter at which the profile\nfunction equals one mean free path. In Fig.\\ \\ref{fig1} the profile\nfunction for $^{12}$C, $^{40}$Ca and $^{208}$Pb are pictured, and the\narrows indicate approximately how far into a nucleus a pion of the\nlabeled energy can penetrate. The pion in the energy region from 500\nMeV to 1 GeV is one of the most penetrating of the strongly interacting\nparticles.\n\nThe second implication of the weaker two-body cross section is\nthat multiple scattering theory for the optical potential becomes\nincreasingly convergent. A simple estimate of the convergence\nis obtained by comparing a typical second--order term in the optical\npotential to the first--order term.  For the case of short--range\ncorrelations, one obtains \\cite{erns77}\n\\begin{equation}\nR\\equiv {U^{(2)}\\over U^{(1)}}=\\sqrt{\\sigma}{\\ell_c\\over k}\\rho\\,\\,\\,,\n\\end{equation}\nwhere $\\sigma$ is the total two--body cross section, $\\ell_c$ is the\ncorrelation length, and $\\rho$ is the nuclear density at which the\npion interacts (See Fig. 1). The\nfactor of $1\/k$, where $k$ is the incident pion momentum, reflects the\nsuppression of the pion propagator in $U^{(2)}$ with increasing energy.  On\nresonance, we find $R\\approx 0.1$, with $R < 1$ only\nbecause $\\rho$ is so small there.  At 500 MeV, we\nfind $R\\approx 0.04$ and at 1 GeV, $R\\approx 0.02$.\nThus differences between the data and a carefully calculated result\nutilizing a first--order optical potential would be a strong indication\nof the presence of unconventional phenomena such as mesonic current\ncontributions, modified nucleon properties in the medium, or other yet to be\nthought of effects.\n\nOne option for calculating high--energy\npion scattering is a momentum--space optical--potential approach.  This\npresents the\nopportunity \\cite{gieb88} to\ncalculate the scattering from a lowest--order optical potential\nincluding the following features 1.) exact\nFermi--averaging integration, 2.) fully covariant kinematics \\cite{erns80},\nnormalizations, and phase--space factors, 3.) invariant\namplitudes \\cite{gieb88,erns90} and 4.) finite--range, physically motivated\ntwo--body off--shell \\cite{erns90} amplitudes. Within the multiple\nscattering theory developed in Refs.\\ \\cite{john92,john83} this is an exact\ncalculation of the lowest--order optical potential.  A brief review of\nthis approach will be given in Sec.\\ \\ref{sec:formal}.\n\nA practical difficulty arises when one wants to use the momentum--space\noptical model to study high--energy pion--nucleus scattering. As the\nprojectile energy increases, more partial waves are needed in the\npion--nucleon\ntwo--body amplitudes. Below 300 MeV, only S--waves and P--waves are needed.\nAbove 300 MeV, D--waves become important. F--waves become significant\nabove 500 MeV and G--waves and H--waves above 700 MeV. At the same time\nthe number of pion--nucleus partial waves is increasing at a rate\nproportional to the pion momentum. At the high energies the momentum--space\napproach becomes prohibitively computer--intensive, and one would like to\nsearch for a simpler alternative.\n\nThe semi--classical theory immediately comes to mind as an alternative to\nthe momentum--space approach.  It is often used at high energies not only\nbecause it is easier to compute, but also because the simpler character\nof the theory facilitates obtaining physical insights into the reaction.\nHowever, the semi--classical theory is\nexpected to be a good approximation only for local potentials and only when\nthe wavelength is sufficiently short.  Because the exact pion--nucleus\noptical potential appears to be highly\nnonlocal, we examine the role that nonlocalities play in\nhigh--energy pion--nucleus scattering. This is done numerically in momentum\nspace and is presented in Sec. \\ref{subsec:opt}.  We undertake this study\nas a first step in obtaining a more quantitative measure of the validity\nof the semi--classical theory than currently exist in the literature.\n\nIn Sec. \\ref{subsec:eikon} we present our semi--classical model \\cite{semi}.\nIn\norder to have a quantitative as well as a simple model for confronting data, we\ninclude the Coulomb interaction and the first Wallace correction\n\\cite{wallace}.\nWe establish the validity of the semi--classical model both by examining\nthe size of the Wallace correction and by comparing it to the model--exact\nresults obtained in momentum space.  Comparisons of the eikonal model to\nmodel--exact calculations are made in Sec.\\ \\ref{sec:results}.\n\nA previous similar investigation can be found in Ref.\\ \\cite{seki}.\nThere a factorized approximation \\cite{pipit} was used in the\nmomentum--space calculations, so that the full nonlocality of the\noptical potential was not considered. Furthermore, Glauber multiple scattering\ntheory\\ \\cite{glauber} was proposed, whereas we will examine an even\nsimpler model, that of a local optical potential with the scattering\nsolved via an eikonal approximation. A more detailed discussion that\ncompares and contrasts our results with previous work is given in\nSec.\\ \\ref{sec:comp}.\nA summary, conclusions and future prospects are presented in Sec.\\\n\\ref{sec:conc}.\n\\section{SCATTERING THEORY}\n\\label{sec:formal}\nAt the higher--energies where the pion--nucleon amplitude\nbecomes a smooth function of energy and is much weaker than on resonance,\nthe first--order optical potential may well be adequate for describing\nall of the conventional nuclear physics phenomena that enter into the\nreaction dynamics. Whether this is so remains for further investigation.\nIndependent of this question, there is the question of whether simpler\napproaches to the calculation of the lowest--order optical potential\nand the scattering from this potential are reasonably quantitative. It is\nthis question upon which we concentrate here. Below we first review the\nmomentum--space optical potential which serves as our `model--exact'\ncalculation. In momentum--space, we examine the importance of the\nnonlocalities caused by both the resonance propagation and the finite\nrange of the two--body amplitude. Finding that these both can be accurately\napproximated, we present a simple eikonal model.\n\nBoth the optical and the eikonal models use the same target\nwave functions, which are obtained from Hartree--Fock calculations\\\n\\cite{hf,bein75} with spurious center--of--mass motion removed as discussed in\n\\ref{sec:dcm}.  They also use the same on--shell pion--nucleon two--body\namplitudes,\nwhich are constructed from Arndt's\\ \\cite{arndt} and H\\\"ohler's\\\n\\cite{hohler} phase--shifts.\n\\subsection{Optical potential}\n\\label{subsec:opt}\nA complete description of the momentum--space optical potential that we use\ncan be found in Ref.\\ \\cite{gieb88}, and a detailed discussion of\nthe covariant kinematics that we use can be found in Refs.\\ \\cite{erns80}.\nThe formal multiple--scattering theory in which this work is embedded is\ngiven in Refs.\\ \\cite{john92,john83}. Here we provide only a brief\noverview.\n\nThe first--order optical potential in the impulse approximation can be\nwritten as,\n\\begin{eqnarray}\n\\langle \\vec k'_\\pi\\vec k'_A\\,\\vert\\,U_{\\rm mo}(E)\\,\\vert\\,\n\\vec k_\\pi\\vec k_A\\rangle\n=\\sum_\\alpha \\int&&\\,{d^3 k_{A-1}\\over 2\\bar E_{A-1}}\\,{d^3 k'_N\\over 2\\bar\nE'_N}\\,{d^3 k_N\\over 2\\bar E_N}\\,\n\\langle\\Psi'_{\\alpha ,\\vec\nk'_A}\\,\\vert\\,\\vec k'_N\\vec k_{A-1}\\rangle\\nonumber \\\\\n&&~~~~~~\\times\\,\\langle \\vec k'_\\pi\n\\vec k'_N\\,\\vert\\,t(E)\\,\\vert\\,\\vec k_\\pi \\vec k_N \\rangle\\langle \\vec\nk_N\\vec k_{A-1}\\,\\vert\\,\\Psi_{\\alpha ,\\vec k_A}\\rangle\\,\\,,\n\\label{eq:optical}\n\\end{eqnarray}\nwhere $\\vec k_\\pi$, $\\vec k_N$ and $\\vec k_{A-1}$ are the momenta of the\npion, the struck nucleon and the $A-1$ residual nucleons in the\npion--nucleus center--of--momentum frame\nrespectively. $E$ is the incident energy of the pion and the nucleus\n(in the pion--nucleus center--of--momentum frame also), and $\\alpha$\nis a set of quantum numbers that specifically label the nuclear bound state.\n\nThe target wave function $\\langle \\vec k_N\\vec k_{A-1} \\,\\vert\\,\n\\Psi_{\\alpha ,\\vec k_A}\\rangle$ defined covariantly \\cite{gieb88} contains\na momentum--conserving delta function $\\delta (\\vec k_A - \\vec k_N - \\vec\nk_{A-1})$ and is a function of the relative momentum between the nucleon\nand the $A-1$ residual nucleons. The pion--nucleon t-matrix similarly\ncontains a momentum--conserving delta function and is a function\nof the relative momentum between the pion and the nucleon, $\\vec \\kappa$,\n\\begin{equation}\n\\langle \\vec k'_N \\vec k'_\\pi \\,\\vert\\,t(E) \\,\\vert\\, \\vec k_N \\vec\nk_\\pi \\rangle =\n\\delta (\\vec k'_N + \\vec k'_\\pi - \\vec k_N-\\vec k_\\pi)\n\\langle \\vec \\kappa' \\,\\vert\\, t[\\omega_\\alpha (E,\\vec k_\\pi\n,\\vec k_N )] \\,\\vert\\, \\vec \\kappa \\rangle\\,.\n\\end{equation}\nOne of the these three momentum--conserving delta functions leads to overall\nmomentum\nconservation; the others allow one to perform two of the three\nintegrals in Eq.~\\ref{eq:optical}. The remaining integral is the Fermi\naveraging integration which\nmust be performed numerically. The details of how we perform the integration\ncan be found in \\cite{gieb88}.\n\nThe energy at which one evaluates the two--body t--matrix, $\\omega_\\alpha\n(E, \\vec k_\\pi, \\vec k_N )$, must be carefully chosen \\cite{erns85} if a\nconvergent perturbation theory is to result for calculations at and below the\n$\\Delta{33}$ resonance region. The energy $\\omega_\\alpha$ is defined\ncovariantly by\nfirst defining the energy available to the pion--nucleon two--body subsystem\n\\begin{equation}\nE_{\\pi N}=E -\\sqrt{(\\vec k_\\pi+\\vec k_N)^2+m_{A-1}^2}\\,\\,,\n\\end{equation}\nand then defining the invariant center-of-momentum energy for this system\n\\begin{equation}\n\\omega_\\alpha^2=E_{\\pi N}^2-(\\vec k_\\pi+\\vec k_N)^2\\,\\,.\n\\label{eq:cmen}\n\\end{equation}\nThe mass of the $A-1$ system, $m_{A-1}$, differs from the mass of the\n$A$--body target, $m_A$, by a nucleon mass and a binding energy,\n$m_A=m_{A-1}+m_N+E_b$. This energy must then be shifted by a `mean-spectral\nenergy', $E_{ms}$, which is a calculated number \\cite{erns85} that\napproximately accounts for the interaction of the intermediate $\\Delta_{33}$\n(or the intermediate nucleon or other hadronic resonance) with the residual\nnucleus. This first--order potential produces results for energies\nnear the $\\Delta_{33}$ resonance that are in remarkable agreement with the\ndata. This is because \\cite{john92} the sum of the second--order\neffects (Pauli exclusion, true absorption and correlation corrections)\ncancel amongst themselves. The use of invariant normalizations,\ninvariant phase\nspace and invariant amplitudes produces the phase-space factors that are\npresent in Eq. \\ref{eq:optical}.\n\nThe optical potential so defined is then inserted into the Klein--Gordon\nequation which is solved numerically to produce our model--exact calculation.\n\n\\subsection{Eikonal model}\n\\label{subsec:eikon}\nThe eikonal model we propose to examine results from first replacing\nthe optical potential of Eq.\\ \\ref{eq:optical} by a local potential and\nthen solving for the scattering amplitude arising from the use of the local\npotential and the eikonal propagator. In order to include the Coulomb\ninteraction, we divide the scattering amplitude as\n\\begin{equation}\nF(q)=F_{pt}(q)+F_{CN}(q)\n\\end{equation}\nwhere $q$ is the momentum transferred to the pion, $\\vec q=\\vec k'_\\pi-\\vec\nk_\\pi$, $F_{pt}$ is the point Coulomb scattering amplitude,\nand $F_{CN}$ is the scattering amplitude calculated from the sum of the\nstrong nuclear\npotential plus a Coulomb correction, which is the difference between the\npoint Coulomb interaction and the Coulomb interaction of a finite charge\ndistribution. The eikonal approximation gives for $F_{CN}(q)$\n\\begin{equation}\nF_{CN}(q)=ik_\\pi \\int_0^\\infty\nbdbJ_0(qb)e^{i\\chi_{pt}(b)}\\Gamma_{CN}(E,b),\n\\end{equation}\nwhere $k_\\pi$ is the momentum of the incident pion in the\npion--nucleus center--of--momentum frame, $b$ is the impact parameter,\n$\\chi_{pt}$ is the point Coulomb phase (in the eikonal approximation), and\n$\\Gamma_{CN}$ is the profile function defined below.\n\nIn order to incorporate the distortion and energy shift caused by the\nCoulomb interaction, we follow the prescription given in Ref.\\ \\cite{coul}\nand write the profile function as\n\\begin{eqnarray}\n\\Gamma_{CN}(E,b)&&=1-{\\rm exp}\\lbrace\ni\\chi_{CN}(E,b)\\rbrace\\nonumber \\\\\n&&=1-{\\rm exp}\\lbrace i\\chi_N\\lbrack E,\nb(1+EV_c(b)\/k_\\pi^2)\\rbrack\n+i\\chi_c(b)-i\\chi_{pt}(b)\\rbrace,\n\\end{eqnarray}\nwhere $V_c$ is the Coulomb potential of a uniformly\ncharged sphere,\n$\\chi_c$ is the phase shift caused by that potential, and the nuclear phase\nshift $\\chi_N$ is written as\n\\begin{equation}\n\\chi_N(E,b)=-{1\\over 2k_\\pi}\\,\\int_{-\\infty}^\\infty dz\\,\nU_{\\rm eik}\\lbrack E-V_c(r),r\\rbrack,\n\\end{equation}\nwhere\n\\begin{equation}\nr=(b^2+z^2)^{1\/2},\n\\end{equation}\nand $U_{\\rm eik}$ is the strong potential obtained as follows.\nFirst we construct a local potential $U_o$ from\nthe pion--nucleon two--body scattering amplitude and the target density by\n\\begin{equation}\nU_o(E,r)= -4\\pi Z\\lbrack f_p(0)\\rho_p +{f'_p(0)\\over\n2k_\\pi^2}\\nabla^2\\rho_p\\rbrack\n-4\\pi N\\lbrack f_n(0)\\rho_n+{f'_n(0)\\over\n2k_\\pi^2}\\nabla^2\\rho_n\\rbrack,\n\\label{eq:eiku}\n\\end{equation}\nwhere $Z$ and $N$ are the proton number and the neutron number\nrespectively. The scattering amplitude $f_p(0)$ [$f_n(0)$] is the\npion--proton [pion--neutron] scattering amplitude in the forward direction\nin the pion--nucleus center--of--momentum frame. The finite range of the\npion--nucleon interaction is included through lowest nonvanishing order\nand produces the additional terms in Eq.\\ \\ref{eq:eiku} which are\nproportional to the derivatives of the scattering amplitude as shown in\nAppendix \\ref{sec:dir}.\n\nIn order to investigate the importance of the lowest--order corrections to\nthe eikonal propagator as derived by Wallace \\cite{wallace}, next we\nreplace the potential $U_o (E,r)$ by the potential\n\\begin{equation}\nU_{\\rm eik}(E,r)=U_o+{U^2_o\\over 4k_\\pi^2}\\Big( 1+{2b^2\\over r}{d\\over dr}\n{\\rm ln}U_o\\Big).\n\\label{eq:wallace}\n\\end{equation}\nThe details of the derivation of this correction and higher order\ncorrections can be found in Ref.~\\cite{wallace}.\n\\section{ RESULTS}\n\\label{sec:results}\nIn this section we investigate the importance of some of the ingredients of\nour model--exact calculation of high--energy pion--nucleus scattering. We\nwill then examine how well our simple eikonal theory\nreproduces the results of the model--exact theory.\n\n\\subsection{Contributions from nonlocalities in the optical potential}\n\\label{subsec:nonloc}\nThe most difficult part of evaluating the optical potential in the\nmodel--exact theory is performing the Fermi--averaging integration.  This\nis known to be very important in the region of the $\\Delta_{33}$ resonance,\nwhere proper accounting for the recoil and propagation of the pion--nucleus\nresonance requires a careful treatment of the dependence of the\nenergy $\\omega_\\alpha$ in Eq.~\\ref{eq:cmen} on the momentum of the nucleon.\nResonance propagation manifests itself as a nonlocality in the optical\npotential, and this represents the first source of nonlocality whose\nimportance we want to assess for high--energy pions.