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+{"text":"\\section{#1}\\setcounter{equation}{0}}\n\\newcommand{\\ra}{\\rangle}\n\\newcommand{\\la}{\\langle}\n\\newcommand{\\p}{\\partial}\n\\newcommand{\\hp}{{\\Phi}}\n\\newcommand{\\hq}{{Q_B}}\n\\newcommand{\\he}{{\\eta_0}}\n\\newcommand{\\ha}{{{A}}}\n\\newcommand{\\rrr}{\\big\\rangle\\big\\rangle}\n\\newcommand{\\lllb}{\\Bigl\\langle\\Bigl\\langle}\n\\newcommand{\\rrrb}{\\Bigr\\rangle\\Bigr\\rangle}\n\\allowdisplaybreaks\n\n\\makeatother\n\n\\usepackage{babel}\n\\begin{document}\n{}~ \\hfill\\vbox{\\hbox{CTP-SCU\/2019008}}\\break\n\\vskip 3.0cm\n\\centerline{\\Large \\bf  Are nonperturbative AdS vacua possible in bosonic string theory?}\n\n\n\\vspace*{10.0ex}\n\\centerline{\\large Peng Wang, Houwen Wu and Haitang Yang}\n\\vspace*{7.0ex}\n\\vspace*{4.0ex}\n\\centerline{\\large \\it College of physics}\n\\centerline{\\large \\it Sichuan University}\n\\centerline{\\large \\it Chengdu, 610065, China} \\vspace*{1.0ex}\n\\vspace*{4.0ex}\n\n\n\\centerline{pengw@scu.edu.cn, iverwu@scu.edu.cn, hyanga@scu.edu.cn}\n\\vspace*{10.0ex}\n\\centerline{\\bf Abstract} \\bigskip \\smallskip\nIn this paper, following the work of Hohm and Zwiebach [arXiv:1905.06583], we show that in bosonic string theory nonperturbative anti-de Sitter (AdS) vacua could exist with all $\\alpha^{\\prime}$ corrections included. We also discuss the possibility of the coexistence of  nonperturbative dS and AdS vacua.\n\n\\vfill\n\\eject\n\\baselineskip=16pt\n\\vspace*{10.0ex}\n\nWhether bosonic string theory permits stable de Sitter (dS) or anti-de Sitter (AdS) vacua is a long-standing unsolved problem.\nThere are conjectures that superstring theory does not have solutions of dS vacua \\cite{Obied:2018sgi,Agrawal:2018own,Garg:2018reu}.\nAnother conjecture states that there is no stable nonsupersymmetric AdS vacuum with fluxes \\cite{Ooguri:2016pdq}. Incredibly, by analyzing the nonperturbative properties of the spacetime action of closed string theory (also known as the low-energy effective string theory), Hohm and Zwiebach \\cite{Hohm:2019ccp,Hohm:2019jgu}\nrecently showed that \\emph{nonperturbative} dS vacua are possible in bosonic string theory. The most important ingredients in their arguments are   $O(d,d)$ symmetry  and the classification of all of the $\\alpha'$ corrections for particular configurations.\n\nIt is well known from the work of  Meissner and Veneziano  \\cite{Veneziano:1991ek} in 1991 that, at the zeroth order of $\\alpha'$, when all  fields depend only on time, the $D=d+1$-dimensional spacetime action of closed string theory reduces to an $O(d,d)$-invariant \\emph{reduced action}. Soon after that, in Refs. \\cite{Sen:1991zi,Sen:1991cn} Sen extended this result  to full string field theory. Specifically, by considering an exact solution of the string field that was independent of $m$-dimensional spacetime coordinates ($m\\le d$)\\footnote{We concentrate on noncompact configurations here.} , Sen proved the following. (i) The space of such solutions has an $O(m,m)$ symmetry. In the language of low energy effective theory, the reduced action derived from such solutions   possesses an $O(m,m)$ symmetry to all orders in $\\alpha'$. (ii) The $m$ coordinates could be all spacelike or include one timelike coordinate, as explained  in Ref. \\cite{Sen:1991cn}. (iii) In the solution space, inequivalent solutions are connected by nondiagonal $O(m)\\otimes O(m)$ transformations [$O(m-1,1)\\otimes O(m-1,1)$ if one of the $m$ coordinates is timelike]. (iv) Other generators of $O(m,m)$ outside of the nondiagonal  $O(m)\\otimes O(m)$ [or $O(m-1,1)\\otimes O(m-1,1)$] generate gauge transformations accompanied by a shift of the dilaton, and thus equivalent solutions. On the other hand, in Ref. \\cite{Meissner:1991zj}, from the perspective of  $\\sigma$ model expansion, since the nilpotency of the BRST operator $Q$ is not altered by an $O(d,d)$ transformation, it was argued that the  $O(d,d)$ symmetry should persist at all orders in $\\alpha'$ for the reduced action. It is expected that, in terms of the standard fields, the $O(d,d)$ transformations  receive higher-order $\\alpha'$ corrections when introducing higher-derivative terms to the reduced action. For configurations depending only on time, to the first order in $\\alpha'$, in Ref. \\cite{Meissner:1996sa}, Meissner demonstrated that one can trade it with standard $O(d,d)$ transformations in terms of $\\alpha'$-corrected fields. In the appendix we show that this is also true for configurations that only depend on one spatial coordinate $x$, i.e., the case we study in this paper.\n\n\nAs for the yet unknown higher-order $\\alpha'$ corrections, some important progress has been made recently using the formalism of double field theory \\cite{Hohm:2013jaa,Hohm:2014xsa,Marques:2015vua,Hohm:2016lge,Hohm:2015doa,Baron:2018lve}. Remarkably, in Ref. \\cite{Hohm:2019jgu, Hohm:2015doa} Hohm and Zwiebach demonstrated that, for cosmological, purely time-dependent configurations, the $O(d,d)$-covariant closed string spacetime action can be expressed in a very simple form. All orders of  $\\alpha^{\\prime}$ corrections do not include\nthe trivial dilaton and can be constructed using even powers of\n$\\partial_{t}\\mathcal{S}$, where $\\mathcal{S}$ is the spatial part\nof the generalized metric defined in Eq. (\\ref{M}). This surprising simplification of the $\\alpha^{\\prime}$ corrections enabled them to discuss the nonperturbative solutions.  The most interesting result they obtained is that nonperturbative dS vacua are possible for bosonic string theory \\cite{Hohm:2019ccp,Hohm:2019jgu}, which possibly provides a cornerstone for the connection between string theory and our real world. In this paper, following their derivations, we show that nonperturbative AdS vacua are also possible with all  $\\alpha'$ corrections for bosonic string theory.\n\nIt is worth noting that we work in the string frame and not the Einstein frame, the same in Hohm and Zwiebach's work \\cite{Hohm:2019ccp,Hohm:2019jgu}. It is still unclear if there could be dS or AdS solutions in the Einstein frame, since when we substitute the\nsolutions with a constant dilaton field $\\phi$ and Hubble parameter\n$\\bar{H}_{0}$ back into the Einstein frame, $\\bar{H}_{0}^{E}$ goes to\nzero and the metric becomes flat.    Another issue is that in order to completely determine the dS\/AdS vacua, we still need to know all of the $\\alpha'$ corrections. One of the purposes of this paper is to deny the nonexistence of AdS vacua in bosonic string theory, rather than provide exact solutions.\n\n\nFor the sake of completeness, let us briefly summarize Hohm and Zwiebach's work on  nonperturbative dS vacua. Details can be found in Ref. \\cite{Hohm:2019jgu}. To the zeroth order of $\\alpha'$, the $D=d+1$-dimensional spacetime action of closed string theory is\n\n\n\n\n\n\n\n\n\\begin{equation}\nI_{0}\\equiv\\int d^{D}x\\sqrt{-g}e^{-2\\phi}\\left[R+4\\left(\\partial_{\\mu}\\phi\\right)^{2}-\\frac{1}{12}H_{ijk}H^{ijk}\\right],\n\\end{equation}\n\n\\noindent where $g_{\\mu\\nu}$ is the string metric, $\\phi$ is the\ndilaton and $H_{ijk}=3\\partial_{\\left[i\\right.}b_{\\left.jk\\right]}$\nis the field strength of the antisymmetric Kalb-Ramond $b_{ij}$\nfield. For cosmological backgrounds, choosing the synchronous\ngauge $g_{tt}=-1$, $g_{ti}=b_{t\\mu}=0$,\n\n\\begin{equation}\ng_{\\mu\\nu}=\\left(\\begin{array}{cc}\n-1 & 0\\\\\n0 & G_{ij}\\left(t\\right)\n\\end{array}\\right),\\qquad b_{\\mu\\nu}=\\left(\\begin{array}{cc}\n0 & 0\\\\\n0 & B_{ij}\\left(t\\right)\n\\end{array}\\right),\\qquad\\phi=\\phi\\left(t\\right),\n\\end{equation}\n\n\\noindent and defining the $O\\left(d,d\\right)$ dilaton $\\Phi$ as\n\n\\noindent\n\\begin{equation}\ne^{-\\Phi}=\\sqrt{g}e^{-2\\phi},\n\\end{equation}\n\n\\noindent the action can be rewritten as\n\n\\noindent\n\\begin{equation}\nI_{0}=\\int dte^{-\\Phi}\\left[-\\dot{\\Phi}^{2}-\\frac{1}{8}\\mathrm{Tr}\\left(\\dot{\\mathcal{S}}^{2}\\right)\\right],\\label{ST action}\n\\end{equation}\n\n\\noindent with\n\n\\noindent\n\\begin{equation}\nM=\\left(\\begin{array}{cc}\nG^{-1} & -G^{-1}B\\\\\nBG^{-1} & G-BG^{-1}B\n\\end{array}\\right),\\qquad\\mathcal{S}=\\eta M=\\left(\\begin{array}{cc}\nBG^{-1} & G-BG^{-1}B\\\\\nG^{-1} & -G^{-1}B\n\\end{array}\\right),\\label{M}\n\\end{equation}\n\n\\noindent where $M$, a $2d\\times 2d$ matrix, is the spatial part of the generalized metric $\\mathcal{H}$ of\ndouble field theory,   $\\dot{A}\\equiv\\partial_{t}A$, and $\\eta$ is the invariant metric of the $O\\left(d,d\\right)$\ngroup\n\n\\noindent\n\\begin{equation}\n\\eta=\\left(\\begin{array}{cc}\n0 & I\\\\\nI & 0\n\\end{array}\\right).\n\\end{equation}\n\n\n\n\\noindent Noticing that $M$ is symmetric and $\\mathcal{S}=\\mathcal{S}^{-1}$,  this action\nis manifestly invariant under the $O\\left(d,d\\right)$ transformations\n\n\\noindent\n\\begin{equation}\n\\Phi\\longrightarrow\\Phi,\\qquad\\mathcal{S}\\longrightarrow\\tilde{\\mathcal{S}}=\\Omega^{T}\\mathcal{S}\\Omega,\\label{O(d,d) trans}\n\\end{equation}\n\n\\noindent where $\\Omega$ is a constant matrix, satisfying\n\n\\noindent\n\\begin{equation}\n\\Omega^{T}\\eta\\Omega=\\eta.\n\\end{equation}\n\n\n\n\\noindent If we choose the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, $G_{ij}=\\delta_{ij}a^{2}\\left(t\\right)$, and a\nvanishing Kalb-Ramond field $B=0$:\n\n\\begin{equation}\nds^{2}=-dt^{2}+a^{2}\\left(t\\right)\\delta_{ij}dx^{i}dx^{j},\n\\label{FLRW}\n\\end{equation}\n\n\\noindent the matrix $\\mathcal{S}$ becomes\n\n\\noindent\n\\begin{equation}\n\\mathcal{S}=\\left(\\begin{array}{cc}\n0 & G\\\\\nG^{-1} & 0\n\\end{array}\\right).\n\\end{equation}\n\n\\noindent Applying the $O(d)\\otimes O(d)$ transformation, we\nobtain a new solution,\n\n\\begin{equation}\n\\tilde{\\mathcal{S}}=\\left(\\begin{array}{cc}\n0 & G^{-1}\\\\\nG & 0\n\\end{array}\\right),\n\\end{equation}\n\n\\noindent which implies that the action is invariant under $a\\left(t\\right)\\rightarrow a^{-1}\\left(t\\right)$, which is known as the scale-factor duality in traditional string\ncosmology. The next step is to include the $\\alpha^{\\prime}$\ncorrections. Benefiting from the $O\\left(d,d\\right)$ invariance, the corrections\nare classified by  even powers of $\\dot{\\mathcal{S}}$ only:\n\n\\begin{eqnarray}\nI & = & \\int d^{D}x\\sqrt{-g}e^{-2\\phi}\\left(R+4\\left(\\partial\\phi\\right)^{2}-\\frac{1}{12}H^{2}+\\frac{1}{4}\\alpha^{\\prime}\\left(R^{\\mu\\nu\\rho\\sigma}R_{\\mu\\nu\\rho\\sigma}+\\ldots\\right)+{\\alpha'}^2(\\ldots)+\\ldots \\right),\\label{eq:original action with alpha}\\\\\n & = & \\int dte^{-\\Phi}\\left(-\\dot{\\Phi}^{2}+ {\\sum_{k=1}^{\\infty}}\\left(\\alpha^{\\prime}\\right)^{k-1}c_{k}\\mathrm{tr}\\left(\\dot{\\mathcal{S}}^{2k}\\right)\\right).\\label{eq:Odd action with alpha}\n\\end{eqnarray}\n\n\\noindent Eq. (\\ref{eq:original action with alpha}) is the action for the general background\nwith all $\\alpha^{\\prime}$ corrections. Eqn (\\ref{eq:Odd action with alpha})  is\nthe $O\\left(d,d\\right)$-covariant action  applied to the metric (\\ref{FLRW}) with  $B=0$,\nwhere $c_{1}=-\\frac{1}{8}$\nto recover Eq. (\\ref{ST action}) and $c_{k\\geq2}$ are\nundetermined constants. Using the action   (\\ref{eq:Odd action with alpha}), in Refs. \\cite{Hohm:2019ccp,Hohm:2019jgu} Hohm and Zwiebach showed that nonperturbative\ndS vacua are permitted for infinitely many classes of  $c_{k\\geq2}$, namely, $a(t)=e^{H_0 t}$ with $H_0\\not= 0$.\n\nNow we want to investigate if nonperturbative AdS vacua are also allowed. To address this question, an appropriate ansatz is crucial. We take the ansatz\n\n\\begin{equation}\nds^{2}=-a^{2}\\left(x\\right)dt^{2}+dx^{2}+a^{2}\\left(x\\right)\\left(dy^{2}+dz^{2}+\\ldots\\right),\\label{eq:BH metric}\n\\end{equation}\n\n\\noindent whose metric components depend on a single space direction, say, $x$. The dimensionality is still $D=d+1$. As we explained earlier, in Refs. \\cite{Sen:1991zi,Sen:1991cn}  Sen proved that the reduced action based on such solutions also maintains $O(d,d)$ symmetry to all orders in $\\alpha'$. The coset for the space of such solutions is $O(d-1,1)\\otimes O(d-1,1)\/O(d-1,1)$, in contrast to $O(d)\\otimes O(d)\/O(d)$ for the cosmological solutions (\\ref{FLRW}). In the Appendix we explicitly show that for this ansatz the spacetime action (\\ref{eq:original action with alpha}) also possesses the standard $O(d,d,R)$ symmetry and can be reduced to\n\n\\begin{equation}\n\\bar{I}= -\\int dxe^{-\\Phi}\\left(-\\Phi^{\\prime2}+ {\\sum_{k=1}^\\infty}\\left(\\alpha^{\\prime}\\right)^{k-1}\\bar{c}_{k}\\mathrm{tr}\\left(\\mathcal{M}^{\\prime2k}\\right)\\right),\\label{eq: Odd with alpha x}\n\\end{equation}\n\n\\noindent where $A^{\\prime}\\equiv\\partial_{x}A$. Note the overall minus sign and the fact that we have a new set of undetermined coefficients $\\bar c_k$ other than the $c_k$'s in Eq. (\\ref{eq:Odd action with alpha}).  It turns out that\n\n\\begin{equation}\n\\bar{c}_{2k-1}=c_{2k-1},\\quad\\quad \\bar{c}_{2k}=-c_{2k},\\quad\\quad \\mathrm{for}\\quad k=1,2,3\\ldots \\label{eq:coeff relation}\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\n\\mathcal{M}=\\left(\\begin{array}{cccc}\n0 & 0 & -a^{2}\\left(x\\right) & 0\\\\\n0 & 0 & 0 & a^{2}\\left(x\\right)\\delta_{ij}\\\\\n-a^{-2}\\left(x\\right) & 0 & 0 & 0\\\\\n0 & a^{-2}\\left(x\\right)\\delta_{ij} & 0 & 0\n\\end{array}\\right),\n\\end{equation}\n\n\\noindent where $i,j=y,z,\\ldots$. The equations of motion (EOM) of (\\ref{eq: Odd with alpha x}) can be calculated directly,\n\n\\begin{eqnarray}\n\\Phi^{\\prime\\prime}+\\frac{1}{2}\\bar{H}\\bar{f}\\left(\\bar{H}\\right) & = & 0,\\nonumber \\\\\n\\frac{d}{dx}\\left(e^{-\\Phi}\\bar{f}\\left(\\bar{H}\\right)\\right) & = & 0,\\nonumber \\\\\n(\\Phi^{\\prime})^2+\\bar{g}\\left(\\bar{H}\\right) & = & 0,\\label{eq:EOM}\n\\end{eqnarray}\n\n\\noindent where\n\n\\begin{eqnarray}\n\\bar{H}\\left(x\\right) & = & \\frac{a^{\\prime} \\left(x\\right)}{a\\left(x\\right)},\\nonumber \\\\\n\\bar{f}\\left(\\bar{H}\\right) & = & d{\\sum_{k=1}^\\infty}\\left(-\\alpha^{\\prime}\\right)^{k-1} 2^{2\\left(k+1\\right)}k\\bar{c}_{k}\\bar{H}^{2k-1},\\nonumber \\\\\n\\bar{g}\\left(\\bar{H}\\right) & = & d {\\sum_{k=1}^\\infty} \\left(-\\alpha^{\\prime}\\right)^{k-1} 2^{2k+1}\\left(2k-1\\right)\\bar{c}_{k} \\bar{H}^{2k}.\\label{eq:EOM fh gh}\n\\end{eqnarray}\n\n\\noindent It is easy to see that $\\bar g'(\\bar H)=\\bar H \\bar f'(\\bar H)$. Note that $\\bar{H}\\left(x\\right)$ is not the Hubble parameter since our background is space dependent.\n\nNow, let us check whether there is a solution $a^{2}\\left(x\\right)=e^{2\\bar{H}_{0}x}$\nfor the EOM (\\ref{eq:EOM}) such that\n\n\\begin{equation}\nds^{2}=-e^{2\\bar{H}_{0}x}dt^{2}+dx^{2}+e^{2\\bar{H}_{0}x}\\left(dy^{2}+dz^{2}+\\ldots\\right).\\label{eq:ads solution}\n\\end{equation}\n\n\\noindent The scalar curvature of this metric is\n\n\\begin{equation}\nR=-D\\left(D-1\\right)\\bar{H}_{0}^{2},\n\\label{eq:Ricci Scalar}\n\\end{equation}\n\n\\noindent which implies that the metric (\\ref{eq:ads solution}) is\nan AdS background for constant  $\\bar{H}_{0}\\neq0$. To see this more clearly, we apply the transformation $x \\to - \\log[\\bar H_0 \\xi]\/\\bar H_0$ and recover the familiar Poincare coordinate\n\n\\begin{equation}\nds^2 = \\frac{1\/\\bar H_0^2}{\\xi^2}\\big(-dt^2 +d\\xi^2 + dy^2+dz^2+\\ldots\\big).\n\\end{equation}\nSo, we have  $\\bar H_0=1\/R_{AdS}$, consistent with Eq.(\\ref{eq:Ricci Scalar}). If we do not include the $\\alpha^{\\prime}$ corrections,\n$\\bar{f}\\left(\\bar{H}\\right)\\sim \\bar H_0$ is a constant. From the second equation\nof (\\ref{eq:EOM}), we can figure out that $\\Phi$ is a constant.\nTherefore, to satisfy the first equation of (\\ref{eq:EOM}), we must have\n$\\bar{H}_{0}=0$, and thus the metric (\\ref{eq:ads solution}) becomes\nflat and there is no AdS solution. One thus concludes that there\nis no $D$-dimensional AdS vacuum without fluxes and $\\alpha^{\\prime}$\ncorrections.\n\nOur aim is to search for solutions with constant $\\bar{H}_{0}\\neq0$.  Considering\nthe effects of $\\alpha^{\\prime}$ corrections, if there is a nonvanishing\n$\\bar{H}_{0}$ solution, $\\bar{f}\\left(\\bar{H}_{0}\\right)$ is\na constant and then $\\Phi$ is also a constant from the second equation\nof (\\ref{eq:EOM}). Finally, from the first and third equations of (\\ref{eq:EOM}), we  obtain the condition for a nonvanishing $\\bar{H}_{0}$ solution:\n\n\\begin{equation}\n\\bar{f}\\left(\\bar{H}_{0}\\right)=\\bar{g}\\left(\\bar{H}_{0}\\right)=0.\\label{eq:fh condition}\n\\end{equation}\n\n\nLet us determine the general form of $\\bar{f}\\left(\\bar{H}\\right)$ with specific choices for $c_{k\\geq2}$\nthat satisfy the condition (\\ref{eq:fh condition}). Instead of  $\\bar{f}\\left(\\bar{H}\\right)$,\nit is better to consider its integral,\n\n\\begin{equation}\n\\bar{F}\\left(\\bar{H}\\right)\\equiv\\int_{0}^{\\bar{H}}f\\left(\\bar{H}^{\\prime}\\right)d\\bar{H}^{\\prime}.\n\\end{equation}\n\n\\noindent The condition $\\bar{f}\\left(\\bar{H}_{0}\\right)=\\bar{g}\\left(\\bar{H}_{0}\\right)=0$\nis replaced by\n\n\\begin{equation}\n\\bar{F}\\left(\\bar{H}_{0}\\right)=\\bar{F}^{\\prime}\\left(\\bar{H}_{0}\\right)=0.\n\\end{equation}\n\n\\noindent It is then easy to understand that the general form\n\n\\begin{equation}\n\\bar{F}\\left(\\bar{H}\\right)=-d\\bar{H}^{2}\\left(1+ {\\sum_{p=1}^{\\infty}}\\bar{d}_{p}\\left(\\alpha^{\\prime}\\right)^{p} \\bar{H}^{2p}\\right){\\prod_{i=1}^k}\\left(1-\\left(\\frac{\\bar{H}}{\\bar{H}_{0}^{\\left(i\\right)}}\\right)^{2}\\right)^{2}, \\label{eq:Fform}\n\\end{equation}\n\n\\noindent admits  $2k$ solutions of AdS vacua: $\\pm\\bar{H}_{0}^{\\left(1\\right)},\\ldots,\\pm\\bar{H}_{0}^{\\left(k\\right)}$, for an arbitrary integer $k>0$. So, the question is: do the coefficients $\\bar c_k$ support the functional form (\\ref{eq:Fform}) ?  Although it appears impossible to obtain a definite answer, the bottom line is that  $\\alpha^{\\prime}$ corrections do support the possibility of nonperturbative AdS vacua.  It is worth noting that ``nonperturbative''\nhere means that we  use all $\\alpha^{\\prime}$ corrections to\nobtain the solution but not to obtain the solution from the two-derivative\nequations and then be $\\alpha^{\\prime}$ corrected.\nThere  exists another ``stronger'' version of ``nonperturbative'', namely\n$\\bar{F}\\left(\\bar{H}\\right)$ cannot be expressed by a series expansion of $\\alpha'$  \\cite{Krishnan:2019mkv}. The same scenario occurs for the dS vacua as explained in Refs. \\cite{Hohm:2019ccp,Hohm:2019jgu}.\n\n\nHowever, the real story may   be intriguing. As an illustration, let us assume that all orders of $\\alpha^{\\prime}$ corrections  have a very special form that gives\n\n\\begin{equation}\n\\bar{f}\\left(\\bar{H}\\right)=-\\frac{2d}{\\sqrt{\\alpha^{\\prime}}}\\sin\\left(\\sqrt{\\alpha^{\\prime}}\\bar{H}\\right)=-2d {\\sum_{k=1}^{\\infty}}\\left(\\alpha^{\\prime}\\right)^{k-1}\\frac{1} {\\left(2k-1\\right)!}H^{2k-1}.\\label{eq:fh sin}\n\\end{equation}\n\n\\noindent This functional form is a valid candidate for Eq. (\\ref{eq:EOM fh gh}) for\nspecial choices of $\\bar c_{k\\geq2}$. It includes all orders of $\\alpha'$ corrections and evidently is nonperturbative.\nThe solutions  satisfying the condition $\\bar{f}\\left(\\bar{H}_{0}\\right)=0$ are\n\\begin{equation}\n\\sqrt{\\alpha^{\\prime}}\\bar{H}_{0}=2\\pi,4\\pi,\\ldots\n\\end{equation}\n\n\\noindent It is easy to check that $\\bar{g}\\left(\\bar{H}_{0}\\right)=0$ is also satisfied for these solutions, leading to a discrete infinity of AdS vacua. However, note the coefficient relations (\\ref{eq:coeff relation}) between dS from Eq. (\\ref{eq:Odd action with alpha}) and AdS from Eq. (\\ref{eq: Odd with alpha x}):\n$\\bar{c}_{2k+1}=c_{2k+1}$, $\\bar{c}_{2k}=-c_{2k}$. We can immediately see that if $\\bar{f}\\left(\\bar{H}\\right)\\sim\\sin\\left(\\sqrt{\\alpha^{\\prime}}\\bar{H}\\right)$ for AdS, then the corresponding function\n$f\\left(H\\right)\\sim\\sinh\\left(\\sqrt{\\alpha^{\\prime}}H\\right)$ in dS, and vice versa. But the $sinh$ function has no nontrivial zero. So,  for the trial function (\\ref{eq:fh sin}), AdS or dS vacua cannot coexist and only one of them survives.\n\n\n\nThis looks like merely a coincidence since\nin any case, one could use a general form of Eq. (\\ref{eq:Fform}) to\npermit  AdS or dS vacua. But we have some reasons to conjecture that by plugging the dS (AdS) metric into the yet unknown\ninfinite $\\alpha^{\\prime}$ expansion, one could sum the series\ninto an expression including a  factor that is very close to the trial function of Eq. (\\ref{eq:fh sin})\n\\footnote{We want to emphasize that the real functional form of $\\bar{f}\\left(\\bar{H}\\right)$ could be more complicated than Eq. (\\ref{eq:fh sin}). We simply use this toy model to discuss the coexistence of nonperturbative dS and AdS vacua.}. In Ref. \\cite{Wang:2017mpm}, we showed that,\nwhen expressed in Riemann normal coordinates, the AdS (dS) metric\ncan be expressed in a simple form, which is called the $J$-factor by some mathematicians.\nTo see this explicitly, by considering the nonlinear sigma model of\nstring theory\n\n\\begin{equation}\nS=-\\frac{1}{4\\pi\\alpha'}\\int_{\\Sigma}g_{ij}(X)\\partial_{\\alpha}X^{i}\\partial^{\\alpha}X^{j},\n\\end{equation}\n\n\\noindent we can expand $X^{i}$ at some point $\\bar{x}$, say, $X^{i}\\left(\\tau,\\sigma\\right)=\\bar{x}^{i}+\\sqrt{\\alpha^{\\prime}}\\mathbb{Y}^{i}\\left(\\tau,\\sigma\\right)$,\nwhere the $\\mathbb{Y}^{i}$'s are dimensionless fluctuations. Locally\naround any point, one can always pick Riemann normal coordinates\n\n\\begin{eqnarray}\ng_{ij}\\left(X\\right) & = & \\eta_{ij}+\\frac{\\ell_{s}^{2}}{3}R_{iklj}\\mathbb{Y}^{k}\\mathbb{Y}^{l}+\\frac{\\ell_{s}^{3}}{6}D_{k}R_{ilmj}\\mathbb{Y}^{k}\\mathbb{Y}^{l}\\mathbb{Y}^{m}\\nonumber \\\\\n &  & +\\frac{\\ell_{s}^{4}}{20}\\left(D_{k}D_{l}R_{imnj}+\\frac{8}{9}R_{iklp}R_{\\;mnj}^{p}\\right)\\mathbb{Y}^{k}\\mathbb{Y}^{l}\\mathbb{Y}^{m}\\mathbb{Y}^{n}+\\ldots.\n\\end{eqnarray}\n\n\\noindent When the background is  maximally symmetric, the expansion is greatly simplified and can\nbe summed  into a closed form.  For dS, we have\n\n\\begin{equation}\nS_{dS}=-\\frac{1}{4\\pi}\\int_{\\Sigma}\\partial\\mathbb{Y}^{i}\\partial\\mathbb{Y}^{j}\\left[\\frac{\\sin^{2}\\left(\\frac{\\sqrt{\\alpha'}}{R_{dS}}\\mathbb{W}\\right)}{\\left(\\frac{\\sqrt{\\alpha'}}{R_{dS}}\\mathbb{W}\\right)^{2}}\\right]^{a}\\,_{i}\\,\\eta_{aj}\\,,\\qquad\\left(\\mathbb{W}^{2}\\right)_{\\quad b}^{a}\\equiv\\delta_{b}^{a}\\mathbb{Y}^{2}-\\mathbb{Y}^{a}\\mathbb{Y}_{b}.\n\\end{equation}\n\n\\noindent If the background is AdS, we get\n\n\\begin{equation}\nS_{AdS}=-\\frac{1}{4\\pi}\\int_{\\Sigma}\\partial\\mathbb{Y}^{i}\\partial\\mathbb{Y}^{j}\\left[\\frac{\\sinh^{2}\\left(\\frac{\\sqrt{\\alpha'}}{R_{AdS}}\\mathbb{W}\\right)}{\\left(\\frac{\\sqrt{\\alpha'}}{R_{AdS}}\\mathbb{W}\\right)^{2}}\\right]^{a}\\,_{i}\\,\\eta_{aj}\\,,\\qquad\\left(\\mathbb{W}^{2}\\right)_{\\quad b}^{a}\\equiv\\delta_{b}^{a}\\mathbb{Y}^{2}-\\mathbb{Y}^{a}\\mathbb{Y}_{b}.\n\\end{equation}\n\n\\noindent Noting that $H_{0}\\sim1\/R_{dS}$ and $\\bar{H}_{0}\\sim1\/R_{AdS}$,\nthe results strongly suggest that the beta functions or EOMs of\nthese two actions $S_{dS}$ and $S_{AdS}$ may behave very similarly to $f\\left(H\\right)\\sim\\sin\\left(\\sqrt{\\alpha^{\\prime}}H\\right)$\nand $\\bar{f}\\left(\\bar{H}\\right)\\sim\\sinh\\left(\\sqrt{\\alpha^{\\prime}}\\bar{H}\\right)$,\nor, equivalently speaking, there are  nonperturbative dS vacua but not\nnonperturbative AdS vacua, or vice versa. So it looks like we still need more information about the $\\alpha'$ corrections to give a definite answer.\n\nFinally, we wish to remark that we have only considered the string\nmetric. The relation between the Einstein metric $g_{\\mu\\nu}^{E}$ and\nstring metric $g_{\\mu\\nu}$ is $g_{\\mu\\nu}^{E}=e^{-\\frac{4\\phi}{D-2}}g_{\\mu\\nu}$.\nWhen we substitute our solution with a constant $\\phi$ and $\\bar{H}_{0}= 0$\nback into the Einstein frame, $\\bar{H}_{0}^{E}$  goes to zero and the metric\nbecomes flat. This implies that there is no dS or AdS vacuum when $\\phi$\nis a constant in the Einstein frame without $\\alpha'$ corrections.\n\n\\vspace{5mm}\n\n\\noindent {\\bf Acknowledgements}\nWe are deeply indebted to Olaf Hohm and Barton Zwiebach for illuminating discussions and advice.  We are also  grateful to Hiroaki Nakajima, Bo Ning, Shuxuan Ying for very helpful discussions and suggestions. This work is supported in part by the National Natural Science Foundation of China (Grants No. 11875196, 11375121 and 11005016).\n\n\n\n\\section*{Appendix}\n\nThis appendix has two purposes. The first is to explicitly show that, at the leading order in $\\alpha'$, for our ansatz (\\ref{eq:BH metric}) the $O(d,d)$ symmetry of the spacetime action can be expressed in the standard form in terms of $\\alpha'$-corrected fields. The derivations follow the same pattern  as the calculations in Refs. \\cite{Veneziano:1991ek,Meissner:1996sa}, except for some minus signs in particular places that account for the difference between time and space coordinates.\n\nThe second purpose is to briefly demonstrate that, based on our ansatz,  the closed string spacetime action reduces to Eqs. (\\ref{eq: Odd with alpha x}-\\ref{eq:coeff relation}). The derivations are completely parallel to those in Ref. \\cite{Hohm:2019jgu}. Extra minus signs show up in the coefficients $c_k$ of the $\\alpha'$ expansion.\n\n\n\n\n\\subsection*{Zeroth order  of $\\alpha^{\\prime}$}\n\n\\noindent We start with the tree-level closed string spacetime action without\n$\\alpha^{\\prime}$ corrections\n\n\n\\begin{equation}\nI_{0}\\equiv\\int d^{D}x\\sqrt{-g}e^{-2\\phi}\\left[R+4\\left(\\partial_{\\mu}\\phi\\right)^{2}-\\frac{1}{12}H_{ijk}H^{ijk}\\right],\\label{eq:Polyakov}\n\\end{equation}\n\n\\noindent where $g_{\\mu\\nu}$ is the string metric, $\\phi$ is the\ndilaton and $H_{ijk}=3\\partial_{\\left[i\\right.}b_{\\left.jk\\right]}$\nis the field strength of the antisymmetric Kalb-Ramond field $b_{ij}$.\nThe ansatz  we use is\n\n\\begin{equation}\nds^{2}=-a^{2}\\left(x\\right)dt^{2}+dx^{2}+a^{2}\\left(x\\right)\\left(dy^{2}+dz^{2}+\\ldots\\right),\\quad b_{x\\mu}=0,\\label{eq:our ansatz}\n\\end{equation}\n\n\\noindent or\n\n\\begin{equation}\ng_{\\mu\\nu}=\\left(\\begin{array}{ccc}\n-a^{2}\\left(x\\right) & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & a^{2}\\left(x\\right)\\delta_{ab}\n\\end{array}\\right),\\qquad b_{\\mu\\nu}=\\left(\\begin{array}{ccc}\n0 & 0 & b_{0b}\\left(x\\right)\\\\\n0 & 0 & 0\\\\\nb_{a0}\\left(x\\right) & 0 & b_{ab}\\left(x\\right)\n\\end{array}\\right),\\qquad\\phi=\\phi\\left(x\\right),\n\\label{eq:our ansatz matrix}\n\\end{equation}\n\n\\begin{comment}\n\\begin{equation}\nb_{\\mu\\nu}=\\left(\\begin{array}{cccc}\n0 & 0 & 0 & 0\\\\\n0 & 0 & b_{23}\\left(t\\right) & b_{24}\\left(t\\right)\\\\\n0 & b_{32}\\left(t\\right) & 0 & b_{34}\\left(t\\right)\\\\\n0 & b_{42}\\left(t\\right) & b_{43}\\left(t\\right) & 0\n\\end{array}\\right)\n\\end{equation}\n\n\\begin{equation}\nb_{\\mu\\nu}=\\left(\\begin{array}{cccc}\n0 & 0 & b_{13}\\left(x\\right) & b_{14}\\left(x\\right)\\\\\n0 & 0 & 0 & 0\\\\\nb_{12}\\left(x\\right) & 0 & 0 & b_{34}\\left(x\\right)\\\\\nb_{12}\\left(x\\right) & 0 & b_{43}\\left(x\\right) & 0\n\\end{array}\\right)\n\\end{equation}\n\\end{comment}\n\n\\noindent where $a,b=2,3,\\ldots$ Mimicking the metric of cosmological backgrounds, we choose $b_{x\\mu}=0$  in our ansatz.