diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjsyu" "b/data_all_eng_slimpj/shuffled/split2/finalzzjsyu"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjsyu"
@@ -0,0 +1,5 @@
+{"text":"\\subsection{ Zeroth order effective field equations with ${\\bf G(\\Box)}$}\n\\hspace*{\\parindent}\n\\label{sec:field_gbox}\n\nA scale-dependent Newton's constant is expected to lead to small modifications\nof the standard cosmological solutions to the Einstein field equations.\nHere we will summarize what modifications are\nexpected from the effective field equations on the basis of $G(\\Box)$.\nThe starting point is the quantum effective nonlocal field equations\nof Eq.~(\\ref{eq:field1}), \nwith $G(\\Box)$ defined in Eq.~(\\ref{eq:grun_box}).\nIn the Friedmann-Lema\\^itre-Robertson-Walker (FLRW) framework these are\napplied to the standard homogeneous isotropic metric\n\\beq\nd \\tau^2 \\; = \\;  dt^2 - a^2(t) \\left \\{ { dr^2 \\over 1 - k\\,r^2 } \n+ r^2 \\, \\left ( d\\theta^2 + \\sin^2 \\theta \\, d\\varphi^2 \\right )  \\right \\} \\;\\;\\;\\; k =0, \\pm1 \\; .\n\\label{eq:frw}\n\\eeq\nIn the following, we will only consider the case $k=0$ (spatially flat universe).\nIt should be noted that there are in fact {\\it two} related quantum contributions to the\neffective covariant field equations. \nThe first one arises because of the presence of a nonvanishing \ncosmological constant $\\lambda \\simeq 3 \/ \\xi^2 $, caused by the\nnonperturbative quantum vacuum condensate $<R> \\, \\neq 0$ \\cite{loo07}.\nAs in the case of the standard FLRW cosmology, this is expected to be \nthe dominant contributions at large times $t$, and gives an exponential\n(for $\\lambda>0$), or cyclic (for $\\lambda < 0$) expansion of the scale factor.\nThe second contribution arises because of the explicit running of $G (\\Box)$ in the \neffective field equations.\n\nThe next step therefore is a systematic examination of the nature of\nthe solutions to the full effective field equations,\nwith $G ( \\Box )$ involving the relevant covariant d'Alembertian operator\n\\beq\n\\Box \\; = \\; g^{\\mu\\nu} \\, \\nabla_\\mu \\nabla_\\nu \n\\label{eq:box}\n\\eeq\nacting on second rank tensors as in the case of $T_{\\mu\\nu}$.\nTo start the process, we will assume that $T_{\\mu\\nu}$ is described by the perfect fluid form, \n\\beq\nT_{\\mu \\nu} = \\left [ \\, p(t) + \\rho(t) \\, \\right ] u_\\mu \\, u_\\nu + g_{\\mu \\nu} \\, p(t)\n\\label{eq:tmunu_perf}\n\\eeq\nfor which one needs to compute the action of $\\Box^n$ on $T_{\\mu\\nu}$,  and \nthen analytically continues the answer to negative fractional values of $n = -1\/2 \\nu $.\nThe results of \\cite{hw05,hw06,lop07,ht10}  then show  that a \nnonvanishing pressure contribution is generated in \nthe effective field equations, even if one initially assumes a pressureless fluid, $p(t)=0$.\nAfter a somewhat lengthy derivation one obtains for a universe filled with nonrelativistic \nmatter ($p$=0) the following set of effective Friedmann equations,\n\\bea\n{ k \\over a^2 (t) } \\, + \\,\n{ \\dot{a}^2 (t) \\over a^2 (t) }  \n& = & { 8 \\pi \\, G(t) \\over 3 } \\, \\rho (t) \\, + \\, { \\lambda \\over 3 }\n\\nonumber \\\\\n& = & { 8 \\pi \\, G_0 \\over 3 } \\, \\left [ \\,\n1 \\, + \\, c_t \\, ( t \/ \\xi )^{1 \/ \\nu} \\, + \\, \\dots \\, \\right ]  \\, \\rho (t)\n\\, + \\, { \\lambda \\over 3 }\n\\label{eq:fried_tt}\n\\eea\nfor the $tt$ field equation, and\n\\bea\n{ k \\over a^2 (t) } \\, + \\, { \\dot{a}^2 (t) \\over a^2 (t) }\n\\, + \\, { 2 \\, \\ddot{a}(t) \\over a(t) } \n& = & - \\, { 8 \\pi \\, G_0 \\over 3 } \\, \\left [ \\, c_t \\, ( t \/ \\xi )^{1 \/ \\nu} \n\\, + \\, \\dots \\, \\right ] \\, \\rho (t) \n\\, + \\, \\lambda\n\\label{eq:fried_rr}\n\\eea\nfor the $rr$ field equation.\nIn the above expressions, the running of $G$ appropriate for \nthe Robertson-Walker metric is\n\\beq\nG (t) \\, \\equiv \\, G_0 \\left ( 1 + { \\delta G(t) \\over G_0 } \\, \\right ) \n\\, = \\, G_0 \\left [ 1 + c_t \\, \n\\left ( { t \\over \\xi } \\right )^{1 \/ \\nu} \\, + \\, \\dots \\right ] \n\\label{eq:grun_t}\n\\eeq\nwith $c_t$ of the same order as $c_0$ in Eq.~(\\ref{eq:grun_k}) \\cite{hw05}\n(in the quoted reference the estimate $c_t \\simeq 0.450 \\; c_0$ was given for the tensor box operator).\nFrom the above form of $\\delta G(t)$ one sees that the amplitude of the quantum correction\nis actually proportional  to the combination $c_0 \/ \\xi^3$ for $\\nu=1\/3$.\nNote also that the running of $G$ induces an effective pressure term in the second \n($rr$) equation, due to the presence of a relativistic fluid whose origin is in\nthe vacuum-polarization contribution.\nAnother noteworthy general feature of the new field equations is the additional power-law\nacceleration contribution, on top of the standard one due to the $\\lambda$ term.\n\n\n\n\\vskip 40pt\n\\subsection{Introduction of the ${\\bf w_{vac}}$ parameter}\n\\hspace*{\\parindent}\n\\label{sec:w_vac}\n\nIt was noted in \\cite{hw05,ht10} that the field equations with a running $ G $, Eqs.~(\\ref{eq:fried_tt})\nand (\\ref{eq:fried_rr}), can be recast in an equivalent, but slightly more appealing, \nform by defining a vacuum-polarization pressure $p_{vac}$ and density \n$\\rho_{vac}$, such that for the FLRW background one has\n\\beq\n\\rho_{vac} (t) = {\\delta G(t) \\over G_0} \\, \\rho (t)  \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \np_{vac} (t) = { 1 \\over 3} \\, {\\delta G(t) \\over G_0} \\, \\rho (t) \\; .\n\\label{eq:rhovac_t}\n\\eeq\nFrom this viewpoint, the inclusion of a vacuum-polarization contribution in the FLRW \nframework seems to amount to a replacement \n\\beq\n\\rho(t) \\rightarrow \\rho(t) + \\rho_{vac} (t)     \n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\np(t) \\rightarrow p(t) +  p_{vac} (t)\n\\label{eq:vac}\n\\eeq\nin the original field equations.\nJust as one introduces the parameter $w$, describing the matter equation of state, \n\\beq\np (t) = w \\,  \\rho(t)\n\\label{eq:w_def}\n\\eeq\nwith $ w=0 $ for nonrelativistic matter, one can do the same for the remaining  contribution\nby setting\n\\beq\np_{vac} (t) = w_{vac} \\; \\rho_{vac} (t) \\; .\n\\label{eq:wvac_def}\n\\eeq\nWe should remark here that the original calculations \\cite{hw05}, and more recently\n\\cite{ht10} which included metric perturbations, also indicate that \n\\beq\nw_{vac}= \\third\n\\label{eq:wvac}\n\\eeq\nis obtained {\\it generally} for the given class of $G(\\Box)$ considered, and is \nnot tied therefore to a specific choice of $\\nu$, such as $\\nu=\\third$.\n\nThe previous, slightly more compact, notation allows one to rewrite the field \nequations for the FLRW background in an equivalent form, which we will describe next.\nWe note here that, when dealing with density perturbations, we will have to distinguish\nthe background, which will involve a background pressure ($\\bar p$) and background\ndensity ($\\bar \\rho$), from the corresponding perturbations which will be denoted here\nby $\\delta p$ and $\\delta \\rho$.\nWith this notation and for constant $G_0$,\n the FLRW field equations for the background are written as\n\\bea\n3 \\, {{\\dot{a}}^2 (t) \\over {a}^2 (t)} \n& = & \n8 \\pi \\, G_0  \\, \\bar{\\rho} (t) + \\lambda \n\\nonumber \\\\\n{{\\dot{a}}^2 (t) \\over {a}^2 (t)} + 2\\, {\\ddot{a} (t) \\over a (t)} \n& = &\n- 8 \\pi \\, G_0 \\, {\\bar p}  (t) + \\lambda \\; .\n\\label{eq:fried_0}\n\\eea\nThen in the presence of a running $G(\\Box)$,  and in accordance with the results \nof Eqs.~(\\ref{eq:fried_tt}) and (\\ref{eq:fried_rr}), \nthe modified FLRW equations for the background read\n\\bea\n3 \\, {{\\dot{a}}^2 (t) \\over {a}^2 (t)} \n& = & \n8 \\pi \\, G_0 \\left ( 1 + {\\delta G(t) \\over G_0} \\right ) \\, \\bar{\\rho}(t) + \\lambda \n\\nonumber \\\\\n{{\\dot{a}}^2 (t) \\over {a}^2 (t)} + 2\\, {\\ddot{a} (t) \\over a (t)} \n& = & \n- 8 \\pi \\, G_0 \\, \\left ( w + w_{vac}  {\\delta G(t) \\over G_0} \\right )\\, \\bar{\\rho} (t) + \\lambda \\; ,\n\\label{eq:fried_run}\n\\eea\nusing the definitions in Eqs.~(\\ref{eq:w_def}) and (\\ref{eq:wvac_def}),\nhere with $\\bar{p}_{vac} (t) = w_{vac} \\,  \\bar{\\rho}_{vac}(t)$.\n\nOf course the procedure of defining a $\\rho_{vac}$ and a $p_{vac}$ contribution,\narising from quantum vacuum-polarization effects, is not necessarily\nrestricted to the FLRW background metric case.\nIn general one can decompose the full source term in the effective nonlocal\nfield equations of Eq.~(\\ref{eq:field1}), making use of\n\\beq\nG(\\Box) = G_0 \\, \\left ( 1 \\, +  {\\delta G(\\Box) \\over G_0} \\right ) \n\\;\\;\\;\\;\\;\\;  {\\rm with} \\;\\;\\;\\;\\;\n{\\delta G(\\Box) \\over G_0} \\equiv c_0 \\left ( { 1 \\over \\xi^2 \\Box } \\right )^{1 \/ 2 \\nu}  \\; ,\n\\label{eq:grun_box_1}\n\\eeq\n as two contributions,\n\\beq\n{ 1 \\over G_0 } \\, G(\\Box) \\, T_{\\mu\\nu}  \\, = \\, \n\\left ( 1 + {\\delta G(\\Box) \\over G_0}  \\right ) \\, T_{\\mu\\nu}  \\, = \\,\nT_{\\mu\\nu}   +  T_{\\mu\\nu}^{vac} \\; .\n\\label{eq:tmunu_vac}\n\\eeq\nThe latter involves the nonlocal part\n\\beq\nT_{\\mu\\nu}^{vac} \\, \\equiv \\,  {\\delta G(\\Box) \\over G_0} \\, T_{\\mu\\nu} \\; .\n\\label{eq:tmunu_vac1}\n\\eeq\nConsistency of the full nonlocal field equations requires that the sum\nbe conserved,\n\\beq\n\\nabla^\\mu \\left ( T_{\\mu\\nu}   +  T_{\\mu\\nu}^{vac}  \\right ) = 0 \\; .\n\\eeq\nIn general one cannot expect that the contribution $ T_{\\mu\\nu}^{vac} $\nwill always be expressible in the perfect fluid form of Eq.~(\\ref{eq:tmunu_perf}), even if the\noriginal $ T_{\\mu\\nu} $ for matter (or radiation) has such a form.\nThe former will in general contain, for example, nonvanishing shear stress contributions, \neven if they were originally absent in the matter part.\n\n\n\n\\vskip 40pt\n\\newsection{Relativistic treatment of matter density perturbations}\n\\hspace*{\\parindent}\n\\label{sec:pert}\n\n\nBesides the modified cosmic scale factor evolution just discussed, \nthe running of $G(\\Box)$, as given in Eq.~(\\ref{eq:grun_box}),\nalso affects the nature of matter density perturbations on large scales.\nIn computing these effects, it is customary to introduce a perturbed metric of\nthe form\n\\beq\n{d\\tau}^2 = {dt}^2 - a^2 \\left ( \\delta_{ij} + h_{ij} \\right ) dx^i dx^j \\; ,\n\\label{eq:pert_metric}\n\\eeq\nwith $a(t)$ the unperturbed scale factor and $ h_{ij} ({\\bf x},t)$ a small\nmetric perturbation, and $h_{00}=h_{i0}=0$ by choice of coordinates.\nAfter decomposing the matter fields into background and fluctuation contribution, \n$\\rho = \\bar{\\rho}+\\delta \\rho$, $p = \\bar{p}+\\delta p $, and ${\\bf v} = \\bar{\\bf v}+\\delta {\\bf v}$, \nit is customary in these treatments to expand the density, pressure and metric\nperturbations in spatial Fourier modes,\n\\bea\n\\delta \\rho ({\\bf x},t) & = & \\delta \\rho_{\\bf q} (t) \\, e^{i \\, {\\bf q} \\, \\cdot \\, {\\bf x}}\n\\;\\;\\;\\;\\;\\;\\;\\;\n\\delta p ({\\bf x},t) = \\delta p_{\\bf q} (t) \\, e^{i \\, {\\bf q} \\,\\cdot \\, {\\bf x}}\n\\nonumber \\\\\n\\delta {\\bf v} ({\\bf x},t) & = & {\\delta {\\bf v}}_{\\bf q} (t)  \\, e^{i \\, {\\bf q} \\, \\cdot \\, {\\bf x}}\n\\;\\;\\;\\;\\;\\;\\;\\;\nh_{ij} ({\\bf x},t) = h_{ {\\bf q} \\, ij} (t)\\, e^{i \\, {\\bf q} \\, \\cdot \\, {\\bf x}} \n\\label{eq:fourier}\n\\eea\nwith ${\\bf q}$ the comoving wave number.\nOnce the Fourier coefficients have been determined, the original perturbations can\nlater be obtained from\n\\beq\n\\delta \\rho ({\\bf x},t) \\, = \\, \\int { d^3 {\\bf x} \\over ( 2 \\pi )^{3\/2} }\n\\, e^{ - i \\, {\\bf q} \\, \\cdot \\, {\\bf x}} \\, \\delta \\rho_{\\bf q} (t) \n\\eeq\nand similarly for the other fluctuation components.\nThen the field equations with a constant $G_0$ [Eq.~(\\ref{eq:field})]\nare given to zeroth order in the perturbations by\nEq.~(\\ref{eq:fried_0}),  which fixes the three background fields \n$a(t)$, $\\bar{\\rho} (t)$, and $\\bar{p} (t) $.\nNote that in a comoving frame the four-velocity appearing in Eq.~(\\ref{eq:tmunu_perf})\nhas components $ u^i = 1, \\; u^0 = 0 $.\nWithout $G(\\Box)$, to first order in the perturbations and in the limit \n${\\bf q} \\rightarrow 0$ the field equations give\n\\bea\n{\\dot{a} (t) \\over a (t)}\\, \\dot{h} (t) & = & 8 \\pi \\, G_0  \\, \\bar{\\rho} (t) \\, \\delta (t) \n\\nonumber \\\\\n\\ddot{h} (t) + 3 \\, {\\dot{a} (t) \\over a (t)}\\, \\dot{h} (t) \n& = & - \\, 24 \\pi \\, G_0  \\, w \\, \\bar{\\rho}(t) \\, \\delta (t)\n\\eea\nwith the matter density contrast defined as $\\delta (t) \\equiv \\delta \\rho (t) \/ \\bar{\\rho} (t) $, \n$h(t) \\equiv h_{ii} (t)$ the trace part of $h_{ij}$, and $w=0$ for nonrelativistic matter.\nWhen combined together, these last two equations then yield \na single equation for the trace of the metric perturbation,\n\\beq\n\\ddot{h} (t) + 2 \\, {\\dot{a} (t) \\over a (t)} \\, \\dot{h} (t) \\; = \\; \n - \\, 8 \\pi \\, G_0 ( 1 + 3\\, w ) \\, \\bar{\\rho}(t) \\, \\delta (t) \\; .\n\\eeq\nFrom first order energy conservation, one has \n$ - {1 \\over 2}\\, \\left ( 1 + w \\right )\\, h (t) = \\delta (t) $, \nwhich then allows one to eliminate $h(t)$ in favor of $\\delta(t)$, which then\nallows one to obtain a single second order equation for the density contrast $\\delta(t)$.\nIn the case of a running $G(\\Box)$, the above equations need to be rederived\nfrom the effective covariant field equations of Eq.~(\\ref{eq:field1}), and lead to several\nadditional terms not present at the classical level \\cite{ht10}.\n\n\n\n\n\\vskip 40pt\n\\subsection{Zeroth order energy-momentum conservation}\n\\hspace*{\\parindent}\n\\label{sec:enmom_zeroth}\n\nAs a first step in computing the effects of density matter perturbations, one\nneeds to examine the consequences of energy and momentum conservation,\nto zeroth and first order in the relevant perturbations.\nIf one takes the covariant divergence of the field equations in Eq.~(\\ref{eq:field1}),\nthe left-hand side has to vanish identically because of the Bianchi identity. \nThe right-hand side then gives \n$\\nabla^\\mu \\left (  T_{\\mu\\nu} + T_{\\mu\\nu}^{vac} \\right ) =0 $,\nwhere the fields in $ T_{\\mu \\nu}^{vac} $ can be expressed, at least to lowest order,\nin terms of the $p_{vac}$ and $\\rho_{vac}$ fields defined in Eqs.~(\\ref{eq:rhovac_t}) and (\\ref{eq:wvac_def}).\nThe first equation one obtains is the zeroth (in the fluctuations) order energy conservation\nin the presence of $G(\\Box)$, which reads\n\\beq\n3 \\, {\\dot{a} (t) \\over a (t)} \\, \n\\left [ \\left (1+w \\right ) + \\left (1+ w_{vac} \\right )\\,{\\delta G(t) \\over G_0} \n\\right ]  \\bar{\\rho} (t) \n +   { \\dot{\\delta G}(t) \\over G_0} \\, \\bar{\\rho} (t)\n + \\left ( 1 + {\\delta G(t) \\over G_0} \\right )\\, \\dot{\\bar{\\rho}} (t)  = 0 \\; .\n\\label{eq:encons_zeroth}\n\\eeq\nIn the absence of a running $G$ these equations reduce to the ordinary mass\nconservation equation for $w=0$,\n\\beq\n\\dot{\\bar{\\rho}}(t) = - 3 \\, { \\dot{a}(t) \\over a(t)} \\, \\bar{\\rho}(t) \\; .\n\\label{eq:encons_frw}\n\\eeq\nIt is often convenient to solve the energy conservation\nequation not for ${\\bar{\\rho}} (t)$, but instead for ${\\bar{\\rho}} (a)$.\nThis requires that, instead of using the expression for $G(t)$ in Eq.~(\\ref{eq:grun_t}),\none uses the equivalent expression for $G(a)$\n\\beq\nG (a) = G_0 \\left ( 1 + {\\delta G (a) \\over G_0}  \\right ) \\; ,\n\\label{eq:grun_a}\n\\eeq\nwhich is easily obtained once the relationship between $t$ and $a(t)$ is known (see discussion later).\nNote for example that the solution to Eq.~(\\ref{eq:encons_zeroth}) can be written as\n\\beq\n\\bar{\\rho} (a)= {\\rm const. } \\; \\exp \\left \\{\n- \\int  { d a \\over a} \\;\n\\left (  3 + { \\delta G (a) \\over G_0 } + a \\, { \\delta G' (a) \\over G_0  } \\right ) \\,\n\\right \\} \\; .\n\\label{eq:rho_zeroth_sim}\n\\eeq\n\n\n\n\n\n\\vskip 40pt\n\\subsection{Effective energy-momentum tensor involving ${\\bf \\rho_{vac}}$ and ${\\bf p_{vac}}$}\n\\hspace*{\\parindent}\n\\label{sec:enmom_vac}\n\nThe next step consists in obtaining the equations which govern the effects of \nsmall field perturbations.\nThese equations will involve, apart from the metric perturbation $h_{ij}$, the matter \nand vacuum-polarization contributions.\nThe latter arise from [see Eq.~(\\ref{eq:tmunu_vac})]\n\\beq\n\\left ( 1 + {\\delta G(\\Box) \\over G_0}  \\right ) \\, T_{\\mu\\nu}  \\, = \\,\nT_{\\mu\\nu}   +  T_{\\mu\\nu}^{vac} \n\\eeq\nwith a nonlocal \n$ T_{\\mu\\nu}^{vac} \\equiv ( \\delta G(\\Box) \/ G_0 ) \\, T_{\\mu\\nu} $.\nFortunately to zeroth order in the fluctuations the results of Ref. \\cite{hw05} indicated \nthat the modifications from the nonlocal vacuum-polarization term could \nsimply be accounted for by the substitution\n\\beq\n\\bar{\\rho} (t) \\rightarrow \\; \\bar{\\rho} (t) + {\\bar\\rho}_{vac} (t)     \n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\bar{p} (t) \\rightarrow \\; \\bar{p} (t) +  {\\bar p}_{vac} (t) \\; .\n\\label{eq:sub0}\n\\eeq\nHere we will apply this last result to the small field fluctuations as well, and set\n\\beq \n\\delta \\rho_{\\bf q} (t) \\rightarrow \\; \\delta \\rho_{\\bf q} (t) + \\delta \\rho_{{\\bf q} \\, vac} (t)\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\delta p_{\\bf q}  (t) \\rightarrow \\; \\delta p_{\\bf q}  (t) +  \\delta p_{{\\bf q} \\, vac} (t) \\; .\n\\label{eq:sub1}\n\\eeq\nThe underlying assumption is of course that the equation of state for the vacuum fluid still\nremains roughly correct when a small perturbation is added.\nFurthermore, just like we had $ {\\bar p} (t) = w \\,  \\bar{\\rho} (t) $ [Eq.~(\\ref{eq:w_def})]\nand  $\\bar{p}_{vac} (t) = w_{vac} \\,  \\bar{\\rho}_{vac}(t) $ [Eq. ~(\\ref{eq:wvac_def})]\nwith $ w_{vac} = \\third $, we now write for the fluctuations\n\\beq\n\\delta p_{\\bf q}  (t) = w \\, \\delta \\rho_{\\bf q}   (t) \\;\\;\\;\\;\\;\\;\\;\\;\\; \n\\delta p_{{\\bf q} \\, vac} (t) = w_{vac} \\, \\delta \\rho_{{\\bf q} \\, vac} (t) \\; ,\n\\label{eq:wvac_fluc}\n\\eeq\nat least to leading order in the long wavelength limit, ${\\bf q} \\rightarrow 0 $.\nIn this limit we then have simply\n\\beq\n\\delta p (t) = w \\, \\delta \\rho   (t) \\;\\;\\;\\;\\;\\;\\;\\;\\; \n\\delta p_{vac} (t) = w_{vac} \\, \\delta \\rho_{vac} (t) \\equiv \nw_{vac}  \\, {\\delta G(t) \\over G_0}  \\delta \\rho (t) \\; ,\n\\label{eq:wvac_fluc1}\n\\eeq\nwith $G(t)$ given in Eq.~(\\ref{eq:grun_t}), and we have used Eq.~(\\ref{eq:rhovac_t}), \nnow applied to the fluctuation $\\delta \\rho_{vac} (t)$,\n\\beq\n\\delta \\rho_{vac} (t) \\, = \\, {\\delta G(t) \\over G_0}  \\, \\delta \\rho (t) + \\dots\n\\label{eq:delta_rhovac_t}\n\\eeq\nwhere the dots indicate possible additional $O(h)$ contributions.\nA bit of thought reveals that the above treatment is incomplete,\nsince $G(\\Box)$ in the effective field equation of Eq.~(\\ref{eq:field1}) \ncontains, for the perturbed Robertson-Walker metric of Eq.~(\\ref{eq:pert_metric}), \nterms of order $h_{ij}$,  which need to be accounted for in the effective $T^{\\mu\\nu}_{vac}$.\nConsequently the covariant d'Alembertian operator \n$ \\Box \\; = \\; g^{\\mu\\nu} \\, \\nabla_\\mu \\nabla_\\nu  $\nacting here on second rank tensors, such as $T_{\\mu\\nu}$,\n\\bea\n\\nabla_{\\nu} T_{\\alpha\\beta} \\, = \\, \\partial_\\nu T_{\\alpha\\beta} \n- \\Gamma_{\\alpha\\nu}^{\\lambda} T_{\\lambda\\beta} \n- \\Gamma_{\\beta\\nu}^{\\lambda} T_{\\alpha\\lambda} \\, \\equiv \\, I_{\\nu\\alpha\\beta}\n\\nonumber\n\\eea\n\\beq \n\\nabla_{\\mu} \\left ( \\nabla_{\\nu} T_{\\alpha\\beta} \\right )\n= \\, \\partial_\\mu I_{\\nu\\alpha\\beta} \n- \\Gamma_{\\nu\\mu}^{\\lambda} I_{\\lambda\\alpha\\beta} \n- \\Gamma_{\\alpha\\mu}^{\\lambda} I_{\\nu\\lambda\\beta} \n- \\Gamma_{\\beta\\mu}^{\\lambda} I_{\\nu\\alpha\\lambda}  \\; ,\n\\label{eq:box_on_tensors}\n\\eeq\nneeds to be Taylor expanded in the small field perturbation $h_{ij}$,\n\\beq\n\\Box (g) \\, = \\, \\Box^{(0)}  + \\Box^{(1)} (h) + O (h^2) \\; .\n\\eeq\nOne then obtains for $G(\\Box)$ itself\n\\beq\nG(\\Box) \\, = \\, G_0 \\,  \\left [ \n1 + \\, { c_0 \\over \\xi^{1 \/ \\nu} } \\,  \n\\left ( { 1 \\over \\Box^{(0)} + \\Box^{(1)} (h) + O(h^2) } \\right )^{1\/ 2 \\nu}  + \\dots\n\\right ]  \\; ,\n\\label{eq:gbox_h}\n\\eeq\nwhich requires the use of the binomial expansion for the operator\n$ (A+B)^{-1} = A^{-1} - A^{-1} B A^{-1}  + \\dots $.\nThus for sufficiently small perturbations it should be adequate to expand $G(\\Box)$ \nentering the effective field equations in powers of the metric perturbation $h_{ij} $.\nNext we turn to a discussion of the above results in different gauges.\n\n\n\n\n\n\\vskip 40pt\n\\newsection{Gauge choices and corresponding transformations}\n\\hspace*{\\parindent}\n\\label{sec:gauge}\n\nThe previous discussion and summary focused exclusively on the comoving gauge\nchoice for the metric, implicit in the definition of Eq.~(\\ref{eq:frw}).\nNext we will consider some additional gauges.\nIn this paper we will specifically refer to {\\it three} choices for the metric: the\ncomoving, synchronous and conformal Newtonian forms.\nThe first two are closely related to each other, and were used to obtain\npart of the results presented in our previous work \\cite{hw05,ht10}, \nwhich was summarized in the previous section.\nNote that in our previous work \\cite{ht10} we did not include the effects of\na stress field $s$, since it was not necessary for the discussion of density perturbations;\nnew terms arising from such a field are included below.\nThe third form of the metric is the primary focus of the present discussion;\nthe results obtained later on in this paper will either be derived for this metric,\nor transformed to it by relying on results obtained previously in the other gauges.\n\n\n\\vskip 40pt\n\\subsection{Comoving, synchronous and conformal Newtonian gauges}\n\\hspace*{\\parindent}\n\\label{sec:metric}\n\n\nThe {\\it comoving} metric has the form\n\\beq\ng_{\\mu \\, \\nu} = \\bar{g}_{\\mu \\, \\nu} + h_{\\mu \\, \\nu} \\; ,\n\\eeq\nwith background metric\n\\beq\n\\bar{g}_{\\mu \\, \\nu} = {\\rm diag} \\left(-1, a^2, a^2, a^2 \\right) \\; .\n\\eeq\nFor the fluctuation one sets\n\\beq\nh_{0i} = h_{i0} = 0 \\; ,\n\\eeq\nand decomposes the remaining $h_{ij}$ as\n\\beq\nh_{ij} ({\\bf k},t) \\; = \\;  a^2\n\\left[ \\, { 1 \\over 3 }\\,  h \\, \\delta_{ij} \n+ \\left( {1 \\over 3} \\, \\delta_{ij} - {k_i \\, k_j \\over k^2} \\right) \\, s  \\right]\n\\label{eq:stress_def}\n\\eeq\nso that $ Tr(h_{ij}) = a^2 \\, h $.\nBesides the scale factor $a$, the metric is therefore parametrized in terms of the\ntwo functions $s$ and $h$.\n\nOn the other hand, in the {\\it synchronous} gauge one sets again \n$ g_{\\mu \\, \\nu} = \\bar{g}_{\\mu \\, \\nu} + h_{\\mu \\, \\nu} $ \nnow with background metric\n\\beq\n\\bar{g}_{\\mu \\, \\nu} =a^2 \\, {\\rm diag} \\left(-1, 1, 1, 1 \\right) \\; .\n\\eeq\nFor the fluctuation one sets again \n$ h_{0i} = h_{i0} = 0 $ and \n\\beq\nh_{ij} ({\\bf k},t)\n\\; = \\; a^2\n\\left[ \\, {k_i \\, k_j \\over k^2} \\, h_{sync} \n+ \\left( {k_i \\, k_j \\over k^2} - {1 \\over 3} \\, \\delta_{ij} \\right) \\, 6 \\, \\eta \n\\right]  \\; ,\n\\eeq\nso that now $ Tr(h_{ij} ) = a^2 \\, h_{sync} $.\nHere, besides the overall scale factor $a$, the metric is parametrized in terms of the\ntwo functions $\\eta$ and $h_{sync}$.\nFrom a comparison of the two gauges (comoving and synchronous) one has\n\\beq\n2 \\, \\eta \\; = \\; - {1 \\over 3} \\left( h + s \\right)\n\\eeq\nand \n\\beq\n h_{sync} + 6 \\, \\eta \\; = \\; - s  \\; .\n\\eeq\n\nFinally the {\\it conformal Newtonian} gauge is in turn described by two scalar potentials \n$ \\psi $ and $\\phi $.\nIn this case the line element is given by\n\\beq\nd\\tau^2 = - g_{\\mu \\, \\nu} dx^\\mu \\, dx^\\nu = a^2 \\, \\Big\\lbrace \\left( 1 + 2 \\, \\psi \\right) \\, dt^2 - \\left(1 - 2 \\, \\phi \\right) \\, dx_i \\, dx^i \\Big\\rbrace \\; .\n\\eeq\nTherefore for the metric itself one writes again\n$ g_{\\mu \\, \\nu} = \\bar{g}_{\\mu \\, \\nu} + h_{\\mu \\, \\nu} $\nwith\n$ \\bar{g}_{\\mu \\, \\nu} =a^2 \\, diag \\left(-1, 1, 1, 1 \\right) $\nas for the synchronous case,  and furthermore \n$ h_{0i} = h_{i0} = 0 $ as before, and now\n\\beq\nh_{00} = a^2 \\, \\left(- \\, 2 \\, \\psi \\right)\n\\eeq\n\\beq\nh_{ij} = a^2 \\, \\left(- \\, 2 \\phi \\right) \\, \\delta_{ij} \\; .\n\\eeq\nA suitable set of gauge transformations then allows one to go from the synchronous, or comoving, to the conformal Newtonian gauge \\cite{mab95}.\n\n\n\n\n\\vskip 40pt\n\\subsection{Tensor box in the comoving gauge}\n\\hspace*{\\parindent}\n\\label{sec:tensorbox}\n\n\nTo compute higher order contributions from the $h_{ij}$ 's appearing in the \ncomoving  gauge metric, one needs to expand $G(\\Box)$ in the various metric\nperturbations,\n\\beq\nG(\\Box) = G_0 \\, \\left [ \n1 + \\, { c_0 \\over \\xi^{1 \/ \\nu} } \\, \n\\left (\n\\left ( { 1 \\over \\Box^{(0)} } \\right )^{1 \/ 2 \\nu} \n- {1 \\over 2 \\, \\nu} \\, { 1 \\over \\Box^{(0)} } \\cdot \\Box^{(1)} (h,s) \\cdot\n\\left ( { 1 \\over \\Box^{(0)} } \\right )^{1 \/ 2 \\nu} \\, + \\dots\n\\right )\n\\right ] \\; ,\n\\label{eq:gbox_hs_expanded}\n\\eeq\nwhere the superscripts $(0)$ and $(1)$ refer to zeroth and first order in this expansion, respectively.\nTo get the correction of $O(h,s)$ to the field equations, one therefore needs to consider\nthe relevant term in the expansion of $ ( 1 + \\delta G(\\Box) \/ G_0 ) \\, T_{\\mu \\nu} $,\n\\beq\n- {1 \\over 2 \\, \\nu} \\, { 1 \\over \\Box^{(0)} } \\cdot \\Box^{(1)} (h,s) \\cdot\n{ \\delta G ( \\Box^{(0)} ) \\over G_0 } \\cdot T_{\\mu \\nu}\n\\; = \\; \n- {1 \\over 2 \\, \\nu} \\, { c_0 \\over \\xi^{1 \/ \\nu} } \\,\n{ 1 \\over \\Box^{(0)} } \\cdot \\Box^{(1)} (h,s) \\cdot \n\\left ( { 1 \\over \\Box^{(0)} } \\right )^{1 \/ 2 \\nu} \\cdot T_{\\mu \\nu} \\; .\n\\label{gbox_hs_tensor}\n\\eeq\nThis last form allows us to use the results obtained previously \nfor the FLRW case, namely\n\\beq\n{ \\delta G ( \\Box^{(0)} ) \\over G_0 } \\, T_{\\mu \\nu} \\; = \\; T_{\\mu \\nu}^{vac}\n\\eeq\nwith here\n\\beq\nT_{\\mu \\nu}^{vac} \\; = \\;\n\\left [ p_{vac} (t) + \\rho_{vac} (t) \\right ] u_\\mu \\, u_\\nu + g_{\\mu \\nu} \\, p_{vac} (t)\n\\label{eq:tmunu_vac_2}\n\\eeq\nto zeroth order in $h$, and\n\\beq\n\\rho_{vac} (t) = {\\delta G(t) \\over G_0} \\, \\bar{\\rho} (t) \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \np_{vac} (t) = w_{vac} \\, {\\delta G(t) \\over G_0} \\, \\bar{\\rho} (t) \\; .\n\\label{eq:rhovac_t_1}\n\\eeq\nand $w_{vac} = 1\/3$.\nTherefore, in light of the results of Ref. \\cite{hw05}, the problem has been \nreduced to computing the more tractable expression\n\\beq\n- {1 \\over 2 \\, \\nu} \\, \n{ 1 \\over \\Box^{(0)} } \\cdot \\Box^{(1)} (h,s) \\cdot T_{\\mu \\nu}^{vac} \\; .\n\\label{gbox_hs_tensor_vac}\n\\eeq\nTo make progress, we will assume a harmonic time dependence for both\nthe perturbations $h(t) = h_0 \\, e^{i \\omega t}$ and $s(t) = s_0 \\, e^{i \\omega t}$, and \nfor the background quantities \n$a(t)=a_0 \\, e^{i \\Gamma t}$, $\\rho (t)= \\rho_0 \\, e^{i \\Gamma t}$, and \\\n$\\delta G (t)= \\delta G_0 \\, e^{i \\Gamma t}$.\nFrom now on we shall consider both $\\omega$ and $\\Gamma$ as slowly varying\nfunctions (indeed constants),\nwith the time scale of variations for the perturbation much shorter\nthan the time scale associated with all the background quantities.\nA more sophisticated treatment will be reserved for future work.\nTherefore we will take here $\\omega \\gg \\Gamma$ or $ \\dot{h} \/ h \\gg \\dot{a} \/ a $,\nwhich is the same approximation that was used in obtaining the results of Ref. \\cite{ht10}.\n\nLet us now list, in sequence, the required matrix elements needed for the present calculation.\nFor the tensor box $tt$ matrix element \n$ ( - {1 \\over 2 \\, \\nu} \\, \n{ 1 \\over \\Box^{(0)} } \\cdot \\Box^{(1)} (h,s) \\cdot T^{vac} )_{00} $ one obtains\n\\beq\n+ \\, {1 \\over 2 \\nu} \\, {11 \\over 3} \\, {\\delta G(t) \\over G_0} \\, \\rho(t) \\, {\\Gamma \\over \\omega} \\, h \n+ {\\mathcal{O}} (k^2) \\; .\n\\eeq\nFor the tensor box $ti$ matrix element \n$ ( - {1 \\over 2 \\, \\nu} \\, \n{ 1 \\over \\Box^{(0)} } \\cdot \\Box^{(1)} (h,s) \\cdot T^{vac} )_{0i} $ one obtains\n\\beq\n- \\, i \\, k_i \\, {1 \\over 2 \\nu} \\, {2 \\over 9} \\, {\\delta G(t) \\over G_0} \\, \\rho(t) \n\\, {1 \\over i \\, \\omega} \\, \\left( h - 2 \\, s \\right) + {\\mathcal{O}} (k^2) \\; .\n\\eeq\nFor the tensor box $ii$ matrix element, summed over $i$, \n$ ( - {1 \\over 2 \\, \\nu} \\, \n{ 1 \\over \\Box^{(0)} } \\cdot \\Box^{(1)} (h,s) \\cdot T^{vac} )_{ii} $, one obtains\n\\beq\n3 \\, \\left(+ \\, {1 \\over 2 \\nu} \\, w_{vac} \\, { 11 \\over 3 } \\, a^2 \\, {\\delta G(t) \\over G_0} \\, \\rho(t) \\, {\\Gamma \\over \\omega} \\, h \\right) + {\\mathcal{O}} (k^2) \\; .\n\\eeq\nFor the tensor box $ii$ matrix element, not summed over $i$, \n$ ( - {1 \\over 2 \\, \\nu} \\, \n{ 1 \\over \\Box^{(0)} } \\cdot \\Box^{(1)} (h,s) \\cdot T^{vac} )_{ii} $, one obtains\n\\beq\n+ \\, {1 \\over 2 \\nu} \\, a^2 \\, {\\delta G(t) \\over G_0} \\, \\rho(t) \n\\, \\left[w_{vac} \\, {11 \\over 3} \\, {\\Gamma \\over \\omega} \\, h \n+ {8 \\over 9} \\, \\left(1 - 3 \\, {k_i \\over k^2} \\, \\right) \\, {\\Gamma \\over \\omega} \\, s \\right]\n+ {\\mathcal{O}} (k^2) \\; .\n\\eeq\nFinally for the tensor box $ij$ matrix element,\n$ (- {1 \\over 2 \\, \\nu} \\, \n{ 1 \\over \\Box^{(0)} } \\cdot \\Box^{(1)} (h,s) \\cdot T^{vac} )_{ij} $, one obtains\n\\beq\n- \\, {k_i \\, k_j \\over k^2} \\, {1 \\over 2 \\nu} \\, a^2 \\, {8 \\over 3} \\, {\\delta G(t) \\over G_0} \\, \\rho(t) \\, {\\Gamma \\over \\omega} \\, s + {\\mathcal{O}} (k^2) \\; .\n\\eeq\nThe above expressions are now inserted in the general effective field equations of\nEq.~(\\ref{eq:field1}), and will give rise to a set of effective field equations appropriate for\nthis particular gauge, to first order in the field perturbation and with the effects\nof $G(\\Box)$ included.