\n\nTo investigate this we calculate elastic scattering of $\\pi^-$ from\n$^{40}$Ca  first with full Fermi averaging and then\nutilizing a closure approximation.\nThe closure approximation that we use results from taking\n\\begin{equation}\n\\omega_\\alpha^2= W_{\\pi N}^2-k_\\pi^2\\,\\,\\,{\\rm and}\\,\\,\\,W_{\\pi\nN}=E-\\sqrt{k_\\pi^2+m_{A-1}^2}\n\\label{eq:closure}\n\\end{equation}\nThis is equivalent to setting the momentum of\nthe struck nucleon to zero in the pion--nucleus c.m. frame.\n\nOn the delta resonance, it has been found \\cite{dhole,liurev,erns83}\nthat the closure approximation is totally\ninadequate and even the `optimally-factorized' approximation \\cite{erns83}\nis not quantitative. At these higher energies the\nconclusion is not {\\it \\'a priori} clear. The pion--nucleon amplitude\nis, partial wave by partial wave, rather energy dependent as the individual\npartial waves are resonant. The total amplitude, however, exhibits\nonly two very broad smooth peaks on a large background.\n\nThe resulting differential cross sections for pions at 500 MeV scattering\nfrom $^{40}$Ca are shown in Fig.\\ \\ref{fig2}. We see that the correct\ntreatment of the recoil of the two--body pion--nucleon system only affects\nthe depth of the minimum but not to a very significant degree.\n\nThe behavior of the pion--nucleon scattering amplitude off--shell is taken\nfrom the doorway--resonance model of Ref.\\ \\cite{erns90}.\nAmplitudes which are resonant are separable in their dependence on the\nrelative momenta, hence maximally nonlocal. This is the second\nsource of nonlocality whose importance we want to investigate.  In the\ndoorway--resonance model, in each angular momentum, spin and isospin channel,\n\\begin{equation}\n\\langle \\kappa'\\,\\vert\\,t_{JLI} (\\omega_\\alpha )\\,\\vert\\, \\kappa\n\\rangle={v(\\kappa')\\over v(\\kappa_o)}\\,\\langle\\kappa_o\n\\,\\vert\\, t_{JLI} (\\omega_\\alpha )\\,\\vert\\,\\kappa_o\\rangle\\,\n{v(\\kappa)\\over v(\\kappa_o)},\n\\end{equation}\nwhere $\\kappa_o$ is the on--shell momentum, i.e. the momentum\ncorresponding to the center--of--momentum energy $\\omega_\\alpha $.\nThis nonlocality, which is also factored in the delta--hole \\cite{dhole}\nmodel, was found quite\nimportant\\cite{erns83} for energies on and below the $\\Delta_{33}$\nresonance.  For numerical convenience, we take the form factor to be a\nGaussian, $v(\\kappa )=\\exp (-\\kappa^2\/\\beta^2)$, and we vary $\\beta$ from\n500 MeV to 4 GeV to investigate the importance of this nonlocality.  The\ndifferential cross sections for $\\pi^-$--$^{40}$Ca scattering calculated\nwith this variation in the form--factor range are shown in\nFig. \\ref{fig3}. We see that the dependence on the off--shell range is\nweak at this energy.\n\nThere exists some confusion in the literature concerning the\npossible values for $\\beta$. First, if one does not include\ninvariant phase space factors explicitly, they will appear effectively in\nwhat one might wish to call the form factor. This difference is then one of\nsemantics. However, in deciding what might be a reasonable range over which\nto vary the form factor, one should treat the kinematic factors explicitly.\nIn the $P$--wave channels, an artificial and incorrect increase\n(in coordinate space) in the range of the form factor will also result if\nthe pion--nucleon pole term \\cite{erns78} is not included in the model of\nthe pion--nucleon amplitude. For our purposes here, the difference of\ninterest is in examining  how the cross section changes in going from the\nphysically motivated doorway--model form factors to the higher momentum\ncutoff. The 4 GeV cutoff produces an  approximately  zero--range interaction\nin coordinate space. Figure \\ref{fig3} demonstrates that the nonlocalities\ncontained in a finite range interaction do not alter significantly the\npredicted elastic cross sections for these high energy pions.\n\n\\label{subsec:comp}\nWe have found that the nonlocalities in the optical potential at the high\nenergies do not have an important effect. This suggests that the simple\neikonal approach outlined above could prove to be an adequate model\nfor quantitative work at these energies. To investigate this, we\npicture the differential cross sections predicted by each model\nfor $\\pi^\\pm$--$^{12}$C and $\\pi^\\pm$--$^{40}$Ca at 800 MeV\/c in\nFig.\\ \\ref{fig4} and Fig.\\ \\ref{fig5}. The data are from Ref.\\\n\\cite{marl84}. As stated earlier, we construct the target density from the\nsame wave\nfunctions as were used in the momentum--space calculation and use the\nsame on--shell pion--nucleon scattering amplitude. For $^{40}$Ca, both the\nlocation of the minima and the\nmagnitude of the cross section at the forward angles are in good agreement.\nThe agreement is, however, not as quantitative for\n$^{12}$C where the minima are slightly shifted. This difference is\npresumably caused in part by the fact that the kinematics\nof target recoil enter explicitly in the momentum--space formulation\nbut not in the eikonal approach.\nThe eikonal model is thus more quantitative for\nthe heavier nuclei. We are examining the eikonal model to see if it\ncan be further improved for the light nuclei.\n\n\\subsection{Important features in the eikonal approximation}\n\\label{subsec:imeik}\nIn this section we investigate the role of the Coulomb interaction\nand the first Wallace correction, which are both necessary if one wants\nto use the eikonal model to quantitatively approximate the results of the\nsolution of the model--exact theory.\n\nFirst, we find that the Coulomb interaction plays an important role in\ndetermining the depth of the minima in the differential cross section. To\nsee exactly how important Coulomb--nuclear interference is, we\ncompare the calculation of the $\\pi^-$--$^{40}$Ca differential cross\nsection at a series of energies as shown in Fig.\\ \\ref{fig6} with and\nwithout the Coulomb interaction. The Wallace correction is not included in\neither the solid or the dashed cureves. We find that the first minimum\nin the differential cross sections becomes very deep when the Coulomb\ninteraction is included, falling well below $10^{-2}$ mb\/sr in the region\nof 500 to 600 MeV. Without the Coulomb interaction, the deepest minimum is\nmuch shallower and occurs at 780 MeV. The inclusion of the Coulomb\ninteraction in the theory is necessary if a quantitative comparison with the\ndata for $\\pi^-$--$^{40}$Ca is to be made. Coulomb--nuclear interference is\ndestructive for $\\pi^-$ and constructive for $\\pi^+$, which produces a less\ndramatic effect for $\\pi^+$--$^{40}$Ca scattering in this energy region.\n\nWallace has shown in Ref.~\\cite{wallace} that the semi-classical\napproximation can be improved if higher order corrections (known as\n``Wallace'' corrections) are included. We examine in Fig.~\\ref{fig7} the\nimportance of the first Wallace correction, Eq.~\\ref{eq:wallace}, for\n$\\pi^-$--$^{40}$Ca scattering. We see in Fig.~\\ref{fig7}, in the absence of\nthe Coulomb interaction, that the Wallace correction is dramatic on\nresonance, $T_\\pi= 180$ MeV, where it fills in the deep minimum. At the\nhigher energies, the correction interferes constructively until about 700\nMeV filling in the minima somewhat. At 780 MeV, however, the Wallace\ncorrection interferes destructively with $U_o$ and produce\na much deeper minimum.\n\nIn Fig.~\\ref{fig8} we turn on both the Coulomb interaction and the Wallace\ncorrection, we find a fascinating interplay between the two. Notice that at\n400 MeV, for example, there two corrections are out of phase and tend to\ncancel. At 680 MeV, however, they are in phase and interfere destructively\nwith $U_o$ producing the very deep minimum. By 780 MeV they\nare both out of phase with each other but in phase with the lowest order\nterm and thus fill in the minima. Both the Coulomb and the Wallace\ncorrection are necessary if there is to be a deep minima at 680 MeV, and not\nat the neighboring energies. We note that the data of Ref.~\\cite{marl84} was\ntaken at a pion lab momentum of 800 MeV\/c which is very near $T_\\pi= 680$\nMeV. The depth of the first minimum of the differential cross section for\n$\\pi^-$ scattering over the energy range 400 to 800 MeV should prove a very\nsensitive test of the existence, or lack thereof, of higher order\ncorrections in the reaction dynamics.\n\n \\section{COMPARISON TO OTHER WORK}\n\\label{sec:comp}\nOther work on high--energy pion--nucleus elastic scattering, which utilizes\nthe full Glauber theory, can be found in Refs.\\ \\cite{seki,oset}.\nIn particular, a comparison between the Glauber theory and the optical\nmodel on $\\pi^\\pm$--$^{12}$C has been made in Ref.\\ \\cite{seki} and the two\napproaches\nwere found to be in good agreement at the higher energies. We earlier\nnoted that the\noptical potential model there did not include the full Fermi averaging,\nand it was not clear to what extent the agreement found\nwas caused by the approximate treatment of the\nnon--localities in the optical potential. Our work clarifies that.\nWe also note that the Glauber theory they used is considerably more\ncomplicated than our eikonal model.  To our knowledge, the Wallace\ncorrections have not been included in any of the numerical tests\nof the eikonal model in this energy region.  Hence, the important\ninterplay between the Wallace and Coulomb corrections that we find has not\nbeen\npreviously noted.  We also found that the center--of--mass\ncorrections in the target wave functions are important for $^{12}$C but\nnot for $^{40}$Ca, which is in agreement with Ref.\\ \\cite{seki}. Our work\nstrongly\nsupports their conclusion that pion scattering at the higher energies becomes\ntheoretically and calculationally simpler to treat at a quantitative\nlevel.\n\\section{ SUMMARY, CONCLUSIONS, AND FUTURE PROSPECTS}\n\\label{sec:conc}\nWe have compared the scattering from the fully microscopic and\nnonlocal lowest--order optical potential to the scattering from a simple\nlocal potential in the eikonal approximation. We have investigated\nthe importance of several features of the full model.\nWithin the framework of our\n``model--exact'' optical model, these include Fermi\naveraging and the off--shell behavior of the pion--nucleon\nscattering amplitude.\nFor the eikonal theory, we examine the importance of the first Wallace\ncorrection and find that this correction is responsible for\na noticeable improvement of the eikonal theory in comparison to the\n``model--exact'' theory. In all cases, the Coulomb--nuclear interference\nis important and must be included if one is to compare with data.\n\nThis semi--classical eikonal theory put forth here is very simple, even\nsimpler than the Glauber theory utilized by others.\nIt appears to be quantitatively\nvalid at high energy, at least for the first several minima in the\ndifferential elastic cross section. This makes possible a much simpler\nreaction theory than has been needed at energies at and below\nthe $\\Delta_{33}$ resonance.\n\nAt high energies and\/or for heavy nuclei, use of the full\nmomentum--space optical model becomes very time--consuming and sometimes\ncomputationally impossible.\nIn contrast, the eikonal theory is relatively simple and fast on the\ncomputer. It therefore becomes a matter of considerable practical\nimportance to realize that the physics is faithfully reproduced by the\nsimple version of the theory out to the position of the second minimum\nin the differential cross section.\n\nAn examination of the pion--nucleon cross section shows the\ncomplex interplay\nof resonances. Ultimately, it is interesting to explore how resonances behave\nin the nucleus and how the specific features of the optical potential\nare suited for nuclear structure studies. In this pursuit, it will be very\nhelpful to\nhave a simple version of the theory as developed here. We will pursue these\nquestions in subsequent work.\n\nThe momentum space optical model has been extended to treat kaon-nucleus\nscattering \\cite{chen92}. Results show qualitative  agreement with previous\nwork \\cite{sieg84}. Modifications to the  kaon-nucleon amplitudes\nin the\nnuclear medium are needed to eliminate discrepancies between the theory and\nthe data. On the quantitative level, results from the momentum space\napproach give larger discrepancies between theory and data than do those\nfrom the coordinate space\ncalculations used in Ref.\\ \\cite{sieg84}. Our eikonal model, which is also a\ncoordinate space approach and been shown to be a good approximation\nto the ``model-exact'' momentum space theory for high-energy pion\nscattering, can also be modified to treat the kaon scattering. Results from\nthe eikonal model may reveal the sources of the discrepancies between the\nmomentum space and the previous coordinate space calculations.\nWork on the kaon-nucleus scattering utilizing the eikonal model is in\nprogress.\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introd}\n\nThe phase diagram of strongly interacting matter in the temperature ($T$) - \nbaryon density plane remains a challenge for theoretical physics. Although there\nis little doubt that it features a low-temperature hadronic phase, with \nbroken chiral symmetry, and a high-temperature deconfined phase,\nwith restored chiral symmetry, the question about the precise location\nand the exact nature of the transition between these two phases is still open. \nYet, the answer to this question has many phenomenological implications:\nthe region of the phase diagram with high $T$ and small baryon density is\nrelevant for the physics of the early Universe, whereas the region of \nlow $T$ and high baryon density is interesting for the astrophysics of \nsome compact objects, but other corners of the phase diagram are not less \ninteresting (see Ref.~\\cite{Stephanov:2007fk} for an overview). \n\nRelativistic heavy-ion collisions provide us with a unique opportunity\nto infer properties of the transition: depending on the energy of the ion beams\nand on the mass number of the colliding ions, the fireball generated in the \ncollision could fulfill the temperature and baryon density conditions under\nwhich the deconfined phase appears as a transient state, before the\nsystem freezes out into hadrons, which are then detected. Thermal-statistical \nmodels, assuming approximate chemical equilibrium at the chemical freeze-out\npoint, are able to describe the particle yields at a given collision energy\nin terms of two parameters only, the freeze-out temperature $T$ and the baryon \nchemical potential $\\mu_B$. The collection of freeze-out parameters extracted\nfrom experiments with different collision energy lie on a curve in \nthe $(T,\\mu_B)$-plane, extending up to $\\mu_B\\lesssim$ 800 MeV (see Fig.~1 of \nRef.~\\cite{Cleymans:2005xv}, or Ref.~\\cite{Becattini:2012xb} for a recent \nre-analysis of experimental data). \n\nQuantum ChromoDynamics (QCD) is widely accepted as the theory of strong \ninteractions and, as such, must encode all the information needed to\nprecisely draw the phase diagram in the $(T,\\mu_B)$-plane. As a matter of\nfact, only some corners of it can be accessed by first-principle applications\nof QCD, in the perturbative or in the nonperturbative regime. Here we \nfocus on the lattice approach of QCD, based on the idea of discretizing\nthe theory on a Euclidean space-time lattice and simulating it by Monte Carlo \nnumerical simulations as a statistical system, with Boltzmann weight \ngiven by $\\exp(-S_E)$, where $S_E$ is the QCD Euclidean action. The region\nof the phase diagram where $\\mu_B\/(3 T)\\lesssim 1$ is within the reach of this \napproach and one can therefore address, at least inside this region, the \nproblem of determining the shape taken by the QCD pseudocritical line \nseparating the hadronic from the deconfined phase.