\nIt turns out that this gauge is crucial to preserving the $O(d,d)$ symmetry. In order to obtain the reduced\naction by using the ansatz (\\ref{eq:our ansatz matrix}), we rotate between the time-like $t$ and the first space-like\n$x$ directions and rewrite the metric and $b_{\\mu\\nu}$ as\n\n\\begin{equation}\ng_{\\mu\\nu}=\\left(\\begin{array}{cc}\n1 & 0\\\\\n0 & G_{ij}\\left(x\\right)\n\\end{array}\\right),\\qquad b_{\\mu\\nu}=\\left(\\begin{array}{cc}\n0 & 0\\\\\n0 & B_{ij}\\left(x\\right)\n\\end{array}\\right),\\label{eq:set up 1}\n\\end{equation}\n\n\\noindent where\n\n\\begin{equation}\nG_{ij}\\left(x\\right)\\equiv\\left(\\begin{array}{cc}\n-a^{2}\\left(x\\right) & 0\\\\\n0 & a^{2}\\left(x\\right)\\delta_{ab}\n\\end{array}\\right),\\qquad B_{ij}\\left(x\\right)\\equiv\\left(\\begin{array}{cc}\n0 & b_{0b}\\left(x\\right)\\\\\nb_{a0}\\left(x\\right) & b_{ab}\\left(x\\right)\n\\end{array}\\right).\\label{eq:set up 2}\n\\end{equation}\n\n\\noindent So $g_{00}\\equiv g_{xx}$, $g_{11}\\equiv g_{tt}$ and $b_{00}\\equiv b_{xx}$, $b_{11}\\equiv b_{tt}$. Henceforth, we will use Eqs. (\\ref{eq:set up 1}) and (\\ref{eq:set up 2}) as the definitions for $g_{\\mu\\nu}$ and $b_{\\mu\\nu}$.\n\nSince we only need to use $G_{ij}$ as a whole to discuss the $O(d,d)$ symmetry and not its components, the time-like minus sign  $G_{11}\\left(x\\right)=-a^{2}\\left(x\\right)$ does not show up until we calculate the reduced action.   Straightforwardly, the Ricci tensor is\n\n\\begin{eqnarray}\nR_{x}^{\\;\\;x} & = & -\\frac{1}{4}\\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right)^{2}-\\frac{1}{2}\\mathrm{Tr}\\left(G^{-1}G^{\\prime\\prime}\\right)-\\frac{1}{2}\\mathrm{Tr}\\left(G^{\\prime}{}^{-1}G^{\\prime}\\right),\\nonumber \\\\\nR_{t}^{\\;\\;t} & = & -\\frac{1}{2}\\left(G^{-1}G^{\\prime\\prime}\\right)_{t}^{\\;\\;t}-\\frac{1}{4}\\left(G^{-1}G^{\\prime}\\right)_{t}^{\\;\\;t}\\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right)+\\frac{1}{2}\\left(G^{-1}G^{\\prime}G^{-1}G^{\\prime}\\right)_{t}^{\\;\\;t},\\nonumber \\\\\nR_{a}^{\\;\\;b} & = & -\\frac{1}{2}\\left(G^{-1}G^{\\prime\\prime}\\right)_{a}^{\\;\\;b}-\\frac{1}{4}\\left(G^{-1}G^{\\prime}\\right)_{a}^{\\;\\;b}\\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right)+\\frac{1}{2}\\left(G^{-1}G^{\\prime}G^{-1}G^{\\prime}\\right)_{a}^{\\;\\;b},\\label{eq:set up 3}\n\\end{eqnarray}\n\n\\noindent and\n\n\\begin{equation}\nH_{\\mu\\nu\\alpha}H^{\\mu\\nu\\alpha}=3H_{0ij}H^{0ij}=3B_{ij}^{\\prime}\\left(G^{-1}B^{\\prime}G^{-1}\\right)^{ij}=-3\\mathrm{Tr}\\left(G^{-1}B^{\\prime}\\right)^{2}.\\label{eq:set up 4}\n\\end{equation}\n\n\\noindent where we used the notation\n\n\\begin{equation}\nG'^{-1} \\equiv \\frac{d}{dx}(G^{-1}),\\quad\\quad \\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right)=g^{\\mu\\nu}g_{\\mu\\nu}^{\\prime}.\n\\end{equation}\n\n\\noindent We then introduce the $O\\left(d,d\\right)$-invariant dilaton $\\Phi$, defined  as\n\n\\begin{eqnarray}\n\\Phi & \\equiv & 2\\phi-\\frac{1}{2}\\ln\\left|\\det g_{\\mu\\nu}\\right|,\\nonumber \\\\\n\\Phi^{\\prime} & = & 2\\phi^{\\prime}-\\frac{1}{2}\\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right).\\label{eq:sign 1}\n\\end{eqnarray}\n\n\\noindent Therefore, the action (\\ref{eq:Polyakov}) can be rewritten\nas\n\n\\begin{eqnarray}\n\\bar I_{0} & = & \\int dx\\, e^{-\\Phi}\\left[{\\Phi'}^2+\\frac{1}{4}\\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right)^{2}-\\frac{1}{2}\\mathrm{Tr}\\left(G^{\\prime}{}^{-1}G^{\\prime}\\right)+\\frac{1}{4}\\mathrm{Tr}\\left(G^{-1}B^{\\prime}\\right)\\right.\\nonumber \\\\\n &  & \\left.-\\mathrm{Tr}\\left(G^{-1}G^{\\prime\\prime}\\right)+{\\Phi'}\\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right)\\right],\n\\end{eqnarray}\n\n\\noindent where we replaced $\\bar I_0$ with $I_0$ to indicate that we are working with the ansatz (\\ref{eq:our ansatz}).  Using integration by parts,\n\\begin{equation}\n\\frac{d}{dx}\\left[e^{-\\Phi}\\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right)\\right]=e^{-\\Phi}\\left[\\mathrm{Tr}\\left(G^{-1}G^{\\prime\\prime}\\right)+\\mathrm{Tr}\\left(G^{\\prime-1}G^{\\prime}\\right)-\\Phi^{\\prime}\\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right)\\right],\n\\end{equation}\nthe action becomes\n\n\\begin{equation}\n\\bar{I}_{0}=\\int dxe^{-\\Phi}\\left[\\Phi^{\\prime2}-\\frac{1}{4}\\mathrm{Tr}\\left(G^{-1}G^{\\prime}\\right)^{2}+\\frac{1}{4}\\mathrm{Tr}\\left(G^{-1}B^{\\prime}\\right)^{2}\\right].\n\\end{equation}\n\n\\noindent  Moreover, we want to point out  the  sign differences\nbetween our ansatz (\\ref{eq:our ansatz}) and the time-dependent FLRW\nmetric:\n\n\\begin{equation}\n\\mathrm{sign}\\left[R_{xx}\\right]=\\mathrm{sign}\\left[\\tilde{R}_{tt}\\right],\\qquad\\mathrm{sign}\\left[R_{tt}\\right]=-\\mathrm{sign}\\left[\\tilde{R}_{xx}\\right],\\qquad\\mathrm{sign}\\left[R_{ab}\\right]=-\\mathrm{sign}\\left[\\tilde{R}_{ab}\\right].\\label{eq:sign 2}\n\\end{equation}\n\n\\begin{equation}\n\\mathrm{sign}\\left[H^{2}\\right]=-\\mathrm{sign}\\left[\\tilde{H}^{2}\\right],\\qquad\\mathrm{sign}\\left[H_{\\mu\\nu}^{2}\\right]=-\\mathrm{sign}\\left[\\tilde{H}_{\\mu\\nu}^{2}\\right],\\qquad\\mathrm{sign}\\left[\\left(\\partial\\phi\\right)^{2}\\right]=-\\mathrm{sign}\\left[\\left(\\partial\\tilde{\\phi}\\right)^{2}\\right],\\label{eq:sign 3}\n\\end{equation}\n\n\\noindent where $\\tilde{A}$ represents the quantities calculated\nin the time-dependent FLRW background. For the Ricci scalar,\nwe have\n\n\\begin{equation}\n\\mathrm{sign}\\left[R\\right]=-\\mathrm{sign}\\left[\\tilde{R}\\right].\\label{eq:sign 4}\n\\end{equation}\n\n\\noindent Finally, the tree-level action (\\ref{eq:Polyakov}) becomes\n\\begin{equation}\n\\bar{I}_{0}=\\int dxe^{-\\Phi}\\left[\\Phi^{\\prime2}+\\frac{1}{8}\\mathrm{Tr}\\left(M^{\\prime}\\eta\\right)^{2}\\right].\\label{eq:0th action}\n\\end{equation}\n\n\\noindent where\n\n\\begin{equation}\nM=\\left(\\begin{array}{cc}\nG^{-1} & -G^{-1}B\\\\\nBG^{-1} & G-BG^{-1}B\n\\end{array}\\right),\\qquad\\eta=\\left(\\begin{array}{cc}\n0 & I\\\\\nI & 0\n\\end{array}\\right).\n\\end{equation}\n\n\\noindent Since $\\mathrm{Tr}\\left(M^{\\prime}\\eta\\right)^{2}=\\mathrm{Tr}\\left(M^{\\prime}M^{\\prime}{}^{-1}\\right)$, it is easy to see that the tree-level action (\\ref{eq:0th action}) is invariant under  $O\\left(d,d,R\\right)$\ntransformations,\n\n\\begin{equation}\n\\Phi\\rightarrow\\Phi,\\quad M\\rightarrow\\tilde{M}=\\Omega^{T}M\\Omega,\n\\end{equation}\n\n\\noindent where $\\Omega$ satisfies $\\Omega^{T}\\eta\\Omega=\\eta$.  Considering the gravitational background with $B_{ij}=0$ and a global\ntransformation by  $\\eta\\in O\\left(d,d,R\\right)$, we have\n\n\\begin{equation}\nM=\\left(\\begin{array}{cc}\nG^{-1} & 0\\\\\n0 & G\n\\end{array}\\right)\\rightarrow\\tilde{M}=\\eta M\\eta=\\left(\\begin{array}{cc}\nG & 0\\\\\n0 & G^{-1}\n\\end{array}\\right).\n\\end{equation}\n\n\\noindent This is the space-dependent duality   corresponding to the scale-factor duality in the time-dependent FLRW background.\n\n\\subsection*{First-order correction of $\\alpha^{\\prime}$}\n\nWe now demonstrate that  the closed string spacetime action with the first-order $\\alpha^{\\prime}$ correction also possesses the standard $O(d,d)$ symmetry for our ansatz  (\\ref{eq:our ansatz}). The action with the first-order $\\alpha^{\\prime}$ correction and vanishing Kalb-Ramond field is\n\n\\begin{equation}\nI=\\int d^{D}x\\sqrt{-g}e^{-2\\phi}\\left[R+4\\left(\\partial_{\\mu}\\phi\\right)^{2}-\\alpha^{\\prime}\\lambda_{0} R_{\\mu\\nu\\sigma\\rho}R^{\\mu\\nu\\sigma\\rho}+\\mathcal{O}\\left(\\alpha^{\\prime2}\\right)\\right].\n\\end{equation}\n\n\\noindent With some $\\alpha'$-corrected field  redefinitions  as in \\cite{Meissner:1996sa}, it can be expressed with first-order derivatives as\n\n\n\\begin{eqnarray}\nI & &=  \\int d^{D}x\\sqrt{-g}e^{-2\\phi}\\left[R+4\\left(\\partial_{\\mu}\\phi\\right)^{2}\\right]\\nonumber \\\\\n &  & -\\alpha^{\\prime}\\lambda_{0}\\int d^{D}x\\sqrt{-g}e^{-2\\phi}\\left[-R_{GB}^{2}+16\\left(R^{\\mu\\nu}-\\frac{1}{2}g^{\\mu\\nu}R\\right)\\partial_{\\mu}\\phi\\partial_{\\nu}\\phi-16\\partial^{2}\\phi \\left(\\partial\\phi\\right)^{2}+16\\left(\\partial\\phi\\right)^{4}\\right] +\\mathcal{O} \\left(\\alpha^{\\prime2}\\right),\n\\end{eqnarray}\n\n\\noindent where $R_{GB}^{2}$ is the Gauss-Bonnet term:\n\n\\begin{equation}\nR_{GB}^{2}=R_{\\mu\\nu\\sigma\\rho}R^{\\mu\\nu\\sigma\\rho}-4R_{\\mu\\nu}R^{\\mu\\nu}+R^{2}.\n\\end{equation}\n\n\\noindent By using our ansatz (\\ref{eq:our ansatz}), as one can check directly,  the action turns out to be\n\n\\begin{equation}\n\\bar{I}=-\\int dxe^{-\\Phi}\\left\\{ -\\Phi^{\\prime2}-\\frac{1}{8}\\mathrm{Tr}\\mathcal{M}^{\\prime2}+\\alpha^{\\prime}\\lambda_{0}\\left[\\frac{1}{16}\\mathrm{Tr}\\mathcal{M}^{\\prime4}-\\frac{1}{64}\\left(\\mathrm{Tr}\\mathcal{M}^{\\prime2}\\right)^{2}-\\frac{1}{4}\\Phi^{\\prime2}\\mathrm{Tr}\\mathcal{M}^{\\prime2}-\\frac{1}{3}\\Phi^{\\prime4}\\right]\\right\\}+\\mathcal{O} \\left(\\alpha^{\\prime2}\\right),\n\\label{eq:1st order O(d,d)}\n\\end{equation}\n\n\\noindent with\n\n\\begin{equation}\n\\mathcal{M}\\equiv M\\eta=\\left(\\begin{array}{cc}\n0 & G^{-1}\\\\\nG & 0\n\\end{array}\\right).\n\\end{equation}\n\n\n\n\n\\noindent Now, let us introduce the Kalb-Ramond field to the action\nwith the first-order $\\alpha^{\\prime}$ correction, which is\ngiven by\n\n\\begin{eqnarray}\nI & = & \\int d^{D}x\\sqrt{-g}e^{-2\\phi}\\left[R+4\\left(\\partial_{\\mu}\\phi\\right)^{2}-\\frac{1}{12}H^{2}\\right]\\nonumber \\\\\n &  & -\\alpha^{\\prime}\\lambda_{0}\\int d^{D}x\\sqrt{-g} e^{-2\\phi}\\left[-R_{GB}^{2}+16\\left(R^{\\mu\\nu}-\\frac{1}{2}g^{\\mu\\nu}R\\right)\\partial_{\\mu}\\phi\\partial_{\\nu} \\phi-16\\partial^{2}\\phi\\left(\\partial\\phi\\right)^{2}+16\\left(\\partial\\phi\\right)^{4}\\right.,\\nonumber \\\\\n &  & +\\frac{1}{2}\\left(R^{\\mu\\nu\\sigma\\rho}H_{\\mu\\nu\\alpha} H_{\\sigma\\rho}^{\\quad\\alpha}-2R^{\\mu\\nu}H_{\\mu\\nu}^{2}+\\frac{1}{3} RH^{2}\\right)-2\\left(D^{\\mu}\\partial^{\\nu}\\phi H_{\\mu\\nu}^{2}-\\frac{1}{3}\\partial^{2}\\phi H^{2}\\right)\\nonumber \\\\\n &  & \\left.-\\frac{2}{3}H^{2}\\left(\\partial\\phi\\right)^{2} - \\frac{1}{24}H_{\\mu\\nu\\lambda}H_{\\quad\\rho\\alpha}^{\\nu} H^{\\rho\\sigma\\lambda}H_{\\sigma}^{\\quad\\mu\\alpha}+\\frac{1}{8}H_{\\mu\\nu}^{2}H^{2\\mu\\nu}-\\frac{1}{144}\\left(H^{2}\\right)^{2}\\right]+\\mathcal{O} \\left(\\alpha^{\\prime2}\\right).\n\\end{eqnarray}\n\n\\noindent Using Eqs. (\\ref{eq:set up 1}), (\\ref{eq:set up 2}),\n(\\ref{eq:set up 3}),  (\\ref{eq:set up 4}) and (\\ref{eq:sign 1}), the action above can be expressed in the $O\\left(d,d\\right)$-invariant form (\\ref{eq:1st order O(d,d)}), but with\n\n\\begin{equation}\n\\mathcal{M}=\\left(\\begin{array}{cc}\nBG^{-1} & G-BG^{-1}B\\\\\nG^{-1} & -G^{-1}B\n\\end{array}\\right),\n\\end{equation}\nas one can verify by applying the EOM of $\\mathcal{M}$ and $\\phi$.\n\n\\noindent The action (\\ref{eq:1st order O(d,d)}) can be further simplified by using the tree-level EOM of $\\Phi$  from Eq. (\\ref{eq:0th action}), which is\n\n\\begin{equation}\n\\Phi^{\\prime2}+\\frac{1}{8}\\mathrm{Tr}\\left(M^{\\prime}\\eta\\right)^{2}=0\\rightarrow\\Phi^{\\prime2}=-\\frac{1}{8}\\mathrm{Tr}\\mathcal{M}^{\\prime2}.\\label{eq:redefine}\n\\end{equation}\n\n\\noindent Then the action is reduced to\n\n\\begin{equation}\n\\bar{I}=-\\int dxe^{-\\Phi}\\left\\{ -\\Phi^{\\prime2}-\\frac{1}{8}\\mathrm{Tr}\\mathcal{M}^{\\prime2}+\\alpha^{\\prime}\\lambda_{0}\\left[\\frac{1}{16}\\mathrm{Tr}\\mathcal{M}^{\\prime4}+\\frac{1}{96}\\left(\\mathrm{Tr}\\mathcal{M}^{\\prime2}\\right)^{2}\\right]\\right\\}.\n\\end{equation}\n\n\\noindent This action manifests the invariance under the $O\\left(d,d,R\\right)$\ntransformation\n\\begin{equation}\n\\Phi\\rightarrow\\Phi,\\quad M\\rightarrow\\tilde{M}=\\Omega^{T}M\\Omega.\n\\end{equation}\n\n\\subsection*{Higher-order corrections of $\\alpha^{\\prime}$}\nFor our ansatz, we have shown that the zeroth order and the first\norder in $\\alpha^{\\prime}$  can be rewritten in a standard $O\\left(d,d\\right)$-invariant form. Following the same logic as Refs. \\cite{Hohm:2019ccp,Hohm:2019jgu}, it is reasonable to assume that this is also true for all orders in $\\alpha'$.\n\nFollowing the derivations in Ref. \\cite{Hohm:2019jgu}, we now show that for our ansatz the action can be put into the reduced of (\\ref{eq: Odd with alpha x}). From the definition $\\mathcal{M}$, it is easy to get\n\n\n\\begin{equation}\n\\mathrm{Tr}\\mathcal{M}=\\mathrm{Tr}\\mathcal{M}^{\\prime}=\\mathrm{Tr}\\mathcal{M}^{\\prime\\prime}=0.\n\\end{equation}\n\n\\noindent Moreover,  $\\mathcal{M}\\mathcal{M}=1$ leads to   $\\mathcal{M}\\mathcal{M}^{\\prime}+\\mathcal{M}^{\\prime}\\mathcal{M}=0$ and then\n\n\n\\begin{equation}\n2\\mathcal{M}^{\\prime}\\mathcal{M}^{\\prime}+\\mathcal{M}^{\\prime\\prime}\\mathcal{M}+ \\mathcal{M}\\mathcal{M}^{\\prime\\prime}=0\\quad{\\rm and} \\quad \\mathcal{M}\\mathcal{M}^{\\prime2k+1}=-\\mathcal{M}^{\\prime2k+1}\\mathcal{M}.\n\\end{equation}\n\n\n\n\\noindent  Multiplying  by $\\left(\\mathcal{M}^{\\prime}\\right)^{2k+1}$   and taking traces,  one finds\n\n\n\\begin{equation}\n\\mathrm{Tr}\\left(\\mathcal{M}^{\\prime2k+1}\\right)=0,\\qquad k=0,1,\\ldots.\n\\end{equation}\n\n\\noindent Second, by using the equations of motion of the action\n(\\ref{eq:0th action}), higher space-dependent derivatives of $\\mathcal{M}$\ncan be written in the terms of $\\mathcal{M}$ and $\\mathcal{M}^{\\prime}$.\nTherefore, higher-order corrections of $\\alpha^{\\prime}$ can be\nbuilt from $\\mathcal{M}$ and $\\mathcal{M}^{\\prime}$. Third, if the\nterms of the higher-order $\\alpha^{\\prime}$ corrections take the form\n$\\mathrm{Tr}\\left(\\mathcal{M}^{m}\\mathcal{M}^{\\prime k}\\right)$,\nby using $\\mathcal{M}\\mathcal{M}=1$ we  get\n\n\\begin{equation}\n\\mathrm{Tr}\\left(\\mathcal{M}\\mathcal{M}^{\\prime k}\\right)=-\\mathrm{Tr}\\left(\\mathcal{M}\\mathcal{M}^{\\prime k}\\right)=0.\n\\end{equation}\n\n\\noindent Finally, due to Eq. (\\ref{eq:redefine}),\nthe dilation could be replaced by $\\mathrm{Tr}\\mathcal{M}^{\\prime2}$.\nIn summary, the higher-order corrections of $\\alpha^{\\prime}$ are\nconstructed using $\\mathcal{M}^{\\prime2}$. For example,\n\n\\begin{eqnarray}\n\\mathcal{O}\\left(\\alpha^{\\prime}\\right): &  & a_{1}\\mathrm{Tr}\\mathcal{M}^{\\prime4}+a_{2}\\left(\\mathrm{Tr}\\mathcal{M}^{\\prime2}\\right)^{2},\\nonumber \\\\\n\\mathcal{O}\\left(\\alpha^{\\prime2}\\right): &  & b_{1}\\mathrm{Tr}\\mathcal{M}^{\\prime6}+b_{2}\\mathrm{Tr}\\mathcal{M}^{\\prime4}\\mathrm{Tr}\\mathcal{M}^{\\prime2}+b_{3}\\left(\\mathrm{Tr}\\mathcal{M}^{\\prime2}\\right)^{3},\\nonumber \\\\\n & \\vdots\\nonumber \\\\\n\\mathcal{O}\\left(\\alpha^{\\prime k-1}\\right): &  & d_{1}\\mathrm{Tr}\\mathcal{M}^{\\prime2k}+d_{2}\\mathrm{Tr}\\mathcal{M}^{\\prime2k-2}\\mathrm{Tr}\\mathcal{M}^{\\prime2}+d_{3}\\mathrm{Tr}\\mathcal{M}^{\\prime2k-4}\\left(\\mathrm{Tr}\\mathcal{M}^{\\prime2}\\right)^{2}+\\ldots.\n\\end{eqnarray}\n\n\\noindent Furthermore, considering the action at zeroth order Eq.(\\ref{eq:0th action}),\nthe variation for $g_{xx}$ gives\n\n\\begin{equation}\n\\delta\\bar{I}_{0}=\\int dxe^{-\\Phi}\\left[\\Phi^{\\prime2}+\\frac{1}{8}\\mathrm{Tr}\\left(\\mathcal{M}^{\\prime}\\right)^{2}\\right]\\delta g_{xx}.\\label{eq:delta 1}\n\\end{equation}\n\n\\noindent This variation can be generalized to the higher orders with\n$X_{2k}\\left(\\mathcal{M}^{\\prime}\\right)=\\mathrm{Tr}\\left[\\left(\\mathcal{M}^{\\prime}\\right)^{2k_{1}}\\right]\\cdots\\mathrm{Tr}\\left[\\left(\\mathcal{M}^{\\prime}\\right)^{2k_{n}}\\right]$,\n$k=k_{1}+k_{n}$:\n\n\\begin{equation}\n\\delta\\bar{I}_{k}=\\frac{\\beta k}{2\\left(4k-1\\right)}\\int dxe^{-\\Phi}X_{2k}\\left(\\mathcal{M}^{\\prime}\\right)\\mathrm{Tr}\\left(\\mathcal{M}^{\\prime}\\right)^{2},\\label{eq:delta 2}\n\\end{equation}\n\n\\noindent where\n\n\\begin{equation}\n\\delta g_{xx}=\\beta\\alpha^{\\prime k}X_{2k}\\left(\\mathcal{M}^{\\prime}\\right).\\label{eq: rede gxx}\n\\end{equation}\n\n\\noindent If we substitute the redefinition (\\ref{eq: rede gxx})\nback into Eq. (\\ref{eq:delta 1}) and set $\\frac{\\beta k}{2\\left(4k-1\\right)}=-1$,\nwe find that the terms with $\\mathrm{Tr}\\mathcal{M}^{\\prime2}$ can\nbe eliminated when we sum Eqs. (\\ref{eq:delta 1}) and (\\ref{eq:delta 2}).\nIn other words, we can safely set  $\\mathrm{Tr}\\mathcal{M}^{\\prime2}=0$ for $\\alpha'$-corrected terms in the action\nand obtain\n\n\\begin{eqnarray}\n\\mathcal{O}\\left(\\alpha^{\\prime}\\right): &  & a_{1}\\mathrm{Tr}\\mathcal{M}^{\\prime4},\\nonumber \\\\\n\\mathcal{O}\\left(\\alpha^{\\prime2}\\right): &  & b_{1}\\mathrm{Tr}\\mathcal{M}^{\\prime6},\\nonumber \\\\\n & \\vdots\\nonumber \\\\\n\\mathcal{O}\\left(\\alpha^{\\prime k-1}\\right): &  & d_{1}\\mathrm{Tr}\\mathcal{M}^{\\prime2k}+d_{4}\\mathrm{Tr}\\mathcal{M}^{\\prime2k-4}\\mathrm{Tr}\\mathcal{M}^{\\prime4}\\ldots.\n\\end{eqnarray}\n\n\\noindent The action with higher-order $\\alpha^{\\prime}$ corrections\nthen reduces to\n\n\\begin{equation}\n\\bar{I}\\equiv-\\int dxe^{-\\Phi}\\left(-\\Phi^{\\prime2}+{\\sum_{k=1}^{\\infty}}\\left(\\alpha^{\\prime}\\right)^{k-1}\\bar{c}_{k}\\mathrm{Tr}\\left(\\mathcal{M}^{\\prime2k}\\right)+\\mathrm{multitraces}\\right).\n\\end{equation}\n\n\\noindent After extracting the overall minus sign of the action above,\nthe even orders of $\\alpha^{\\prime}$ corrections acquire a\nminus sign. Since $\\bar{c}_{k}$ is the coefficient of $\\mathrm{Tr}\\mathcal{M}^{\\prime2k}$,\nwhich is not modified by   $\\Phi^{\\prime k}\\simeq\\frac{1}{8}\\left(\\frac{k-1}{k-3}\\right)\\Phi^{\\prime k-2}\\mathrm{Tr}\\mathcal{M}^{\\prime2}$,\nthe values of $\\left|c_{k}\\right|$ and $\\left|\\bar{c}_{k}\\right|$\nare the same. Moreover,\nby using Eqs. (\\ref{eq:sign 1}), (\\ref{eq:sign 2}), and (\\ref{eq:sign 3})\nto all orders, we  find $\\bar{c}_{1}=c_{1}=-\\frac{1}{8}$, $\\bar{c}_{2k+1}=c_{2k+1}$, and\n$\\bar{c}_{2k}=-c_{2k}$.\nIt is worth noting that the relationships between $\\bar{c}_{k}$ and $c_{k}$ are not changed after including the contributions of the multitrace terms.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nQueueing models with many-servers are prevalent in modeling call\ncenters and other large-scale service systems. They are used for\noptimizing staffing and making dynamic control decisions. The\ncomplexity of the underlying queueing model renders such optimization\nproblems intractable for exact analysis, and one needs to resort to\napproximations. A prominent mode of approximate analysis is to study\nsuch systems in the so-called Halfin--Whitt (HW) heavy-traffic regime;\ncf. \\cite{HaW81}. Roughly speaking, the analysis of a queueing system\nin the HW regime proceeds by scaling up the number of servers and the\narrival rate of customers in such a way that the system load approaches\none asymptotically. To be more specific, instead of considering a\nsingle system, one considers a sequence of (closely related) queueing\nsystems indexed by a parameter $n$ along which the arrival rates and\nthe number of servers scale up so that the system traffic intensity~$\\rho^n$\nsatisfies\n\\begin{equation}\\label{eqHTcond}\n\\sqrt{n} (1-\\rho^n)\\rightarrow\\beta\\qquad\\mbox{as } n\\tinf.\n\\end{equation}\n\nIn the context of dynamic control, passing to a formal limit of the\n(properly scaled) system dynamics equations as $n\\tinf$ gives rise to\na \\textit{limit} diffusion control problem, which is often more tractable\nthan the original dynamic control problem it approximates.\nThe approximating diffusion control problem typically provides useful\nstructural insights and guides the design of good policies for the\noriginal system. Once a candidate policy is proposed for the original\nproblem of interest, its asymptotic performance can be studied in the\nHW regime. The ultimate goal is to establish that the proposed policy\nperforms well. To this end, a useful criterion is the notion of\nasymptotic optimality, which provides assurance that the optimality gap\nassociated with the proposed policy vanishes asymptotically \\textit{under\ndiffusion scaling} as $n\\tinf$. Hence, asymptotic optimality in this\ncontext is equivalent to showing that the optimality gap is $o(\\sqrt{n})$.\n\nA central reference for our purposes is the recent paper by Atar,\nMandelbaum and Reiman \\cite{AMR02}, where the authors apply all steps\nof the above scheme to the important class of problems of dynamically\nscheduling a multiclass queue with many identical servers in the HW\nregime. Specifically, \\cite{AMR02} considers a sequence of systems\nindexed by the number of servers $n$, where the number of servers and\nthe arrival rates of the various customer classes increase with $n$\nsuch that the heavy-traffic condition holds; cf. equation (\\ref\n{eqHTcond}). Following the scheme described above, the authors derive\nan approximate diffusion control problem through a formal limiting\nargument. They then show that the diffusion control problem admits an\noptimal Markov policy, and that the corresponding HJB equation (a\nsemilinear elliptic PDE) has a unique classical solution. Using the\nMarkov control policy and the HJB equation, the authors propose\nscheduling control policies for the original (sequence of) queueing\nsystems of interest. Finally, they prove that the proposed sequence of\npolicies is asymptotically optimal under diffusion scaling. Namely, the\noptimality gap of the proposed policy for the $n$th system is\n$o(\\sqrt{n})$. A similar approach is applied to more general networks\nin \\cite{atar2005scheduling}. In this paper, we study a similar\nqueueing system (see Section \\ref{secmodel}). Our goal, however, is\nto provide an improved optimality gap which, in turn, requires a\nsubstantially different scheme than the one alluded to above.\n\nApproximations in the HW regime for performance analysis have been used\nextensively for the study of fixed policies. Given a particular policy,\nit may often be difficult to calculate various performance measures in\nthe original queueing system. Fortunately, the corresponding\napproximations in the HW regime are often more tractable. The machinery\nof strong approximations (cf. Cs\\\"{o}rgo and Horv\\'{a}th \\cite{csorgo})\noften plays a central role in such analysis. In the context\nof many-server heavy-traffic analysis, with strong approximations, the\narrival and service processes (under suitable assumptions on the\ninter-arrival and service times) can be approximated by a diffusion\nprocess so that the approximation error on finite intervals is $O(\\log\nn)$ (where $n$ is the number of servers as before). Therefore, it is\nnatural to expect that, under a given policy, the error in the\ndiffusion approximations of the various performance metrics is $O(\\log\nn)$, which is indeed verified for various settings in the literature\n(see, e.g., \\cite{MMR98}).\n\nA natural question is then whether one can go beyond the analysis of\nfixed policies and achieve an optimality gap that is logarithmic in $n$\nalso under dynamic control, improving upon the usual optimality gap of\n$o(\\sqrt{n})$. More specifically, can one propose a sequence of\npolicies (one for each system in the sequence) where the optimality gap\nfor the policy (associated with the $n$th system) is logarithmic\nin~$n$? While one hopes to get logarithmic optimality gaps as suggested by\nstrong approximations, it is not a priori clear if this can be achieved\nunder dynamic control. The purpose of this paper is to provide a\nresolution to this question. Namely, we study whether one can establish\nsuch a strong notion of asymptotic optimality and if so, then how\nshould one go about constructing policies which are asymptotically\noptimal in this stronger sense.\n\nOur results show that such strengthened bounds on optimality gaps can\nbe attained. Specifically, we construct a sequence of asymptotically\noptimal policies, where the optimality gap is logarithmic in $n$. Our\nanalysis reveals that identifying (a sequence of) candidate policies\nrequires a new approach. To be specific, we advance a sequence of\ndiffusion control problems (as opposed to just one) where the diffusion\ncoefficient in each system depends on the state and the control. This\nis contrary to the existing work on the asymptotic analysis of queueing\nsystems in the HW regime. In that stream of literature, the diffusion\ncoefficient is typically a (deterministic) constant. Indeed, Borkar\n\\cite{borkar2005controlled} views the constant diffusion coefficient\nas a characterizing feature of the problems stemming from the\nheavy-traffic approximations in the HW regime. Interestingly, it is\nessential in our work to have the diffusion coefficient depend on the\nstate and the control for achieving the logarithmic optimality gap. In\nessence, incorporating the impact of control on the diffusion\ncoefficient allows us to track the policy performance in a more refined manner.\n\nWhile the novelty of having the diffusion coefficient depend on the\ncontrol facilitates better system performance, it also leads to a more\ncomplex diffusion control problem. In particular, the associated HJB\nequation is fully nonlinear; it is also nonsmooth under a linear\nholding cost structure. In what follows, we show that each of the HJB\nequations in the sequence has a unique smooth solution on bounded\ndomains and that each of the diffusion control problems (when\nconsidered up to a stopping time) admits an optimal Markov control\npolicy. Interpreting this solution appropriately in the context of the\noriginal problem gives rise to a policy under which the optimality gap\nis logarithmic in $n$. As in the performance analysis of fixed\npolicies,\\vadjust{\\goodbreak} strong approximations will be used in the last step, where we\npropose a sequence of controls for the original queueing systems, and\nshow that we achieve the desired performance. However, it is important\nto note that strong approximation results alone are not sufficient for\nour results. Rather, for the improved optimality gaps we need the\nrefined properties of the solutions to the HJB equations. Specifically,\ngradient estimates for the sequence of solutions to the HJB equations\n(cf. Theorem \\ref{thmHJB1sol}) play a central role in our proofs.