\n\n\n\n\n\n\\vskip 40pt\n\\subsection{Field equations in the comoving, synchronous and conformal Newtonian gauges}\n\\hspace*{\\parindent}\n\\label{sec:efe}\n\nAs a result of the previous manipulations one obtains in the comoving gauge\nwith fields $ (h, s) $ the following $tt$, $ti$, $ii$ (or $xx+yy+zz$), and $ij$ field\nequations\n\\beq\n{k^2 \\over 3 \\, a^2} \\left(h + s \\right) \n+ {\\dot{a} \\over a } \\, {\\dot{h}} \n= 8 \\pi G_0 \\left(1 + {\\delta G \\over G_0} \\right) \\, {\\bar{\\rho}} \\, \\delta \n+ 8 \\pi G_0 \\, {\\delta G \\over G_0} \\, {c_h \\over 2 \\nu} \\, h \\, {\\bar{\\rho}} \n+ {\\mathcal{O}} (k^2)\n\\label{eq:efe_00_comov}\n\\eeq\n\\beq\n- {1 \\over 3} \\left( \\dot{h} + \\dot{s} \\right) \n= 8 \\pi G_0 \\, {\\delta G \\over G_0} \\left( - {1 \\over 2 \\nu} \\right) \n\\, {2 \\over 9} \\, {1 \\over i \\omega} \\, \\left( h - 2 s \\right)\\, {\\bar{\\rho}} + {\\mathcal{O}} (k^2)\n\\label{eq:efe_0i_comov}\n\\eeq\n\\beq\n- \\, {1 \\over 3} {k^2 \\over a^2} \\, \\left( h + s \\right) \n- 3 {\\dot{a} \\over a} \\, \\dot{h} - \\ddot{h} \n= 24 \\pi G_0 \\, {\\delta G \\over G_0} \\, w_{vac} \\, \\bar{\\rho} \\, \\delta \n+ 24 \\pi G_0 \\, {\\delta G \\over G_0} \\, w_{vac}\\,{c_h \\over 2 \\nu} \\, h \\bar{\\rho} \n+ {\\mathcal{O}} (k^2)\n\\label{eq:efe_ii_comov}\n\\eeq\n\\beq\n{1 \\over 6} \\, {k^2 \\over a^2} \\, \\left( h + s \\right) \n- {3 \\over 2 } \\, {\\dot{a} \\over a} \\dot{s} \n- {1 \\over 2} \\, \\ddot{s} \n= - \\, 8 \\pi G_0 \\, {\\delta G \\over G_0} \\, {c_s \\over 2 \\nu} \\, s \\, \\bar{\\rho} \n+ {\\mathcal{O}} (k^2) \\; .\n\\label{eq:efe_ij_comov}\n\\eeq\nAs in Ref. \\cite{ht10}, we have found it convenient to here to set in the above\nexpressions \n\\beq\nc_s \\; \\equiv \\; \\, \\left( {8 \\over 3} \\right) \\, {\\Gamma \\over \\omega}\n\\label{eq:cs}\n\\eeq\nand\n\\beq\nc_h \\; = \\; \\equiv (-1) \\, \\left( - {11 \\over 3} \\right) \\, {\\Gamma \\over \\omega} \n\\;  = \\;  \\, {11 \\over 3}\\, {\\Gamma \\over \\omega}  \\; .\n\\label{eq:ch}\n\\eeq\nIn the field equations listed above the terms $ {\\mathcal{O}} (k^2) $ arise because \nof terms ${\\mathcal{O}} (k^2) $ in the expansion of the tensor box operator.\n\n\nThe next step is to convert the left-hand sides of the above field equations, namely\nEqs.~(\\ref{eq:efe_00_comov}), (\\ref{eq:efe_0i_comov}), (\\ref{eq:efe_ii_comov}) and (\\ref{eq:efe_ij_comov}), which are all \nexpressed in the comoving gauge $ (h, s) $, to the synchronous gauge\nwith fields $ (h_{sync}, \\eta)$.\nThe result of this change of gauge is the sequential replacement\n\\bea\n{k^2 \\over 3 \\, a^2} \\left(h + s \\right) \n+ {\\dot{a} \\over a } \\, {\\dot{h}} \n& \\;\\longrightarrow\\; &\n- 2 \\, {k^2 \\over a^2 } \\eta + {1 \\over a^2} \\, {\\dot{a} \\over a} \\, \\dot{h}_{sync} \n\\nonumber \\\\\n- {1 \\over 3} \\left( \\dot{h} + \\dot{s} \\right) \n& \\;\\longrightarrow\\; &\n2 \\, \\dot{\\eta} \n\\nonumber \\\\\n- \\, {1 \\over 3} {k^2 \\over a^2} \\, \\left( h + s \\right) \n- 3 {\\dot{a} \\over a} \\, \\dot{h} - \\ddot{h} \n& \\;\\longrightarrow\\; &\n2 \\, {k^2 \\over a^2 } \\, \\eta \n- {1 \\over a^2} \\, \\ddot{h}_{sync}\n- 2 \\, {1 \\over a^2 }\\, {\\dot{a} \\over a} \\, \\dot{h}_{sync}\n\\nonumber \\\\\n{1 \\over 6} \\, {k^2 \\over a^2} \\, \\left( h + s \\right) \n- {3 \\over 2 } \\, {\\dot{a} \\over a} \\dot{s} \n- {1 \\over 2} \\, \\ddot{s} \n& \\;\\longrightarrow\\; &\n- \\, {k^2 \\over a^2 } \\, \\eta \n+ {1 \\over 2} \\, {1 \\over a^2} \\, \\left( \\ddot{h}_{sync} + 6 \\, \\ddot{\\eta} \\right) \n+ {1 \\over a^2} \\, {\\dot{a} \\over a} \\, \\left( \\dot{h}_{sync} + 6 \\, \\dot{\\eta} \\right) \n\\eea\nThe next step involves one more transformation, this time\nfrom the synchronous $(h_{sync}, \\eta)$ to the desired\nconformal Newtonian $(\\phi,\\psi)$  gauge,\n\\bea\n{1 \\over a^2} \\left[\n- 2 \\, k^2 \\, \\eta + {\\dot{a} \\over a} \\, \\dot{h}_{sync} \n\\right]\n& \\longrightarrow &\n- \\, {2 \\over a^2} \\left[\nk^2 \\, \\phi \n+ 3 \\, {\\dot{a} \\over a} \\, \\left( \\dot{\\phi} \n+ { \\dot{a} \\over a} \\, \\psi \\right) \n\\right]\n\\nonumber \\\\\n2 \\, \\dot{\\eta} \n& \\longrightarrow &\n2 \\, \\left( \\dot{\\phi} + {\\dot{a} \\over a} \\, \\psi \\right)\n\\nonumber \\\\ \n{1 \\over a^2 } \\left[\n2 \\, k^2 \\, \\eta \n- \\ddot{h}_{sync}\n- 2 \\, {\\dot{a} \\over a} \\, \\dot{h}_{sync}\n\\right]\n& \\longrightarrow &\n{6 \\over a^2} \\left[\n\\, \\ddot{\\phi} \n+ {\\dot{a} \\over a} \\, \\left( \\dot{\\psi} + 2 \\, \\dot{\\phi} \\right) \n+ \\left( 2 \\, {\\ddot{a} \\over a}\n- {\\dot{a}^2 \\over a^2} \\right) \\, \\psi \n+ {k^2 \\over 3} \\left( \\phi - \\psi \\right) \n\\right]\n\\nonumber \\\\\n{1 \\over a^2 } \\left[\n- \\, k^2 \\, \\eta \n+ {1 \\over 2} \\, \\left( \\ddot{h}_{sync} + 6 \\, \\ddot{\\eta} \\right) \n+ {\\dot{a} \\over a} \\, \\left( \\dot{h}_{sync} + 6 \\, \\dot{\\eta} \\right)\n\\right] \n& \\longrightarrow &\n- \\, {k^2 \\over a^2} \\, \\left( \\phi - \\psi \\right) \\; .\n\\eea\nEquivalently, the above sequence of two transformations can be described by a single\ntransformation, from comoving $(h,s)$ to conformal Newtonian $(\\phi,\\psi)$ gauge,\nwhich is trivially obtained by combining the previous two.\nThe final outcome of all these manipulations is to achieve a rewrite of\nthe full set of four original field equations, given in \nEqs.~(\\ref{eq:efe_00_comov}), (\\ref{eq:efe_0i_comov}), \n(\\ref{eq:efe_ii_comov}) and (\\ref{eq:efe_ij_comov}),\nnow with the left hand side given in the conformal Newtonian gauge \nand the right hand side left in the original comoving gauge.\nOne obtains\n\\beq\nk^2 \\, \\phi \n+ 3 \\, {\\dot{a} \\over a} \\, \\left( \\dot{\\phi} \n+ { \\dot{a} \\over a} \\, \\psi \\right) \n= - \\, 4 \\pi G_0 \\, a^2 \\, \\left( 1 + {\\delta G \\over G_0} \\right) \\, \\bar{\\rho} \\, \\delta \n- 4 \\pi G_0 \\, a^2 \\, {\\delta G \\over G_0} \\, {c_h \\over 2 \\nu} \\, h \\, \\bar{\\rho} \\;\n+ \\; {\\mathcal{O}} (k^2)\n\\label{eq:efe_00_cN}\n\\eeq\n\\beq\n\\left( \\dot{\\phi} + {\\dot{a} \\over a} \\, \\psi \\right) \n= 4 \\pi G_0 \\, {\\delta G \\over G_0} \\, \\left( - \\, {1 \\over 2 \\nu} \\right) \n\\, {2 \\over 9} \\,{1 \\over i \\omega} \\, \\left(h - 2 \\, s \\right) \\, \\bar{\\rho} + \\; {\\mathcal{O}} (k^2)\n\\label{eq:efe_0i_cN}\n\\eeq\n\\bea\n\\ddot{\\phi} \n+ {\\dot{a} \\over a} \\, \\left( \\dot{\\psi} + 2 \\, \\dot{\\phi} \\right) \n+ \\left( 2 \\, {\\ddot{a} \\over a} \n- {\\dot{a}^2 \\over a^2} \\right) \\, \\psi \n+ { k^2 \\over 3 } \\left( \\phi - \\psi \\right) \n& = & 4 \\pi G_0 \\, a^2 \\, \\left( w + w_{vac} \\, {\\delta G \\over G_0} \\right) \\, \\bar{\\rho} \\, \\delta \\nonumber \\\\ \n& + & 4 \\pi G_0 \\, a^2 \\, {\\delta G \\over G_0} \\, w_{vac} \\, {c_h \\over 2 \\nu} \\, h \\, \\bar{\\rho} \\nonumber \\\\\n& + & \\; {\\mathcal{O}} (k^2)\n\\label{eq:efe_ii_cN}\n\\eea\n\\beq\nk^2 \\, \\left( \\phi - \\psi \\right) \n= + \\, 8 \\pi G_0 \\, a^2 \\, {\\delta G \\over G_0} \\, {c_s \\over 2 \\nu} \\, s \\, \\bar{\\rho} \\;\n+ \\; {\\mathcal{O}} (k^2) \\; .\n\\label{eq:efe_ij_cN}\n\\eeq\nNote that we have, for convenience, multiplied out the first, third and fourth equations\nby a factor of $a^2$.\nThe last equation involves the quantity \n\\beq\n\\sigma \\; = \\; { 2 \\over 3 } \\, {\\delta G \\over G_0} \\, {c_s \\over 2 \\nu} \\cdot  s \\; .\n\\label{eq:sigma}\n\\eeq\nFor the purpose of computing the gravitational slip function $\\eta \\equiv \\psi \/ \\phi - 1$\nit will be useful here to record the following relationship between perturbations in the\ncomoving and conformal Newtonian gauge. \nOne has\n\\beq\n\\psi = - \\, { 1 \\over 2 k^2 } \\, a^2 \n\\, \\left( \\ddot{s} + 2 \\, {\\dot{a} \\over a} \\, \\dot{s} \\right)\n\\label{eq:psi_transf_cN_pbls}\n\\eeq\n\\beq\n\\phi = - \\, {1 \\over 6} \\, \\left( h + s \\right) \n+ {1 \\over 2} \\, {a^2 \\over k^2} \\, {\\dot{a} \\over a} \\, \\dot{s}\n\\label{eq:phi_transf_cN_pbls}\n\\eeq\nUse has been made here of the following relationship between derivatives of\nan arbitrary function $f$ in the synchronous and comoving gauges\n\\beq\n{\\dot{f}}^{sync} = a {\\dot{f}}^{com}\n\\label{eq:dot_sync_pbls_1}\n\\eeq\nand \n\\beq\n{d \\over d \\tau_{sync}} = a \\, {d \\over d \\tau_{com}}\n\\label{eq:dot_sync_pbls_2}\n\\eeq\nso that\n\\beq\n{\\ddot{f}}^{sync} = a^2 \\, \\left( {{\\dot{a}}^{com} \\over a} \\, {\\dot{f}}^{com} + {\\ddot{f}}^{com} \\right) \\; .\n\\label{eq:dotdot_sync_pbls}\n\\eeq\n\n\n\n\\vskip 40pt\n\\newsection{Gravitational slip function}\n\\hspace*{\\parindent}\n\\label{sec:slipfcn}\n\nThe gravitational slip function is commonly defined as \n\\beq\n\\eta \\; \\equiv \\; { \\psi - \\phi \\over \\phi } \\; .\n\\label{eq:eta_def}\n\\eeq\nIn classical GR one has $\\phi = \\psi $ so that $\\eta =0$, which makes the quantity $\\eta$ a\nuseful parametrization for deviations from classical GR, whatever their origin might be.\nUsing the $ij$ field equation given in Eqs.~(\\ref{eq:efe_00_cN}), (\\ref{eq:efe_0i_cN}),\n(\\ref{eq:efe_ii_cN}) and (\\ref{eq:efe_ij_cN}),  and the relationship between the conformal \nNewtonian fluctuation $ \\phi $ and the comoving gauge fluctuations $ h $ and $ s $, \none finally obtains the rather simple result \n\\beq\n\\eta \\equiv { \\psi - \\phi \\over \\phi } \n\\; = \\; - \\, 16 \\pi G_0 \\, {\\delta G \\over G_0} \\, {c_s \\over 2 \\nu} \\, \n{a \\over \\dot{a}} \\, {s \\over \\dot{s}} \\, \\bar{\\rho} \\; .\n\\label{eq:eta_cs}\n\\eeq\nThe last expression contains the quantity\n\\beq\nc_s \\; = \\; \\, \\left( {8 \\over 3} \\right) \\, {i \\Gamma \\over i \\omega_s}\n\\eeq\nwhere $\\omega_s$ is the frequency associated with the $s$ perturbation, and\nwe have made use of $ i \\Gamma \\rightarrow \\dot{a} \/ a $.\nAn equivalent form for the expression in Eq.~(\\ref{eq:eta_cs}) is \n\\beq\n\\eta \\; = \\; \n - \\, 16 \\pi \\, G_0 \\, {\\delta G \\over G_0} \\, {1 \\over 2 \\nu} \\, {8 \\over 3} \n\\, {1 \\over i \\omega_s} \\, {s \\over \\dot{s}} \\, \\bar{\\rho} \n\\; = \\; - \\, 16 \\pi \\, G_0 \\, {\\delta G \\over G_0} \\, {1 \\over 2 \\nu} \\, {8 \\over 3} \n\\, { \\int \\! s \\, \\mathrm{d}t \\over \\dot{s}} \\, \\bar{\\rho} \\; .\n\\label{eq:eta_t}\n\\eeq\nIn the last expression we now can make use of the equation of motion for the\nperturbation $s(t)$ to the order we are working, namely\n\\beq\n\\ddot {s}  \\, + \\, 3 \\, { \\dot{a} \\over a } \\, \\dot{s} \\, = \\, 0 \\; .\n\\label{eq:s_t}\n\\eeq\nLet us look here first at the very simple limit of $\\lambda \\simeq 0$; \nthe physically more relevant case of nonzero $\\lambda$ will be discussed a\nbit later.\nNote that, in view of Eq.~(\\ref{eq:xi_lambda}),  this last limit \ncorresponds therefore to a very large $\\xi$.\nThen for a perfect fluid with equation of state $p= w \\rho $ one has\nsimply  $a(t) = a_0 (t\/t_0 )^{2\/3(1+w)} $ and  $\\rho (t) = 1 \/ [6 \\pi G t^2 (1+w)^2 ] $, and\nfrom Eqs.~(\\ref{eq:eta_t}) or (\\ref{eq:eta_a}) one obtains for $w=0$\n\\beq\n\\eta \\; = \\; 4 \\cdot { 8 \\over 3 } \\, c_t \\, \\left ( { t \\over \\xi } \\right )^3 \\, \\ln \\left ( { t \\over \\xi } \\right )\n+ {\\cal O} (t^4) \n\\eeq\nwhereas for $w \\neq 0 $ one has\n\\beq\n\\eta \\; = \\; 2 \\cdot { 8 \\over 3 } \\, { c_t \\over w \\, (1-w) } \\, \\left ( { t \\over \\xi } \\right )^3 \\, \n+ {\\cal O} (t^6)  \\; .\n\\eeq\nAnother extreme, but nevertheless equally simple, case is a pure cosmological constant \nterm (no matter of any type), which can be modeled by the choice $w=-1$.\nIn this case $t$ is related to the scale factor by\n\\beq\n{ a(t) \\over a_0 } \\; = \\; \\exp \\left  \\{  \\sqrt{\\lambda \\over 3 } \\, (t - t_0 ) \\right \\} \\; .\n\\eeq\nThen, using the relation in Eq.~(\\ref{eq:xi_lambda}), one obtains\n\\beq\n{ t \\over \\xi } \\; = \\; 1 + \\ln { a \\over a_\\xi } \\; ,\n\\eeq\nwhere the quantity $a_\\xi$ is therefore related to the time $t_0$ (\"today\", $a_0=1$) \nand the scale $\\xi$ by\n\\beq\n{  t_0 \\over \\xi } \\; = \\; 1 + \\ln { 1 \\over a_\\xi } \\; .\n\\eeq\nSince numerically $t_0$ is close to, but smaller than, $\\xi$, the scale factor\n$a_\\xi$ will be close to, but slightly larger than, one.\n\nTo actually come up with a definite number for $\\eta$ in more realistic cases,\none needs (apart from including\nthe effects of $\\lambda \\neq 0 $, which is done below) a value for the coefficient $c_t$\nappearing in Eq.~(\\ref{eq:grun_t}) for $G(t)$, which in turn is related to the original\nexpression for the running Newton's constant $G(\\Box)$ in Eq.~(\\ref{eq:grun_box}).\nThis issue will be discussed in some detail later, but here let us say the following.\nIn Ref. \\cite{hw05} it was estimated that the values of $c_t$ in Eq.~(\\ref{eq:grun_t})\nand $c_0$ in Eq.~(\\ref{eq:grun_t}) are of the same order of magnitude, $ c_t \\approx 0.62 c_0 $.\nThe most difficult part has been therefore a reliable estimate of $c_0$, which is\nobtained from a lattice computation of invariant curvature correlations at fixed geodesic\ndistance \\cite{cor94}, and which, after reexamination of various systematic uncertainties,\nleads to the recent estimate used in\\cite{ht10} of $c_0 \\approx 33.3 $.\nThat would give $c_t \\approx 20.6 $ which, as we will see later, is still very large.\nNevertheless it is expected that $c_0$ (or $c_t$) enter \n{\\it all} calculations with $G(\\Box)$  with the {\\it same} magnitude and sign.\n\nLet us now go back to the more physical case of $\\lambda \\neq 0$.\nThe relevant expression for $\\eta(t) $ is Eq.~(\\ref{eq:eta_t}), where\nwe use the equation for $ s(t) $, Eq.~(\\ref{eq:s_t}), to eliminate the latter.\nIt is also convenient at this stage to change variables from $t$ to $a(t)$, and use \nthe equivalent equation for $s(a)$, namely\n\\beq\ns'' (a) \\, + \\, \\left ( { H' (a) \\over H(a) } + { 4 \\over a } \\right ) \\,  s' (a) \\; = \\; 0\n\\label{eq:s_a} \\; ,\n\\eeq\nwhere the prime denotes differentiation with respect to the scale factor $a$.\nIn the above equation one can use, for nonrelativistic matter with equation \nof state such that $w=0$, and to the order needed here, the first Friedmann equation\n\\beq\nH(a) \\; = \\; \\sqrt{ { \\lambda \\over 3 } + { 4 \\over 9 \\, a^3 } } \\; .\n\\label{eq:H_a}\n\\eeq\nWe have also made use of the unperturbed result for the background matter \ndensity valid for $ w = 0 $ (which follows from energy conservation), namely\n\\beq\n\\bar{\\rho} = \\bar{\\rho}_0 \\, {1 \\over a^3} \\; .\n\\label{eq:rhobar}\n\\eeq\nNote that the above expression for $ \\bar{\\rho} $ is valid to zeroth order in $ \\delta G $, \nwhich is entirely adequate when substituted into $ \\eta (a) $,  since the rest there is already \nfirst order in $ \\delta G $.\nThis finally gives an explicit solution for $ s(a) $\n\\beq\ns(a) \\; \\propto \\; { 2 \\over 3 a^{3\/2} } \\, \\sqrt { 1 + a^3 \\, \\theta } \\; ,\n\\eeq\\\nwith parameter $\\theta \\equiv \\lambda \/ 8 \\pi G_0 \\, \\bar{\\rho}_0 $.\nThe above solution for $s(a)$ can then be substituted directly in Eq.~(\\ref{eq:eta_t}),\nprovided one changes variables from $t$ to $a(t)$, and in the process uses\nthe following identities\n\\beq\n\\int \\! s(t) \\, \\mathrm{d}t = \\int \\! s(a) \\, {1 \\over a \\, H (a) } \\, \\mathrm{d}a  \\; ,\n\\eeq\nas well as\n\\beq\n\\dot{s} = a \\, H (a) \\, {\\partial s \\over \\partial a} \\; ,\n\\eeq\nwith $H(a)$ given a few lines above.\n\nThe resulting expression, which still involves an integral over the scale factor $a(t)$, can now \nbe readily evaluated, and leads eventually to a rather simple expression for $\\eta$.\nThe general result for nonrelativistic matter ($w=0 $) but $\\lambda \\neq 0 $ is\n\\beq\n\\eta (a) \\; = \\; {16 \\over 3 \\, \\nu} \\, {\\delta G (a) \\over G_0} \\, \\log \\! \\left[{a \\over a_\\xi}\\right]  \\; .\n\\label{eq:eta_a}\n\\eeq\nThis is the main result of the paper.\nThe integration constant $a_\\xi$ has been fixed following the requirement that\nthe scale factor $a \\rightarrow a_\\xi$ for $t \\rightarrow \\xi $ [see Eqs.~(\\ref{eq:grun_box}),\n(\\ref{eq:grun_t}) and (\\ref{eq:grun_a}) for the definitions of $\\xi$].\nIn other words, by switching to the variable $a(t)$ instead of $t$, the quantity $\\xi$ has\nbeen traded for $a_\\xi$. \nIn the next section we will show that in practice the quantity $a_\\xi$ is generally\nexpected to be slightly larger than the scale factor \"today\", i.e. for $t=t_0$.\nAs a result the correction in Eq.~(\\ref{eq:eta_a}) is expected to be negative today.\n\nThe next section will be devoted to establishing the general relationship between\n$t$ and $a(t)$, for nonvanishing cosmological constant $\\lambda$, so that a \nquantitative estimate for the slip function $\\eta$\ncan be obtained from Eq.~(\\ref{eq:eta_a}) in a realistic cosmological context.\nSpecifically we will be interested in the value of $\\eta$ for a current matter fraction \n$\\Omega \\simeq 0.25$, as suggested by current astrophysical measurements.\n\n\n\n\n\n\\vskip 40pt\n\\subsection{Relating the scale factor $a$ to $t$, and vice versa}\n\\hspace*{\\parindent}\n\\label{sec:a0t}\n\nLet us now come back to the general problem of estimating  $\\eta (a)$, using the\nexpression given\nin Eq.~(\\ref{eq:eta_a}),  for $\\lambda \\neq 0 $ and a nonrelativistic fluid with\n$w=0$.\nTo predict the correct value for the slip function $\\eta (a) $ one needs the \nquantity $\\delta G (a)$, which is obtained from the FLRW version of $G(\\Box)$, \nnamely $G(t)$ in Eq.~(\\ref{eq:grun_t}), via the replacement, in this\nlast quantity, of $ t \\rightarrow t(a) $.\nThe last step requires therefore that the correct relationship between $t$ and $a(t)$\nbe established, for any value of $\\lambda$.\nIn the following we will first relate $t$ to $a(t)$, and vice versa, to zeroth order in \nthe quantum correction $\\delta G$ [we will call them $a^{(0)} (t) $ and $t^{(0)} (a) $],\nand then compute the first order correction in $\\delta G$ to the above quantities\n[we will call those $a^{(1)} (t) $ and $t^{(1)} (a) $].\n\nLet us look first at the zeroth order result.\nThe field equations and the energy conservation equation for $ a^{(0)} (t) $,\nwithout a $ \\delta G $ correction, but with the $\\lambda $ term, were already given\nin Eq.~(\\ref{eq:fried_0}),\n\\bea\n3 \\, {{\\dot{a}}^{(0) \\, 2} (t) \\over {a}^{(0) \\, 2} (t)} \n& = & \n8 \\pi \\, G_0 \\, \\bar{\\rho}^{(0)} (t) + \\lambda \n\\nonumber \\\\\n{{\\dot{a}}^{(0) \\, 2} (t) \\over {a^{(0)}}^2 (t)} + 2\\, {\\ddot{a}^{(0)} (t) \\over a^{(0)} (t)} \n& = &\n- 8 \\pi \\, G_0 \\, w \\, \\bar{\\rho}^{(0)} (t) + \\lambda \n\\label{eq:efe_a0t}\n\\eea\nfor a spatially flat universe $ ( k = 0 )$, and\n\\beq\n{\\dot{\\bar{\\rho}}}^{(0)}(t) + 3 \\, ( 1 + w ) \\, \n{ {\\dot{a}}^{(0)} (t) \\over a^{(0)} (t) } \\, \\bar{\\rho}^{(0)} (t) = 0 \\; .\n\\label{eq:encons_a0t}\n\\eeq\nFrom these one can obtain $ a^{(0)}(t) $ and then $ \\bar{\\rho}^{(0)}(t) $.\nAs a result the scale factor is found to be related to time by\n\\beq\nt^{(0)}(a) \n= { 2 \\, \\mathrm{Arcsinh} \\! \\left[ \\, a^{3\/2} \\, \\theta^{1 \\over 2} \\right] \n\\over \\sqrt{3 \\, \\lambda} }\n\\label{eq:t0_a}\n\\eeq\nwhere we have defined the parameter\n\\beq\n\\theta \\; \\equiv \\; { \\lambda \\over 8 \\pi G_0 \\bar \\rho_0 } \\; = \\; { 1 - \\Omega \\over \\Omega }\n\\label{eq:theta}\n\\eeq\nwith $\\bar \\rho_0$ the current ($t=t_0$) matter density, and $\\Omega$ the\ncurrent matter fraction.\nNote that in practice we will be interested in a matter fraction which\ntoday is around $0.25$, giving $\\theta \\simeq 3.0 $, a number which\nis of course quite far from the zero cosmological constant case of $\\theta =0$.\n\nOne can express the time today ($t_0$) in terms of cosmological constant $ \\lambda $,\nand therefore in terms of $ \\theta $, as follows\n\\beq\nt_0^{(0)} = { 2 \\, \\mathrm{Arcsinh}({\\sqrt{\\theta}}) \\over \\sqrt{3 \\lambda} }\n\\label{eq:t0}\n\\eeq\nwith the normalization for $ t^{(0)}(a) $ such that \n$ t^{(0)}(a=0) = 0 $ and $ t^{(0)}(a=1) = t_0 $ \"today\".\nSo here we follow the customary choice of having the scale factor equal to one \"today\".\nThen one has \n\\beq\n{t^{(0)}(a) \\over t_0^{(0)}} \n={\\mathrm{Arcsinh} \\! \\left[ \\sqrt{a^{3} \\, \\theta} \\right] \n\\over \n\\mathrm{Arcsinh}({\\sqrt{\\theta}})} \\; .\n\\label{eq:t0_a_norm}\n\\eeq\nWhen expanded out in $\\theta$, the above result leads to some perhaps\nmore recognizable terms,\n\\beq\n{t^{(0)}(a) \\over t_0^{(0)}} \n= a^{3 \\over 2} \n\\, \\left[ 1 - {1 \\over 6}\\, \\left(-1 + a^3 \\right) \\, \\theta + {1 \\over 360}\\, \\left(-17 - 10 \\, a^3 + 27 \\, a^6 \\right) \\, \\theta^2 \n+ \\cdots \\right] \\; .\n\\label{eq:t0_a_power}\n\\eeq\nConversely, one has for the scale factor as a function of the time\n\\beq\na^{(0)}(t) = \\left({ \\mathrm{Sinh}^2 \\! \\left[{\\sqrt{3 \\, \\lambda} \\over 2} \\, t\\right] \\over \\theta }\\right)^{1 \\over 3} \\; ,\n\\label{eq:a0t_norm}\n\\eeq\nwhich, when expanded out in $\\lambda$ or $t$, gives the more recognizable result\n\\beq\n\\left [ a^{(0)}(t) \\right ]^3 \\; = \\; \n{3 \\, \\lambda \\, t^2 \\over 4 \\theta} \\, \n\\left(1 + {\\lambda \\, t^2 \\over 4} + {\\lambda^2 \\, t^4 \\over 40} + \\cdots \\right) \\; .\n\\eeq\nSimilarly for the pressure one obtains\n\\beq\n\\bar{\\rho}^{(0)}(t) \n= { \\lambda \\, \\mathrm{Csch}^2 \\! \\left[ {\\sqrt{3 \\, \\lambda} \\over 2} \\, t \\right] \\over 8 \\pi G_0} \\; ,\n\\label{eq:rho0t}\n\\eeq\nwhich when expanded out in $\\lambda$ or $t$ gives the more familiar result\n\\beq\n\\bar{\\rho}^{(0)}(t) \n= {1 \\over 6 \\pi G_0 \\, t^2 \n\\, \\left(1 + {t^2 \\, \\lambda \\over 4} + {t^4 \\, \\lambda^2 \\over 40} + {3 \\, t^6 \\, \\lambda^3 \\over 2240} + \\cdots \\right)} \\; .\n\\eeq\nTo be more specific, let us set $\\theta=3$, which corresponds to a matter fraction today\nof $\\Omega \\sim 0.25$.\nIn addition, we will now make use of Eq.~(\\ref{eq:xi_lambda}) and set \n$\\lambda \\rightarrow {3\/\\xi^2} $.\nOne then obtains\n\\beq\nt_0^{(0)} \\left(\\theta = 3 \\right) = 0.878 \\;  \\xi \\; ,\n\\label{eq:t0_theta3}\n\\eeq\nwhich shows that $t_0$ and $\\xi$ are rather close to each other\n(apparently a numerical coincidence).\n\nThen, from the expression for $G(t)$ in Eq.~(\\ref{eq:grun_t}),\n\\beq\n{\\delta G (t) \\over G_0} = c_t \\left( {t \\over \\xi} \\right)^{1 \\over \\nu} \\; ,\n\\eeq\none can obtain $G(a)$ in all generality, by the replacement $t \\rightarrow t(a)$\naccording to the result of Eqs.~(\\ref{eq:t0_a}) or (\\ref{eq:t0_a_norm}).\nFor the special case of pure nonrelativistic matter with equation of state $w=0$ and $\\lambda =0$\none obtains, using Eq.~(\\ref{eq:t0_a_power}),\n\\beq\n{\\delta G (a) \\over G_0} \\; = \\;  c_t \\left( {a \\over a_\\xi} \\right)^{\\gamma_\\nu} \\; ,\n\\eeq\nwith exponent\n\\beq\n\\gamma_\\nu = {3 \\over 2 \\nu} \\; .\n\\eeq\nThe latter is largely the expression used earlier in the matter density perturbation\ntreatment of our earlier work of Ref. \\cite{ht10}.\n\nMore generally one can define $ a_\\xi $ as the value for the scale factor $a$ which\ncorresponds to the scale $ \\xi $,\n\\beq\na_\\xi^{(0)}  \\; \\equiv \\; \n\\left( {1 \\over \\theta} \\right)^{1\\over 3} \\mathrm{Sinh}^{2 \\over 3} \\! \\left[ {3 \\over 2}\\right] \n= 1.655 \\, \\left( {1 \\over \\theta} \\right)^{1\\over 3} \\; ,\n\\label{eq:a_xi}\n\\eeq\nso that in general $ a_\\xi \\neq a_0 $, where $ a_0 =1 $ is the scale factor \"today\".\nThen for the observationally favored case $ \\theta \\simeq 3 $ one obtains\n\\beq\na_\\xi^{(0)}(\\theta = 3) = 1.148 \\; ,\n\\label{eq:axi_a0}\n\\eeq\nwhich clearly  implies $ a_\\xi^{(0)}  > a_0 = 1 $.\n\\footnote{\nLet us give here a few more observational numbers for present and future reference. \nFrom the present age of the Universe\n$t_0 \\approx 13.75 \\, Gyrs \\simeq 4216 \\, Mpc $, whereas from the observed \nvalue of $\\lambda $ (mostly extracted from distant supernovae surveys)\none has following Eq.~(\\ref{eq:xi_lambda}) $ \\xi \\simeq  4890 \\, Mpc $, \nwhich then gives $t_0 \/ \\xi \\simeq 0.862  = 1 \/ 1.160 $.\nThis last ratio is similar to the number we used in Eq.~(\\ref{eq:t0_theta3}),\nby setting there $\\Omega=0.25$ exactly. \n}\nThe above expressions will be used in the next section to obtain a quantitative\nestimate for the slip function $\\eta (a)$, evaluated at today's time $t=t_0$.\n\n\nThe discussion above dealt with the case of $\\delta G=0$.\nLet us now consider briefly the corrections to $a(t)$ and, conversely, $t(a)$\nthat come about when the running of $G$ is included, in other words when \na constant  $G$ is replaced by $G(t)$ or $G(a)$ in the effective field equations.\nIn Eq.~(\\ref{eq:fried_run}) the Friedman equations were given in the presence\nof a running $G$, namely\n\\bea\n3 \\, {{\\dot{a}}^2 (t) \\over {a}^2 (t)} \n& = & \n8 \\pi \\, G_0 \\left ( 1 + {\\delta G(t) \\over G_0} \\right ) \\, \\bar{\\rho}(t) + \\lambda \n\\nonumber \\\\\n{{\\dot{a}}^2 (t) \\over {a}^2 (t)} + 2\\, {\\ddot{a} (t) \\over a (t)} \n& = & \n- 8 \\pi \\, G_0 \\, \\left ( w + w_{vac} {\\delta G(t) \\over G_0} \\right )\\, \\bar{\\rho} (t) + \\lambda  \\; ,\n\\label{eq:fried_run1}\n\\eea\ntogether with the energy conservation equation\n\\beq\n3 \\, {\\dot{a} (t) \\over a (t)} \\, \n\\left [ \\left (1+w \\right ) + \\left (1+ w_{vac} \\right )\\,{\\delta G(t) \\over G_0} \n\\right ] \\bar{\\rho} (t) \n+ { \\dot{\\delta G}(t) \\over G_0} \\, \\bar{\\rho} (t)\n+ \\left ( 1 + {\\delta G(t) \\over G_0} \\right )\\, \\dot{\\bar{\\rho}} (t) = 0 \\; .\n\\label{eq:encons_zeroth_w_1}\n\\eeq\nTo solve these equations to first order in $\\delta G$ we set\n\\beq\na(t) = a^{(0)} (t) \\left [ 1 + c_t \\, a^{(1)}(t) \\right ]\n\\label{eq:a0t_1t}\n\\eeq\n\\beq\n\\bar{\\rho}(t) = \\bar{\\rho}^{(0)} (t) \\left [ 1 + c_t \\,\\bar{\\rho}^{(1)} (t)\\right ]\n\\label{eq:rho0t_1t}\n\\eeq\nwhere $ a^{(0)} (t) $ and $ \\bar{\\rho}^{(0)} (t) $ here represent the solutions\nobtained previously for $\\delta G=0$.\nOne then finds for the correction to the matter density\n\\beq\n\\bar{\\rho}^{(1)} (t) = - \\, \\left( { t \\over \\xi }\\right)^{1 \\over \\nu} \\, \n\\left(1 + w_{vac} \\, {\\nu \\over (1 + \\nu)} \n\\, \\sqrt{3 \\, \\lambda} \\; t \\;\\; \\mathrm{Coth} \\! \\left[{ \\sqrt{3 \\, \\lambda} \\over 2} \\, t \\right] \n\\right)\n\\label{eq:rho1t}\n\\eeq\nand to lowest nontrivial order in $t$ and for $ w_{vac}=1\/3 $ \n\\beq\n\\bar{\\rho}^{(1)} (t) = - \\, {3 + 5 \\nu \\over 3 ( 1 + \\nu)} \\, \\left({t \\over \\xi} \\right)^{1 \\over \\nu} + \\dots \\; .\n\\eeq\nFor the correction to the scale factor one finds\n\\beq\na^{(1)} (t) = - \\, w_{vac} \\, {\\nu \\over (1 + \\nu)} \\, \\lambda \\; \n\\int^t_0 \\! \n{t^{\\prime} \\left({ t^{\\prime} \\over \\xi }\\right)^{1 \\over \\nu} \n\\over \n- 1 + \\mathrm{Cosh} \\! \\left[ \\sqrt{3 \\, \\lambda} \\; t^{\\prime} \\right]} \\, \n\\mathrm{d}t^{\\prime}\n\\label{eq:a1t}\n\\eeq\nand to lowest nontrivial order in $t$ for $ w_{vac}=1\/3 $,\n\\beq\na^{(1)} (t) = - \\, {2 \\nu^2 \\over 9 \\, ( 1 + \\nu)} \\, \\left({t \\over \\xi} \\right)^{1 \\over \\nu} + \\dots \\; .\n\\eeq\nAfter having obtained the relevant formulas for $a(t)$ and $t(a)$ in the general case, i.e. for\nnonzero $\\lambda$, we can return to the problem of evaluating the slip function $\\eta$.\n\n\n\\vskip 40pt\n\\subsection{Quantitative estimate of the slip function $\\eta (z) $}\n\\hspace*{\\parindent}\n\\label{sec:eta_z}\n\n\nThe general expression for the gravitational slip function \n$\\eta (a)$ was given earlier in Eq.~(\\ref{eq:eta_a}) \nfor $w=0$ and $\\lambda \\neq 0$,\n\\beq\n\\eta (a) \\; = \\; {16 \\over 3 \\nu} \\, {\\delta G (a) \\over G_0} \\, \\log \\! \\left[{a \\over a_\\xi}\\right]  \\; .\n\\eeq\nTo obtain $\\delta G(a)$ we now use, from Eq.~(\\ref{eq:grun_t}), \n\\beq\n{\\delta G (t) \\over G_0} = c_t \\, \\left({t \\over \\xi} \\right)^{1 \\over \\nu}\n\\label{eq:grun_t_1}\n\\eeq\nand substitute in the above expression for $\\delta G(t) $ the correct relationship between \n$t$ and $a$, namely $t(a)$ from Eq.~(\\ref{eq:t0_a}), which among other things\ncontains the constant defined in Eq.~(\\ref{eq:a_xi}),\n\\beq\na_\\xi \\; = \\; \n\\left( {1 \\over \\theta} \\right)^{1\\over 3} \\mathrm{Sinh}^{2 \\over 3} \\! \\left[ {3 \\over 2}\\right]  \\; .\n\\label{eq:axi_1}\n\\eeq\nIt will be convenient, at this stage, to also make use of the relationship\nin Eq.~(\\ref{eq:xi_lambda}), namely\n\\beq\n\\lambda \\; \\rightarrow \\; {3 \\over \\xi^2} \\; .\n\\eeq\nThe last step left is to make contact with observationally accessible quantities,\nby expanding  in the redshift $ z $, \nrelated in the usual way to the scale factor $a$ by $ a \\equiv {1 \/(1 + z)} $.\nThen for $ \\nu = 1\/3 $ and $ \\theta = 3 $ (matter fraction $\\Omega=0.25$) one \nfinally obtains for the gravitational slip function\n\\beq\n\\eta(z) \\; = \\; - \\, 1.491 \\,c_t - 6.418 \\, c_t \\, z + 30.074 \\, c_t \\, z^2 + \\cdots\n\\label{eq:eta_z}\n\\eeq\nTo obtain an actual number for $\\eta (z=0) $ one needs to address two more issues.\nThey are (i) to provide a bound on the theoretical uncertainties in the above expression,\nand (ii) to give an estimate for the coefficient $c_t$, which is traced back to Eq.~(\\ref{eq:grun_t})\nand therefore to the original expression for $G(\\Box)$ in Eq.~(\\ref{eq:grun_box}).\nThe latter contains the coefficient $c_0$, but in Ref. \\cite{hw05} the estimate\nwas given $c_t = 0.450 \\; c_0$ for the tensor box operator; thus $c_t$ and $c_0$\ncan safely be assumed to have the same sign, and comparable magnitudes.\n\nTo estimate the level of uncertainty in the magnitude of the correction coefficient in\nEq.~(\\ref{eq:eta_z}) we will consider here an infrared regulated version\nof $G(\\Box)$, where an infrared cutoff is supplied so that in Fourier space $k > \\xi^{-1}$,\nand the spurious infrared divergence at small $k$ is removed.\nThis is quite analogous to an infrared regularization used very successfully \nin phenomenological applications to QCD heavy quark bound states \\cite{ric79,eic81}, and \nwhich has recently found some limited justification in the framework of infrared \nrenormalons \\cite{ben99}.\nAs shown already in the first cited reference, it works much better than expected;\nhere a similar prescription will be used just as a means to provide some estimate\non the theoretical uncertainty in the result of Eq.~(\\ref{eq:eta_z}).\nTherefore, instead of the $G(\\Box)$ in Eq.