\n\nThere is no {\\it a priori} argument for the coincidence of the QCD \npseudocritical line with the chemical freeze-out curve: if the deconfined \nphase is realized in the fireball, in cooling down the system first \nre-hadronizes, then reaches the chemical freeze-out. This implies that the \nfreeze-out curve lies below the pseudocritical line in the $\\mu_B$-$T$ plane. \nIt is a common working hypothesis that the delay between chemical freeze-out and \nrehadronization is so short that the two curves lie close to each other and \ncan therefore be compared.\nUnder the assumptions of charge-conjugation invariance at $\\mu_B=0$ and \nanalyticity around this point, the QCD pseudocritical line, as \nwell as the freeze-out curve, can be parameterized, at low baryon densities, \nby a lowest-order  expansion in the dimensionless  quantity $\\mu_B\/T(\\mu_B)$, as\n\\begin{equation}\n\\frac{T(\\mu_B)}{T_c(0)}=1-\\kappa \\left(\\frac{\\mu_B}{T(\\mu_B)}\\right)^2 \\, + \\, \\ldots \\;,\n\\label{curv}\n\\end{equation}\nwhere $T_c(0)$ and $\\kappa$ are, respectively, the pseudocritical temperature \nand the curvature at vanishing baryon density.\n\nDirect Monte Carlo simulations of lattice QCD at nonzero baryon density\nare hindered by the well known ``sign problem'': $S_E$ becomes complex and \nthe Boltzmann weight loses its sense. Several ways out of this problem\nhave been devised (see Ref.~\\cite{Philipsen:2005mj,*Schmidt:2006us,*deForcrand:2010ys,*Aarts:2013bla} for a review): \nredesigning the Monte Carlo updating algorithms for a complex action~\\cite{Karsch:1985cb,*Aarts:2009uq,*Aarts:2011ax,*Aarts:2011zn,*Seiler:2012wz,*Aarts:2013uxa,*Sexty:2013ica}, \nreweighting from the ensemble at $\\mu_B=0$~\\cite{Barbour:1997bh,*Fodor:2001au,*Fodor:2001pe,*Fodor:2004nz}, \nTaylor expanding the relevant observables around $\\mu_B=0$ and calculating the \nfirst coefficients of the series by simulations at $\\mu_B=0$~\\cite{Gottlieb:1988cq,*Choe:2001cq,*Choe:2002mt,*Allton:2002zi,Allton:2003vx,*Allton:2005gk,*Ejiri:2005uv,Endrodi:2011gv,*Borsanyi:2012cr}, \nusing the canonical formulation~\\cite{Alford:1998sd,*Hasenfratz:1991ax,*Kratochvila:2005mk,*Alexandru:2005ix,deForcrand:2006ec},\nusing the density of states method~\\cite{Bhanot:1986kv,*Karliner:1987cu,*Azcoiti:1990ng,*Ambjorn:2002pz}\nand simulating the theory at imaginary chemical potentials and performing\nthe analytic continuation to real ones~\\cite{Lombardo:1999cz,*deForcrand:2002ci,*deForcrand:2003hx,*D'Elia:2002gd,*D'Elia:2004at,*Azcoiti:2005tv,*deForcrand:2006pv,*deForcrand:2008vr,*deForcrand:2007rq,Cea:2010md,Cea:2010fh,*Cea:2012vi,Cea:2012ev,*Wu:2006su,*Nagata:2011yf,*Giudice:2004se,*Giudice:2004pe,*Papa:2006jv,*Cea:2007wa,*Cea:2007wa,*Cea:2010bp,*Cea:2009ba,*Cea:2010bz,*Karbstein:2006er,Cea:2014xva,Bonati:2014rfa,Bonati:2015bha,Bellwied:2015rza}.\n\nThe numerical evidence gathered so far in QCD with $n_f=2+1$ and physical \nor almost physical quark masses points to a scenario with a smooth\ncrossover between the hadronic and the deconfined (or chirally symmetric)\nphase at $\\mu_B=0$, with a pseudocritical temperature $T_c(0)$ of about \n155~MeV~\\cite{Aoki:2006br,*Aoki:2006we,*Aoki:2009sc,Bazavov:2011nk,*Bhattacharya:2014ara}. This crossover behavior should persist in some neighborhood of\n$\\mu_B=0$, up to the onset of a first-order transition at some value of \n$\\mu_B>0$. \n\nThe state-of-the-art of lattice determinations of the curvature $\\kappa$,\nup to the very recent papers of Ref.~\\cite{Bonati:2015bha,Bellwied:2015rza}, \nis summarized in Fig.~10 of Ref.~\\cite{Bonati:2014rfa}: depending on the\nlattice setup and on the observable used to probe the transition, the value \nof $\\kappa$ can change even by almost a factor of three. The lattice \nsetup dependence stems from the kind of adopted discretization, the lattice \nsize, the choice of quark masses and chemical potentials, the procedure\nto circumvent the sign problem. This dependence would totally disappear if, \nideally, all groups would use the same lattice setup. A contribution to the\nunderstanding of the impact of the lattice setup dependence is provided \nin the Appendix~B of Ref.~\\cite{Bonati:2014rfa}. The dependence on the\nprobe observable is, instead, irreducible: since a smooth crossover is taking\nplace rather than a true phase transition, one cannot define a {\\it bona fide} \norder parameter whose behavior would permit to uniquely locate the transition\npoint; instead, for any adopted surrogate observable, a different transition\npoint should be expected, at least in principle.\n\nOn the side of the determinations of the freeze-out curve, two recent \ndeterminations~\\cite{Cleymans:2005xv,Becattini:2012xb} of $\\kappa$, both\nbased on the thermal-statistical model, but the latter of them including the\neffect of inelastic collisions after freeze-out, give two quite different\nvalues of $\\kappa$, each seeming to prefer a different subset of lattice\nresults (see Fig.~3 of Ref.~\\cite{Cea:2014xva} for a snapshot of the\nsituation). \n\nThe aim of this work is to contribute to a better understanding of the \nsystematics underlying lattice determinations of the curvature $\\kappa$, by \ncorroborating our previous determination~\\cite{Cea:2014xva} with \nan extrapolation to the continuum limit and by comparing it with\nexperimental analyses of the freeze-out curve.\n\nOur lattice setup is as follows.  We simulate the HISQ\/tree action of the \nMILC collaboration with 2+1 staggered fermions on lattices with temporal\nextension $L_t=6$, 8, 10 and 12 and aspect ratio equal to four. We work on \nthe line of constant physics (LCP) as determined in Ref.~\\cite{Bazavov:2011nk},\nwith the strange mass set at the physical value and the light quark mass fixed \nat $m_l=m_s\/20$. \nAs discussed in Ref.~\\cite{Bazavov:2011nk}, this amounts to tune the strange quark mass \nuntil the mass of the fictitious $\\eta_{s\\bar{s}}$ meson matches the lowest order perturbation theory estimate\n$m_{\\eta_{s\\bar{s}}} = \\sqrt{2 m_K^2 - m_\\pi^2}$. Consequently within our simulations the pion mass is \n$m_\\pi \\simeq 160 \\, \\text{MeV}$.\n\nWe perform simulations at imaginary quark chemical potentials, \nassigning the same value to the three quark species, $\\mu_l=\\mu_s\\equiv \\mu$, \nthen extrapolate to real chemical potentials. Our probe observables are the \ndisconnected susceptibility of the light quark chiral condensate and its \nrenormalized counterpart.\nSimulating the theory at imaginary chemical potentials poses no restriction\non the lattice size or in the choice of the couplings. However, the periodicity \nin ${\\rm Im}(\\mu_l)\/T$ of the partition function~\\cite{Roberge:1986mm} implies \nthat the information gathered outside a narrow interval of imaginary chemical\npotentials is redundant. For the setup with $\\mu_l=\\mu_s\\equiv \\mu$, this \ninterval can be chosen as the region $0\\leq {\\rm Im}(\\mu)\/T\\leq \\pi\/3$. A safe\nextrapolation of the critical line to real chemical potentials requires that\nit exhibits a smooth dependence on imaginary chemical potential over this\ninterval, a condition which must be checked to be satisfied {\\it a posteriori} \nby our data. \n\nThe preference to the disconnected susceptibility of the light quark chiral \ncondensate has  multiple motivations~\\cite{Bazavov:2011nk}.  First of all, for \nsmall enough quark masses, its contribution to the chiral susceptibility \ndominates over the connected one, which is harder to compute; then, it shows a \nstrong sensitivity to the transition; finally, it is exempt from additive \nrenormalization, undergoing only a multiplicative one. This translate into\na very precise determination of the critical couplings at imaginary $\\mu$,\nwhich is the main prerequisite of a safe extrapolation to the real values of\n$\\mu$.\n\nThere are two main limitations in our setup. The first is that we work\nwith a physical strange quark mass, but with light quarks a bit heavier\nthan physical ones. Numerical results in $n_f=2$ indicate a mild dependence \nof the curvature on the quark mass (see the discussion in Sec.~III of \nRef.~\\cite{Cea:2012ev}). If the same applies here, as we believe, our result\nfor $\\kappa$ will only slightly underestimate (in absolute value) the true \nphysical curvature. The second limitation is that, for the sake of comparison\nwith the freeze-out curve, our setup of chemical potentials could not be the\none which better reproduces the initial conditions of heavy ion collision.\nIn fact, strangeness neutrality would rather impose $\\mu_s \\alt \\mu_l$.\nIn general, the setup $\\mu_l=\\mu_s= \\mu_B\/3$ approximates strangeness neutrality at low temperatures, while\nthe $\\mu_s= 0$ setup is relevant for high enough temperatures. \\\\\n It is natural  to expect that the effect of taking $\\mu_s=\\mu_l$ instead of $\\mu_s=0$\nbecomes less and less evident when $\\mu_l\/T$ approaches zero, so that the \ncurvature $\\kappa$ at zero baryon density should not differ too much in the\ntwo cases. The numerical analysis of Refs.~\\cite{ Bonati:2014rfa,Bonati:2015bha,Bellwied:2015rza} has shown \nthat this effect is invisible within the accuracy of the lattice setup\nadopted there.\n\nThe paper is organized as follows: in Sec.~II we give some further details\nof our numerical simulations. In Sec.~III we show our numerical results\nfor $\\kappa$. Finally, in Sec.IV we draw our conclusions.\n\n\\section{Simulation details and numerical results}\n\nWe perform simulations of lattice QCD with 2+1 flavors of rooted staggered \nquarks at imaginary quark chemical potential.\nWe have made use of the HISQ\/tree action~\\cite{Follana:2006rc,Bazavov:2009bb,Bazavov:2010ru} \nas implemented in the publicly available MILC code~\\cite{MILC}, \nwhich has been suitably modified by us in order to introduce an imaginary quark\nchemical potential $\\mu = \\mu_B\/3$. \nThat has been done by multiplying all forward and backward \ntemporal links entering the discretized Dirac operator \nby $\\exp (i a \\mu )$ and $\\exp (- i a \\mu )$, respectively: in this way,\nthe fermion determinant is still real and positive, so that standard\nMonte Carlo methods can be applied.\nAs already remarked above, in the present study we have \n$\\mu=\\mu_l=\\mu_s$. This means that the Euclidean partition function of the\ndiscretized theory reads\n\\begin{equation}\n\\label{Z}\nZ=\\int [DU] e^{-S_{\\rm gauge}} \\prod_{q=u,d,s} {\\rm det}(D_q[U,\\mu])^{1\/4} \\;,\n\\end{equation}\nwhere $S_{\\rm gauge}$ is the Symanzik-improved gauge \naction and $D_q[U,\\mu]$ is the staggered Dirac operator, modified as \nexplained above for the inclusion of the imaginary quark chemical\n(see Ref.~\\cite{Bazavov:2009bb} and appendix~A of Ref.~\\cite{Bazavov:2010ru}    for the precise definition of\nthe gauge action and the covariant derivative for highly improved staggered fermions). \n\nAll simulations make use of the rational hybrid Monte Carlo (RHMC) \nalgorithm. The length of each RHMC trajectory has been set to  \n$1.0$ in molecular dynamics time units.\n\nWe have simulated the theory at finite temperature, and for several values of \nthe imaginary quark chemical potential, near the transition temperature, \nadopting lattices of size $16^3\\times 6$, $24^3\\times 6$, $32^3 \\times 8$, \n$40^3 \\times 10$ and $48^3 \\times 12$.  \nWe have discarded typically not less than one thousand trajectories for each \nrun and have collected from {4k to 8k} trajectories for measurements.\n\nThe pseudocritical point $\\beta_c(\\mu^2)$ has been determined as the value \nfor which the renormalized disconnected susceptibility of the light quark chiral condensate\ndivided by $T^2$ exhibits a peak. \\\\\nThe bare disconnected susceptibility is given by:\n\\begin{equation}\n\\label{chi_dis}\n\\chi_{l, \\rm disc} =\n{{n_f^2} \\over 16 L_s^3 L_t}\\left\\{\n\\langle\\bigl( {\\rm Tr} D_q^{-1}\\bigr)^2  \\rangle -\n\\langle {\\rm Tr} D_q^{-1}\\rangle^2 \\right\\}\\;,\n\\end{equation}\n Here $n_f=2$ is the number of light flavors\nand $L_s$ denotes the lattice size in the space direction.  The renormalized chiral susceptibility\nis defined as: \n\\begin{equation}\n\\label{chi_ren}\n\\chi_{l, \\rm ren} =   \\frac{1}{Z_m^2} \\;  \\chi_{l, \\rm disc} .\n\\end{equation}\nThe multiplicative renormalization factor $Z_m$  can be deduced\nfrom an analysis of the line of constant physics for the light quark masses.\nMore precisely, we have~\\cite{Bazavov:2010ru}:\n\\begin{equation}\n\\label{zeta}\nZ_m(\\beta) \\;  =  \\;  \\frac{m_l(\\beta)}{m_l(\\beta^*)}  \\; ,\n\\end{equation}\nwhere the  renormalization point $\\beta^*$ is chosen such that:\n\\begin{equation}\n\\label{beta*}\n\\frac{r_1}{a(\\beta^*)} \\; =  \\;  2.37 \\; ,\n\\end{equation}\nwhere the function  $a(\\beta)$ is discussed below.\nIn Fig.~\\ref{fig_zetam} is shown the multiplicative renormalization \nfactor $Z_m$ determined in the case  when $r_1$ is used to set the\nscale (see below).\n\\begin{figure}[tb]\n\\includegraphics*[width=1.\\columnwidth]\n{.\/zetam_plot.eps}\n\\caption{The multiplicative renormalization factor $Z_m$ in the case of \n$r_1$-scale. The renormalization point is $\\beta^*=6.54706$.}\n\\label{fig_zetam}\n\\end{figure}\nTo precisely  localize the peak in $\\chi_{l, \\rm ren}\/T^2$, a Lorentzian fit has \nbeen used. For illustrative purposes, \nin Fig.~\\ref{fig_chiral_light_suscep} we display our determination of the \npseudocritical couplings at $\\mu\/(\\pi T)=0.2i$ for all lattices\nconsidered in this work. The complete collection of results for\nthe disconnected susceptibility of the light quark chiral condensate obtained\nin this work is presented in the Appendix.\n\n\\begin{figure}[tb]\n\\includegraphics*[width=1.\\columnwidth]\n{.\/Figure_1.eps}\n\\caption{The real part of the renormalized susceptibility of the light quark \nchiral condensate over $T^2$ on the lattices $16^3\\times 6$,\n$32^3\\times 8$, $40^3\\times 10$ and $48^3\\times 12$ at $\\mu\/(\\pi T)=0.2i$. \nFull lines give the Lorentzian fits near the peaks. The temperature\nhas been determined from the $r_1$ scale.}\n\\label{fig_chiral_light_suscep}\n\\end{figure}\n\nTo get the ratios $T_c(\\mu)\/T_c(0)$, we fix the lattice spacing through the \nobservables $r_1$ and $f_K$, following the discussion in the Appendix~B of \nRef.~\\cite{Bazavov:2011nk}. \\\\\n For the  $r_1$ scale  the lattice spacing\nis given in terms of the $r_1$ parameter as:\n\\begin{equation}\n\\label{scale-r1}\n\\frac{a}{r_1}(\\beta)_{m_l=0.05m_s}=\n\\frac{c_0 f(\\beta)+c_2 (10\/\\beta) f^3(\\beta)}{\n1+d_2 (10\/\\beta) f^2(\\beta)} \\; ,\n\\end{equation}\nwith $c_0=44.06$, $c_2=272102$, $d_2=4281$, $r_1=0.3106(20)\\ {\\text{fm}}$. \\\\\nOn the other hand, in the case of  the  $f_K$ scale we have:\n \\begin{equation}\n\\label{scale-fk}\n a f_K(\\beta)_{m_l=0.05m_s}=\n\\frac{c_0^K f(\\beta)+c_2^K (10\/\\beta) f^3(\\beta)}{\n1+d_2^K (10\/\\beta) f^2(\\beta)} \\; ,\n\\end{equation}\nwith $c_0^K=7.66$, $c_2^K=32911$, $d_2^K=2388$, $r_1f_K \\simeq 0.1738$. \nIn Eqs.~(\\ref{scale-r1}) and (\\ref{scale-fk}), $f(\\beta)$ is the two-loop beta \nfunction,\n\\begin{equation}\n\\label{beta function}\nf(\\beta)=(b_0 (10\/\\beta))^{-b_1\/(2 b_0^2)} \\exp(-\\beta\/(20 b_0))\\;,\n\\end{equation}\n$b_0$ and $b_1$ being its universal coefficients. \\\\\n\nOur results are summarized in Table~\\ref{summary}.