\n\nOur analysis restricts attention to a linear holding cost structure.\nHowever, we expect the analysis to go through for some other cost\nstructures including convex holding costs. Indeed, the analysis of the\nconvex holding cost case will probably be simpler as one tends to get\n``interior'' solutions in that case as opposed to the corner solutions\nin the linear cost case, which causes nonsmoothness. One could also\nenrich the model by allowing abandonment. We expect the analysis to go\nthrough with no major changes in these cases as well; see the\ndiscussion of possible extensions in Section \\ref{secconclusions}.\nFor purposes of clarity, however, we chose not to incorporate these\nadditional\/alternative features because we feel that the current set-up\nenables us to focus on and clearly communicate the main idea: the use\nof a novel Brownian model with state\/control dependent diffusion\ncoefficient to obtain improved optimality gaps.\n\n\\subsection*{Organization of the paper} Section \\ref{secmodel}\nformulates the model and states the main result. Section \\ref\n{secresult} introduces a (sequence of) Brownian control problem(s),\nwhich are then analyzed in Section \\ref{secADCP}. A performance\nanalysis of our proposed policy appears in Section \\ref{sectracking}.\nThe major building blocks of the proof are combined to establish the\nmain result in Section \\ref{seccombining} and some concluding remarks\nappear in Section~\\ref{secconclusions}.\n\n\\section{Problem formulation}\\label{secmodel}\n\nWe consider a queueing system with a single server-pool consisting of\n$n$ identical servers (indexed from 1 to $n$) and a set $\\I=\\{1,\\ldots\n,I\\}$ of job classes as depicted in Figure \\ref{figv}. Jobs of\n\\begin{figure}\n\n\\includegraphics{777f01.eps}\n\n\\caption{A multiclass queue with many servers.}\n\\label{figv}\n\\end{figure}\nclass-$i$ arrive according to a Poisson process with rate $\\lambda_i$\nand wait in their designated queue until their service begins. Once\nadmitted to service, the service time of a class-$i$ job is distributed\nas an exponential random variable with rate $\\mu_i>0$. All service and\ninterarrival times are mutually independent.\n\n\\subsection*{Heavy-traffic scaling}\n\nWe consider a sequence of systems indexed by the number of servers $n$.\nThe superscript $n$ will be attached to various processes and\nparameters to make the dependence on $n$ explicit. (It will be omitted\nfrom parameters and other quantities that do not change with $n$.) We\nassume\\vspace*{1pt} that $\\lambda_i^n=a_i\\lambda^n$ for all~$n$, where $\\lambda\n^n$ is the total arrival rate and $a_i>0$ for $i\\in\\I$ with $\\sum\n_{i}a_i=1$. This assumption is made for simplicity of notation and\npresentation. Nothing changes\\vspace*{1pt} in our results if one assumes, instead,\nthat $\\lambda_i^n\/n\\rightarrow\\lambda_i$ and $\\sqrt{n}(\\lambda\n_i^n\/n-\\lambda_i)\\rightarrow\\hat{\\lambda}_i$ as $n\\tinf$ where\n$\\lambda_i\/\\sum_{k\\in\\I}\\lambda_k=a_i>0$.\\vadjust{\\goodbreak}\n\nThe nominal load in the $n$th system is then given by\n\\[\nR^n=\\sum_{i}\\frac{\\lambda_i^n}{\\mu_i}=\\lambda^n\\sum_{i}\\frac\n{a_i}{\\mu_i},\n\\]\nso that defining $\\bar{\\mu}=[\\sum_{i}a_i\/\\mu_i]^{-1}$ we have that\n$R^n=\\lambda^n\/\\bar{\\mu}$, which corresponds to the nominal number\nof servers required to handle all the incoming jobs. The heavy-traffic\nregime is then imposed by requiring that the number of servers deviates\nfrom the nominal load by a term that is a square root of the nominal\nload. Formally, we impose this by assuming that $\\lambda^n$ is such\nthat\n\\begin{equation} \\label{eqHTstaff}\nn= R^n+\\beta\\sqrt{R^n}\n\\end{equation}\nfor some\n$\\beta\\in(-\\infty,\\infty)$ that does not scale with $n$. Also, we\ndefine the relative load imposed on the system by class-$i$ jobs,\ndenoted by $\\nu_{i}$, as follows:\n\\begin{equation}\\label{eqrhodefin}\n\\nu_i =\\frac{a_i\/\\mu_i}{\\sum\n_{k\\in\\I}a_k\/\\mu_k}.\n\\end{equation}\nNote that $\\sum_{i\\in\\I}\\nu_i=1$, and $\\nu_i n$ can be interpreted\nas a first-order (fluid) estimate for the number of servers busy\nserving class-$i$ customers.\n\n\\subsection{System dynamics}\n\nLet $Q_i\\lam(t)$ and $X_i\\lam(t)$ denote the number of class-$i$ jobs\nin the queue and in the system, respectively, at time $t$ in the $n$th\nsystem. Similarly, let $Z_i\\lam(t)$ denote the number of servers\nworking on class-$i$ jobs at time $t$. Clearly, for all $i, n,\nt$, the following holds:\n\\[\nX_i\\lam(t)=Z_i\\lam(t)+Q_i\\lam(t).\\vadjust{\\goodbreak}\n\\]\n\nIn our setting, a control corresponds to determining how many of the\n\\mbox{class-$i$} jobs currently in the system are placed in queue and in\nservice for $i\\in\\I$. We take the process $Z^n$ as our control in the\n$n$th system. Note that one can equivalently take the queue length\nprocess $Q^n$ as the control. (The knowledge of either process is\nsufficient to pin down the evolution of the system given the arrival,\nservice processes and the initial conditions.) Clearly, the control\nprocess must satisfy certain requirements for admissibility, including\nthe usual nonanticipativity requirement. We defer a precise\nmathematical definition of admissible controls for now (see Definition\n\\ref{definadmissiblecontrols}). However, it should be clear that,\ngiven the process $Z^n$, one can construct the other processes of interest.\n\nTo be specific, consider a complete probability space $(\\Omega\n,\\mathcal{F},\\mathbb{P})$ and $2I$ mutually independent\n\\textit{unit-rate} Poisson processes $(\\mathcal{N}_i^a(\\cdot), \\mathcal\n{N}_i^d(\\cdot), i\\in\\I)$ on that space. Given the \\textit{primitives}\n$(\\mathcal{N}_i^d(\\cdot),\\mathcal{N}_i^a(\\cdot),X_i\\lam(0),Z_i\\lam\n(0);i\\in\\I)$ and the control process~$Z^n$, we construct the\nprocesses $X^n, Q^n$ as follows: for $t \\geq0$ and $i \\in\\mathcal{I}$\n\\begin{eqnarray} \\label{eqdynamics1}\nX_i\\lam(t)&=&X_i\\lam(0)+\\mN_i^a(\\lambda_i^n t)-\\mN_i^d\\biggl( \\mu\n_i\\int_0^t Z_i\\lam(s)\\,ds \\biggr),\\\\\nQ_i\\lam(t)&=&X_i\\lam(t)-Z_i\\lam(t).\n\\end{eqnarray}\nThe processes $Z^n, Q^n, X^n$ must jointly satisfy the constraints\n\\begin{equation}\\label{eqnon-negativity}\n(Q\\lam(t), X\\lam(t),Z\\lam(t))\\in\\mathbb{Z}_+^{3I},\\qquad\ne\\cdot Z\\lam(t)\\leq n,\n\\end{equation}\nwhere $e$ is the $I$-dimensional vector of ones.\n\nControls can be preemptive or nonpreemptive. Under a nonpreemptive\ncontrol, a job that is assigned to a server keeps the server busy until\nits service is completed. In particular, given a nonpreemptive control\n$Z^n$, the process $Z_i^n$ can decrease only through service\ncompletions of class-$i$ jobs. In contrast, the class of preemptive\ncontrols is broader. While it includes nonpreemptive policies, it also\nincludes controls that (occasionally) may preempt a job's service. The\npreempted job is put back in the queue and its service is resumed at a\nlater time (possibly by a different server). Hence, the class of\npreemptive controls subsumes the class of nonpreemptive ones (which is\nalso immediate from Definition 1 in \\cite{AMR02}) and the cost of an\noptimal policy among preemptive ones gives a lower bound for that among\nthe nonpreemptive ones.\n\nIn what follows, we will largely focus on preemptive controls, which\nare easier to work with, and derive a specific policy which is near\noptimal in that class. The specific policy we derive is, however,\nnonpreemptive, and therefore, is near optimal among the nonpreemptive\npolicies as well. More specifically, the policy we propose belongs to a\nclass which we refer to as \\textit{tracking policies.}\n\nTo facilitate the definition of tracking policies, define $\\mathcal{U}\n\\subset\\mathbb{R}_+^I$ as\n\\begin{equation}\\label{eqmUdefin}\n\\mathcal{U}=\\biggl\\{u\\in\\bbR_+^I\\dvtx\n\\sum_{i}u_i=1\\biggr\\}.\n\\end{equation}\nAlso, for all $i$ and $t\n\\geq0$, let\n\\begin{equation} \\label{eqtildeXdefin}\n\\check{X}_i\\lam(t)=X_i\\lam(t)-\\nu_in.\n\\end{equation}\nHence, the process $\\check{X}_i^n$ captures the\noscillations of the process $X_i\\lam$ around its ``fluid''\napproximation $\\nu_i n$. Throughout our analysis, for $x\\in\\bbR$ we\nlet $(x)^+=\\max\\{0,x\\}$ and $(x)^-=\\max\\{0,-x\\}$.\n\\begin{defin}\\label{defintracking}\nGiven a function $h\\dvtx\\bbR^I\\to\\mathcal{U}$, an $h$-tracking policy\nmakes resource allocation decisions in the $n$th system as follows:\n\\begin{longlist}\n\\item It is nonpreemptive. That is, once a server starts\nworking on a job, it continues without interruption until that job's\nservice is completed.\n\\item It is work conserving. That is, the number of busy\nservers satisfies $e\\cdot Z^n(t)=(e\\cdot X^n(t))\\wedge n$ for all $t>\n0$. In particular, no server is idle as long as there are $n$ or more\njobs in the system.\n\\item When a class-$i$ job arrives to the system it joins the\nqueue of class $i$ if all servers are busy processing other jobs.\nOtherwise, the lowest-indexed idle server starts working on that job.\n\\item A server that finishes processing a job at a time $t$,\nidles if all queues are empty. Otherwise, she starts working on a job\nof class $i\\in\\mathcal{K}(t-)$ with probability $\\lambda_i^n\/\\sum\n_{k\\in\\mathcal{K}(t-)}\\lambda_k^n$, where, for $t>0$, the set\n$\\mathcal{K}(t-)$ is defined by\n\\begin{equation}\\label{eqmathKdefin}\n\\mathcal{K}(t-)=\\bigl\\{k\\in\\I\\dvtx Q_k(t)-h_k(\\check\n{X}^n(t-))\\bigl(e\\cdot\\check{X}^n(t-)\\bigr)^+>0\\bigr\\}.\n\\end{equation}\nFinally, if $(e\\cdot\\check{X}^n(t-))^+>0$ and $\\mathcal\n{K}(t-)=\\varnothing$, she picks for service a customer from the lowest\nindex nonempty queue.\n\\end{longlist}\n\\end{defin}\n\\begin{rem}\\label{remrandomization}\nFor our optimality-gap bounds and, in particular, for the proof of\nTheorem \\ref{thmSSC} it is important that the policy be such that\neach of the job classes in the set $\\mathcal{K}(t)$ gets a sufficient\nshare of the capacity. This prevents excessive oscillation of the\nqueues that may compromise the optimality gaps. Such oscillations could\narise if, for example, the policy chooses for service a~job of class\n\\[\ni=\\min\\argmax_{k\\in\\I}\\bigl\\{Q_k(t-)-h_k(\\check{X}^n(t-))\\bigl(e\\cdot\n\\check{X}^n(t-)\\bigr)^+\\dvtx Q_k^n(t-)>0\\bigr\\}.\n\\]\nRandomization is just one way\nto overcome such oscillations and, as the proofs (specifically that\nof Theorem \\ref{thmSSC}) reveal, any choice rule that guarantees a\nsufficient share of the capacity to a class in $\\mathcal{K}(t-)$ will\nsuffice.\n\\end{rem}\n\nOur main result shows that a (nonpreemptive) tracking policy can\nachieve a near optimal performance among preemptive policies. Note that\nin our setting under\\vadjust{\\goodbreak} preemption, one can restrict attention to\nwork-conserving policies, that is, policies under which the servers\nnever idle as long as there are jobs to work on.\\footnote{By a coupling\nargument, this can be shown to hold with general queueing costs\nprovided that there are no abandonments and that the service times are\nexponential; see, for example, the coupling argument on page 1126 of\n\\cite{AMR02}.} More precisely, a control is work conserving if the\nfollowing holds for all $t>0$:\n\\begin{equation}\\label{eqworkconservation}\ne\\cdot Q\\lam(t)=\\bigl(e\\cdot\\check{X}\\lam(t)\\bigr)^+.\n\\end{equation}\n\nHereafter, we focus on work-conserving controls. Each such control can\nbe mapped into a ratio control, which specifies what fraction of the\ntotal number of jobs in queue belongs to each class. To that end, let\n\\begin{equation}\\label{eqUQmap}\nU_i\\lam(t)=\\frac{Q_i\\lam(t)}{(e\\cdot Q\\lam(t))\\vee1} .\n\\end{equation}\nNote that the original control $Z^n$ can be recovered from the ratio\ncontrol~$U^n$ as follows:\n\\[\nZ_{i}^{n}(t) = X_i^n(t) - U_{i}^{n}(t) \\bigl(e\\cdot\\check{X}^n(t)\\bigr)^+ .\n\\]\nEquations (\\ref{eqdynamics1})--(\\ref{eqnon-negativity}) can then be\nreplaced by\n\\begin{eqnarray}\n\\label{eqdynamics2}\nX_i\\lam(t)&=&X_i\\lam(0)+\\mN_i^a(\\lambda_i^n t) \\nonumber\\\\[-8pt]\\\\[-8pt]\n&&{} -\\mN_i^d\\biggl(\\mu_i \\int_0^t \\bigl(\nX_i\\lam(s)-U_i\\lam(s) \\bigl(e\\cdot\\check{X}^n(t)\\bigr)^+ \\bigr) \\,ds\\biggr), \\nonumber\\\\\nQ_i\\lam(t)&=&U_i\\lam(t)\\bigl(e\\cdot\\check{X}^n(t)\\bigr)^+, \\\\\nZ_i\\lam(t)&=&X_i\\lam(t)-Q_i\\lam(t),\\\\\n\\check{X}_i\\lam(t)&=&X_i\\lam(t)-\\nu_i n,\\\\\n\\label{eqnon-negativity2}\nU\\lam(t)&\\in&\\mathcal{U},\\qquad Q\\lam(t)\\in\\mathbb{Z}_+^I,\\qquad X\\lam(t)\\in\n\\mathbb{Z}_+^I.\n\\end{eqnarray}\n\nDefine the filtration\n\\[\n\\bar{\\mathcal{F}}_t=\\sigma\\{\\mN_i^a(s),\\mN_i^d(s);i\\in\\I, s\\leq\nt\\}\n\\]\nand the $\\sigma$-field\n\\begin{equation}\\label{eqcheckFdefin}\n\\bar{\\mathcal{F}}_{\\infty}=\\bigvee\n_{t\\geq0} \\bar{\\mathcal{F}}_t.\n\\end{equation}\nInformally,\n$\\bar{\\mathcal{F}}_{\\infty}$ contains the information about the\nentire evolution of the processes $(\\mN_i^a,\\mN_i^d,i\\in\\I)$. A\nnatural notion of admissibility requires that the control is\nnonanticipative so that it only uses historical information about the\nprocess $X^n$ and about the arrivals and service completions up to the\ndecision epoch. To accommodate randomized policies (as the $h$-tracking\npolicy) we allow the control to use other information too as long as\nthis information is independent of $\\bar{\\mathcal{F}}_{\\infty}$.\n\\begin{defin}\\label{definadmissiblecontrols}\nA process $U=(U_i(t), t\\geq0, i \\in\\mathcal{I})$\nis a ratio control for the $n$th system if there exists a process\n$\\mathbb{X}\\lam=(X\\lam,Q\\lam,Z\\lam,\\check{X}\\lam)$ such that,\ntogether with the primitives, $(\\mathbb{X}\\lam,U)$ satisfies\n(\\ref{eqdynamics2})--(\\ref{eqnon-negativity2}). The process\n$U$ is an admissible ratio control if, in addition, it is adapted to\nthe filtration $\\mathcal{G}\\vee\\mathcal{F}_t\\lam$ where\n\\begin{eqnarray*}\n\\mathcal{F}_t\\lam&=&\\sigma\\biggl\\{\\mN_i^a(\\lambda\n_i^n s),X_i\\lam(s),\\mu_i\\int_0^s Z_i\\lam(u)\\,du,\\\\\n&&\\hphantom{\\sigma\\biggl\\{}\n\\mN_i^d\\biggl(\\mu_i\\int_0^s Z_i\\lam(u)\\,du\\biggr); i\\in\n\\I,0\\leq s\\leq t\\biggr\\},\n\\end{eqnarray*}\nand $\\mathcal{G}$ is a $\\sigma$-field that is independent of $\\bar\n{\\mathcal{F}}_{\\infty}$. The process $\\mathbb{X}^n$ is then said to\nbe the queueing process associated with the ratio control $U$. We let\n$\\Pi\\lam$ be the set of admissible ratio controls for the $n$th\nsystem.\n\\end{defin}\n\nRatio controls are work conserving by definition, but they need not be\nnonpreemptive in general. However, note that given a function $h\\dvtx \\bbR\n^I\\to\\mathcal{U}$, the (nonpreemptive) $h$-tracking policy\ncorresponds to a ratio control $U_h$, which is nonpreemptive. To be\nspecific, given the primitives and the $h$-tracking policy, one can\nconstruct the corresponding queueing process $\\mathbb\n{X}^n=(X^n,Q^n,Z^n,\\check{X}^n)$ (see the construction after Lemma\n\\ref{lemstrongappbounds}). Then the ratio control $U_h$ is\nconstructed using the relation (\\ref{eqUQmap}) so that $\\mathbb\n{X}^n$ and $U_h$ jointly satisfy (\\ref{eqdynamics2})--(\\ref\n{eqnon-negativity2}). Hence, one can speak of the ratio control and\nthe queueing process associated with an $h$-tracking policy. Note that\nsince the tracking policy makes resource allocation decisions using\nonly information on the state of the system at the decision epoch\n(together with a randomization that is independent of the history), the\nresulting ratio control is admissible in the sense of Definition \\ref\n{definadmissiblecontrols}. The terms ratio control and $h$-tracking\npolicy appear in several places in the paper. It will be clear from the\ncontext whether we refer to an arbitrary ratio control or to one\nassociated with an $h$-tracking policy.\n\nWe close this section by stating the main result of the paper. To that\nend, let\n\\begin{equation}\\label{eqmathXdefin}\n\\mathcal{X}^n=\\{(x,q)\\in\\mathbb{Z}_+^{2I}\\dvtx\nq=u(e\\cdot\nx-n)^+\\mbox{ for some } u\\in\\mathcal{U}\\}.\n\\end{equation}\nThat is, $\\mathcal{X}^n$ is the set on which $(X^n, Q^n)$ can\nobtain values under work conservation. In this set $e\\cdot q=(e\\cdot\nx-n)^+$ so that positive queue and idleness do not co-exist. We let\n$\\Ex_{x,q}^{U}[\\cdot]$ denote the expectation with respect to the\ninitial condition $(X^n(0),Q^n(0))=(x,q)$ and an admissible ratio\ncontrol $U$. Given a ratio control $U$ and initial conditions $(x,q)$,\nthe expected infinite horizon discounted cost in the $n$th system\nis given by\n\\begin{equation}\\label{eqcost1}\nC\\lam(x,q,U)=\\Ex_{x,q}^{U}\\biggl[\\int_0^{\\infty\n}e^{-\\gamma s}\nc\\cdot Q\\lam(s)\\,ds\\biggr],\n\\end{equation}\nwhere $c=(c_1,\\ldots,c_I)'$ is the strictly positive vector of holding\ncost rates and $\\gamma> 0$ is the discount rate. For $(x,q) \\in\n\\mathcal{X}^n$, the value function is given by\n\\[\nV\\lam(x,q)=\\inf_{U\\in\\Pi\\lam}\\Ex_{x,q}^{U}\\biggl[\\int_0^{\\infty\n} e^{-\\gamma s}c\\cdot Q\\lam(s)\\,ds\\biggr].\n\\]\n\nWe next state our main result.\n\\begin{theorem}\\label{thmmain} Fix a sequence $\\{(x^n,q^n),n\\in\\bbZ_+\\}\n$ such that\n\\mbox{$(x^n,q^n)\\in\\mathcal{X}^n$} and $|x^n-\\nu n| \\leq M \\sqrt{n}$ for\nall $n$ and some $M>0$. Then, there exists a sequence of tracking\nfunctions $\\{h^n,n\\in\\bbZ_+\\}$ together with constants $C,k>0$ (that\ndo not depend on $n$) such that\n\\[\nC\\lam(x^n,q^n,U_h^n)\\leq V\\lam(x^n,q^n)+C\\log^{k} n \\qquad\\mbox{for all\n} n,\n\\]\nwhere $U_h^n$ is the ratio control associated with the $h^{n}$-tracking\npolicy.\n\\end{theorem}\n\nThe constant $k$ in our bound may depend on all system and cost\nparameters but not on $n$. In particular, it may depend on $(\\mu\n_i,c_i,a_i;i\\in\\I)$ and $\\beta$. Its value is explicitly defined\nafter the statement of Theorem \\ref{thmHJB1sol}.\n\nTheorem \\ref{thmmain} implies, in particular, that the optimal\nperformance for nonpreemptive policies is close to that among the\nlarger family of preemptive policies. Indeed, we identify a\nnonpreemptive policy (a tracking policy) in the queueing model whose\ncost performance is close to the optimal value of the preemptive\ncontrol problem.\n\nThe rest of the paper is devoted to the proof of Theorem \\ref{thmmain},\nwhich proceeds by studying a sequence of auxiliary Brownian\ncontrol problems. The next subsection offers a heuristic derivation and\na justification for the relevance of the sequence of Brownian control\nproblems to be considered in later sections.\n\n\\subsection{Toward a Brownian control problem}\nWe proceed by deriving a sequence of approximating Brownian control\nproblems heuristically, which will be instrumental in deriving a\nnear-optimal policy for our original control problem. It is important\nto note that we derive an approximating Brownian control problem for\neach $n$ as opposed to deriving a single approximating problem (for the\nentire sequence of problems). This distinction is crucial for achieving\nan improved optimality gap for $n$ large because it allows us to tailor\nthe approximation to each element of the sequence of systems.\n\nTo this end, let\n\\[\nl_i^n=\\lambda_i^n-\\mu_i\\nu_i n \\qquad\\mbox{for } i\\in\\I.\n\\]\nFixing an admissible control $U^n$ for the $n$th system [and\ncentering as in (\\ref{eqtildeXdefin})], we can then write (\\ref\n{eqdynamics2}) as\n\\begin{equation}\\label{eqcheckXdynamics}\n\\check{X}_i\\lam(t)=\\check{X}_i\\lam(0)+l_i\\lam t-\\mu\n_i\\int_0^t\n\\bigl( \\check{X}_i\\lam(s)-U^n_i(s)\\bigl( e\\cdot\\check{X}\\lam(s)\\bigr)^+\n\\bigr) \\,ds+\\check{W}_i\\lam(t),\\hspace*{-30pt}\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{eqWtildedefin}\n\\check{W}_i\\lam(t)&=&\\mN_i^a(\\lambda_i^n t)-\\lambda_i^n\nt+\\mu_i\\int_0^t \\bigl( \\check{X}_i\\lam(s)-U^n_i(s)\\bigl( e\\cdot\\check\n{X}\\lam(s)\\bigr)^+ \\bigr) \\,ds\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&{}-\n\\mN_i^d\\biggl(\\mu_i\\int_0^t \\bigl( \\check{X}_i\\lam(s)+\\nu_i\nn-U^n_i(s)\\bigl( e\\cdot\\check{X}\\lam(s)\\bigr)^+ \\bigr) \\,ds \\biggr).\\nonumber\n\\end{eqnarray}\nIn words, $\\check{W}_i\\lam(t)$ captures the deviations of the Poisson\nprocesses from their means. It is\\vspace*{1pt} natural to expect that an\napproximation result of the following form will hold: $(\\check\n{X}_i^n,\\check{W}_i^n;i\\in\\I)$ can be approximated by $(\\hX\n_i^n,\\hW_i^n;i\\in\\I)$ where\n\\begin{eqnarray*}\n\\hat{X}_i\\lam(t)&=&\\hat{X}_i\\lam(0)+l_i^n t -\\mu_i\\int_0^t \\bigl(\n\\hat{X}_i\\lam(s)-U^n_i(s)\\bigl(e\\cdot\\hat{X}\\lam(s)\\bigr)^+ \\bigr) \\,ds+\\hat\n{W}_i\\lam(t),\n\\\\\n\\hat{W}_i(t)&=&\\tilde{B}_i^a(\\lambda_i^n t)+\\tilde{B}_i^S\\biggl(\\mu\n_i\\int_0^t \\bigl( \\hat{X}_i\\lam(s)+\\nu_i n-U^n_i(s)\\bigl( e\\cdot\\hat\n{X}\\lam(s)\\bigr)^+ \\bigr) \\,ds\\biggr)\n\\end{eqnarray*}\nand $\\tilde{B}^a,\\tilde{B}^s$ are $I$-dimensional independent\nstandard Brownian motions. Moreover, by a time-change argument we can\nwrite (see, e.g., Theorem 4.6 in \\cite{KaS91})\n\\begin{eqnarray}\\label{eqhatX}\n\\hat{X}_i\\lam(t) &=&\\hat{X}_i\\lam\n(0)+l_i^n t -\\mu_i\\int_0^t \\hat{X}_i\\lam(s)-U^n_i(s)\\bigl(e\\cdot\\hat\n{X}\\lam(s)\\bigr)^+ \\,ds\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&{}+\\int_0^t \\sqrt{\\lambda_i^n+\\mu_i\\bigl( \\hat\n{X}_i\\lam(s)+\\nu_i n-U^n_i(s)\\bigl( e\\cdot\\hat{X}\\lam(s)\\bigr)^+\n\\bigr)}\\,dB_i(s),\\nonumber\\hspace*{-30pt}\n\\end{eqnarray}\nwhere $B$ is an $I$-dimensional standard Brownian motion constructed by setting\n\\begin{eqnarray*}B_i(t)&=& \\int_0^t\\frac{d\\tilde{B}_i^{S}(\\mu\n_i\\int_0^s ( \\hat{X}_i\\lam(u)+\\nu_i n-U^n_i(u)( e\\cdot\\hat\n{X}\\lam(u))^+ ) \\,du)}{\\sqrt{\\mu_i ( \\hat\n{X}_i\\lam(s)+\\nu_i n-U^n_i(s)( e\\cdot\\hat{X}\\lam(s))^+\n)}}\\\\\n&&{} + \\frac{\\tilde{B}_i^a(\\lambda_i^nt)}{\\lambda_i^n t}.\n\\end{eqnarray*}\n\nTaking a leap of faith and arguing heuristically, we next consider a\nBrownian control problem with the system dynamics\n\\begin{equation}\\label{eqnbmdefn1}\n\\hat{X}\\lam(t)=x+\\int_0^t b\\lam(\\hat{X}\\lam\n(s),\\hat\n{U}^n(s))\\,ds+\\int_0^t \\sigma\\lam(\\hat{X}\\lam(s),\\hat\n{U}^n(s))\\,dB(t),\\hspace*{-30pt}\n\\end{equation}\nwhere $\\hat{U}^n$ will be\nan admissible control for the Brownian system and\n\\begin{equation}\n\\label{eqnbmdefn30}\nb_i\\lam(x,u)=l_i\\lam-\\mu_i\\bigl(x_i-u_i(e\\cdot x)^+\\bigr)\n\\end{equation}\nand\n\\begin{equation}\n\\label{eqnbmdefn3}\n\\sigma_i\\lam(x,u)=\\sqrt{\\lambda_i^n+\\mu_i \\nu_in +\\mu\n_i\\bigl(x_i-u_i(e\\cdot x)^+\\bigr)}.\n\\end{equation}\nNote that the Brownian control problem will only be used to propose a~candidate policy, whose near optimality will be verified from first\nprinciples without relying on the heuristic derivations of this section.\n\nTo repeat, the preceding definition is purely formal and provided only\nas a means of motivating our approach. In what follows, we will\ndirectly state and analyze an auxiliary Brownian control problem\nmotivated by the above heuristic derivation. The analysis of the\nauxiliary Brownian control problem lends itself to constructing near\noptimal policies for our original control problem. To be more specific,\nthe system dynamics equation (\\ref{eqnbmdefn1}), and in particular,\nthe fact that its variance is state and control dependent, is crucial\nto our results. Indeed, it is this feature of the auxiliary Brownian\ncontrol problems that yields an improved optimality gap.\n\nNeedless to say, one needs to take care in interpreting (\\ref\n{eqnbmdefn1})--(\\ref{eqnbmdefn3}), which are meaningful only up\nto a suitably defined hitting time. In particular, to have $\\sigma^n$\nwell defined, we restrict attention to the process while it is within\nsome bounded domain. Actually, it suffices for our purposes to fix\n$\\kappa>0$ and $m\\geq3$ and consider the Brownian control problem\nonly up to the hitting time of a ball of the form\n\\begin{equation}\\label{eqBkappadefin}\n\\mB_{\\kappa}^n=\\bigl\\{x\\in\\mathbb{R}^I\\dvtx |x|< \\kappa\n\\sqrt{n}\\log^m\nn\\bigr\\},\n\\end{equation}\nwhere \\mbox{$|\\cdot|$} denotes the Euclidian\nnorm. We will fix the constant $m$ throughout and suppress the\ndependence on $m$ from the notation. Setting\n\\begin{equation}\\label{eqnkappa}\nn(\\kappa)=\\inf\\{n\\in\n\\bbZ_+\\dvtx \\sigma^n(x,u)\\geq1 \\mbox{ for all } x\\in\\mB_{\\kappa}^n,\nu\\in\\mathcal{U}\\},\n\\end{equation}\nthe diffusion coefficient is strictly positive for all $n \\geq n(\\kappa\n)$ and $x\\in\\mB_{\\kappa}^n$. Note that, for all $i\\in\\I$, $x\\in\n\\mB_{\\kappa}^n$ and $u\\in\\mathcal{U}$,\n\\[\n(\\sigma_i^n(x,u))^2\\geq\\lambda_i^n+\\mu_i\\nu_in-2\\mu_i\\kappa\\sqrt\n{n}\\log^mn,\n\\]\nso that $(\\sigma_i^n(x,u))^2\\geq\\mu_i\\nu_in\/2\\geq1$ for all\nsufficiently large $n$ and, consequently, $n(\\kappa)<\\infty$.\n\\begin{rem} In what follows, and, in particular, through the proof of\nTheorem \\ref{thmmain}, the reader should note that while choosing the\nsize of the ball to be $\\epsilon n$ (with $\\epsilon$ small enough)\nwould suffice for the nondegeneracy of the diffusion coefficient, that\nchoice would be too large for our optimality gap proofs.\n\\end{rem}\n\n\\section{An approximating diffusion control problem (ADCP)}\\label{secresult}\n\nMotivated by the discussion in the preceding section, we define\nadmissible systems as follows.\n\\begin{defin}[(Admissible systems)]\\label{definadmissiblesystemBrownian}\nFix $\\kappa>0$, $n\\in\\bbZ\n_+$ and $x\\in\\mathbb{R}^I$. We refer to\n$\\theta=(\\Omega,\\mathcal{F},(\\mathcal{F}_t),\\mathbb{P},\\hat\n{U},B)$ as an admissible $(\\kappa,n)$-system if:\n\\begin{longlist}[(a)]\n\\item[(a)] $(\\Omega,\\mathcal{F},(\\mathcal{F}_t),\\mathbb{P})$ is\na complete filtered probability space.\\vadjust{\\goodbreak}\n\\item[(b)] $B(\\cdot)$ is an $I$-dimensional standard Brownian motion\nadapted to $(\\mF_t)$.\n\\item[(c)] $\\hat{U}$ is $\\mathcal{U}$-valued, $\\mF$-measurable\nand $(\\mF_t)$ progressively measurable.\n\\end{longlist}\n\nThe process $\\hat{U}$ is said to be the control associated with\n$\\theta$. We also say that $\\hat{X}$ is a controlled process\nassociated with the initial data $x$ and an admissible system $\\theta$\nif $\\hat{X}$ is a continuous $(\\mF_t)$-adapted process on $(\\Omega\n,\\mathcal{F},\\mathbb{P})$ such that, almost surely, for $t\\leq\\hat\n{\\tau}_{\\kappa}^n$,\n\\[\n\\hat{X}(t)=x+\\int_0^t b\\lam(\\hat{X}(s),\\hat{U}(s))\\,ds+\\int_0^t\n\\sigma\\lam(\\hat{X}(s),\\hat{U}(s))\\,d\\tilde{B}(t),\n\\]\nwhere $b^n(\\cdot,\\cdot)$ and $\\sigma^n(\\cdot,\\cdot)$ are as\ndefined in (\\ref{eqnbmdefn30}) and (\\ref{eqnbmdefn3}),\nrespectively, and\n$\\hat{\\tau}_{\\kappa}^n=\\inf\\{t\\geq0\\dvtx\\hat{X}(t)\\notin\\mB_{\\kappa\n}^n\\}$. Given $\\kappa>0$ and $n\\in\\bbZ_+$, we let $\\Theta(\\kappa\n,n)$ be the set of admissible $(\\kappa,n)$-systems.