~(\\ref{eq:grun_box}),\nwhich in momentum space corresponds to \n\\begin{equation}\nG(k^2) \\; \\simeq \\; G_0 \\, \\left [ \n\\, 1 \\, + \\, c_0 \\, \\left ( { 1 \\over \\xi^2 \\, k^2 } \\right )^{ 1\/ 2 \\nu }\n\\, + \\, \\dots \\right ] \\; ,\n\\label{eq:grun_k} \n\\end{equation}\nwe will consider a corresponding infrared regulated version,\n\\beq\nG(k^2) \\; \\simeq \\; G_0 \\left [ \\; 1 \\, \n+ \\, c_0 \\left ( { \\xi^{-2} \\over k^2 \\, + \\, \\xi^{-2} } \\right )^{1 \/ 2 \\nu} \\, \n+ \\, \\dots \\; \\right ] \\; .\n\\label{eq:grun_k_reg}\n\\eeq\nOf course the small distance, $  k \\gg \\xi^{-1} $ or $ r \\ll \\xi $, behavior is unchanged,\nwhereas for large distances $r \\gg \\xi$  the gravitational coupling\nno longer exhibits the spurious infrared divergence;\ninstead  it approaches a finite value $G_\\infty \\simeq ( 1 + c_0 + \\dots ) \\, G_0 $.\nNow, in momentum space the infrared regulated $ \\delta G (k) $ reads\n\\beq\n{\\delta G(k^2) \\over G_0} = c_0 \\, \\left({m^2 \\over k^2 + m^2} \\right)^{1 \/ 2 \\, \\nu} \\; ,\n\\label{eq:grun_k_reg_1}\n\\eeq\nwith $m= 1\/ \\xi $, and in position space the corresponding form is\n\\beq\n{\\delta G(\\Box) \\over G_0} = c_0 \\, \\left({1 \\over - \\, \\xi^2 \\Box + 1 } \\right)^{1 \/ 2 \\, \\nu} \\; .\n\\label{eq:grun_box_reg}\n\\eeq\nFollowing the results of Ref. \\cite{hw05}, if the above differential operator \nacts on functions of $ t $ only, then one obtains for $\\delta G (t) $\n\\beq\n{\\delta G (t) \\over G_0} \\; = \\;  c_0 \\, \\left ( \n{1 \\over \\left({c_0 \\over c_t}\\right)^{2 \\, \\nu} \\, \\left({\\xi \\over t}\\right)^2 + 1} \n\\right)^{1 \\over 2 \\, \\nu}\n\\label{eq:grun_t_reg}\n\\eeq\nwith again $ c_t \/ c_0 \\approx 0.62 $ \\cite{hw05}.\nNote that the expression in Eq.~(\\ref{eq:grun_t_reg}) could also have been obtained\ndirectly from Eq.~(\\ref{eq:grun_t}), by a direct regularization.\n\nOne can then repeat the whole calculation for $\\eta (a)$ with the regulated\nversion of $\\delta G(t)$ given in Eq.~(\\ref{eq:grun_t_reg}).\nThe result is \n\\beq\n\\eta (z) = - 0.766 \\, c_t - 4.109 \\, c_t \\, z + 12.188 \\, c_t \\, z^2 + \\cdots \\; .\n\\label{eq:eta_z_reg}\n\\eeq\nIt seems that the effect of the infrared regularization has been to\nreduce the magnitude of the effect (at $z=0$) by about a factor of 2.\nIt is encouraging that, at this stage of the calculation,\nthe negative trend in $ \\eta (z) $ due to the running of \n$ G $ appears unchanged.\nFurthermore, in all cases we have looked so far, the value $\\eta (z=0)$ is found to\nbe negative.\n\n\n\\vskip 40pt\n\\subsection{Slip function $\\eta(z)$ for stress perturbation $s = 0$}\n\\hspace*{\\parindent}\n\\label{sec:eta_s_0}\n\n\nIn Ref. \\cite{ht10} a preliminary estimate of the magnitude of the slip\nfunction $\\eta$ was given.\nThe calculation there neglected the stress field $s$ in Eq.~(\\ref{eq:stress_def})\nand only included the\nmetric perturbation $h$ in the comoving gauge.\nThe main reason was that nonrelativistic matter density perturbations,\nand therefore the growth exponents, are unaffected by the stress field contribution.\nWe will show here that in this case one still obtains a nonvanishing $ \\eta $, \nwhose value we will discuss below.\nThe results will be useful, since now a direct comparison can be done\nwith the full answer (including the stress field) for $\\eta (z) $ given in the previous section.\n\nIn the absence of stress ($s=0$) and finite $k$, the $tt$ and $xx + yy + zz $ field equations\nread\n\\beq\n- 2 \\, {k^2 \\over a^2} \\phi \n- 8 \\pi G_0 {c_h \\over 2 \\nu} \\, {\\delta G \\over G_0} \\, \\rho \\; \\delta \\, \\left( - {2 \\over 1 + w} \\right)\n= 8 \\pi G_0 \\, \\left(1 + {\\delta G \\over G_0} \\right) \\, \\rho \\, \\delta\n\\eeq\n\\beq\n2 {k^2 \\over a^2} \\, \\left(\\psi - \\phi \\right) \n+ 24 \\pi G_0 {c_h \\over 2 \\nu} \\, w_{vac} \\,{\\delta G \\over G_0} \\, \\rho \\; \\delta \\, \\left( - {2 \\over 1 + w } \\right)\n= - \\, 24 \\pi G_0 \\, \\left(w + w_{vac} \\, {\\delta G \\over G_0} \\right) \\, \\rho \\, \\delta  \\; .\n\\eeq\nIn both equations we have made use of zeroth order (in ${\\delta G\/ G_0}$) energy conservation,\nwhich leads to $ h = - {2 \\over ( 1 + w )} \\, \\delta $, where $\\delta$ is the matter fraction.\nOne can then take the ratio of the two equations given above, and obtain \nagain an expression for the slip function $ \\eta = { \\left(\\psi - \\phi\\right)\/ \\phi } $. \nFor $ w =0 $ (nonrelativistic matter), after expanding in $ {\\delta G\/G_0} $, one finds the\nrather simple result\n\\beq\n\\eta \\; = \\; {\\psi - \\phi \\over \\phi}  \\; = \\; \n3 \\, w_{vac}  \\, \\left( 1 - {c_h \\over \\nu} \\right) \\, {\\delta G \\over G_0} \\; .\n\\eeq\nHere the quantity $c_h$ is the same as in Eq.~(\\ref{eq:ch}), and depends on the\nchoices detailed below.\nIn the following we will continue to use $w_{vac} = 1\/3 $ [see Eqs.~(\\ref{eq:wvac_def}) \nand (\\ref{eq:wvac})] \\cite{hw05,ht10}, which is the correct value associated with\n $G (\\Box)$ in the FLRW background metric.\n\nIn Ref. \\cite{ht10} we used the {\\it scalar box} value $c_h = 1\/2$, which then gives\n\\beq\n\\eta \\; = \\; \\left( 1 - {1 \\over 2 \\, \\nu} \\right) \\, {\\delta G \\over G_0} \\; = \\; \n\\left( 1 - {1 \\over 2 \\, \\nu} \\right) \\, c_t \\, \\left({t \\over \\xi}\\right)^{1 \\over \\nu} \n+ \\cdots\n\\eeq\nIn this last case it is then easy to recompute the slip function in terms of the redshift, \njust as was done in the previous section, and one finds, \nunder the same conditions as before [$ \\nu = 1\/3 $, $\\theta=3$,\nand $t_0\/ \\xi$ as given in Eq.~(\\ref{eq:t0_theta3})] the following result\n\\beq\n\\eta \\simeq - \\, 0.338 \\, c_t \\; + \\; O( z) \\; .\n\\eeq\nFor the infrared regulated version of ${\\delta G\/G_0}$ given in Eq.~(\\ref{eq:grun_t_reg})\none obtains instead the slightly smaller value\n\\beq\n\\eta \\simeq - \\, 0.174 \\, c_t \\; + \\; O( z) \\; .\n\\eeq\nFor the {\\it tensor box} case (also discussed extensively in \\cite{ht10}, where it\nwas shown that this is in fact the correct way of doing the calculation) one finds a\nsignificantly larger value $ c_h \\simeq 7.927 $, so that in this case the slip \nfunction $ \\eta $ becomes\n\\beq\n\\eta \\simeq \\left( 1 - {7.927 \\over \\nu} \\right) \\, { \\delta G \\over G_0 } \n= \\left( 1 - {7.927 \\over \\nu} \\right) \\, c_t \\, \\left({t \\over \\xi}\\right)^{1 \\over \\nu} \n+ \\cdots\n\\label{eq:eta_h}\n\\eeq\nAlso in this case one can recompute the slip function in terms of the redshift, \nand one finds, under the same conditions as before,\n\\beq\n\\eta \\simeq - \\, 15.42 \\, c_t \\; + \\; O(z) \\; .\n\\label{eq:eta_h_reg}\n\\eeq\nFor the infrared regulated $ {\\delta G\/G_0} $ given in Eq.~(\\ref{eq:grun_t_reg})\none finds instead \n\\beq\n\\eta \\simeq - \\, 7.919 \\, c_t \\; + \\; O(z) \\; ,\n\\eeq\nwhich is again about a factor of 2 smaller than the unregulated value.\n\nWe conclude from the above exercise of calculating $\\eta$ with vanishing\nstress field $s=0$ three things.\nThe first is that using the scalar box result on the trace of the \nenergy-momentum tensor (which ultimately is not an entirely correct, or at least \nan incomplete,  procedure, given the tensor nature of the matter \nenergy-momentum tensor) underestimates the effects of $G(\\Box)$ on \nthe slip function $\\eta (z=0)$ by a factor that can be as large as an\norder of magnitude.\n\nThe second lesson is that the stress field ($s$) contribution is indeed important,\nsince it reduces the size of the quantum correction significantly \n[Eqs.~(\\ref{eq:eta_z}) and (\\ref{eq:eta_z_reg})], compared to the $s=0$ \nresult [Eqs.~(\\ref{eq:eta_h}) and (\\ref{eq:eta_h_reg})], again by almost\nan order of magnitude, which would imply some degree of cancellation\nbetween the $s$ and $h$ contributions.\n\nThe third observation is that in all cases we have looked at so far the quantum\ncorrection to the slip function is {\\it negative} at $z=0$.\n\n\n\n\n\n\\vskip 40pt\n\\newsection{Conclusions}\n\\hspace*{\\parindent}\n\\label{sec:conclusion}\n\nIn the previous sections we computed corrections to the gravitational \nslip function $\\eta = \\psi \/ \\phi -1 $\narising from the renormalization-group motivated running $G(\\Box)$, as \ngiven in Eq.~(\\ref{eq:grun_box}).\nThe relevant result was presented in Eqs.~(\\ref{eq:eta_z}) and (\\ref{eq:eta_z_reg}), \nthe first expression representing the answer for an unregulated $G(\\Box)$, \nand the second answer found for an infrared regulated version of the same.\nIt should be noted that, so far, in the treatment of metric and matter perturbations \nwe have considered only the ${\\bf k} \\rightarrow 0 $ limit [see Eq.~(\\ref{eq:fourier})].\nLet us focus here for definiteness on the first of the two results \n[Eq.~(\\ref{eq:eta_z})], which is\n\\beq\n\\eta (z) \\; \\simeq \\;  - 1.491 \\, c_t \\; + \\; O(z)\n\\label{eq:eta_z_1}\n\\eeq\nat $z \\simeq 0$.\nWe now come to the last issue, namely an estimate for the magnitude of the constant $c_t$.\nAs already discussed previously in Sec. \\ref{sec:eta_z}, to get an actual number for $\\eta (z=0) $ \none needs a number for $c_t$, whose appearance is traced back to Eq.~(\\ref{eq:grun_t}),\nand therefore to the original expression for $G(\\Box)$ in Eq.~(\\ref{eq:grun_box}),\nwith $c_t  \\approx 0.450 \\times c_0$ for the relevant tensor box operator \\cite{hw05} .\n\nThe value of the constant $c_0$ has to be extracted from a nonperturbative lattice \ncomputation of invariant curvature correlations at fixed geodesic distance \\cite{cor94};\nit relates the physical correlation length $\\xi$ to the bare lattice coupling $G$, and is\ntherefore a genuinely nonperturbative amplitude.\nAfter a reexamination of various systematic uncertainties, these lead to the recent \nestimate used in \\cite{ht10} of $c_0 \\approx 33.3 $.\nThat would give for the amplitude $c_t \\approx 20.6 $ which still seems rather large.\nNevertheless, based on experience with other field-theoretic models which\nalso exhibit nontrivial fixed points such as the nonlinear sigma model, \nas well as QCD and non-Abelian gauge theories,\none would expect this amplitude to be of order unity; \nvery small or very large numbers would seem rather atypical and un-natural.\n\nAs far as astrophysical observations are concerned,\ncurrent estimates for $\\eta (z=0)$ obtained from CMB measurements\ngive values around $0.09 \\pm 0.7 $ \\cite{ame07,dan09},\nwhich would then imply an observational bound $c_t \\lsim 0.3 $.\n\nIndeed a similar problem of magnitudes for the theoretical amplitudes\nwas found in our recent calculation of matter \ndensity perturbations with $G(\\Box)$, where again the corrections \nseemed rather large \\cite{ht10} in view of the above quoted value of $c_t$.\nLet us briefly summarize those results here.\nSpecifically, in Ref. \\cite{ht10} a value for the density perturbation growth \nindex $\\gamma$ was obtained in the presence of $G(\\Box)$.\nThe quantity $\\gamma$ is in general obtained from the growth index $ f (a) $ \\cite{pee93}\n\\beq\nf (a) \\equiv { \\partial \\ln \\delta (a) \\over \\partial \\ln a }  \\; ,\n\\label{eq:fa_def}\n\\eeq\nwhere $\\delta (a)$ is the matter density contrast.\nOne is mainly interested in the neighborhood of the present era, $a (t) \\simeq a_0 \\simeq 1 $,\nwhich leads to the definition of the growth index parameter $ \\gamma $ via\n\\beq\n\\gamma \\equiv  \\left. { \\ln f \\over \\ln \\Omega  }  \\right \\vert_{a=a_0} \\; .\n\\label{eq:gamma_def}\n\\eeq\nThe latter has been the subject of increasingly accurate cosmological observations, for\nsome recent references see \\cite{smi09,vik09,rap09}.\n\\footnote{\nFor a recent detailed review on the many tests of general relativity on astrophysical scales,\nand a much more complete set of references, see for example \\cite{uza02,uza09}.}\n\nOn the theoretical side, for the tensor box one finds \\cite{ht10}, \nfor a matter fraction $\\Omega = 0.25$,\n\\beq \n\\gamma =  0.556 - 106.4 \\, c_t + O(c_t^2 ) \\; .\n\\label{eq:gamma_ten}\n\\eeq\nwhere the first contribution is the classical GR value from the relativistic\ntreatment of matter density perturbations \\cite{pee93}.\nThe result presented above is in fact a slight improvement over the answer quoted in our \nearlier work \\cite{ht10},\nsince now the improved relationship between $t$ and $a$ given in Eq.~(\\ref{eq:t0_a})\nhas been used, which reduces the magnitude of the correction proportional\nto $c_t$.\nNevertheless, it should be emphasized that the above result has been obtained\nin the ${\\bf k} \\rightarrow 0 $ limit of the perturbation Fourier modes in Eq.~(\\ref{eq:fourier}).\n\nRecent observational bounds on x-ray studies of large galactic clusters at distance scales \nof up to about $1.4$ to $8.5 Mpc$ \n(comoving radii of $\\sim 8.5 Mpc $ and  viral radii of $\\sim 1.4 Mpc$) \\cite{vik09} \nfavor values for $\\gamma= 0.50 \\pm 0.08 $, and more recently $\\gamma = 0.55+0.13-0.10 $\n\\cite{rap09}.\nThis would then constrain the amplitude $c_t$ in Eq.~(\\ref{eq:gamma_ten}) \n{\\it at that scale} to $c_t \\lsim 5 \\times 10^{-4}$.\nThe latter bound from density perturbations\nseems a much more stringent bound than the one\ncoming from the observed slip function.\nIndeed with the bound on $c_t$ coming from the observed density perturbation\nexponents one\nwould conclude that, according to Eq.~(\\ref{eq:eta_z_1}), the correction to the slip\nfunction at $z \\simeq 0 $ must indeed be very small, $ \\eta \\simeq O(10^{-3}) $, \nwhich is a few orders of magnitude below the observational limit quoted above, \n$\\eta \\simeq 0.09 \\pm 0.7 $.\n\nIt is of course possible that the galactic clusters in question are not large enough \nyet to see the quantum effect of $G(\\Box)$, since after all the relevant scale in \nEq.~(\\ref{eq:grun_box}) is related to $\\lambda$ and is\nsupposed therefore to be very large, $\\xi \\simeq 4890 Mpc $.\n\\footnote{\nOne might perhaps think that the running of $G$ envisioned here might lead to small \nobservable consequences on much shorter, galactic length scales.\nThat this is not the case can be seen, for example, from the following argument.\nFor a typical galaxy one has a size $\\sim 30 \\, kpc$, giving for the quantum correction\nthe estimate, from the potential obtained in the\nstatic isotropic metric solution with $G(\\Box)$ \\cite{hw06} which gives \n$\\delta G(r) \\sim (r\/\\xi)^{1\/\\nu}$, \n$( 30 \\, kpc \/ 4890 \\times 10^3 \\, kpc )^3 \\sim 2.31 \\times 10^{-16} $ which is tiny given\nthe large size of $\\xi$.\nIt is therefore unlikely that such a correction will be detectable at these length scales,\nor that it could account for large anomalies in the galactic rotation curves.\nThe above argument also implies a certain sensitivity of the results to the value of the scale $\\xi$;\nthus an increase in $\\xi$ by a factor of two tends to reduce the effects of $G(\\Box)$ by \nroughly $2^3 = 8$, as can be seen already from Eq.~(\\ref{eq:grun_box}) with $\\nu=1\/3$.\nMore specifically, the amplitude of the quantum correction is proportional, \nin the non-infrared improved case, to the combination $c_0 \/ \\xi^3$.}\nBut most likely the theoretical uncertainties in the value of $c_t$ have also been underestimated\nin \\cite{cor94}, and new, high precision lattice calculation will be required to \nsignificantly reduce the systematic errors.\n\nNevertheless it seems clear that the non-perturbative coefficient $c_0$ (or $c_t$) \nenters {\\it all} calculations involving $G(\\Box)$ with the {\\it same} magnitude and sign.\nThis is simply a consequence of $c_0$ being part of the renormalization\ngroup $G(\\Box)$ which enters the covariant effective field equations of Eq.~(\\ref{eq:field1}).\nConsequently, one should be able to relate one set of physical results to another, \nsuch as the value of the slip function $\\eta (z=0)$ in \nEq.~(\\ref{eq:eta_z}) to the corrections to the density perturbation growth \nexponent $\\gamma$  computed in \\cite{ht10}, and given here in Eq.~(\\ref{eq:gamma_ten}).\nThen the amplitude $c_t$ can be made to conveniently drop out when computing\nthe ratio of $G(\\Box)$ corrections to two different physical processes.\nThe resulting predictions are then entirely independent of the theoretical\nuncertainty in the amplitude $c_0$, and remain sensitive only to the uncertainties in\nthe two other quantum parameters $\\xi$ and $\\nu$, which are expected to be \nsignificantly smaller.\nOne then obtains for the ratio of the corrections to the growth exponent $\\gamma$\nto the slip function $\\eta (z=0)$  at $ z \\simeq 0 $ \n\\beq\n{ \\delta \\, \\gamma \\over \\delta \\, \\eta } \\; \\simeq \\; \n{ - 106.4 \\, c_t \\over  - 1.491 \\, c_t } \\; \\simeq \\;  + \\, 71.4 \n\\eeq\nfor the infrared unimproved case.\nOne conclusion that one can draw from the numerical value of the above ratio is\nthat it might be significantly harder to see the $G(\\Box)$ correction\nin the slip function than in the matter density growth exponent, by almost 2\norders of magnitude in relative magnitude.\nHopefully increasingly accurate astrophysical measurements of the latter will\nbe done in the not too distant future.\nOf particular interest would be any trend in the growth exponents as a function of\nthe maximum galactic cluster size.\n\n\n\n \\newpage\n\n\\vspace{20pt}\n\n{\\bf Acknowledgements}\n\nOne of the authors (HWH) wishes to thank Thibault Damour and Gabriele Veneziano\nfor inspiration and discussions leading to the present work, and\nAlexey Vikhlinin for correspondence regarding astrophysical measurements \nof structure growth indices.\nHWH wishes to thank Thibault Damour and the\nI.H.E.S. in Bures-sur-Yvette for a very warm hospitality. \nThe work of HWH was supported in part by the I.H.E.S. and\nthe University of California.\nThe work of RT was supported in part by a DoE GAANN student fellowship.\n\n\\vspace{20pt}\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Main Results}\n\\ \\ \\ \\ \\ In \\cite{r3} and [5], M. Willem  and Mawhin studied the following\nsecond order Hamiltonian system\n\\begin{equation}\\label{1}\n\\ddot{u}(t)=-\\nabla F(t,u(t))=-F'(t,u(t))\n\\end{equation}\nwhere $F: [0,T]\\times R^{N}\\rightarrow R ,\\nabla F(t,u(t))=F'(t,u(t))$\nis the gradient of $F(t,u(t))$ with respect to $u$.\nWe assume $F(t,u(t))$ satisfies the following\nassumption:\n\n(A). $F(t,x)$ is measurable in $t$ for each $x\\in R^{N}$,\ncontinuously differentiable in $x$ for a.e. $t\\in [0,T]$, and there\nexist $a\\in C(R^{+},R^{+})$ and $b\\in L^{1}(0,T;R^{+})$ such that\n$$|F(t,x)|\\leq a(|x|)b(t),$$\n$$|\\nabla F(t,x)|\\leq a(|x|)b(t)$$ for all\n$x\\in R^{N}$ and a.e. $t\\in [0,T]$.\n\n  M. Willem (\\cite{r3}) got the following  theorem :\n\\vspace{0.4cm}\n\n\\textbf{Theorem 1.1}\\\n(\\cite{r3} and [5]) Assume $F$ satisfies\n condition (A) and for the canonical basis $\\{e_{i}|1\\leq i\\leq N\\}$\n of $R^{N}$, there exist $T_{i}>0$ such that for $\\forall x\\in\n R^{N}$ and a.e. $t\\in [0,T]$,\n\\begin{equation}\\label{2}\nF(t,x+T_{i}e_{i})=F(t,x), \\ \\ \\ \\ 1\\leq i\\leq N\n\\end{equation}\nThen (1.1) has at least one solution which minimizes\n$$f(u)=\\int_{0}^{T}[\\frac{1}{2}|\\dot{u}(t)|^{2}-F(t,u(t))]dt$$\non\n$H_{T}^{1}=\\{u|u, \\dot{u}\\in L^{2}[0,T], u(t+T)=u(t)\\}$.\n\\vspace{0.4cm}\n\nIn order to cover the forced pendulum equation:\n\\begin{equation}\\label{3}\n\\ddot{u}(t)=-a\\sin u+e(t),\n\\end{equation}\n\nMawhin-Willem [5] also study the following forced equation:\n\\begin{equation}\\label{1}\n\\ddot{u}(t)=-\\nabla F(t,u(t))-e(t)=-F'(t,u(t))-e(t)\n\\end{equation}\nthey got the following Theorem:\n\n\\textbf{Theorem 1.2}\\\n(\\cite{r3} and [5]) Assume $F$ satisfies\n the conditions of Theorem 1.1,and $e(t)\\in L^{1}(0,T;R^N)$ verifying\n $$\\int_{0}^{T}e(t)dt=0,$$\n then (1.4) has at least one solution which minimizes on $H_{T}^{1}$ the following functional:\n$$f(u)=\\int_{0}^{T}[\\frac{1}{2}|\\dot{u}(t)|^{2}-F(t,u(t))-e(t)u(t)]dt$$\n   We notice that the potential $F(t,x)=-(a\\cos x+e(t)x)$ does not satisfy (\\ref{2}). But\nif $\\int_{0}^{T}e(t)dt=0$, then $F(t,x)=-(a\\cos x+e(t)x)$ does satisfy\n\\begin{equation}\\label{4}\n\\int_{0}^{T}F(t,x+2\\pi)dt=\\int_{0}^{T}F(t,x)dt.\n\\end{equation}\nSo instead of (\\ref{2}) we only assume the weaker integral condition:\n\\begin{equation}\\label{5}\n\\int_{0}^{T}F(t,x+T_{i}e_{i})dt=\\int_{0}^{T}F(t,x)dt \\ \\ \\ \\\ni=1,2,...,N\n\\end{equation}\nWe obtain the following results:\n\n \\vspace{0.4cm}\\textbf{Theorem 1.3}\\ Assume $F: R\\times R^{N}\\rightarrow\n R$ satisfies condition (A) and\\\\\n (F1). $F(t+T,x)=F(t,x)$, $\\forall (t,x)\\in R\\times R^{N}$,\\\\\n (F2). $F$ satisfies (\\ref{5}).\\\\\n (F3). There exist $0<C_{1}<\\frac{1}{2}(\\frac{2\\pi}{T})^2$, $C_2>0$ such that\n $$|F(t,x)|\\leq C_1|x|^2+C_2$$\n\n  Then (1.1) has at least one $T$-periodic\n solution.\n\n\\vspace{0.4cm}\n\n\\textbf{Corollary 1.1} (J. Mawhin, M. Willem\n\\cite{r5})\\ For the pendulum equation (\\ref{3}), the potential\n$F(t,x)=a\\cos x+e(t)x$ satisfies all conditions in Theorem 1.3\nprovided $e(t+T)=e(t)$ and $\\int_{0}^{T}e(t)dt=0$. In this case,\n(\\ref{3}) has at least one $T$-periodic solution.\n\n\\vspace{0.4cm}\n\n\n\\textbf{Theorem 1.4}\\ Suppose $F: R\\times\nR^N\\rightarrow R$ satisfies conditions (A), (F1), (F2) and\\\\\n (F4).\nThere are $\\mu_1<2$, $\\mu_2\\in R$ such that\n$$F'(t,x)\\cdot x\\leq \\mu_1F(t,x)+\\mu_2,$$\\\\\n(F5). There is $\\delta>0$ such that for $ t\\in R$, $F(t,x)>\\delta$, as\n$|x|\\rightarrow +\\infty$,\\\\\n(F6). $F(t,x)\\leq b|x|^2$.\\\\\n\n Then if $T<\\sqrt{\\frac{2}{b}}\\pi$, (1.1) has a $T-$periodic solution;\nfurthermore, if $\\forall x\\in R^N$, $\\int_0^T F(t,x)dt\\geq 0$, then\n(\\ref{1}) has a non-constant $T-$periodic solution.\n\n\n\n\\section{Some Important Lemmas}\n\n\n\\ \\ \\ \\ \\ \\ \\ \\ {\\bf Lemma 2.1}\\ (Eberlin-Smulian\\cite{ES}) A\nBanach space X is reflexive if and only if any bounded sequence in X\nhas a weakly convergent subsequence.\n\\vspace{0.3cm}\n\n {\\bf Lemma 2.2}\\\n(\\cite{e1},\\cite{r1},\\cite{e15})\\ Let $q\\in W^{1,2}(R\/TZ,R^{n})$ and\n$\\int^T_0q(t)dt=0$, then\n\n(i). We have Poincare-Wirtinger's inequality\n$$\\int^T_0|\\dot{q}(t)|^2dt\\geq (\\frac{2\\pi}{T})^2\\int_0^T|q(t)|^2dt$$\n\n(ii). We have Sobolev's inequality\n$$\\max_{0\\leq t\\leq T}|q(t)|=\\|q\\|_{\\infty}\\leq\n\\sqrt{\\frac{T}{12}}(\\int^T_0|\\dot{q}(t)|^2dt)^{1\/2}$$\n\\vspace{0.2cm}\n\n\nWe define the equivalent norm in $H^{1}_{T}=H^{1}=W^{1,2}(R\/TZ,R^n):$\n$$\\|q\\|_{H^1}=(\\int_0^T|\\dot{q}|^2)^{1\/2}+|\\int_0^T q(t)dt|$$\n\n{\\bf Lemma 2.3}(\\cite{r1})\\ Let $X$ be a reflexive Banach space,\n$M\\subset X$ a weakly closed subset, and $f: M\\rightarrow R\\cup\n\\{+\\infty\\}$ weakly lower semi-continuous. If the minimizing\nsequence for $f$ on $M$ is bounded, then $f$ attains its infimum on\n$M$.\n\\vspace{0.4cm}\n\n{\\bf Definition 2.1}(\\cite{e4})\\ Suppose $X$ is a Banach space and $f\\in C^{1}(X,R)$ and\n$\\{q_{n}\\}\\subset X$ satisfies\n$$f(q_{n})\\rightarrow C,\\ \\ \\ \\ (1+\\|q_{n}\\|)f'(q_{n})\\rightarrow 0.$$\nThen we say $\\{q_n\\}$ satisfies the $(CPS)_C$ condition.\n\\vspace{0.4cm}\n\n {\\bf Lemma 2.4}(Rabinowitz's Saddle Point Theorem\\cite{e12},\nMawhin-Willem\\cite{r1})\\ Let $X$ be a Banach space with $f\\in\nC^1(X,R)$. Let $X=X_1\\oplus X_2$ with\n$$dim X_1<+\\infty$$\nand\n$$\\sup_{S^1_R} f<\\inf_{X_2} f,$$\nwhere $S^1_R=\\{u\\in X_1||u|=R\\}$.\n\nLet $B_R^1=\\{u\\in X_1, |u|\\leq R\\}$, $M=\\{g\\in C(B^1_R,X)|g(s)=s,\ns\\in S^1_R\\}$\n$$C=\\inf_{g\\in M}\\max_{s\\in B^1_R}f(g(s)).$$\nThen $C>\\inf_{X_2}f$, and if $f$ satisfies $(CPS)_{C}$ condition, then $C$ is\na critical value of $f$.\n\\vspace{0.4cm}\n\n\\section{The Proof of Theorem 1.3}\n\n\\vspace{0.4cm}\n\n\\textbf{Lemma 3.1} (Morrey \\cite{r2}, M-W \\cite{r3})\\\nLet $L: [0,T]\\times R^{N}\\times R^{N}\\rightarrow R$,\n$(t,x,y)\\rightarrow L(t,x,y)$ be measurable in $t$ for each\n$(x,y)\\in R^{N}\\times R^{N}$ and continuously differentiable in\n$(x,y)$ for a.e. $t\\in [0,T]$. Suppose there exists $a\\in\nC(R^{+},R^{+})$, $b\\in L^{1}(0,T; R^{+})$ and $c\\in L^{q}(0,T;\nR^{+})$, $1<q<\\infty$, such that for a.e. $t\\in [0,T]$ and every\n$(x,y)\\in R^{N}\\times R^{N}$ one has\n$$|L(t,x,y)|\\leq a(|x|)(b(t)+|y|^{q}),$$\n$$|D_{x}L(t,x,y)|\\leq a(|x|)(b(t)+|y|^{p}),$$\n$$|D_{y}L(t,x,y)|\\leq\na(|x|)(c(t)+|y|^{p-1}).$$\n where $\\frac{1}{p}+\\frac{1}{q}=1$, then the\nfunctional\n$$\\varphi(u)=\\int_{0}^{T}L(t,u(t),\\dot{u}(t))dt$$\nis continuously differentiable on the Sobolev space\n\\begin{equation}\\label{6}\nW^{1,p}=\\{u\\in L^{p}(0,T), \\dot{u}\\in L^{p}(0,T)\\}\n\\end{equation}\nand\n\\begin{equation}\\label{7}\n<\\varphi'(u),v>=\\int_{0}^{T}[<D_{x}L(t,u,\\dot{u}),v>+D_{y}L(t,u,\\dot{u})\\cdot\\dot{v}]dt\n\\end{equation}\n\nFrom Lemma 3.1 and the assumptions (A), we know that the variational\nfunctional\n\\begin{equation}\\label{8}\nf(u)=\\int_{0}^{T}[\\frac{1}{2}|\\dot{u}|^{2}-F(t,u(t))]dt\n\\end{equation}\nis $C^{1}$ on $W_{T}^{1,2}=H^{1}_{T}$, and the critical point is\njust the periodic solution for the system (1.1).\n\nFurthermore, if (F1) and (F2) are satisfied, we will prove the\nfunctional $f(u)$ attains its infimum on $H_{T}^{1}$; in fact,\n\\begin{equation}\\label{10}\nH_{T}^{1}=X\\oplus R^{N},\n\\end{equation}\nwhere\n\\begin{equation}\\label{11}\n X=\\{x\\in H^{1}_{T}: \\bar{x}\\triangleq\\frac{1}{T}\\int_{0}^{T}x(t)dt=0\\}\n\\end{equation}\n and $\\forall u\\in H_{T}^{1}$, we have $\\widetilde{u}\\in X$ and\n$\\overline{u}\\in R^{N}$, such that $u=\\widetilde{u}+\\overline{u}$.\n\\vspace{0.2cm}\n\n\nBy Poincare-Wirtinger's inequality,\n\\begin{equation}\\label{12}\n  \\begin{aligned}\nf(\\tilde{u})&\\geq\\frac{1}{2}\\int_{0}^{T}|\\dot{\\tilde{u}}|^{2}dt-C_{1}\\int_{0}^{T}|\\tilde{u}|^{2}dt-C_2T\\\\\n&\\geq[\\frac{1}{2}-C_1(\\frac{2\\pi}{T})^{-2}]\\int_{0}^{T}|\\dot{\\widetilde{u}}|^{2}dt-C_2T;\n \\end{aligned}\n\\end{equation}\nhence, $f$ is coercive on $X$.\n\nLet $\\{u_{k}\\}$ be a minimizing sequence for $f(u)$ on $H_{T}^{1}$,\n$u_{k}=\\widetilde{u}_{k}+\\overline{u}_{k}$, where\n$\\widetilde{u}_{k}\\in X$, $\\overline{u}_{k}\\in R^{N}$, then by\n(\\ref{12}) we have\n\\begin{equation}\\label{14}\n\\|\\widetilde{u}_{k}\\|_{H^{1}_{T}}\\leq C.\n\\end{equation}\nBy condition (F2), we have\n\\begin{equation}\\label{15}\nf(u+T_{i}e_{i})=f(u), \\ \\ \\ \\ \\forall u\\in H_{T}^{1}, \\ \\ \\ \\ 1\\leq\ni\\leq N.\n\\end{equation}\nSo if $\\{u_{k}\\}$ is a minimizing sequence for $f$, then\n$$(\\widetilde{u}_{k}\\cdot e_{1}+\\overline{u}_{k}\\cdot e_{1}+k_{1}T_{1},...,\\widetilde{u}_{k}\\cdot e_{N}+\\overline{u}_{k}\\cdot e_{N}+k_{N}T_{N})$$\nis also a minimizing sequence of $f(u)$, and so we can assume\n\\begin{equation}\\label{16}\n0\\leq\\overline{u}_{k}\\cdot e_{i}\\leq T_{i}, \\ \\ \\ \\ 0\\leq i\\leq N.\n\\end{equation}\nBy (\\ref{14}) and (\\ref{16}), we know $\\{u_{k}\\}$ is a bounded\nminimizing sequence in $H_{T}^{1}$, and it has a weakly convergent\nsubsequence; furthermore, $f$ is weakly lower semi-continuous since\n$f$ is the sum of a convex continuous function and a weakly\ncontinuous function. We can conclude that $f$ attains its infimum on\n$H_{T}^{1}$. The\ncorresponding minimizer is a periodic solution of (\\ref{1}).\\\\\n\n\n\n\n\\section{The Proof of Theorem 1.4}\n\n\n{\\bf Lemma 4.1}:\\ If conditions (A), (F1), (F2) and (F4) in Theorem\n1.4 hold, then $f(q)$ satisfies the $(CPS)_C$ condition on $H^1$.\n\n{\\bf Proof}: For any $C$, let $\\{u_n\\}\\subset H^1$ satisfy\n\\begin{equation}\\label{17'}\nf(u_n)\\rightarrow C, \\ \\ \\ \\ (1+\\|u_n\\|)f'(u_n)\\rightarrow 0.\n\\end{equation}\nWe claim $\\|\\dot{u}_n\\|_{L^2}$ is bounded; in fact, by\n$f(u_n)\\rightarrow C$, we have\n\\begin{equation}\\label{18'}\n\\frac{1}{2}\\|\\dot{u}_n\\|^2_{L^2}-\\int_0^T F(t,u_n)dt\\rightarrow C.\n\\end{equation}\nBy (F4) we have\n\\begin{equation}\\label{19'}\n\\begin{aligned}\n <f'(u_n),u_n>&=&\\|\\dot{u}_n\\|^2_{L^2}-\\int_0^T\n (<F'(t,u_n),u_n>)dt\\\\\n &\\geq&\\|\\dot{u}_n\\|^2_{L^2}-\\int_0^T [\\mu_2+\\mu_1F(t,u_n)]dt.\n \\end{aligned}\n\\end{equation}\nBy (\\ref{18'}) and (\\ref{19'}), we see that\n\n\\begin{equation}\\label{20'}\n0\\leftarrow<f'(u_n),u_n>\\geq a\\|\\dot{u}_n\\|^2_{L^2}+C_1+\\delta,\nn\\rightarrow +\\infty\n\\end{equation}\nwhere $C_1=C\\mu_1-T\\mu_2+\\delta, \\delta>0, a=1-\\frac{\\mu_1}{2}>0.$\\\\\nWe have shown that $\\|\\dot{u}_n\\|_{L^2}$ is bounded.\n\nBy condition (F2), we have\n\\begin{equation}\\label{15''}\nf(u+T_{i}e_{i})=f(u), \\ \\ \\ \\ \\forall u\\in H_{T}^{1}, \\ \\ \\ \\ 1\\leq\ni\\leq N.\n\\end{equation}\nHence, if $\\{u_{k}\\}$ is a $(CPS)_C$ sequence for $f$, then\n$$(\\widetilde{u}_{k}\\cdot e_{1}+\\overline{u}_{k}\\cdot e_{1}+k_{1}T_{1},...,\\widetilde{u}_{k}\\cdot e_{N}+\\overline{u}_{k}\\cdot e_{N}+k_{N}T_{N})$$\nis also a $(CPS)_C$ sequence of $f(u)$, so we can assume\n\\begin{equation}\\label{16''}\n0\\leq\\overline{u}_{k}\\cdot e_{i}\\leq T_{i}, \\ \\ \\ \\ 0\\leq i\\leq N.\n\\end{equation}\nBy (\\ref{16''}), we know $|\\bar{u}_{k}|$ is bounded, and so $\\|u_n\\|=\\|\\dot{u}_n\\|_{L^2}+|\\int_0^T u_n(t)dt|$ is bounded.\n\nThe rest of he lemma can be completed in a now standard fashion.\n\nWe finish the proof of \\textbf{Theorem 1.4.} In Rabinowitz's Saddle Point\nTheorem, we take\n$$X_1=R^N, X_2=\\{u\\in W^{1,2}(R\/TZ,R^N),\\int_0^T u dt=0\\}.$$\n\nFor $u\\in X_2$, we may use the Poincare-Wirtinger inequality, and so by Lemma\n2.2 and (F6), we have\n\\begin{eqnarray*}\\label{22}\nf(u)&\\geq&\\frac{1}{2}\\int_0^T|\\dot{u}|^2 dt-b\\int_0^T|u|^2 dt\\\\\n&\\geq&[\\frac{1}{2}-b(2\\pi)^{-2}T^2]\\int_0^T|\\dot{u}|^2dt\\\\\n&\\geq& 0.\n\\end{eqnarray*}\nOn the other hand, if $u\\in R^N$, then by $(F5)$ we have\n$$f(u)=-\\int_0^T F(t,u)dt\\leq -\\delta, |u|=R\\rightarrow\n+\\infty.$$\n\nThe proof of Theorem 1.4 is concluded by calling upon Rabinowitz's Saddle Point Theorem.\nIn fact,there is a critical point $\\bar{u}$ such that $f(\\bar{u})=C>\\inf_{X_2} f(u)\\geq 0$,\nwhich is nonconstant since otherwise $f(\\bar{u})=-\\int_0^TF(\\bar u,t)dt\\leq 0,$ which is a contradiction.\\\\\n\nThe authors would like to thank the referees for their valuable suggestions.