\nFor all lattice sizes but $24^3\\times6$ (where we have only one value of $\\mu$),\nthe behavior of $T_c(\\mu)\/T_c(0)$ can be nicely fitted with a linear function \nin $\\mu^2$,\n\\begin{equation}\n\\label{linearfit}\n\\frac{T_c(\\mu)}{T_c(0)} = 1 + R_q \\left(\\frac{i \\mu}{\\pi T_c(\\mu)}\\right)^2 \\;,\n\\end{equation}\nwhich gives us access to the curvature $R_q$ and, hence, to the curvature\nparameter $\\kappa=-R_q\/(9\\pi^2)$ introduced in Eq.~(\\ref{curv}). \nOn the $24^3\\times6$ lattice the linearity in $\\mu^2$ has been assumed to hold, \nin order to extract $R_q$ from the only available determination at \n$\\mu\/(\\pi T)=0.2i$.\n\n\\begin{table}[tb]\n\\setlength{\\tabcolsep}{0.9pc}\n\\centering\n\\caption[]{Summary of the values of the ratio $T_c(\\mu)\/T_c(0)$ for \nthe imaginary quark chemical potentials $\\mu$ considered in this work.\nThe data for $\\mu=0$ on the $24^3\\times6$, $32^3\\times68$ and $48^3\\times12$ \nlattices have been estimated from the disconnected chiral susceptibilities \nreported respectively on Tables~X, XI and XII of Ref.~\\cite{Bazavov:2011nk}.\nThe datum for $\\mu=0$ on the $40^3\\times10$ lattice has been estimated from \nthe disconnected chiral susceptibilities reported on Table~XI of \nRef.~\\cite{Bazavov:2014pvz}.\nThe values of $T_c(\\mu)\/T_c(0)$ evaluated fixing the lattice scale by\n$r_1$ and $f_K$ are reported, respectively, in the third and in the \nfourth column of the table.}\n\\begin{tabular}{cllll}\n\\hline\n\\hline\nlattice & $\\mu\/(\\pi T)$ & $T_c(\\mu)\/T_c(0)$  & $T_c(\\mu)\/T_c(0)$   \\\\\n        &               & \\ ($r_1$ scale)      & \\ ($f_K$ scale) \\\\\n\\hline\n16$^3\\times 6$ & 0.15$i$ & 1.038(13) & 1.043(14) \\\\\n               & 0.2$i$  & 1.063(15) & 1.070(15) \\\\\n               & 0.25$i$ & 1.085(16) & 1.095(18) \\\\\n\\hline\n24$^3\\times 6$ & 0.2$i$  & 1.061(9)  & 1.067(10) \\\\\n\\hline\n32$^3\\times 8$ & 0.15$i$ & 1.054(7)  & 1.059(8) \\\\\n               & 0.2$i$  & 1.066(10) & 1.071(11) \\\\\n               & 0.25$i$ & 1.117(10) & 1.126(10) \\\\\n\\hline\n40$^3\\times 10$ & 0.15$i$ & 1.023(23)  & 1.024(24) \\\\\n                & 0.2$i$  & 1.075(14)  & 1.079(15) \\\\\n                & 0.25$i$ & 1.102(15)  & 1.107(15) \\\\\n\\hline\n48$^3\\times 12$ & 0.15$i$  & 1.013(31)  & 1.013(33) \\\\\n                & 0.20$i$  & 1.051(14)  & 1.052(15) \\\\\n                & 0.25$i$  & 1.094(26)  & 1.097(25) \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{summary}\n\\end{table}\n\nFor the sake of the extrapolation to the continuum limit, in \nFig.~\\ref{curvature_vs_Nt} we report our determinations of $R_q$ on the \nlattices $24^3\\times 6$, $32^3\\times 8$, $40^3\\times 10$, $48^3\\times 12$,  \nand from the two different methods to set the scale, {\\it versus} $1\/L_t^2$.\n\n\nWithin our accuracy, cutoff effects on $R_q$ are negligible, so that a \nconstant fit works well over the whole region ($\\chi^2_r \\simeq 0.99$), thus \nincluding also the smallest $24^3\\times6$ lattice. Taking into account\nthe uncertainties due to the continuum limit extrapolation, \n\\begin{equation}\n\\label{curvature}\n\\kappa = 0.020(4) \\;.\n\\end{equation}\nOur estimate of the uncertainties for the curvature given in Eq.~(\\ref{curvature}) takes into account both the error in the fit minimization\nand the choice of the minimization function.\nWe stress, however, that if we exclude from the fit the value on the lattice\nwith the smallest $L_t$, {\\it i.e.} the rightmost points in \nFig.~\\ref{curvature_vs_Nt}, the extrapolation to the continuum becomes largely\nundetermined. Indeed, with the values of $R_q$ obtained in the present work\n(see Table~\\ref{Rq}), the fit with a constant is rather stable even if\n$L_t=6$ is excluded, but the fit with a linear function in $1\/L_t^2$ in the \nlatter case gives a much smaller value of the curvature $\\kappa$, though\nwith a large uncertainty (see Table~\\ref{fit}).\n\\begin{table*}[tb]\n\\setlength{\\tabcolsep}{0.9pc}\n\\centering\n\\caption[]{Summary of determinations of the curvature $R_q$ for all values of \n$L_t$ considered in this work, and from the two different methods to set the \nscale.}\n\\begin{tabular}{lcccc}\n\\hline\n\\hline\n$L_t$ & $6$ & $8$  & $10$  & $12$\\\\\n\\hline\n$R_q$ ($r_1$ scale) & $-1.466(306)$ & $-1.902(192)$ & $-1.685(294)$ \n& $-1.337(410)$ \\\\\n$\\kappa$ ($r_1$ scale) & $0.017(3)$ &  $0.021(2) $  & $0.019(3)$ & $0.015(5)$ \\\\\n\\hline\n$R_q$ ($f_K$ scale) & $-1.646(336)$ & $-2.041(206)$ & $-1.769(309)$ \n& $-1.394(415)$ \\\\\n$\\kappa$ ($r_K$ scale) & $0.019(4)$ & $0.023(2)$ & $0.020(3)$ & $0.016(5)$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{Rq}\n\\end{table*}\n\\begin{table*}[tb]\n\\setlength{\\tabcolsep}{1.5pc}\n\\centering\n\\caption[]{Summary of the fit of the curvature $\\kappa$ with the \nfunction $\\kappa(L_t) = \\kappa + A\/L_t^2$, with $A$ taken equal to zero or left\nfree. The first column specifies the values of $L_t$ included in the fit,\nthe last column the reduced $\\chi^2$. The uncertainties on the fit parameters\nare obtained with 70\\% confidence level.}\n\\begin{tabular}{lllc}\n\\hline\n\\hline\n$L_t$ included & \\hspace{0.6cm} $\\kappa$ & \\hspace{0.6cm} $A$ & $\\chi^2_{\\rm r}$ \n\\\\\n\\hline\n6, 8, 10, 12 & 0.01991(114) & 0.            & 0.76 \\\\ \n6, 8, 10, 12 & 0.02014(449) & \\hspace{-0.35cm} $-$0.015(259)  & 0.89 \\\\ \n\\hline\n8, 10, 12 & 0.02048(127) & 0.         & 0.80 \\\\ \n8, 10, 12 & 0.01182(748) & 0.669(559) & 0.14 \\\\ \n\\hline\n\\hline\n\\end{tabular}\n\\label{fit}\n\\end{table*}\n\\begin{figure}[tb]\n\\includegraphics*[width=1.\\columnwidth]\n{.\/Figure_2.eps}\n\\caption{Determinations of the curvature $R_q$ on the lattices $24^3\\times 6$, \n$32^3 \\times 8$, $40^3 \\times 10$, $48^3 \\times 12$, and from the two different \nmethods to set the scale, {\\it versus} $1\/L_t^2$. Data points related with \nthe $f_K$ scale setting have been slightly shifted along the horizontal axis \nfor better readability. The dashed horizontal line gives the result of \nthe fit to all data with a constant; the solid horizontal lines indicate the \nuncertainty on this constant (95\\% confidence level).}\n\n\\label{curvature_vs_Nt}\n\\end{figure}\n\n\\section{Conclusions and discussion}\n\nWe have studied QCD with $n_f = 2+1$ flavors discretized in the HISQ\/tree \nrooted staggered fermion formulation and in the presence of an imaginary \nbaryon chemical potential, with a physical strange quark mass and a \nlight-to-strange mass ratio $m_l\/m_s = 1\/20$, and $\\mu=\\mu_l=\\mu_s$.\n\nWe have estimated, by the method of analytic continuation, the continuum limit\nof the curvature of the pseudocritical line in the temperature - baryon \nchemical potential, defined in Eq.~(\\ref{curv}). The observable adopted\nto identify, for each fixed $\\mu$, the crossover temperature has been\nthe disconnected part of the renormalized susceptibility of the light quark \nchiral condensate, in units of the squared temperature. This observable is\nconvenient for many reasons: it dominates, for small enough quark masses, the\nwhole light chiral susceptibility, which would be much harder to implement;\nit undergoes only a multiplicative renormalization; it is strongly sensitive \nto the transition, thus allowing precise determinations of the pseudocritical \ntemperatures.\n\nWe have found that, within the accuracy of our determinations, cutoff effects\non the curvature are negligible already on the lattice with temporal size\n$L_t=6$. Our determination of the curvature parameter, $\\kappa$=0.020(4),\nis indeed compatible with the value quoted in our previous \npaper~\\cite{Cea:2014xva}, $\\kappa$=0.018(4), without the extrapolation to\nthe continuum.\n\n\\begin{figure}[tb]\n\\includegraphics*[height=0.295\\textheight,width=1.\\columnwidth]\n{.\/Figure_3.eps}\n\\caption{$T_c(\\mu_B)$ versus $\\mu_B$ (units in GeV). \nExperimental values of $T_c(\\mu_B)$ are taken\nfrom Fig.~1 of Ref.~\\cite{Cleymans:2005xv} (black circles) and from\nFig.~3 of Ref.~\\cite{Alba:2014eba} (green triangles),  for the \nstandard  hadronization model and for the  susceptibilities of conserved charges respectively. \nThe dashed line is a parametrization  corresponding to\n$T_c(\\mu_B) = T_c(0) - b \\mu_B^2$ with $T_c(0)=0.154(9)\\, \\text{GeV}$ and \n$b=0.128(25)\\,{\\text{GeV}}^{-1}$ . \nThe solid lines represent the corresponding  error band.}\n\\label{Tcmu}\n\\end{figure}\n\nIt is interesting to extrapolate the critical line as determined in this work \nto the region of real baryon density and compare it with the freeze-out\ncurves resulting from a few phenomenological analyses of relativistic\nheavy-ion collisions. This is done in Fig.~\\ref{Tcmu}, where we report\ntwo different estimates. The first is from the analysis of \nRef.~\\cite{Cleymans:2005xv}, based on the standard statistical hadronization \nmodel, where the freeze-out curve is parametrized as \n\\begin{equation}\n\\label{cleymans}\nT_c(\\mu_B) = a - b \\mu_B^2 - c \\mu_B^4\\;,\n\\end{equation}\nwith $a=0.166(2) \\ {\\text{GeV}}$, $b=0.139(16) \\ {\\text{GeV}}^{-1}$, \nand $c=0.053(21) \\ {\\text{GeV}}^{-3}$.\nThe second estimate is from Ref.~\\cite{Alba:2014eba} and is based on the analysis\nof susceptibilities of the (conserved) baryon and electric charges.\nIn fact, our critical line is in nice agreement with all the freeze-out points of\nRefs.~\\cite{Cleymans:2005xv,Alba:2014eba}. In particular, using our estimate\nof the curvature,  Eq.~(\\ref{curvature}), we get   \n$b=0.128(25) \\ {\\text{GeV}}^{-1}$, in very good agreement with the\nquoted phenomenological  value. The significance of the comparison\npresented in Fig.~\\ref{Tcmu}, with special reference to the question\nwhether the pseudocritical line lies indeed above the freeze-out curve, can be \nincreased at the (nonnegligible) price of reducing the uncertainties on \n$T_c(0)$ and on $\\kappa$. \\\\\n\nSome {\\it caveats} are in order here. We do not expect our critical line \nto be reliable too far from $\\mu=0$: as a rule of thumb, we can trust it up \nto real quark chemical potentials of the same order of the modulus\nof the largest imaginary chemical potential included in the \nfit~(\\ref{linearfit}), {\\it i.e.} $|\\mu|\/(\\pi T)=0.25$. This translates\nto real baryon chemical potentials in the region $\\mu_B \\lesssim 0.4\\,\\text{GeV}$.\nMoreover, the effect of taking $\\mu_s=\\mu_l$ instead of $\\mu_s < \\mu_l$\nshould become visible on the shape of the critical line as we move away\nfrom $\\mu=0$ in the region of real baryon densities, thus reducing\nfurther the region of reliability of our critical line. So, from a prudential\npoint of view, the agreement shown in Fig.~\\ref{Tcmu} could be considered\nthe fortunate combination of different kinds of systematic effects. We cannot\nhowever exclude the possibility that the message from Fig.~\\ref{Tcmu} is to be \ninterpreted in positive sense, {\\it i.e.} the setup we adopted and the\nobservable we considered may catch better some features of the crossover\ntransition, thus explaining the nice comparison with freeze-out data.\nIndeed,  our  result for the  continuum extrapolation of the curvature $\\kappa$ is\nin fair agreements with the recent estimates in  Ref.~\\cite{Bonati:2015bha}, where\nboth setup  $\\mu_s=\\mu_l$ and $\\mu_s=0$ were adopted,\nand Ref.~\\cite{ Bellwied:2015rza},  where the strangeness neutral trajectories\nwere determined from lattice simulations  by imposing $\\langle n_S \\rangle=0$.\n\n\\section*{Acknowledgments}\nThis work was in part based on the MILC collaboration's public lattice gauge \ntheory code. See {\\url{http:\/\/physics.utah.edu\/~detar\/milc.html}}.\nThis work has been partially supported by the INFN SUMA project.\nSimulations have been performed on BlueGene\/Q at CINECA \n(Projects Iscra-B\/EXQCD and INF14\\_npqcd), on the BC$^2$S cluster in Bari and on the CSNIV \nZefiro cluster in Pisa.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $I\\subset k[x_0,\\dots,x_N]$ be a homogeneous ideal.  For $r\\geq 0$, the $r$-th symbolic power of $I$ is defined to be \\[I^{(r)}=\\bigcap_{p\\in\\mbox{Ass}(R\/I)}(I^rR_p\\cap R).\\]  Symbolic powers of ideals are interesting for a number of reasons, not least of which is that, for a radical ideal $I$,the $r$-th symbolic power $I^{(r)}$ is the ideal of all polynomials vanishing to order at least $r$ on $V(I)$ (by the Zariski-Nagata theorem).\n\nContainment relationships between symbolic and ordinary powers are a source of great interest. As an immediate consequence of the definition, $I^r\\subseteq I^{(r)}$ for all $r$.  However, the other type of containment, namely that of a symbolic power in an ordinary power is much harder to pin down.  It has been proved by Ein-Lazarsfeld-Smith \\cite{ELS} and Hochster-Huneke \\cite{HH} that $I^{(m)}\\subseteq I^r$ for all $m\\geq Nr$, but as of yet there are no examples in which this bound is sharp.\\\\\n\nIt was conjectured by Harbourne in \\cite[Conjecture 8..4.3]{PSC} (and later in  \\cite[Conjecture 4.1.1]{HaHu} in the case $e=N-1$) that $I^{(m)}\\subseteq I^r$ for all $m\\geq er-(e-1)$, where $e$ is the codimension of $V(I)$.  While this conjecture holds in a number of important cases, some counterexamples have also been found.  Notably, the main counterexamples come from singular points of hyperplane arrangements \\cite{BNAL}. One particular family is known in the  literature under the name of Fermat configurations of points cf. $\\mathbb P^2$ \\cite{BNAL, MS}. These have been recently generalized to Fermat-like configurations of lines in $\\mathbb P^3$ in \\cite{MS}. The $\\rm{Ceva}(n)$ arrangement of hyperplanes in $\\mathbb{P}^N$ is defined by the linear factors of \\[F_{N,n}=\\prod_{0\\leq i<j\\leq N}(x_i^n-x_j^n),\\] where $n\\geq 3$ is an integer.\n\nIn \\cite{DST} Dumnicki, Szemberg, and Tutaj-Gasinska showed that, for\nthe ideal $I_{2,3}$ corresponding to all triple intersection points of\nthe lines defined by linear factors of $F_{2,3}$ in $\\mathbb{P}^2$,\n$F_{2,3}\\not\\in I_{2,3}^2$, but $F_{2,3}\\in I_{2,3}^{(3)}$.  This was the first\ncounterexample to the above mentioned conjecture. Later, in \\cite{MS} Malara and\nSzpond generalized this construction to $\\mathbb{P}^3$, by showing\nthat for the ideal $I_{3,n}$, corresponding to all triple intersection\nlines of the planes defined by the linear factors of $F_{3,n}$,\n$F_{3,n}\\not\\in I_{3,n}^2$, but $F_{3,n}\\in I_{3,n}^{(3)}$.  In the following, the\nconstruction of counterexamples to $I^{(3)}\\subseteq I_2$ from\nFermat arrangements is generalized to $\\mathbb{P}^N$ for all $N\\geq\n2$.\n\n\\section{Main result}\nLet $n\\in\\mathbb{N}$.  Let $k$ be a field\nwhich contains a primitive $n$-th root of unity, $\\varepsilon$.  For each $N\\in\\mathbb{N}$, let $S_N:=k[x_0,x_1,\\dots,x_N]$, and define \\[F_{N,n}:=\\prod_{0\\leq i<j\\leq N}(x_i^n-x_j^n).\\]\nLet \\[C_{N,n}:=\\bigcap_{0\\leq i<j\\leq N}(x_i,x_j)\\]\\[J_{N,n}:=\\bigcap_{\\substack{0\\leq i<j<l\\leq N \\\\ 0\\leq a,b< n}}(x_i-\\varepsilon^a x_j, x_i-\\varepsilon^b x_l),\\]\nand let\n\\[I_{N,n}:=J_{N,n}\\cap C_{N,n}.\\]\nWe show in Lemma \\ref{lem1} that $I_{N,n}$ is the ideal of the $N-2$ dimensional flats arising from triple intersection of hyperplanes corresponding to linear factors of $F_{N,n}$.\n\n\\begin{theorem}\\label{main}\nFor all $N\\geq 2$, $I_{N,n}^{(3)}\\not\\subseteq I_{N,n}^2$.\n\\end{theorem}\nBefore we can prove this, we must introduce a few lemmas.\n\n\\begin{lemma}\\label{lem1}\nThe ideal $I_{N,n}$ defined above defines the union of all the $N-2$ dimensional linear spaces that are intersections of at least three hyperplanes corresponding to linear factors of $F_{N,n}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $0\\leq a,b<n$, and let $0\\leq i<j\\leq l\\leq N$.  