\n\\end{defin}\n\nThe Brownian control problem then corresponds to optimally choosing an\nadmissible $(\\kappa,n)$-system with associated control $(\\hat\n{U}(t),t\\geq0)$ that achieves the minimal cost in the optimization problem\n\\begin{equation}\\label{eqoptbrownian}\n\\hat{V}\\lam(x,\\kappa)=\\inf_{\\theta\\in\\Theta\n(\\kappa,n)}\\Ex\n_{x}^{\\theta}\\biggl[\\int_0^{\\hat{\\tau}_{\\kappa}^n} e^{-\\gamma s}\n\\sum_{i\\in\\I}c_i \\hat{U}_i(s)\\bigl(e\\cdot\\hat{X}(s)\\bigr)^+\\,ds\n\\biggr],\n\\end{equation}\nwhere $\\Ex_x^{\\theta}[\\cdot]$ is the\nexpectation operator when the initial state is $x\\in\\mathbb{R}^I$ and\nthe admissible system $\\theta$. Hereafter, we refer to (\\ref\n{eqoptbrownian}) as the \\textit{ADCP on} $\\mB_{\\kappa}^n$.\nThe following lemma shows that the Definition \\ref\n{definadmissiblesystemBrownian} is not vacuous. The proof appears in\nthe \\hyperref[app]{Appendix}.\n\\begin{lem} \\label{lemexistenceofcontrolled}\nFix the initial state $x\\in\\mathbb{R}^I$, $\\kappa>0$,\n$n\\geq n(\\kappa)$ and an admissible $(\\kappa,n)$-system $\\theta$.\nThen, there exists a unique controlled process $\\hat{X}$ associated\nwith $x$ and $\\theta$.\n\\end{lem}\n\nTo facilitate future analysis, note from the definition of $\\hat{\\tau\n}_k^n$ and (\\ref{eqoptbrownian}) that\n\\begin{equation}\\label{eqvaluebound}\n\\hat{V}^n(x,\\kappa)\\leq\n\\frac{1}{\\gamma}(e\\cdot c)\\kappa\\sqrt{n}\\log^m n.\n\\end{equation}\n\\begin{defin}[(Markov controls)]\n\\label{definmarkoviancontrols} We say that an admissible\n$(\\kappa,n)$-system $\\theta=(\\Omega,\\mathcal{F},(\\mathcal\n{F}_t),\\mathbb{P},\\hat{U},B)$ with the associated controlled process\n$\\hat{X}^n$ induces a Markov control if there exists a function\n$g^n(\\cdot)\\dvtx \\mathcal{B}_{\\kappa}^n \\to\\mathcal{U}$ such that\n$\\hat{U}(t)=g^n(\\hat{X}^n(t))$ for $t \\leq\\hat{\\tau}_{\\kappa}^n$.\nWe extend the function $g^n$ to $\\mathbb{R}^I$ as follows:\n\\begin{equation}\nh^n(x)= \\cases{\ng^n(x), &\\quad$x \\in\\mathcal{B}_{\\kappa}^{n}$, \\cr\ne_1, & \\quad otherwise,}\n\\end{equation}\nwhere $e_1$ is the $I$-dimensional vector whose first component is $1$\nwhile the others are $0$. We refer to $h^n(\\cdot)$ as the tracking\nfunction associated with the admissible system $\\theta$.\n\\end{defin}\n\nIn what follows, a policy $\\hat{U}$ will be called optimal for the\napproximating diffusion control problem (ADCP) on $\\mB_{\\kappa}^n$ if\nthere exists an admissible $(\\kappa,n)$-system $\\theta=(\\Omega\n,\\mathcal{F},(\\mathcal{F}_t),\\mathbb{P},\\hat{U},B)$ such that\n\\[\n\\hat{V}\\lam(x,\\kappa)=\\Ex_{x}^{\\theta}\\biggl[\\int_0^{\\hat{\\tau\n}_{\\kappa}^n} e^{-\\gamma s} \\sum_{i\\in\\I}c_i \\hat{U}_i(s)\\bigl(e\\cdot\n\\hat{X}(s)\\bigr)^+\\,ds\\biggr].\n\\]\n\nRecall that $X$ and $U$ are used to denote performance relevant\nstochastic processes in both the Brownian model and the original\nqueueing model, and that we add a hat, that is, we use $\\hat{X}$ and\n$\\hat{U}$ in the context of the Brownian model. To avoid confusion,\nthe reader should keep in mind that hat-processes correspond to the\nADCP while the ones with no hats correspond to the original queueing model.\n\n\\subsection*{Roadmap for the remainder of the paper} The main result in\nTheorem~\\ref{thmmain} builds on the following steps:\n\\begin{enumerate}\n\\item In Section \\ref{secADCP} we show that for each $n$, the HJB\nequation associated with the ADCP has a unique\\vadjust{\\goodbreak} and sufficiently smooth\nsolution. Using that solution we advance an optimal Markov control for\nthe ADCP together with the corresponding tracking function. We also\nidentify useful gradient bounds on the solutions to the sequence of HJB\nequations; cf. Theorem~\\ref{thmHJB1sol}.\n\n\\item In Section \\ref{sectracking} we conduct a performance analysis\nof $h$-tracking policies in the queueing system; cf. Theorem \\ref{thmSSC}.\n\n\\item The result of Theorem \\ref{thmSSC} together with the gradient\nestimates in Theorem \\ref{thmHJB1sol} are combined in a\nTaylor expansion-type argument in Section \\ref{seccombining} to\ncomplete the proof of Theorem \\ref{thmmain}.\n\\end{enumerate}\n\nAs a convention, throughout the paper we use the capital letter $C$ to\ndenote a constant that does not depend on $n$. The value of $C$ may\nchange from line to line within the proofs but it will be clear from\nthe context.\n\n\\section{Solution to the ADCP} \\label{secADCP} This section provides\na solution for the ADCP on $\\mB_{\\kappa}^n$ for each $n\\in\\bbZ$ and\n$\\kappa>0$. The HJB equation is a fully nonlinear and nonsmooth PDE.\nAs such, it requires extra care when compared with the usual semilinear\nPDEs that arise in the analysis of \\textit{asymptotically} optimal\ncontrols in the Halfin--Whitt regime. We will build on existing results\nin the theory of PDEs and proceed through the following steps: (a)\nestablish the existence and uniqueness of classical solutions; (b)\nrelate this unique solution to the value function of the ADCP and (c)\nestablish useful gradient estimates on the solution for the HJB\nequation. The last step is not necessary for existence and uniqueness\nbut is important for the analysis of optimality gaps.\n\nIn what follows, we fix $\\kappa>0$ and $n\\geq n(\\kappa)$ and suppress\nthe dependence of the solution to the HJB equation on $n$ and $\\kappa\n$. The following notation is needed to introduce the HJB equation.\nGiven a twice continuously differentiable function $\\phi$, define\n\\[\n\\phi_i=\\frac{\\partial\\phi}{\\partial x_i}\\quad\\mbox{and}\\quad \\phi_{ii}=\n\\frac{\\partial^2 \\phi}{\\partial x_i^2} .\n\\]\nAlso, define the operator $A^n_u$ for $u\\in\\mathcal{U}$ as follows:\n\\begin{equation}\\label{eqgendefin}\nA_{u}\\lam\\phi= \\sum_{i\\in\\I} b_i\\lam(\\cdot\n,u)\\phi_i+\\frac\n{1}{2}\\sum_{i\\in\\I} (\\sigma_i\\lam(\\cdot,u))^2 \\phi_{ii}.\n\\end{equation}\nDefining\n\\[\nL(x,u)=\\sum_{i\\in\\I}c_iu_i (e\\cdot x)^+\n\\]\nfor $x\\in\\bbR_+^I$ and $u\\in\\mathcal{U}$, the HJB equation is given by\n\\begin{equation} \\label{eqHJB0}\n0=\\inf_{u \\in\\mathcal{U}}\\{\nL(x,u)+A_{u}\\lam\\phi\n(x)-\\gamma\\phi(x)\\}.\n\\end{equation}\nSubstituting $b\\lam(\\cdot,\\cdot)$ and $\\sigma\\lam(\\cdot\n,\\cdot)$ into (\\ref{eqHJB0}) gives\n\\begin{eqnarray} \\label{eqHJB1simp}\n0&=& -\\gamma\\phi(x) + (e\\cdot x)^+\\cdot\\min_{i\\in\\I}\\biggl\\{\nc_i+\\mu_i\\phi_i(x)-\\frac{1}{2}\\mu_i\\phi_{ii}(x)\\biggr\\}\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&{}+\\sum_{i\\in\\I} (l_i\\lam-\\mu_ix_i)\\phi_i(x)\n+\\frac{1}{2}\\sum_{i\\in\\I} \\bigl(\\lambda_i^n+\\mu_i(\\nu\n_in+x_i)\\bigr)\\phi_{ii}(x).\\nonumber\n\\end{eqnarray}\n\nOur analysis of the HJB equation (\\ref{eqHJB1simp}) draws on existing\nresults on fully nonlinear PDEs, and, in particular, the results on\nBellman--Pucci type equations; cf. Chapter 17 of \\cite{TandG}.\n\nIn what follows, fixing a set $\\mB\\subseteq\\mathbb{R}_+^I$, $\\mC\n^2(\\mB)$ denotes the space of twice continuously differentiable\nfunctions from $\\mB$ to $\\mathbb{R}$. For $u\\in\\mC^{2}(\\mB)$, we\nlet~$Du$ and $D^2u$ denote the gradient and the Hessian of $u$,\nrespectively. The space $\\mC^{2,\\alpha}(\\mB)$ is then the subspace\nof $\\mC^{2}(\\mB)$ members of which also have second derivatives that\nare H\\\"{o}lder continuous of order $\\alpha$. That is, a twice\ncontinuously differentiable function $u\\dvtx \\mathbb{R}^I\\to\\mathbb{R}$\nis in $\\mC^{2,\\alpha}(\\mB)$ if\n\\[\n\\sup_{x,y\\in\\mB, x\\neq y} \\frac{|D^2u(x)-D^2u(y)|}{|x-y|^{\\alpha\n}}<\\infty,\n\\]\nwhere \\mbox{$|\\cdot|$} denotes the Euclidian norm. We define\n$d_x\\,{=}\\,\\operatorname{dist}(x,\\partial\\mB)\\,{=}\\,\\inf\\{|x\\,{-}\\,y|,\\allowbreak\ny\\,{\\in}\\,\\partial\\mB\\}$ where\n$\\partial\\mB$ stands for the boundary of $\\mB$ and we let\n$d_{x,z}\\,{=}\\,\\min\\{d_x,d_z\\}$. Also, we define\n\\begin{equation}\\label{equstardefin}\n|u|^*_{2,\\alpha,\\mB}=\\sum_{j=0}^2 [u]_{j,\\mB}^*+\\sup_{x,y\\in\\mB,x\\neq\ny}d_{x,y}^{2+\\alpha}\\frac{|D^2u(x)-D^2 u(y)|}{|x-y|^{\\alpha}},\n\\end{equation}\nwhere $[u]_{j,\\mB}^*=\\sup_{x\\in\\mB}d_x^j\n|D^j u(x)|$ for $j=0,1,2$. Note that $d_x^j$ denote the $j$th power\nof $d_x$ and, similarly, $d_{x,y}^{2+\\alpha}$ is the $(2+\\alpha)$th\npower of $d_{x,y}$. Finally, we let $|u|^*_{0,\\mB}=[u]_{0,\\mB\n}^*=\\sup_{x\\in\\mB}|u(x)|$.\n\nIn the statement of the following theorem, $e_j$ is the $I$-dimensional\nvector with $1$ in the $j$th place and zeros elsewhere. Also, $\\mB\n_{\\kappa}^n$, $m$ and $n(\\kappa)$ are as defined in (\\ref\n{eqBkappadefin}) and (\\ref{eqnkappa}), respectively.\n\\begin{theorem}\\label{thmHJB1sol}\nFix $\\kappa>0$ and $n\\geq n(\\kappa)$. Then, there\nexists $0<\\alpha\\leq1$ (that does not depend on $n$) and a unique\nclassical solution $\\phi_{\\kappa}\\lam\\in\\mC^{0,1}(\\bar{\\mB}\n_{\\kappa}^n)\\cap\\mC^{2,\\alpha}(\\mB_{\\kappa}^n)$ to the HJB\nequation (\\ref{eqHJB1simp}) on $\\mB_{\\kappa}^n$ with the\nboundary condition $\\phi_{\\kappa}\\lam=0$ on $\\partial\\mB_{\\kappa\n}^n$. Furthermore, there exists a constant $C>0$ (that does not depend\non $n$) such that\n\\begin{equation}\\label{eqgradients0}\n|\\phi_{\\kappa}\\lam|^*_{2,\\alpha,\\mB_{\\kappa\n}^n}\\leq C\\sqrt\n{n}\\log^{k_0} n,\n\\end{equation}\nwhere $k_0=4m(1+1\/\\alpha)$. In turn, for any $\\vartheta<1$,\n\\begin{equation}\\label{eqgradients1}\n\\sup_{x\\in\\mB_{\\vartheta\\kappa}^n}|D\\phi_{\\kappa\n}^n(x)|\\leq\n\\frac{C}{1-\\vartheta}\\log^{k_1} n \\quad\\mbox{and}\\quad\\sup_{x\\in\\mB\n_{\\vartheta\\kappa}^n}|D^2\\phi_{\\kappa}^n(x)|\\leq\\frac\n{C}{1-\\vartheta}\\frac{\\log^{k_2}n} {\\sqrt{n}}\\hspace*{-26pt}\n\\end{equation}\nwith $k_1=k_0-m$ and $k_2=k_0-2m$. Also,\n\\begin{eqnarray}\\label{eqgenbound}\n&&\\sup_{u\\in\\mathcal{U}}\\biggl|\\sum_{i\\in\\I}\\bigl((\\phi_{\\kappa\n}^n)_{ii}(y)-(\\phi_{\\kappa}^n)_{ii}(x)\\bigr)(\\sigma_i^n(x,u))^2\n\\biggr|\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\leq\\frac{C}{1-\\vartheta}\\log^{k_1}n\\nonumber\n\\end{eqnarray}\nfor all $x,y\\in\\mB_{\\vartheta\\kappa}^n$ with $|x-y|\\leq1$.\n\\end{theorem}\n\nNote that (\\ref{eqgradients1}) follows immediately from (\\ref\n{eqgradients0}) through the definition of the operation \\mbox{$|\\cdot\n|^*_{2,\\alpha,\\mB_{\\kappa}^n}$} in (\\ref{equstardefin}).\nHenceforth, we will use $k_i,i=0,1,2$ for the values given in the\nstatement of Theorem \\ref{thmHJB1sol}. Moreover, the constant $k$\nappearing in the statement of Theorem \\ref{thmmain} is equal to $k_0+3$.\n\nTheorem \\ref{thmHJB1sol} facilitates a verification result, which we\nstate next followed by the proof of Theorem \\ref{thmHJB1sol}. Below,\n$\\hat{V}^n(x,\\kappa)$ is the value function of the ADCP; cf. equation\n(\\ref{eqoptbrownian}).\n\\begin{theorem}\\label{thmBrownianverification}\nFix $\\kappa>0$ and $n\\geq n(\\kappa)$. Let $\\phi\n_{\\kappa}^n$ be the unique solution to the HJB equation (\\ref\n{eqHJB1simp}) on $\\mB_{\\kappa}^n$ with the boundary condition $\\phi\n_{\\kappa}\\lam=0$ on $\\partial\\mB_{\\kappa}^n$. Then,\n$\\phi_{\\kappa}^n(x)=\\hat{V}^n(x,\\kappa)$ for all $x\\in\\mB_{\\kappa\n}^n$. Moreover, there exists a Markov control which is optimal for the\nADCP on $\\mB_{\\kappa}^n$. The tracking function $h_{\\kappa}^{*,n}$\nassociated with this optimal Markov control is defined by $h_{\\kappa\n}^{*,n}(x)=e_{i^n(x)}$, where\n\\begin{equation}\\label{eqixdefin}\ni^n(x)=\\min\\mathop{\\argmin}_{i\\in\\I} \\biggl\\{ \\biggl(c_i+\\mu\n_i(\\phi\n_{\\kappa}\\lam)_i(x)-\\frac{1}{2}\\mu_i(\\phi_{\\kappa}\\lam\n)_{ii}(x)\\biggr)(e\\cdot x)^+\\biggr\\}.\\hspace*{-25pt}\n\\end{equation}\n\\end{theorem}\n\nThe HJB equation (\\ref{eqHJB1simp}) has two sources of\nnondifferentiability. The first source is the minimum operation and the\nsecond is the nondifferentiability of the term $(e\\cdot x)^+$. The\nfirst source of nondifferentiability is covered almost entirely by the\nresults in \\cite{TandG}. To deal with the nondifferentiability of the\nfunction $(e\\cdot x)^+$, we use a construction by approximations. The\nproof of existence and uniqueness in Theorem \\ref{thmHJB1sol} follows\nan approximation scheme where one replaces the nonsmooth function\n$(e\\cdot x)^+$ by a smooth (parameterized by $a$) function $f_a(e\\cdot\nx)$. We show that the resulting ``perturbed'' PDE has a unique\nclassical solution and that as $a\\tinf$ the corresponding sequence of\nsolutions converges, in an appropriate sense, to a solution to~(\\ref\n{eqHJB1simp}) which will be shown to be unique. Note that this argument\nis repeated for each fixed $n$ and $\\kappa$.\n\nTo that end, given $a>0$, define\n\\begin{equation}\\label{eqfdefin}\nf_a(y)=\\cases{\ny, &\\quad$\\displaystyle y\\geq\\frac{1}{4a}$,\\vspace*{2pt}\\cr\n\\displaystyle ay^2+\\frac{1}{2}y+\\frac{1}{16a}, &\\quad$\\displaystyle -\\frac{1}{4a}\\leq y\\leq\\frac\n{1}{4a}$,\\vspace*{2pt}\\cr\n0, & \\quad otherwise.}\n\\end{equation}\nReplacing $(e\\cdot x)^+$ with $f_a(e\\cdot x)$ in (\\ref{eqHJB1simp})\ngives the following equation:\n\\begin{eqnarray} \\label{eqHJB2}\n0&=& - \\gamma\\phi(x)+f_a(e\\cdot x)\\cdot\\min_{i\\in\n\\I} \\biggl\\{ c_i+\\mu_i\\phi_i(x)-\\frac{1}{2}\\mu_i\\phi_{ii}(x)\n\\biggr\\} \\nonumber\\\\[-8pt]\\\\[-8pt]\n&&{}+\\sum_{i\\in\\I} (l_i\\lam-\\mu_ix_i)\\phi_i(x)\n+\\frac{1}{2}\\sum_{i\\in\\I} \\bigl(\\lambda_i^n+\\mu_i(\\nu\n_in+x_i)\\bigr)\\phi_{ii}(x).\\nonumber\n\\end{eqnarray}\n\nTo simplify this further, let $\\Gamma= \\mB_{\\kappa}^n\\times\\mathbb\n{R}_+\\times\\mathbb{R}^I\\times\\mathbb{R}^{I\\times I}$ and for all\n$y\\in\\Gamma$, define the function\n\\begin{equation}\\label{eqFForm}\nF_a^k[y]=\\min\\{F_a^1[y],\\ldots, F_a^I[y]\\},\n\\end{equation}\nwhere for $k\\in\\I$ and $y=(x,z,p,r)\\in\\Gamma$,\n\\begin{eqnarray}\\label{eqFdefin}\nF^k_a[y]&=&f_a(e\\cdot x)\n\\biggl[c_k+\\mu_kp_k-\\frac{1}{2}\\mu_kr_{kk}\\biggr]+\\sum_{i\\in\\I\n}(l_i\\lam-\\mu_ix_i)p_i\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&{}+\\frac{1}{2}\\sum_{i\\in\\I} \\bigl(\\lambda_i^n+\\mu_i(\\nu\n_in+x_i)\\bigr)r_{ii}-\\gamma z.\\nonumber\n\\end{eqnarray}\nThen, (\\ref{eqHJB2}) can be rewritten as\n\\begin{equation}\\label{eqFFormPDe}\nF_a[x,u(x),Du(x),D^2u(x)]=0.\n\\end{equation}\n\nIn the following statement we use the gradient notation introduced at\nthe beginning of this section.\n\\begin{prop}\\label{propsolPHJB} Fix $\\kappa>0$, $n\\geq n(\\kappa)$ and $a>0$. A unique\nclassical solution $\\phi_{\\kappa,a}^n\\in\\mC^{0,1}(\\bar{\\mB}\n_{\\kappa}^n)\\cap\\mC^{2,\\alpha}(\\mB_{\\kappa}^n)$ exists for\nthe PDE (\\ref{eqHJB2}) on $\\mB_{\\kappa}^n$ with the boundary\ncondition $\\phi_{\\kappa,a}^n=0$ on $\\partial\\mB_{\\kappa}^n$. Moreover,\n\\begin{equation}\\label{eqgradients}\n|\\phi^n_{\\kappa,a}|^*_{2,\\alpha,\\mB_{\\kappa\n}^n}\\leq C |\\phi\n^n_{\\kappa,a}|^*_{0,\\mB_{\\kappa}^n}\\log^{k_0}n \\leq\\tilde{C}\n\\end{equation}\nfor $k_0=4m(1+1\/\\alpha)$ where\n$0<\\alpha\\leq1$ and $C>0$ do not depend on $a$ and $n$ and $\\tilde\n{C}$ does not depend on $a$. Also, $\\phi^n_{\\kappa,a}$ is Lipschitz\ncontinuous on the closure $\\bar{\\mB}_{\\kappa}^n$ with a\nLipschitz constant that does not depend on $a$ (but can depend on\n$\\kappa$ and $n$).\n\\end{prop}\n\nWe postpone the proof of Proposition \\ref{propsolPHJB} to the \\hyperref[app]{Appendix}\nand use it to complete the proof of Theorem \\ref{thmHJB1sol},\nfollowed by the proof of Theorem \\ref{thmBrownianverification}.\n\n\\subsection*{\\texorpdfstring{Proof of Theorem \\protect\\ref{thmHJB1sol}}{Proof of Theorem 4.1}}\n\nSince we fix $n$ and $\\kappa$, they will be suppressed below. We\nproceed to show the existence by an approximation argument. To that\nend, fix a sequence $\\{a^k;k\\in\\bbZ\\}$ with $a^k\\tinf$ as $k\\tinf$\nand let $\\phi_{a^k}$ be the unique solution to (\\ref{eqHJB2}) as\ngiven by Proposition \\ref{propsolPHJB}. The next step is to show\nthat $\\phi_{a^k}$ has a subsequence that converges in an appropriate\nsense to a function $\\phi$, which is, in fact, a solution to the HJB\nequation (\\ref{eqHJB1simp}). To that end, let\n\\begin{equation}\\label{eqCstar}\n\\mC_{*}^{2,\\alpha\n}(\\mB)=\\{u\\in\\mC^{2,\\alpha}(\\mB)\\dvtx |u|_{2,\\alpha,\\mB}^*<\\infty\\}.\n\\end{equation}\nThen, $\\mC_{*}^{2,\\alpha}(\\mB)$ is a Banach\nspace (see, e.g., Exercise 5.2 in \\cite{TandG}). Since the bound in\n(\\ref{eqgradients}) is independent of $a$, we have that $\\{\\phi\n_{a^k}\\}$ is a bounded sequence in $C_{*}^{2,\\alpha}(\\mB)$ and hence,\ncontains a convergent subsequence. Let $u$ be a limit point of the\nsequence $\\{\\phi_{a^k}\\}$. Since the gradient estimates in Proposition\n\\ref{propsolPHJB} are independent of $a$, they hold also for the\nlimit function $u$, that is,\n\\begin{equation}\\label{equ-bound}\n|u|_{2,\\alpha,\\mB}^*\\leq C|u|_{0,\\mB\n}^*\\log^{k_0} n\\leq\\tilde{C}\n\\end{equation}\nfor constants\n$\\alpha$ and $C$ that are independent of $n$. Proposition \\ref\n{propsolPHJB} also guarantees that the global Lipschitz constant is\nindependent of $a$ so that we may conclude that $u\\in\\mC\n^{0,1}(\\bar{\\mB})$ and that $u=0$ on $\\partial\\mB$.\n\nWe will now show that $u$ solves (\\ref{eqHJB1simp}) uniquely. To show\nthat $u$ solves (\\ref{eqHJB1simp}), we need to show that $F[u]=0$\n(where $F[\\cdot]$ is defined similar to $F_a[\\cdot]$ with $(e\\cdot\nx)^+$ replacing $f_a(e\\cdot x)$). To that end, let $\\{a^k,k\\in\\bbZ\\}\n$ be the corresponding convergent subsequence [i.e., such that $\\phi\n_{a^k}\\rightarrow u$ in $\\mC_*^{2,\\alpha}(\\mB)$]. Henceforth, to\nsimplify notation, we write\n\\[\nF_{\\akl}[\\phi_{\\akl}(x)]=F_{\\akl}[x,\\phi_{\\akl}(x),D\\phi_{\\akl\n}(x),D^2\\phi_{\\akl}(x)]\n\\]\n(and similarly for $F[\\cdot]$). Fix $\\delta\\,{>}\\,0$ and let $\\mB(\\delta\n)\\,{=}\\,\\{x\\,{\\in}\\,\\bbR^I\\dvtx |x|\\,{<}\\,\\kappa\\sqrt{n}\\log^m n\\,{-}\\,\\delta\\}$. Note that\nsince $\\phi_{\\akl}\\rightarrow u$ in $\\mC_*^{2,\\alpha}(\\mB)$ we\nhave, in particular, the convergence of $(\\phi_{\\akl}(x),D\\phi_{\\akl\n}(x),D^2\\phi_{\\akl}(x))\\rightarrow(u(x),Du(x),D^2u(x))$ uniformly in\n$x\\in\\mB(\\delta)$. The equicontinuity of the function\n$F^a[\\cdot]$\\vadjust{\\goodbreak}\non $\\Gamma$ guarantees then that\n\\begin{equation}\\label{eqinterim11}\n|F_{\\akl}[\\phi_{\\akl}(x)]-F_{\\akl}[u(x)]|\\leq\n\\epsilon\n\\end{equation}\nfor all $l$ large enough and $x\\in\\mB(\\delta)$.\nNote that $\\sup_{x\\in\\bbR^I}|f_{a}(e\\cdot x)-(e\\cdot x)^+|\\leq\n\\epsilon$ for all $a$ large enough so that,\n\\begin{equation}\\label{eqinterim12}\n\\sup_{x\\in\\mB}|F_{\\akl}[u(x)]-F[u(x)]|\\leq\\epsilon\n\\end{equation}\nfor all $l$ large enough. Combining (\\ref{eqinterim11}) and (\\ref\n{eqinterim12}), we then have\n\\[\n\\sup_{x\\in\\mB}|F_{\\akl}[\\phi_{\\akl}(x)]-F[u(x)]|\\leq2\\epsilon\n\\]\nfor all $l$ large enough and $x\\in\\mB(\\delta)$. By definition\n$F^{a^k}[\\phi_{\\akl}(x)]=0$ for all $x\\in\\mB$ and since $\\epsilon$\nwas arbitrary we have that $F[u(x)]=0$ for all $x\\in\\mB(\\delta)$.\nFinally, since $\\delta$ was arbitrary we have that $F[u(x)]=0$ for all\n$x\\in\\mB$. We already argued that $u=0$ on $\\partial\\mB$, so that\n$u$ solves (\\ref{eqHJB1simp}) on~$\\mB$ with $u=0$ on~$\\partial\\mB\n$. This concludes the proof of existence of a solution to (\\ref\n{eqHJB1simp}) that satisfies the gradient estimates (\\ref{eqgradients0}).\n\nFinally, the uniqueness of the solution to (\\ref{eqHJB1simp}) follows\nfrom Corollary 17.2 in \\cite{TandG} noting that the function\n$F[x,z,p,r]$ is indeed continuously differentiable in the $(z,p,r)$\narguments and it is decreasing in $z$ for all $(x,p,r)$.\n\nUsing Theorem \\ref{thmBrownianverification} [which only uses the\nexistence and uniqueness of the solution $\\phi_{\\kappa}^n(x)$ that we\nalready established] together with (\\ref{eqvaluebound}) we have that\n\\[\n|\\phi_{\\kappa}^n|_{0,\\mB_{\\kappa}^n}=\\sup_{x\\in\\mB_{\\kappa\n}^n}\\hat{V}^n(x,\\kappa)\\leq\\frac{1}{\\gamma}\\kappa\\sqrt{n}\\log^m n.\n\\]\nThe bounds (\\ref{eqgradients0}) and (\\ref{eqgradients1}) now follow\nfrom (\\ref{equ-bound}) and we turn to prove (\\ref{eqgenbound}).\n\nTo that end, since $\\phi_{\\kappa}^n$ solves\n(\\ref{eqHJB1simp}), fixing $x,y\\in\\mB_{\\kappa}^n$ we have\n\\begin{eqnarray}\\label{eqinternational}\\quad\n&&\\biggl|\\frac{1}{2}\\sum_{i\\in\\I}\n\\bigl(\\lambda_i^n+\\mu_i(\\nu_in+x_i)\\bigr)(\\phi_{\\kappa}^n)_{ii}(x)-\n\\frac{1}{2}\\sum_{i\\in\\I} \\bigl(\\lambda_i^n+\\mu_i(\\nu\n_in+y_i)\\bigr)(\\phi_{\\kappa}^n)_{ii}(y)\\biggr|\\nonumber\\\\\n&&\\qquad \\leq\\gamma|\\phi_{\\kappa}^n(x)-\\phi_{\\kappa\n}^n(y)|\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\quad{} +\\biggl|(e\\cdot x)^+\\cdot\\min\n_{i\\in\\I}\\biggl\\{ c_i+\\mu_i(\\phi_{\\kappa}^n)_i(x)-\\frac{1}{2}\\mu\n_i(\\phi_{\\kappa}^n)_{ii}(x)\\biggr\\}\\nonumber\\\\\n&&\\qquad\\quad\\hphantom{{}+\\biggl|}{} -(e\\cdot y)^+\\cdot\\min_{i\\in\\I}\\biggl\\{\nc_i+\\mu_i(\\phi_{\\kappa}^n)_i(y)-\\frac{1}{2}\\mu_i(\\phi_{\\kappa\n}^n)_{ii}(y)\\biggr\\}\\biggr|.\\nonumber\n\\end{eqnarray}\nWe will now bound each of the elements on the right-hand side. To that\nend, let $i(x)$ be as defined in (\\ref{eqixdefin}) and for each\n$x,z\\in\\mB_{\\vartheta\\kappa}^n$ define\n\\[\nM_{i(x)}^n(z)=c_{i(x)}+\\mu_{i(x)}(\\phi_{\\kappa}^n)_{i(x)}(z)-\\tfrac\n{1}{2}\\mu_{i(x)}(\\phi_{\\kappa}^n)_{i(x)i(x)}(z).\n\\]\nUsing (\\ref{eqgradients1}), we have by the mean value theorem that\n\\begin{equation}\\label{eqinterinter}\n|\\phi_{\\kappa}^n(x)-\\phi_{\\kappa}^n(y)|\\leq\n{|x-y|\\max_{i\\in\\I\n}\\sup_{z\\in\\mB_{\\vartheta\\kappa}^n}}|(\\phi_{\\kappa}^n)_i(z)|\\leq\nC\\log^{k_1} n\\vadjust{\\goodbreak}\n\\end{equation}\nfor all $x,y\\in\\mB\n_{\\vartheta\\kappa}^n$ with $|x-y|\\leq1$, and we turn to bound the\nsecond element on the right-hand side of (\\ref{eqinternational}).\nHere, there are two cases to consider. Suppose first that\n$i(x)=i(y)=i$. Then, using (\\ref{eqgradients1}) and the mean value\ntheorem we have\n\\[\n|(\\phi_{\\kappa}^n)_{i}(x)-(\\phi_{\\kappa}^n)_{i}(y)|\\leq{|x-y|\\max\n_{i\\in\\I}\\sup_{z\\in\\mB_{\\vartheta\\kappa}^n}}|(\\phi_{\\kappa\n}^n)_{ii}(z)|\\leq C\\frac{\n\\log^{k_2} n}{\\sqrt{n}}\n\\]\nand, in turn, that\n\\begin{equation} \\label{eqMbound}\n|M_{i}^n(x)-M_{i}^n(y)|\\leq C\\frac{\\log^{k_2}\nn}{\\sqrt{n}}\n\\end{equation}\nfor all $x,y\\in\\mB_{\\vartheta\\kappa}^n$ with $|x-y|\\leq1$. Now,\n$|x|\\vee|y|\\leq\\kappa\\sqrt{n}\\log^m n$ for all $x,y\\in\\mB\n_{\\vartheta\\kappa}^n$ and, by (\\ref{eqgradients1}), $\\sup_{z\\in\n\\mB_{\\vartheta\\kappa}^n}|(\\phi_{\\kappa}^n)_{ii}(z)|\\vee|(\\phi\n_{\\kappa}^n)_{i}(z)|\\leq C\\log^{k_1}n$ so that\n\\begin{eqnarray}\\label{eqinterim222222}\n&& |(e\\cdot x)^+ M_i^n(x)-(e\\cdot y)^+\nM_i^n(y)|\\nonumber\\\\\n&&\\qquad\n\\leq\n\\kappa\\sqrt{n}\\log^m n | M_i^n(x)-M_i^n(y)|+\\sup_{z\\in\n\\mB_{\\vartheta\\kappa}^n}\n|M_i^n(z)|\\\\\n&&\\qquad\\leq C\\log^{k_1}n.\\nonumber\n\\end{eqnarray}\nIf, on the other hand, $i(x)\\neq i(y)$ then by the definition\nof $i(\\cdot)$,\n\\begin{eqnarray*}\n&&c_{i(x)}+\\mu_{i(x)}(\\phi_{\\kappa\n}^n)_{i(x)}(x)-\\tfrac{1}{2}\\mu_{i(x)}(\\phi_{\\kappa\n}^n)_{i(x)i(x)}(x)\\\\\n&&\\qquad \\leq\nc_{i(y)}+\\mu_{i(y)}(\\phi_{\\kappa}^n)_{i(y)}(x)-\\tfrac{1}{2}\\mu\n_{i(y)}(\\phi_{\\kappa}^n)_{i(y)i(y)}(x)\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n&& c_{i(y)}+\\mu_{i(y)}(\\phi_{\\kappa\n}^n)_{i(y)}(y)-\\tfrac{1}{2}\\mu_{i(y)}(\\phi_{\\kappa\n}^n)_{i(y)i(y)}(y)\\\\\n&&\\qquad \\leq\nc_{i(x)}+\\mu_{i(x)}(\\phi_{\\kappa}^n)_{i(x)}(y)-\\tfrac{1}{2}\\mu\n_{i(x)}(\\phi_{\\kappa}^n)_{i(x)i(x)}(y).\n\\end{eqnarray*}\nThat is,\n\\begin{equation} \\label{eqyetonemore1}\nM_{i(x)}^n(x)\\leq M_{i(y)}^n(x) \\quad\\mbox{and}\\quad\nM_{i(y)}^n(y)\\leq M_{i(x)}^n(y).\n\\end{equation}\nUsing\n(\\ref{eqgradients1}) as before we have for $x,y\\in\\mB_{\\vartheta\n\\kappa}^n$ with $|x-y|\\leq1$ and $i(x)\\neq i(y)$ that\n\\[\n\\bigl| M_{i(x)}^n(x)-M_{i(x)}^n(y)\\bigr|+\\bigl|\nM_{i(y)}^n(x)-M_{i(y)}^n(y)\\bigr| \\leq C\\frac{\\log^{k_2} n}{\\sqrt{n}}.\n\\]\nBy (\\ref{eqyetonemore1}) we then have that\n\\begin{eqnarray*}\n\\bigl| M_{i(x)}^n(x)-M_{i(y)}^n(y)\\bigr|&\\leq&\\bigl|\nM_{i(x)}^n(x)-M_{i(x)}^n(y)\\bigr|\\\\\n&&{}+ \\bigl|\nM_{i(y)}^n(x)-M_{i(y)}^n(y)\\bigr| \\\\\n&\\leq& C\\frac{\\log^{k_2}n}{\\sqrt{n}}\n\\end{eqnarray*}\nfor all such $x$ and $y$. In turn, since $|x|\\vee|y|\\leq\\kappa\\sqrt\n{n}\\log^m n$,\n\\begin{equation}\\label{eqinterim222224}\n\\bigl| (e\\cdot x)^+M_{i(x)}^n(x)- (e\\cdot\ny)^+M_{i(y)}^n(y)\\bigr|\\leq C\\log^{k_1} n\n\\end{equation}\nfor $x,y\\in\\mB_{\\vartheta\\kappa}^n$ with $|x-y|\\leq1$ and\n$i(x)\\neq i(y)$. Plugging (\\ref{eqinterinter}), (\\ref{eqinterim222222})\nand (\\ref{eqinterim222224}) into the right-hand\nside of (\\ref{eqinternational}) we get\n\\begin{eqnarray}\\label{eqtheinternational}\\qquad\n&&\\biggl|\\frac{1}{2}\\sum_{i\\in\\I} \\bigl(\\lambda\n_i^n+\\mu_i(\\nu_in+x_i)\\bigr)(\\phi_{\\kappa}^n)_{ii}(x)-\n\\frac{1}{2}\\sum_{i\\in\\I} \\bigl(\\lambda_i^n+\\mu_i(\\nu\n_in+y_i)\\bigr)(\\phi_{\\kappa}^n)_{ii}(y)\\biggr|\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\n\\leq C \\log^{k_1} n\\nonumber\n\\end{eqnarray}\nfor all $x,y\\in\\mB_{\\vartheta\\kappa}^n$ with $|x-y|\\leq1$.\nFinally, recall that\n\\[\n\\sigma_i\\lam(x,u)=\\sqrt{\\lambda_i^n+\\mu_i \\nu_in +\\mu\n_i\\bigl(x_i-u_i(e\\cdot x)^+\\bigr)}\n\\]\nso that for all $u\\in\\mathcal{U}$,\n\\begin{eqnarray*}\n\\hspace*{-4pt}&&\\biggl|\\sum_{i\\in\\I}\\bigl((\\phi_{\\kappa\n}^n)_{ii}(y)-(\\phi_{\\kappa}^n)_{ii}(x)\\bigr)(\\sigma_i^n(x,u))^2\\biggr|\n\\\\\n\\hspace*{-4pt}&&\\qquad =\\biggl|\\sum_{i\\in\\I}(\\phi_{\\kappa}^n)_{ii}(y)\\bigl(\\lambda\n_i^n+\\mu_i \\nu_in +\\mu_i\\bigl(x_i-u_i(e\\cdot x)^+\\bigr)\\bigr)\\\\\n\\hspace*{-4pt}&&\\qquad\\quad\\hspace*{3.5pt}{}\n-(\\phi_{\\kappa}^n)_{ii}(x)\\bigl(\\lambda_i^n+\\mu_i \\nu_in +\\mu\n_i\\bigl(x_i-u_i(e\\cdot x)^+\\bigr)\\bigr)\\biggr|\n\\\\\n\\hspace*{-4pt}&&\\qquad\\leq\\biggl|\\frac{1}{2}\\sum_{i\\in\\I} \\bigl(\\lambda_i^n+\\mu\n_i(\\nu_in+x_i)\\bigr)(\\phi_{\\kappa}^n)_{ii}(x)\\\\\n\\hspace*{-4pt}&&\\qquad\\quad\\hspace*{2pt}{}-\n\\frac{1}{2}\\sum_{i\\in\\I} \\bigl(\\lambda_i^n+\\mu_i(\\nu\n_in+y_i)\\bigr)(\\phi_{\\kappa}^n)_{ii}(y)\\biggr|\\\\\n\\hspace*{-4pt}&&\\qquad\\quad{} +\\biggl|\\frac{1}{2}\\sum_{i\\in\\I} (\\phi_{\\kappa\n}^n)_{ii}(x)\\mu_iu_i(e\\cdot x)^+-(\\phi_{\\kappa}^n)_{ii}(y)\\mu\n_iu_i(e\\cdot y)^+\\biggr|\\\\\n\\hspace*{-4pt}&&\\qquad\\quad{} +\\biggl|\\frac{1}{2}\\sum_{i\\in\n\\I} (\\phi_{\\kappa}^n)_{ii}(y)\\mu_i(x_i-y_i)\\biggr|.\n\\end{eqnarray*}\nThe last two terms above are bounded by $C\\log^{k_1}n$ by (\\ref\n{eqgradients1}) and using $|x|\\vee|y|\\leq\\kappa\\sqrt{n}\\log^mn$.\nTogether with (\\ref{eqtheinternational}) this establishes (\\ref\n{eqgenbound}) and concludes the proof of the theorem.