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe purpose of this study is to update our earlier study \\cite{bs_old} on the \npossibility to observe the rare $B^{0}_{S} \\rightarrow \\mu \\mu$ decay with \nthe CMS detector. As well known, this channel with a very small Standard\nModel branching ratio is a sensitive probe of a ``new physics'' affecting\nFCNC \\cite{br_rat}, and only the large b production rate of a hadron collider\npossibly allows to obtain the needed level of sensitivity. The reasons which \nmight modify the previous results are the following (for details see the \nnext chapters) :\n\n- detailed simulation of the TDR version of the CMS detector, tracker in\nparticular;\n\n- more sophisticated algorithms for track and vertex reconstruction and \ntighter selection criteria;\n\n- new version of the PYTHIA Monte-Carlo generator \\cite{pythia};\n\n- improved statistical precision of the results due to a much bigger sample \nof generated events;\n \n- new theoretical estimate of the branching ratio of the decay in the\nStandard Model $(3.5 \\pm 1.0) \\times 10^{-9}$ \\cite{br_rat};\n\nIn the previous study \\cite{bs_old} we established that the dominant \nbackground is due to direct muons from $b \\bar{b}$ pairs produced \nthrough the gluon splitting mechanism. All other sources of background are \nfound  to be about\none order of magnitude smaller than this one. In the present work we thus \nconcentrated only on this source of background. The higher level trigger and\nkinematics selection criteria were taken as in \\cite{bs_old}:\n\n\\begin{equation}\np_{t}^{\\mu} > 4.3~GeV,~~~|\\eta| < 2.4\n\\end{equation}\n\\begin{equation}\n0.4 < \\Delta R_{\\mu \\mu} < 1.2,~~~p_{t}^{\\mu \\mu} > 12~GeV\n\\end{equation}\n\nwhere $R_{\\mu \\mu}$ is the distance between the two muons in $\\eta$, \n$\\phi$ space and $p_{t}^{\\mu \\mu}$ is the transverse momentum of the di-muon \npair. We assume that there is no significant loss of dimuon events at the\nfirst trigger level. This is justified for running at luminosity \n$10^{33}cm^{-2}s^{-1}$ but should be examined in more detail when \napproaching $10^{34}cm^{-2}s^{-1}$. \nTo suppress the background we have exploited as in our previous study \nthe good dimuon mass resolution of CMS, the precise two-muon vertex \nreconstruction, and isolation criteria with the tracker and calorimeters.\n\nWe have also investigated sensitivity to  \n$B^{0}_{d} \\rightarrow \\mu \\mu$ decays.\n\n\n\\section{Event generation}\n\n\\subsection{Kinematics and cross-sections}\n\nBackground events have been generated by PYTHIA5.7 with default CTEQ2L \nstructure functions and the default choice of the fragmentation function,\nwhich is\nthe Lund symmetric fragmentation function modified for heavy endpoint\nquarks (see references in \\cite{pythia}). Pairs of $b\\bar{b}$ quarks have\nbeen extracted from MSEL=1 QCD 2 $\\rightarrow$ 2 processes where gluons\nare produced: process 28 (f+g $\\rightarrow$ f+g) and 68 \n(g+g $\\rightarrow$ g+g). Data selection kinematics cuts (1), (2) on muons \nhave been applied at the generation level. We have generated 10000 of di-muon \nbackground events passing through these cuts. \n\nThe normalisation has been done on the $b\\bar{b}$ production cross-section \nobtained by counting the number of events where at least one $b\\bar{b}$ \npair was produced. We give in Table 1 the PYTHIA output on the \ncross-sections of MSEL=1 subprocesses and cross-section of $b\\bar{b}$ pairs \nproduction with the processes 11-13, 53 and 28,68. The total fraction of  \nevents with $b\\bar{b}$ pairs is $7.4 \\times 10^{-3}$ and the corresponding \ncross-section is 409 $\\mu$b. According to the data in Table 1, we have \nnormalised the background from gluon splitting on 282 $\\mu$b. In the previous\nstudy we normalised gluon splitting background on 370 $\\mu$b given by\nPYTHIA5.6 with EHLQ1 structure functions (PYTHIA5.6 default).\n   \nSignal $B^{0}_{S} \\rightarrow \\mu \\mu$ events have been extracted from all\nMSEL=1 subprocesses and are present therefore a mixture of gluon \nsplitting and gluon fusion production mechanisms.\n\nWe estimate that after trigger and kinematic selections the number\nof signal and background events, for one year running at luminosity\n$L=10^{33}cm^{-2}s^{-1}$ (i.e. $10^{4}pb^{-1}$), is 66 and \n$2.9 \\times 10^{7}$ respectively.\n\nWe checked that choosing the Peterson fragmentation function instead of the\ndefault one does not make a difference in the spectra of B-hadrons and charged \nparticles around the B-hadron and therefore does not affect the efficiency of \nthe kinematics selections and isolation criteria (see below). \nIn Figure~\\ref{fragm_func} are shown the spectra of the B-hadron \n(Figure~\\ref{fragm_func}a) and of charged \nparticles (Figure~\\ref{fragm_func}b) with $p_{t}>0.9$ GeV (not including \ndecay products of B-hadron) in cone $\\Delta R <$0.4 around the B-hadron for \nthe default choice of the fragmentation function (solid line) and the \nPeterson fragmentation \nfunction \\cite{peterson} (dashed line). For this plot, $b \\bar{b}$ pairs have \nbeen generated by a gluon fusion mechanism $gg \\rightarrow b\\bar{b}$ with \n$\\hat{p_{t}} >$4 GeV. There is almost no difference between the two choices \nof the fragmentation function.\n \n\\begin{verbatim}\n\nTable 1. Cross-sections for MSEL=1 subprocesses and corresponding\n         cross-sections for bb pair production in these processes.\n  -------------------------------------------------------------------------\n         Subprocesses             Cross-section     Cross-section of \n                                     (mb)         bb pairs production (mb)\n  -------------------------------------------------------------------------\n   I  11 f + f' -> f + f' (QCD) I  1.528E+00   I\n   I  12 f + f~ -> f' + f~'     I  2.209E-02   I        0.002\n   I  13 f + f~ -> g + g        I  2.086E-02   I        \n  -------------------------------------------------------------------------\n   I  53 g + g -> f + f~        I  8.553E-01   I        0.125\n  -------------------------------------------------------------------------\n   I  28 f + g -> f + g         I  1.659E+01   I\n   I  68 g + g -> g + g         I  3.621E+01   I        0.282\n  -------------------------------------------------------------------------\n   I  All included subprocesses I  5.522E+01   I        0.409       \n  -------------------------------------------------------------------------\n\n\\end{verbatim}\n\n\\subsection{CMS detector simulation}\n\nDetector simulation has been done with the CMSIM package \\cite{cms114} which\nis based on GEANT3 and simulates properly the response of the CMS subdetectors\nused for the reconstruction. The TDR designs for the calorimetry and \ntracker are taken. Two options of the tracker geometry have been simulated, \ncorresponding to the low and high luminosity running. The important\ndifference between these two designs is the position of the pixel vertex \ndetector. The radii of the two barrel pixel cylinders are 4 and 7 cm \nfor the low luminosity design, and 7 and 11 cm for the high luminosity case.\nThe low luminosity configuration provides a better vertex reconstruction \nprecision over the entire tracker acceptance \\cite{tracker_tdr}.   \n    \nBoth signal and background events were\npassed through the detailed detector simulation and reconstruction procedure \nto obtain\nthe efficiency of the selection criteria. Energy thresholds on particle \ntracking is taken as 100 KeV (1 MeV) for electrons and photons, and \n1 MeV (10 MeV) for hadrons in the Tracker (Calorimetry).  \n\n\\section{Event selections}\n\n\\subsection{Track and vertex reconstruction}\n\nThe CM\\_FKF track finder \\cite{tracker_tdr} was used to reconstruct\nall tracks with $p_{t} \\geq 0.9$ GeV, $\\mid \\eta \\mid \\leq 2.5$ with at \nleast 6 hits and $\\chi^2 \/ ndf \\leq 7$.\nIn these conditions, the global \ntrack reconstruction efficiency is about 91 \\% both for signal and\nbackground samples. After track reconstruction the GVF vertex finder\n\\cite{gvf} was implemented. The GVF allows in principle to reconstruct all \n\"interesting\"\nvertices in a given event, including primary, secondary and two prong\nparticle decay vertices. However, in our particular case, a simplified\nversion of the algorithm is used, tuned to reconstruct just the dimuon \nsecondary vertices. A given reconstructed track was considered to \nbe a muon and was taken into account in the vertex reconstruction,\nif it had at least two pixel hits and was associated with the true\nMonte Carlo muon, i.e. no realistic tracker-muon system matching has been \nimplemented yet. We consider that the efficiency for this matching should be\nvery high as the b-jets are relatively soft i.e. open and of moderate\nmultiplicity while the muons are typically will be of $p_{t}^{\\mu}\\geq$ 6 GeV. \nTo understand the vertex quality cuts \nused below one has to keep in\nmind that the GVF is a two-step procedure. In this case it creates first\nthe vertex seeds\ntrying all dimuon combinations. A given track pair is accepted as a secondary\nvertex seed if the minimal 3D distance (called M2D below) \nbetween the two spirals is small enough. At this stage some tracks can \nbelong, in principle, to several vertex seeds. At a second stage of the \nalgorithm all accepted seeds are processed by the dedicated algorithm, \nwhich tries to solve ambiguities and to fit simultaneously the \nvertex positions and the track parameters.\n\nSeveral event samples have been used to optimise the selection criteria\nand the signal to background ratio. \n     \n\\begin{enumerate}\n\n\\item $\\sim$ 400 signal events.\n\n\\item 3000 signal events with muons only. Other particles in the event \nwere not propagated through the detector. This sample has been used to tune \nthe vertex reconstruction and selection criteria.\n\n\\item 10000 background events.\n\n\\item 10000 same background events, but bearing only the muons.\n\n\\end{enumerate}\n\nThe samples with muons only have been used mostly to evaluate the high \nluminosity scenario, since the processing of a full event (bunch crossing),\nwith pile-up, takes enormous CPU and memory resources at the moment.\n\nThe basic variables which characterise the performance of the low luminosity \ntracker configuration are shown in Figure~\\ref{global_var}. In this figure\none can see dimuon mass resolution, secondary vertex resolution in X\/Y and \nZ coordinates and proper time resolution.\n\n\\subsection{Selections criteria based on the tracker}\n\n\\textbf{i) mass cut}\n\nThe first background suppression criterion is a \n$B^{0}_{s} \\rightarrow \\mu \\mu$ mass cut against the continuum dimuon\nbackground population. To search for signal events we have chosen a di-muon \nmass window of $\\pm$ 40 MeV centred on the $B_{S}$ mass of 5.369 GeV. \nIn Figure~\\ref{isol_mass_corr}a \nthe di-muon mass spectrum for the background is shown after the trigger and \ninitial kinematics selections. The 80 MeV mass window is also shown in \nthe same figure. Only $1.1 \\pm 0.1~\\%$ of background events is retained\nin this mass window. As full simulations have shown, the high luminosity\ntracker configuration provides nearly the same di-muon mass resolution.\nFor high luminosity running we thus expect the same selectivity of \nthe dimuon mass cut both for the signal and background as for the \nlow luminosity case.\n\n\\textbf{ii) isolation}\n\n$B^{0}_{s} \\rightarrow \\mu \\mu$ decays produced in soft b-jets are\nsemi-isolated and significantly more isolated than dimuons from accidental\n$g \\rightarrow b(\\rightarrow \\mu) \\bar{b}(\\rightarrow \\mu)$ associations.\nIn our previous study \\cite{bs_old} it was found that the rejection \nfactor based on the tracker isolation depends strongly on the lower cutoff \non the transverse momentum of the charged particles in the isolation cone \naround the di-muon momentum. The detailed tracker pattern recognition \nstudies show that charged tracks with transverse momenta above 0.9 GeV can \nbe reconstructed with an efficiency exceeding 90 \\% within the tracker \nacceptance \\cite{tracker_tdr}.\nIn this work we have used a slightly different tracker\nisolation definition than in \\cite{bs_old}. We required no charged\ntracks with $p_{t} >$ 0.9 GeV in a cone \n$\\Delta R~=~0.5 \\times \\Delta R_{\\mu \\mu}~+~0.4$ around the di-muon \nmomentum direction. Such a criterion gives an efficiency of 0.49 for\nthe signal and $(3.0 \\pm 0.2) \\times 10^{-2}$ for the background in \nconditions of low luminosity running.\n\nAssuming a charged track reconstruction efficiency 0.9 for tracks \nwith $p_{t}>$ 0.9 GeV, we reproduced at the particle-level simulations \nthe efficiency of the tracker isolation criterion obtained with full \nsimulation and pattern recognition: $(2.7 \\pm 0.2) \\times 10^{-2}$ at the \nparticle level as compared to $(3.0 \\pm 0.2) \\times 10^{-2}$  with full \nsimulation and pattern recognition. This then allows us to obtain the \nefficiency of the tracker isolation criterion for the high luminosity case \nwhere we didn't make the full detector simulation.\nFigures~\\ref{isol_ll}a,c\nand Figure~\\ref{isol_hl}a,c show the tracker isolation parameter - the\nnumber of charged tracks with  $p_{t} >$ 0.9 GeV in a cone\n$\\Delta R~=~0.5 \\times \\Delta R_{\\mu \\mu}~+~0.4$ around the di-muon \nmomentum direction obtained with particle-level simulation as explained\nabove for the case of low (Figure~\\ref{isol_ll}a,c) and high \n(Figure~\\ref{isol_hl}a,c) luminosity.\n\n\\textbf{iii) secondary vertex selection}\n\nA third set of cuts is based on the secondary vertex reconstruction\nquality and primary-secondary vertex separation.\nThe minimal cuts on the reconstructed secondary vertex applied at the event \nreconstruction level allows to keep almost all signal events ($\\sim\n95$\\%), but the background rejection factor is then about 4. \nThe di-muon vertex reconstruction algorithm\nprovides however a number of parameters which can be used to improve the \nvertexing rejection power.  After the preliminary analysis, 5 variables were \nchosen:\n\n\\begin{enumerate}\n\n\\item M2D - minimal distance of approach in space between the two tracks to \ncreate the secondary vertex seed (Figure~\\ref{m2d}).\n\n\\item M2D\\_rel - relative minimal distance between the two tracks \nto create the secondary vertex seed (Figure~\\ref{m2d_rel}).\n\n\\item VTR\\_rel - relative transverse distance (transverse flight path) of \nthe reconstructed secondary vertex to the primary vertex (by relative we \nmean here the variable measured in terms of\nits errors ). It is shown in Figure~\\ref{vrt_rel}\n\n\\item VTR\\_err - absolute error on the reconstructed secondary vertex \nin the transverse plane (Figure~\\ref{vrt_err}). \n\n\\item COS\\_pr - cosine of the angle between the vector pointing to the\nsecondary vertex position (flight path direction) and the two-tracks \nmomentum ($B_{s}^{0}$ momentum) in the transverse plane (Figure~\\ref{cos_pr}).\n\\end{enumerate}\n\nTo reject effectively the false background vertices one has to \nselect high quality reconstructed secondary vertices which are rather far \naway from the primary interaction vertex in the transverse plane.\nOptimising signal observability, we find that a rejection \nfactor of order $10^{-4}$ can be obtained. The following set of cuts\nallows to achieve such a rejection:    \n\n\\begin{enumerate}\n\\item $VTR\\_rel \\geq 12$ (as we checked this cut is nearly equivalent to a\nsharp cut on the vertex distance in the transverse plane of about 820 $\\mu$, \nhowever, we prefer to use a cut on a variable calculated on an event by event \nbasis) \n\\item $COS\\_pr \\geq 0.9997$ \n\\item $VTR\\_err \\leq 80 \\mu$ \n\\item $M2D \\leq 50 \\mu$ \n\\item $M2D\\_rel \\leq 2$ \n\\end{enumerate}\n\nThis in fact allows to eliminate ALL events from the 10K background sample \nfor the low luminosity tracker geometry, still keeping about 30 \\% of the \nsignal. We conclude that the vertexing rejection power is better than\n$2.3 \\times 10^{-4}$ at the 90\\% C.L. level.\n\nTo obtain the same rejection power for the high luminosity tracker geometry\none has to use tighter cuts, in particular changing the cut on the relative \ntransverse distance  $VTR\\_rel \\geq 12$ to $VTR\\_rel \\geq 15$ \nwe can still reject all background events. However, a factor of \n$\\approx 1.8$ in signal efficiency is lost. Other combinations of cuts \ninvestigated for the high luminosity geometry lead to comparable loss \nin signal efficiency.      \n\n\\subsection{Selection criterion based on calorimeters}\n\nWe have also included calorimeter isolation criteria to get an additional\nbackground suppression factor. We required no transverse energy in the ECAL \nplus HCAL above 4 GeV (6 GeV) at $L=10^{33}cm^{-2}s^{-1}$ \n($L=10^{34}cm^{-2}s^{-1}$) in the same cone as for the tracker isolation.\nAn additional rejection factor of $2.3 \\pm 0.2$ (both for low and high\nluminosity) to the tracker isolation only has been obtained at the \nparticle-level simulation and checked on the restricted sample of background \nevents with full GEANT simulation. The efficiency of this additional \ncriterion for the signal is $0.94 \\pm 0.08$ for low luminosity and\n$0.91 \\pm 0.08$ for the high luminosity case.  The isolation energy \ndistribution in the calorimeter obtained at particle level is shown in \nFigure~\\ref{isol_ll}b,d for low luminosity running and in \nFigure~\\ref{isol_hl}b,d for the high luminosity case.\n\n\\subsection{Correlation of the mass and isolation selections}\n\nWe have checked the correlation between the isolation of the dimuon pair \n(in the tracker and calorimeter) and the dimuon mass, to check whether it \nwas correct to evaluate independently the efficiency of the mass cut and the \nisolation criteria.\n\nIn Figure~\\ref{isol_mass_corr}b the efficiency of the isolation criteria \napplied to different regions of di-muon mass is shown. One can clearly \nsee a correlation. However, the efficiency of the isolation criterion \nobtained with full \nstatistics over the whole di-muon mass interval \n(shown in the Figure~\\ref{isol_mass_corr} b as open marks) coincides within \nthe statistical errors with the one obtained for the proper interval of \nmass 4-6 GeV where the $B_{S}$ mass is located. One can also see in \nFigure~\\ref{isol_mass_corr} b that the calorimeter isolation gives a \nsufficient additional suppression factor over the entire interval of dimuon \nmass.\n\n\\section{Conclusion}\n\nTable 1 summarised the efficiency of the selection criteria and the number\nof signal $B^{0}_{S} \\rightarrow \\mu \\mu$\nand background events after successive selection steps, for one\nyear running at both low and high luminosity. Due to insufficient Monte-Carlo\nstatistics we give for the number of background events an upper \nlimit at 90\\% C.L. As one can obtain from Table 1, the \n$B^{0}_{S} \\rightarrow \\mu \\mu$ decay should be seen with significance \n4 after 3 year of running at a luminosity of\n$L=10^{33}cm^{-2}s^{-1}$ even if the background were\nunderestimated by a factor 2. With $3 \\times 10^{4} pb^{-1}$ data at low \nluminosity ($10^{33}cm^{-2}s^{-1}$) and $10^{5}pb^{-1}$ at high luminosity\n($10^{34}cm^{-2}s^{-1}$) significance 6.3 can be achieved. \nThis more optimistic expectation when \ncompared to our previous study \\cite{bs_old} comes from the more \nsophisticated algorithms for the track and vertex reconstruction and \nthe tighter vertex selection criteria. \n\nTaking into account the $B^{0}\/B_{S}^{0}$ = 0.40\/0.11 production ratio and\nthe expected Standard Model \n$Br(B^{0}_{d} \\rightarrow \\mu \\mu)=(1.5 \\pm 0.9) \\times 10^{-10}$, \nwe also estimated that we should get for one year running at low luminosity \n$(1.1 \\pm 0.7)$ of $B^{0}_{d} \\rightarrow \\mu \\mu$ decays again on essentially\nno background.\n\nWe should mention again that in the evaluation of the number of the signal and \nbackground events we have assumed 100 \\% efficiency of the High Level \nTriggers for this di-muon channel. This might not be exactly the case \nalthough only a minor loss is expected, but a special study of the High Level \nTriggers efficiency for this mode is now under way.\nAlso we assumed full efficiency for tracker-muon chamber muon track matching; \npreliminary studies \\cite{genchev} indicate that it is $\\geq$90 \\%.\n \n\n\\begin{table}[htb]\n\\begin{center}\n{Table 1. Selection efficiencies and number of signal and background\nevents before and after selections for $10^{4}pb^{-1}$ running at \nlow ($L=10^{33}cm^{-2}s^{-1}$) and $10^{5}pb^{-1}$ running at\nhigh ($L=10^{34}cm^{-2}s^{-1}$) luminosity.}\n\n\\medskip\n\n\\begin{tabular}{|c|c|c|} \n\\hline\n  & Signal & Background \\\\\n\\hline\n\\hline \nnumber of events after trigger and kinematics selections & \n$66$ &  $2.9 \\times 10^{7}$ \\\\\n\\hline\n\\hline\ntracker isolation. Low luminosity &\n$0.49$ & $3.0 \\times 10^{-2}$ \\\\\n\\hline\ntracker isolation. High luminosity &\n$0.34$ & $2.0 \\times 10^{-2}$ \\\\\n\\hline\ntracker+calo isolation. Low luminosity &\n$0.46$ & $1.3 \\times 10^{-2}$ \\\\\n\\hline\ntracker+calo isolation. High luminosity &\n$0.31$ & $0.87 \\times 10^{-2}$ \\\\\n\\hline\n$2-\\mu$ rec. + sec.vertex selections. Low luminosity &\n$0.32$ & $\\leq 2.3 \\times 10^{-4}$ \\\\\n\\hline\n$2-\\mu$ rec. + sec.vertex selections. High luminosity &\n$0.18$ & $\\leq 2.3 \\times 10^{-4}$ \\\\\n\\hline\nmass window 80 MeV &\n$0.72$ & $1.1 \\times 10^{-2}$ \\\\\n\\hline\n\\hline\nnumber of events after cuts. Low luminosity &\n$7.0$ & $\\leq 1.0$ at 90\\% C.L. \\\\\n\\hline\nnumber of events after cuts. High luminosity &\n$26.0$ & $\\leq 6.4$ at 90\\% C.L. \\\\\n\\hline\n\n\\end{tabular}\n\\end{center}\n\\end{table}\n \n\\section{Acknowledgements}\n\nWe are thankful to D. Denegri, Y. Lemoigne and all members of the CMS B-physics\ngroup for discussions and comments. We specially thank to D. Denegri\nfor careful reading of this note and very useful corrections.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sect1}\nThe simplest variant of the Gross-Neveu (GN) model family \\cite{L1} consists of $N$ species of massless Dirac fermions in 1+1 dimensions, \ninteracting via a scalar-scalar four-fermion interaction,\n\\begin{equation}\n{\\cal L} =  \\sum_{k=1}^N \\bar{\\psi}_k i\\partial \\!\\!\\!\/ \\psi_k + \\frac{g^2}{2} \\left( \\sum_{k=1}^N \\bar{\\psi}_k \\psi_k \\right)^2.\n\\label{I1}\n\\end{equation}\nThis model has asymptotic freedom, no scale, and a discrete chiral symmetry $\\psi \\to \\gamma_5 \\psi$ which gets broken spontaneously in \nthe vacuum, yielding a dynamical fermion mass $m$ via dimensional transmutation. Throughout this paper we will be exclusively dealing with \nthe 't~Hooft limit $N\\to \\infty, Ng^2 =$ const. \\cite{L2}. As is well known, the attractive interaction gives rise to a marginally bound scalar\nfermion-antifermion state (the $\\sigma$-meson) with mass 2$m$ and to a rich variety of multi-fermion bound states (baryons) \\cite{L3,L4}. \nMoreover, the model features a non-trivial phase diagram as a function of temperature and chemical potential with three distinct phases \n(massless and massive Fermi gas, baryon crystal) meeting at a tricritical point \\cite{L5}. It is of interest not only as a toy model for strong \ninteraction particle physics, but also because of its almost literal recurrence in condensed matter systems such as conducting polymers,\ncarbon nanotubes or quasi-one-dimensional superconductors \\cite{L6}. In the large $N$ limit, baryons as well as baryonic matter and the\nphase diagram can be determined with semiclassical methods, notably the relativistic Hartree-Fock (HF) method \\cite{L7}. Relativity enters\nin two ways --- use of the Dirac equation instead of the Schr\\\"odinger equation, and taking into account the filled, interacting Dirac sea. \nRecently we have started to address time-dependent questions by generalizing this approach to time-dependent Hartree-Fock (TDHF) \n\\cite{L8}. In a first step, the boosted baryon was considered, demonstrating a covariant energy-momentum relation and deriving exact \nstructure functions as fermion momentum distributions in the infinite momentum frame \\cite{L9}. In the present work, we proceed to the next\nlevel of complication and study baryon-baryon scattering, an issue which has not yet been discussed in any detail in the large $N$ limit.\nThis may well be the first concrete realization of Witten's vision about baryon-baryon interactions in the large $N$ limit, originally developed\nin a non-relativistic context \\cite{L9a}. Aside from the genuine interest in the scattering problem of composite, relativistic bound states, we hope\nto get further insights into the mathematical structure of the GN model by enlarging the spectrum of questions addressed in the framework of the\nTDHF approach.\n\nBefore embarking on this problem, it may be worthwhile to recapitulate some of the experience gained with previous applications of the\nHF approach. The basic mathematical problem can be formulated in a single line as\n\\begin{equation}\n\\left( i \\partial \\!\\!\\!\/ - S\\right) \\psi_{\\alpha} =0, \\qquad S=-g^2 \\sum_{\\beta}^{\\rm occ} \\bar{\\psi}_{\\beta}\\psi_{\\beta},\n\\label{I2}\n\\end{equation}\nwhere the $\\psi_{\\alpha}$ are $c$-number spinors (single particle wave functions).\nSince the sum over occupied states includes the Dirac sea, this is an infinite system of coupled, non-linear partial differential equations.\nAt finite temperature and chemical potential, a similar formula applies, but the sum includes thermal occupation numbers.\nExcept in the case of a homogeneous mean field $S$ which acts like a dynamical mass, the solution of Eqs.~(\\ref{I2}) is highly non-trivial.\nNevertheless, for the model defined in Eq.~(\\ref{I1}), closed analytical solutions have been found in all cases studied so far. If one examines\nin detail how self-consistency is achieved, one notices that all known solutions fall into two classes: Type I solutions are such that the \ncontribution from every occupied single particle state is either zero or proportional to the full mean field $S$,\n\\begin{equation}\n\\bar{\\psi}_{\\alpha} \\psi_{\\alpha} = \\lambda_{\\alpha} S.\n\\label{I3}\n\\end{equation}\nThe self-consistency condition then becomes space-time independent,\n\\begin{equation}\n- Ng^2 \\sum_{\\alpha}^{\\rm occ} \\lambda_{\\alpha} = 1.\n\\label{I4}\n\\end{equation}\nAside from the trivial case of the vacuum, type I solutions have been found for the kink at rest \\cite{L3,L10} or in flight \\cite{L9} and the kink \ncrystal at zero temperature \\cite{L11}  relevant \nfor the ground state at finite density. In type II solutions, the scalar condensate for each occupied state involves two different space-time \nfunctions. Without loss of generality, this can be written as\n\\begin{equation}\n\\bar{\\psi}_{\\alpha} \\psi_{\\alpha} = \\lambda_{\\alpha} S + \\lambda_{\\alpha}' S',\n\\label{I5}\n\\end{equation}\nwhere $S$ is the self-consistent potential and $S'$ a second, different function. This yields two space-time independent self-consistency\nconditions,\n\\begin{equation}\n- Ng^2 \\sum_{\\alpha}^{\\rm occ} \\lambda_{\\alpha} = 1, \\quad \\sum_{\\alpha}^{\\rm occ} \\lambda_{\\alpha}' = 0.\n\\label{I6}\n\\end{equation}\nKnown type II solutions are the kink-antikink baryons \\cite{L3,L10}, the kink crystal at finite temperature \\cite{L12} and the time-dependent\nbreather \\cite{L3} to be discussed below in more detail. Incidentally, all HF solutions of the massive GN model (i.e., Lagrangian (\\ref{I1})\nsupplemented by a term $-m_0 \\sum_{k=1}^N \\bar{\\psi}_k \\psi_k$) are also of type II \\cite{L13,L14,L15}, and no\nsolution of type III or higher is known in the GN model which would require more than two functions and hence more than two space-time\nindependent\nself-consistency conditions. As will become clear later on, this classification of HF solutions is useful if one wants to relate the quantum \ntheory to classical GN models with a small number of flavors. \n \nIn this paper, we shall focus on the dynamics of kinks, supplementing the known single kink and kink crystal solutions by kink-antikink\nscattering and briefly commenting on the generalization to $N$ kinks and antikinks. It turns out that this restriction to kink dynamics \nis at the same time a restriction to \ntype I HF solutions. For this class of particularly simple (though exact) solutions of the large $N$, massless GN model, we will identify an\neffective bosonic theory whose soliton solutions are closely related to self-consistent HF potentials, using methods developed by Neveu and \nPapanicolaou in their proof of integrability of the classical one- and two-flavor GN models \\cite{L16}. The relevant equation is the sinh-Gordon\nequation, the hyperbolic version of the more familiar sine-Gordon equation. This in turn is the key for mapping type I solutions of the GN model\nonto classical string theory in 3-dimensional anti de Sitter space (AdS$_3$), following recent work of Jevicki and collaborators\n\\cite{L17,L18,L19}. In \nthis way we hope to show that in spite of the simplicity of the GN model, its mature age and the considerable amount of work devoted to it, \nthere is still room for new insights and surprises.\n\nThis paper is organized as follows. In Sec.~\\ref{sect2}, we solve the kink-antikink scattering problem with the help of the known breather.\nSec.~\\ref{sect3} aims at identifying the sinh-Gordon equation as underlying effective bosonic theory. In Sec.~\\ref{sect4}, we show how to map\nsolutions of the GN model onto strings moving in AdS$_3$ and illustrate this mapping with a few examples. In Sec.~\\ref{sect5} we summarize\nour findings and identify promising directions for future work.\n\n\\section{Kink-antikink scattering}\\label{sect2}\nOur starting point is the kink-antikink breather solution of the GN model discovered by Dashen, Hasslacher and Neveu (DHN) \n\\cite{L3}. Since the inverse scattering method is not available for time-dependent semi-classical solutions, these authors \nhad to guess the scalar potential by analogy with the well-known sine-Gordon breather. They then show that \na self-consistent solution can be found whose quantized fluctuations yield the spectrum of kink-antikink type baryons.