Then \\[(x_i-\\varepsilon^ax_j,x_i-\\varepsilon^bx_l)=(x_i-\\varepsilon^ax_j,x_i-\\varepsilon^bx_l,x_j-\\varepsilon^{b-a}x_l)\\] defines the intersection of the three hyperplanes corresponding to $(x_i-\\varepsilon^ax_j)$, $(x_i-\\varepsilon^bx_l)$, and $(x_j-\\varepsilon^{b-a}x_l)$. \nFurthermore $$(x_i,x_j)=(x_i-\\varepsilon^ax_j : a=0,1,\\dots, n),$$ so $(x_i,x_j)$ defines the intersection of  $n$ hyperplanes corresponding to linear factors of $x_i^n-x_j^n$. \n\nIt remains to be seen that all $N-2$ dimensional linear spaces that arise as intersections of at least three hyperplanes corresponding to linear factors of $F_{N,n}$ are accounted for above. Let $L$ be the ideal defining such a linear space. then $L$ contains three linearly dependent binomials of the form $x_i-\\varepsilon^ax_j, x_k-\\varepsilon^bx_l, x_u-\\varepsilon^cx_v$. Without loss of generality (after multiplication by appropriate powers of $\\varepsilon$) this yields $i=k$ and $\\{j,l\\}=\\{u,v\\}$. If $j\\neq l$ then $L=(x_i-\\varepsilon^ax_j, x_i-\\varepsilon^bx_l)$ is one of the primes appearing in the decomposition of $J_{N,n}$ and if $j=l$ then $L=(x_i,x_j)$ is one of the primes appearing in the decomposition of $C_{N,n}$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem2}\nLet $R$ and $S$ be finitely generated graded-local Noetherian rings. Let $m$ be the homogeneous maximal ideal of $R$. Let  $I\\subset R$ be a homogeneous ideal, and suppose  $F\\not\\in I^r$ for some $r\\in\\mathbb{N}$.  Let $J\\subset S$ be an ideal, and let $\\pi:S\\rightarrow R$ be a (not necessarily homogeneous) ring homomorphism such that $\\pi(J)\\subseteq I$.\nIf  $G\\in R$ is such that $\\pi(G)=Fg$, where $g\\not\\in m$, then $G\\not\\in J^r$.\n\\end{lemma}\n\\begin{proof}\nSuppose by way of contradiction that such a $G$ exists and $G\\in J^r$.  Then \\[Fg=\\pi(G)\\in\\pi(J^r)\\subseteq(\\pi(J))^r\\subseteq I^r.\\]\nThus $Fg\\in I^r$.\nLet $I^r=Q_1\\cap\\dots\\cap Q_t$ be a primary decomposition.  Then, $Fg\\in Q_i$ for each $i\\in\\{1,\\dots,t\\}$. Suppose that, for some $i$, $F\\not\\in Q_i$.  Then $g^s\\in Q_i$ for some $s\\in\\mathbb{N}$.  However, $I^r$ is a homogeneous ideal, so $Q_i\\subseteq m$ for all $i\\in\\{1,2,\\dots, t\\}$.  But since $g\\not\\in m$, we know that $g^s\\not\\in m$, a contradiction.  Thus $F\\in Q_i$ for all $i$, which yields $F\\in I^r$, a contradiction.\n\\end{proof}\n\n\nThis lemma  allows us to construct an inductive argument for the main theorem.\n\n\\begin{proof}[Proof of Theorem \\ref{main}]\nBy Lemma \\ref{lem1} $F_{N,n}$ must vanish to order 3 or greater on each of the linear spaces whose union is $V(I_{N,n})$,  thus $F_{N,n}\\in I_{N,n}^{(3)}$. To finish the proof, it suffices to show that for all $N\\geq 2$, $F_{N,n}\\not\\in I_{N,n}^2$.\n\nWe argue by induction on $N$.  For $N=2$, this is proved in the paper of Dumnicki, Szemberg, and Tutaj-Gasinska \\cite{DST}.\n\nFor $N>3$, assume that $F_{N-1,n}\\not\\in I_{N-1,n}^2$ and consider the evaluation homomorphism $\\pi:S_N\\rightarrow S_{N-1}$ defined by $\\pi(x_N)=1$ and $\\pi(x_i)=x_i$ for $i \\leq N-1$.  Then:\n\\[\\pi(I_{N,n})\\subseteq C_{N-1,n}\\cap\\left(\\bigcap_{0\\leq i<N}(x_i,1)\\right)\\cap J_{N-1,n}\\cap \\left(\\bigcap_{\\substack{0\\leq i<j<N\\\\0\\leq a,b<n}}(x_i-\\varepsilon^a,x_i-\\varepsilon^b)\\right)\\subseteq I_{N-1,n}.\\]\n\nWe note that $\\pi(F_{N,n})=F_{N-1,n}g$ where $g=\\prod_{0\\leq i<N}(x_i^n-1)$.  Since $g\\not\\in (x_0,\\dots,x_{N-1})$, we conclude by Lemma \\ref{lem2} that $F_{N,n}\\not\\in I_{N,n}^2$.\n\\end{proof}\n\n\\section{Concluding Remarks}\nAnother proof for the noncontainment noncontainment  $I_{N,n}^{(3)}\\subseteq I_{N,n}^2$ has been found by Grzegorz Malara and Justinya Szpond and can bee seen in their upcoming paper \\cite{MS2}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\nQuantum-Chromo-Dynamics (QCD) has become increasingly important to astrophysics during the last few decades. QCD is becoming especially ubiquitous in the study of compact objects, most specifically in the study of neutron stars (NSs). \nThe hypothesis that quark matter (specifically, matter  made of up, down, and strange quarks) could be the most stable state of matter in the universe \\cite{PhysRevD.4.1601,PhysRevD.30.272} has revealed the possibility for the existence of compact objects  made entirely of deconfined quark matter namely, quark stars.  The extremely dense cores of NSs could allow for the possibility of QCD phase transitions  \\cite{cabibbo1975exponential} leading to the formation of these quark stars.\n\n If the phase transition is an exothermic process, the energy could manifest itself observationally as an energetic event that could have drastic consequences to high-energy astrophysics.  The exothermic release of energy by a QCD phase transition in a NS, resulting in a quark star, may lead to a highly explosive astrophysical phenomenon known as the Quark Nova (QN) \\cite{ouyed2002quark}. Calculations have revealed that the energy released could be as high as $10^{53}$ ergs \\cite{ouyed2002quark}. Such excesses of energy could account for the mechanism behind poorly understood phenomena such as super-luminous supernovae \\cite{ouyed2013sn}. Furthermore, the energetic ejecta of such events could trigger nuclear reactions leading to heavy element nucleo-synthesis \\cite{jaikumar2007nucleosynthesis}, p-process element synthesis \\cite{ouyed2014unified}, and spallation processes \\cite{ouyed2014spallation,ouyedprl2011}. Thus proving that the transition from a NS to a quark star is an explosive one will have transformative impact to astrophysics.\n\n\nHowever,  making use of QCD concepts  in the context of compact objects faces challenges. For one, the thermodynamic regimes appropriate for QCD in astrophysics are non-perturbative. For example, equations of states (EoSs) of nuclear\/quark matter  that include QCD physics in  studies of compact stars cannot be constructed from first principles. Instead phenomenological EoSs are used.  Although these equations have been relatively successful in modelling certain aspects of compact stars,  different EoSs models have important distinctions that would manifest in different dynamics  in astrophysics, in particular in the interior of NSs. \n\nHow do we choose amongst all the possible phenomenological equations of state? Perhaps astrophysics might be of help. Astrophysical observations that correlate with specific EoS models could constraint the correct equations of state. This in turn, would have consequences not only in astrophysics but in particle physics in general, for it would shed light onto the true nature of the non-perturbative regime of QCD. \n\nPerhaps, the first step in finding out the true equation of state would be in simulating the micro-physics behind the phase transition of a NS into a quark star. Such a simulation would cover the parameter space related to disparate EoS and different initial conditions. In doing so, one could accomplish two things: (i) simulate astrophysical signatures that could help discriminate between the parameters, and ultimately the equations of state, and (ii) shed light  on whether a QCD phase transition inside a NS could reach an explosive regime or the phase transition simmers quietly and slowly. \n\nHere we describe the Burn-UD code which has been built and used by the Quark-Nova group to investigate the physics of this transition.\nBurn-UD \\cite{niebergal2010numerical} is a hydrodynamic code that simulates the micro-physics of the hadronic-quark matter phase transition. The work in general explores the effects of different EoSs states on the transition,  the importance of the initial\nconditions (prior to the transition), and the hydrodynamics behind the combustion process.  Other points are organized as\nfollows: Section \\ref{model} lays out the assumptions used in our model. Section 3 is a very brief introduction to the Burn-UD code\/software\n which is presented as a unifying platform for the QCD community  exploring the phases of Quark Matter and astrophysicists working on Compact Stars. Some preliminary results of simulations run using Burn-UD are discussed and examined in  Section \\ref{results}. Section 5 looks at future work and finally   we  presents our conclusions in Section 6.\n\n\\section{Model Assumptions}\\label{model}\n\nBefore we describe Burn-UD, we describe\n our assumptions concerning the  the stability of {\\it uds} matter, introduction of the {\\it s}-quark seeding, the fluidity of quarks, the reaction rates of the weak interaction, and the EoSs used.  \n\n\\subsection{Strange Matter Hypothesis}\n\nThe strange matter hypothesis states that matter made  of up, down and strange quarks could  be the most stable\nstate of matter \\cite{PhysRevD.30.272}.  If true, then it would be interesting to investigate the\n feasibility of the  phase transition from hadronic to {\\it uds} matter in the universe.   If this transition\n is energetic and if it leads to the formation of stable {\\it uds} stars, the implications to astrophysics are undeniable.\n Here, we assume that the strange matter hypothesis is true. This is dynamically important because  it implies that once a critical amount of strange quarks are seeded in the core of a NS, the whole system will try to lower its energy by converting to the more stable strange matter. \n\n\n\n\n\\subsection{Seeding of Strange Matter}\n\n\nThere are numerous theories of how a {\\it s}-quark  seed could be introduced to the core of a NS. Seeding has to appear in such a way that a critical number of strange quarks are created instantaneously.\nIn our model, we assume an initial seed. The cause of the seeding itself is left as an open question. Numerous studies have pushed forward different seeding models, such as clustering of hyperons in the core, and neutrino sparking \\cite{alcock1986strange}. Another model explored for seeding is the creation of strange quarks through DM self-annihilation \\cite{perez2013,perez2010dark}. This  seeding mechanism \nis further discussed in Section \\ref{dmseed}.\n\n \n\\subsection{Asymptotic Freedom and the Fluidity of Quarks}\n\n\nBecause our model instantiates the combustion process in the core of the NS, we already assume a critical density for asymptotic freedom, i.e. hadrons in the core have deconfined into a fluid made of quarks. We therefore model the combustion of a fluid consisting  (ud)\nmatter  into a fluid made of  (uds) matter. Moreover, the timescale of the strong force, which governs quark confinement, is much shorter than the weak interaction, which is responsible of turning ud matter into uds matter. Thus, it is reasonable to assume that hadrons at the vicinity of the flame have already dissolved into their constituent up and down quarks \\cite{niebergal2010numerical}.\n\nThe conversion process of ud into uds matter is modelled as a combustion process; the ud matter acts as a fuel that is burnt into a uds ash. However, usually combustion studies usually deal with a pre-mixed fuel, i.e. the fuel is mixed with the oxidant. In such a pre-mixed combustion, the speed of the flame is dictated by the thermal conductivity, and the hydrodynamics (diffusion and advection) become unimportant compared to the conduction of heat across the medium. \n\nHowever, in the case of the combustion of ud matter into uds matter, the oxidant (strange quarks instantiated in the core of the NS) is not pre-mixed. Thus, combustion is only activated when the strange matter oxidant diffuses into the ud fuel. When the strange quarks diffuse into the ud fuel, the up, down and strange quark mix will attempt to achieve chemical equilibrium, converting up and down quarks into strange quarks, releasing heat in the process.\n\nBecause the dynamics of combustion are dominated by a hydrodynamic process, i.e. diffusion of the strange quark oxidant into ud fuel, and not by heat transfer, it is necessary to solve numerically the full system of hydrodynamic equations. We solve the 1-D hydrodynamical equations:\n\n\\begin{equation}\n\\frac{\\partial U}{\\partial t}= - \\nabla F (U) + S(U).\n\\end{equation}\n\nwhere $ \\nabla F (U)$ are the advection components, and $S(U)$ the source components. We refer the interested reader  to \\cite{niebergal2010numerical} for the details of the hydrodynamics of the ud to uds combustion. \n\n\\subsection{Reactions}\\label{rates}\n\nOnce a strange matter seed is introduced, the following reactions will try to achieve equilibrium. This creates a net surplus of strange quarks, resulting in the conversion of the ud fluid into uds matter:\n\n\\begin{align}\n d &\\rightarrow u + e^- + \\overline{\\nu}_e \n\\\\ u + e^- &\\rightarrow d + \\nu_e\n\\\\ s &\\rightarrow u + e^{-} + \\overline{\\nu}_e\n\\\\ u + e^- &\\rightarrow s + \\overline{\\nu_e}\n\\\\ u + d &\\rightarrow u + s\n\\end{align}\n\nWe take the rates for these reactions from \\cite{anand1997burning}. See also \\cite{niebergal2010numerical}.\n\n\n \n\n\n\\subsection{Neutrinos}\\label{neutrino}\n\nThe {\\it uds} mixture  will attempt to reach equilibrium through the weak interaction reactions. As such, the reactions will release neutrinos which  act as an energy sink that lowers the temperature behind the burning interface (see reactions above). The sink is parameterized and  is proportional to the neutrino mean free path. Se also \\cite{niebergal2010numerical}. \n\n\n\\subsection{Equations of State and Color Superconductivity}\n  \n\n We assume that both the ud and uds matter near the burning interface are deconfined. In its early version, Burn-UD\n  utilizes the MIT bag model \\cite{johnson1975bag} for both sides of the interface. The EoS used is:\n\n\\begin{align} \\label{miteos}\nP&= \\frac{13}{36}\\pi^2 T^4 + \\frac{T^2}{2}\\sum_i \\mu_i^2+ \\frac{1}{4 \\pi ^2}\\sum_i \\mu_i^4-B \n\\\\ h &= 4(P+B)\n\\end{align}\n\nWhere $P$ is pressure, $T$ is temperature, $\\mu_i$ is the chemical potential for quark specie $i$, $h$ is enthalpy, and $B$ is the Bag constant. \n\nIn addition, as the {\\it uds} phase (the ashes in our simulation) cools we assume that it becomes superconducting and\nenters a  Color-Flavor-Locked (CFL) phase \\cite{alford1999color}. \nWe model the CFL phase through the MIT Bag Model EoS with the following prescription \\cite{paulucci2008color}.\n\n\\begin{align}\n\\\\ \\Delta_0 &= 100 \\text{ MeV}  \n\\\\ T_c &= 2^{1\/3}\\times 0.57 \\Delta_0\n\\\\ \\Delta &=2^{-1\/3} \\Delta_0 \\sqrt{1- (T\/T_c)^2}\n\\\\ P &=P_{\\text{non-CFL}} + \\frac{3 \\Delta ^2 \\mu^2}{\\pi^2} \\label{eoscfl}\n\\\\ h &= 4(P+B) - 2\\times \\frac{3 \\Delta ^2 \\mu^2}{\\pi^2} \n\\end{align}\n\n\\noindent where $T, P, h$, are temperature, pressure, enthalpy. $T_c$ is the critical temperature, where $T<T_c$ is the temperature for phase transition to CFL.  $ \\Delta$ is the CFL energy gap,   and $ \\Delta_0$ is the energy gap for CFL at $T=0$.   $P_{\\text{non-CFL}}$ is defined in equation \\eqref{miteos}. \n\nThus, the CFL state stiffens the EoS by adding a gap term $ \\frac{3 \\Delta ^2 \\mu^2}{\\pi^2} $, and therefore effectively lowers the energy density of the system. This in turn releases energy that energizes the burning front. \n\nPrevious studies, e.g. \\cite{paulucci2008color}, usually model CFL in equilibrium requiring $\\mu_s=\\mu_u=\\mu_d$ for the massless quark case. However we model the moment of the phase transition as a non-equilibrated system where $\\mu_s <\\mu_u$. Thus we add the following condition that dictates when the non-CFL to CFL transition happens:\n\\begin{align}\n \\mu_u-\\mu_s &< \\Delta \\label{mucond} \\ ,\n\\\\ T &< T_c \\label{tccond} \\ .