\n\n\\subsection*{\\texorpdfstring{Proof of Theorem \\protect\\ref{thmBrownianverification}}{Proof of Theorem 4.2}}\nFix an initial condition $x\\in\\mB_{\\kappa}^n$ and an admissible\n$(\\kappa,n)$-system $\\theta=(\\Omega,\\mathcal{F},(\\mathcal\n{F}_t),\\mathbb{P},\\hat{U},B)$ and let $\\hat{X}^n$ be the associated\ncontrolled process.\\vadjust{\\goodbreak} Using It\\^{o}'s lemma for the function $\\varphi\n(t,x)=e^{-\\gamma t} \\phi_{\\kappa}\\lam(x)$ in conjunction with the inequality\n\\[\nL(x,u)+A_{u}\\phi_{\\kappa}\\lam(x)-\\gamma\\phi_{\\kappa}\\lam(x)\\geq\n0 \\qquad\\mbox{for all } x\\in\\mB_{\\kappa}^n, u\\in\\mathcal{U}\n\\]\n[recall that $\\phi_{\\kappa}^n$ solves (\\ref{eqHJB1simp})] we have that\n\\begin{eqnarray}\\label{eqinterim3}\\qquad\n\\phi_{\\kappa}\\lam(x)&\\leq& \\Ex_x^{\\theta}\\int_0^{t\\wedge\\hat\n{\\tau}_{\\kappa}^n}e^{-\\gamma s} L(\\hat{X}^n(s),\\hat{U}(s))\\,ds+\\Ex\n_{x}^{\\theta}e^{-\\gamma(t\\wedge\\hat{\\tau}_{\\kappa}^n)}\\phi\n_{\\kappa}\\lam\\bigl(\\hat{X}^n(t\\wedge\\hat{\\tau}_{\\kappa}^n)\\bigr)\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&{}-\\Ex_{x}^{\\theta}\\sum_{i\\in\\I}\\int_0^{t\\wedge\\hat{\\tau\n}_{\\kappa}^n}\ne^{-\\gamma s} (\\phi_{\\kappa}\\lam)_i(\\hat{X}^n(s))\\sigma_i^n(\\hat\n{X}^n(s),\\hat{U}(s)) \\,dB(s).\\nonumber\n\\end{eqnarray}\nHere, $\\hat{\\tau}_{\\kappa}^n$ is as defined in Definition \\ref\n{definadmissiblesystemBrownian} and it is a stopping time with respect\nto $(\\mathcal{F}_t)$ because of the continuity of $\\hat{X}^n$. We now\nclaim that\n\\[\n\\Ex_{x}^{\\theta}\\bigl[e^{-\\gamma t\\wedge\\hat{\\tau}_{\\kappa\n}^n}\\phi_{\\kappa}\\lam\\bigl(\\hat{X}^n(t\\wedge\\hat{\\tau}_{\\kappa\n}^n)\\bigr)\\bigr]\\rightarrow0 \\qquad\\mbox{as } t\\tinf.\n\\]\nIndeed, as $\\phi_{\\kappa} \\lam$ is bounded on $\\mB_{\\kappa}^n$, on\nthe event $\\{\\hat{\\tau}_{\\kappa}^n=\\infty\\}$ we have that\n\\[\ne^{-\\gamma(t\\wedge\\hat{\\tau}_{\\kappa}^n)}\\phi_{\\kappa}\\lam\n\\bigl(X(t\\wedge\\hat{\\tau}_{\\kappa}^n)\\bigr)\\rightarrow0 \\qquad\\mbox{as }t\\tinf.\n\\]\nOn the event $\\{\\hat{\\tau}_{\\kappa}^n<\\infty\\}$ we have $\\hat\n{X}^n(\\hat{\\tau}_{\\kappa}^n)\\in\\partial\\mB$ and, by the\ndefinition of $\\hat{\\tau}_{\\kappa}^n$, that $\\phi_{\\kappa}\\lam\n(\\hX^n(\\hat{\\tau}_{\\kappa}^n))=0$. The convergence in expectation\nthen follows from the bounded convergence theorem (using again the\nboundedness of $\\phi_{\\kappa}\\lam$ on $\\mB_{\\kappa}^n$). The last\nterm in (\\ref{eqinterim3}) equals zero by the optional stopping\ntheorem.\\vadjust{\\goodbreak}\n\nLetting $t\\tinf$ in (\\ref{eqinterim3}) and applying the monotone\nconvergence theorem, we then have\n\\[\n\\phi_{\\kappa}\\lam(x)\\leq\\Ex_x^{\\theta} \\biggl[\\int_0^{\\hat{\\tau\n}_{\\kappa}^n}e^{-\\gamma s} L(\\hat{X}^n(s),\\hat{U}(s))\\,ds\\biggr].\n\\]\n\nSince the admissible system $\\theta$ was arbitrary, we have that $\\phi\n_{\\kappa}\\lam(x)\\leq\\hat{V}\\lam(x,\\kappa)$. To show that this\ninequality is actually an equality, let\n\\begin{equation}\\label{eqratiofunchouce}\nh_{\\kappa\n}^n(x)=e_{i^n(x)},\n\\end{equation}\nwhere $e_{i^n(x)}$ is\nas defined in the statement of the theorem.\n\nThe continuity of $\\phi_{\\kappa}^n$ guarantees that the function\n$i^n(x)$ is Lebesgue measurable, and so is, in turn, $h_{\\kappa\n}^n(\\cdot)$. Consider now the autonomous SDE:\n\\begin{equation}\\label{eqautonomous}\n\\hX^n(t)=x+\\int\n_0^t \\hat{b}^n(\\hX^n(s))\\,ds+\\int_0^t \\hat{\\sigma}^n(\\hX\n^n(s))\\,dB(s),\n\\end{equation}\nwhere $\\hat\n{b}^n(y)=b^n(y,h_{\\kappa}^n(y))$ and $\\hat{\\sigma}^n(y)=\\sigma\n^n(y,h_{\\kappa}^n(y))$ on $\\mB_{\\kappa}^n$. Then, $\\hat{b}^n$ and\n$\\hat{\\sigma}^n$ are bounded and measurable on the bounded domain\n$\\mB_{\\kappa}^n$. Also, as the matrix $\\hat{\\sigma}^n$ is diagonal\nand the elements on the diagonal are strictly positive on $\\mB_{\\kappa\n}^n$, it is positive definite there. Hence, a weak solution exists for\nthe autonomous SDE (see, e.g.,\\vspace*{2pt} Theorem 6.1 of \\cite\n{krylov2008controlled}). In particular, there exists a probability\\vadjust{\\goodbreak}\nspace $(\\tilde{\\Omega},\\mathcal{G},\\tilde{\\Pd})$, a filtration\n$(\\mathcal{G}_t)$ that satisfies the usual conditions, a Brownian\nmotion $B(t)$ and a continuous process $\\hX^n$---both adapted to\n$(\\mathcal{G}_t)$, so that $\\hX^n$ satisfies the autonomous SDE (\\ref\n{eqautonomous}). Finally, since~$\\hX^n$ has continuous sample paths\nand it is adapted, it is also progressively measurable (see,\ne.g.,\\vspace*{1pt}\nProposition 1.13 in \\cite{KaS91}) and, by measurability of $h_{\\kappa\n}^n(\\cdot)$, so is the process $\\hat{U}(t)=h_{\\kappa}^n(\\hX^n(t))$.\nConsequently, $\\theta=(\\tilde{\\Omega},\\mathcal{G},\\mathcal\n{G}_t,\\tilde{\\Pd},\\hat{U},B)$ is an admissible system in the sense\nof Definition \\ref{definadmissiblesystemBrownian} and $\\hX^n$ is the\ncorresponding controlled process.\n\nTo see that $\\theta$ is optimal for the ADCP on $\\mB_{\\kappa}^n$,\nnote that for $s<\\hat{\\tau}_{\\kappa}^n$, we have by the HJB equation\n(\\ref{eqHJB0}) that\n\\[\nL(\\hX^n(s),\\hat{U}(s))+A_{\\hat{U}(s)}\\phi_{\\kappa}\\lam(\\hX\n^n(s))-\\gamma\\phi_{\\kappa}\\lam(\\hX^n(s))=0.\n\\]\nApplying It\\^{o}'s rule as before, together with the bounded and\ndominated convergence theorems, we then have that\n\\[\n\\phi_{\\kappa}\\lam(x)=\\Ex_x^{\\theta}\\biggl[\\int_0^{\\hat{\\tau\n}_{\\kappa}^n}e^{-\\gamma s} L(\\hX^n(s),\\hat{U}(s))\\,ds\\biggr]\n\\]\nand the proof is complete.\n\n\\section{The performance analysis of tracking policies}\\label{sectracking}\n\nThis section shows that given an optimal Markov control policy for the\nADCP together with its associated tracking function $h_{\\kappa\n}^{*,n}$, the nonpreemptive tracking policy imitates, in a particular\nsense, the performance of the Brownian system.\n\\begin{theorem} \\label{thmSSC}\nFix $\\kappa$ and $\\kappa'<\\kappa$ as well as a\nsequence $\\{(x^n,q^n),n\\in\\bbZ_+\\}$ such that $(x^n,q^n)\\in\\mathcal\n{X}^n$, and $|x^n-\\nu n| \\leq M \\sqrt{n}$ for all $n$ and some\n$M>0$. Let $\\phi_{\\kappa}^n$ and $h_{\\kappa}^{*,n}$ be as in Theorem\n\\ref{thmBrownianverification} and define\n\\[\n\\psi^n(x,u)=L(x,u)+A^n_u \\phi_{\\kappa}^n(x)-\\gamma\\phi_{\\kappa\n}^n(x) \\qquad\\mbox{for }x\\in\\mB_{\\kappa}^n,u\\in\\mathcal{U}.\n\\]\nLet $U_h^n$ be the ratio control associated with the $h_{\\kappa\n}^{*,n}$-tracking policy and let $\\mathbb{X}^n=(X^n,Q^n,Z^n,\\check\n{X}^n)$ be the associated queueing process with the initial conditions\n$Q^n(0)=q^n$ and $\\check{X}^n(0)=x^n-\\nu n$ and define\n\\[\n\\tau_{\\kappa',T}^n=\\inf\\{t\\geq0\\dvtx\\check{X}^n(t)\\notin\\mB_{\\kappa\n'}^n\\}\\wedge T\\log n.\n\\]\nThen,\n\\[\n\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} |\\psi\n^n(\\check{X}^n(s),U_h^n(s))-\\psi^n(\\check{X}^n(s),h_{\\kappa\n}^{*,n}(\\check{X}^n(s)))|\\,ds\\biggr]\\leq C\\log^{k_0+3}n\n\\]\nfor a constant $C$ that does not depend on $n$.\n\\end{theorem}\n\nTheorem \\ref{thmSSC} is proved in the \\hyperref[app]{Appendix}. The proof builds on the\ngradient estimates in Theorem \\ref{thmHJB1sol} and on a state-space\ncollapse-type result for certain sub-intervals of $[0,\\tau_{\\kappa',T}^n]$.\n\\begin{rem} \\label{remSSC} Typically one establishes a stronger\nstate-space collapse result showing that the actual queue and the\ndesired queue values are close in supremum norm. The difficulty with\nthe former approach is that the tracking functions here are nonsmooth.\nWhile it is plausible that one can smooth these functions appropriately\n(as is done, e.g., in \\cite{AMR02}), such smoothing might\ncompromise the optimality gap. Fortunately, the weaker integral\ncriterion implied by Theorem\n\\ref{thmSSC} suffices for our purposes.\n\\end{rem}\n\n\\section{Proof of the main result}\\label{seccombining}\n\nFix $\\kappa>0$ and let $\\phi_{\\kappa}^n$ be the solution to (\\ref\n{eqHJB1simp}) on $\\mB_{\\kappa}^n$ (see Theorem \\ref{thmHJB1sol}).\nWe start with the following lemma where $b_i^n(\\cdot,\\cdot)$ and\n$\\sigma_i^n(\\cdot,\\cdot)$ are as in (\\ref{eqnbmdefn30}) and\n(\\ref{eqnbmdefn3}), respectively.\n\\begin{lem} \\label{lemito} Let $U^n$ be an admissible ratio control\nand let $\\mathbb{X}^n=(X^n,Q^n$, $Z^n,\\check{X}^n)$ be the queueing\nprocess associated with $U^n$. Fix $\\kappa'<\\kappa$ and $T>0$ and let\n\\[\n\\tau_{\\kappa',T}^n=\\inf\\{t\\geq0\\dvtx\\check{X}^n(t)\\notin\\mB_{\\kappa\n'}^n\\}\\wedge T\\log n.\n\\]\nThen, there exists a constant $C$ that does not depend on $n$ (but may\ndepend on $T$, $\\kappa$ and $\\kappa'$) such that\n\\begin{eqnarray*}\n\\Ex[e^{-\\gamma\\tau_{\\kappa',T}^n}\\phi_{\\kappa}^n(\\check\n{X}^n(\\tau_{\\kappa',T}^n))]&\\leq&\\phi_{\\kappa}^n(\\check{X}^n(0))+\n\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s}\nA_{U^n(s)}^n\\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds\\biggr]\\\\\n&&{}-\\gamma\\Ex\n\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} \\phi_{\\kappa\n}^n(\\check{X}^n(s))\\,ds\\biggr] +C\\log^{k_1+1} n\\\\\n& \\leq&\n\\Ex[e^{-\\gamma\\tau_{\\kappa',T}^n}\\phi_{\\kappa}^n(\\check\n{X}^n(\\tau_{\\kappa',T}^n))]+2C\\log^{k_1+1}n.\n\\end{eqnarray*}\n\\end{lem}\n\nWe will also use the following lemma where $c=(c_1,\\ldots, c_I)$ are\nthe cost coefficients (see Section \\ref{secmodel}).\n\\begin{lem}\\label{lemafterstop}\nLet $(x^n,q^n)$ be as in the conditions of Theorem \\ref\n{thmmain}. Then, there exists a constant $C$ that does not depend on\n$n$ such that\n\\begin{equation}\\label{eqafterstop2}\n\\Ex_{x^n,q^n}^{U}\\biggl[\\int_{\\tau_{\\kappa\n',T}^n}^{\\infty} e^{-\\gamma s} (e\\cdot c)\\bigl(e\\cdot\\check\n{X}^n(s)\\bigr)^+\\,ds\\biggr]\\leq C\\log^2n\n\\end{equation}\nand\n\\begin{equation}\\label{eqafterstop1}\n\\Ex_{x^n,q^n}^{U}[e^{-\\gamma\\tau_{\\kappa\n',T}^n}\\phi\n_{\\kappa}^n(\\check{X}^n(\\tau_{\\kappa',T}^n))]\\leq C\\log^2\nn\n\\end{equation}\nfor all $n$ and any admissible ratio control\n$U$.\n\\end{lem}\n\nWe postpone the proof of Lemma \\ref{lemito} to the end of the section\nand that of Lemma \\ref{lemafterstop} to the \\hyperref[app]{Appendix} and proceed now to\nprove the main result of the paper.\n\n\\subsection*{\\texorpdfstring{Proof of Theorem \\protect\\ref{thmmain}}{Proof of Theorem 2.1}} Let $h_{\\kappa\n}^{*,n}$ be the ratio function associated with the optimal Markov\ncontrol for the ADCP (as in Theorem \\ref{thmHJB1sol}). Since $\\kappa\n$ is fixed we omit the subscript $\\kappa$ and use $h^n=h_{\\kappa\n}^{*,n}$. Let $U_h^n$ be the ratio associated with the $h^n$-tracking policy.\n\nThe proof will proceed in three main steps. First, building on Theorem~\\ref{thmSSC} we will show that\n\\begin{equation} \\label{eqinterim2}\n\\Ex\\biggl[\\int_0^{\\tau_{\\kappa\n',T}^n} e^{-\\gamma s} L(\\check{X}^n(s),U_h^n(s))\\,ds\\biggr] \\leq\\phi\n_{\\kappa}^n(\\check{X}^n(0))+C\\log^{k_0+3} n.\n\\end{equation}\nUsing Lemma \\ref{lemafterstop}, this implies\n\\begin{eqnarray}\\label{eqinterim13}\nC\\lam(x^n,q^n,U_h^n)&=&\\Ex\\biggl[\\int_0^{\\infty} e^{-\\gamma s} L(\\check\n{X}^n(s),U_h^n(s))\\,ds\\biggr]\\nonumber\\\\[-8pt]\\\\[-8pt]\n&\\leq&\\phi_{\\kappa}^n(\\check\n{X}^n(0))+C\\log^{k_0+3} n.\\nonumber\n\\end{eqnarray}\nFinally, we will\nshow that for any ratio control $U^n$,\n\\begin{equation}\\label{eqinterim303}\n\\phi_{\\kappa}^n(\\check\n{X}^n(0))\\leq\\Ex\\biggl[\\int_0^{\\infty} e^{-\\gamma s} L(\\check\n{X}^n(s),U^n(s))\\,ds\\biggr]+C\\log^{k_1+1}n,\n\\end{equation}\nwhere we recall that $k_1=k_0-m$. In turn,\n\\[\nV^n(x^n,q^n)\\geq\\phi_{\\kappa}^n(x^n-\\nu n)-C\\log^{k_1+1} n \\geq\nC\\lam(x^n,q^n,U_h^n)-2C\\log^{k_1+1}n,\n\\]\nwhich establishes the statement of the theorem.\\vadjust{\\goodbreak}\n\nWe now turn to prove each of (\\ref{eqinterim2}) and (\\ref{eqinterim303}).\n\n\\subsection*{\\texorpdfstring{Proof of (\\protect\\ref{eqinterim2})}{Proof of (61)}} To simplify\nnotation we fix $\\kappa>0$ throughout and let $h^n(\\cdot\n)=h_{\\kappa}^{*,n}$. Using Lemma \\ref{lemito} we have\n\\begin{eqnarray}\\label{eqinterim1}\n&&\\Ex[e^{-\\gamma\\tau_{\\kappa',T}^n}\\phi_{\\kappa}^n(\\check\n{X}^n(\\tau_{\\kappa',T}^n))]\\nonumber\n\\\\\n&&\\qquad\\leq \\phi_{\\kappa}^n(\\check{X}^n(0))+\n\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s}\nA_{U_h^n(s)}^n\\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds\\biggr]\\\\\n&&\\qquad\\quad{}-\\gamma\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} \\phi\n_{\\kappa}^n(\\check{X}^n(s))\\,ds\\biggr]+ C\\log^{k_1+1} n .\\nonumber\n\\end{eqnarray}\nFrom the definition of $h^n$ as a minimizer in the HJB equation we have that\n\\begin{eqnarray*}\n0&=& \\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s}\nA_{h^n(\\check{X}^n(s))}^n\\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds\\biggr]\\\\\n&&{}-\\gamma\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} \\phi\n_{\\kappa}^n(\\check{X}^n(s))\\,ds\\biggr]\\\\\n&&{}+\\Ex\\biggl[\\int\n_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} L(\\check{X}^n(s),h^n(\\check\n{X}^n(s)))\\,ds\\biggr] .\n\\end{eqnarray*}\nBy Theorem \\ref{thmSSC} we then have that\n\\begin{eqnarray}\\label{eqinterim101}\nC\\log^{k_0+3}n&\\geq& \\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n}\ne^{-\\gamma s} A_{U_h^n(s)}^n\\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds\n\\biggr]\\nonumber\\\\[-2pt]\n&&{} -\\gamma\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n}\ne^{-\\gamma s}\n\\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds\\biggr]\\nonumber\\\\[-9pt]\\\\[-9pt]\n&&{} +\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} L(\\check\n{X}^n(s),U_h^n(s))\\,ds\\biggr]\\nonumber\\\\[-2pt]\n&\\geq&0.\\nonumber\n\\end{eqnarray}\nSince $\\phi_{\\kappa}^n$ is nonnegative, combining (\\ref{eqinterim1})\nand (\\ref{eqinterim101}) we have that\n\\[\n\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} L(\\check\n{X}^n(s),U_h^n(s))\\,ds\\biggr]\\leq\\phi_{\\kappa}^n(\\check\n{X}^n(0))+C\\log^{k_0+3} n,\n\\]\nwhich concludes the proof of (\\ref{eqinterim2}).\n\n\\subsection*{\\texorpdfstring{Proof of (\\protect\\ref{eqinterim303})}{Proof of (63)}} We now show that\n$V^n(x,q)\\geq\\phi_{\\kappa}^n(\\check{X}^n(0))-C\\log^{k_1+1} n$. To\nthat end, fix an arbitrary ratio control $U^n$ and recall that by the\nHJB equation,\n\\[\nA_u^n\\phi_{\\kappa}^n(x)-\\gamma\\phi_{\\kappa\n}^n(x)+L(x,u)\\geq0\n\\]\nfor all $u\\in\\mathcal{U}$ and $x\\in\\mB\n_{\\kappa}^n$. In turn, using the second inequality in Lemma \\ref\n{lemito} we have that\n\\begin{eqnarray*}\n&&\\Ex[e^{-\\gamma\\tau_{\\kappa\n',T}^n}\\phi_{\\kappa}^n(\\check{X}^n(\\tau_{\\kappa',T}^n))\n]\\\\[-2pt]\n&&\\qquad\\geq \\phi_{\\kappa}^n(\\check{X}^n(0))\n-\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} L(\\check\n{X}^n(s),U^n(s))\\,ds\\biggr]\\\\[-2pt]\n&&\\qquad\\quad{} -2C\\log^{k_1+1} n.\n\\end{eqnarray*}\nUsing Lemma \\ref{lemafterstop}, we have, however, that\n\\[\n\\Ex[e^{-\\gamma\\tau_{\\kappa',T}^n}\\phi_{\\kappa}^n(\\check\n{X}^n(\\tau_{\\kappa',T}^n))]\\leq C\\log^{2} n\n\\]\nfor a redefined constant $C$ so that\n\\begin{eqnarray*}C\\log^{2} n &\\geq& \\phi\n_{\\kappa}^n(\\check{X}^n(0))-\n\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} L(\\check\n{X}^n(s),U^n(s))\\,ds\\biggr]\\\\[-2pt]\n&&{}-2C\\log^{k_1+1} n\\\\&\\geq& \\phi_{\\kappa\n}^n(\\check{X}^n(0))-\n\\Ex\\biggl[\\int_0^{\\infty} e^{-\\gamma s} L(\\check\n{X}^n(s),U^n(s))\\,ds\\biggr]\\\\[-2pt]\n&&{}-2C\\log^{k_1+1} n\n\\end{eqnarray*}\nand, finally,\n\\[\n\\phi_{\\kappa}^n(\\check{X}^n(0))\\leq\\Ex\\biggl[\\int_0^{\\infty}\ne^{-\\gamma s}\nL(\\check{X}^n(s),U^n(s))\\,ds\\biggr]+C\\log^{k_1+1}n\\vadjust{\\goodbreak}\n\\]\nfor a redefined constant $C>0$. This concludes the proof of (\\ref\n{eqinterim303}) and of the theorem.\n\nWe end this section with the proof of Lemma \\ref{lemito} in which the\nfollowing auxiliary lemma will be of use.\n\\begin{lem}\\label{lemmartingales}\nFix $\\kappa>0$ and an admissible ratio control $U^n$ and\nlet $\\mathbb{X}\\lam=(X\\lam,Q\\lam,Z\\lam,\\check{X}\\lam)$ be the\ncorresponding queueing process. Let\n\\[\n\\tau_{\\kappa,T}^n=\\inf\\{t\\geq0\\dvtx\\check{X}^n(t)\\notin\\mB_{\\kappa\n}^n\\}\\wedge T\\log n,\n\\]\nand $(\\check{W}_i^n,i\\in\\I)$ be as defined in (\\ref{eqWtildedefin}).\nThen, for each $i\\in\\I$, the process $\\check\n{W}_i^n(\\cdot\\wedge\\tau_{\\kappa,n}^n)$ is a square integrable\nmartingale w.r.t to the filtration $(\\mathcal{F}_{t\\wedge\\tau\n_{\\kappa,T}^n}^n)$ as are the processes\n\\[\n\\mathcal{M}_i^n(\\cdot)=\\bigl(\\check{W}_i^n(\\cdot\\wedge\\tau_{\\kappa\n,T}^n)\\bigr)^2-\\int_0^{\\cdot\\wedge\\tau_{\\kappa,T}^n} (\\sigma\n_i^n(\\check{X}^n(s),U^n(s)))^2\\,ds\n\\]\nand\n\\[\n\\mathcal{V}_i^n(\\cdot)=\\bigl(\\check{W}_i^n(\\cdot\\wedge\\tau_{\\kappa\n,T}^n)\\bigr)^2-\\sum_{s\\leq\\cdot\\wedge\\tau_{\\kappa,T}^n} (\\Delta\\check\n{W}_i^n(s))^2.\n\\]\n\\end{lem}\n\nLemma \\ref{lemmartingales} follows from basic results on martingales\nassociated with time-changes of Poisson processes. The detailed proof\nappears in the \\hyperref[app]{Appendix}.\n\n\\subsection*{\\texorpdfstring{Proof of Lemma \\protect\\ref{lemito}}{Proof of Lemma 6.1}}\nNote that, as in\n(\\ref{eqcheckXdynamics}), $\\check{X}^n$ satisfies\n\\[\n\\check{X}_i\\lam(t)=\\check{X}_i\\lam(0)+\\int_0^t b_i^n(\\check\n{X}^n(s),U^n(s))\\,ds+\\check{W}_i\\lam(t),\n\\]\nand is a semi martingale. Applying It\\^{o}'s formula for\nsemimartingales (see, e.g., Theorem 5.92 in \\cite{vandervaart}) we\nhave for all $t\\leq\\tau_{\\kappa',T}^n$, that\n\\begin{eqnarray*}\ne^{-\\gamma t}\\phi_{\\kappa}^n(\\check{X}^n(t))&=&\\phi_{\\kappa\n}^n(\\check{X}^n(0))\\\\\n&&{}+\n\\sum_{s\\leq t\\dvtx |\\Delta\\check{X}^n(s)|> 0} e^{-\\gamma s}[\\phi\n_{\\kappa}^n(\\check{X}^n(s))-\\phi_{\\kappa}^n(\\check{X}^n(s-))]\\\\\n&&{}-\n\\sum_{i\\in\\I}\\sum_{s\\leq t\\dvtx |\\Delta\\check{X}^n(s)|> 0} e^{-\\gamma\ns}(\\phi_{\\kappa})_i^n(\\check{X}^n(s))\\Delta\\check{X}_i^n(s)\\\\\n&&{}+\n\\sum_{i\\in\\I}\\int_0^t e^{-\\gamma s} (\\phi_{\\kappa}^n)_i(\\check\n{X}^n(s-))b_i^n(\\check{X}^n(s),U^n(s))\\,ds\\\\\n&&{}-\\gamma\\int_0^t\ne^{-\\gamma s} \\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds\n\\end{eqnarray*}\nand, after rearranging terms, that\n\\begin{eqnarray*}\n&&e^{-\\gamma t}\\phi_{\\kappa}^n(\\check{X}^n(t))\\\\\n&&\\qquad=\n\\phi_{\\kappa}^n(\\check{X}^n(0))\n+ \\frac{1}{2}\\sum_{i\\in\\I\n}\\sum_{s\\leq t\\dvtx |\\Delta\\check{X}^n(s)|>0}e^{-\\gamma s} (\\phi\n_{\\kappa}^n)_{ii}(\\check{X}^n(s-))(\\Delta\\check{X}_i^n(s))^2\\\\\n&&\\qquad\\quad{}+\n\\sum_{i\\in\\I}\\int_0^t e^{-\\gamma s} (\\phi_{\\kappa}^n)_i(\\check\n{X}^n(s-))b_i(\\check{X}^n(s),U^n(s))\\,ds\\\\\n&&\\qquad\\quad{}+ C^n(t)-\\gamma\\int_0^t\ne^{-\\gamma s} \\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds,\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*} C^n(t)&=& \\sum_{s\\leq t\\dvtx |\\Delta\\check\n{X}^n(s)|>0}e^{-\\gamma s}  \\biggl[\\phi_{\\kappa}^n(\\check\n{X}^n(s))-\\phi_{\\kappa}^n(\\check{X}^n(s-))\\\\\n&&\\hphantom{\\sum_{s\\leq t\\dvtx |\\Delta\\check\n{X}^n(s)|>0}e^{-\\gamma s}  \\biggl[}{}-\\sum_{i\\in\\I}(\\phi\n_{\\kappa}^n)_i(\\check{X}^n(s-)) \\Delta\\check{X}_i^n(s) \\\\\n&&\\hphantom{\\sum_{s\\leq t\\dvtx |\\Delta\\check\n{X}^n(s)|>0}e^{-\\gamma s}  \\biggl[}{}-\\frac\n{1}{2}\\sum_{i\\in\\I} (\\phi_{\\kappa}^n)_{ii}(\\check\n{X}^n(s-))(\\Delta\\check{X}_i^n(s))^2\\biggr].\n\\end{eqnarray*}\nSetting $t=\\tau_{\\kappa',T}^n$ as defined in the statement of the\nlemma and taking expectations on both sides we have\n\\begin{eqnarray}\\label{eqinterim404}\n&&\\Ex[e^{-\\gamma\\tau_{\\kappa',T}^n}\\phi_{\\kappa}^n(\\check\n{X}^n(t))]\\nonumber\\\\\n&&\\qquad=\\phi_{\\kappa}^n(\\check{X}^n(0))\n+\n\\sum_{i\\in\\I}\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s}\n(\\phi_{\\kappa}^n)_i(\\check{X}^n(s-)) b_i^n(\\check\n{X}^n(s),U^n(s))\\,ds\\biggr]\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\quad{}+\\frac{1}{2}\\sum\n_{i\\in\\I}\\Ex\\biggl[\\sum_{s\\leq t\\dvtx |\\Delta\\check\n{X}^n(s)|>0}e^{-\\gamma s} (\\phi_{\\kappa}^n)_{ii}(\\check\n{X}^n(s-))(\\Delta\\check{X}_i^n(s))^2\\biggr]\\nonumber\\\\\n&&\\qquad\\quad{}+ \\Ex\n[C^n(\\tau_{\\kappa',T}^n)]-\\gamma\\Ex\\biggl[\\int_0^{\\tau_{\\kappa\n',T}^n} e^{-\\gamma s} \\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds\n\\biggr].\\nonumber\n\\end{eqnarray}\n\nWe will now examine each of the elements on the right-hand side of\n(\\ref{eqinterim404}). First, note that $\\Delta\\check\n{X}_i^n(s)=\\Delta\\check{W}_i^n(s)$ and, in particular,\n\\begin{eqnarray*}\n&&\\Ex\n\\biggl[\\sum_{s\\leq\\tau_{\\kappa',T}^n\\dvtx |\\Delta\\check\n{X}^n(s)|>0}e^{-\\gamma s} (\\phi_{\\kappa}^n)_{ii}(\\check\n{X}^n(s-))(\\Delta\\check{X}_i^n(s))^2\\biggr]\\\\\n&&\\qquad=\n\\Ex\n\\biggl[\\sum_{s\\leq\\tau_{\\kappa',T}^n\\dvtx |\\Delta\\check\n{X}^n(s)|>0}e^{-\\gamma s} (\\phi_{\\kappa}^n)_{ii}(\\check\n{X}^n(s-))(\\Delta\\check{W}_i^n(s))^2\\biggr].\n\\end{eqnarray*}\nUsing the fact that $\\mathcal{V}_i^n$, as defined in Lemma \\ref\n{lemmartingales}, is a martingale as well as the fact that $\\phi\n_{\\kappa}^n(\\check{X}^n(s))$ and its derivative processes are bounded\nup to~$\\tau_{\\kappa'}^n$, we have that the processes\n\\begin{equation}\\label{eqbarV}\n\\bar{\\mathcal{V}}_i^n(\\cdot):=\\int_0^{\\cdot\\wedge\n\\tau\n_{\\kappa',T}^n} e^{-\\gamma s}(\\phi_{\\kappa}^n)_{ii}(\\check\n{X}^n(s-))\\,d\\mathcal{V}_i^n(s)\n\\end{equation}\nand\n\\begin{equation}\\label{eqbarM}\n\\bar{\\mathcal{M}}_i^n(\\cdot):=\\int_0^{\\cdot\\wedge\n\\tau\n_{\\kappa',T}^n} e^{-\\gamma s}(\\phi_{\\kappa}^n)_{ii}(\\check\n{X}^n(s-))\\,d\\mathcal{M}_i^n(s)\n\\end{equation}\nare themselves\nmartingales with $\\bar{\\mathcal{V}}_i^n(0)=\\bar{\\mathcal\n{M}}_i^n(0)=0$ and in turn, by optional stopping, that\n$\\Ex[\\bar{\\mathcal{V}}_i^n(\\tau_{\\kappa',T}^n)]=\\Ex[\\bar\n{\\mathcal{M}}_i^n(\\tau_{\\kappa',T}^n)]$ (see,\\vspace*{1pt} e.g., Lemma 5.45 in\n\\cite{vandervaart}). In turn, by the definition of $\\mathcal\n{M}_i^n(\\cdot)$ and $\\mathcal{V}_i^n(\\cdot)$ we have\n\\begin{eqnarray*}\n&&\\Ex\\biggl[\\sum_{s\\leq\\tau_{\\kappa',T}^n\\dvtx\n|\\Delta\\check{X}^n(s)|>0}e^{-\\gamma s} (\\phi_{\\kappa\n}^n)_{ii}(\\check{X}^n(s-))(\\Delta\\check{W}_i^n(s))^2\\biggr]\\\\\n&&\\qquad=\\Ex\\biggl[\\int_0^t (\\phi_{\\kappa}^n)_{ii}(\\check\n{X}^n(s-))\\,d(\\check{W}_i^n(s))^2\\biggr]\\\\\n&&\\qquad=\\Ex\\biggl[\\int\n_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} (\\phi_{\\kappa}^n)_{ii}(\\check\n{X}^n(s-))(\\sigma_i^n(\\check{X}^n(s),U^n(s)))^2\\,ds\\biggr].\n\\end{eqnarray*}\nPlugging this back into (\\ref{eqinterim404}) we have that\n\\begin{eqnarray*}\n&&\\Ex[e^{-\\gamma\\tau_{\\kappa',T}^n}\\phi_{\\kappa}^n(\\check\n{X}^n(t))]\\\\\n&&\\qquad=\\phi_{\\kappa}^n(\\check{X}^n(0))\n+\\sum_{i\\in\\I}\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s}\n(\\phi_{\\kappa}^n)_{i}(\\check{X}^n(s-)) b_i^n(\\check\n{X}^n(s),U^n(s))\\,ds\\biggr]\\\\\n&&\\qquad\\quad{}+\n\\frac{1}{2}\\sum_{i\\in\\I}\n\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} (\\phi_{\\kappa\n}^n)_{ii}(\\check{X}^n(s-))(\\sigma_i^n(\\check{X}^n(s),U^n(s)) )^2\n\\,ds\\biggr]\\\\\n&&\\qquad\\quad{}-\\gamma\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n}\ne^{-\\gamma s} \\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds\\biggr]+ \\Ex\n[C^n(\\tau_{\\kappa',T}^n)],\n\\end{eqnarray*}\nwhich, using the definition of $A_u^n$ in (\\ref{eqgendefin}), yields\n\\begin{eqnarray*}\n&&\\Ex[e^{-\\gamma\\tau_{\\kappa',T}^n}\\phi_{\\kappa}^n(\\check\n{X}^n(t))]\\\\\n&&\\qquad=\\phi_{\\kappa}^n(\\check{X}^n(0))\n+\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s}\nA_{U^n(s)}^n\\phi_{\\kappa}^n(\\check{X}^n(s))\\,ds\\biggr]\\\\\n&&\\qquad\\quad{}-\\gamma\\Ex\n\\biggl[\\int_0^{\\tau_{\\kappa',T}^n} e^{-\\gamma s} \\phi_{\\kappa\n}^n(\\check{X}^n(s))\\,ds\\biggr]\\\\\n&&\\qquad\\quad{}+ \\Ex[C^n(\\tau_{\\kappa',T}^n)].\n\\end{eqnarray*}\nTo complete the proof it then remains only to show that there\nexists a~constant $C$ such that\n\\[\n|\\Ex[C^n(\\tau_{\\kappa',T}^{n})]|\\leq C\\log\n^{k_1+1} n.\n\\]\nTo that end, note that by Taylor's\nexpansion,\n\\begin{eqnarray*}\\phi_{\\kappa}^n(\\check{X}^n(s))&=&\\phi_{\\kappa\n}^n(\\check{X}^n(s-))\n+\\sum_{i\\in\\I}(\\phi_{\\kappa}^n)_i(\\check{X}^n(s-))\\Delta\n\\check{X}_i^n(s)\\\\\n&&{}\n+\\frac{1}{2}\\sum_{i\\in\\I}(\\phi_{\\kappa}^n)_{ii}\\bigl(\\check\n{X}^n(s-)+\\eta_{\\check{X}^n(s-)}\\bigr)\\Delta\\check{X}_i^n(s),\n\\end{eqnarray*}\nwhere\n$\\eta_{\\check{X}^n(s-)}$ is such that $\\check{X}^n(s-)+\\eta_{\\check\n{X}^n(s-)}$ is between $\\check{X}^n(s-)$ and $\\check{X}^n(s-)+\\Delta\n\\check{X}^n(s)$. In turn, adding and subtracting a term, we have that\n\\begin{eqnarray}\\label{eqCninterim}\n&&\\phi_{\\kappa}^n(\\check{X}^n(s))-\\phi_{\\kappa\n}^n(\\check{X}^n(s-))-\\sum_{i\\in\\I}(\\phi_{\\kappa}^n)_i(\\check\n{X}^n(s-)) \\Delta\\check{X}_i^n(s) \\nonumber\\\\\n&&\\quad{}-\\frac{1}{2}\\sum_{i\\in\\I}(\\phi_{\\kappa}^n)_{ii}(\\check\n{X}^n(s-))(\\Delta\\check{X}_i^n(s))^2\\\\\n&&\\qquad = \\sum_{i\\in\\I}\\frac\n{1}{2}\\bigl((\\phi_{\\kappa}^n)_{ii}\\bigl(\\check{X}^n(s-)+\\eta_{\\check\n{X}^n(s-)}\\bigr) -(\\phi_{\\kappa}^n)_{ii}(\\check{X}^n(s-))\\bigr)(\\Delta\n\\check{X}_i^n(s))^2.\\nonumber\n\\end{eqnarray}\nSince the jumps are of size $1$ and, with probability 1, there are no\nsimultaneous jumps, we have that $|\\eta_{\\check{X}^n(s-)}|\\leq1$.\nAdding the discounting, summing and taking expectations we have\n\\begin{eqnarray}\\label{eqinterim2222}\\quad\n&&\\Ex[C^n(t)]\\nonumber\\\\\n&&\\qquad\\leq\\Ex\\biggl[\\sum_{s\\leq t\\dvtx\n|\\Delta\\check{X}^n(s)|>0} e^{-\\gamma s} \\sum_{i\\in\\I}\\frac\n{1}{2}\\max_{y\\dvtx|y|\\leq1}\\bigl((\\phi_{\\kappa}^n)_{ii}\\bigl(\\check\n{X}^n(s-)+y\\bigr)\\\\\n&&\\qquad\\quad\\hspace*{163.6pt}{} -(\\phi_{\\kappa\n}^n)_{ii}(\\check{X}^n(s-))\\bigr)(\\Delta\\check{X}_i^n(s))^2\n\\biggr],\\nonumber\n\\end{eqnarray}\nand a lower bound can be created by minimizing over $y$ instead of\nmaximizing. Using again the fact that $\\Delta\\check{X}_i^n(t)=\\Delta\n\\check{W}_i^n(t)$ and that $\\bar{\\mathcal{M}}_i^n$ and $\\bar\n{\\mathcal{V}}_i^n$ as defined in (\\ref{eqbarM}) and (\\ref{eqbarV})\nare martingales, we have that\n\\begin{eqnarray}\\label{eqinterim505}\n\\Ex[C^n(t)]&\\leq&\\Ex\\biggl[\\int_0^t \\sum_{i\\in\\I\n}\\frac{1}{2}\\max_{y\\dvtx|y|\\leq1}\\bigl((\\phi_{\\kappa}^n)_{ii}\\bigl(\\check\n{X}^n(s-)+y\\bigr) \\nonumber\\\\\n&&\\hspace*{92.3pt}{} -(\\phi_{\\kappa}^n)_{ii}(\\check{X}^n(s-))\\bigr)\\\\\n&&\\hspace*{60.6pt}{}\\times(\\sigma\n_i^n(\\check{X}^n(s),U^n(s)))^2\\,ds\\biggr].\\nonumber\n\\end{eqnarray}\nFrom (\\ref{eqgenbound}) we have that\n\\begin{equation}\\label{eqinterim1111}\n\\frac{1}{2}\\biggl|\\sum_{i\\in\\I}\\bigl((\\phi_{\\kappa\n}^n)_{ii}(y)-(\\phi_{\\kappa}^n)_{ii}(x)\\bigr)(\\sigma_i^n(x,u))^2\n\\biggr|\\leq C\\log^{k_1} n\n\\end{equation}\nfor all $u\\in\\mathcal{U}$ and $x,y\\in\\mB_{\\kappa'}^n$ with\n$|x-y|\\leq1$. The proof is then concluded by plugging (\\ref\n{eqinterim1111}) into (\\ref{eqinterim505}), setting $t=\\tau_{\\kappa\n',T}^n$ and recalling that we can repeat all the above steps to obtain\na lower bound in (\\ref{eqinterim505}) by replacing $\\max_{y\\dvtx|y|\\leq\n1}$ with $\\min_{y\\dvtx|y|\\leq1}$ in~(\\ref{eqinterim2222}).