\nAt the end of their paper, DHN mention that the imaginary choice $\\epsilon= i\/v$ of a certain parameter governing the frequency of the\nbreather should describe scattering of a kink-antikink pair with velocities $\\pm v$ in their center-of-mass (cm) frame. We take\nup this suggestion here and analyze the scattering of a kink and an antikink in greater depth, using the language of the TDHF approach.\nBefore doing this however, let us review the results for the DHN breather which are relevant for the analytic continuation in $\\epsilon$.\nThis is also necessary because the formulae given in \\cite{L3} do not seem to be fully consistent, either due to typos or to some unspecified\nconventions. \n\nThe breather is a solution of the Dirac equation\n\\begin{equation}\n\\left( i \\partial \\!\\!\\!\/ - S \\right) \\psi = 0\n\\label{1}\n\\end{equation}\nand will be given in the representation\n\\begin{equation}\n\\gamma^0 = - \\sigma_1, \\quad \\gamma^1 =  i\\sigma_3, \\quad \\gamma_5 = \\gamma^0 \\gamma^1 = - \\sigma_2\n\\label{2}\n\\end{equation}\nof the $ \\gamma$ matrices. Units in which the vacuum fermion mass is 1 will be used throughout this work. \nThe scalar potential can be written as\n\\begin{equation}\nS = 1 + \\xi f_2 + \\eta f_4\n\\label{3}\n\\end{equation}\nwith\n\\begin{eqnarray}\nf_2 & = & f_4 \\cos \\Omega t ,\n\\nonumber \\\\\nf_4 & = & (\\cosh Kx + a \\cos \\Omega t + b)^{-1}.\n\\label{4}\n\\end{eqnarray}\nThe parameters appearing here are defined and related as follows,\n\\begin{eqnarray}\n\\Omega & = & \\frac{2}{\\sqrt{1+\\epsilon^2}}, \\quad K \\ = \\ \\epsilon \\Omega,\n\\nonumber \\\\\n\\xi & = &  -2a, \\quad \\eta \\ = \\ - \\frac{1}{2} b K^2.\n\\label{5}\n\\end{eqnarray}\n$a$ is the solution of the equation\n\\begin{equation}\n0 = \\eta^2+K^2 \\left(1-b^2 \\right) + \\Omega^2 a^2,\n\\label{6}\n\\end{equation}\nwhere we choose the positive square root. $S$ is fully specified if the parameters $\\epsilon$ and $b$ are given.\nThe Dirac equation (\\ref{1}) with potential (\\ref{3}) has two types of solutions.\nFirst, there is a continuum of wave-like solutions of the form\n\\begin{equation}\n\\psi_k = \\left( \\begin{array}{c} u_k \\\\ v_k \\end{array} \\right) e^{i(kx-\\omega t)} \n\\label{7}\n\\end{equation}\nwith $\\omega = \\pm \\sqrt{k^2+1}$. Like in the single baryon problem, these scattering states have no \nreflected wave, showing that also the time-dependent potential is transparent.\nDHN give the explicit form of $\\psi_k$ in terms of $f_2,f_4$ from Eq.~(\\ref{4}) and two further functions\n\\begin{equation}\nf_1 = f_4 \\sinh Kx, \\quad f_3 = f_4 \\sin \\Omega t .\n\\label{8}\n\\end{equation}\nWe have redetermined the coefficients with the following results,\n\\begin{eqnarray}\nu_k & = & {\\cal N}_k \\left( 1 + \\frac{ iK}{2k} f_1 - \\frac{ ia}{k} f_2 + \\frac{a \\Omega}{2k} \\frac{1+ik}{\\omega} f_3\n+ \\frac{ i\\eta}{2k} f_4 \\right),\n\\nonumber \\\\\nv_k & = & {\\cal N}_k  \\left( - \\frac{1+ik}{\\omega} - \\frac{K}{2k}\\frac{i-k}{\\omega} f_1- \\frac{a}{k}\\frac{i-k}{\\omega}f_2 \\right.\n\\nonumber \\\\\n& &  \\left. + \\frac{a\\Omega}{2k} f_3 + \\frac{\\eta}{2k}\\frac{ i-k}{\\omega} f_4 \\right).\n\\label{9}\n\\end{eqnarray} \nThey deviate from Eq.~(4.6) of Ref.~\\cite{L3}, but one can easily check that Eqs.~(\\ref{7},\\ref{9}) do solve the Dirac equation. The second type of\nsolution are bound states. DHN find two distinct states which can be expressed in terms of the functions\n\\begin{eqnarray}\n\\varphi_1 & = & f_4 \\cos \\Omega t\/2 \\cosh Kx\/2 ,\n\\nonumber \\\\\n\\varphi_2 & = & f_4 \\sin \\Omega t\/2 \\sinh Kx\/2 ,\n\\nonumber \\\\\n\\varphi_3 & = & f_4 \\sin \\Omega t\/2 \\cosh Kx\/2 ,\n\\nonumber \\\\\n\\varphi_4 & = & f_4 \\cos \\Omega t\/2 \\sinh Kx\/2 .\n\\label{10}\n\\end{eqnarray}\nTwo orthogonal solutions of the Dirac equation are \n\\begin{eqnarray}\n\\psi_0^{(1)} & = & {\\cal N}_0 \\left( \\begin{array}{r} \\varphi_1 + c_4 \\varphi_4\n \\\\ c_2 \\varphi_2 + c_3 \\varphi_3\n \\end{array} \\right),\n\\nonumber \\\\\n\\psi_0^{(2)} & = & {\\cal N}_0 \\left( \\begin{array}{r}  -c_2 \\varphi_2 + c_3 \\varphi_3\n   \\\\  \\varphi_1 - c_4 \\varphi_4    \\end{array} \\right).\n\\label{11}\n\\end{eqnarray}\nHere, our coefficients\n\\begin{eqnarray}\nc_2 & = & - i\\epsilon \\frac{1-b-\\eta\/2}{a-\\eta\/2},\n\\nonumber \\\\\nc_3 & = & \\frac{ i\\Omega}{2} \\frac{1+b-a}{1+b+\\eta\/2},\n\\nonumber \\\\\nc_4 & = & - \\frac{K}{2} \\frac{1-a-b}{a-\\eta\/2},\n\\label{12}\n\\end{eqnarray}\ndiffer from those of DHN in the overall signs of $c_3$ and $c_4$. \nSelf-consistency of this solution can be established as follows: DHN find that $\\bar{\\psi}_k\\psi_k$ for each occupied continuum state yields \ntwo contributions, one proportional to $S$ and one proportional to $f_4$. The discrete states yield only a single contribution proportional\nto $f_4$. By relating the parameter $b$ to the occupation fraction of the discrete states, one gets a self-consistent result where the\n$f_4$ terms cancel. According to the classification of TDHF solutions given in the introduction, this shows that the breather is a type II\nsolution of the GN model. Only for the value $b=0$, the contributions proportional to $f_4$ would vanish and the breather would be \na type I solution. However $b=0$ is ruled out since it is incompatible with a real potential $S$.\n\nWe now turn to the kink-antikink scattering problem. Following the suggestion of DHN, we analytically continue the above breather\nsolution to the value $\\epsilon= i\/v$. This gives rise to the following changes in our formulae for the potential and the spinors,\n\\begin{eqnarray}\nK & = & \\frac{2}{\\sqrt{1-v^2}},\n\\nonumber \\\\\n\\Omega & = & -\\frac{2 i v}{\\sqrt{1-v^2}},\n\\nonumber \\\\\n\\cos \\Omega t & = & \\cosh \\frac{2vt}{\\sqrt{1-v^2}},\n\\nonumber \\\\\n\\sin \\Omega t & = & - i \\sinh \\frac{2vt}{\\sqrt{1-v^2}},\n\\nonumber \\\\\na & = & \\sqrt{\\frac{1}{v^2}+ \\frac{b^2}{1-v^2}}.\n\\label{13}\n\\end{eqnarray}\nThe continuum states now have to be normalized to free spinors at $t\\to - \\infty$ using\n\\begin{equation}\n{\\cal N}_k=\\sqrt{\\frac{2 k^2}{4 k^2+K^2}}.\n\\label{15}\n\\end{equation}\nThe discrete states are square integrable and normalized to 1 by \n\\begin{equation}\n{\\cal N}_0 =  \\sqrt{\\frac{K(1+a+b)}{2}}.\n\\label{16}\n\\end{equation}\nLet us now consider the issue of self-consistency for the scattering problem. The contribution from the negative energy\n($\\omega<0$) continuum to the scalar potential is found to be \n\\begin{eqnarray}\n- g^2 \\sum_k \\bar{\\psi}_k \\psi_k & = &  S Ng^2  \\int_{-\\Lambda\/2}^{\\Lambda\/2} \\frac{ dk}{2\\pi} \\frac{1}{\\sqrt{k^2+1}} \n\\nonumber \\\\\n& &  + f_4 Ng^2  \\int_{-\\infty}^{\\infty}\n\\frac{ dk}{2\\pi} h(k)\n\\label{17}\n\\end{eqnarray}\nwith\n\\begin{equation}\nh(k) = \\frac{2bv^2}{\\sqrt{1+k^2}(1-v^2)(1+k^2-v^2k^2)}.\n\\label{18}\n\\end{equation}\nThe first term by itself gives the self-consistent result owing to the vacuum gap equation,\n\\begin{equation}\n1 = Ng^2 \\int_{-\\Lambda\/2}^{\\Lambda\/2} \\frac{ dk}{2\\pi} \\frac{1}{\\sqrt{k^2+1}} .\n\\label{19}\n\\end{equation}\nThis observation is the same as for the breather.  Differences arise in the treatment of the discrete states. Since there\nare two orthogonal bound states, there is an ambiguity how to fill these states in the TDHF approach. DHN argue that in the case of \nthe breather one should take the linear combinations $\\psi_0^{(1)}\\pm \\psi_0^{(2)}$ and fill them with $N$ and $n_0$ fermions, respectively.\nWe have to reconsider this prescription in the light of the scattering problem. Appropriate initial conditions for the TDHF equation \nare an incoming kink and antikink with prescribed baryon numbers. In order to prepare such an initial state, we need to know\nwhich linear combinations of the two bound states goes over into the kink or antikink (zero-energy) valence state for $t\\to - \\infty$.\nLikewise, in order to be able to interpret the final state of the scattering process, we need the corresponding analysis at\n$t\\to \\infty$.\nConsider an arbitrary (normalized) linear combination of $\\psi_0^{(1)}, \\psi_0^{(2)}$,\n\\begin{equation}\n\\psi_0 = \\frac{\\lambda \\psi_0^{(1)}+ \\mu \\psi_0^{(2)}}{\\sqrt{|\\lambda|^2+|\\mu|^2}}.\n\\label{20}\n\\end{equation}\nIn order to exhibit the asymptotic behavior of this wave function for $x \\to \\pm \\infty, t \\to \\pm \\infty$, we introduce\nlabels for the incoming\/outgoing wave from\/to the left\/right as follows,\n\\begin{eqnarray}\n{\\rm il} & \\simeq & t\\to - \\infty, x \\to - \\infty\n\\nonumber \\\\ \n{\\rm ir} & \\simeq & t\\to - \\infty, x \\to + \\infty\n\\nonumber \\\\\n{\\rm ol} & \\simeq & t\\to + \\infty, x \\to - \\infty\n\\nonumber \\\\\n{\\rm or} & \\simeq & t\\to + \\infty, x \\to + \\infty\n\\label{21}\n\\end{eqnarray}\nUsing this shorthand notation, we find the asymptotic expressions (omitting a common normalization factor ${\\cal N}_0\/4\\sqrt{a}\\,$)\n\\begin{eqnarray}\n\\psi_{0,{\\rm il}} & = & \\frac{1}{\\cosh z_+} \\left( \\begin{array}{r} \\lambda(1-c_4)+ i\\mu(c_2+c_3) \\\\ -  i\\lambda(c_2-c_3)+\\mu (1+c_4) \n\\end{array} \\right),\n\\nonumber \\\\\n\\psi_{0,{\\rm ir}} & = & \\frac{1}{\\cosh y_-} \\left( \\begin{array}{r}   \\lambda(1+c_4)-  i\\mu(c_2-c_3)      \\\\     i\\lambda (c_2+c_3)+\\mu(1-c_4) \n\\end{array} \\right),\n\\nonumber \\\\\n\\psi_{0,{\\rm ol}} & = & \\frac{1}{\\cosh y_+} \\left( \\begin{array}{r} \\lambda(1-c_4)-  i\\mu(c_2+c_3)         \\\\    i\\lambda (c_2-c_3)+\\mu(1+c_4)    \n   \\end{array} \\right),\n\\nonumber \\\\\n\\psi_{0,{\\rm or}} & = & \\frac{1}{\\cosh z_-} \\left( \\begin{array}{r} \\lambda(1+c_4)+  i\\mu(c_2-c_3)        \\\\   -  i\\lambda (c_2+c_3) + \\mu(1-c_4) \n      \\end{array} \\right),\n\\label{22}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\ny_{\\pm} &=& \\frac{1}{2} \\left(K(x+vt) \\pm \\ln a \\right) ,\n\\nonumber \\\\\nz_{\\pm} & = & \\frac{1}{2} \\left(K(x-vt) \\pm \\ln a \\right) .\n\\label{23}\n\\end{eqnarray}\nWe introduce two orthonormal discrete states with parameters ($\\lambda_1,\\mu_1$) and ($\\lambda_2,\\mu_2$) and\noccupation $N_1,N_2$. In order to match the initial conditions of the kink-antikink scattering problem, we require that state 1 \nhas only incident fermions from the left and state 2 only incident fermions from the right. Eqs.~(\\ref{22}) then imply the two pairs\nof homogeneous, linear equations \n\\begin{eqnarray}\n0 & = & \\lambda_1 (1+c_4) -  i\\mu (c_2-c_3)\n\\nonumber \\\\\n0 & = & i\\lambda_1(c_2+c_3)+\\mu_1(1-c_4)\n\\nonumber \\\\\n0 & = & \\lambda_2(1-c_4)+  i\\mu_2 (c_2+c_3)\n\\nonumber \\\\\n0 & = & -  i\\lambda_2(c_2-c_3)+\\mu_2(1+c_4)\n\\label{24}\n\\end{eqnarray}\nThe conditions for the existence of a non-trivial solution,\n\\begin{eqnarray}\n0 & = & \\det \\left( \\begin{array}{cc} 1+c_4 & -  i(c_2-c_3) \\\\ i(c_2+c_3) & 1-c_4 \\end{array} \\right)  ,\n\\nonumber \\\\\n0 & = & \\det \\left( \\begin{array}{cc} 1-c_4 & i(c_2+c_3) \\\\ -  i(c_2-c_3) & 1+c_4 \\end{array} \\right)  ,\n\\label{25}\n\\end{eqnarray}\nare indeed satisfied for arbitrary $v,b$. We can then determine the ratios $\\mu_i\/\\lambda_i$,\n\\begin{eqnarray}\n\\frac{\\mu_1}{\\lambda_1} & = & -  i \\frac{1+c_4}{c_2-c_3},\n\\nonumber \\\\\n\\frac{\\mu_2}{\\lambda_2}  & = &  i\\frac{1-c_4}{c_2+c_3}.\n\\label{26}\n\\end{eqnarray}\nThe right-hand side is purely imaginary. This implies a vanishing contribution to the chiral condensate from the discrete states\nwhich is proportional to $\\lambda^* \\mu + \\mu^* \\lambda$, as can easily be checked. Self-consistency in the scattering problem \nis therefore only possible for $b=0$. In this particular case, the coefficients $c_i$ are \n\\begin{equation}\nc_2=1, \\quad c_3 = - c_4 = - \\frac{1-v}{\\sqrt{1-v^2}},\n\\label{26a}\n\\end{equation}\nso that Eqs.~(\\ref{26}) simplify to\n\\begin{equation}\n\\frac{\\mu_1}{\\lambda_1} = -  i = - \\frac{\\mu_2}{\\lambda_2}.\n\\label{27}\n\\end{equation}\nA consistent choice of parameters is therefore $b=0$ and\n\\begin{equation}\n\\lambda_1=\\lambda_2 = 1, \\quad \\mu_1 = - \\mu_2 = -  i.\n\\label{28}\n\\end{equation}\nFor these parameters, one finds that state 1 has only an incoming wave from the left and an outgoing wave to the right, state 2 only \nan incoming wave from the right and an outgoing wave to the left. The valence fermions are exchanged between \nthe two scatterers, presumably a consequence of the fact that the potential is transparent. These valence fermions do not play any role\nin the self-consistency issue. We can prescribe the number of valence quarks independently for kink\nand antikink as $N_1,N_2$ where $0 \\leq N_i \\leq N$. In the final state, $N_1,N_2$ are then simply exchanged.\nWe recall that the kink (or antikink) with $N_i$ valence quarks carries fermion number $N_f=N_i-N\/2$ as a result of induced fermion number in\na topologically nontrivial background potential \\cite{L19a,L19b}.\nExpressed in terms of \nreduced fermion number $N_f\/N=N_i\/N-1\/2=\\nu_i-1\/2$, the scattering process thus reads\n\\begin{equation}\nK(\\nu_2-1\/2) + \\bar{K}(\\nu_1-1\/2) \\rightarrow K(\\nu_1 -1\/2) + \\bar{K}(\\nu_2 - 1\/2).\n\\label{28a}\n\\end{equation}\nThe time delay\n\\begin{equation}\n\\Delta t = \\frac{\\ln v}{v}\\sqrt{1-v^2}\n\\label{29}\n\\end{equation}\nis independent of the fermion numbers and negative, indicating a repulsive interaction between kink and antikink.\nNote that the value $b=0$ implies that the kink-antikink scattering solution is of type I and hence significantly simpler than the breather. \nThis is also reflected in the form of the scalar potential, \n\\begin{equation}\nS= \\frac{v \\cosh Kx - \\cosh Kvt}{v \\cosh Kx+ \\cosh Kvt}, \\quad K= \\frac{2}{\\sqrt{1-v^2}}.\n\\label{30}\n\\end{equation}\nIncidentally, Eq.~(\\ref{30}) agrees with a result quoted in Ref.~\\cite{L16} without derivation.  \nThe fermion density for the discrete states is given by\n\\begin{equation}\n\\rho_{1,2} = \\frac{vK\\left[ v+\\cosh K(x\\pm vt) \\right]}{2(v\\cosh Kx+ \\cosh Kvt)^2},\n\\label{32}\n\\end{equation}\nwhere the $+$ sign belongs to state 2 moving to the left, the $-$ sign to state 1 moving to the right, cf. Eq.~(\\ref{28}).\nExpressions (\\ref{32}) have been normalized to 1 and have to be multiplied by the occupation numbers to get the true fermion density.\nThe density $\\psi_k^{\\dagger}\\psi_k$ of the negative energy continuum states is\n\\begin{eqnarray}\n\\rho_k &=& 1 + \\frac{2v\\left( v+\\cosh Kx \\cosh Kvt \\right)}{(k^2v^2-k^2-1)\\left( v \\cosh Kx+\\cosh Kvt \\right)^2}\n\\label{34} \\\\\n&- &  \\frac{2kv^2 \\sinh Kx \\sinh Kvt}{\\sqrt{1+k^2}(k^2v^2-k^2-1)\\left(v \\cosh Kx+\\cosh Kvt\\right)^2}.\n\\nonumber\n\\end{eqnarray}\nIntegrating over $k$ and subtracting the vacuum contribution yields the simple, finite result\n\\begin{equation}\n\\int_{-\\infty}^{\\infty}   \\frac{ dk}{2\\pi} (\\rho_k-1) = - \\frac{1}{2} (\\rho_1+\\rho_2).\n\\label{35} \n\\nonumber\n\\end{equation}\nThe total fermion density per flavor is thus given by\n\\begin{equation}\n\\rho_f  =  \\left(\\nu_1- \\frac{1}{2} \\right) \\rho_1 +\\left(  \\nu_2 - \\frac{1}{2} \\right) \\rho_2 .\n\\label{36} \n\\end{equation}\nIntegration over $x$ finally yields the (conserved) total fermion number,\n\\begin{equation}\n\\frac{N_f}{N} = \\int_{-\\infty}^{\\infty} dx \\rho_f(x,t) = \\nu_1 + \\nu_2 -1\\ .\n\\label{37}\n\\end{equation}\nThe kink-antikink scattering process is illustrated in Figs.~\\ref{fig1}--\\ref{fig3}. Fig.~\\ref{fig1} shows, by means of the scalar potential $S$, \nthat kink and antikink approach each other, are repelled and bounce back. This picture is independent of the baryon numbers involved.\nFigs.~\\ref{fig2} and \\ref{fig3} exhibit the baryon density for baryon-antibaryon and baryon-baryon scattering. In the first case one clearly \nsees the exchange of fermions during the collision.\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=Fig1.eps,height=8cm,width=6.4cm,angle=270}\n\\caption{Scalar potential for kink-antikink scattering at $v=0.5$, showing a repulsive interaction.} \n\\label{fig1}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=Fig2.eps,height=8cm,width=6.4cm,angle=270}\n\\caption{Fermion density for kink-antikink scattering. Parameters: $v=0.5, \\nu_1=1, \\nu_2=0$ (baryon-antibaryon collision).} \n\\label{fig2}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=Fig3.eps,height=8cm,width=6.4cm,angle=270}\n\\caption{Fermion density for kink-antikink scattering. Parameters: $v=0.5, \\nu_1=1, \\nu_2=1$ (baryon-baryon collision).} \n\\label{fig3}\n\\end{center}\n\\end{figure}\n\\section{Effective bosonic field theory}\\label{sect3}\nA big hurdle in applying the TDHF approach is to find the self-consistent potential. Obviously it would be extremely nice if one could\nwrite down a closed equation satisfied by the potential. This has been achieved approximately in the case of the massive chiral \nGN model near the chiral limit, using the derivative expansion \\cite{L13}. Such classical field equations may be regarded as equations of motion\nof an effective scalar theory where the fermions have been ``integrated out\", in the language of the path integral. The derivative expansion\nis of little use in the\ndiscrete chiral GN model where the baryons remain localized in the chiral limit. Here we address the question whether one can \nnevertheless identify an effective bosonic theory for the non-chiral GN model.\n\nSince the tanh kink appears in $\\phi^4$ theory, one may be tempted to postulate the equation\n\\begin{equation}\n\\partial_{\\mu}\\partial^{\\mu} S + 2 S^3 - 2 S = 0.\n\\label{39}\n\\end{equation}\nIt is indeed solved by the boosted kink $\\tanh(\\gamma(x-vt))$. However, this cannot be correct because Eq.~(\\ref{39}) has no solitons in the\nstrict sense of the word: If one scatters a kink and an antikink, a numerical study shows that they interact inelastically and do not\nkeep their $\\tanh$ shape \\cite{L20}. A simple bosonic theory which does not have this problem can be found for type I solutions only, to which\nwe\nrestrict ourselves in this section. These solutions are specific for kink dynamics (kink, kink crystal, kink-antikink scattering), and therefore\nonly exist in the massless GN model. For type I solutions we can write\n\\begin{equation}\nS = \\ell_{\\alpha} \\bar{\\psi}_{\\alpha} \\psi_{\\alpha} \n\\label{40}\n\\end{equation} \nfor every occupied state $\\alpha$ with nonvanishing $\\bar{\\psi}_{\\alpha}\\psi_{\\alpha}$. If we pick any such state, the TDHF problem \nevidently reduces to the $N=1$ classical GN model, i.e., a non-linear Dirac equation \n\\begin{equation}\n\\left( i \\partial \\!\\!\\!\/ - \\ell_{\\alpha} \\bar{\\psi}_{\\alpha} \\psi_{\\alpha} \\right) \\psi_{\\alpha} = 0\n\\label{41}\n\\end{equation}\nwith $c$-number spinors. We suppress the label $\\alpha$ from now on. Neveu and Papanicolaou \\cite{L16} have proven\nthe integrability of the classical $N=1,2$ GN models long ago. The part of their work dealing with $N=1$ in fact contains the answer\nto the question about the effective bosonic theory. To demonstrate this fact we closely follow their work in the present section.\n\nTo this end it is useful to switch to a different representation of the $\\gamma$-matrices in which the upper and lower components of the\nDirac spinor $\\psi_1, \\psi_2$ have definite chirality,\n\\begin{equation}\n\\gamma^0 = \\sigma_1, \\quad \\gamma^1 = i \\sigma_2, \\quad \\gamma_5 = - \\sigma_3 \\ .\n\\label{42}\n\\end{equation}\nUsing light-cone coordinates\n\\begin{equation}\nz=x-t, \\quad \\bar{z} = x+t\n\\label{43}\n\\end{equation}\nand the comma-notation for partial derivatives, the non-linear Dirac equation assumes the simple form\n\\begin{eqnarray}\n-2i\\psi_{1,z} & = & S \\psi_2,\n\\nonumber \\\\\n2i \\psi_{2,\\bar{z}} & = & S \\psi_1,\n\\label{44}\n\\end{eqnarray}\nwhere\n\\begin{equation}\nS = \\ell \\left( \\psi_1^*\\psi_2+ \\psi_2^* \\psi_1 \\right).\n\\label{45}\n\\end{equation}\nThe Dirac equation expresses $\\psi_{1,z}$ and $\\psi_{2,\\bar{z}}$ in terms of $\\psi_1,\\psi_2$. What about the other derivatives,\n$\\psi_{1,\\bar{z}}$ and $\\psi_{2,z}$? Following Neveu and Papanicolaou, one first derives the identities\n\\begin{eqnarray}\nS \\psi_{1,\\bar{z}}- S_{,\\bar{z}}\\psi_1 & = & -i h_1 \\ell \\psi_2,\n\\nonumber \\\\\nS \\psi_{2,z}-S_{,z}\\psi_2 & = & -i h_2 \\ell \\psi_1,\n\\label{46}\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\nh_1 & = & i \\left( \\psi_1^* \\psi_{1,\\bar{z}} - \\psi_1 \\psi_{1,\\bar{z}}^* \\right),\n\\nonumber \\\\\nh_2 & = &  i \\left( \\psi_2^* \\psi_{2,z} - \\psi_2 \\psi_{2,z}^* \\right).\n\\label{47}\n\\end{eqnarray}\nSince the right-hand side depends again on the unknown derivatives $\\psi_{1,\\bar{z}},\\psi_{2,z}$ and their complex conjugates, it looks\nas if the goal of expressing these derivatives through $\\psi_1,\\psi_2$ had not been reached. However, simple algebra shows that\n\\begin{equation}\nh_{1,z} = 0 = h_{2,\\bar{z}}\n\\label{48}\n\\end{equation}\nso that $h_{1,2}$ can only depend on either $z$ or $\\bar{z}$. If these functions would not be constant, they would describe fermion bilinears\npropagating at the speed of light. In the absence of massless particles in the model we conclude that $h_1,h_2$ are constant\nfor physically sensible solutions. This enables us to express all derivatives of $\\psi$ in terms of $\\psi$ itself as follows,\n\\begin{equation}\n\\psi_{,\\bar{z}} = C_1 \\psi, \\qquad \\psi_{,z} = C_2 \\psi,\n\\label{49}\n\\end{equation}\nwith the matrices\n\\begin{eqnarray}\nC_1 & = &  \\left( \\begin{array}{cc} S_{,\\bar{z}}S^{-1} & -i h_1 \\ell S^{-1} \\\\ -i S\/2 & 0 \\end{array} \\right), \n\\nonumber \\\\\nC_2 & = &  \\left( \\begin{array}{cc} 0  & i S\/2 \\\\ -ih_2 \\ell S^{-1} & S_{,z} S^{-1} \\end{array} \\right).\n\\label{50}\n\\end{eqnarray}\nThe integrability condition for the system (\\ref{49}) reads\n\\begin{equation}\nC_{1,z} - C_{2,\\bar{z}} + \\left[ C_1,C_2 \\right] = 0\n\\label{51}\n\\end{equation}\nor equivalently\n\\begin{equation}\nS S_{,z \\bar{z}} - S_{,z}S_{,\\bar{z}} - \\frac{1}{4} S^4 = h_1 h_2 \\ell^2\n\\label{52}\n\\end{equation}\nAs the right-hand side is constant, we have succeeded in deriving a non-linear partial differential equation for\nthe self-consistent TDHF potential, at least for type I solutions. If we disregard the kink crystal for the moment and focus\non localized solutions where $S$ reaches its vacuum value asymptotically, we can even determine the constant.\nIn units where $S=m=1$ in the vacuum, we infer from Eq.~(\\ref{52}) that\n\\begin{equation}\nh_1 h_2 \\ell^2 = - \\frac{1}{4}\\ .\n\\label{53}\n\\end{equation}\nIn normal coordinates, the final equation for $S$ reads\n\\begin{equation}\nS \\partial_{\\mu} \\partial^{\\mu}S - \\partial_{\\mu}S \\partial^{\\mu}S + S^4-1 = 0.\n\\label{53a}\n\\end{equation}\nIt is satisfied by the kink and the kink-antikink solutions in arbitrary Lorentz frames, but also by the kink crystal for which we have not derived\nthe value of the constant, Eq.~(\\ref{53}). One can write down a simple (although singular) Lagrangian which yields Eq.~(\\ref{53a}) as\nEuler-Lagrange equation, namely\n\\begin{equation}\n{\\cal L} = \\frac{1}{S^2} \\left( \\partial_{\\mu}S \\partial^{\\mu}S - S^4 -1 \\right).\n\\label{53b}\n\\end{equation}\nHowever, this should not be interpreted as effective bosonic field theory for the GN model. If one derives the Hamiltonian density \n${\\cal H}$ from ${\\cal L}$ in the usual way, one does not get the correct energy density. The reason is presumably the fact that we already\nmade use of properties of the solution when deriving Eq.~(\\ref{53a}), notably the fact that we are dealing with a type I solution of the TDHF \nproblem. In this way we are not really able to integrate out the fermions in full generality and construct the effective bosonic action\nfor arbitrary scalar fields.\n\nEq.~(\\ref{53a}) does not yet resemble any of the well-known equations with solitonic solutions. \nThe closest we could come to a more familiar looking form was by means of the change of variables \\cite{L16}\n\\begin{equation}\nS^2=e^{\\theta}, \\qquad \\theta= \\ln S^2,\n\\label{54}\n\\end{equation}\nwhich reduces Eq.~(\\ref{53a}) to the sinh-Gordon equation,\n\\begin{equation}\n\\partial_{\\mu} \\partial^\\mu \\theta + 4  \\sinh \\theta = 0.\n\\label{55}\n\\end{equation}\nHowever we loose the information about the sign of $S$ and hence also the Z$_2$ chiral symmetry in this nonlinear transformation -- \nthe two vacua $S=\\pm 1$ are mapped onto the same value $\\theta=0$. Since the zero's of $S$ give rise to singularities of $\\theta$, it is easy \nto reconstruct a solution $S$ of Eq.~(\\ref{53a}) from a singular solitonic solution $\\theta$ of the sinh-Gordon equation, so that the mapping \nis nevertheless quite useful.  With this caveat, the matrices $C_1,C_2$ and the linear equations (\\ref{49}) can then be identified with the \nLax pair of the sinh-Gordon equation. \n\nNotice that the coefficient 4 in Eq.~(\\ref{55}) has a simple physical interpretation: The linearized sinh-Gordon equation, \n\\begin{equation}\n\\left( \\partial_{\\mu} \\partial^{\\mu}  + 4 \\right) \\theta = 0,\n\\label{56}\n\\end{equation}\nyields the Klein-Gordon equation for a scalar meson with mass 2 in units of the fermion mass. This can be identified with the $\\sigma$ meson \nof the massless GN model. The relation between kink, $\\sigma$ meson and sinh-Gordon equation in the massless\nGN model is analogous to the relation between the light baryon, the $\\pi$ meson and the sine-Gordon equation in\nthe massive 2d Nambu--Jona-Lasinio model (NJL$_2$) close to chiral limit (derivative expansion \\cite{L13}). In the latter case, this was\ninterpreted as the simplest\nrealization of the Skyrme model in the sense that baryons are topologically non-trivial excitations of the pion field \\cite{L21}. In the GN\nmodel, we have now identified a similar picture for the case of a discrete chiral symmetry, the baryon emerging as a large amplitude \nexcitation of the $\\sigma$ field.\nThe similarity \nbetween the baryons in both cases is particularly striking if one writes the sine-Gordon and sinh-Gordon solitons in the following form:\n\\begin{enumerate}\n\\item\nMassless GN model (exact, $m_{\\sigma}=2$),\n\\begin{eqnarray}\nS^2 & = &  e^{\\theta}, \\qquad \\theta \\ =\\  - 4 {\\rm \\ artanh\\ }e^{-m_{\\sigma}x},\n\\nonumber \\\\\n0 & = & \\partial_{\\mu}\\partial^{\\mu} \\theta + m_{\\sigma}^2 \\sinh \\theta  .\n\\label{59}\n\\end{eqnarray}\n\\item\nMassive NJL$_2$ model (leading order derivative expansion, $m_{\\pi}=2\\sqrt{\\gamma}$ with $\\gamma$ the confinement parameter \\cite{L13}),\n\\begin{eqnarray}\nS-iP & = & e^{i\\phi}, \\qquad \\phi \\ =\\  4 \\arctan e^{m_{\\pi}x},\n\\nonumber \\\\\n0 & = &  \\partial_{\\mu}\\partial^{\\mu} \\phi + m_{\\pi}^2 \\sin \\phi.  \n\\label{60}\n\\end{eqnarray}\n\\end{enumerate}\nWe can also write down a common formula for topological baryon number,\n\\begin{equation}\n\\ln \\Phi(\\infty) - \\ln \\Phi(-\\infty) = 2\\pi i N_B,\n\\label{61}\n\\end{equation}\nwith $\\Phi=S$ for the GN model and $\\Phi=S-iP$ for the NJL$_2$ model. However, in the GN case, this only determines the non-integer part\nof the induced baryon number \\cite{L22} (kink and antikink give $\\mp 1\/2$ although they have the same value of induced baryon number of \n$-1\/2$) so that the analogy should perhaps not be overrated. \n \nThe kink belongs to the class of ``traveling wave solutions\" of the sinh-Gordon equation. Kink-antikink scattering\nis an example of a  ``functional separable solution\" \\cite{L23}\n\\begin{equation}\n\\theta(x,t)  =   4 \\,{\\rm artanh} \\left[ f(t)g(x) \\right]\n\\label{57}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n(f_{,t})^2  & = &   A f^4 + B f^2 + C,\n\\nonumber \\\\\n- (g_{,x})^2 & = &  C g^4 + (B+4)g^2 + A.\n\\label{58}\n\\end{eqnarray}\nThe general $N$ soliton solution is also known for the sinh-Gordon equation  \\cite{L19,L24} and is a likely candidate for the \nTDHF solution of the GN model with $N$ kinks and antikinks. Since the Lax pair including the spinor wave functions are\nknown, all what one would have to do is find the bound state solutions and verify self-consistency. From the point of view of\nparticle physics, this is perhaps somewhat academic because it describes a scattering problem with $N$ incident baryons. Nevertheless, it \nwould be challenging to solve the relativistic $N$-baryon problem exactly and analytically in terms of the elementary fermion constituents,\nincluding the Dirac sea effects.\n\\section{Relation to strings in AdS$_3$}\\label{sect4}\nJevicki et al. have explored the close relationship between sinh-Gordon theory on the one hand and classical strings in AdS$_3$ on the other \nhand \\cite{L17,L18,L19}. Since the GN model can also be related to the sinh-Gordon model, this should enable us to map type I solutions of the \nGN model onto solutions of\nclassical string theory. From the physics point of view, a stringy interpretation is not obvious for a non-gauge theory like the GN model. \nIt may be of interest to see explicitly how such a mapping works.\n\nLet us briefly review the required string theory background  following Ref.~\\cite{L17}. The AdS$_3$ target space is parametrized by the\nembedding coordinates $Y_a (a=-1,0,1,2)$ in R$^{2,2}$,\n\\begin{equation}\n\\vec{Y} \\cdot \\vec{Y} := - Y_{-1}^2-Y_0^2+Y_1^2+Y_2^2 = -1.\n\\label{62}\n\\end{equation}\nThe string equation of motion in conformal gauge reads ($\\partial=\\partial_z, \\bar{\\partial}=\\partial_{\\bar{z}}$)\n\\begin{equation}\n\\partial {\\bar{\\partial}} \\vec{Y} - ( \\partial \\vec{Y} \\cdot \\bar{\\partial} \\vec{Y}) \\vec{Y}=0\n\\label{63}\n\\end{equation}\nand has to be supplemented by the Virasoro constraints\n\\begin{equation}\n( \\partial \\vec{Y} )^2 = 0 =( \\bar{\\partial}\\vec{Y} )^2 .\n\\label{64}\n\\end{equation}\nThe crucial instrument in relating strings in AdS$_3$ to the sinh-Gordon equation is the Pohlmeyer reduction \\cite{L25}, using the\nfactorization of the AdS$_3$ isometry group SO(2,2) into SO(2,1)$\\times$SO(2,1).\nThe string coordinates are expressed through two auxiliary spinors $\\phi,\\chi$\nas follows,\n\\begin{eqnarray}\nZ_1 & = & Y_{-1} + i Y_0 \\ = \\ \\phi_1^*\\chi_1 - \\phi_2^*\\chi_2,\n\\nonumber \\\\\nZ_2 & = & Y_1+iY_2 \\ = \\ \\phi_2^*\\chi_1^* - \\phi_1^*\\chi_2^*.