\n\\end{align}\n\nThus, in a non-equilibrium situation,  $ \\frac{3 \\Delta ^2 \\mu^2}{\\pi^2} $ cannot simply be absorbed into an effective bag constant, because the CFL transition only appears when the conditions in equations \\eqref{mucond} and \\eqref{tccond} are met. \n\nCFL is triggered as follows: first the instantiated strange quark seed interacts with the ud fuel, raising $\\mu_s$  but lowering $\\mu_u$ and $\\mu_d$. After some time at a point near the interface, the chemical potentials will almost equal each other, and condition \\eqref{mucond} is met. At this point the EoS stiffens because the component   $ \\frac{3 \\Delta ^2 \\mu^2}{\\pi^2} $  is added to the pressure equation, which creates a large pressure differential across the interface that in turn, triggers an advection wave travelling from the uds ash to the ud fuel. \n\n\n\n\n\\section{Burn-UD} \\label{code}\n\nBurn-UD is a hydrodynamic combustion code used to model the phase transition of hadronic to quark matter. The most recent version of the code is written in java (the original was written in Fortran 95, \\cite{niebergal2010numerical}) and will be publicly available soon on all major platforms (windows, linux, Macintosh).\n\n\nBeyond the micro-physics modelled by Burn-UD that we described in the preceding sections, what makes Burn-UD distinct is that it has a user-friendly graphical user interface (GUI). The user can modify in a transparent and user-friendly way many  of the initial conditions, including initial temperature, distribution of the strange seed and initial total density. Fig. 1 shows a screenshot of the Burn-UD GUI in its most recent version. Furthermore, Burn-UD's GUI allows the user to change the EoS and turn on and off specific hydrodynamical effects such as advection or diffusion. \n\n Burn-UD also contains graphic modules that plot in real time the different variables of the hydrodynamic simulation. The user can select to store the data for specific time intervals in order to seamlessly create an animation of the results. It also gives the user the ability to store the data in ascii files. \n\n\t\\begin{figure}\n \\centering\n         \\includegraphics[width=0.9\\textwidth]{burnuddesk.pdf}\n    \\includegraphics[width=0.9\\textwidth]{burnudgraph.pdf} \n  \\caption{ \\label{code} Screenshots of the Burn-UD GUI. Upper window is the dashboard and the panel in the left is where the input parameters can be changed. Lower window is the graphing module where the chemical potentials of the strange, up, and down quarks are being plotted.}\n\\end{figure}\n\n\n\n \n \n\\section{Results}\\label{results}\n\n\\subsection{Preliminary Results}\n\nEarly results using the first version of Burn-UD \\cite{niebergal2010numerical} showed: (i) typical speeds for the burning front are 0.002c - 0.04c where c is the speed of light. (ii) Cooling due to escaped neutrinos slows down the burning front, and in some cases can halt it completely. \n\nThese results hint towards an explosive regime. The details are explored in \\cite{niebergal2010numerical} but can be summarized as follow: Neutrino cooling slows down the burning front unevenly across a multidimensional surface, which  wrinkles the interface. This wrinkling will be unstable, forming a deleptonization instability. This instability could either: (i) collapse the hadronic layers into the quark core and thus trigger a core collapse explosion or (ii) accelerate the burning front further and cause it to achieve supersonic speeds which would lead the system into a defragation to detonation regime, triggering an explosion.\n\n\\subsection{The CFL and Burning Speeds}\n\nFor the case of quark matter at a particular temperature and chemical potential, the CFL EoS has a lower energy density than the non-superconducting Bag equation of state used in our previous studies. Thus by simple energy conservation considerations, the transition of non-superconducting uds matter to CFL matter would release energy. This energy excess could also aid in leading the phase transition to an explosion.  Another important aspect of a CFL phase transition that could accelerate the burning is the release of a photon fireball \\cite{ouyed2005fireballs}. Photons have momentum and thus, they will transfer momentum to the external hadronic layers that would in turn accelerate the burning.  \n\n\nSeveral tests for the CFL case have been run, and the results seem promising. Figure \\ref{fig:cfl} shows the \nposition of the burning front versus time in nano-seconds. \nCFL stiffens the EoS which in turn creates an advection wave moving towards the unburnt ud fuel, which accelerates the burning front at even higher speeds.  The increase in speed at time $\\sim 3$ nano-seconds in Figure \\ref{fig:cfl} occurs when the\ncondition $(\\mu_{\\rm u} - \\mu_{\\rm s})< \\Delta$  (see eq. (15)) is met. Furthermore, in a multidimensional situation,  the onset of CFL transition could further exacerbate the deleptonization wrinkling instability by further increasing the pressure gradient across the interface, by the simple fact that a CFL transition would increase the pressure in the cooler regions of the interface. Tests for the CFL case are ongoing to verify these claims.  \n\n\n\n\t \n\n\n\n \\begin{figure}\n \\centering\n         \\includegraphics[width=1.0\\textwidth]{corepos.pdf}\n  \\caption{ \\label{code} Position vs Time of the flame for a non-CFL (eq. \\eqref{miteos}) EoS and a CFL (eq. \\eqref{eoscfl}) EoS. Position is measured as the distance of the flame from the center of the NS.   The CFL flame is much faster than the non-CFL case. }\n  \t\\label{fig:cfl}\n\\end{figure}\n\n\n\\section{Future Considerations} \\label{future}\n\n\\subsection{Dark Matter Annihilation and Strange Quark Seeding}\\label{dmseed}\n\nA possible source of the strange quark seed may be dark matter (DM) annihilation \\cite{perez2013,perez2010dark}. The distribution of DM  in galaxies allows for it to be accreted gravitationally onto a dense and massive astrophysical object such as a NS \\cite{dmcapture}.  Figure \\ref{dmaccretion} shows a schematic diagram representing the stages of DM accretion and self-annihilation to produce strange quarks.\nOne scenario explores the self-annihilation of  WIMPs (weakly interacting massive particles) inside NSs. They \n  inject the NS with energetic photons that in turn could deconfine hadrons into bubbles of ud quark plasma. Then non-leptonic weak processes could create strange quarks from these bubbles, giving rise to uds seeds \\cite{perez2013,perez2010dark}. \n   Whether the amount of {\\it s}-quark seeding is enough (i.e. reaches a critical mass) to trigger the hadronic-to-quark-matter phase\n   transition remains to be confirmed making Burn-UD an ideal tool\/software for such investigations.\n \n\n \n \n\\begin{figure}\n\\centering\n\\includegraphics[width= 6 in]{DMseeding.pdf}\n\\caption{Four main stages of DM capture by NSs. (I) Shows the DM energy loss by scattering with nucleons at high density inside the core of the NSs. (II) After losing energy, the DM particles are accreted in the core of the star and can't escape the gravitational pull of the star. (III) The density of DM has increased enough so that self-annihilation of DM happens in the core of the star. One of the possible products of this self-annihilation is the strange quark. Finally in (IV) the density of strange quarks increases, effectively seeding the hadronic-to-quark-matter phase transition process. }\n\\label{dmaccretion}\n\\end{figure}\n\n\n\n\n\nBurn-UD is flexible enough to be used as a tool to study the strange quark seeding produced by DM annihilation. Burn-UD allows the user to input their own function for strange matter initial distribution.  Different seeding models might imply different functions for the initial distribution of the strange matter seed. The fact that Burn-UD allows the user to input their own   strange matter seed initial distribution makes Burn-UD useful for the study of many seeding models, including models for DM. \n\n\n\\subsection{Mass of the Strange Quark}\n\nUp and down quarks have similar masses, yet the strange quark has a mass that is an order of magnitude higher. This difference between the up, down and strange mass should have consequences in the dynamics of the combustion.\nAt the moment, the EoS used for Burn-UD assumes the quarks are massless. The only dependency in Burn-UD to the quark mass lies in the rates mentioned in section \\ref{results}, where a strange quark mass of $150 \\text{ MeV}$  only affects the results very weakly as explored in \\cite{niebergal2010numerical}.\n\nDifferent masses for quarks would affect the binding energy necessary for the assembling of quarks. Therefore the creation of massive quarks would deplete the burning front of energy and slow down the burning speed.  It is important to include the quark masses in future studies to investigate how the results of the Burn-UD simulations are affected.\n\n\\subsection{More Sophisticated Equations of State}\n\nBurn-UD models the flame micro-physics by imposing the MIT Bag EoS on both sides of the interface, i.e. for both the uds ash and the ud fuel. Thus the EoS at both sides of the interface shares the $B$ parameter as well. This means that at the region near the interface, where quarks have already deconfined,  the behaviour of the fluid becomes independent of the $B$ parameter. The speed of the burning front then becomes a function of the binding energy released during the creation and destruction of quarks. \n\nThe MIT Bag Model is very popular because despite its simplicity, it manages to  successfully reproduce some of the properties of hadrons. It also models asymptotic freedom in an intuitive way making it useful for simulating the properties of bulk quark matter.\nYet such simplifications might miss some important dynamics.  One of them is chiral symmetry breaking.  A popular model that includes chirality is the Nambu-Jonas-Lasinio (NJL) model for QCD. One important difference in the NJL picture, is that strange quark matter is not the ground state of all matter \\cite{buballa2005njl}. However, the NJL model, like  the MIT Bag Model, is phenomenological and as such its predictions are not completely conclusive either. In future releases of Burn-UD, the code will incorporate several EoS's, including NJL, to compare their effects on the simulation.\n\nFinally, a more realistic simulation would include a mixed phase EoS as well. At the moment, deconfined quarks that act as a fluid are only considered. However, the dissolution of hadrons into quarks, if it is a first order phase transition, would change the pressure and therefore preshock the ud fuel to a certain degree, which would change the dynamics of the combustion simulation. \n\n\\subsection{More Sophisticated Neutrino Transport}\n\nAs shown in \\ref{neutrino}, Burn-UD currently models neutrino transport  as a parameterized energy sink that saps the matter near the interface of energy. Although this model addresses many of the larger and more qualitative aspects of neutrino transport, a more accurate model would actually implement the time dependent Boltzmann equations of neutrino transport. In the Boltzmann equations, neutrino flux is modelled as a density function that evolves across time and space. \n\nHaving an accurate model of the neutrino transport is vital because the dynamics of these neutrinos are directly related to the deleptonization instability occurring at the burning interface. A sophisticated neutrino transport model will be one of the deciding ingredients in the discovery of whether the deleptonization instability leads to supersonic deflagration to detonation burning, or core-collapse Quark-Nova. \n\n\n  \n\n\\section{Conclusion}\\label{conclusion}\n\nWe hope that Burn-UD will evolve into a platform\/software to be used and shared by  the QCD \ncommunity exploring the phases of Quark Matter and astrophysicists working on Compact Stars. \nBurn-UD offers the possibility of studying the time-dependent phase transition of hadronic matter to quark matter. This software models hydrodynamically the combustion front (i.e. the flame's dynamics) which gives a physical window to diagnose whether the combustion process will simmer quietly and slowly, or transitions from deflagration to detonation. However, it is important to note that there exists solutions where the burning front is halted due to the deleptonization instability which make the system undergo a core-collapse QN explosion.  \nWhether via a Deflagration-Detonation-Transition  or core-collapse, such an energetic phase transition would have many consequences in high-energy astrophysics and could aid in our understanding of many still enigmatic astrophysical transients. Furthermore, having a precise understanding of the phase transition dynamics for different EoSs could aid further in constraining the nature of the non-perturbative regimes of QCD in general.  \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec_1}\n\nThe relay channel problem was first studied by van der Meulen \\cite{van_der}, where the source transmits a message to the destination with the help of a relay node. In \\cite{cover}, the authors formalized the problem and proposed several effective full-duplex relay strategies, which are the so-called decode-and-forward (DF) and compress-and-forward (CF) schemes. Subsequently, the corresponding schemes in the time division half-duplex (TDD) mode were studied in \\cite{anders}. The upper bound on the capacity and various achievable rates were established in \\cite{cover} and \\cite{anders}, while the general capacity problem is still open.\n\nOne of the important applications of relaying is in low-power communication networks (such as wireless sensor networks), where the transmitting SNRs are usually low such that relays are needed to enhance the performance at the destination nodes. In \\cite{verdu}, the author presented two important performance criteria in the low-SNR regime: One is the minimum energy per bit required for reliable communication; and the other is the slope of the spectral efficiency. Such criteria have been adopted to study the relay channel in the low-SNR regime~\\cite{gamal,xiaodong}. In particular, the time-sharing CF scheme and the traditional DF scheme were investigated in \\cite{gamal} for both the full-duplex and frequency division half-duplex (FDD) modes, where the authors derived the DF-based upper bound on the minimum energy per bit, and the lower bound using the max-flow min-cut capacity theorem. In \\cite{xiaodong}, the fading relay channel was studied, where the upper and lower bounds on the minimum energy per bit were derived using the same method as in \\cite{gamal}.\n\n\nIn the low-SNR regime, the CF scheme is much more appealing than the DF scheme since the relay cannot decode much information such that the DF strategy at the relay would cause the bottleneck effect. In this paper, we thus focus on the time-sharing CF scheme in the low-SNR regime under both the full-duplex and half-duplex TDD modes. We show that the time-sharing CF scheme improves the achievable rate under the TDD mode when the SNR is low. Furthermore, the lower bound on the minimum energy per bit is derived by applying the max-flow min-cut theorem for the channel capacity, and the upper bound is derived by deploying the time-sharing CF scheme. The aforementioned upper bound can be further tightened by letting the active phase duration decrease to zero at the same rate as the SNR value.