\n\n\\section{Concluding remarks}\\label{secconclusions}\n\nThis paper proposes a novel approach for solving problems of dynamic\ncontrol of queueing systems in the Halfin--Whitt many-server\nheavy-traffic regime. Its main contribution is the use of Brownian\napproximations to construct controls that achieve optimality gaps that\nare logarithmic in the system size. This should be contrasted with the\noptimality gaps of size $o(\\sqrt{n})$ that are common in the\nliterature on asymptotic optimality. A distinguishing feature of our\napproach is the use of a \\textit{sequence} of Brownian control problems\nrather than a single (limit) problem. Having an entire sequence of\napproximating problems allows us to perform a more refined analysis,\nresulting in the improved optimality gap.\n\nIn further contrast with the earlier literature, in each of these\nBrownian problems the diffusion coefficient depends on both the system\nstate and the control. Incorporating the impact of control on diffusion\ncoefficients allows us to track the performance of the policy better\nbut, at the same time, it leads to a more complex diffusion control\nproblem in which the associated HJB equation is fully nonlinear and\nnonsmooth. For \\textit{each} Brownian problem, we show that the HJB\nequation has a sufficiently smooth solution that coincides with the\nvalue function and that admits an optimal Markov policy. Most\nimportantly, we derive useful gradient estimates that apply to the\nwhole sequence and bound the growth rate of the gradients with the\nsystem size. These bounds are crucial for controlling the approximation\nerrors when analyzing the original queueing system under the proposed\ntracking control.\n\nThe motivating intuition behind our approximation scheme is that the\nvalue functions of each queueing system and its corresponding Brownian\ncontrol problem ought to be close. In particular, the optimal control\nfor the Brownian problem should perform well for the queueing system.\nMoreover, the optimal Markov control of the Brownian problem can be\napproximated by a ratio (or tracking) control for the queueing system.\nWhile these observations are ``correct'' at a high level, they need to\nbe qualified further. Our analysis underscores two sources of\napproximation errors that need to be addressed in order to obtain the\nrefined optimality gaps.\nFirst, the value function of the Brownian control problem may be\nsubstantially different than that of the (preemptive) optimal control\nproblem for the queueing system. This difference must be quantified\nrelative to the system size, which we do indirectly through the\ngradient estimates for the value function of the Brownian control\nproblem; this is manifested, for example, in the proof of Lemma \\ref{lemito}.\n\nThe second source of error is in trying to imitate the optimal ratio\ncontrol of the approximating Brownian control by a tracking control in\nthe corresponding queueing system. The error arises because we insist\non having a~nonpreemptive control for the queueing system. Whereas\nunder a preemptive control, one may be able to rearrange the queues\ninstantaneously to match the tracking function of the Brownian system,\nthis is not possible with nonpreemptive controls. Instead, we carefully\nconstruct and analyze the performance of the proposed nonpreemptive\ntracking policy. In doing so, we prove that the tracking control\nimitates closely the Brownian system with respect to a specific\nintegrated functional of the queueing dynamics (see Theorem \\ref\n{thmSSC} and Remark \\ref{remSSC}). Here too, the gradient estimates\nfor the value function of the Brownian system play a key role.\n\nWhile the focus of this paper has been a relatively simple model\nto illustrate the key ideas behind our approach and the important steps\nin the analysis, we expect that similar results can be established in\nthe cases of impatient customers, more general cost structures as well\nas more general network structures.\n\nAs suggested by the preceding analysis, the viability of these\nextensions and others will depend on whether it is possible to (a)\nsolve the\nsequence of Brownian control problems and establish the necessary\ngradient estimates and (b) establish the corresponding approximation\nresult for the nonpreemptive tracking control.\n\nWhile we expect that the results of \\cite{TandG} on fully nonlinear\nelliptic PDEs can be invoked for the more general settings, extending\nour analysis which builds on those results may not be always straightforward.\nIn particular, it is not immediately obvious how to generalize the\nproof of the tracking result in Theorem \\ref{thmSSC} to more general settings.\n\nNevertheless, we can make some observations about the extensions\nmentioned above:\n\\begin{itemize}\n\\item\\textit{General convex costs.} As discussed in the\n\\hyperref[sec1]{Introduction}, the analysis of the convex holding cost case will\nprobably be simpler as one tends to get ``interior'' solutions in that\ncase as opposed to the corner solutions in the linear cost case, which\ncauses nonsmoothness.\nWe expect that the enhanced smoothness (relative to the linear holding\ncost case) will simplify the analysis of the HJB equations as well as\nthat of the tracking performance.\n\n\\item\\textit{Abandonment.} Our starting point in the analysis is that,\namong preemptive policies, work conserving policies are optimal. This\nis not, in general, true when customers are impatient and may abandon\nwhile waiting (see the discussion in Section 5.1 of \\cite{AMR02}). As is\nthe case in \\cite{AMR02}, our analysis will go through also for the\ncase of impatient customers provided that the cost structure is such\nthat work conservation is optimal among preemptive policies.\n\\item\\textit{General networks.} Inspired by the generalization of\n\\cite{AMR02}, by Atar \\cite{atar2005scheduling}, to tree-like networks,\nwe expect, for example, that such a generalization is viable in our\nsetting as well. Indeed, we expect that the analysis of the (sequence\nof) HJB equations and the sequence of ADCPs be fairly similar for the\ntree-like network setting. We expect that, in that more general\nsetting, it would be more complicated to bound the performance of the\ntracking policies as in Theorem~\\ref{thmSSC}.\n\\end{itemize}\n\n\\begin{appendix}\\label{app}\n\\section*{Appendix}\n\n\\subsection*{\\texorpdfstring{Proof of Lemma \\protect\\ref{lemexistenceofcontrolled}}{Proof of Lemma 3.1}}\n\nUp to $\\tau_{\\kappa}^n$, both functions $b^n(\\cdot,u)$ and $\\sigma\n^n(\\cdot,u)$ are bounded and Lipschitz continuous (uniformly in $u$).\nWith these conditions satisfied, strong existence and uniqueness follow\nas in Appendix D of \\cite{FlemingSoner}. Specifically, strong\nexistence follows by successive approximations as in the proof of\nTheorem~2.9 of \\cite{KaS91} and uniqueness follows as in Theorem 2.5\nthere.\n\n\\subsection*{\\texorpdfstring{Proof of Proposition \\protect\\ref{propsolPHJB}}{Proof of Proposition 4.1}} Fix\n$\\kappa>0$, $n\\in\\bbZ_+$ and $a>0$. Recall that (\\ref{eqHJB2})\ncorresponds to finding $\\phi_{\\kappa,a}^n\\in\\mC^2(\\mB)$ such that\n\\begin{equation}\\label{eqHJB1min}\n0=F_a[x,\\phi_{\\kappa,a}^n(x),D\\phi_{\\kappa\n,a}^n(x),D^2\\phi\n_{\\kappa,a}^n(x)],\\qquad x\\in\\mB,\n\\end{equation}\nand so that\n$\\phi_{\\kappa,a}^n=0$ on $\\partial\\mB$ where \\mbox{$F_a[\\cdot]$} is as\ndefined in (\\ref{eqFForm}). Then, Proposition~\\ref{propsolPHJB}\nwill follow from Theorem 17.18 in \\cite{TandG} upon verifying certain\nconditions. The gradient estimates will also follow from \\cite{TandG}\nby carefully tracing some constants to identify their dependence on\n$\\kappa,n$ and $a$.\n\nTo that end, note that the function $F_a^i(x,z,p,r)$ [as defined in\n(\\ref{eqFdefin})] is linear in the $(z,p,r)$ arguments for all $k\\in\n\\I$ and $x\\in\\mB$. In turn, this function is concave in these\narguments. Hence, to apply Theorem 17.18 of \\cite{TandG} it remains to\nestablish that condition (17.53) of \\cite{TandG} is satisfied for each\nof these functions. In the following we suppress the constant $a>0$\nfrom the notation. It suffices to show that there exist constants\n$\\underbar{\\Lambda}\\leq\\bar{\\Lambda}$ and $\\eta$ such that\nuniformly in $k\\in\\I$, $y=(x,z,p,r)\\in\\Gamma$, and $\\xi\\in\\mathbb{R}^I$\n\\begin{eqnarray}\\label{eqFcond1}\n&\\displaystyle\n0< \\underbar{\\Lambda}|\\xi^2|\\leq\\sum_{i,j}F^k_{i,j}[y] \\xi_i\\xi\n_j\\leq\\bar{\\Lambda} |\\xi|^2,&\\\\\n\\label{eqFcond2}\n&\\displaystyle \\max\\{\n|F^k_p[y]|,|F^k_z[y]|,|F^k_{rx}[y]|,|F^k_{px}[y]|,|F^k_{zx}[y]|\\}\n\\leq\\eta\\underbar{\\Lambda},&\n\\\\\n\\label{eqFcond3}\n&\\displaystyle\n\\max\\{ |F^k_x[y]|,|F^k_{xx}[y]|\\}\\leq\\eta\\underbar\n{\\Lambda}(1+|p|+|r|),&\n\\end{eqnarray}\nwhere\n\\[\nF^k_{i,j}(x,z,p,r)=\\frac{\\partial}{\\partial r_{ij}}F^k(x,z,p,r),\\qquad\nF^k_{x_l}(x,z,p,r)=\\frac{\\partial}{\\partial x_{l}}F^k(x,z,p,r)\\vadjust{\\goodbreak}\n\\]\nand\n\\[\n(F^k_{rx}(x,z,p,r))_{ilj}=\\frac{\\partial^2}{\\partial r_{il}\\,\\partial\nx_{j}}F^k(x,z,p,r).\\vspace*{-2pt}\n\\]\nThe other cross-derivatives are defined similarly. We will show that we\ncan choose $\\underbar{\\Lambda}=\\varepsilon_0 n$, $\\bar{\\Lambda}=\n\\varepsilon_1 n$, $\\eta=\\varepsilon_2$ for constants $\\varepsilon\n_0,\\varepsilon_1$ and $\\varepsilon_2$ that do not depend on $n$ and\n$a$---this will be important in establishing the aforementioned\ngradient estimates. To establish (\\ref{eqFcond1}) note that, given\n$\\xi\\in\\mathbb{R}^I$,\n\\begin{equation}\\label{eqFijDeriv}\\quad\nF^{k}_{ij}\\xi_i\\xi_j=\\cases{\n\\frac{1}{2}\\bigl(\\lambda_i^n+\\mu_i(\\nu_in+x_i)\\bigr)\\xi\n_i^2, &\\quad for $i=j, i\\neq k$,\\vspace*{1pt}\\cr\n\\frac{1}{2}\\bigl(\\lambda_i^n+\\mu_i(\\nu_in+x_i)\\bigr)\\xi_i^2 -\\frac\n{1}{2}f(e\\cdot x), &\\quad for $i=j=k$,\\vspace*{1pt}\\cr\n0, &\\quad otherwise.}\\vspace*{-2pt}\n\\end{equation}\nHence,\n\\[\n\\sum_{i,j}F_{ij}^k \\xi_i\\xi_j=\\frac{1}{2}\\sum_{i\\in\\I}\\bigl(\\lambda\n_i^n+\\mu_i(\\nu_in+x_i)\\bigr)\\xi_i^2-\\frac{1}{2}f(e\\cdot x)\\xi_k^2.\\vspace*{-2pt}\n\\]\n\nConsequently, for $(x,z,r,p)\\in\\Gamma$ we have that\n\\[\n\\sum_{i,j}F_{ij}^k \\xi_i\\xi_j\\leq I \\bigl( \\lambda+\\mu_{\\max\n}n+\\mu_{\\max}\\kappa\\sqrt{n}\\log^mn\\bigr)\\sum_{i\\in\\I}\\xi_i^2+\n\\frac{1}{2}\\kappa\\sqrt{n}\\log^m n\\xi_k^2,\\vspace*{-2pt}\n\\]\nwhere $\\mu_{\\max}=\\max_{k}\\mu_k$. In particular, we can choose\n$\\varepsilon_1>0$ so that for all $n\\in\\bbZ$,\n\\[\n\\sum_{i,j}F_{ij}^k \\xi_i\\xi_j\\leq\\varepsilon_1 n.\\vspace*{-2pt}\n\\]\nTo obtain the lower bound note that, for $y\\in\\Gamma$,\n\\[\n\\sum_{i,j}F_{ij}^k \\xi_i\\xi_j\\geq\\frac{1}{2}\\Bigl(\\min_{i\\in\\I\n}\\lambda_i^n+\\min_{i\\in\\I}\\mu_i\\kappa\\sqrt{n}\\log^m n\n\\Bigr)\\sum_{i\\in\\I}\\xi_i^2\n-\\frac{1}{2}\\xi_k^2\\kappa\\sqrt{n}\\log^mn.\\vspace*{-2pt}\n\\]\nHence, we can find $\\varepsilon_0>0$ such that for all n,\n\\[\n\\sum_{i,j}F_{ij}^k \\xi_i\\xi_j\\geq\\varepsilon_0 n.\\vspace*{-2pt}\n\\]\nNote that above $\\varepsilon_0$ and $\\varepsilon_1$ can depend on\n$\\kappa$ but they do not depend on $n$ and $a$. Hence, we have\nestablished (\\ref{eqFcond1}) and we turn to (\\ref{eqFcond2}). To\nthat end, note that\n\\begin{eqnarray}\\label{eqFp}\nF_{p_k}^k(x,z,p,r)&=&f(e\\cdot x)+l_k\\lam-\\mu\n_kx_k \\quad\\mbox{and }\\nonumber\\\\[-9pt]\\\\[-9pt]\nF_{p_i}^k(x,z,p,r)&=&l_i\\lam-\\mu_ix_i \\qquad\\mbox{for }\ni\\neq k.\\nonumber\\vspace*{-2pt}\n\\end{eqnarray}\nTherefore,\n\\begin{eqnarray*} |F_{p}^k|&\\leq& (e\\cdot x)^+ +1+\\sum_{i}|l_i\\lam\n+\\mu_i x_i|\\\\[-2pt]\n&\\leq&\nI\\kappa\\sqrt{n}\\log^mn+1+I\\max_{i}\\bigl(|l_i\\lam|+\\mu_i\\kappa\\sqrt\n{n}\\log^m n\\bigr),\\vspace*{-2pt}\\vadjust{\\goodbreak}\n\\end{eqnarray*}\nwhere we used the simple observation that $f(e\\cdot x)\\leq(e\\cdot\nx)^++1$. Clearly, we can choose $\\varepsilon_2$ so that $|F_p^k|\\leq\n\\varepsilon_2\\varepsilon_0\\sqrt{n}\\log^m n$. Also $F_z^k=-\\gamma$\nand \\mbox{$F_{zx}=0$} so that by re-choosing $\\varepsilon_2$ large enough we\nhave $\\max\\{|F^k_z[y]|,|F^k_{zx}[y]|\\}\\leq\\varepsilon_2\\varepsilon\n_0\\sqrt{n}\\log n$. Finally, by (\\ref{eqFijDeriv}) we have that\n\\begin{eqnarray*}\nF^k_{r_{ij}x_l}&=&0 \\qquad\\mbox{for } i\\neq j,\\\\\nF^k_{r_{ii}x_j}&=&0 \\qquad\\mbox{for } i\\neq k, i\\neq\nj,\\\\\nF^k_{r_{ii}x_i}&=&\\frac{1}{2}\\mu_i \\qquad\\mbox{for } i\\neq k,\\\\\nF^k_{r_{ii}x_i}&=&\\frac{1}{2}\\mu_i \\qquad\\mbox{for } i\\neq\nk,\\\\\nF^k_{r_{kk}x_k}&=&\\frac{1}{2}\\mu_k-\\frac{1}{2}\\,\\frac{\\partial\n}{\\partial x_k}f(e\\cdot x),\\\\\nF^k_{r_{kk}x_j}&=&\\frac{1}{2}\\,\\frac{\\partial}{\\partial x_k}f(e\\cdot\nx) \\qquad\\mbox{for } j\\neq k.\n\\end{eqnarray*}\nThus,\n\\[\n|F_{rx}^k|^2\\leq\\sum_{l\\in\\I}\\frac{1}{2}\\biggl|\\frac{\\partial\n}{\\partial x_l}f(e\\cdot x)\\biggr|^2+\\frac{1}{2}\\mu_{\\max}\\leq\\frac\n{1}{2}(1+\\mu_{\\max}),\n\\]\nwhere we used the fact that $f(\\cdot)$ is continuously differentiable\nwith Lipschitz constant $1$ (independently of $a$). Finally,\n\\[\nF_{x_i}^k=\\frac{\\partial}{\\partial x_i}f(e\\cdot x)\\biggl(c_k+\\mu\n_kp_k-\\frac{1}{2}\\mu_kr_{kk}\\biggr)-\\mu_ip_i+\\frac{1}{2}\\mu_ir_{ii},\n\\]\nso that\n\\begin{equation}\\label{eqFx}\n|F_{x_i}^k|\\leq|c_k|+\\mu_k|p|+\\tfrac{1}{2}\\mu\n_k|r|+\\mu\n_i|p|+\\tfrac{1}{2}\\mu_i |r|.\n\\end{equation}\nAlso, note that\n\\[\nF_{x_ix_j}^k=\\frac{\\partial}{\\partial x_i\\,\\partial x_j}f(e\\cdot\nx)\\biggl(c_k+p_k-\\frac{1}{2}r_{kk}\\biggr),\n\\]\nso that\n\\[\nF_{x_ix_j}^k=\\cases{\n2\\bigl[c_k+\\mu_kp_k-\\frac{1}{2}\\mu_kr_{kk}\\bigr], &\\quad\nif $|e\\cdot x|\\leq\\frac{1}{4}$,\\vspace*{2pt}\\cr\n0, &\\quad otherwise.}\n\\]\nCombining the above gives\n\\[\n|F_{xx}^k|\\leq\\varepsilon_2\\varepsilon_0(1+|p|+|r|)\n\\]\nfor suitably\\vspace*{1pt} redefined $\\varepsilon_2$ which concludes the proof that\nthe conditions (\\ref{eqFcond1})--(\\ref{eqFcond3}) hold with $\\bar\n{\\Lambda}=\\varepsilon_1n$, $\\underbar{\\Lambda}=\\varepsilon_0n$\nand $\\eta=\\varepsilon_2$. Having verified these conditions, the\nexistence and uniqueness of the solution $\\phi_{\\kappa,a}^n$ to\n(\\ref{eqHJB2}) now follows from Theorem~17.18 in \\cite{TandG}.\n\nTo obtain the gradient estimates in (\\ref{eqgradients})\nwe first outline how the solution~$\\phi\n_{k,a}^n$ is obtained in \\cite{TandG} as a limit of solutions to\nsmoothed equations (we refer the reader to~\\cite{TandG}, page 466, for\nthe more elaborate description). To that end, let\n$F_a^i$ be as defined in (\\ref{eqFdefin}) and for $y\\in\\Gamma$ define\n\\begin{equation}\\label{eqGhPDE}\nF^h[y]=G_{h}(F^1_a[y],\\ldots,F^I_a[y]),\n\\end{equation}\nwhere\n\\[\nG_h(y)=h^{-I}\\int_{\\bar{y}\\in\\bbR^I}\\rho\\biggl(\\frac{y-\\bar\n{y}}{h}\\biggr)G_0(\\bar{y})\\,d\\bar{y}\n\\]\nand $G_0(x)=\\min_{i\\in\\I}x_i$ and $\\rho(\\cdot)$ is a mollifier on\n$\\bbR^I$ (see \\cite{TandG}, page 466). $F^h$ satisfies all the bounds\nin (\\ref{eqFcond1})--(\\ref{eqFcond3}) uniformly in $h$; cf. \\cite\n{TandG}, page 466. Then, there exists a unique solution $u^h$ for the equations\n\\begin{equation} \\label{eqGhPDE2}\nF^h[u^h]=0\n\\end{equation}\non $\\mB_{\\kappa}^n$ with\n$u^h=0$ on $\\partial\\mB_{\\kappa}^n$.\n\nThe solution $\\phi_{\\kappa,a}^n$ is now obtained as a limit of $\\{\nu^h\\}$ in the space $C_*^{2,\\alpha}(\\mB)$ as defined in (\\ref\n{eqCstar}). Moreover, since the gradient bounds are shown in \\cite\n{TandG} to be independent of $h$, it suffices for our purposes to fix\n$h$ and focus on the construction of the gradient bounds.\n\nOur starting point is the bound at the bottom of page 461 of \\cite\n{TandG} by which\n\\begin{equation}\\label{eqonemoreinterim}\n|u^h|^*_{2,\\alpha,\\mB_{\\kappa}^n}\\leq\\check\n{C}(a,n)(1+|u^h|^*_{2,\\mB_{\\kappa}^n}),\n\\end{equation}\nwhere\\vspace*{-1pt} $|u^h|^*_{2,\\mB_{\\kappa}^n}=\\sum_{j=0}^2 [u^h]_{j,\\mB}^*$ and\n$[\\cdot]_{j,\\mB}^*, j=0,1,2$, are as defined in Section~\\ref{secADCP}.\nThe constant $\\alpha(a,n)$ depends only on the number of\nclasses $I$ and on $\\bar{\\Lambda}\/\\underbar{\\Lambda}$ (see~\\cite\n{TandG}, top of page 461) and this fraction equals, in our context, to\n$\\varepsilon_1\/\\varepsilon_0$ and is thus constant and independent of\n$n$ and $a$.\n\nWe will\\vspace*{1pt} address the constant $\\check{C}(a,n)$ shortly. We first argue\nhow one proceeds from (\\ref{eqonemoreinterim}). Fix $0<\\delta< 1$,\nlet $\\epsilon=\\delta\/\\check{C}(a,n)$ and $C(\\epsilon)=2\/(\\epsilon\n\/8)^{1\/\\alpha}$ (see \\cite{TandG}, top of page 132). Then, applying\nan interpolation inequality (see~\\cite{TandG}, bottom of page 461 and\nLemma 6.32 on page 130), it is obtained that\n\\[\n|u^h|^*_{2,0,\\mB_{\\kappa}^n}\\leq C(\\epsilon)|u^h|^*_{0,\\Omega\n}+\\epsilon|u^h|^*_{2,\\alpha,\\mB_{\\kappa}^n}.\n\\]\nPlugging this back into (\\ref{eqonemoreinterim}) one then has\n\\[\n|u^h|^*_{2,\\alpha,\\mB_{\\kappa}^n}\\leq\\check{C}(a,n)\\biggl(1+\\bar\n{C}\\check{C}(a,n)^{1\/\\alpha}|u^h|^*_{0,\\mB_{\\kappa}^n}\n+\\frac{\\delta}{\\check{C}(a,n)}|u^h|^*_{2,\\alpha,\\mB_{\\kappa\n}^n}\\biggr)\n\\]\nfor a constant $\\bar{C}$ that depends only on $\\delta$ and $\\alpha$.\nIn turn,\n\\[\n|u^h|^*_{2,\\alpha,\\mB_{\\kappa}^n}\\leq\\bar{C}(\\check\n{C}(a,n))^{1+1\/\\alpha}|u^h|^*_{0,\\mB_{\\kappa}^n}\n\\]\nfor a constant $\\bar{C}$ that\\vadjust{\\goodbreak} does not depend on $a$ or $n$.\n\nHence, to obtain the required bound in (\\ref{eqgradients})\nit remains only to\\break bound~$\\check{C}(a,n)$. Following \\cite{TandG},\nbuilding on equation (17.51) of \\cite{TandG}, $\\check{C}(a,n)$ is the\n(minimal) constant\nthat satisfies\n\\begin{equation}\\label{eqCandefin}\nC(1+M_2)(1+\\tilde{\\mu}R_0+\\bar{\\mu}R_0^2)\\leq\n\\check{C}(a,n)(1+|u^h|^*_{2,\\mB}),\n\\end{equation}\nwhere (as stated in \\cite{TandG}, bottom of page 460) the (redefined)\nconstant $C$ depends only on the number of class $I$ and on $\\bar\n{\\Lambda}\/\\underbar{\\Lambda}=\\varepsilon_1\/\\varepsilon_0$.\nThe constants~$\\tilde{\\mu}$ and~$\\bar{\\mu}$ are defined in \\cite{TandG}\nand we will explicitly define them shortly. Here one should not\nconfuse $\\bar{\\mu}$ with the average service rate in our system. In\nwhat follows $\\bar{\\mu}$ will only be used as the constant in \\cite\n{TandG}. We now bound constants~$\\tilde{\\mu}$ and~$\\bar{\\mu}$.\nThese are defined by\n\\begin{eqnarray*}\n\\tilde{\\mu}&=&\\frac{D_0}{\\underbar{\\Lambda\n}(1+M_2)},\\qquad\n\\bar{\\mu} =\\frac{C(I)}{\\underbar{\\Lambda}}\n\\biggl(\\frac{A_0^2}{\\underbar{\\Lambda}\\epsilon}+\\frac{B_0}{1+M_2}\\biggr),\n\\\\\nD_0&=&\\sup_{x,y\\in\\mB}\\{|F^h_x(y,u^h(y),Du^h(y),D^2u^h(x))|\n\\\\\n&&\\hphantom{\\sup_{x,y\\in\\mB}\\{}{}+|F^h_z(y,u^h(y),Du^h(y),D^2u^h(x))||Du^h(y)|\\\\\n&&\\hphantom{\\sup_{x,y\\in\\mB}\\{}{}+|F^h_p(y,u^h(y),Du^h(y),D^2u^h(x))||D^2u^h(y)|\\},\n\\\\\nA_0&=&\\sup_{\\mB}\\{|F^h_{rx}|+|F^h_p|\\}, \\\\\nB_0&=& \\sup_{\\mB} \\{|F_{px}||D^2 u^h|+ |F_z||D^2\nu^h|+|F_{zx}||Du^h|+|F_{xx}|\\},\n\\end{eqnarray*}\nwhere $C(I)$ is a constant that depends only on the number of classes\n$I$, $\\epsilon\\in(0,1)$ is arbitrary and fixed (independent of $n$\nand $a$) and $M_2=\\sup_{\\mB}|D^2u^h|$. The constants $\\bar{\\mu}$,\n$\\tilde{\\mu}$ and $M_2$ are defined in \\cite{TandG}, pages 456--460,\nand $A_0$ and~$B_0$ are as on page 461 there.\n\nWe note that $F^h_z$ is a constant, $F^h_p$ is bounded by $\\bar\n{C}\\sqrt{n}\\log^m n$ for some constant $\\bar{C}$ [see (\\ref\n{eqgradients})] that depends only on $\\kappa$ and, by (\\ref{eqFx}),\n$|F_x^h|\\leq\\varepsilon_2\\varepsilon_0(1+|p|+|r|)$. In turn,\n$D_0\\leq4\\varepsilon_2\\varepsilon_0\\sqrt{n}\\log^m n\\sup_{\\mB\n}(1+|Du^h|+|D^2u^h|)$. Arguing similarly for $A_0$ and $B_0$ we find\nthat there exists a constant $\\bar{C}$ (that does not depend on $n$\nand $a$) such that\n\\[\nA_0\\leq\\bar{C}\\sqrt{n}\\log^mn\\quad\\mbox{and}\\quad B^0\\leq\\bar{C} \\sup\n_{\\mB}(1+|Du^h|+|D^2u^h|),\n\\]\nwhich in turn implies the existence of a redefined constant $\\bar{C}$\nsuch that\n\\[\n\\tilde{\\mu}\\leq\\frac{\\bar{C}\\log^{m} n}{\\sqrt{n}(1+M_2)}\\sup\n_{\\mB}(1+|Du^h|+|D^2u^h|)\n\\]\nand\n\\[\n\\bar{\\mu}\\leq\\frac{\\bar{C}\\log^{2 m} n}{n}+\\frac{\\bar\n{C}}{n(1+M_2)}\\sup_{\\mB}(1+|Du^h|+|D^2u^h|).\n\\]\nThe proof of the bound is concluded by plugging these back into (\\ref\n{eqCandefin}) and setting $R_0=\\kappa\\sqrt{n}\\log^m n$ there to get\nthat\n\\[\n\\check{C}(a,n)\\leq C\\log^{4m(1+{1}\/{\\alpha})}n\n\\]\nfor some $C$ that does not depend on $a$ and $n$.\n\nThe constant $\\tilde{C}$ on the right-hand side of (\\ref{eqgradients})\n(which can depend on $n$ but does not depend on $a$) is\nargued as in the proof of Theorem 17.17 in~\\cite{TandG} and we\nconclude the proof by noting that the global Lipschitz constant (that\nwe allow to depend on~$n$) follows from Theorem 7.2 in\n\\cite{trudinger1983fully}.\n\nWe next turn to proof of Theorem \\ref{thmSSC}. First, we will\nexplicitly construct the queueing process under the $h$-tracking policy\nand state a lemma that will be of use in the proof of the theorem.\nDefine $A_i^n(t)=\\mN_i^a(\\lambda_i^n t)$ so that~$A_i^n$ is the\narrival process of class-$i$ customers. Given a ratio control $U^n$ and\nthe associated queueing process $\\mathbb{X}^n=(X^n,Q^n,Z^n,\\check{X}^n)$,\n$\\check{W}^n$ is as defined in~(\\ref{eqWtildedefin}). Also, we define\n\\[\nD^n(t)=\\sum_{i\\in\\I}\\mN_i^d\\biggl( \\mu_i\\int_0^t Z_i\\lam(s)\\,ds\n\\biggr).\n\\]\nThat is, $D^n(t)$ is the total number of service completions by time\n$t$ in the $n$th system.\n\nFor the construction of the queueing process under the tracking policy\nwe define a family of processes\n$\\{\\mathcal{A}_{i,\\mH}^n, i\\in\\I,\\mathcal{H}\\subset\\I\\}$ as follows:\nlet $\\{\\xi_{\\mK}^l; l\\in\\bbZ_+,\\mK\\subset\\I\\}$ be a family of i.i.d\nuniform $[0,1]$ random variables independent of\n$\\bar{\\mathcal{F}}_{\\infty}$ as defined in~(\\ref{eqcheckFdefin}). For\neach $\\mathcal{K}\\subset\\I$, define the processes\n$(\\mathcal{A}_{i,\\mH}^n, i\\in\\I)$ by\n\\begin{equation}\\label{eqmAdefin}\n\\mathcal{A}_{i,\\mH}^n(t)=\\sum_{l=1}^{D^n(t)}1\\biggl\\{\\frac{\\sum_{k<\ni,k\\in\\mH}\\lambda_k}{1\\vee\\sum_{k\\in\\mH}\\lambda_k} < \\xi_{\\mK\n}^l\\leq\n\\frac{\\sum_{k\\leq i,k\\in\\mH}\\lambda_k}{1\\vee\n\\sum_{k\\in\\mH}\\lambda_k}\\biggr\\}.\n\\end{equation}\n\nWe note that for any strict subset $\\mH\\subset\\I$ and $i\\in\\mK$,\nthe probability that a jump of $D^n(t)$ results in a jump of $\\mathcal\n{A}_{i,\\mH}^n$ is equal to $\\lambda_i^n\/\\sum_{k\\in\\mH}\\lambda\n_k^n=a_i\/\\sum_{k\\in\\mH}a_k$ and is strictly greater than $\\lambda\n_i^n\/\\sum_{k\\in\\I}\\lambda_k^n=a_i$. We define\n\\begin{equation}\\label{eqepsilondefin}\n\\epsilon_i=\\min\n_{\\mathcal{H}\\subset\\I}a_i-\\frac{a_i}{\\sum_{k\\in\\mH}a_k},\n\\end{equation}\nand note that $\\epsilon_i>0$ by our assumption\nthat $a_i>0$ for all $i\\in\\I$ (see Section~\\ref{secmodel}). Let\n$\\bar{\\epsilon}=\\min_{i}\\epsilon_i\/4$.\n\nNote that at time intervals in which $i\\in\\mK(\\cdot)=\\mH$ (see\nDefinition \\ref{defintracking}) for some $\\varnothing\\neq\\mH\\subset\n\\I$, the process $\\mathcal{A}_{i,\\mH}^n$ jumps with probability\n$\\lambda_i^n\/\\sum_{k\\in\\mH}\\lambda_k$ whenever a server becomes\navailable (i.e., upon a jump of $D^n$). In turn, we will use the\nprocesses $\\{\\mathcal{A}_{i,\\mH}^n, i\\in\\I,\\mathcal{H}\\subset\\I\\}\n$ to generate (randomized) admissions to service of class-$i$ customers\nunder the $h$-tracking policy.\n\nMore specifically, under the $h$-tracking policy (see Definition \\ref\n{defintracking}) a customer from the class-$i$ queue enters service in\nthe following events:\n\\begin{longlist}\n\\item A class-$i$ customer that arrives at time $t$ enters\nservice immediately if there are idle servers, that is, if $(e\\cdot\n\\check{X}\\lam(t-))^{-}>0$.\n\\item If a server becomes available at time $t$ (corresponding\nto a jump of~$D^n$) and $t$ is such that $i\\in\\mathcal{K}(t-)=\\mH\n\\subset\\I$, then a customer from the class-$i$ queue is admitted to\nservice at time $t$ with probability $\\lambda_i^n\/\\sum_{k\\in\\mH\n}\\lambda_k$. This admission to service corresponds to a jump of the\nprocess $\\mathcal{A}_{i,\\mH}^n$ as defined in (\\ref{eqmAdefin}).\n\\item If a server becomes available at time $t$ (corresponding\nto a jump of~$D^n$) and $t$ is such that $\\mathcal{K}(t-)=\\varnothing$\nand $i=\\min\\{k\\in\\I\\dvtx Q_i^n(t)>0\\}$, then a~class-$i$ customer is\nadmitted to service.\n\\end{longlist}\nFormally, the queueing process $\\mathbb{X}\\lam=(X\\lam,Q\\lam,Z\\lam\n,\\check{X}\\lam)$ satisfies\n\\begin{eqnarray*} Z_i\\lam(t)&=&Z_i\\lam(0)+\\int_0^t 1\\bigl\\{\\bigl(e\\cdot\n\\check{X}^n(s)\\bigr)^->0\\bigr\\}\\,dA_i^n(s)\\\\\n&&{}+\\sum_{\\mH\\subset\\I}\\int\n_0^t1\\{i\\in\\mathcal{K}(s-),\\mathcal{K}(s-)=\\mathcal{H}\\}\\,\nd\\mathcal{A}_{i,\\mH}\\lam(s)\\\\\n&&{}+ \\int_0^t 1\\bigl\\{\\mK\n(s-)=\\varnothing,i=\\min\\{k\\in\\I\\dvtx Q_k^n(s-)>0\\}\\bigr\\}\\,dD^n(s)\\\\\n&&{}-\\mN\n_i^d\\biggl( \\mu_i\\int_0^t Z_i\\lam(s)\\,ds \\biggr),\\qquad i\\in\\I,\\\\\nX_i\\lam(t)&=&X_i\\lam(0)+A_i^n(t)-\\mN_i^d\\biggl(\\mu_i \\int_0^t\nZ_i\\lam(s) \\,ds \\biggr),\\qquad i\\in\\I, \\\\ Q_i\\lam(t)&=&X_i\\lam(t)-Z_i\\lam\n(t),\\qquad i\\in\\I.\n\\end{eqnarray*}\n\nThe second, third and fourth terms on the right-hand side of the\nequation for $Z_i^n$ correspond, respectively, to the events described\nby items (i)--(iii) above. Finally, $\\check{X}\\lam$ is defined from\n$X\\lam$ as in (\\ref{eqtildeXdefin}). The fact that the above system\nof equations has a unique solution is proved by induction on arrival\nand service completions times (see, e.g., the proof of Theorem 9.2 of\n\\cite{MMR98}). Clearly, $\\mathbb{X}^n$ satisfies (\\ref\n{eqdynamics2})--(\\ref{eqnon-negativity2}) with $U_i\\lam$ there\nconstructed from $Q\\lam$ using~(\\ref{eqUQmap}).\n\nWe note that, with this construction, the tracking policy is admissible\nin the sense of Definition \\ref{definadmissiblecontrols}. Also, it\nwill be useful for the proof of Theorem~\\ref{thmSSC} to note that\nwith this construction, if $[s,t]$ is an interval such that \\mbox{$i\\in\n\\mathcal{K}(u)\\subset\\I$} for all $u\\in[s,t]$ then\n\\begin{equation}\\label{eqQtrack}\\qquad\nQ_i^n(t)-Q_i^n(s)=A_i^n(t)-A_i^n(s)-\\sum_{\\mH\\subset\n\\I}\\int\n_s^t1\\{\\mathcal{K}(u-)=\\mathcal{H}\\}\\,d\\mathcal{A}_{i,\\mH\n}\\lam(u).\n\\end{equation}\n\nBefore proceeding to the proof of Theorem \\ref{thmSSC} the following\nlemma provides preliminary bounds for arbitrary ratio controls.