\n\\label{65}\n\\end{eqnarray}\nHere, $\\phi$ and $\\chi$ are normalized according to\n\\begin{equation}\n1 = \\phi_1^*\\phi_1-\\phi_2^*\\phi_2 = \\chi_1^*\\chi_1-\\chi_2^*\\chi_2\n\\label{66}\n\\end{equation}\nand satisfy the linear system of equations\n\\begin{eqnarray}\n\\phi_{,\\bar{z}} & = & A_1 \\phi, \\qquad \\phi_{,z} \\ = \\  A_2 \\phi,\n\\nonumber \\\\\n\\chi_{,\\bar{z}} & = & B_1 \\chi, \\qquad \\chi_{,z} \\ = \\ B_2 \\chi,\n\\label{67}\n\\end{eqnarray}\nwhere now $z,\\bar{z}$ are light-cone coordinates derived from worldsheet space-time coordinates. \nThe matrices depend on two arbitrary functions $u(\\bar{z}),v(z)$ which we choose as $u=2,v=-2$ for the sake of simplicity.\nFor these particular values the normalization of the light-cone coordinates agrees with Eq.~(\\ref{43}) if we identify $x,t$ with the worldsheet\ncoordinates. The matrices introduced in Eq.~(\\ref{67}) are then given by\n\\begin{eqnarray}\nA_1 & = & \\frac{1}{4} \\left( \\begin{array}{cc} -i\\lambda c_+ & i \\alpha_{,\\bar{z}} - \\lambda c_- \\\\ -i \\alpha_{, \\bar{z}} - \\lambda c_- & i \\lambda c_+ \n    \\end{array}\\right) ,\n\\nonumber \\\\ \nA_2 & = &  \\frac{1}{4}\\left( \\begin{array}{cc}  i c_+\/\\lambda & -i \\alpha_{,z}- c_-\/\\lambda \\\\ i \\alpha_{,z}-  c_-\/\\lambda  &  -i c_+ \/\\lambda   \\end{array}\n\\right),\n\\nonumber \\\\\nB_1 & = & \\frac{1}{4} \\left( \\begin{array}{cc} -i\\lambda c_- & i \\alpha_{,\\bar{z}} - \\lambda c_+ \\\\ -i \\alpha_{, \\bar{z}} - \\lambda c_+ & i \\lambda c_-  \n   \\end{array}\\right) ,\n\\nonumber \\\\\nB_2  & = & \\frac{1}{4}\\left( \\begin{array}{cc}  i c_-\/\\lambda & -i \\alpha_{,z}- c_+\/\\lambda \\\\ i \\alpha_{,z}-  c_+\/\\lambda  &  -i c_- \/\\lambda   \\end{array}\n\\right),\n\\label{68}\n\\end{eqnarray}\nwith\n\\begin{equation}\nc_{\\pm} = e^{-\\alpha\/2} \\pm e^{\\alpha\/2}.\n\\label{69}\n\\end{equation}\n$\\alpha(z,\\bar{z})$ is a function which is expressible in terms of $\\phi$, but this relation will not be needed here. \nThe integrability condition for both systems of linear equations (\\ref{67}) yields the sinh-Gordon equation,\n\\begin{equation}\n\\alpha_{,z \\bar{z}} = \\sinh \\alpha.\n\\label{70}\n\\end{equation}\nIt is straightforward to verify that a pair of spinors $\\phi,\\chi$ satisfying the normalization conditions (\\ref{66}) and the linear sytem\n (\\ref{67}) yields\na solution of the string equations (\\ref{62}--\\ref{64}), the link between the two problems being given by Eqs.~(\\ref{65}).\n\nThe way to relate the GN model to strings in AdS$_3$ is to find a gauge transformation between two different representations\nof the sinh-Gordon Lax pair, Eqs.~(\\ref{49},\\ref{50}) and Eqs.~(\\ref{67},\\ref{68}), where we have to expect complications due to the\nfact that the mapping from the GN model to the sinh Gordon model was not one-to-one. Using the parametrization\n\\begin{equation}\nS = \\pm e^{\\theta\/2}\n\\label{70a}\n\\end{equation}\ndepending on the sign of $S$, let us first consider the case $S>0$.\nThe gauge transformation from\n($C_1,C_2,\\psi$) to ($A_1,A_2,\\phi$) is given by \n\\begin{eqnarray}\nA_1 & = & \\Omega \\left( C_1 - \\bar{\\partial} \\right) \\Omega^{-1},\n\\nonumber \\\\\nA_2 & = & \\Omega \\left( C_2 - \\partial \\right) \\Omega^{-1},\n\\nonumber \\\\\n\\phi & = & \\Omega \\psi.\n\\label{71}\n\\end{eqnarray}\n$A_{1,2}$ depend on $\\alpha$ and a scale parameter $\\lambda$, $C_{1,2}$ depend on $S=e^{\\theta\/2}$ and the constants $h_1\\ell, h_2\\ell$ \nrelated\nthrough Eq.~(\\ref{53}). For future convenience, we introduce a parameter $\\zeta$ (to be identified with the spectral parameter of the GN model\nTDHF solutions) via \n\\begin{equation}\nh_1 \\ell = 2 \\zeta^2, \\quad h_2 \\ell = - \\frac{1}{8 \\zeta^2}.\n\\label{72}\n\\end{equation}\n$\\Omega$ can be found in 2 successive steps: First, make a local  Abelian gauge transformation to render $C_1,C_2$ traceless ($A_1,A_2$\nare in the SO(2,1) Lie algebra). This is achieved by the choice\n\\begin{equation}\n\\Omega_1 = S^{-1\/2}.\n\\label{73}\n\\end{equation}\nSecondly, perform a global non-Abelian gauge transformation\n\\begin{equation}\n\\Omega_2 = \\left( \\begin{array}{cc} \\lambda^{-1} & 1 \\\\ i\\lambda^{-1}  & -i \\end{array} \\right) .\n\\label{74}\n\\end{equation}\nThe product \n\\begin{equation}\n\\Omega= \\Omega_2 \\Omega_1 =  \\left( \\begin{array}{cc} \\lambda^{-1} & 1 \\\\ i\\lambda^{-1}  & -i \\end{array} \\right) S^{-1\/2} \\qquad (S>0)\n\\label{75}\n\\end{equation}\nthen transforms $C_1,C_2$ onto $A_1,A_2$ provided we relate the field variables ($\\theta, \\alpha$)\nand spectral parameters ($\\zeta, \\lambda$) as follows,\n\\begin{equation}\n\\alpha = - \\theta, \\qquad \\lambda^2 = 4 \\zeta^2.\n\\label{76}\n\\end{equation}\nAs will be seen later on in concrete examples, the correct sign for negative energy continuum states is $\\lambda=-2 \\zeta$. Hence the\nmapping from the GN solution $\\psi$ to the first string $\\sigma$ model spinor $\\phi$ is\n\\begin{equation}\n\\phi = \\Omega  \\psi(\\zeta=- \\lambda\/2).\n\\label{77}\n\\end{equation}\nThe normalization condition (\\ref{66}) then tells us what the correct normalization of the GN model spinor (which differs from the one used in\nthe standard TDHF approach) should be, namely\n\\begin{equation}\n1 = \\phi^{\\dagger} \\sigma_3 \\phi = \\frac{2}{\\lambda S} \\bar{\\psi}\\psi =  - \\frac{1}{\\zeta S} \\bar{\\psi}\\psi.\n\\label{78}\n\\end{equation}\nComparison with Eq.~(\\ref{45}) shows that $\\ell=- 1 \/ \\zeta$.\nWe now turn to the second spinor $\\chi$ and the gauge transformation from $C_{1,2}$ to $B_{1,2}$. The same gauge transformation\n$\\Omega$ can \nbe used once again, but Eq.~(\\ref{76}) is replaced by\n\\begin{equation}\n\\alpha = - \\theta, \\qquad \\lambda^2 = - 4 \\zeta^2.\n\\label{79a}\n\\end{equation}\nSince $\\lambda$ on the string side is real, we have to analytically continue the TDHF spinors to purely imaginary spectral parameter \n$\\zeta= \\pm i \\lambda\/2$. Which sign is the correct one? It turns out \nthat the normalization condition (\\ref{66}) for $\\chi$ can only be satisfied by taking a linear combination of\nspinors with both signs \\cite{L19}. The norm vanishes if we keep only one sign. We choose\n\\begin{equation}\n\\chi =\\Omega \\frac{1}{\\sqrt{2}} \\left[ \\psi(\\zeta=- i\\lambda\/2) +\ni \\psi(\\zeta=i \\lambda\/2) \\right]\n\\label{80}\n\\end{equation}\nwhere $\\psi(\\zeta)$ is normalized according to (\\ref{78}) before the analytic continuation. The detailed justification is given in the appendix.\nThese algebraic manipulations are rooted in the symmetries of the GN model on the one hand and AdS$_3$ space on the other hand. \nNeveu and Papanicolaou have identified a dynamical SO(2,1) symmetry of the classical $N=1$ GN model which is also relevant for\ntype I solutions of the large $N$ quantum theory. To match the SO(2,1)$\\times$SO(2,1) symmetry of AdS$_3$, one needs two independent \nspinor solutions -- this is the non-trivial part of the mapping. One of them is given directly by the (appropriately normalized)\nTDHF solution, the other involves an analytic continuation of the first one to imaginary spectral parameters. This is apparently what it\ntakes to embed the fermionic quantum field theory in a higher dimensional, classical string theory.\nWe can now write down the string coordinates in compact form. Introduce the basic GN TDHF spinor and the \nanalytically continued one as [see Eqs.~(\\ref{77},\\ref{80})]\n\\begin{eqnarray}\n\\psi_a & = & \\Omega^{-1} \\phi \\ = \\ \\psi (\\zeta=-\\lambda\/2),\n\\label{81} \\\\\n\\psi_b & = &  \\Omega^{-1} \\chi = \\frac{1}{\\sqrt{2}} \\left[ \\psi(\\zeta=-i\\lambda\/2) + i \\psi(\\zeta=i \\lambda\/2) \\right].\n\\nonumber\n\\end{eqnarray}\nnormalized according to (cf. Eq.~(\\ref{78}) and the appendix)\n\\begin{equation}\n\\bar{\\psi}_a \\psi_a = \\bar{\\psi}_b \\psi_b =  \\frac{\\lambda S}{2}.\n\\label{82}\n\\end{equation}\nThen, using the above gauge transformation,  the string coordinates (\\ref{65}) can be expressed in the concise form\n\\begin{equation}\nZ_1 =  \\frac{\\bar{\\psi}_a\\psi_b}{\\bar{\\psi}_a \\psi_a}, \\quad Z_2 = -  \\frac{\\bar{\\psi}_a i \\gamma_5 \\psi_b^*}{\\bar{\\psi}_a \\psi_a}.\n\\label{83}\n\\end{equation}\nHere it looks as if the normalization of the TDHF spinors would simply drop out, provided we use $\\bar{\\psi}_a\\psi_a = \\bar{\\psi}_b\\psi_b$.\nHowever Eq.~(\\ref{82}) for $\\psi_a$ is important \nbecause it teaches us how to normalize the GN spinor before doing the analytic continuation. Otherwise, the whole procedure would be ill\ndefined, since the normalization factor will in general depend on $\\zeta$.\n\nAll of these manipulations were done under the tacit assumption $S>0$. If $S<0$, we can repeat the same procedure with the following \nchanges. \nThe gauge transformation defined in Eq.~(\\ref{75}) has to be replaced by\n\\begin{equation}\n\\Omega=  \\left( \\begin{array}{cc} \\lambda^{-1} & -1 \\\\ i\\lambda^{-1}  & i \\end{array} \\right) (-S)^{-1\/2} \\qquad (S<0).\n\\label{83a}\n\\end{equation} \nOne can easily check that all other equations, in particular the final result (\\ref{83}), then remain valid.\n\nUsing Eqs.~(\\ref{82},\\ref{83}) it is straightforward to verify that the string equations (\\ref{62}--\\ref{64}) are satisfied. One needs the\nfact that $\\psi_a,\\psi_b$ satisfy the Lax system (\\ref{49},\\ref{50}) with the parameters $h_1\\ell=\\lambda^2\/2$ for $\\psi_a$ and\n$h_1\\ell=- \\lambda^2\/2$ for $\\psi_b$; $h_2 \\ell$ then follows from (\\ref{53}). The AdS$_3$ condition (\\ref{62}) can be shown via the Fierz \nidentity\n\\begin{equation}\n - |\\bar{\\psi}_a \\psi_b |^2 + |\\bar{\\psi}_a  \\gamma_5 \\psi_b^*|^2 = - \\bar{\\psi}_a\\psi_a \\bar{\\psi}_b\\psi_b.\n\\label{85}\n\\end{equation}  \nThe Virasoro constraints (\\ref{64}) follow trivially from the useful identities\n\\begin{eqnarray}\nZ_{1,z} & = & - \\frac{2i}{\\lambda^3S^2} \\psi_{a1}^*\\psi_{b1}, \\quad Z_{2,z} \\ = \\ \\frac{2}{\\lambda^3 S^2} \\psi_{a1}^*\\psi_{b1}^*\n\\nonumber \\\\\nZ_{1,\\bar{z}} & = & \\frac{2i \\lambda}{S^2} \\psi_{a2}^* \\psi_{b2}, \\quad Z_{2,\\bar{z}} \\ = \\ \\frac{2\\lambda}{S^2} \\psi_{a2}^*\\psi_{b2}^*\n\\label{86}\n\\end{eqnarray}\nFinally, the equation of motion (\\ref{63}) can be shown with the help of\n\\begin{equation}\n\\partial \\vec{Y} \\cdot \\bar{\\partial}\\vec{Y} = \\frac{1}{2S^2}.\n\\label{87}\n\\end{equation}\nLet us illustrate this mapping between type I solutions of the GN model and classical strings in AdS$_3$ by means of examples\ninvolving zero, one and two solitons, respectively.\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=Fig4.eps,height=4cm,width=4cm,angle=270}\n\\caption{String from zero soliton TDHF solution (vacuum) in AdS$_3$ for fixed global time. The coordinates on the Poincare disk are given in\nEq.~(\\ref{91}). The string is spinning rigidly counterclockwise around its center. }\n\\label{fig4}\n\\end{center}\n\\end{figure}\n\n{\\em Zero soliton solution.} In the vacuum, $S=m=1$ and the correctly normalized HF spinor in the present representation reads\n\\begin{equation}\n\\psi(\\zeta) = \\left( \\begin{array}{c} \\zeta \\\\ -1\/2 \\end{array} \\right) e^{i(\\bar{z} \\zeta - z\/4\\zeta)}, \\qquad \\zeta= (k-\\omega)\/2.\n\\label{88}\n\\end{equation}\nThe spectral parameter $\\zeta$ is closely related to the light-cone momentum or energy, depending on conventions. \nThe string coordinates (\\ref{83}) are given by \n\\begin{eqnarray}\nZ_1 & = &  \\sqrt{i} e^{iA_-} \\cosh A_+,\n\\nonumber \\\\\nZ_2 & = &  \\sqrt{-i} e^{iA_-} \\sinh A_+,\n\\nonumber \\\\\n\\quad A_{\\pm} & = &  \\frac{\\bar{z}\\lambda}{2} \\pm \\frac{z}{2\\lambda}.\n\\label{89}\n\\end{eqnarray}\nTo exhibit the motion of the string, we first introduce global coordinates in AdS$_3$ through\n\\begin{equation}\nZ_1 = e^{it} \\cosh \\xi, \\qquad Z_2 = e^{i\\phi} \\sinh \\xi\n\\label{90}\n\\end{equation}\nwhere $t$ is the global time and ($\\xi,\\phi$) are coordinates on the time slice, the hyperbolic plane H$_2$. \nThe string at fixed $t$ is most easily visualized on the Poincare disk of radius 1 by the choice of coordinates \n\\begin{equation}\nX= \\rho \\cos \\phi, \\quad Y = \\rho \\sin \\phi, \\quad \\rho = \\sqrt{\\frac{\\cosh \\xi -1}{\\cosh \\xi +1}}. \n\\label{91}\n\\end{equation}\nThe circumference of the disk is the boundary of AdS$_3$ space.\nSince we have two independent variables $z, \\bar{z}$ while keeping only $t$ fixed, Eqs.~(\\ref{91}) define a one-dimensional curve. In the case of the vacuum,\nthe string is evidently just a straight line along a diameter of the disk uniformly spinning around its center, see Fig.~\\ref{fig4}.\nThis agrees with the findings of \\cite{L17} where the sinh-Gordon vacuum was mapped onto string solutions.   \n\\begin{figure}\n\\begin{center}\n\\epsfig{file=Fig5.eps,height=8cm,width=7.5cm,angle=270}\n\\caption{Motion of the string for the one-soliton TDHF solution (kink at rest, $\\lambda=-2$) in AdS$_3$. The parameter $t$ is the global time\naccording to the parametrization (\\ref{90}). The spike corresponds to the zero of $S$ and must lie on the boundary of the disk.} \n\\label{fig5}\n\\end{center}\n\\end{figure}\n\n{\\em One soliton solution.} A kink moving with velocity $v$ has the scalar potential \n\\begin{equation}\nS(x,t) = \\tanh \\gamma (x-vt), \\qquad \\gamma= \\frac{1}{\\sqrt{1-v^2}}.\n\\label{92}\n\\end{equation}\nThe upper and lower components of the continuum spinors belonging to this potential are\n\\begin{eqnarray}\n\\psi_1 &=& {\\cal N} \\left( 1+ \\frac{i Y S}{2\\zeta} \\right) e^{i{\\cal A}},\n\\nonumber \\\\\n\\psi_2 & = & {\\cal N} \\left( -  \\frac{ iY}{4 \\zeta^2}- \\frac{S}{2\\zeta}\\right)  e^{ i{\\cal A}},\n\\label{93}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n{\\cal A} & = & kx-\\omega t \\ = \\ \\zeta \\bar{z}-\\frac{z}{4\\zeta},\n\\nonumber \\\\\n{\\cal N} & = & - \\frac{2 i\\zeta^2}{Y-2 i\\zeta},\n\\nonumber \\\\\nY & = &  \\sqrt{\\frac{1-v}{1+v}}.\n\\label{94}\n\\end{eqnarray}\nHere, the normalization \n\\begin{equation}\n\\bar{\\psi} \\psi = - \\zeta S\n\\label{95}\n\\end{equation}\nhas been chosen in accordance with Eq.~(\\ref{78}).\n$Z_1$ and $Z_2$ can be constructed using Eqs.~(\\ref{81}--\\ref{83}) above, but the result is not very transparent and will not be given here.\nIn order to map out the motion of the string, we proceed as follows: After choosing an initial global time $t$, we construct a trajectory in the ($z,\\bar{z}$) plane\nby numerically solving the transcendental equation\n\\begin{equation}\nt = \\arctan \\frac{Y_{0}}{Y_{-1}}.\n\\label{96}\n\\end{equation}\nAlong this trajectory we then evaluate $Z_1,Z_2$ and convert them to the coordinates $(X,Y)$ according to Eqs.~(\\ref{90},\\ref{91}). We then plot the string\nand repeat the procedure for a sequence of time steps. We recall that we had to choose different gauge transformations depending on the sign of $S$, \ncf. Eqs.~(\\ref{75}), (\\ref{83a}). When gluing together the corresponding solutions, we impose continuity on the string coordinates in order to\nspecify the relative phase between the spinors $\\psi_a,\\psi_b$, which would otherwise be undetermined. In this way we find that we have to change the sign of \n$Z_1,Z_2$ whenever $S$ crosses zero. Fig.~\\ref{fig5} shows the result of such a calculation, using the parameters $v=0$ (kink at rest)\nand $\\lambda=-2$. As expected, the endpoints of the string lie on the boundary.\nSince $S=0$ is a singular point, the fold in the string also touches the boundary, unlike in Ref.~\\cite{L17}.\nThis behavior can easily be traced back to the factor $1\/\\sqrt{|S|}$ in the gauge transformation $\\Omega$, see Eqs.~(\\ref{75}) and (\\ref{83a}).\nComparing Figs.~\\ref{fig4} and \\ref{fig5} we observe that the kink string resembles the vacuum string at the beginning and the end of the motion shown, reflecting\nthe characteristic behavior of the kink potential which also interpolates between two vacua. In order to better understand the shape of the strings, we\nalso recall that geodesics on the Poincare disk are either straight lines through the center or circles intersecting the boundary at right angles. \n\n{\\em Two soliton solution.} It is straightforward to repeat such a calculation for the kink-antikink scattering solution discussed above, at least using Maple.\nSince $S$ now changes sign twice, the string is obtained by patching together three different solutions. The resulting string now connects two points \non the boundary of the disk and touches this boundary in cusps at two intermediate points corresponding to the zeros of $S$. Figs.~\\ref{fig6},\\ref{fig7} show an\nexample derived from kink-antikink scattering at $v=\\tanh 1 \\approx 0.76$ (rapidity 1) and $\\lambda=-2$. Since the motion is already fairly complicated, we have \nchosen finer time steps than for the one soliton case. Each of the three parts of the string is infinitely long due to the Poincare metric and qualitatively follows\nthe geodesics.\n\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=Fig6.eps,height=8cm,width=7.5cm,angle=270}\n\\caption{Time evolution of string derived from kink-antikink scattering at $v=0.76, \\lambda=-2$. The representation is the same as in Fig.~\\ref{fig5}, but \nnow there are two cusps on the boundary, reflecting the two zeros of $S$.} \n\\label{fig6}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=Fig7.eps,height=8cm,width=7.5cm,angle=270}\n\\caption{Continuation of Fig.~\\ref{fig6} to larger times.} \n\\label{fig7}\n\\end{center}\n\\end{figure}\n\n\\section{Summary and discussion}\\label{sect5}\n\nIn this paper, we have reconsidered the simplest version of the GN model in the large $N$ limit --- the massless model with discrete chiral symmetry,\nEq.~(\\ref{I1}). Following a\nsuggestion of Dashen, Hasslacher and Neveu we have first studied kink-antikink scattering, using the TDHF language.\nA full analytical solution has uncovered an interesting interplay between the scalar potentials of kink and antikink which repell each other\nand the fermion constituents which travel in one direction only, being exchanged during the collision. From a technical point of view, we have found that\nunlike our starting point, the DHN breather, the scattering solution belongs to the simple class of type I solutions in which each single\nparticle orbit contributes a term proportional to the full self-consistent potential $S(x,t)$ to the chiral condensate. This complements earlier \nknown solutions of type I, the vacuum, the kink and the kink crystal. Type I solutions are singled out by the fact that the TDHF equations\nreduce to a non-linear Dirac equation. In this way we were able to make contact with earlier studies of the classical $N=1$ GN model\nand take advantage of the techniques which have been developed there. We have reproduced the finding of Neveu and Papanicolaou\nthat the square of the self-consistent potential is related to the sinh-Gordon equation. The linearized version of the latter equation is nothing but the\nKlein-Gordon equation for the $\\sigma$ meson of the GN model. This supports a Skyrme-type picture of the baryon in the non-chiral GN model.\nAll presently known type I solutions can be identified (up to a trivial sign ambiguity) with well-known solitons of\nthe sinh-Gordon equation, an observation which would have saved a lot of guesswork, had it been known before. Even more interesting is perhaps the fact that\n$N$ soliton solutions and the Lax representation of the associated linear problem are known explicitly for the sinh-Gordon equation. This \nshould enable us to generalize the two-soliton solution of the present paper to the scattering problem of $N$ composite, relativistic bound states.\nAnother possible generalization of the present work would be to consider type II solutions, thereby covering all analytically known HF and TDHF\nsolutions of both the massless and the massive (non-chiral) GN models. The methods of Ref.~\\cite{L16} are also developed for the $N=2$ classical GN model,\nrelevant for type II solutions of the large $N$ quantum theory. Here however, the formalism is significantly more involved, and one apparently\nneeds coupled, non-linear differential equations for several functions, including the scalar potential.\n\nSo far we have not yet mentioned  the chiral GN model or Nambu--Jona-Lasinio model in 2 dimensions (NJL$_2$). In the massless NJL$_2$ model, transparent\npotentials appear in the context of the massless baryon \\cite{L21} and the Shei bound state \\cite{L26}. Generalizations to periodic, finite gap potentials include\nthe chiral spiral \\cite{L27} and the twisted kink crystal \\cite{L28}. Basar and Dunne have identified the non-linear Schr\\\"odinger equation as the \nrelevant equation for the HF potential, using completely different techniques from the ones employed in the present paper. In the massive NJL$_2$ model,\napproximate results from the derivative expansion indicate that the sine-Gordon equation is relevant near the chiral limit, but needs to be replaced by\nincreasingly complicated, higher order differential equations with increasing bare fermion mass \\cite{L13}. Here, the baryon potentials are apparently\nnot transparent, so that we are dealing with neither type I nor type II solutions, nor with solutions of any finite type, for that matter.\nThe common theme of all of these efforts is the search for a closed, classical theory of the self-consistent\npotential, the relevant saddle point in the functional integral approach. In a sense, one might identify this effort with the search for Witten's \n``master field\" in the large $N$ limit of quantum chromodynamics \\cite{L29} where unfortunetely little progress has been made until now.\n\nThe close relationship between the GN model and the sinh-Gordon equation gave us the clue for yet another kind of mapping. It is known that the sinh-Gordon\nmodel and strings in AdS$_3$ are related by some kind of gauge transformation, using the Pohlmeyer reduction. In the same vein, we have constructed a gauge\ntransformation mapping type I GN solutions onto classical strings in AdS$_3$. Although this has nothing to do with the celebrated AdS\/CFT correspondence,\nit is rather intriguing. When mapping the sinh-Gordon model to string theory, one has to use artificial\nspinor fields from the associated Lax pair. In our case, the spinors have a more direct physical meaning as TDHF solutions. Another difference to\nthe string\/sinh-Gordon correspondence is a singularity in the gauge transformation at the zeros of $S$. This yields folded strings whose spikes \nmust always lie on the boundary of AdS$_3$. The existence of such a mapping seems to be closely related to the symmetries\nof the models. However, the fact that two models have the same symmetry is in general not sufficient to conclude that their dynamics is the same.\nHere we have shown how to map a fermionic quantum field theory in the large $N$ limit onto a classical string theory in a \nfully explicit manner. Whether this is of any practical use remains to be seen. If nothing else, it provides us with a novel way of visualizing TDHF\nsingle particle spinors, which contain more detailed information than the self-consistent potential.\n\\vskip 0.5cm\n\\section*{Acknowledgement}\n\\vskip 0.2cm\nM. T. thanks Antal Jevicki and Kewang Jin for a clarifying correspondence about their work, and Johanna Erdmenger and Hans-J\\\"urgen Pirner for\nhelpful discussions. This work has been supported in part by the DFG under grant TH 842\/1-1.\n\n\\vskip 0.5cm\n{\\bf Appendix: Definition and normalization of $\\psi_b$}\n\\vskip 0.2cm \nHere we motivate the particular choice for the analytically continued TDHF spinor $\\psi_b$, Eq.~(\\ref{81}). Consider first a real spectral parameter \n$\\zeta$. The $N$ soliton solution of the TDHF equation can be factored as\n\\begin{equation}\n\\psi = \\varphi e^{i(\\bar{z}\\zeta-z\/4\\zeta)}\n\\label{D1}\n\\end{equation}\nwhere $\\varphi$ is a solution of the reduced Dirac equation\n\\begin{eqnarray}\n-2i \\varphi_{1,z} - \\frac{1}{2\\zeta} \\varphi_1 & = &  S \\varphi_2 ,\n\\nonumber \\\\\n2i \\varphi_{2,\\bar{z}} - 2 \\zeta \\varphi_2 & = & S \\varphi_1.\n\\label{D2}\n\\end{eqnarray}\nThis  merely expresses the fact that the potentials are reflectionless.  If $\\varphi$ solves Eq.~(\\ref{D2}) with spectral parameter $\\zeta$, then\n\\begin{equation}\n\\gamma_5 \\varphi^* = \\left( \\begin{array}{c} -\\varphi_1^* \\\\ \\varphi_2^* \\end{array} \\right)\n\\label{D3}\n\\end{equation}\nsolves the corresponding equation with spectral parameter $-\\zeta$. Since we wish to introduce imaginary $\\zeta$ parameters, we avoid \ncomplex conjugation and write the scalar condensate and the reflection property under $\\zeta \\to - \\zeta$ in the analytic form\n\\begin{eqnarray}\n\\bar{\\varphi}(\\zeta) \\varphi(\\zeta) & = & \\varphi_1(\\zeta,-i)\\varphi_2(\\zeta,i) + \\varphi_2(\\zeta,-i)\\varphi_1(\\zeta,i),\n\\nonumber \\\\\n\\varphi_1(-\\zeta,i) & = & - \\varphi_1(\\zeta,-i),\n\\nonumber \\\\\n\\varphi_2(-\\zeta,i) & = & \\varphi_2(\\zeta,- i).\n\\label{D4}\n\\end{eqnarray}\nWe also need the normalization condition (\\ref{82}) for $\\psi_a$ which translates into\n\\begin{equation}\n\\bar{\\varphi}(\\zeta) \\varphi(\\zeta) = - \\zeta S.\n\\label{D5}\n\\end{equation}\nFrom Eqs.~(\\ref{D4}) one deduces the relations\n\\begin{eqnarray}\n\\bar{\\varphi}(-\\zeta) \\varphi(\\zeta) & = & \\bar{\\varphi}(\\zeta) \\varphi(-\\zeta) \\ = \\ 0,\n\\nonumber \\\\\n\\bar{\\varphi}(-\\zeta) \\varphi(-\\zeta) & = & - \\bar{\\varphi}(\\zeta) \\varphi(\\zeta).\n\\label{D6}\n\\end{eqnarray}\nIn this form, all reflection properties also hold for complex $\\zeta$. $\\psi_b$ can be written as\n\\begin{equation}\n\\psi_b = \\frac{1}{\\sqrt{2}} \\left[ \\varphi(\\zeta=-i\\lambda\/2) e^{\\cal B} + i \\varphi(\\zeta=i\\lambda\/2)e^{-{\\cal B}}\\right]\n\\label{D7}\n\\end{equation}\nwith \n\\begin{equation}\n{\\cal B} = \\frac{\\bar{z} \\lambda}{2} + \\frac{z}{2\\lambda}.\n\\label{D8}\n\\end{equation}\nUsing Eqs.~(\\ref{D5}) and (\\ref{D6}) for imaginary $\\zeta$, the condensate of $\\psi_b$ becomes\n\\begin{eqnarray}\n\\bar{\\psi}_b \\psi_b & = & \\frac{1}{2} \\left[ \\bar{\\varphi}(i\\lambda\/2) \\varphi(-i\\lambda\/2) e^{2{\\cal B}}\n+ \\bar{\\varphi}(-i\\lambda\/2)\\varphi(i\\lambda\/2) e^{-2{\\cal B}} \\right.\n\\nonumber \\\\\n& &  \\left. + i  \\bar{\\varphi}(i\\lambda\/2)\\varphi(i\\lambda\/2)\n- i \\bar{\\varphi}(-i\\lambda\/2) \\varphi(-i \\lambda\/2) \\right]\n\\nonumber \\\\\n& = & i \\bar{\\varphi}(i\\lambda\/2) \\varphi(i\\lambda\/2)\n\\nonumber \\\\\n& = & \\frac{\\lambda}{2}S,\n\\label{D9}\n\\end{eqnarray}\nproving the 2nd half of Eq.~(\\ref{82}).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{L}{ow} latency has been one of the key features that next wireless systems, such as 5G and beyond networks, are designed to fulfill in a wide range of scenarios. Nevertheless, this requisite is probably one of the most challenging from the quality of service (QoS) provision point of view. \n\nOn the one hand, achieving small block error rates with latencies of the order of few milliseconds requires the specific design of the radio interface functions, specially the radio scheduler. On the other hand, when the system is handling a traffic load close to its capacity, the packet delay can greatly increase leading to packet loss and an unacceptable latency. These two aspects pose an interesting challenge to the design of the scheduling algorithms that must minimize the average delay while providing user fairness. \n\nIn this sense, the use of hybrid control systems based on machine learning (ML) is considered as a promising approach that enables intelligent and adaptive decision making to the variant conditions of the network.\nIn \\cite{Pedersen18}, the essential role played by the radio scheduler in 5G is tackled. From a more theoretical point of view, reinforcement learning (RL) is presented in \\cite{Sutton14} as a promising solution that allows dealing with the dynamic radio environment and learning mapping functions over time. For instance, research advances described in \\cite{Comsa19} and \\cite{Comsa19bis} analyze the user fairness in terms of throughput for a modified proportional fair (PF) strategy, finding a balance between the system spectral efficiency and the fairness among users. However, to the best of the author's knowledge, user fairness in terms of delay has not been addressed yet. This is a critical issue in the emerging low latency services expected for 5G and beyond. Therefore, it is meaningful to investigate this issue and propose smart resource allocation algorithms that keep a balance between user fairness and system performance in terms of delay. \n\nIn this paper, an intelligent controller is proposed to customize a new version of the modified largest weighted delay first (M-LWDF) scheduling strategy \\cite{LDF} at each Transmission Time Interval (TTI). In particular, we propose a novel utility function called $\\beta$-M-LWDF that is able to adjust dynamically the experienced delay by means of a parameter $\\beta$ that is managed by the controller. It is important to remark that other scheduling algorithms such as largest delay first (LDF) \\cite{LDF}, PF and M-LWDF can be viewed as special cases of our proposed utility function for $\\beta \\to \\infty$, $\\beta = 0$ and $\\beta = 1$, respectively. Besides, with our proposed ML scheme, the $\\beta$ parameter is selected at each TTI to achieve an appropriate balance between user fairness and system performance in terms of delay. Such a balance is defined as a feasible-fair region on the cumulative distribution function (CDF) of the normalized delay.  This allows us to identify \\textit{unfair} states, where the difference of the delay between different users is excessive; \\textit{over-fair} states, where the difference is small, but the average delay increases; and \\textit{feasible-fair} states which reach an appropriate balance between latency fairness and average delay.  \nTo cope with the high  complexity of this approach, the controller implements a RL algorithm known as deep Q-learning (DQL) as introduced in \\cite{MNIH15}, which makes use of a neural network to estimate the Q-Function that approximates the best decisions at each TTI. \n\n\n\\section{System Model}\n\\label{sec:system_model}\nWe consider the downlink (DL) of the 5G NR (New Radio) system, which is based on Orthogonal Frequency Division Multiple Access (OFDMA). The available bandwidth is divided into equal resource blocks (RBs), where one RB consists on 12 sub-carriers and it represents the minimum resource unit that can be assigned in the frequency domain. The sub-carrier spacing (SCS) can be expressed as $\\Delta f = 2^\\mu 15$ kHz, where $\\mu$ is the numerology and it ranges from $0$ to $3$ for data channels.  The time domain resources are divided in slots, also named TTIs.\nIt is considered adaptive modulation and coding (AMC), and thus the transmission rate, i.e., the modulation and coding scheme (MCS), is chosen to maximize the throughput while guaranteeing a block error rate (BLER) below a target value, $\\mathrm{BLER}_T = 10 \\%$ \\cite{Martin21}. The set of MCSs are taken from 5G specs (Table 5.1.3.1-1 of 38.214 v16.7.0). \nLet $\\mathcal{B} = \\{b_1, b_2,\u2026, b_N\\}$ be the set of RBs for a given bandwidth, where $N$ represents the total number of RBs. Additionally, let $\\mathcal{U} = \\{u_1, u_2,\u2026, u_M\\}$ be the set of users to be scheduled.\nA constant bit rate (CBR) traffic model is assumed for each user where a packet sizes of $S_\\mathrm{CBR}$ bytes reach the buffer every $T_\\mathrm{CBR}$ seconds.  \nIt is assumed that the packets are stored in a unbounded buffer and the packets are segmented according to the transport block (TB) size that is determined by the scheduler and the selected MCS. \nThe average signal-to-noise ratio (SNR) of each user, $\\bar{\\gamma_u}$, is drawn randomly according to a Log-Normal distribution with expected value $\\mu_\\gamma$ and standard deviation $\\sigma_\\gamma$. In addition, the multi-path fading follows a realistic Tapped Delay Line Channel (TDL-A) model where the channel complex samples, \n$H_{u,k}[n]$, are correlated in time and frequency domains. Thus, the instantaneous SNR of user $u$ at RB $k$ and TTI $n$ is computed as $\\gamma_{u,k}[n] = \\bar{\\gamma_u} |H_{u,k}[n]|^2$.\n\n\n\\subsection{Scheduling algorithms}\n\\label{sec:alg}\nThe aim of the scheduler is to allocate each RB $b_i \\in B$ to a particular user $u_i \\in U$ at each TTI in order to meet a predefined QoS requirement. We have evaluated four different scheduling policies:\n\n\\begin{enumerate}\n    \\item \n    The well known Proportional Fair (PF) uses the following utility function at each TTI $n$, $U[n]$, as decision criterion to assign the user $\\widehat u$ to the RB $\\widehat k$ on the time instant $n$:\n\\begin{equation}\n\\label{eq:PF}\n\\small\n\\left\\{ {\\widehat u[n],\\widehat k[n]} \\right\\} = \\arg \\mathop {\\max }\\limits_{u,k} \\left\\{ {\\frac{{r_{u,k} [n]}}{{\\overline {r_u }[n] }}} \\right\\},\n\\end{equation}\n\\noindent \nwhere $r_{u,k}[n]$ represents the potential rate for user $u$ on the RB $k$ and TTI $n$, and $\\overline {r_u }[n]$ is a weighted moving average of the data rate values reached in previous TTIs, which is computed as \\cite{Musleh15}\n\\begin{equation}\n\\label{eq:r_PF}\n\\small\n\\overline {r_u } \\left[n \\right] = \\left( {1 - \\frac{1}{{T_{PF} }}} \\right)\\overline {r_u } \\left[ {n - 1} \\right] + \\frac{1}{{T_{PF} }}r_u \\left[ {n - 1} \\right],\n\\end{equation}\n\\noindent \nwhere $\\overline {r_u }[n-1]$ represents the weighted average data rate reached up to the TTI $n-1$, $r_u [n-1]$ is the data rate reached in TTI $n-1$, and $T_{PF}> 0$ is the average window size. \n\n    \\item \nLDF algorithm \\cite{LDF} is adaptive to the delay, providing the turn to the user that has been suffering the largest delay in its queue, as follows:\n\\begin{equation}\n\\small\n\\label{eq:LDF}\n\\widehat u[n] = \\arg \\mathop {\\max}\\limits_{u} \\left\\{ { W_u [n] } \\right\\}.\n\\end{equation}\n\\noindent \nwhere $W_u [n]$ is the delay experienced in the queues, which is computed as the sum of the delays of every packet that is stored in the queue. \n    \\item \nThe algorithm M-LWDF considers the waiting time in the queues, the instantaneous capacity of the channels and a parameter related to the delay tolerance \\cite{Andrews2002}:\n\\begin{equation}\n\\small\n\\label{eq:M-LWDF}\n\\left\\{ {\\widehat u[n],\\widehat k[n]} \\right\\} = \\arg \\mathop {\\max} \\limits_{u,k} \\left\\{ {g_u [n] \\cdot W_u [n] \\cdot r_{u,k} [n]} \\right\\},\n\\end{equation}\n\\noindent\nwhere $g_u [n] \\in [0, \\infty)$ is a QoS factor with a value of $g_u [n] = a_u \/ \\overline {r_u } [n]$, with $\\overline {r_u } [n]$ as defined in \\eqref{eq:r_PF} and $a_u  =  - \\log \\left( {\\delta _u } \\right) \\cdot T_u$, being $\\delta_u \\in [0, 1]$ the desired probability of fulfilling the delay requirement $T_u$ \\cite{Andrews2002}.\n\n\n    \\item \nWe propose a new version of the M-LWDF, called $\\beta$-M-LWDF, in which the impact of the delay term can be adjusted by a parameter $\\beta$ that is selected by the intelligent controller at each TTI:\n\\begin{equation}\n\\small\n\\label{eq:bmlwdf}\n\\left\\{ {\\widehat u[n],\\widehat k[n]} \\right\\} = \\arg \\mathop {\\max}\\limits_{u,k} \\left\\{ {g_u  \\cdot W_u^{\\beta} [n] \\cdot r_{u,k} [n]} \\right\\}.\n\\end{equation}\n\\end{enumerate}\n\\begin{remark}\n\\label{rem:special cases}\nThe utility functions of PF, LDF and M-LWDF that are given with \\eqref{eq:PF}, \\eqref{eq:LDF} and \\eqref{eq:M-LWDF} are special cases of the proposed $\\beta$-M-LWDF with $\\beta = 0$, $\\beta \\to \\infty$ and $\\beta = 1$, respectively. \n\\end{remark}\n\\begin{proof}\nIt is clear that $\\beta$-M-LWDF reduces to M-LWDF algorithm when $\\beta=1$. For $\\beta=0$, $\\beta$-M-LWDF reduces to PF as long as the parameter $a_u$ is equal for all users (as considered in this work). The proof for the case of LDF when $\\beta \\to \\infty$ is given in Appendix \\ref{proof:remark}. \n\\end{proof}\n\\vspace{-3mm}\n\\begin{remark}\n\\label{rem:beta behaviour}\nAs it is shown in \\cite{LDF}, LDF provides the highest fairness in terms of delay while sacrificing system performance (i.e., average delay), whereas M-LWDF achieves a higher system performance at the expense of fairness. This is due to the fact that LDF only accounts for the user delay whereas M-LWDF also accounts for the instantaneous rate. In view of remark \\ref{rem:special cases},  this involves that increasing $\\beta$ increases the fairness while decreasing $\\beta$ reduces the fairness in terms of delay. \n\\end{remark}\n\\section{Proposed RL Framework}\n\\label{sec:framework}\n\\subsection{RL Framework Description}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{RL_Framework.eps}\n\\caption{Proposed RL framework}\n\\label{fig:framework}\n\\end{figure}\nOur proposed RL framework is shown in Fig. \\ref{fig:framework}. At TTI $n$, the \\textit{channel} block generates the multi-path fading samples.\nThen the instantaneous SNR, $\\gamma_{u,k}[n]$, is computed. This SNR is used by the \\textit{AMC} block to select the MCS that maximizes the potential rate, $r_{u,k}[n]$, while keeping a BLER below $\\mathrm{BLER}_T$. On the other hand, packets are generated by the \\textit{traffic source} and stored in the \\textit{buffer} queue. This block performs segmentation of the stored packets according to the TB size determined by the \\textit{scheduler} and it computes the queue delay of each user $W_u[n]$.\nAfterwards, the scheduler updates its $\\beta[n]$ parameter according to the actions made by the agent and it allocates resources based on the proposed $\\beta$-M-LWDF utility function. Once the scheduler has performed the resource allocation, the \\textit{QoS} block estimates the empirical CDF of the normalized user delay, which is used to compute the reward and state variables. Those variables are used by the agent to make the next action in the following TTI. This action involves determining the step $\\Delta \\beta [n] \\in [-1,+1]$ to increase or decrease the beta parameter as $\\beta[n] = \\beta[n-1] + \\Delta \\beta [n]$.\n\\vspace{-3mm}\n\\subsection{Latency Fairness Criteria}\nThe Next Generation Mobile Networks (NGMN) Alliance defines an impartiality requirement in terms of throughput. This criteria is defined so that a system is seen as fair if at least the $(100-x)\\%$ of active users reach at least $x\\%$ of the normalized user data rate \\cite{NGNM}. \n\nIn this work we propose a RL framework intended to fulfill a predefined user fairness criteria in terms of latency. However, up to date there is no standardized fairness requirement in terms of delay. For that reason, in this paper we propose an impartiality requirement with the following $3$ desired features: i) half of the users have a delay smaller than the average; ii) there are no users with a delay $50\\%$ higher than the average; and iii) there are no users with a delay $50\\%$ smaller than the average. \nThese $3$ desired features lead to a CDF requirement as an straight line that passes through the points $(0.5, 0)$ and $(1.5, 1)$ as follows\n\\vspace{-3mm}\n\\begin{equation}\n\\small\n\\label{eq:CDF requirement}\ny = f_R(w) = \n    \\begin{cases}\n        1 & w > 1.5 \\\\\n        w - 0.5 & 0.5 \\leq w \\leq 1.5 \\\\\n        0 & w < 0.5\n    \\end{cases}\n\\end{equation}\n\\noindent \nwhere $w = f^{-1}_R (y) = y + 0.5$ stands for the inverse CDF requirement. \nIt can be noticed that such a CDF requirement  can be also read as $c\\%$ of the users to be below the $(50-c)\\%$ of the normalized user delay, $\\tilde{W}_u[n]$, which is defined as\n\\begin{equation}\n\\small\n\\label{eq:normalized delay}\n\\tilde{W}_u[n] = \\frac{W_u[n]}{\\frac{1}{M} \\sum_{u^\\prime = 1}^M W_{u^\\prime}[n]} \n\\end{equation}\n\nAs it can be noticed, $\\tilde{W}_u[n]=1$ involves that the delay of the user at TTI $n$ is equal to the average user delay. In a perfectly fair system all the users would have the same delay, and thus, all of them would have a unitary normalized delay.\n\nNevertheless, as it is discussed in \\cite{Sadr09, LDF} there is a trade-off between fairness and system performance. This involves that increasing the delay fairness is at the expense of increasing the average delay. For that reason, our proposed CDF requirement aims at identifying those cases where the fairness severely degrades the average delay, and those cases where the difference of delay between users is excessive. The former case is labeled as \\textit{over-fair (OF)} and the latter is labeled as \\textit{unfair (UF)}. Those cases that do not fall on the two aforementioned cases are labeled \\textit{feasible-fair (FF)}.  \nThe goal of the agent is to maximize the number of TTIs where the system is in FF state since this case leads to an appropriate balance between fairness and average delay. \n\nThe empirical CDF of the normalized delay at TTI $n$ is expressed below\n\\vspace{-3mm}\n\\begin{equation}\n\\small\n    \\label{eq:empirical CDF}\n    \\hat{F}_{\\tilde{W}[n]}(w) = \\frac{1}{M} \\sum_{u=1}^M \\mathbbm{1} \\left( \\tilde{W}_u[n] \\leq w  \\right) \n\\end{equation}\n\\noindent\nwhere $\\tilde{W}[n]$ represents a randomly chosen normalized user delay at TTI $n$. $ \\mathbbm{1}\\left( \\mathcal{E}  \\right)$ stands for the indicator function, which is $1$ if the event $\\mathcal{E}$ is true and $0$ otherwise. Let us represent the sorted vector of normalized delays as $\\mathbf{W}^\\uparrow[n]=\\left(W^\\uparrow_1[n], .., W^\\uparrow_M[n]\\right)$, where $W^\\uparrow_j[n]$ is the $j$-th delay in ascending order. Then, if we evaluate the empirical CDF from \\eqref{eq:empirical CDF} on $\\mathbf{W}^\\uparrow[n]$ we get $M$ equally spaced samples between $0$ and $1$ as follows $\\mathbf{Y}[n]=(Y_1[n],..,Y_M[n]) = \\hat{F}_{\\tilde{W}[n]}\\left(\\mathbf{W}^\\uparrow\\right[n])$ where $Y_j[n] = \\frac{j}{M} = \\hat{F}_{\\tilde{W}[n]}\\left(W^\\uparrow_j[n]\\right)$. Therefore, we can obtain a required normalized delay, $w^{(R)}_j$, for each of the sorted users as $w^{(R)}_j=f^{-1}_R \\left(\\frac{j}{M} \\right) = \\frac{j}{M} + 0.5$ with $j\\in [1,M] \\subset \\mathbb{N}$. A given sorted user $j$ is said to fulfill the delay requirement if $\\Delta W_j[n] = W^\\uparrow_j[n] - w^{(R)}_j \\leq \\xi$, where $\\xi$ is a confidence factor.\n\n\\begin{algorithm}\n\\small\n   \\hspace*{\\algorithmicindent} \\textbf{Input:} $\\mathbf{\\tilde{W}}[n]=\\left(\\tilde{W}_1[n], .., \\tilde{W}_M[n] \\right)$ \\\\\n   \\hspace*{\\algorithmicindent} \\textbf{Output:} $\\mathcal{C}[n]$, fairness case  \n\\begin{algorithmic}[1]   \n\\STATE   Sort the vector of normalized delays, $\\mathbf{\\tilde{W}}[n]$, to get $\\mathbf{W}^\\uparrow[n]$ \n   \\IF {$ \\left[ \\sum_{j=1}^{\\lceil \\lambda M \\rceil} \\mathbbm{1} \\left( W^\\uparrow_j > w^{(R)}_j + \\xi \\right) \\right] > 1$ }\n   {\n      \\STATE $\\mathcal{C}[n] = \\mathrm{OF}$ \n   }\n   \\ELSIF {$ \\left[ \\sum_{j= \\lceil \\lambda M \\rceil +1}^{M - \\lceil \\psi M \\rceil} \\mathbbm{1} \\left( W^\\uparrow_j > w^{(R)}_j + \\xi \\right) \\right] > 1$ }\n   {\n      \\STATE $\\mathcal{C}[n] = \\mathrm{UF}$  \n   }\n   \\ELSE\n   {\n       \\STATE $\\mathcal{C}[n] = \\mathrm{FF}$  \n   }\n   \\ENDIF\n\\end{algorithmic}\n\\caption{{Selection of fairness cases}}\n\\label{alg:case selection}\n\\end{algorithm}\n\nAlgorithm \\ref{alg:case selection} is used to identify in which of the three cases $\\mathcal{C}[n] \\in \\{\\mathrm{OF}, \\mathrm{UF}, \\mathrm{FF}\\}$ is the system working at a given TTI, $n$. It can be observed from line $2$ that the system is labeled as OF if any of the $\\lceil \\lambda M \\rceil$ best users (i.e., users with smallest delay) have a normalized delay greater than the required delay plus the confidence factor. The parameter $\\lambda \\in [0, 1]$ represents the percentage of best users. An OF situation means that the best users have a delay closer to the average delay, which increases the fairness. \nHowever, this happens at the expense of a significant reduction on the average delay.\n\nFrom line $4$ of algorithm \\ref{alg:case selection} it is observed that the system is labeled as UF if any of the worst users do not fulfill the delay requirement. The agent aims at avoiding this case in order to reduce the normalized delay of the worst users. \nAs it can be seen, the $\\lceil \\psi M \\rceil$ users with the greatest delays are not considered because they are treated as outliers, where $\\psi \\in [0, 1]$. This means that those users have such a poor channel condition that do not represent the conditions of the vast majority of users and thus, they are not considered.\nIf all the users (excluding the outliers) fulfill the delay requirement (including the factor $\\xi$), then the system is in FF case. \n\nFig. \\ref{fig:cdf_req} illustrates an example of the OF and UF cases by means of their estimated CDF of the normalized delay. As it is seen with algorithm \\ref{alg:case selection}, the criteria to select the fairness case involves checking if any of the best or worst users do not fulfill the delay requirement. Nevertheless, this criteria is equivalent to check if the empirical CDF at some TTI $n$ falls within the OF, UF or FF regions respectively. Those regions are shown in Fig. \\ref{fig:cdf_req} with blue, red and green colors. For instance, as it can be observed, the sorted user $u=36$, which makes the empirical CDF to reach $0.6$, has a normalized delay of $W_u^\\uparrow=1.466$, which is greater than the related CDF requirement plus the confidence factor, i.e., $w^{(R)}+\\xi = 1.1+0.1$. Since this user, and others that belong to the worst users set, do not fulfill the delay requirement, the system is labeled as UF in that case. \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.71\\columnwidth]{fig_cdf_req2.pdf}\n\\caption{Empirical CDF of the normalized delay for UF (red) and OF (blue) cases after defining $40\\%$ of the users as best users (i.e., $\\lambda=0.4$) and $10\\%$ of the users as outliers ($\\xi=0.1$) for a cell with $M=60$ users.}\n\\label{fig:cdf_req}\n\\end{figure}\n\n\\vspace{-3mm}\n\\subsection{States and Actions}\nLet $\\mathcal{S}$ be the state space and let $s[n] \\in \\mathcal{S}$ be the instantaneous state at TTI $n$. The state $s[n]$ can be seen as a union of two sub-states $s[n] = \\left(s_c[n], s_u[n] \\right)$, the controllable ($s_c[n]$) and uncontrollable ($s_u[n]$) sub-states. \nThe controllable sub-state is comprised of $s_c[n] = \\left(\\beta[n-1], {d}^{(\\mathrm{inf})}[n], {d}^{(\\mathrm{sup})}[n]\\right)$ where ${d}^{(\\mathrm{inf})}[n]$ and ${d}^{(\\mathrm{sup})}[n]$ represent the distance of the normalized delay to the requirement for the best and worst users sets as described in algorithm \\ref{alg:distance}.\n\n\\begin{algorithm}\n\\small\n   \\hspace*{\\algorithmicindent} \\textbf{Input:} $\\mathbf{\\Delta W}[n] = \\left(\\Delta W_1[n],.., \\Delta W_M[n]\\right)$  \\\\\n   \\hspace*{\\algorithmicindent} \\textbf{Output:} ${d}^{(\\mathrm{inf})}[n], {d}^{(\\mathrm{sup})}[n]$\n\\begin{algorithmic}[1]   \n\n   \\IF {$ \\mathcal{C}[n] = \\mathrm{OF}$ }\n   {\n      \\STATE ${d}^{(\\mathrm{inf})}[n] = \\max \\left(\\mathbf{\\Delta W}_{\\left\\{1:\\lceil \\lambda M \\rceil \\right\\}}[n] \\right)$  \n      \\STATE ${d}^{(\\mathrm{sup})}[n] = \\min \\left(\\mathbf{\\Delta W}_{\\{\\lceil \\lambda M \\rceil+1:M\\}}[n] \\right)$ \n   }\n   \\ELSIF {$ \\mathcal{C}[n] = \\mathrm{UF}$ }\n   {\n      \\STATE ${d}^{(\\mathrm{inf})}[n] = \\min \\left(\\mathbf{\\Delta W}_{\\{1:\\lceil \\lambda M \\rceil\\}}[n] \\right)$  \n      \\STATE ${d}^{(\\mathrm{sup})}[n] = \\max \\left(\\mathbf{\\Delta W}_{\\{\\lceil \\lambda M \\rceil+1:M\\}}[n] \\right)$  \n   }\n   \\ELSE\n   {\n      \\STATE ${d}^{(\\mathrm{inf})}[n] = \\min \\left(\\mathbf{\\Delta W}_{\\{1:\\lceil \\lambda M \\rceil\\}}[n] \\right)$\n      \\STATE ${d}^{(\\mathrm{sup})}[n] = \\max \\left(\\mathbf{\\Delta W}_{\\{\\lceil \\lambda M \\rceil+1:M\\}}[n] \\right)$  \n   }\n   \\ENDIF\n\\end{algorithmic}\n\\caption{{Selection of state values}}\n\\label{alg:distance}\n\\end{algorithm}\n\n\nFor the OF case $d^{\\rm (inf)}$ is the maximum of the distances of the best users while $d^{\\rm (sup)}$ is the minimum of the distances of the worst users. The reasoning of this assignment is to give more importance to the distances of worst users, since that set is the one that do not fulfill the requirement in the OF case. Similarly, in the UF case, the assignment using the $\\max()$ function is for the worst users, which are the users that do not fulfill the requirement in such a case. \n\nThe sub-state $s_u[n]$ is comprised of $\\scriptsize s_u[n] = \\Big( \\hat{\\mathbb{E}}\\left[\\mathbf{\\tilde{W}}[n]\\right], \\hat{\\mathbb{S}}\\left[\\mathbf{\\tilde{W}}[n]\\right], \\hat{\\mathbb{E}}\\Big[\\mathbf{I}_\\mathrm{MCS}[n]\\Big], \\hat{\\mathbb{S}}\\Big[\\mathbf{I}_\\mathrm{MCS}[n]\\Big] \\Big) $, where $\\scriptsize \\hat{\\mathbb{E}}\\left[\\bullet \\right]$ and $\\scriptsize \\hat{\\mathbb{S}}\\left[\\bullet \\right]$ stands for the empirical mean and standard deviation respectively. $\\mathbf{\\tilde{W}}[n]$ represents the vector of normalized delays of all users and \n$\\scriptsize \\mathbf{I}_\\mathrm{MCS}[n]= \\Big({I}_{\\mathrm{MCS},1}[n],..,{I}_{\\mathrm{MCS},M}[n] \\Big)$ is the vector of the MCS indexes reported by all users.\n\n\nWe have considered a discrete action space $\\mathcal{A} = \\Big\\{0,\\pm 10^{-4}, \\pm 10^{-3}, \\pm 10^{-2}, \\pm 5\\cdot10 ^ {- 2}, \\pm 10 ^ {-1}\\Big\\}$. At each time step $n$, the action taken by the agent will select the step size $\\Delta \\beta [n] \\in \\mathcal{A}$ that maximizes the expected cumulative reward.\n\n\\subsection{Reward Function}\nThe proposed reward function encourages the agent to stay in the FF case by taking into account two aspects: (a) the fairness state; and (b) the action taken by the agent. The value of the reward function is defined in (\\ref{eq:reward_general}), (\\ref{eq:reward_uf}) and (\\ref{eq:reward_of}):\n\\vspace{-2mm}\n\\begin{equation}\n\\small\n\\label{eq:reward_general}\n    {r}[n+1] = \n    \\begin{cases}\n      {r}_{\\mathrm{UF}}[n+1] & \\mathrm{if} \\;  \\mathcal{C}[n] = \\mathrm{UF}\\\\\n      1 & \\mathrm{if} \\;  \\mathcal{C}[n] = \\mathrm{FF}\\\\\n      {r}_{\\mathrm{OF}}[n+1] & \\mathrm{if} \\;  \\mathcal{C}[n] = \\mathrm{OF}\n    \\end{cases}\n\\end{equation}\n\\vspace{-2mm}\n\\begin{equation}\n\\small\n\\label{eq:reward_uf}\n  {r}_{\\mathrm{UF}}[n+1] = \n    \\begin{cases}\n      \\Delta\\beta[n] & \\mathrm{if} \\;  \\Delta \\beta[n] > 0\\\\\n      -1 & \\mathrm{if} \\; \\Delta\\beta[n] \\le 0\n    \\end{cases}\n\\end{equation}\n\\vspace{-2mm}\n\\begin{equation}\n\\small\n\\label{eq:reward_of}\n  {r}_{\\mathrm{OF}}[n+1] = \n    \\begin{cases}\n      -\\Delta\\beta[n] & \\mathrm{if} \\; \\Delta \\beta[n] < 0\\\\\n      -1 & \\mathrm{if} \\; \\Delta\\beta[n] \\ge 0\n    \\end{cases}\n\\end{equation}\nThe reasoning of \\eqref{eq:reward_general} is to encourage the agent to stay in FF state with a maximal positive reward (i.e., $1$). If the fairness case is UF, the agent should increase the $\\beta$ parameter to augment the fairness as per remark \\ref{rem:beta behaviour}. Thus, according to \\eqref{eq:reward_uf}, the reward is positive if $\\Delta \\beta[n]$ is positive, which means that the action made in previous instant was adequate; otherwise the rewards in negative to penalize the action of decreasing $\\beta$ that was made through a negative $\\Delta \\beta[n]$. Analogously, for OF case the agent is encouraged to decrease the fairness through the reward function as given with \\eqref{eq:reward_of}. \n\n\\section{Numerical Results and Discussions}\n \\label{sec:results}\n \n\nTo simulate the environment we have assumed a single cell scenario with $M =$ 60 active users.\nA detailed list of the network parameters is shown in Table \\ref{tab:net_param}.\n\n\n\\begin{table}[t]\n\\scriptsize\n\\caption{Network parameters setting}\n\\label{tab:net_param}\n\\begin{tabular}{|l|l|l|l|}\n\\noalign{\\hrule height 1pt}\n\\textbf{Parameter} & \\textbf{Value} & \\textbf{Parameter} & \\textbf{Value} \\\\ \n\\noalign{\\hrule height 1pt}\n $N$                   & $100$ RBs     &      Carrier frequency (GHz)                & $5$                       \\\\\nUser speed  (Km\/h)                     & $5$         & Delay Spread ($\\mu s$)                      & $100$                      \\\\\nTTI (ms) & $1$                   & Number of users                  & $60$                          \\\\\n$S_\\mathrm{CBR}$ (bytes)                    &  $850$           & $T_\\mathrm{CBR}$ (ms)                    &  $6$                             \\\\\n$\\mu_\\gamma$ (dB)                & $15$  & $\\sigma_\\gamma$ (dB) & $3$  \\\\\n$\\mathrm{BLER}_T$           & $0.1$                   & PF window size ($T_{PF}$)                & $100$ \\\\ \n$\\delta_u$ & 0.05& $T_u$ (ms) & $100$\\\\\n\\noalign{\\hrule height 0.5pt}\n\\end{tabular}\n\\end{table}\n\nThe agent implements a DQL with a  neural network comprised of $L$ layers, being $N_\\ell$ the number of nodes configured for each layer where  $\\ell \\in [1,L]$. It should be noted  that the number of nodes in both the input and output layers is fixed, corresponding to the dimensions of the state and action spaces respectively. Based on \\cite{Comsa19}, the configuration $\\{L = 3, N_2 = 60\\}$ is selected to find a balance between flexibility and complexity of our learning system. \nThe agent is trained with a Decayed $\\epsilon$-greedy policy. During the training stage the agent was trained during $2\\cdot 10^5$ steps (i.e. TTIs).\n\n\n\n\n\n\n\n\nFig. \\ref{fig:cdf} shows the CDF of the normalized user delay for different scheduling policies.\nIt can be observed that PF provides the most unfair results as its utility function does not consider the delay. On the contrary, LDF gives strict priority to the delay, thus showing a clear over-fair behaviour, since the best users tend to have delays close to the average delay (i.e., unit normalized delay). M-LWDF provides intermediate results although it still presents an unfair behaviour, since the worst users tend to have high normalized delays in statistical terms. Finally, our proposed algorithm, $\\beta$-M-LWDF, is able to fulfill the delay requirement for most of the users. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.71\\columnwidth]{fig_cdf.pdf}\n\\caption{Comparison between the CDF of the normalized user delay of different resource allocation algorithms. The $\\beta$-M-LWDF considers $\\lambda=20\\%$ of the users as best users and $\\psi=10\\%$ of outliers.}\n\\label{fig:cdf}\n\\end{figure}\n\n\\begin{table}[t]\n\\scriptsize\n\\caption{Average Delay results of different resource allocation algorithms}\n\\label{tab:av_delay}\n\\begin{tabular}{|l|l|l|}\n\\noalign{\\hrule height 1pt}\n\\textbf{Algorithm} & \\textbf{Average delay (ms)} & \\textbf{Max. average delay (ms)}  \\\\ \n\\noalign{\\hrule height 1pt}\n    LDF & $163.2$ & $4192.7$ \\\\\n    M-LWDF & $37.3$ & $134.0$ \\\\\n    $\\beta$-M-LWDF & $53.6$ & $95.0$ \\\\\n    PF & $1228.3$ & $3964.5$ \\\\\n\\noalign{\\hrule height 0.5pt}\n\\end{tabular}\n\\end{table}\n\nIn table \\ref{tab:av_delay} it is shown the average delay, which is averaged in time and user domains, and the maximum average delay on time domain. \nIt is observed that LDF leads to the highest delay fairness, but also to the highest average delay.  This is due to the over-fair behaviour of such algorithm.\nIt is observed that M-LWDF achieves the smallest average delay. Nevertheless, the average delay of the worst user is clearly greater than with our $\\beta$-M-LWDF algorithm. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.71\\columnwidth]{fig_cdf_xi.pdf}\n\\caption{CDF of the normalized delay with $\\beta$-M-LWDF for different outliers percentages, $\\psi=\\{10 \\%, 15 \\%, 20 \\%\\}$.}\n\\label{fig:cdf_xi}\n\\end{figure}\n\n\\begin{table}[t]\n\\scriptsize\n\\caption{Percentage of time in different fairness cases}\n\\label{tab:percentage}\n\\begin{tabular}{|l|l|l|l|l|}\n\\noalign{\\hrule height 1pt}\n\\textbf{$\\psi$} & \\textbf{Av. delay (ms)} & FF time ($\\%$) & UF time ($\\%$) & OF time ($\\%$) \\\\ \n\\noalign{\\hrule height 1pt}\n    $10\\%$ & $53.60$ & $90.47$ & $6.60$ & $2.93$ \\\\\n    $15\\%$ & $47.03$ & $86.68$ & $12.91$ & $0.41$ \\\\\n    $20\\%$ & $39.76$ & $87,17$ & $12.83$ & $0.0$ \\\\\n\\noalign{\\hrule height 0.5pt}\n\\end{tabular}\n\\end{table}\n\nFig. \\ref{fig:cdf_xi} shows the CDF of normalized delay with $\\beta$-M-LWDF algorithm for different outliers percentages, $\\psi$. It can be observed that lower $\\psi$ values increase the delay fairness since the CDF tend to be more centered at the unit normalized delay ($w=1$). Nevertheless, reducing $\\psi$ also tends to increase the average delay as shown with table \\ref{tab:percentage}. Such a table represents the percentage of time that the system is on each of the fairness cases. It can be observed that the proposed algorithm achieves a high percentage of time on the desired FF case. \n\\vspace{-1mm}\n\\section{Conclusions}\n \\label{sec:conclusions}\nWe have proposed a novel framework based on deep RL to provide an adequate latency fairness. Our proposal includes a new scheduling policy, named as $\\beta$-M-LWDF, which is able to adjust instantaneously the allocation criteria based on the experienced delay of the users at each TTI. Simulation results show that our proposal ourperforms other well known scheduling solutions like PF, LDF or M-LWDF in terms of latency fairness and average delay. \n\n\\appendices\n\\section{}\\label{proof:remark}\nWhen $\\beta \\to \\infty$, \\eqref{eq:bmlwdf} can be expressed as follows\n\\begin{equation}\n\\small\n\\arg \\mathop {\\max}\\limits_{u,k} \\left\\{ \\lim\\limits_{\\beta \\to \\infty} {\\log(g_u)  + \\beta \\log \\left(W_u [n]\\right) + \\log \\left(r_{u,k} [n]\\right)} \\right\\}, \n\\end{equation}\n\\noindent where it has be used the fact that any strictly monotonic function does not change the result of the $\\arg \\max$ operator. Finally, the proof is completed after applying the following two facts: (i)  the limit when $\\beta \\to \\infty$ only depends on the term multiplied by $\\beta$, and (ii) any positive scalar that multiplies a function does not change the result of the $\\arg \\max$ operator.\n\n\n\\section{Introduction}\n\\IEEEPARstart{L}{ow} latency has been one of the key features that next wireless systems, such as 5G and beyond networks, are designed to fulfill in a wide range of scenarios. Nevertheless, this requisite is probably one of the most challenging from the quality of service (QoS) provision point of view. \n\nOn the one hand, achieving small block error rates with latencies of the order of few milliseconds requires the specific design of the radio interface functions, specially the radio scheduler. On the other hand, when the system is handling a traffic load close to its capacity, the packet delay can greatly increase leading to packet loss and an unacceptable latency. These two aspects pose an interesting challenge to the design of the scheduling algorithms that must minimize the average delay while providing user fairness. \n\nIn this sense, the use of hybrid control systems based on machine learning (ML) is considered as a promising approach that enables intelligent and adaptive decision making to the variant conditions of the network.