\n\nThe rest of the paper is organized as follows. The channel model is discussed in Section \\ref{sec_2}. In Section \\ref{sec_3} we present the achievable rates of the time-sharing CF scheme under both the full-duplex and half-duplex TDD modes, and subsequently we derive the bounds on the minimum energy per bit for reliable communication. The numerical results are presented in Section \\ref{sec_4}. Finally, the paper is concluded in Section \\ref{sec_5}.\n\n\\section{System Model}\\label{sec_2}\nConsider the Gaussian relay channel as shown in Fig. \\ref{fig_relay_channel_model}, where the channel amplitude gains are assumed fixed and denoted as $h_{21}$, $h_{31}$, and $h_{32}$, respectively. Next, we introduce the signal models for both the traditional relaying schemes (full-duplex and half-duplex TDD), and the time-sharing relaying scheme. Note that in this paper we only consider real signals since the extension to complex cases is straightforward.\n\n\\subsection{Traditional Relaying}\n\n\\subsubsection{Full-Duplex Mode}\nSuppose the relay can transmit and receive signals at the same time. The channel inputs at the source and relay are represented as $X_1$ and $X_2$ with average power values $P_1$ and $P_2$, respectively. The noises at the relay and destination are denoted by $Z_1$ and $Z$, which are independent additive white Gaussian noises (AWGNs) with distributions $\\mathcal{N}(0,N_1)$ and $\\mathcal{N}(0,N)$, respectively.\n\nDenoting the received signals at the relay and destination by $Y_1$ and $Y$, respectively, we can fully describe the channel as:\n\\begin{eqnarray*}\nY_1 &=& h_{21}X_{1}+Z_1,\\\\\nY &=& h_{31}X_{1} + h_{32}X + Z.\n\\end{eqnarray*}\n\n\\subsubsection{Half-Duplex Mode}\nWe also consider the half-duplex TDD mode, where at each transmission slot the relay listens to the source during the first $\\lambda$ fraction of time, and transmits to the destination in the remaining $1-\\lambda$ fraction.\n\nIn the first $\\lambda$ fraction, the transmitted signal at the source is denoted by $X_{1}^{(1)}$ with an average power constraint $P_{1}^{(1)}$. The received signals at the relay and destination are given by:\n\\begin{eqnarray*}\nY_{1} &=& h_{21}X_{1}^{(1)} + Z_{1},\\\\\nY^{(1)} &=& h_{31}X_{1}^{(1)} +\nZ^{(1)},\n\\end{eqnarray*}\nrespectively, where $Z_{1} \\sim \\mathcal{N}(0,N_{1})$ and $Z^{(1)} \\sim \\mathcal{N}(0,N)$ are independent AWGNs.\n\nIn the remaining $1-\\lambda$ fraction, the transmitted signals at the source and relay are denoted as $X_{1}^{(2)}$ and $X_{2}$,\nwith average power constraints $P_{1}^{(2)}$ and $P_{2}$, respectively. Accordingly, the received signal at the destination is given by:\n\\begin{eqnarray*}\nY^{(2)} &=& h_{31}X_{1}^{(2)} +\nh_{32}X_{2} + Z^{(2)},\n\\end{eqnarray*}\nwhere $Z^{(2)} \\sim \\mathcal{N}(0,N)$ is the AWGN noise at the destination.\n\n\\subsection{Relaying with Time-Sharing}\nIt is known that for low-SNR point-to-point Gaussian channels, low-duty-cycle on-off keying is as good as Gaussian inputs~\\cite{verdu}. Along a similar line, for the low-SNR relay channels, the authors in~\\cite{gamal} proposed a time-sharing CF scheme based on the argument that the CF achievable rate is not a concave function over the power allocation factor, where the underlying idea is that the relay can ``hear\" the source more clearly if we first deploy a silent phase in each slot and increase the peak power in the following active phase.\n\nWith the above discussions, in this paper we consider the time-sharing relaying scheme, where in each time slot all the nodes keep silent in the first $1-\\alpha$ fraction (called the silent phase), and deploy the traditional relaying scheme in the remaining $\\alpha$ fraction (called the active phase), which could be either full-duplex or half-duplex as introduced in the previous subsection. As such, the signal model in the active phase follows the previously introduced model except that the power at the source and relay should be scaled up accordingly to meet the average power constraint. For example, if full-duplex relaying is deployed, the power values at the source and relay become $P_1\/\\alpha$ and $P_2\/\\alpha$, respectively.\n\n\n\n\\section{Achievable Rates and Minimum Energy Per Bit}\\label{sec_3}\nIn this section, the performance of the time-sharing CF scheme is investigated under both the full-duplex and half-duplex TDD modes. Specifically, we derive the achievable rates and the upper\/lower bounds on the minimum energy per bit in the low-SNR regime. For convenience, we define the following function:\n\\[ \\Gamma \\left(x\\right) =\n\\frac{1}{2} \\log_{2} \\left(1 + x\\right),\n\\]\nand denote the \\emph{normalized channel gains} of the corresponding links\nas:\n\\begin{equation}\n\\nonumber \\gamma_{21} = \\frac{h_{21}^{2}}{N_{1}},\\quad \\gamma_{32} =\n\\frac{h_{32}^{2}}{N}, \\quad \\gamma_{31} = \\frac{h_{31}^{2}}{N}.\n\\end{equation}\n\n\n\\subsection{Capacity Upper Bound and Achievable Rates}\n\\subsubsection{Full-Duplex Mode}\nWe first give the upper bound on the capacity by applying the max-flow min-cut theorem \\cite{cover}:\n\\begin{equation}\nC^{+}_{\\textrm{full}} = \\max_{0\\leq \\rho_x \\leq\n1}\\min\\left\\{C^{+}_{\\textrm{1-full}}(\\rho_x),\nC^{+}_{\\textrm{2-full}}(\\rho_x)\\right\\}, \\label{eq_full_upperbound}\n\\end{equation}\nwhere $\\rho_x$ is the correlation coefficient between the source and relay inputs, and\n\\begin{eqnarray}\n\\nonumber C^{+}_{\\textrm{1-full}}(\\rho_x) &=&\n\\Gamma\\left(\\gamma_{31}P_1 +\n\\gamma_{32}P_2\\right), \\\\\n\\nonumber C^{+}_{\\textrm{2-full}}(\\rho_x) &=&\n\\Gamma\\left(P_1(1-\\rho_x^2)(\\gamma_{21}+\\gamma_{31}) \\right).\n\\end{eqnarray}\n\nNow we first consider the traditional CF strategy, where the relay compresses the received signal with Wyner-Ziv coding~\\cite{wyner}, and then forwards the binning index to the destination. The following achievable rate is established with the CF scheme \\cite{cover}:\n\\[\nR_{\\textrm{cf}} = \\sup_{p(\\cdot)\\in\\mathcal{P}^{*}} I(X_1;Y,\\hat{Y}_1|X_2),\n\\]\nsubject to the constraint\n\\[\nI(X_2;Y) \\geq I(Y_1;\\hat{Y}_1|X_2,Y).\n\\]\nCorrespondingly, the achievable rate with CF over the Gaussian relay channel can be written as:\n\\begin{equation}\nR_{\\textrm{cf}} = \\Gamma \\left(\\gamma_{31}P_1 + \\frac{\\gamma_{32}\\gamma_{21}P_1P_2}{1+\\gamma_{21}P_1+\\gamma_{31}P_1+\\gamma_{32}P_2}\\right).\n\\end{equation}\nAs we discussed before, to improve the performance in the low-SNR regime, the authors in~\\cite{gamal} proposed the time-sharing CF scheme based on the argument that the achievable rate is not a concave function of the power. Specifically, the relay channel is only utilized for $\\alpha$ portion of each time slot.  In this $\\alpha$ portion, the power values of the source and relay are $P_1\/\\alpha$ and $P_2\/\\alpha$, respectively. Accordingly, the achievable rate for the time-sharing CF scheme is given as~\\cite{gamal}:\n\\begin{equation}\nR_{\\textrm{ts}} = \\max_{0<\\alpha\\leq 1} \\alpha\\Gamma \\left(\\frac{\\gamma_{31}P_1}{\\alpha} + \\frac{\\gamma_{32}\\gamma_{21}P_1P_2}{\\alpha^2+\\alpha(\\gamma_{21}P_1+\\gamma_{31}P_1+\\gamma_{32}P_2)}\\right).\n\\end{equation}\nIt is also shown in \\cite{gamal} that the above achievable rate is higher than the one with the traditional CF scheme \\cite{cover} in the low-SNR regime.\n\n\\subsubsection{Half-Duplex Mode}\nIn practice, full-duplex relaying is technically difficult to implement. Hence, we now consider the more practical TDD mode. We first give the max-flow min-cut bound on the capacity \\cite{anders}:\n\\begin{equation}\nC^{+}_{\\textrm{half}} = \\max_{0 \\leq \\rho_{x} \\leq 1} \\min\n\\{C_{\\textrm{1-half}}^{+}\\left(\\rho_{x}\\right),\nC_{\\textrm{2-half}}^{+}\\left(\\rho_{x}\\right)\\},\\label{eq_half_upperbound}\n\\end{equation}\nwhere $\\rho_x$ is the correlation coefficient between $X_1^{(2)}$ and $X_2$, and\n\\begin{eqnarray*}\nC_{\\textrm{1-half}}^{+}\\left(\\rho_{x}\\right)&=& \\lambda \\Gamma\n\\left(\\gamma_{31}P_1^{(1)}\\right) + \\\\  &&\\left(1-\\lambda\\right) \\Gamma\n\\bigg(\\gamma_{31}P_1^{(2)} +\n\\gamma_{32}P_2 + 2\\rho_{x}\\sqrt{\\gamma_{31}P_1^{(2)}\\gamma_{32}P_2}\\bigg); \\nonumber \\\\\nC_{\\textrm{2-half}}^{+}\\left(\\rho_{x}\\right)&=& \\lambda \\Gamma\n\\left(\\left(\\gamma_{21}+\\gamma_{31}\\right)P_1^{(1)}\\right)+\\nonumber\n\\\\&&\\ \\left(1-\\lambda\\right) \\Gamma\n\\left(\\left(1-\\rho_x^2\\right)\\gamma_{31}P_1^{(2)}\\right).\n\\end{eqnarray*}\n\n\nFurthermore, the traditional CF achievable rate without time-sharing under the half-duplex TDD mode is given by \\cite{anders}:\n\\begin{equation}\nR_{\\textrm{cf\\_h}} = \\lambda \\Gamma \\left(\\gamma_{31}P_1^{(1)} + \\frac{\\gamma_{21}P_1^{(1)}}{1+N_{\\textrm{w}}\/N_1}\\right) + (1-\\lambda)\\Gamma \\left(\\gamma_{31}P_1^{(2)}\\right),\n\\end{equation}\nwhere $N_{\\textrm{w}}$ is the power of the quantization noise at the relay, with\n\\begin{equation}\n\\nonumber N_{\\textrm{w}} = N_1\\frac{1+\\gamma_{21}P_1^{(1)}+\\gamma_{31}P_1^{(1)}}{\\left(1+\\gamma_{31}P_1^{(1)}\\right)\\left(\\left(1+\\frac{\\gamma_{32}P_2}{1+\\gamma_{31}P_1^{(2)}}\\right)^{\\frac{1-\\lambda}{\\lambda}}-1\\right)}.\n\\end{equation}\n\n\n\nIn the TDD mode, the above achievable rate function is not a concave function of the power either. Thus, by a similar argument as that in \\cite{gamal}, we can apply time-sharing in the low-SNR regime to improve the achievable rate. In particular, the source and relay keep silent in the first $1-\\alpha$ portion of each time slot; then the source sends the message to the relay and destination in the following $\\alpha\\lambda$ portion with power $P_1^{(1)}\/\\alpha$. The relay compresses the received signal and sends it to the destination over the last $\\alpha(1-\\lambda)$ fraction of time with power $P_2\/\\alpha$. The source also utilizes the last $\\alpha(1-\\lambda)$ portion to send signal to the destination with power $P_1^{(2)}\/\\alpha$. The achievable rate of the above time-sharing CF scheme under the half-duplex TDD mode can be computed as:\n\n\\begin{equation}\n\\scriptsize\\label{eq_half_cf_ts_rate}\n\\nonumber R_{\\textrm{ts\\_h}} = \\max_{0<\\alpha\\leq 1}\\quad \\alpha\\lambda \\Gamma \\left(\\frac{\\gamma_{31}P_1^{(1)}}{\\alpha} + \\frac{\\gamma_{21}P_1^{(1)}}{\\alpha+\\alpha \\frac{N_{\\textrm{ts\\_w}}}{N_1}}\\right)   +\\  \\alpha(1-\\lambda)\\Gamma \\left(\\frac{\\gamma_{31}P_1^{(2)}}{\\alpha}\\right),\n\\end{equation}}\nwhere $N_{\\textrm{ts\\_w}}$ is the power of the quantization noise at the relay, with\n\\begin{equation}\n\\nonumber N_{\\textrm{ts\\_w}} = N_1\\frac{\\alpha+\\gamma_{21}P_1^{(1)}+\\gamma_{31}P_1^{(1)}}{\\left(\\alpha+\\gamma_{31}P_1^{(1)}\\right)\\left(\\left(1+\\frac{\\gamma_{32}P_2}{\\alpha+\\gamma_{31}P_1^{(2)}}\\right)^{\\frac{1-\\lambda}{\\lambda}}-1\\right)}.\n\\end{equation}\n\nThe achievable rate improvement in the low-SNR regime under both the full-duplex and half-duplex modes will be further illustrated in Section \\ref{sec_4}, by numerical comparisons against the schemes without time-sharing.\n\n\\subsection{Minimum Energy Per Bit}\nIn \\cite{verdu}, the author presented two important performance criteria in the low-SNR regime: One is the minimum energy per bit required for reliable communication, and the other is the slope of the spectral efficiency. In this paper, we focus on the first criterion. Denoting the channel capacity as $C(P)$ with $P$ the total power consumed at the source and the relay, the minimum energy per bit is defined as~\\cite{verdu}:\n\\begin{equation}\n\\left(\\frac{E_b}{N_0}\\right)_{\\min}=\\lim_{P \\to 0}\\frac{P}{C(P)}=\\frac{\\log_e 2}{C'(0)},\n\\end{equation}\nwhere $C'(0)$ is the first-order derivative at $P = 0$. Note that here the energy per bit is referring to the total energy consumed at both the source and the relay for each bit transmitted over the relay channel.\n\n\nIn the rest of this section, we derive the upper and lower bounds on the minimum energy per bit under both the full-duplex and half-duplex modes for the Gaussian relay channel under consideration.\n\n\\subsubsection{Full-Duplex Mode}\nWe skip the derivation of the lower bound on the minimum energy per bit under the full-duplex mode, which is obtained by applying the max-flow min-cut upper bound on the channel capacity. Interested readers can refer to \\cite{gamal} and \\cite{xiaodong} for related results. Now we derive the upper bound on the minimum energy per bit.\n\nAssume that the sum power across the source and the relay is subject to a total power constraint $P$. Denote the power at the source as $\\beta P$ and the power at the relay as $(1-\\beta)P$ for $0<\\beta\\leq 1$, i.e., $\\{P_1,P_2\\} = \\{\\beta P, (1-\\beta) P\\}$. The achievable rate of the time-sharing CF scheme can be calculated as:\n\\begin{equation}\nR_{\\textrm{ts}} = \\max_{0<\\alpha\\leq 1} \\alpha\\Gamma\\left(\\frac{\\gamma_{31}\\beta P}{\\alpha}+\\frac{\\gamma_{32}\\gamma_{21}\\beta(1-\\beta)P^2}{\\alpha^2+\\alpha GP}\\right),\n\\end{equation}\nwhere $G:=\\gamma_{21}\\beta+\\gamma_{31}\\beta+\\gamma_{32}(1-\\beta)$. For a given active fraction $\\alpha$, the upper bound on the minimum energy per bit can be evaluated as:\n\\begin{eqnarray}\n\\nonumber\\left(\\frac{E_b}{N_0}\\right)_{\\min}^{+} &=& \\lim_{P \\to 0} \\frac{P}{R_{\\textrm{ts}}(P)}\\\\\n&=& \\lim_{P \\to 0} \\frac{P}{\\alpha}\\frac{1}{\\Gamma\\left(\\gamma_{31}\\beta P\/\\alpha+\\frac{\\gamma_{32}\\gamma_{21}\\beta(1-\\beta)P^2\/\\alpha^2}{1+GP\/\\alpha}\\right)}.\\label{eq_full_ebn0_upper}\n\\end{eqnarray}\nTo determine the limit, now we discuss several interesting cases based on the behavior of the parameter $\\alpha$. When the power $P$ decreases to $0$, while $\\alpha$ keeps constant or does not scale down as fast as $P$, the upper bound on the minimum energy per bit in \\eqref{eq_full_ebn0_upper} is equal to $2\\log_e 2\/(\\gamma_{31}\\beta)$, which is minimized at $\\beta=1$. In this case, the power is all utilized by the source, and the upper bound on the minimum energy per bit is the same as that of the point-to-point communication channel. The above result also implies that such an upper bound corresponding to the traditional CF scheme with a fixed $\\alpha=1$ is equal to $2\\log_e 2\/\\gamma_{31}$. Furthermore, if the active fraction $\\alpha$ goes to $0$ faster than $P$, $\\left(E_b\/N_0\\right)_{\\min}^{+}$ in \\eqref{eq_full_ebn0_upper} goes to infinity, which leads to a non-meaningful upper bound on the minimum energy per bit. However, if we keep the ratio $P\/\\alpha$ constant (denoted by $A$) when $P$ and $\\alpha$ decrease to zero, we could achieve a tighter upper bound (against the case of $\\alpha=1$) on the minimum energy per bit.\nTherefore, we have the following theorem describing the asymptotic upper bound on the minimum energy per bit using the time-sharing CF scheme.\n\n\\begin{Theorem}\nUnder the full-duplex mode, when both $P$ and $\\alpha$ tend to zero with $\\frac{P}{\\alpha}=A$, the minimum energy per bit on the relay channel has the following asymptotic upper bound:\n\\begin{equation}\\label{eq_full_ebn0_upper_theorem}\n\\left(\\frac{E_b}{N_0}\\right)_{\\min}^{+} = \\min_{0<\\beta\\leq 1, A\\geq0} \\frac{A}{\\Gamma\\left(\\gamma_{31}\\beta A+ \\frac{\\gamma_{32}\\gamma_{21}\\beta(1-\\beta)A^2}{1+GA}\\right)}.\n\\end{equation}\n\\end{Theorem}\n\nIt can be shown that under the full-duplex mode, the above upper bound converges to the lower bound on the minimum energy per bit when $\\gamma_{32}\\to\\infty$~\\cite{gamal} since both the upper and lower bounds go to $2\\log_e 2\/(\\gamma_{31}+\\gamma_{21})$.\nThe upper bound improvement of the minimum energy per bit due to time-sharing will be further illustrated in Section \\ref{sec_4} by numerical comparisons in the low-SNR regime.