\n\\begin{lem}\\label{lemstrongappbounds}\nFix $\\kappa,T>0$ and a ratio control $U^n$, let $\\mathbb\n{X}^n=(X^n,Q^n,\\allowbreak Z^n,\\check{X}^n)$ be the associated queueing process\nand define\n\\[\n\\tau_{\\kappa,T}^n=\\inf\\{t\\geq0\\dvtx\\check{X}^n(t)\\notin\\mB_{\\kappa\n}^n\\}\\wedge T\\log n.\n\\]\nThen, there exist constants $C_1,C_2,K_0>0$ (that depend on $T$ and\n$\\kappa$ but that do not depend on $n$ or on the ratio control $U^n$)\nsuch that for all $K>K_0$ and all $n$ large enough,\n\\begin{eqnarray}\\label{eqstrapp1}\n&&\\Pd\\Bigl\\{\\sup_{0\\leq t\\leq2T\\log n}|\\check\n{W}^n(t)|> K\\sqrt\n{n}\\log n\\Bigr\\}\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\leq C_1e^{-C_2K\\log n},\\nonumber\\\\\n\\label{eqstrapp2}\n&&\\Pd\\bigl\\{|\\check{X}^n(t)-\\check{X}^n(s)|> \\bigl((t-s)+(t-s)^2\n\\bigr)K\\sqrt{n}\\log n+K\\log n,\n\\nonumber\\\\\n&&\\hspace*{163pt}\n\\mbox{ for some } s<t\\leq2T\\log n\\bigr\\}\\\\\n&&\\qquad\\leq C_1 e^{-C_2K\\log n},\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eqstrapp3}\n&&\\Pd\\{|A_i^n(t)-A_i^n(s)-\\lambda_i^n(t-s)|\n> \\bar{\\epsilon}n(t-s)+K\\log n\\nonumber\\\\\n&&\\hspace*{139.3pt} \\mbox{ for some }\ns<t\\leq\\tau_{\\kappa,T}^n\\}\\\\\n&&\\qquad\\leq C_1e^{-C_2K\\log n},\\qquad i\\in\\I,\\nonumber\\\\\n\\label{eqstrapp4}\n&&\\Pd\\biggl\\{\n\\biggl|D^n(t)-D^n(s)-\\sum_{i}\\mu_i\\nu_i n(t-s)\\biggr|\n> \\bar{\\epsilon}n(t-s)+K\\log n\\nonumber\\\\\n&&\\hspace*{174pt}\\mbox{ for some } s<t\\leq\\tau_{\\kappa,T}^n\\biggr\\}\n\\\\\n&&\\qquad\\leq C_1e^{-C_2K\\log n},\\nonumber\\\\\n\\label{eqstrapp5}\n&&\\Pd\\biggl\\{\\mathcal{A}_{i,\\mH}^n(t)-\\mathcal{A}_{i,\\mH\n}^n(s)-\\lambda_i^n(t-s)\\leq\\frac{\\epsilon_i}{2} n(t-s)-K\\log\nn \\nonumber\\\\\n&&\\hspace*{156.6pt}\\mbox{ for some } s<t\\leq\\tau\n_{\\kappa,T}^n\\biggr\\}\\\\\n&&\\qquad\\leq C_1e^{-C_2K\\log n},\\qquad i\\in\\I,\\mK\\subset\\I\n.\\nonumber\n\\end{eqnarray}\n\\end{lem}\n\\begin{pf}\nEquation (\\ref{eqstrapp1}) follows from strong approximations (see,\ne.g., Lem\\-ma~2.2. in \\cite{csorgo}) and known bounds on the supremum of\nBrownian motion (see, e.g., equation 2.1.53 in \\cite{csorgo}).\nEquation (\\ref{eqstrapp2}) then follows using this bound together\nwith (52) in \\cite{AMR02} but with $X^n(t)-X^n(s)$ instead of\n$X^n(t)$ (in the notation of \\cite{AMR02} $\\check{W}^n$ is $\\hat\n{W}^n$). Equations (\\ref{eqstrapp3})--(\\ref{eqstrapp5}) follow by\ncarefully constructing and bounding the increments. We outline the\nproof of (\\ref{eqstrapp3}) and the others follow similarly. To\nthat\\vadjust{\\goodbreak}\nend, note that given $K$ and for all $n$ large enough\n\\begin{eqnarray*}\n&& \\{|A_i^n(t)-A_i^n(s)-\\lambda_i^n(t-s)|\\leq\n\\bar\n{\\epsilon}n(t-s)+K\\log n, \\\\\n&&\\hspace*{109pt}\n\\mbox{ for all }\n0\\leq s\\leq t\\leq2T\\log n\\} \\\\\n&&\\qquad \\supseteq\n\\biggl\\{\\max_{l\\leq N_i^n}\\max_{j\\geq0\\dvtx j\\\/\\log^l n\\leq3T\\log\nn}\\frac{|A_i^{j,l,n}-\\lambda_i^n\/\\log^l n|}{\\sqrt{\\lambda_i^n\/\\log\n^l n}}\\leq K\\sqrt{\\log n}\\biggr\\},\n\\end{eqnarray*}\nwhere\n$A_i^{j,l,n}=A_i^n((j+1)\/\\log^ln)-A_i^n(j\/\\log^ln)$ and $N=\\max\\{l\\dvtx\n\\log^ln \\leq\\lambda_i^n\/\\log n\\}$. Indeed, given an interval $[s,t)$\nwe can construct it from smaller intervals. Starting with $l=0$, we fit\nas many intervals of size $1$ into $[s,t)$, we then continue to fit as\nmany intervals of size $1\/\\log n$ to the uncovered part of the interval\nand continue sequentially in $l$. We omit the simple and detailed\nconstruction. Note that with such construction, given an interval\n$[s,t)$, its covering uses at most $\\log n$ intervals of size $\\log^l\nn$ for each $l\\geq0$. Also, note that $N_i^n\\leq C\\log n$ for all $n$\nand some constant $C$. From here, using strong approximations (or\nbounds for Poisson random variables as in \\cite{glynn1987upper}) we\nhave, for each $j$ and $l$, that\n\\[\n\\Pd\\biggl\\{\\frac{|A_i^{j,l,n}-\\lambda_i^n\/\\log^l n|}{\\sqrt{\\lambda\n_i^n\/\\log^l n}}> K\\sqrt{\\log n}\\biggr\\}\\leq C_1e^{-C_2 K\\log n}.\n\\]\nSince the number of intervals considered is of the order of $n\\log n$,\nthe bound follows with redefined constants $C_1$ and $C_2$.\n\\end{pf}\n\n\\subsection*{\\texorpdfstring{Proof of Theorem \\protect\\ref{thmSSC}}{Proof of Theorem 5.1}} Since $\\kappa$\nis fixed throughout we use $h^n(\\cdot)=h_{\\kappa}^{*,n}(\\cdot)$. As\nin the statement of the theorem, let\n\\[\n\\psi^n(x,u)=L(x,u)+A^n_u\\phi_{\\kappa}^n(x)-\\gamma\\phi_{\\kappa\n}^n(x)\\qquad \\mbox{for } x\\in\\mB_{\\kappa}^n, u\\in\\mathcal{U}\n,\\vadjust{\\goodbreak}\n\\]\nso that by the definition of $A^n_u(x)$ we have\n\\begin{eqnarray}\\qquad\n\\psi^n(x,u)&=&-\\gamma\\phi_{\\kappa}^n(x)\\nonumber\\\\\n&&{} + (e\\cdot\nx)^+\\cdot\\sum_{i\\in\\I}u_i\\biggl\\{ c_i+\\mu_i(\\phi_{\\kappa\n}^n)_i(x)-\\mu_i\\frac{1}{2}(\\phi_{\\kappa}^n)_{ii}(x)\\biggr\\}\n\\\\\n&&{}+\\sum_{i\\in\\I} (l_i\\lam-\\mu_ix_i)(\\phi_{\\kappa}^n)_i(x)\n+\\frac{1}{2}\\sum_{i\\in\\I} \\bigl(\\lambda_i^n+\\mu_i(\\nu\n_in+x_i)\\bigr)(\\phi_{\\kappa}^n){ii}(x).\\nonumber\n\\end{eqnarray}\nDefining, as before,\n\\[\nM_{i}^n(z)=c_{i}+\\mu_i(\\phi_{\\kappa}^n)_{i}(z)-\\tfrac{1}{2}\\mu\n_i(\\phi_{\\kappa}^n)_{ii}(z),\n\\]\nwe have that\n\\[\n\\psi^n(x,u)-\\psi^n(x,v)=(e\\cdot x)^+\\biggl(\\sum\n_{i\\in\\I}v_iM_i^n(x)-\\sum_{i\\in\\I}u_iM_i^n(x)\\biggr).\n\\]\nLet $U^n$ be the ratio control associated with the $h^n$-tracking\npolicy, let $\\mathbb{X}^n=(X^n,Q^n,Z^n,\\check{X}^n)$ be the\nassociated queueing process and define\n\\begin{eqnarray}\\label{eqpsicheckdefin}\n\\check{\\psi}^n(s)&=&\\psi^n(\\check\n{X}^n(s),U^n(s))-\\psi^n(\\check{X}^n(s),h^n(\\check{X}^n(s)))\\nonumber\n\\\\\n&=&\n\\bigl(e\\cdot\\check{X}^n(s)\\bigr)^+\\sum_{i\\in\\I}h_i^n(\\check\n{X}^n(s))M_i^n(\\check{X}^n(s)) \\\\\n&&{}-\\bigl(e\\cdot\n\\check{X}^n(s)\\bigr)^+ \\sum_{i\\in\\I}U_i^n(s)M_i^n(\\check\n{X}^n(s)).\\nonumber\n\\end{eqnarray}\nRecall that, by construction, $Q_i^n(s)=(e\\cdot\\check\n{X}^n(s))^+U_i^n(s)$ so that (\\ref{eqpsicheckdefin}) can be\nre-written as\n\\begin{eqnarray*} \\check{\\psi}^n(s)&=&\\psi^n(\\check\n{X}^n(s),U^n(s))-\\psi^n(\\check{X}^n(s),h^n(\\check{X}^n(s)))\n\\\\&=&\n\\bigl(e\\cdot\\check{X}^n(s)\\bigr)^+\\sum_{i\\in\\I}h_i^n(\\check\n{X}^n(s))M_i^n(\\check{X}^n(s)) \\\\\n&&{}-\\sum_{i\\in\\I\n}Q_i^n(s)M_i^n(\\check{X}^n(s)).\n\\end{eqnarray*}\nThe theorem will be proved if we show that\n\\begin{equation}\\label{eqwhatneed}\n\\Ex\\biggl[\\int_0^{\\tau_{\\kappa',T}^n}e^{-\\gamma s}\n|\\check\n{\\psi}^n(s)|\\,ds\\biggr]\\leq C\\log^{k_0+3}n.\n\\end{equation}\nTo\nthat end, define a sequence of times $\\{\\tau_{l}^n\\}$ as follows:\n\\[\n\\tau_{l+1}^n=\\inf\\{t> \\tau_l^n\\dvtx h^n(\\check{X}^n(t))\\neq h^n(\\check\n{X}^n(\\tau_l^n))\\}\\wedge\\tau_{\\kappa',T}^n \\qquad\\mbox{for } l\\geq0,\n\\]\nwhere $\\tau_0^n=\\eta^n\\wedge\\tau_{\\kappa',T}^n$ and\n\\begin{equation}\\label{eqetandefin}\n\\eta\n^n=t_0\\frac{\\log^mn}{\\sqrt{n}}\n\\end{equation}\nfor\\vspace*{1pt}\n$t_0=4\\kappa\/\\epsilon_i$ with $\n\\epsilon_i=\\min_{\\mathcal{H}\\subset\\I}a_i-\\frac{a_i}{\\sum_{k\\in\n\\mH}a_k}$ as in (\\ref{eqepsilondefin}). Finally, we define $r^n=\\sup\n\\{l\\in\\bbZ_+\\dvtx \\tau_{l}^n\\leq\\tau_{\\kappa',T}^n\\}$ and set $\\tau\n_{r^n+1}^n=\\tau_{\\kappa',T}^n$. We then have\n\\begin{eqnarray*}\n&&\\int_0^{\\tau\n_{\\kappa',T}^n}e^{-\\gamma s} |\\check{\\psi}^n(s)|\\,ds\\\\\n&&\\qquad=\\sum\n_{l=1}^{r^n+1} \\int_{\\tau_{l-1}^n}^{\\tau_l^n}e^{-\\gamma s} |\\check\n{\\psi}^n(s)|\\,ds\\\\\n&&\\qquad=\\sum_{l=1}^{r^n+1}\\biggl(\\int_{\\tau\n_{l-1}^n}^{(\\tau_{l-1}^n+\\eta^n)\\wedge\\tau_{l}^n}e^{-\\gamma s}\n|\\check{\\psi}^n(s)|\\,ds+\\int_{\\tau_{l-1}^n+\\eta^n}^{\\tau_l^n\\vee\n(\\tau_{l-1}^n+\\eta^n) }e^{-\\gamma s} |\\check{\\psi}^n(s)|\\,ds\n\\biggr).\n\\end{eqnarray*}\nThe proof is now divided into three parts. We will show that,\nunder the conditions of the theorem,\n\\begin{eqnarray}\\label{eqSSC1}\n\\Ex\\Bigl[\\sup_{1\\leq l\\leq\nr^n+1}\\sup_{\\tau_{l-1}^n\\leq s< (\\tau_{l-1}^n+\\eta^n)\\wedge\\tau\n_l^n}|\\check{\\psi}^n(s)|\\Bigr]&\\leq& C\\log^{k_0+2}n,\n\\\\\n\\label{eqSSC2}\n\\Ex\\Bigl[\\sup_{1\\leq l\\leq r^n+1}\\sup_{(\\tau\n_{l-1}^n+\\eta\n^n)\\leq s< \\tau_{l}\\vee(\\tau_{l-1}^n+\\eta^n)}\n|\\check{\\psi}^n(s)|\\Bigr]&\\leq& C\\log^{k_0+2} n,\n\\end{eqnarray}\nwhere we define $\\sup_{(\\tau_{l-1}^n+\\eta^n)\\leq s< \\tau_{l}^n\\vee\n(\\tau_{l-1}^n+\\eta^n)}\n|\\check{\\psi}^n(s)|=0$ if $\\tau_l^n\\leq\\tau_{l-1}^n+\\eta^n$.\nFinally, we will show that\n\\begin{equation}\\label{eqSSC3}\n\\Ex\\biggl[\\int_0^{\\eta^n\\wedge\\tau_{\\kappa\n',T}^n}|\\check\n{\\psi}^n(s)\\,ds|\\biggr]\\leq C\\log^{k_0}n.\n\\end{equation}\nThe proof of (\\ref{eqSSC1}) hinges on the fact that, sufficiently\nclose to a change point~$\\tau_l^n$, all the customer classes, $i$, for\nwhich $h_i^n(\\check{X}^n(s))=1$ for some $s$ in a~neighborhood of\n$\\tau_l^n$, will have similar values of $M_i^n(\\check{X}^n(s))$. This\nwill follow from our gradient estimates for $\\phi_{\\kappa}^n$. The\nproof of (\\ref{eqSSC2}) hinges on the fact that,~$\\eta^n$ time units\nafter a change point $\\tau_l^n$ the queues of all the classes for\nwhich $h^n(\\check{X}^n(\\tau_l^n))=0$ are small because, under the\ntracking policy, these classes receive a significant share of the capacity.\n\nToward formalizing this intuition, define the following event on the\nunderlying probability space:\n\\begin{eqnarray*} \\tilde{\\Omega}(K)&=&\\bigl\\{|\\check{X}^n(t)-\\check\n{X}^n(s)|\\leq K \\sqrt{n}\\log^2 n(t-s)+K\\log n,\\\\[-0.8pt]\n&&\\hspace*{135.3pt}\n\\mbox{ for all } s<t\\leq\\tau_{\\kappa,T}^n\\bigr\\}\\\\[-0.8pt]\n&&{}\\cap_{\\mH\\subset\\I} \\biggl\\{\\mathcal{A}_{i,\\mH\n}^n(t)-\\mathcal{A}_{i,\\mH}^n(s)-\\lambda_i^n(t-s)\\geq\\frac{\\epsilon\n_i }{2} n(t-s)-K\\log n \\\\[-0.8pt]\n&&\\hspace*{183.5pt}\\hspace*{12.1pt} \\mbox{for all } s<t\\leq\\tau\n_{\\kappa,T}^n\\biggr\\}\\\\[-0.8pt]\n&&{}\\cap_{i\\in\\I} \\{\n|A_{i}^n(t)-A_{i}^n(s)-\\lambda_i^n(t-s)|\\leq\\bar{\\epsilon\n}n(t-s)+K\\log n \\\\[-0.8pt]\n&&\\hspace*{171.7pt}\\mbox{for all } s<t\\leq\\tau\n_{\\kappa,T}^n\\}.\n\\end{eqnarray*}\nFor each $0\\leq t\\leq\\tau_{\\kappa',T}^n$ and $i\\in\\I$ let\n\\begin{eqnarray}\n\\label{eqchecktaudefin}\n\\check{\\varsigma}_i^n(t)&=&\\sup\\{s\\leq t\\dvtx h_i^n(\\check{X}^n(s))=1\\},\n\\\\\n\\label{eqtildetaudefin}\n\\tilde{\\varsigma}_i^n(t)&=&\\inf\\{\ns\\geq t\\dvtx h_i^n(\\check{X}^n(s))=1\\}\\wedge\\tau_{\\kappa',T}^n\n\\end{eqnarray}\nand\n\\begin{equation}\\label{eqydefin}\n\\hat{\\varsigma}_i^n(t)=\n\\cases{\\check{\\varsigma}_i^n(t)+\\eta^n,&\\quad if\n$Q_i^n(t)>4K\\log n$,\\cr\nt, &\\quad otherwise.}\n\\end{equation}\nThen, we claim that on $\\tilde{\\Omega\n}(K)$ and for all $t$ with $\\tilde{\\varsigma}_i^n(t)> \\hat{\\varsigma\n}_i^n(t)$,\n\\begin{eqnarray}\\label{eqSSCinterim}\n&&\\sup_{\\hat{\\varsigma}_i^n(t)\\leq s< \\tilde\n{\\varsigma}_i^n(t)}\\bigl|\\bigl(e\\cdot\\check{X}^n(s)\\bigr)^+U_i^n(s)-\\bigl(e\\cdot\\check\n{X}^n(s)\\bigr)^+h_i^n(\\check{X}^n(s))\\bigr|\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\leq12K\\log n.\\nonumber\n\\end{eqnarray}\nNote that since $h_i^n(\\cdot)\\in\\{0,1\\}$, the\nabove is equivalently written as\n\\begin{equation}\\label{eqSSCinterim2}\n\\sup_{\\hat{\\varsigma}_i^n(t)\\leq s< \\tilde\n{\\varsigma}_i^n(t)}Q_i^n(s)\\leq12K\\log n.\n\\end{equation}\nIn words,\nwhen the process $\\check{X}^n(t)$ enters a region in which\n$h^n_i(\\check{X}^n(\\cdot))=0$ the queue of class $i$ will be drained\nup to $12 K\\log n$ within at most $\\eta^n$ time units and it will\nremain there up to $\\tilde{\\varsigma}_i^n(t)$. We postpone the proof\nof (\\ref{eqSSCinterim}) and use it in proceeding with the proof of\nthe theorem.\n\nTo that end, fix $l\\geq0$ and let\n\\[\nj^*_{l}=\\min\\mathop{\\argmin}_{i\\in\\I}M_i^n(\\check{X}^n(\\tau_l^n)).\n\\]\nThen, by the definition of the function $h^n$ in (\\ref{eqixdefin}) we\nhave that $h_{j^*_l}^n(\\check{X}^n(\\tau_l^n))=1$ and $h_i(\\check\n{X}^n(\\tau_l^n))=0$ for all $i\\neq j^*_l$. In particular,\n\\begin{eqnarray*}\n\\check{\\psi}^n(s)&=&\\bigl(e\\cdot\\check\n{X}^n(s)\\bigr)^+h_{j^*_l}^n(\\check{X}^n(s))M_{j^*_l}^n(\\check{X}^n(s))\n\\\\\n&&{}-\\sum_{i\\in\\I}Q_i^n(s)M_i^n(\\check{X}^n(s))\n\\end{eqnarray*}\nfor all $s\\in[\\tau_{l}^n,(\\tau_{l}^n+\\eta^n)\\wedge\\tau_{l+1}^n)$.\nLet\n\\[\n\\mathcal{J}(\\tau_l^n)=\\{i\\in\\I\\dvtx Q_i^n(\\tau_l^n-)>4K\\log n\\}.\n\\]\nThen, simple manipulations yield\n\\begin{eqnarray}\\label{eqcheckpsiinterim}\n|\\check{\\psi}^n(s)|&\\leq&\\sum_{i\\notin\\mathcal{J}(\\tau\n_l^n)\\cup\\{j^*_l\\}}Q_i^n(s)|M_i^n(\\check{X}^n(s))|\\nonumber\\\\\n&&{}+\n|M_{j_l^*}^n(\\check{X}^n(s)|\\biggl|\\bigl(e\\cdot\\check\n{X}^n(s)\\bigr)^+-\\sum_{i\\in\\mathcal{J}(\\tau_l^n)\\cup\\{j^*_l\\}\n}Q_i^n(s)\\biggr|\\\\\n&&{}+\\sum_{i\\in\\mathcal\n{J}(\\tau_l^n)}Q_i^n(s)|M_i^n(\\check{X}^n(s))-M_{j_l^*}^n(\\check\n{X}^n(s))|.\\nonumber\n\\end{eqnarray}\nWe turn to bound each of the elements on the right-hand side of (\\ref\n{eqcheckpsiinterim}). First, note that for all $i\\notin\\mathcal\n{J}(\\tau_l^n)\\cup\\{j^*_l\\}$ it follows from (\\ref{eqSSCinterim2}) that\n\\[\n\\sup_{\\tau_{l}^n\\leq s<(\\tau_{l}^n+\\eta^n)\\wedge\\tau\n_{l+1}^n}Q_i^n(s)\\leq12K\\log n.\n\\]\nAlso, by (\\ref{eqgradients1}) we have for all $i\\in\\I$ that\n\\begin{equation}\\label{eqMboun4}\n\\sup_{0\\leq s\\leq\\tau_{\\kappa',T}^n}|M_i^n(\\check{X}^n(s))|\\leq\nC\\log^{k_1}n,\\vadjust{\\goodbreak}\n\\end{equation}\nso that\n\\begin{equation}\\label{eqitai1}\n\\sum_{i\\notin\\mathcal{J}(\\tau_l^n)\\cup\\{j^*_l\\}}Q_i^n(s)|M_i^n(\\check\n{X}^n(s))|\\leq12IKC\\log^{k_1+1}n\n\\end{equation}\nfor all $s\\in\n[\\tau_{l}^n,(\\tau_{l}^n+\\eta^n)\\wedge\\tau_{l+1}^n)$ and a constant\n$C$ that does not depend on $n$. From (\\ref{eqSSCinterim2}) and from\nthe fact that $\\sum_{i\\in\\I}Q_i^n(s)=(e\\cdot\\check{X}^n(s))^+$ we\nsimilarly have that\n\\begin{equation}\\label{eqitai2}\n|M_{j_l^*}^n(\\check{X}^n(s)|\n\\biggl|\\bigl(e\\cdot\\check\n{X}^n(s)\\bigr)^+-\\sum_{i\\in\\mathcal{J}(\\tau_l^n)\\cup\\{j^*_l\\}\n}Q_i^n(s)\\biggr|\\leq12IC K\\log^{k_1+1}n.\\hspace*{-30pt}\n\\end{equation}\nTo\nbound the last element on the right-hand side of (\\ref\n{eqcheckpsiinterim}) note that for each $i\\in\\mathcal{J}(\\tau_l^n)$\nthere exists $\\tau_l^n-\\eta^n\\leq t\\leq\\tau_{l}^n$ such that\n$h_j^n(\\check{X}^n(t))=1$. Otherwise, we would have a contradiction to\n(\\ref{eqSSCinterim}). We now claim that for each \\mbox{$i\\in\\mathcal\n{J}(\\tau_l^n)$},\n\\begin{equation}\\label{eqinterim747}\n|M_i^n(\\check{X}^n(s))-M_{j^*_l}^n(\\check\n{X}^n(s))|\\leq\\frac{C\\log^{k_1+2} n}{\\sqrt{n}}\n\\end{equation}\nfor all $s$ in $[\\tau_l^n-\\eta^n,\\tau_l^n+\\eta\n^n]$. Indeed, by the definition of $\\tilde{\\Omega}(K)$, we have that\n$|\\check{X}^n(t)-\\check{X}^n(s)|\\leq C\\log^{m+2}n$ for all $s,t$ in\n$[\\tau_l^n-\\eta^n,\\tau_l^n+\\eta^n]$. As in the proof of (\\ref\n{eqgenbound}) [see, e.g., (\\ref{eqMbound})] we have that\n\\begin{equation}\\label{eqMbound2}\n|M_i^n(x)-M_i^n(y)|\\leq\\frac{C\\log^{k_2+m+2}n}{\\sqrt{n}},\\qquad i\\in\\I,\n\\end{equation}\nfor $x,y\\in\\mB_{\\kappa'}^n$ with $|x-y|\\leq\nC\\log^{m+2}n$. In turn,\n\\begin{equation}\\label{eqMbound3}\n|M_i^n(\\check{X}^n(t))-M_i^n(\\check\n{X}^n(s))|\\leq\\frac{C\\log^{k_2+m+2} n}{\\sqrt{n}}=\\frac{C\\log\n^{k_1+2} n}{\\sqrt{n}}\n\\end{equation}\nfor all $i\\in\\I$ and all\n$s,t\\in[\\tau_l^n-\\eta^n,\\tau_l^n+\\eta^n]$ where we used the fact\nthat \\mbox{$k_1=k_2+m$}. Since, for each $j\\in\\mathcal{J}(\\tau_l^n)$, there\nexists $\\tau_l^n-\\eta^n\\leq t\\leq\\tau_{l}$ such that $h_j^n(\\check\n{X}^n(t))=1$ we have, by the definition of $h^n$ that $j\\in\\argmin\n_{i\\in\\I} M_i^n(\\check{X}^n(t))$ for such $t$ so that (\\ref\n{eqinterim747}) now follows from (\\ref{eqMbound3}). Finally, recall\nthat\\break $\\sum_{i\\in\\I}Q_i^n(t)=(e\\cdot\\check{X}^n(s))^+\\leq\\kappa\n\\sqrt{n}\\log^mn$ for all $s\\leq\\tau_{\\kappa',T}^n$ and that\n$k_0=k_1+m$ so that by~(\\ref{eqinterim747})\n\\[\n\\sum_{i\\in\\mathcal{J}(\\tau_l^n)}Q_i^n(s)|M_i^n(\\check\n{X}^n(s))-M_{j_l^*}^n(\\check{X}^n(s))|\\leq C\\log^{k_0+2}n.\n\\]\nPlugging this into (\\ref{eqcheckpsiinterim}) together with (\\ref\n{eqitai1}) and (\\ref{eqitai2}) we then have that, on $\\tilde{\\Omega}(K)$,\n\\[\n\\sup_{\\tau_{l-1}^n\\leq s<(\\tau_{l-1}^n+\\eta\n^n)\\wedge\\tau_l^n}|\\check{\\psi}^n(s)| \\leq C\\log\n^{k_1+m+2}n=CK\\log^{k_0+2}n.\n\\]\nThis argument\\vspace*{1pt} is repeated for each $l$. To complete the proof of (\\ref\n{eqSSC1}) note that, using (\\ref{eqMboun4}) together\\vspace*{-1pt} with ${\\sup\n_{0\\leq s\\leq\\tau_{\\kappa',T}^n}}|e\\cdot\\check{X}^n(s)|\\leq\\kappa\n\\sqrt{n}\\log^mn$, we have that ${\\sup_{0\\leq s\\leq\\tau_{\\kappa\n',T}^n}}|\\check{\\psi}^n(s)|\\leq C\\sqrt{n}\\log^{k_1+m}n$. Applying H\\\"\n{o}lder's inequality we have that\n\\begin{eqnarray*}\n&&\\Ex\\Bigl[\\sup_{1\\leq l\\leq\nr^n+1}\\sup_{\\tau_{l-1}^n\\leq s< (\\tau_{l-1}^n+\\eta^n)}|\\check{\\psi\n}^n(s)|\\Bigr]\\\\\n&&\\qquad\\leq\n\\Ex\\Bigl[\\sup_{1\\leq l\\leq r^n+1}\\sup_{\\tau_{l-1}^n\\leq s< (\\tau\n_{l-1}^n+\\eta^n)}|\\check{\\psi}^n(s)|1\\{\\tilde{\\Omega}(K)\\}\n\\Bigr]\\\\\n&&\\qquad\\quad{}+\n\\Ex\\Bigl[\\max_{1\\leq l\\leq r^n+1}\\sup_{\\tau_{l-1}^n\\leq s< (\\tau\n_{l-1}^n+\\eta^n)}|\\check{\\psi}^n(s)|1\\{\\tilde{\\Omega}(K)^c\\}\n\\Bigr]\\\\\n&&\\qquad\\leq\nC\\log^{k_0+2} n + C\\sqrt{n}\\log^{k_1+m}n C_1e^{-(C_2\n{K}\/{2})\\log n}\n\\end{eqnarray*}\nfor redefined constants $C_1,C_2$ and (\\ref{eqSSC1}) now follows by\nchoosing $K$ large enough.\n\nWe turn to prove (\\ref{eqSSC2}). Rearranging terms in (\\ref\n{eqpsicheckdefin}) we write\n\\[\n\\check{\\psi}^n(s) = \\sum_{i\\in\\I}M_i^n(\\check\n{X}^n(s))\\bigl(\\bigl(e\\cdot\\check{X}^n(s)\\bigr)^+h_i^n(\\check{X}^n(s))-\\bigl(e\\cdot\n\\check{X}^n(s)\\bigr)^+ U_i^n(s)\\bigr),\n\\]\nso that equation (\\ref{eqSSC2}) now follows directly from (\\ref\n{eqSSCinterim}) and (\\ref{eqMboun4}) through an application of H\\\"\n{o}lder's inequality\n\nFinally, to establish (\\ref{eqSSC3}), note that from the definition\nof $\\tau_{\\kappa',T}^n$,\n\\begin{eqnarray*}\n\\sup_{0\\leq t\\leq\\eta^n\\wedge\\tau_{\\kappa',T}^n\n} |\\check{\\psi}^n(s)|&\\leq& I\\sup_{0\\leq t\\leq\\eta^n\\wedge\\tau\n_{\\kappa',T}^n}|\\check{X}^n(t)|\\sum_{i\\in\\I}M_i^n(\\check{X}^n(t))\n\\\\&\\leq&\nI\\sup_{0\\leq t\\leq\\eta^n\\wedge\\tau_{\\kappa',T}^n}C\\log\n^{k_1}n|\\check{X}^n(t)|\\leq C \\kappa\\sqrt{n}\\log^{k_1+m}n.\n\\end{eqnarray*}\nIn turn,\n\\begin{equation}\\label{eqhowmanymoreinterims}\n\\Ex\\biggl[\\int_0^{\\tau_0^n}e^{-\\gamma t} |\\check\n{\\psi\n}^n(t)|\\,dt\\biggr]\\leq C\\log^{k_1+m}n=C\\log^{k_0}n.\n\\end{equation}\n\nWe have thus proved (\\ref{eqSSC1})--(\\ref{eqSSC3}) and to conclude\nthe proof of the theorem it remains only to establish (\\ref\n{eqSSCinterim}). To that end, let $\\check{\\varsigma}_i^n(t)$,\n$\\tilde\n{\\varsigma}_i^n(t)$ and $\\hat{\\varsigma}_i^n(t)$ be as in~(\\ref\n{eqchecktaudefin})--(\\ref{eqydefin}). Fix an interval $[l,s)\\in\n(\\hat\n{\\varsigma}_i^n(t),\\tilde{\\varsigma}_i^n(t))$ such that\n$Q_i^n(u)>2K\\log n $ for all $u\\in[l,s)$. By the\\vadjust{\\goodbreak} definition of the\ntracking policy, (\\ref{eqQtrack}) holds on this interval so that, on\n$\\omega\\in\\tilde{\\Omega}(K)$,\n\\begin{eqnarray}\\label{equntilwhen3}\nQ_i^n(l)-Q_i^n(s)&\\leq& \\bar{\\epsilon}n(t-s)\nn-\\frac{\\epsilon_i}{2}n(t-s)+2K\\log n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&\\leq& -\\frac{\\epsilon_i}{4}n(t-s)+2K\\log n.\\nonumber\n\\end{eqnarray}\nEquation (\\ref{eqSSCinterim}) now follows directly from (\\ref\n{equntilwhen3}). Indeed, note for all $t\\leq\\tau_{\\kappa',T}$,\n$Q_i^n(t)\\leq(e\\cdot\\check{X}^n(t))^+\\leq|\\check{X}^n(t)|\\leq\n\\kappa\\sqrt{n}\\log^mn$. Hence, $Q_i^n(\\check{\\varsigma}_i(t))\\leq\n\\kappa\\sqrt{n}\\log^mn$. In turn, using (\\ref{equntilwhen3}) and\nassuming that $\\tilde{\\varsigma}_i^n(t)\\geq\\check{\\varsigma\n}_i^n(t)+\\eta^n$ we have that $Q_i^n(\\varsigma_{0,i}^n(t))\\leq4K\\log\nn$ for some time $\\varsigma_{0,i}^n(t)\\leq\\check{\\varsigma\n}_i^n(t)+\\eta^n$ with $\\eta^n$ as defined in (\\ref{eqetandefin}).\nAlso, let\n\\[\n\\varsigma_{2,i}^n(t)=\\inf\\{t\\geq\\varsigma_{0,i}^n(t)\\dvtx Q_i^n(t)\\geq\n12K\\log n \\}\n\\]\nand\n\\[\n\\varsigma_{1,i}^n(t)=\\sup\\{t\\leq\\varsigma_{2,i}^n(t)\\dvtx Q_i^n(t)\\leq\n8K\\log n \\}.\n\\]\nNote that (\\ref{equntilwhen3}) applies to any subinterval $[l,s)$ of\n$[\\varsigma_{1,i}^n(t),\\varsigma_{2,i}^n(t))$. In turn, $\\varsigma\n_{2,i}^n(t)\\leq\\tilde{\\varsigma}_i^n(t)$ would constitute\\vspace*{1pt} a\ncontradiction to (\\ref{equntilwhen3}) so that we must have that\n$Q_i^n(s)\\leq12K\\log n$ for all $s\\in[\\varsigma_{0,i}^n(t),\\tilde\n{\\varsigma}^n(t))$ with $\\varsigma_{0,i}^n(t)\\leq\\tilde{\\varsigma\n}^n(t)+\\eta^n$. Finally, note that $\\varsigma_{0,i}^n(t)$ can be\ntaken to be $t$ if $Q_i^n(t)\\leq4K\\log n$.\n\nThis concludes the proof of (\\ref{eqSSCinterim}) and, in turn, the\nproof of the theorem.\n\n\\subsection*{\\texorpdfstring{Proof of Lemma \\protect\\ref{lemafterstop}}{Proof of Lemma 6.2}} Let $T$,\n$\\tau_{\\kappa',T}^n$ and $(x^n,q^n)$ be as in the statement of the\nlemma. We first prove (\\ref{eqafterstop2}). To that end, we claim\nthat, for all $T$ large enough,\n\\begin{equation}\\label{eqthisisjustonemore}\n\\Ex_{x^n,q^n}\\biggl[\\int_{T\\log\nn}^{\\infty} e^{-\\gamma s} (e\\cdot c)\\bigl(e\\cdot\\check{X}^n(s)\\bigr)^+\\,ds\n\\biggr]\\leq C\\log^2n\n\\end{equation}\nfor some $C>0$ and all\n$n\\in\\bbZ$. This is a direct consequence of Lemma 3 in~\\cite{AMR02}\nthat, in our notation, guarantees that\n\\[\n\\Ex_{x^n,q^n}[|\\check{X}^n(t)|]\\leq C\\bigl(1+|x^n|+ \\sqrt{n}(t+t^2)\\bigr)\n\\]\nfor all $t\\geq0$ and some constant $C>0$. We use (\\ref\n{eqthisisjustonemore}) to prove Lemma \\ref{lemafterstop}. The\nassertion of the lemma will be established by showing that\n\\[\n\\Ex_{x^n,q^n}\\biggl[\\int_{\\tau_{\\kappa',T}^n}^{2T\\log n} e^{-\\gamma\ns} (e\\cdot c)\\bigl(e\\cdot\\check{X}^n(s)\\bigr)^+\\,ds\\biggr]\\leq C\\log^2n.\n\\]\nTo that end, applying H\\\"{o}lder's\ninequality, we have\n\\begin{eqnarray}\\label{eqinterim8}\n&&\\Ex_{x^n,q^n}\\biggl[\\int_{\\tau_{\\kappa\n',T}^n}^{2T\\log n} e^{-\\gamma s} (e\\cdot c)\\bigl(e\\cdot\\check\n{X}^n(s)\\bigr)^+\\,ds\\biggr]\\nonumber\\\\\n&&\\qquad\\leq\\Ex_{x^n,q^n}\\Bigl[(2T\\log\nn-\\tau_{\\kappa',T}^n)^+\\sup_{0\\leq t\\leq2T\\log n}(e\\cdot c)\\bigl(e\\cdot\n\\check{X}^n(t)\\bigr)^+\\,ds\\Bigr]\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\leq\n\\sqrt{\\Ex_{x^n,q^n}\\bigl[\\bigl((2T\\log n-\\tau_{\\kappa\n',T}^n)^+\\bigr)^2\\bigr]}\\nonumber\\\\\n&&\\qquad\\quad{}\\times\\sqrt{\n\\Ex_{x^n,q^n}\\Bigl[\\Bigl(\\sup_{0\\leq t\\leq2T\\log n}(e\\cdot\nc)\\bigl(e\\cdot\\check{X}^n(t)\\bigr)^+\\,ds\\Bigr)^2\\Bigr]}.\\nonumber\n\\end{eqnarray}\n\nUsing Lemma \\ref{lemstrongappbounds} we have that\n\\begin{equation}\\label{eqinterim9}\n\\Ex\n_{x^n,q^n}\\Bigl[\\Bigl(\\sup_{0\\leq t\\leq2T\\log n}(e\\cdot c)\\bigl(e\\cdot\n\\check{X}^n(t)\\bigr)^+\\,ds\\Bigr)^2\\Bigr]\\leq Cn\\log^6n\n\\end{equation}\nfor some $C>0$ (that can depend on $T$). Also, since\n$m\\geq3$,\n\\[\n\\Pd\\{\\tau_{\\kappa',T}^n<2T\\log n\\}\\leq\\Pd\n\\Bigl\\{\\sup_{0\\leq t\\leq2T\\log n}|\\check{X}^n(t)|> \\kappa'\\sqrt{n}\\log\n^3 n -M\\sqrt{n}\\Bigr\\}.\n\\]\nChoosing $\\kappa'$ (and in turn\n$\\kappa$ large enough) we then have, using Lemma \\ref\n{lemstrongappbounds}, that\n\\begin{equation}\\label{eqtauprobbound}\n\\Pd\\{\\tau_{\\kappa',T}^n<2T\\log n\\}\n\\leq\\frac{C}{n^2}\n\\end{equation}\nand hence, that\n\\begin{equation}\\label{eqinterim10}\n\\Ex\n_{x^n,q^n}\\bigl[\\bigl((2T\\log n-\\tau_{\\kappa',T}^n)^+\n\\bigr)^2\\bigr]\\leq C.\n\\end{equation}\nPlugging (\\ref{eqinterim9}) and (\\ref{eqinterim10}) into (\\ref\n{eqinterim8}) we\nthen have that\n\\begin{equation}\\label{eqafterstopbound1}\n\\Ex_{x^n,q^n}\\biggl[\\int_{\\tau_{\\kappa\n',T}^n}^{2T\\log n}\ne^{-\\gamma s} (e\\cdot c)\\bigl(e\\cdot\\check{X}^n(s)\\bigr)^+\\,ds\\biggr]\\leq C\\log\n^2 n.\n\\end{equation}\n\nTo conclude the proof we will show that (\\ref{eqafterstop1}) follows\nfrom our analysis thus far. Indeed,\n\\begin{eqnarray*}\n&&\\Ex[e^{-\\gamma\\tau_{\\kappa',T}^n}\\phi\n_{\\kappa}^n(\\check{X}^n(\\tau_{\\kappa',T}^n))]\\\\\n&&\\qquad\\leq\n\\Ex_{x^n,q^n}^{U}\\biggl[\\int_{\\tau_{\\kappa',T}^n}^{2T\\log\nn}e^{-\\gamma s} \\sup_{0\\leq s\\leq2T\\log n}(e\\cdot c)\\bigl(e\\cdot\\check\n{X}^n(s)\\bigr)^+ \\,ds\\biggr].\n\\end{eqnarray*}\nThe right-hand side here is bounded by $C\\log^2n$ by the same argument\nthat leads to (\\ref{eqafterstopbound1}).\n\n\\subsection*{\\texorpdfstring{Proof of Lemma \\protect\\ref{lemmartingales}}{Proof of Lemma 6.3}} Recall that\n$\\check{W}^n$ is defined by $\\check\n{W}_i^n(t)=M_{i,1}^n(t)-M_{i,2}^n(t)$, where\n\\begin{eqnarray*}\nM_{i,1}^n(t)&=&\\mN_i^a(\\lambda_i^n t)-\\lambda_i^n t,\n\\\\\nM_{i,2}^n(t)&=&\n\\mN_i^d\\biggl(\\mu_i\\int_0^t \\bigl( \\check{X}_i\\lam(s)+\\nu_i\nn-U^n_i(s)\\bigl( e\\cdot\\check{X}\\lam(s)\\bigr)^+ \\bigr) \\,ds \\biggr)\\\\\n&&{}-\\mu\n_i\\int_0^t \\bigl( \\check{X}_i\\lam(s)-U^n_i(s)\\bigl( e\\cdot\\check\n{X}\\lam(s)\\bigr)^+ \\bigr) \\,ds.\n\\end{eqnarray*}\nThe fact that each of the processes\n$M_{i,1}^n(t)$ and $M_{i,2}^n(t)$ are square integrable martingales\nwith respect to the filtration $(\\mathcal{F}_t^n)$ follows as in\nSection 3 of \\cite{pang2007martingale} and specifically as in Lemma\n3.2 there.\n\nSince, with probability 1, there are no simultaneous jumps of $\\mathcal\n{N}_i^a$ and $\\mathcal{N}_i^d$, the quadratic variation process satisfies\n\\begin{eqnarray*}\n[\\check{W}_i^n]_t&=&[M_{i,1}^n]_t+[M_{i,2}^n]_t\\\\\n&=&\\sum_{s\\leq\nt}(\\Delta M_{i,1}^n(s))^2+\\sum_{s\\leq t}(\\Delta M_{i,2}^n(s))^2,\n\\end{eqnarray*}\nwhere the last equality follows again from Lemma 3.1 in \\cite\n{pang2007martingale} (see also Example~5.65 in \\cite{vandervaart}).\nFinally, the predictable quadratic variation process satisfies\n\\begin{eqnarray*}\n\\langle\\check{W}_i^n\\rangle_t&=&\\langle\nM_{i,1}^n\\rangle\n_t+\\langle M_{i,2}^n\\rangle_t\\\\&=&\\lambda_i^nt + \\mu_i\\int_0^t\n\\bigl( \\check{X}_i\\lam(s)+\\nu_i n-U^n_i(s)\\bigl( e\\cdot\\check{X}\\lam\n(s)\\bigr)^+ \\bigr) \\,ds\\\\&=&\n\\int_0^t (\\sigma_i^n(\\check{X}^n(s),U^n(s)))^2\\,ds,\n\\end{eqnarray*}\nwhere the second equality follow again follows from Lemma 3.1 in \\cite\n{pang2007martingale} and the last equality from the definition of\n$\\sigma_i^n(\\cdot,\\cdot)$ [see (\\ref{eqnbmdefn3})]. By Theorem\n3.2 in~\\cite{pang2007martingale} $((\\check{W}_i^n(t))^2-[\\check\n{W}_i^n]_t,t\\geq0])$ and $((\\check{W}_i^n(t))^2-[\\check\n{W}_i^n]_t,t\\geq0)$ are both martingales with respect to\n$(\\mathcal{F}_t^n)$. In turn, by the optional stopping theorem so are\nthe processes $\\mathcal{M}_i^n(\\cdot)$ and $\\mathcal{V}_i^n(\\cdot)$\nas defined in the statement of the lemma. Finally, it is easy to verify\nthat these are square integrable martingales using the fact the time\nchanges are bounded for all finite $t$.