\nIn \\cite{Pedersen18}, the essential role played by the radio scheduler in 5G is tackled. From a more theoretical point of view, reinforcement learning (RL) is presented in \\cite{Sutton14} as a promising solution that allows dealing with the dynamic radio environment and learning mapping functions over time. For instance, research advances described in \\cite{Comsa19} and \\cite{Comsa19bis} analyze the user fairness in terms of throughput for a modified proportional fair (PF) strategy, finding a balance between the system spectral efficiency and the fairness among users. However, to the best of the author's knowledge, user fairness in terms of delay has not been addressed yet. This is a critical issue in the emerging low latency services expected for 5G and beyond. Therefore, it is meaningful to investigate this issue and propose smart resource allocation algorithms that keep a balance between user fairness and system performance in terms of delay. \n\nIn this paper, an intelligent controller is proposed to customize a new version of the modified largest weighted delay first (M-LWDF) scheduling strategy \\cite{LDF} at each Transmission Time Interval (TTI). In particular, we propose a novel utility function called $\\beta$-M-LWDF that is able to adjust dynamically the experienced delay by means of a parameter $\\beta$ that is managed by the controller. It is important to remark that other scheduling algorithms such as largest delay first (LDF) \\cite{LDF}, PF and M-LWDF can be viewed as special cases of our proposed utility function for $\\beta \\to \\infty$, $\\beta = 0$ and $\\beta = 1$, respectively. Besides, with our proposed ML scheme, the $\\beta$ parameter is selected at each TTI to achieve an appropriate balance between user fairness and system performance in terms of delay. Such a balance is defined as a feasible-fair region on the cumulative distribution function (CDF) of the normalized delay.  This allows us to identify \\textit{unfair} states, where the difference of the delay between different users is excessive; \\textit{over-fair} states, where the difference is small, but the average delay increases; and \\textit{feasible-fair} states which reach an appropriate balance between latency fairness and average delay.  \nTo cope with the high  complexity of this approach, the controller implements a RL algorithm known as deep Q-learning (DQL) as introduced in \\cite{MNIH15}, which makes use of a neural network to estimate the Q-Function that approximates the best decisions at each TTI. \n\n\n\\section{System Model}\n\\label{sec:system_model}\nWe consider the downlink (DL) of the 5G NR (New Radio) system, which is based on Orthogonal Frequency Division Multiple Access (OFDMA). The available bandwidth is divided into equal resource blocks (RBs), where one RB consists on 12 sub-carriers and it represents the minimum resource unit that can be assigned in the frequency domain. The sub-carrier spacing (SCS) can be expressed as $\\Delta f = 2^\\mu 15$ kHz, where $\\mu$ is the numerology and it ranges from $0$ to $3$ for data channels.  The time domain resources are divided in slots, also named TTIs.\nIt is considered adaptive modulation and coding (AMC), and thus the transmission rate, i.e., the modulation and coding scheme (MCS), is chosen to maximize the throughput while guaranteeing a block error rate (BLER) below a target value, $\\mathrm{BLER}_T = 10 \\%$ \\cite{Martin21}. The set of MCSs are taken from 5G specs (Table 5.1.3.1-1 of 38.214 v16.7.0). \nLet $\\mathcal{B} = \\{b_1, b_2,\u2026, b_N\\}$ be the set of RBs for a given bandwidth, where $N$ represents the total number of RBs. Additionally, let $\\mathcal{U} = \\{u_1, u_2,\u2026, u_M\\}$ be the set of users to be scheduled.\nA constant bit rate (CBR) traffic model is assumed for each user where a packet sizes of $S_\\mathrm{CBR}$ bytes reach the buffer every $T_\\mathrm{CBR}$ seconds.  \nIt is assumed that the packets are stored in a unbounded buffer and the packets are segmented according to the transport block (TB) size that is determined by the scheduler and the selected MCS. \nThe average signal-to-noise ratio (SNR) of each user, $\\bar{\\gamma_u}$, is drawn randomly according to a Log-Normal distribution with expected value $\\mu_\\gamma$ and standard deviation $\\sigma_\\gamma$. In addition, the multi-path fading follows a realistic Tapped Delay Line Channel (TDL-A) model where the channel complex samples, \n$H_{u,k}[n]$, are correlated in time and frequency domains. Thus, the instantaneous SNR of user $u$ at RB $k$ and TTI $n$ is computed as $\\gamma_{u,k}[n] = \\bar{\\gamma_u} |H_{u,k}[n]|^2$.\n\n\n\\subsection{Scheduling algorithms}\n\\label{sec:alg}\nThe aim of the scheduler is to allocate each RB $b_i \\in B$ to a particular user $u_i \\in U$ at each TTI in order to meet a predefined QoS requirement. We have evaluated four different scheduling policies:\n\n\\begin{enumerate}\n    \\item \n    The well known Proportional Fair (PF) uses the following utility function at each TTI $n$, $U[n]$, as decision criterion to assign the user $\\widehat u$ to the RB $\\widehat k$ on the time instant $n$:\n\\begin{equation}\n\\label{eq:PF}\n\\small\n\\left\\{ {\\widehat u[n],\\widehat k[n]} \\right\\} = \\arg \\mathop {\\max }\\limits_{u,k} \\left\\{ {\\frac{{r_{u,k} [n]}}{{\\overline {r_u }[n] }}} \\right\\},\n\\end{equation}\n\\noindent \nwhere $r_{u,k}[n]$ represents the potential rate for user $u$ on the RB $k$ and TTI $n$, and $\\overline {r_u }[n]$ is a weighted moving average of the data rate values reached in previous TTIs, which is computed as \\cite{Musleh15}\n\\begin{equation}\n\\label{eq:r_PF}\n\\small\n\\overline {r_u } \\left[n \\right] = \\left( {1 - \\frac{1}{{T_{PF} }}} \\right)\\overline {r_u } \\left[ {n - 1} \\right] + \\frac{1}{{T_{PF} }}r_u \\left[ {n - 1} \\right],\n\\end{equation}\n\\noindent \nwhere $\\overline {r_u }[n-1]$ represents the weighted average data rate reached up to the TTI $n-1$, $r_u [n-1]$ is the data rate reached in TTI $n-1$, and $T_{PF}> 0$ is the average window size. \n\n    \\item \nLDF algorithm \\cite{LDF} is adaptive to the delay, providing the turn to the user that has been suffering the largest delay in its queue, as follows:\n\\begin{equation}\n\\small\n\\label{eq:LDF}\n\\widehat u[n] = \\arg \\mathop {\\max}\\limits_{u} \\left\\{ { W_u [n] } \\right\\}.\n\\end{equation}\n\\noindent \nwhere $W_u [n]$ is the delay experienced in the queues, which is computed as the sum of the delays of every packet that is stored in the queue. \n    \\item \nThe algorithm M-LWDF considers the waiting time in the queues, the instantaneous capacity of the channels and a parameter related to the delay tolerance \\cite{Andrews2002}:\n\\begin{equation}\n\\small\n\\label{eq:M-LWDF}\n\\left\\{ {\\widehat u[n],\\widehat k[n]} \\right\\} = \\arg \\mathop {\\max} \\limits_{u,k} \\left\\{ {g_u [n] \\cdot W_u [n] \\cdot r_{u,k} [n]} \\right\\},\n\\end{equation}\n\\noindent\nwhere $g_u [n] \\in [0, \\infty)$ is a QoS factor with a value of $g_u [n] = a_u \/ \\overline {r_u } [n]$, with $\\overline {r_u } [n]$ as defined in \\eqref{eq:r_PF} and $a_u  =  - \\log \\left( {\\delta _u } \\right) \\cdot T_u$, being $\\delta_u \\in [0, 1]$ the desired probability of fulfilling the delay requirement $T_u$ \\cite{Andrews2002}.\n\n\n    \\item \nWe propose a new version of the M-LWDF, called $\\beta$-M-LWDF, in which the impact of the delay term can be adjusted by a parameter $\\beta$ that is selected by the intelligent controller at each TTI:\n\\begin{equation}\n\\small\n\\label{eq:bmlwdf}\n\\left\\{ {\\widehat u[n],\\widehat k[n]} \\right\\} = \\arg \\mathop {\\max}\\limits_{u,k} \\left\\{ {g_u  \\cdot W_u^{\\beta} [n] \\cdot r_{u,k} [n]} \\right\\}.\n\\end{equation}\n\\end{enumerate}\n\\begin{remark}\n\\label{rem:special cases}\nThe utility functions of PF, LDF and M-LWDF that are given with \\eqref{eq:PF}, \\eqref{eq:LDF} and \\eqref{eq:M-LWDF} are special cases of the proposed $\\beta$-M-LWDF with $\\beta = 0$, $\\beta \\to \\infty$ and $\\beta = 1$, respectively. \n\\end{remark}\n\\begin{proof}\nIt is clear that $\\beta$-M-LWDF reduces to M-LWDF algorithm when $\\beta=1$. For $\\beta=0$, $\\beta$-M-LWDF reduces to PF as long as the parameter $a_u$ is equal for all users (as considered in this work). The proof for the case of LDF when $\\beta \\to \\infty$ is given in Appendix \\ref{proof:remark}. \n\\end{proof}\n\\vspace{-3mm}\n\\begin{remark}\n\\label{rem:beta behaviour}\nAs it is shown in \\cite{LDF}, LDF provides the highest fairness in terms of delay while sacrificing system performance (i.e., average delay), whereas M-LWDF achieves a higher system performance at the expense of fairness. This is due to the fact that LDF only accounts for the user delay whereas M-LWDF also accounts for the instantaneous rate. In view of remark \\ref{rem:special cases},  this involves that increasing $\\beta$ increases the fairness while decreasing $\\beta$ reduces the fairness in terms of delay. \n\\end{remark}\n\\section{Proposed RL Framework}\n\\label{sec:framework}\n\\subsection{RL Framework Description}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{RL_Framework.eps}\n\\caption{Proposed RL framework}\n\\label{fig:framework}\n\\end{figure}\nOur proposed RL framework is shown in Fig. \\ref{fig:framework}. At TTI $n$, the \\textit{channel} block generates the multi-path fading samples.\nThen the instantaneous SNR, $\\gamma_{u,k}[n]$, is computed. This SNR is used by the \\textit{AMC} block to select the MCS that maximizes the potential rate, $r_{u,k}[n]$, while keeping a BLER below $\\mathrm{BLER}_T$. On the other hand, packets are generated by the \\textit{traffic source} and stored in the \\textit{buffer} queue. This block performs segmentation of the stored packets according to the TB size determined by the \\textit{scheduler} and it computes the queue delay of each user $W_u[n]$.\nAfterwards, the scheduler updates its $\\beta[n]$ parameter according to the actions made by the agent and it allocates resources based on the proposed $\\beta$-M-LWDF utility function. Once the scheduler has performed the resource allocation, the \\textit{QoS} block estimates the empirical CDF of the normalized user delay, which is used to compute the reward and state variables. Those variables are used by the agent to make the next action in the following TTI. This action involves determining the step $\\Delta \\beta [n] \\in [-1,+1]$ to increase or decrease the beta parameter as $\\beta[n] = \\beta[n-1] + \\Delta \\beta [n]$.\n\\vspace{-3mm}\n\\subsection{Latency Fairness Criteria}\nThe Next Generation Mobile Networks (NGMN) Alliance defines an impartiality requirement in terms of throughput. This criteria is defined so that a system is seen as fair if at least the $(100-x)\\%$ of active users reach at least $x\\%$ of the normalized user data rate \\cite{NGNM}. \n\nIn this work we propose a RL framework intended to fulfill a predefined user fairness criteria in terms of latency. However, up to date there is no standardized fairness requirement in terms of delay. For that reason, in this paper we propose an impartiality requirement with the following $3$ desired features: i) half of the users have a delay smaller than the average; ii) there are no users with a delay $50\\%$ higher than the average; and iii) there are no users with a delay $50\\%$ smaller than the average. \nThese $3$ desired features lead to a CDF requirement as an straight line that passes through the points $(0.5, 0)$ and $(1.5, 1)$ as follows\n\\vspace{-3mm}\n\\begin{equation}\n\\small\n\\label{eq:CDF requirement}\ny = f_R(w) = \n    \\begin{cases}\n        1 & w > 1.5 \\\\\n        w - 0.5 & 0.5 \\leq w \\leq 1.5 \\\\\n        0 & w < 0.5\n    \\end{cases}\n\\end{equation}\n\\noindent \nwhere $w = f^{-1}_R (y) = y + 0.5$ stands for the inverse CDF requirement. \nIt can be noticed that such a CDF requirement  can be also read as $c\\%$ of the users to be below the $(50-c)\\%$ of the normalized user delay, $\\tilde{W}_u[n]$, which is defined as\n\\begin{equation}\n\\small\n\\label{eq:normalized delay}\n\\tilde{W}_u[n] = \\frac{W_u[n]}{\\frac{1}{M} \\sum_{u^\\prime = 1}^M W_{u^\\prime}[n]} \n\\end{equation}\n\nAs it can be noticed, $\\tilde{W}_u[n]=1$ involves that the delay of the user at TTI $n$ is equal to the average user delay. In a perfectly fair system all the users would have the same delay, and thus, all of them would have a unitary normalized delay.\n\nNevertheless, as it is discussed in \\cite{Sadr09, LDF} there is a trade-off between fairness and system performance. This involves that increasing the delay fairness is at the expense of increasing the average delay. For that reason, our proposed CDF requirement aims at identifying those cases where the fairness severely degrades the average delay, and those cases where the difference of delay between users is excessive. The former case is labeled as \\textit{over-fair (OF)} and the latter is labeled as \\textit{unfair (UF)}. Those cases that do not fall on the two aforementioned cases are labeled \\textit{feasible-fair (FF)}.  \nThe goal of the agent is to maximize the number of TTIs where the system is in FF state since this case leads to an appropriate balance between fairness and average delay. \n\nThe empirical CDF of the normalized delay at TTI $n$ is expressed below\n\\vspace{-3mm}\n\\begin{equation}\n\\small\n    \\label{eq:empirical CDF}\n    \\hat{F}_{\\tilde{W}[n]}(w) = \\frac{1}{M} \\sum_{u=1}^M \\mathbbm{1} \\left( \\tilde{W}_u[n] \\leq w  \\right) \n\\end{equation}\n\\noindent\nwhere $\\tilde{W}[n]$ represents a randomly chosen normalized user delay at TTI $n$. $ \\mathbbm{1}\\left( \\mathcal{E}  \\right)$ stands for the indicator function, which is $1$ if the event $\\mathcal{E}$ is true and $0$ otherwise. Let us represent the sorted vector of normalized delays as $\\mathbf{W}^\\uparrow[n]=\\left(W^\\uparrow_1[n], .., W^\\uparrow_M[n]\\right)$, where $W^\\uparrow_j[n]$ is the $j$-th delay in ascending order. Then, if we evaluate the empirical CDF from \\eqref{eq:empirical CDF} on $\\mathbf{W}^\\uparrow[n]$ we get $M$ equally spaced samples between $0$ and $1$ as follows $\\mathbf{Y}[n]=(Y_1[n],..,Y_M[n]) = \\hat{F}_{\\tilde{W}[n]}\\left(\\mathbf{W}^\\uparrow\\right[n])$ where $Y_j[n] = \\frac{j}{M} = \\hat{F}_{\\tilde{W}[n]}\\left(W^\\uparrow_j[n]\\right)$. Therefore, we can obtain a required normalized delay, $w^{(R)}_j$, for each of the sorted users as $w^{(R)}_j=f^{-1}_R \\left(\\frac{j}{M} \\right) = \\frac{j}{M} + 0.5$ with $j\\in [1,M] \\subset \\mathbb{N}$. A given sorted user $j$ is said to fulfill the delay requirement if $\\Delta W_j[n] = W^\\uparrow_j[n] - w^{(R)}_j \\leq \\xi$, where $\\xi$ is a confidence factor.\n\n\\begin{algorithm}\n\\small\n   \\hspace*{\\algorithmicindent} \\textbf{Input:} $\\mathbf{\\tilde{W}}[n]=\\left(\\tilde{W}_1[n], .., \\tilde{W}_M[n] \\right)$ \\\\\n   \\hspace*{\\algorithmicindent} \\textbf{Output:} $\\mathcal{C}[n]$, fairness case  \n\\begin{algorithmic}[1]   \n\\STATE   Sort the vector of normalized delays, $\\mathbf{\\tilde{W}}[n]$, to get $\\mathbf{W}^\\uparrow[n]$ \n   \\IF {$ \\left[ \\sum_{j=1}^{\\lceil \\lambda M \\rceil} \\mathbbm{1} \\left( W^\\uparrow_j > w^{(R)}_j + \\xi \\right) \\right] > 1$ }\n   {\n      \\STATE $\\mathcal{C}[n] = \\mathrm{OF}$ \n   }\n   \\ELSIF {$ \\left[ \\sum_{j= \\lceil \\lambda M \\rceil +1}^{M - \\lceil \\psi M \\rceil} \\mathbbm{1} \\left( W^\\uparrow_j > w^{(R)}_j + \\xi \\right) \\right] > 1$ }\n   {\n      \\STATE $\\mathcal{C}[n] = \\mathrm{UF}$  \n   }\n   \\ELSE\n   {\n       \\STATE $\\mathcal{C}[n] = \\mathrm{FF}$  \n   }\n   \\ENDIF\n\\end{algorithmic}\n\\caption{{Selection of fairness cases}}\n\\label{alg:case selection}\n\\end{algorithm}\n\nAlgorithm \\ref{alg:case selection} is used to identify in which of the three cases $\\mathcal{C}[n] \\in \\{\\mathrm{OF}, \\mathrm{UF}, \\mathrm{FF}\\}$ is the system working at a given TTI, $n$. It can be observed from line $2$ that the system is labeled as OF if any of the $\\lceil \\lambda M \\rceil$ best users (i.e., users with smallest delay) have a normalized delay greater than the required delay plus the confidence factor. The parameter $\\lambda \\in [0, 1]$ represents the percentage of best users. An OF situation means that the best users have a delay closer to the average delay, which increases the fairness. \nHowever, this happens at the expense of a significant reduction on the average delay.\n\nFrom line $4$ of algorithm \\ref{alg:case selection} it is observed that the system is labeled as UF if any of the worst users do not fulfill the delay requirement. The agent aims at avoiding this case in order to reduce the normalized delay of the worst users. \nAs it can be seen, the $\\lceil \\psi M \\rceil$ users with the greatest delays are not considered because they are treated as outliers, where $\\psi \\in [0, 1]$. This means that those users have such a poor channel condition that do not represent the conditions of the vast majority of users and thus, they are not considered.\nIf all the users (excluding the outliers) fulfill the delay requirement (including the factor $\\xi$), then the system is in FF case. \n\nFig. \\ref{fig:cdf_req} illustrates an example of the OF and UF cases by means of their estimated CDF of the normalized delay. As it is seen with algorithm \\ref{alg:case selection}, the criteria to select the fairness case involves checking if any of the best or worst users do not fulfill the delay requirement. Nevertheless, this criteria is equivalent to check if the empirical CDF at some TTI $n$ falls within the OF, UF or FF regions respectively. Those regions are shown in Fig. \\ref{fig:cdf_req} with blue, red and green colors. For instance, as it can be observed, the sorted user $u=36$, which makes the empirical CDF to reach $0.6$, has a normalized delay of $W_u^\\uparrow=1.466$, which is greater than the related CDF requirement plus the confidence factor, i.e., $w^{(R)}+\\xi = 1.1+0.1$. Since this user, and others that belong to the worst users set, do not fulfill the delay requirement, the system is labeled as UF in that case. \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.71\\columnwidth]{fig_cdf_req2.pdf}\n\\caption{Empirical CDF of the normalized delay for UF (red) and OF (blue) cases after defining $40\\%$ of the users as best users (i.e., $\\lambda=0.4$) and $10\\%$ of the users as outliers ($\\xi=0.1$) for a cell with $M=60$ users.}\n\\label{fig:cdf_req}\n\\end{figure}\n\n\\vspace{-3mm}\n\\subsection{States and Actions}\nLet $\\mathcal{S}$ be the state space and let $s[n] \\in \\mathcal{S}$ be the instantaneous state at TTI $n$. The state $s[n]$ can be seen as a union of two sub-states $s[n] = \\left(s_c[n], s_u[n] \\right)$, the controllable ($s_c[n]$) and uncontrollable ($s_u[n]$) sub-states. \nThe controllable sub-state is comprised of $s_c[n] = \\left(\\beta[n-1], {d}^{(\\mathrm{inf})}[n], {d}^{(\\mathrm{sup})}[n]\\right)$ where ${d}^{(\\mathrm{inf})}[n]$ and ${d}^{(\\mathrm{sup})}[n]$ represent the distance of the normalized delay to the requirement for the best and worst users sets as described in algorithm \\ref{alg:distance}.\n\n\\begin{algorithm}\n\\small\n   \\hspace*{\\algorithmicindent} \\textbf{Input:} $\\mathbf{\\Delta W}[n] = \\left(\\Delta W_1[n],.., \\Delta W_M[n]\\right)$  \\\\\n   \\hspace*{\\algorithmicindent} \\textbf{Output:} ${d}^{(\\mathrm{inf})}[n], {d}^{(\\mathrm{sup})}[n]$\n\\begin{algorithmic}[1]   \n\n   \\IF {$ \\mathcal{C}[n] = \\mathrm{OF}$ }\n   {\n      \\STATE ${d}^{(\\mathrm{inf})}[n] = \\max \\left(\\mathbf{\\Delta W}_{\\left\\{1:\\lceil \\lambda M \\rceil \\right\\}}[n] \\right)$  \n      \\STATE ${d}^{(\\mathrm{sup})}[n] = \\min \\left(\\mathbf{\\Delta W}_{\\{\\lceil \\lambda M \\rceil+1:M\\}}[n] \\right)$ \n   }\n   \\ELSIF {$ \\mathcal{C}[n] = \\mathrm{UF}$ }\n   {\n      \\STATE ${d}^{(\\mathrm{inf})}[n] = \\min \\left(\\mathbf{\\Delta W}_{\\{1:\\lceil \\lambda M \\rceil\\}}[n] \\right)$  \n      \\STATE ${d}^{(\\mathrm{sup})}[n] = \\max \\left(\\mathbf{\\Delta W}_{\\{\\lceil \\lambda M \\rceil+1:M\\}}[n] \\right)$  \n   }\n   \\ELSE\n   {\n      \\STATE ${d}^{(\\mathrm{inf})}[n] = \\min \\left(\\mathbf{\\Delta W}_{\\{1:\\lceil \\lambda M \\rceil\\}}[n] \\right)$\n      \\STATE ${d}^{(\\mathrm{sup})}[n] = \\max \\left(\\mathbf{\\Delta W}_{\\{\\lceil \\lambda M \\rceil+1:M\\}}[n] \\right)$  \n   }\n   \\ENDIF\n\\end{algorithmic}\n\\caption{{Selection of state values}}\n\\label{alg:distance}\n\\end{algorithm}\n\n\nFor the OF case $d^{\\rm (inf)}$ is the maximum of the distances of the best users while $d^{\\rm (sup)}$ is the minimum of the distances of the worst users. The reasoning of this assignment is to give more importance to the distances of worst users, since that set is the one that do not fulfill the requirement in the OF case. Similarly, in the UF case, the assignment using the $\\max()$ function is for the worst users, which are the users that do not fulfill the requirement in such a case. \n\nThe sub-state $s_u[n]$ is comprised of $\\scriptsize s_u[n] = \\Big( \\hat{\\mathbb{E}}\\left[\\mathbf{\\tilde{W}}[n]\\right], \\hat{\\mathbb{S}}\\left[\\mathbf{\\tilde{W}}[n]\\right], \\hat{\\mathbb{E}}\\Big[\\mathbf{I}_\\mathrm{MCS}[n]\\Big], \\hat{\\mathbb{S}}\\Big[\\mathbf{I}_\\mathrm{MCS}[n]\\Big] \\Big) $, where $\\scriptsize \\hat{\\mathbb{E}}\\left[\\bullet \\right]$ and $\\scriptsize \\hat{\\mathbb{S}}\\left[\\bullet \\right]$ stands for the empirical mean and standard deviation respectively. $\\mathbf{\\tilde{W}}[n]$ represents the vector of normalized delays of all users and \n$\\scriptsize \\mathbf{I}_\\mathrm{MCS}[n]= \\Big({I}_{\\mathrm{MCS},1}[n],..,{I}_{\\mathrm{MCS},M}[n] \\Big)$ is the vector of the MCS indexes reported by all users.\n\n\nWe have considered a discrete action space $\\mathcal{A} = \\Big\\{0,\\pm 10^{-4}, \\pm 10^{-3}, \\pm 10^{-2}, \\pm 5\\cdot10 ^ {- 2}, \\pm 10 ^ {-1}\\Big\\}$. At each time step $n$, the action taken by the agent will select the step size $\\Delta \\beta [n] \\in \\mathcal{A}$ that maximizes the expected cumulative reward.\n\n\\subsection{Reward Function}\nThe proposed reward function encourages the agent to stay in the FF case by taking into account two aspects: (a) the fairness state; and (b) the action taken by the agent. The value of the reward function is defined in (\\ref{eq:reward_general}), (\\ref{eq:reward_uf}) and (\\ref{eq:reward_of}):\n\\vspace{-2mm}\n\\begin{equation}\n\\small\n\\label{eq:reward_general}\n    {r}[n+1] = \n    \\begin{cases}\n      {r}_{\\mathrm{UF}}[n+1] & \\mathrm{if} \\;  \\mathcal{C}[n] = \\mathrm{UF}\\\\\n      1 & \\mathrm{if} \\;  \\mathcal{C}[n] = \\mathrm{FF}\\\\\n      {r}_{\\mathrm{OF}}[n+1] & \\mathrm{if} \\;  \\mathcal{C}[n] = \\mathrm{OF}\n    \\end{cases}\n\\end{equation}\n\\vspace{-2mm}\n\\begin{equation}\n\\small\n\\label{eq:reward_uf}\n  {r}_{\\mathrm{UF}}[n+1] = \n    \\begin{cases}\n      \\Delta\\beta[n] & \\mathrm{if} \\;  \\Delta \\beta[n] > 0\\\\\n      -1 & \\mathrm{if} \\; \\Delta\\beta[n] \\le 0\n    \\end{cases}\n\\end{equation}\n\\vspace{-2mm}\n\\begin{equation}\n\\small\n\\label{eq:reward_of}\n  {r}_{\\mathrm{OF}}[n+1] = \n    \\begin{cases}\n      -\\Delta\\beta[n] & \\mathrm{if} \\; \\Delta \\beta[n] < 0\\\\\n      -1 & \\mathrm{if} \\; \\Delta\\beta[n] \\ge 0\n    \\end{cases}\n\\end{equation}\nThe reasoning of \\eqref{eq:reward_general} is to encourage the agent to stay in FF state with a maximal positive reward (i.e., $1$). If the fairness case is UF, the agent should increase the $\\beta$ parameter to augment the fairness as per remark \\ref{rem:beta behaviour}. Thus, according to \\eqref{eq:reward_uf}, the reward is positive if $\\Delta \\beta[n]$ is positive, which means that the action made in previous instant was adequate; otherwise the rewards in negative to penalize the action of decreasing $\\beta$ that was made through a negative $\\Delta \\beta[n]$. Analogously, for OF case the agent is encouraged to decrease the fairness through the reward function as given with \\eqref{eq:reward_of}. \n\n\\section{Numerical Results and Discussions}\n \\label{sec:results}\n \n\nTo simulate the environment we have assumed a single cell scenario with $M =$ 60 active users.\nA detailed list of the network parameters is shown in Table \\ref{tab:net_param}.\n\n\n\\begin{table}[t]\n\\scriptsize\n\\caption{Network parameters setting}\n\\label{tab:net_param}\n\\begin{tabular}{|l|l|l|l|}\n\\noalign{\\hrule height 1pt}\n\\textbf{Parameter} & \\textbf{Value} & \\textbf{Parameter} & \\textbf{Value} \\\\ \n\\noalign{\\hrule height 1pt}\n $N$                   & $100$ RBs     &      Carrier frequency (GHz)                & $5$                       \\\\\nUser speed  (Km\/h)                     & $5$         & Delay Spread ($\\mu s$)                      & $100$                      \\\\\nTTI (ms) & $1$                   & Number of users                  & $60$                          \\\\\n$S_\\mathrm{CBR}$ (bytes)                    &  $850$           & $T_\\mathrm{CBR}$ (ms)                    &  $6$                             \\\\\n$\\mu_\\gamma$ (dB)                & $15$  & $\\sigma_\\gamma$ (dB) & $3$  \\\\\n$\\mathrm{BLER}_T$           & $0.1$                   & PF window size ($T_{PF}$)                & $100$ \\\\ \n$\\delta_u$ & 0.05& $T_u$ (ms) & $100$\\\\\n\\noalign{\\hrule height 0.5pt}\n\\end{tabular}\n\\end{table}\n\nThe agent implements a DQL with a  neural network comprised of $L$ layers, being $N_\\ell$ the number of nodes configured for each layer where  $\\ell \\in [1,L]$. It should be noted  that the number of nodes in both the input and output layers is fixed, corresponding to the dimensions of the state and action spaces respectively. Based on \\cite{Comsa19}, the configuration $\\{L = 3, N_2 = 60\\}$ is selected to find a balance between flexibility and complexity of our learning system. \nThe agent is trained with a Decayed $\\epsilon$-greedy policy. During the training stage the agent was trained during $2\\cdot 10^5$ steps (i.e. TTIs).\n\n\n\n\n\n\n\n\nFig. \\ref{fig:cdf} shows the CDF of the normalized user delay for different scheduling policies.\nIt can be observed that PF provides the most unfair results as its utility function does not consider the delay. On the contrary, LDF gives strict priority to the delay, thus showing a clear over-fair behaviour, since the best users tend to have delays close to the average delay (i.e., unit normalized delay). M-LWDF provides intermediate results although it still presents an unfair behaviour, since the worst users tend to have high normalized delays in statistical terms. Finally, our proposed algorithm, $\\beta$-M-LWDF, is able to fulfill the delay requirement for most of the users. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.71\\columnwidth]{fig_cdf.pdf}\n\\caption{Comparison between the CDF of the normalized user delay of different resource allocation algorithms. The $\\beta$-M-LWDF considers $\\lambda=20\\%$ of the users as best users and $\\psi=10\\%$ of outliers.}\n\\label{fig:cdf}\n\\end{figure}\n\n\\begin{table}[t]\n\\scriptsize\n\\caption{Average Delay results of different resource allocation algorithms}\n\\label{tab:av_delay}\n\\begin{tabular}{|l|l|l|}\n\\noalign{\\hrule height 1pt}\n\\textbf{Algorithm} & \\textbf{Average delay (ms)} & \\textbf{Max. average delay (ms)}  \\\\ \n\\noalign{\\hrule height 1pt}\n    LDF & $163.2$ & $4192.7$ \\\\\n    M-LWDF & $37.3$ & $134.0$ \\\\\n    $\\beta$-M-LWDF & $53.6$ & $95.0$ \\\\\n    PF & $1228.3$ & $3964.5$ \\\\\n\\noalign{\\hrule height 0.5pt}\n\\end{tabular}\n\\end{table}\n\nIn table \\ref{tab:av_delay} it is shown the average delay, which is averaged in time and user domains, and the maximum average delay on time domain. \nIt is observed that LDF leads to the highest delay fairness, but also to the highest average delay.  This is due to the over-fair behaviour of such algorithm.\nIt is observed that M-LWDF achieves the smallest average delay. Nevertheless, the average delay of the worst user is clearly greater than with our $\\beta$-M-LWDF algorithm. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.71\\columnwidth]{fig_cdf_xi.pdf}\n\\caption{CDF of the normalized delay with $\\beta$-M-LWDF for different outliers percentages, $\\psi=\\{10 \\%, 15 \\%, 20 \\%\\}$.}\n\\label{fig:cdf_xi}\n\\end{figure}\n\n\\begin{table}[t]\n\\scriptsize\n\\caption{Percentage of time in different fairness cases}\n\\label{tab:percentage}\n\\begin{tabular}{|l|l|l|l|l|}\n\\noalign{\\hrule height 1pt}\n\\textbf{$\\psi$} & \\textbf{Av. delay (ms)} & FF time ($\\%$) & UF time ($\\%$) & OF time ($\\%$) \\\\ \n\\noalign{\\hrule height 1pt}\n    $10\\%$ & $53.60$ & $90.47$ & $6.60$ & $2.93$ \\\\\n    $15\\%$ & $47.03$ & $86.68$ & $12.91$ & $0.41$ \\\\\n    $20\\%$ & $39.76$ & $87,17$ & $12.83$ & $0.0$ \\\\\n\\noalign{\\hrule height 0.5pt}\n\\end{tabular}\n\\end{table}\n\nFig. \\ref{fig:cdf_xi} shows the CDF of normalized delay with $\\beta$-M-LWDF algorithm for different outliers percentages, $\\psi$. It can be observed that lower $\\psi$ values increase the delay fairness since the CDF tend to be more centered at the unit normalized delay ($w=1$). Nevertheless, reducing $\\psi$ also tends to increase the average delay as shown with table \\ref{tab:percentage}. Such a table represents the percentage of time that the system is on each of the fairness cases. It can be observed that the proposed algorithm achieves a high percentage of time on the desired FF case. \n\\vspace{-1mm}\n\\section{Conclusions}\n \\label{sec:conclusions}\nWe have proposed a novel framework based on deep RL to provide an adequate latency fairness. Our proposal includes a new scheduling policy, named as $\\beta$-M-LWDF, which is able to adjust instantaneously the allocation criteria based on the experienced delay of the users at each TTI. Simulation results show that our proposal ourperforms other well known scheduling solutions like PF, LDF or M-LWDF in terms of latency fairness and average delay. \n\n\\appendices\n\\section{}\\label{proof:remark}\nWhen $\\beta \\to \\infty$, \\eqref{eq:bmlwdf} can be expressed as follows\n\\begin{equation}\n\\small\n\\arg \\mathop {\\max}\\limits_{u,k} \\left\\{ \\lim\\limits_{\\beta \\to \\infty} {\\log(g_u)  + \\beta \\log \\left(W_u [n]\\right) + \\log \\left(r_{u,k} [n]\\right)} \\right\\}, \n\\end{equation}\n\\noindent where it has be used the fact that any strictly monotonic function does not change the result of the $\\arg \\max$ operator. Finally, the proof is completed after applying the following two facts: (i)  the limit when $\\beta \\to \\infty$ only depends on the term multiplied by $\\beta$, and (ii) any positive scalar that multiplies a function does not change the result of the $\\arg \\max$ operator.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}