\n\n\n\\subsubsection{Half-Duplex Mode}\nHere we investigate the lower and upper bounds on the minimum energy per bit under the half-duplex mode. The sum of the source power and the relay power is fixed to $P$ as in the full-duplex mode. Denote the power at the source as $\\beta P$ and the power at the relay as $(1-\\beta)P$, respectively. Furthermore, with CF we assume that the source uses the same power during the first and the second transmitting phases, i.e., $P_1^{(1)} = P_1^{(2)}=\\beta P$. Consequently, we have the power allocation as $\\{P_1^{(1)},P_1^{(2)},P_2\\}=\\{\\beta P,\\beta P, (1-\\beta)P\\}$.\n\nFirst we derive the lower bound on the minimum energy per bit, which is given by the following theorem.\n\n\\begin{Theorem}\nUnder the half-duplex TDD mode, the lower bound on the minimum energy per bit can be obtained by one-dimensional searching over $\\lambda \\in (\\gamma_{31}\/(\\gamma_{31}+\\gamma_{21}),1]$ instead of three-dimensional searching over $\\lambda$, $\\beta$, and $\\rho_x$. When the relay-destination link gain $\\gamma_{32}$ goes to infinity, the lower bound converges to $2\\log_e 2\/(\\gamma_{31}+\\gamma_{21})$.\n\\end{Theorem}\n\\begin{proof}\nSee Appendix \\ref{appendix_1}.\n\\end{proof}\n\nNow we consider the upper bound on the minimum energy per bit, which is achieved by the time-sharing CF scheme.\n\n\\begin{Theorem}\nUnder the half-duplex mode, the minimum energy per bit on the relay channel is upper-bounded by:\n\\begin{equation}\\label{eq_half_ebn0_upper}\\small\n\\left(\\frac{E_b}{N_0}\\right)_{\\min}^{+} = \\min_{0<\\lambda\\leq 1,0<\\beta\\leq 1,A\\geq 0}\\frac{A}{\\lambda\\Gamma\\left(\\gamma_{31}\\beta A + \\frac{\\gamma_{21}\\beta A}{1+\\frac{1+\\gamma_{21}\\beta A+\\gamma_{31}\\beta A}{(1+\\gamma_{31}\\beta A)\\left(\\left(1+\\frac{\\gamma_{32}(1-\\beta)A}{1+\\gamma_{31}\\beta A}\\right)^{(1-\\lambda)\/\\lambda}-1\\right)}}\\right)+(1-\\lambda)\\Gamma\\left(\\gamma_{31}\\beta A\\right)},\\ \\textrm{with}\\ A=\\frac{P}{\\alpha}.\n\\end{equation}\nSuch an upper bound is tight (i.e., achieves the lower bound on the minimum energy per bit) asymptotically when $\\gamma_{32}$ goes to infinity.\n\\end{Theorem}\n\n\\begin{proof}\nAs shown in \\eqref{eq_half_cf_ts_rate}, the achievable rate of the time-sharing CF scheme under the half-duplex mode is computed as:\n\\begin{equation}\n\\small\nR_{\\textrm{ts\\_h}} = \\max_{0<\\alpha\\leq 1} \\alpha\\lambda \\Gamma \\left(\\frac{\\gamma_{31}\\beta P}{\\alpha} + \\frac{\\gamma_{21}\\beta P}{\\alpha+\\alpha N_{\\textrm{ts\\_w}}\/N_1}\\right)   +\\  \\alpha(1-\\lambda)\\Gamma \\left(\\frac{\\gamma_{31}\\beta P}{\\alpha}\\right),\\label{eq_half_cf_ts}\n\\end{equation}\nwhere the quantization noise power $N_{\\textrm{ts\\_w}}$ is\n\\begin{equation}\n\\nonumber N_{\\textrm{ts\\_w}} = N_1\\frac{\\alpha+\\gamma_{21}P_1^{(1)}+\\gamma_{31}P_1^{(1)}}{\\left(\\alpha+\\gamma_{31}P_1^{(1)}\\right)\\left(\\left(1+\\frac{\\gamma_{32}P_2}{\\alpha+\\gamma_{31}P_1^{(2)}}\\right)^{\\frac{1-\\lambda}{\\lambda}}-1\\right)}.\n\\end{equation}\nSubsequently, we can have an upper bound by using the definition of the minimum energy per bit. Then as in the full-duplex mode, the above upper bound can be tightened to achieve that in \\eqref{eq_half_ebn0_upper} by letting $\\alpha$ decrease to $0$ at the same rate as $P$, i.e., $P\/\\alpha = A$. Moreover, the upper bound in \\eqref{eq_half_ebn0_upper} converges to $2\\log_e 2\/(\\gamma_{31}+\\gamma_{21})$ when $\\gamma_{32}\\to\\infty$, which implies that the upper bound is asymptotically optimal.\n\\end{proof}\n\nThe comparisons between the time-sharing CF upper bound and the lower bound under both the full-duplex and half-duplex modes are shown in Section \\ref{sec_4}. Note that the minimum energy per bit can be clearly improved by time-sharing; hence we do not need to check the second performance measure: the relative spectral efficiency~\\cite{verdu}, which is related to the second-order derivative of the achievable rate.\n\n\n\\section{Numerical Results} \\label{sec_4}\n\nIn this section, we illustrate the performance improvement due to time-sharing in CF with several numerical results. The achievable rate and the minimum energy bit are evaluated under both the full-duplex and half-duplex modes.\n\nThe following channel model is considered for all the numerical comparisons. The source, relay, and destination are aligned over a line. The distance between the source and the relay is $d$ ($0<d<1$), and the distance between the source and the destination is $1$. The channel amplitude is inversely proportional to the distance, which means:\n\\begin{equation}\nh_{21} = \\frac{1}{d},\\ h_{32} = \\frac{1}{1-d},\\ h_{31} = 1,\\quad d\n\\in (0,1).\n\\end{equation}\n\n\nThe improvement of achievable rate for various $d$ values is shown in Fig. \\ref{fig_full_rate} and Fig. \\ref{fig_half_rate} under the full-duplex and half-duplex modes when the active fraction $\\alpha$ is optimized for the corresponding time-sharing CF strategies. Here we demonstrate the improvement by drawing the relative rate improvement $(R_{\\textrm{ts}}-R_{\\textrm{cf}})\/R_{\\textrm{cf}}$ since the absolute values of the achievable rates are small in the low-SNR regime. As shown in Fig. \\ref{fig_full_rate}, the time-sharing CF scheme improves the achievable rate up to $80\\%$ under the full-duplex mode when $d = 0.25$ or $0.75$, and up to $75\\%$ when $d = 0.5$. Specifically, the relative improvement at $d=0.25$ and $0.75$ are the same since the CF achievable rate under the full-duplex mode is a symmetric function with respect to $d=0.5$. If the relay works under the half-duplex mode, in Fig. \\ref{fig_half_rate} we show that the time-sharing CF scheme improves the achievable rate up to $70\\%$ when $d=0.25$, up to $55\\%$ when $d=0.5$, and up to $49\\%$ when $d=0.75$. Unlike the full-duplex mode, the relative improvement decreases if $d$ is larger than $0.5$.\n\nThe bounds on the minimum energy per bit are given in Fig. \\ref{fig_full_ebn0} and Fig. \\ref{fig_half_ebn0} with respect to the source-relay distance $d$. Under the full-duplex mode as shown in Fig. \\ref{fig_full_ebn0}, the upper bound can be obviously tightened by the time-sharing CF scheme when the relay is closer to the destination than the source ($d\\geq 0.6$). Under the half-duplex mode as shown in Fig. \\ref{fig_half_ebn0}, the time-sharing CF scheme tightens the upper bound when the relay is further away from the source ($d\\geq 0.8$). We can also see that the upper and lower bounds meet under both the full-duplex and half-duplex modes when $d$ goes to $1$, i.e., $\\gamma_{32}\\to \\infty$.\n\n\n\n\n\n\n\n\n\n\\section{Conclusion} \\label{sec_5}\n\nWe investigated the performance of the time-sharing CF scheme in the low-SNR regime over Gaussian relay channels. The achievable rates and the minimum energy per bit were studied under both the full-duplex and half-duplex TDD modes, where we derived the upper and lower bounds on the minimum energy per bit. Furthermore, we showed that the time-sharing CF scheme tightens the upper bound on the minimum energy per bit by letting the active fraction go to zero at the same speed as the transmit power. We also provided some numerical results, which validates the performance improvement due to time-sharing.\n\n\n\\appendices\n\\section{Proof For Theorem 2}\\label{appendix_1}\nBased on the max-flow min-cut bound \\eqref{eq_half_upperbound} on the system capacity, the minimum energy per bit can be lower-bounded by:\n\\begin{equation}\n\\left(\\frac{E_b}{N_0}\\right)_{\\min}^{-} = \\min_{0\\leq\\rho_x\\leq 1, 0<\\beta\\leq 1, 0<\\lambda\\leq 1} \\max \\left\\{\\frac{\\log_e 2}{C_{\\textrm{1-half}}^{+'}},\\frac{\\log_e 2}{C_{\\textrm{2-half}}^{+'}}\\right\\},\\label{eq_half_lowerbound}\n\\end{equation}\nwhere the first-order derivatives of $C_{\\textrm{1-half}}^{+}$ and $C_{\\textrm{2-half}}^{+}$ at $P=0$ can be respectively computed as:\n\\begin{eqnarray}\nC_{\\textrm{1-half}}^{+'} &=& \\frac{1}{2}\\left(\\gamma_{31}\\beta+\\gamma_{32}\\left(1-\\lambda\\right)(1-\\beta)+2\\left(1-\\lambda\\right)\\rho_x\\sqrt{\\gamma_{31}\\gamma_{32}\\beta(1-\\beta)}\\right),\\nonumber\\\\\nC_{\\textrm{2-half}}^{+'} &=& \\frac{1}{2} \\left(\\gamma_{31}\\beta+\\gamma_{21}\\lambda\\beta-\\gamma_{31}\\left(1-\\lambda\\right)\\beta\\rho_x^2\\right).\\label{eq_half_lower_deri}\n\\end{eqnarray}\nWe can equivalently formulate the optimization problem as:\n\\begin{equation}\n\\max_{0\\leq\\rho_x\\leq 1, 0<\\beta\\leq 1, 0<\\lambda\\leq 1} \\min \\left\\{C_{\\textrm{1-half}}^{+'},C_{\\textrm{2-half}}^{+'}\\right\\}.\n\\end{equation}\n\nIf the relay-destination link gain $\\gamma_{32}$ goes to infinity, $C_{\\textrm{2-half}}^{+'}$ is always smaller than $C_{\\textrm{1-half}}^{+'}$ for $\\beta \\in (0,1)$ and $\\lambda \\in (0,1)$, and the lower bound on the energy per bit can be calculated as:\n\\begin{eqnarray}\n\\nonumber\\left(\\frac{E_b}{N_0}\\right)_{\\min}^{-} &=& \\min_{0\\leq\\rho_x\\leq 1, 0<\\beta< 1, 0<\\lambda< 1} \\frac{\\log_e 2}{C_{\\textrm{2-half}}^{+'}}\\\\\n&=& \\frac{2\\log_e 2}{\\gamma_{31}+\\gamma_{21}},\n\\end{eqnarray}\nwhich proves the second part of Theorem 2.\n\nWhen the relay-destination link gain $\\gamma_{32}$ is finite, first we calculate the corresponding optimal $\\beta$ and $\\rho_x$ for an arbitrarily given TDD parameter $\\lambda$.\nIf $\\beta\\leq \\gamma_{32}(1-\\lambda)\/\\left(\\gamma_{32}(1-\\lambda)+\\gamma_{21}\\lambda\\right) \\triangleq \\beta_1$, the optimal $\\rho_x$ is $0$ since $C_{\\textrm{1-half}}^{+'}\\geq C_{\\textrm{2-half}}^{+'}$ and $\\min \\left\\{C_{\\textrm{1-half}}^{+'},C_{\\textrm{2-half}}^{+'}\\right\\} = C_{\\textrm{2-half}}^{+'}$, where $C_{\\textrm{2-half}}^{+'}$ is a monotonically decreasing function over $\\rho_x$. In this case, $C_{\\textrm{2-half}}^{+'}$ is maximized by $\\beta = \\beta_1$.\n\nIf $\\beta \\geq \\gamma_{32}(1-\\lambda)\/\\left(\\gamma_{32}(1-\\lambda) + \\left(\\sqrt{\\gamma_{21}\\lambda}-\\sqrt{\\gamma_{31}(1-\\lambda)}\\right)^2\\right) \\triangleq \\beta_2$, we have $\\min \\left\\{C_{\\textrm{1-half}}^{+'},C_{\\textrm{2-half}}^{+'}\\right\\} = C_{\\textrm{1-half}}^{+'}$ since $C_{\\textrm{2-half}}^{+'}$ is greater than $C_{\\textrm{1-half}}^{+'}$ for all $\\rho_x \\in [0,1]$ (This case only happens when $\\lambda > \\gamma_{31}\/\\left(\\gamma_{31}+\\gamma_{21}\\right)$ to ensure that $C_{\\textrm{2-half}}^{+'}$ is positive.). Now the optimal $\\rho_x$ is $1$ since $C_{\\textrm{1-half}}^{+'}$ is a monotonically increasing function over $\\rho_x$. The optimal $\\beta$ is given as follows by maximizing $C_{\\textrm{1-half}}^{+'}\\big\\vert_{\\rho_x = 1}$, which stands for the value of $C_{\\textrm{1-half}}^{+'}$ given $\\rho_x=1$ where the resulting function is concave over $\\beta$:\n\\begin{equation}\n\\beta^*=\\max \\left(\\beta_2, \\beta_2^* \\triangleq \\frac{1}{2}+\\frac{1}{2}\\frac{\\gamma_{31}-\\gamma_{32}(1-\\lambda)}{\\sqrt{\\left(\\gamma_{31}-\\gamma_{32}(1-\\lambda)\\right)^2+4\\gamma_{31}\\gamma_{32}(1-\\lambda)^2}}\\right) .\n\\end{equation}\n\nIf $\\beta_1 \\leq \\beta \\leq \\beta_2$, we let $C_{\\textrm{1-half}}^{+'}(\\rho_x) = C_{\\textrm{2-half}}^{+'}(\\rho_x)$ in order to solve the min-max problem since there must exist an unique crossing point. The optimal $\\rho_x$ can be obtained as:\n\\begin{equation}\n\\rho_x^* = \\frac{\\sqrt{\\gamma_{21}\\lambda\\beta\/(1-\\lambda)}-\\sqrt{\\gamma_{32}(1-\\beta)}}{\\sqrt{\\gamma_{31}\\beta}}.\\label{eq_half_rho_x}\n\\end{equation}\nBy substituting \\eqref{eq_half_rho_x} into \\eqref{eq_half_lower_deri}, we can maximize $C_{\\textrm{2-half}}^{+'}$ for $\\beta \\in [\\beta_1,\\beta_2]$, which leads to the following optimal $\\beta$:\n\\begin{equation}\n\\beta^*=\\label{eq_half_beta}\\min \\left(\\beta_2,\\beta_3^* \\triangleq \\frac{1}{2}+\\frac{1}{2}\\frac{\\gamma_{31}+\\gamma_{32}(1-\\lambda)}{\\sqrt{(\\gamma_{31}+\\gamma_{32}(1-\\lambda))^2+4\\gamma_{21}\\gamma_{32}\\lambda(1-\\lambda)}}\\right).\n\\end{equation}\nWhen $\\beta_3^*$ is optimal for a given $\\lambda$, the lower bound on the minimum energy per bit can be derived by substituting $\\beta_3^*$ and the corresponding $\\rho_x^*$ into \\eqref{eq_half_lowerbound} and \\eqref{eq_half_lower_deri}:\n\\begin{eqnarray}\\label{eq_half_lowerbound_3}\n\\nonumber\\left(\\frac{E_b}{N_0}\\right)_{\\min}^{-} &=& \\min_{0<\\lambda\\leq 1} \\frac{\\log_e 2}{C_{\\textrm{2-half}}^{+'}}\\\\\n&=& \\min_{0<\\lambda\\leq 1} \\frac{4\\log_e 2}{\\gamma_{31}-\\gamma_{32}(1-\\lambda) + \\sqrt{\\left(\\gamma_{31}+\\gamma_{32}(1-\\lambda)\\right)^2+4\\gamma_{21}\\gamma_{32}\\lambda(1-\\lambda)}},\n\\end{eqnarray}\nwhere the denominator is a concave function over $\\lambda \\in (0,1]$. Therefore, the optimal $\\lambda$ can be obtained by maximizing the denominator in \\eqref{eq_half_lowerbound_3}:\n\\begin{equation}\\label{eq_opt_lambda}\n\\lambda_{\\textrm{opt}} = \\frac{\\gamma_{31}+\\gamma_{32}-2\\gamma_{21}-\\sqrt{\\gamma_{31}^2+\\gamma_{31}\\gamma_{32}+\\gamma_{21}\\gamma_{32}}}{\\gamma_{32}-4\\gamma_{21}}\n\\end{equation}\nCorrespondingly, we can calculate the lower bound on the minimum energy per bit by substituting $\\lambda_{\\textrm{opt}}$ into \\eqref{eq_half_lowerbound_3}:\n\\begin{equation}\\label{eq_half_lowerbound_4}\n\\left(\\frac{E_b}{N_0}\\right)_{\\min}^{-} = \\frac{2(\\gamma_{32}-4\\gamma_{21})\\log_e 2 }{\\gamma_{31}\\gamma_{32}-2\\gamma_{21}\\gamma_{31}+\\gamma_{21}\\gamma_{32}-2\\gamma_{21}\\sqrt{\\gamma_{31}^2+\\gamma_{31}\\gamma_{32}+\\gamma_{21}\\gamma_{32}}}.\n\\end{equation}\n\nNow we summarize the solution by combining all the cases that we discussed above.\n\\begin{enumerate}\n\\item For $\\lambda \\in (0,\\gamma_{31}\/(\\gamma_{31}+\\gamma_{21})]$, $C_{\\textrm{2-half}}^{+'}$ is not ensured to be larger than $C_{\\textrm{1-half}}^{+'}$ for all $\\rho_x \\in [0,1]$. Accordingly, the lower bound is $\\left(\\frac{E_b}{N_0}\\right)_{\\min}^{-}$ in \\eqref{eq_half_lowerbound_4} if $\\lambda_{\\textrm{opt}}\\in (0, \\gamma_{31}\/(\\gamma_{31}+\\gamma_{21})]$. Otherwise, the lower bound is $\\frac{2\\log_e 2}{C_{\\textrm{1-half}}^{+'}}\\big |_{\\lambda=\\gamma_{31}\/(\\gamma_{31}+\\gamma_{21}),\\beta_{3}^*,\\rho_x^*}$.\n\\item For $\\lambda \\in (\\gamma_{31}\/(\\gamma_{31}+\\gamma_{21}), 1]$, one-dimensional searching can be utilized to obtain the lower bound. Specifically, for a given $\\lambda \\in (\\gamma_{31}\/(\\gamma_{31}+\\gamma_{21}), 1]$, we obtain an intermediate lower bound by taking the minimum among the ones derived in the following four cases. Then we search over all possible $\\lambda \\in (\\gamma_{31}\/(\\gamma_{31}+\\gamma_{21}), 1]$ to have the lower bound for case 2).\n    \\begin{itemize}\n    \\item If $\\beta_2\\leq\\beta_2^*,\\ \\beta_3^*$, the corresponding lower bound is $\\frac{2\\log_e 2}{C_{\\textrm{1-half}}^{+'}}\\big |_{\\beta_2^*,\\rho_x=1}$;\n    \\item If $\\beta_3^*<\\beta_2\\leq\\beta_2^*$, the corresponding lower bound is $\\frac{2\\log_e 2}{\\max\\left(C_{\\textrm{1-half}}^{+'}|_{\\beta_2^*,\\rho_x=1}, C_{\\textrm{1-half}}^{+'}|_{\\beta_3^*,\\rho_x^*}\\right)}$;\n    \\item If $\\beta_2^*<\\beta_2\\leq\\beta_3^*$, the corresponding lower bound is $\\frac{2\\log_e 2}{C_{\\textrm{1-half}}^{+'}}\\big |_{\\beta_2,\\rho_x=1}$;\n    \\item Otherwise, $\\beta_2^*,\\ \\beta_3^*<\\beta_2$, the corresponding lower bound is $\\frac{2\\log_e 2}{C_{\\textrm{1-half}}^{+'}}\\big |_{\\beta_3^*,\\rho_x^*}$.\n    \\end{itemize}\n\\item The global lower bound can be calculated as the minimum value between that for case 1) of $\\lambda \\in (0,\\gamma_{31}\/(\\gamma_{31}+\\gamma_{21})]$ and that for case 2) of $\\lambda \\in (\\gamma_{31}\/(\\gamma_{31}+\\gamma_{21}), 1]$.\n\\end{enumerate}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}