\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe third $r$-process peak at $A\\sim195$, the platinum peak, seems particularly sensitive to both the nuclear physics input and the conditions of the stellar environment as can be seen in the detailed sensitivity study reported in Ref.~\\cite{Arcones11}. Radioactive Ion Beam (RIB) facilities have enabled the production and measurement of several waiting-point nuclei directly on the $r$-process path from $N=50$ to $N=82$. However all of the nuclei that lead directly to the formation of the third $r$-process abundance peak still remain in the region of the {\\it terra incognita}. To a large extent this is due to the very low production cross sections, the limited primary beam intensities available at present RIB facilities and also the challenging experimental conditions of large backgrounds and very low production rates. This contribution focuses on the impact of half-lives and beta-delayed neutrons on the formation of the third $r$-process peak, and how present theoretical models compare with the experimental data available.\n\n\n\\section{The r-process paradigm around N=126}~\\label{sec:hl}\n\nThe aim of this section is to describe briefly the latest experimental and theoretical results in this mass region, and to explore the impact of the present uncertainties (or discrepancies) on the nucleosynthesis of the $r$ process. On the experimental side, special focus will be made on the latest experiment performed at GSI, thus providing an update on the present status of the data analysis and its future perspective. \n\nAs mentioned above, the nuclei which are directly in the path of the $r$ process along $N=126$, approximately from gadolinium to tantalum, could not be accessed experimentally yet. Nevertheless, one can try to measure neutron-rich nuclei in the higher $Z$ neighbourhood, on both sides of the $N=126$ shell closure, and use such information as a benchmark for theoretical models. In turn, such models can be applied more reliably to extrapolate the decay properties of the $r$-path nuclei.\nThree independent experiments in the neutron-rich region around $N=126$ have been carried out recently at the GSI facility for heavy ion research (Germany). The FRS fragment separator~\\cite{Geissel92} was used for selection and identification of the ions of interest. The high energy available for primary beams at GSI, of up to 1~GeV\/u, has an advantage over other facilities in order to reduce ion identification difficulties related to the charged states of the secondary fragments. Fragmentation of primary $^{238}$U and\/or $^{208}$Pb beams on a beryllium target was used for the production of secondary neutron-rich ion beams. These experiments are briefly described below. For details the reader is referred to the cited publications. Only the main results from each experiment are summarised here.\n\n\\subsection{$\\beta$-decay experiments around N$\\sim$126}\n\nThe first experiment~\\cite{Alvarez09,Alvarez10} used relativistic fragmentation of both $^{238}$U and $^{208}$Pb projectiles for the production of a large number of new neutron-rich nuclei. Several isotopic species were implanted in a stack of four double-sided silicon-strip detectors (DSSSDs)~\\cite{Kumar09}, which allowed for the position and time measurement of both implant- and decay-events. $\\beta$-Decay half-lives were determined after developing a new numerical method~\\cite{Kurtukian08}, which was needed to account for the high and complex background conditions. Half-lives have been published from $^{194}$Re ($N=119$) up to $^{202}$Ir ($N=125$)~\\cite{Kurtukian07,Kurtukian09}. From this experiment, another isotopes have been also analysed~\\cite{Morales11,Benlliure12}, but they will not be considered in this contribution because they have not been published yet.\n\nNew half-lives have been published recently beyond $N=126$, for $^{219}$Bi and $^{211,212,213}$Tl~\\cite{Benzoni12}. In the latter experiment, in addition to a stack of three DSSDs for measuring ion-implants and $\\beta$-decays, the RISING array~\\cite{Wollersheim05} of 105 HPGe detectors was used to detect $\\gamma$-rays following the $\\beta$-decays. As is nicely illustrated in Ref.~\\cite{Benzoni12}, by means of high-resolution $\\gamma$-ray spectroscopy it was possible to perform a validation of the numerical technique described in Ref.~\\cite{Kurtukian08}, which has been applied in both works~\\cite{Kurtukian09,Benzoni12}.\n\nFinally, the latest measurement in this mass region ($N>126$) used a stack of six SSSDs and three DSSSDs~\\cite{Steiger09} surrounded by a prototype of the BEta deLayEd Neutron (BELEN) detector~\\cite{Gomez11}. The latter allowed for the experimental determination of neutron-branching ratios. Technical details and a few preliminary results have been reported recently in Ref.~\\cite{Caballero13}. In summary, the $N>126$ nuclei $^{208-211}$Hg, $^{211-215}$Tl and $^{214-218}$Pb were implanted with sufficient statistics for a reliable analysis of their half-lives. As first approach, the numerical method~\\cite{Kurtukian08} was applied to the measured thallium isotopes. As reported in Ref.~\\cite{Caballero13}, the preliminary half-lives thus obtained are compatible, within the error bars, with those reported by Benzoni et al.~\\cite{Benzoni12} for the common Tl-nuclei. Alternatively, despite the large and complex background environment, we have investigated the possibility of using the more conventional analytical approach based on determining the half-life from implant-beta time correlations. This seems still feasible in those cases with relatively large implant rates (see Fig.~2 in Ref.~\\cite{Caballero13}). In the case of e.g. $^{212}$Tl, an ion implant-rate of $\\sim$3$\\times 10^{-5}$~counts\/pixel\/s was achieved, being the average rate of $\\beta$-like events of about 4$\\times 10^{-4}$~counts\/pixel\/s. Using this nuclide as example for illustration purposes, the corresponding implant-beta time correlations are shown in Fig.~\\ref{fig:212Tl}. The correlation area comprises the pixel where the ion was implanted, as well as the eight neighbouring pixels.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{hl_212Tl_s410_preliminary.eps}\n\\caption{\\label{fig:212Tl} (Left) $^{212}$Tl Implant-$\\beta$ time correlations in the forward direction (solid symbols) and in the backward direction (open symbols) for all decay events within the implant pixel and the 8 neighbouring pixels. All decay events over a broad time window of several times the expected half-life were considered. (Right) Background subtracted correlation spectrum. The quoted half-life value is preliminary.}\n\\end{center}\n\\end{figure}\n\nIn the spectra shown in Fig.~\\ref{fig:212Tl} each implanted $^{212}$Tl ion was correlated with all following $\\beta$-decay events registered inside the aforementioned correlation area of 9 mm$^2$. The time-window for correlations spanned several times the expected half-life. The background was estimated in a similar way, by performing backward ion-$\\beta$ time-correlations over the same DSSSD area. After background subtraction the resulting time-spectrum was adjusted to a simple exponential decay (the daughter nucleus $^{212}$Pb shows a half-life of 10.64~h\\cite{ensdf}), as shown in Fig.~\\ref{fig:212Tl}. The preliminary value obtained for the half-life of $^{212}$Tl was $t^{^{212}Tl}_{1\/2} = 44(20)$~s. This value, although smaller than the one obtained with the numerical method, is still in perfect agreement (within the quoted error bars) with the half-life reported previously. From the 14 isotopic species implanted, we expect that the analytical analysis method can be applied to derive the half-lives of at least half of them, those showing high implant statistics. The remaining cases will be analysed, most probably, by applying the numerical approach~\\cite{Kurtukian08}. \n\nThe results for the half-lives determined in these three experiments are displayed in Fig.~\\ref{fig:hl}-left. The solid red circle shows the preliminary result discussed above for the half-life of $^{212}$Tl.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{hl_theo_exp_NoMorales.eps}\n\\hfill\n\\includegraphics[width=0.49\\textwidth]{fig_theo_n126.eps}\n\\caption{\\label{fig:hl}(Left) Comparison between theoretical predictions and measurements in the region around $N=126$. Experimental values from $^{194}$Re to $^{202}$Ir are from Ref.~\\cite{Kurtukian09}, values between $^{211}$Tl and $^{219}$Bi are from Ref.~\\cite{Benzoni12}. (Right) Theoretical values published in the literature for the waiting point nuclei along $N=126$ (adapted from Ref.\\cite{Zhi13}).}\n\\end{center}\n\\end{figure}\n\n\\subsection{Theory versus experiment}\n\nAlthough still far from the $r$-process path, these measurements can be used to judge the reliability of theoretical models in the heavy neutron-rich region around $N \\sim 126$. Fig.~\\ref{fig:hl}-left shows a comparison between the aforementioned experimental data, and the two main theoretical calculations available in this region. The blue up-pointing triangles correspond to the model of P.~M\\\"oller~\\cite{Moller03}, commonly used in many $r$-process network calculations. This approach is based on the finite-range droplet mass model (FRDM) and uses the quasiparticle random-phase approximation (QRPA) for the Gamow-Teller (GT) part of the $\\beta$-strength function, whereas the gross-theory is implemented to account for the first-forbidden (FF) part of the $\\beta$-decay. \nThe pink down-pointing triangles correspond to the calculations by I.~Borzov using the Fayans energy-density functional (DF3 version) within a continuum QRPA framework~\\cite{Borzov03,Borzov11}. The latter approach allows for a self-consistent treatment of both allowed GT- and FF-transitions. Although the contribution of FF-transitions can be neglected at the $N=50$ and $N=82$ shell closures, it is expected that around $N = 126$ they contribute remarkably to the $\\beta$-strength distribution and even dominate it above Z$\\geq$70, thus providing a remarkable shortening of the half-lives in this mass region~\\cite{Borzov03}.\n\n\n\nIn comparison with the experimental data displayed on Fig.\\ref{fig:hl}-left the FRDM+QRPA calculations~\\cite{Moller03}, on average overestimate the measured half-lives by a factor of 24 (average ratio of calculation\/experiment) in the region $N\\leq 126$, and underestimate them by less than 40\\% beyond $N=126$ (the average ratio is 0.62). The CQRPA+DF3 model~\\cite{Borzov11} agrees better with the experimental data in the region $N\\leq 126$, showing an average deviation of only a factor of 2 in that region. The discrepancy becomes again $\\lesssim$40\\%, in average, beyond the neutron shell closure. \nAs already noted in Refs.~\\cite{Benlliure12,Benzoni12}, theoretical models seem to overestimate the half-lives before the $N=126$ neutron shell closure, and to underestimate them beyond it. The question, however, is how well these models will perform when one uses them to extrapolate further off stability, into the region of the waiting-point nuclei. To answer this question one can compare the theoretical models among them. In particular shell-model calculations~\\cite{Suzuki12,Zhi13}, which became recently available, seem quite helpful given the absence of any experimental information. Such a comparison is shown on the right-hand side of Fig.~\\ref{fig:hl}, which has been adapted from Ref.~\\cite{Zhi13}. As expected, the agreement of the different models near the neutron shell closure is much better than toward the valley of stability (see Fig.~\\ref{fig:hl}-left), owing to the simplified nature of the shell closure. Indeed, comparing the performance of the FRDM+QRPA model against large-scale shell model calculations~\\cite{Zhi13} one finds discrepancies, which range from $\\sim$30\\% up to a factor of 5, with an average deviation of ``only'' a factor of 2. In summary, one can conclude two things. First, the uncertainties on the beta-decay half-lives along the shell closure itself are expected to be less severe than in the $N<126, \\,\\, Z<82$ mass region covered by the $r$ process during freeze-out. Second, the latest measurements on both sides of $N=126$ provide a very stringent test for theoretical models indeed.\n\n\\subsection{R-process nucleosynthesis}\nThe impact of the variations in half-lives on the final abundances has been studied in several works (see e.g.~\\cite{Borzov08,Suzuki12,Arcones11} and references therein). We have performed new network calculations in order to illustrate the effect of the aforementioned shortening of the half-lives along the $N=126$ waiting-point nuclei, as well as in the $N<126,\\,\\, Z<82$ neighbourhood. In these calculations the so-called cold trajectory of Ref.~\\cite{Arcones11} was used in combination with the single-zone code of \\verb+NucNet Tools+~\\cite{Meyer12,libnucnet}. Rather standard parameters were used for the electron fraction ($Y_e \\approx 0.47$) and the entropy (S$\\approx 200 k_B\/$nuc). In order to accelerate the calculation of the $r$-process phase of the expansions, we\nused Krylov-space iterative matrix solver routines~\\cite{Saad03} as implemented in\nthe freely-available software \\verb+Sparskit2+.  \nThe very sparse network matrix that\nmust be solved during the $r$-process phase is well-conditioned, so such\niterative solvers converge rapidly, and we found roughly a factor of ten\nspeed up over the default sparse matrix solver in \\verb+NucNet Tools+ when we\nused these iterative solvers.\nThe resulting abundances as a function of the mass number are shown in Fig.~\\ref{fig:yvsa}-left. The dotted open circles represent the $r$-process abundances in the solar system. The solid black curve shows a reference calculation performed with the JINA REACLIB library~\\cite{Cyburt10}. In the latter database, beta-decay rates are taken from Ref.~\\cite{Moller03} (FRDM+QRPA) where no data are available. The calculation provides a reasonable description of the solar $r$-process abundance pattern in the region of the second and third $r$-process peaks. Discrepancies such as the width of the peak or the region between both shell closures are not relevant for the present discussion, where the aim is to illustrate rather qualitatively the impact of reasonable half-life variations in this region. Reducing by a factor of two the FRDM+QRPA half-lives of the $N=126$ waiting-point nuclides, from gadolinium up to tantalum, one obtains the abundances represented in Fig.~\\ref{fig:yvsa}-left by the red-dashed line. A small shift towards higher masses is observed, similar to that reported e.g. in Ref.~\\cite{Suzuki12}. This seems to be in contradiction with the solar $r$-process abundances, where the maximum appears at a lower mass number around $A\\sim195$. Furthermore, as discussed above, the uncertainties on the half-lives of the nuclei in the $N<126$ region seem to be much larger than those at the shell closure itself. Obviously, the uncertainty will depend on the complexity of each nucleus and the difficulties of the model in reproducing the beta-strength distribution and the underlying nuclear structure details. Since it is difficult to assess an uncertainty on each particular nucleus, let us see what is the effect of a variation of the half-lives in that region by a constant factor, which is of the same order-of-magnitude as the discrepancies found between experiment and theory (FRDM+QRPA) discussed above. Assuming an average reduction of ``only'' a factor 12 (the average ratio theory\/experiment found above was $\\sim$24) in the half-lives of the nuclei on the left-hand side of $N=126$, from $N=116$ up to N$\\lesssim$ 125, one obtains a much stronger shift of the third $r$-process peak, towards higher masses, as shown in Fig.~\\ref{fig:yvsa}-left by the blue dot-dashed line. Additionally, the third $r$-process peak becomes narrower, an effect that disagrees further with the observed abundance pattern. Thus, one can conclude two things. First, a reduction of the FRDM+QRPA half-lives in the $N\\leq 126$ region, which is indeed expected both from experiment and theory, produces a shift of the third $r$-process peak which seems to go in the wrong direction, i.e. toward higher masses. Secondly, with the data currently available, the impact on the final abundances due to half-life uncertainties for nuclei in the region $N<126$, may be even more important than the uncertainties for the waiting-point nuclei themselves.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{yvsa_thl_npa_v2.eps}\n\\includegraphics[width=0.49\\textwidth]{yvsa_Pn_npa.eps}\n\\caption{\\label{fig:yvsa} Abundance versus mass number. Dotted open circles represent solar $r$-process abundances. The solid black curve corresponds to a reference calculation, where the decay rates are essentially based on FRDM+QRPA~\\cite{Moller03}. (Left panel) The dashed red line shows the same calculation with a factor of 2 reduction on the half-lives of the $N=126$ waiting point nuclei. The blue dash-dotted line shows the abundances obtained when the half-lives of the nuclei on the left-hand side of the waiting points, from $N=116$ up to $N=125$, were shortened by a factor of 12. See text for details. (Right panel) Dashed-dotted line shows the effect of a reduction by a factor of 10 on the neutron emission probabilities for nuclei in the $116 < N \\leq 126$ region. The blue dotted line corresponds to an enhancement, by a factor of 10, of the neutron emission probabilities of the same nuclei. The blue dashed line shows the impact of a reduction by a factor of 2 on the neutron branchings.}\n\\end{center}\n\\end{figure}\nFrom the nuclear physics point of view, the discrepancy between shorter half-lives and the position (and width) of the third $r$-process peak, may be counterbalanced by the effect of $\\beta$-delayed neutron emission. Indeed, for a given $Q_{\\beta}$ value, shorter half-lives usually correlate with a small neutron emission probability. The impact of (smaller) neutron emission probabilities on the final abundances is discussed in Sec.~\\ref{sec:neutrons}.\n\n\n\\section{$\\beta$-Delayed neutron emission around N$\\sim$126}\\label{sec:neutrons}\n\nThe emission of $\\beta$-delayed neutrons plays an important role in the formation of the third $r$-process peak~\\cite{Moller03,Arcones11}. However, the situation in terms of experimental information becomes really critical around $N \\sim 126$, as demonstrated in Fig.~\\ref{fig:pn}-left. \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{nc_Pn_rpath_npa.eps}\n\\includegraphics[width=0.35\\textwidth]{Pn_212Tl.eps}\n\\caption{\\label{fig:pn} (Left) Nuclear chart showing in red nuclei where experimental information on the neutron branching ratio is available~\\cite{ensdf}. (Right) Time spectrum showing implant-$\\beta$-neutron correlations for $^{212}$Tl.}\n\\end{center}\n\\end{figure}\nIn the latter figure, nuclei where any experimental information~\\cite{ensdf} is known about their beta-delayed neutron emission probability are shown in red. For the sake of comparison, the $r$-process path at some stage of a network calculation is also shown in the same figure.\nIt is worth emphasising that beyond $A \\sim 150$ there is practically no experimental information available. The only data available around $N\\sim126$ are on $^{210}$Tl~\\cite{Kogan57}.\n\nMotivated by this situation we used the BELEN neutron detector in the last experiment performed at GSI around $N \\gtrsim 126$. The detector consisted of 30 proportional counters of $^{3}$He embedded in a polyethylene matrix which served as moderator.  More details about this measurement can be found in Ref.~\\cite{Caballero13}. The analysis of the beta-delayed neutron emission probabilities is still ongoing. Following the example shown before for $^{212}$Tl, the spectrum of beta-delayed neutrons registered after the decays of $^{212}$Tl is shown in Fig.~\\ref{fig:pn}-right. These are implant-$\\beta$-neutron correlations, which could be observed mostly due to the high efficiency of BELEN, which was of $\\sim$40\\%. Using this kind of correlation analysis, we expect to obtain neutron branching ratios (or upper limits) for all the implanted species.\n\nSince there is no experimental information on neutron-emission probabilities in the mass-region around $N=126$, it is not possible to study reliably the capability of theoretical models for predicting the effect of neutron emission. However, one can again compare the predictions of different models among themselves. One such comparison is shown in Fig.~17 of Ref.~\\cite{Zhi13} for the three models described above, FRDM+QRPA~\\cite{Moller03}, DF3+CQRPA~\\cite{Borzov11} and LSSM~\\cite{Zhi13}. For the waiting points with Z$<$69, the neutron emission probabilities predicted by the FRDM+QRPA are, on average, a factor of 3 larger than the values calculated with the LSSM. This ratio becomes $\\sim$0.6 for the waiting point nuclei with  Z$\\geq$69, mostly due to the contribution of FF transitions. It is not straight-forward to predict, what the uncertainties in the calculated neutron-emission probabilities would be, in particular for the neutron-rich nuclei involved during the freeze-out phase. In this case, variations of the neutron-branching ratios are rather arbitrary, but they can still be used for a qualitative interpretation of the large uncertainties in these quantities.\nIn this respect, Fig.~\\ref{fig:yvsa}-right shows the effect of neutron-branching variations in the mass region $116 < N \\leq 126$. Final abundances obtained when the neutron emission probabilities are reduced by a factor of 2 are shown by the blue dashed line. The red dash-dotted line shows the effect of a factor of 10 reduction in the neutron branching ratios. The blue dotted line corresponds to a factor of 10 enhancement of the neutron branching ratios.\n\nIn summary, the discrepancy between shorter half-lives and the formation of the third peak discussed in Sec.~\\ref{sec:hl} could be compensated, at least to some extent, with a reduction of the neutron emission probabilities. Clearly, new $\\beta$-delayed neutron emission measurements are needed around $N \\sim 126$ in order to confirm such a hypothesis, and to reduce the still very large contribution of the nuclear physics input uncertainties to the calculated $r$-process abundances.\n\n\\section{Summary and outlook}\nIn summary, both theory and experiment, indicate that half-lives in the $N \\sim 126$ region near the $r$-process path should be much smaller than those commonly used in $r$-process model calculations (FRDM+QRPA model~\\cite{Moller03}). This, however, would imply a shift of the third $r$-process peak towards higher masses, an effect which is in contradiction with the observed $r$-process abundances. From the nuclear physics input, such a discrepancy could be removed if neutron emission probabilities were also smaller. This seems to be the case, as it has been shown by recent large-scale shell-model calculations~\\cite{Zhi13}. In the near future, first experimental values for the neutron emission probabilities of several nuclei beyond $N=126$ will become available from the experiment performed at GSI with the BELEN detector. This will represent a first test of the beta-strength distribution above the neutron separation energy, but clearly new measurements are needed in the neutron-rich heavy-mass region.\n\n\\section{Acknowledgments}\nWe thank the technical staff of the GSI accelerators, the FRS, and the target laboratory for their support during the S410 experiment. This work has been partially supported by the  Spanish Ministry of Economy and Competitivity  under grants FPA2011-24553, FPA 2011-28770-C03-03 and AIC-D-2011-0705. I.D., A.Ev., and M.M. are supported by the Helmholtz association via the Young Investigators group LISA (VH-NG-627).\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\indent \n\nThe theoretical and experimental studying\nthe vector ($W^\\pm$ and $Z^0$) boson production at high energies provide\ninformation about the nature of both the underlying electroweak\ninteraction and the effects of Quantum Chromodynamics (QCD).\nIn many respects these processes have become\none of most important \"standard candles\" in experimental\nhigh energy physics~[1--9]. At the Tevatron, measurements of $W^\\pm$ and $Z^0$\ninclusive cross sections are routinely used to validate\ndetector and trigger perfomance and stability. \nData from gauge boson production also provide\nbounds on parametrizations used to describe the non-perturbative\nregime of QCD processes.\nAt the LHC, such measurements can serve as a useful tool to determine\nthe integrated luminosity and can also be used\nto normalize measurements of other production\ncross sections (for example, cross section of $W + n$-jets or \ndiboson production). \nAdditionally, studying of inclusive vector boson \nproduction is necessary starting point for investigations of \nHiggs or top quark production where many signatures can include these bosons.\n\nAt leading order (LO) of QCD, $W^\\pm$ and $Z^0$ bosons are produced\nvia quark-antiquark annihilation. Beyond the LO Born process,\nvector boson can also be produced by $q + g$ interactions,\nso both the quark and gluon distribution functions of the proton\nplay an important role. Theoretical calculations of the \n$W^\\pm$ and $Z^0$ production cross sections have been \ncarried out at next-to-leading order (NLO) and next-to-next-to-leading\norder (NNLO)~[10--14] of QCD. The NLO cross section is $\\sim 25$\\%\nlarger than the Born-level cross section, and the NNLO cross\nsection is an additional $\\sim 3$\\% higher. \nHowever, these perturbative calculations are reliable at high $p_T$ only\nsince diverge in the small $p_T \\ll m$ region with terms\nproportional to $\\ln m\/p_T$ (appearing due to soft\nand collinear gluon emission). Therefore, the soft gluon\nresummation technique~[15--19] should be used to make QCD predictions at low $p_T$.\nThe traditional calculations combine fixed-order\nperturbation theory with analytic resummation and some matching criterion.\nThe analytic resummation can be performed either\nin the transverse momentum space~[20] or in the Fourier\nconjugate impact parameter space~[21].\nDifferences between the two formalisms are discussed\nin~[22].\n\nAn alternative description can be provided by the\n$k_T$-factorization approach of QCD~[23, 24].\nThis approach is based on the familiar\nBalitsky-Fadin-Kuraev-Lipatov (BFKL)~[25] \nor Catani-Ciafaloni-Fiorani-Marchesini (CCFM)~[26] gluon evolution \nequations and takes into account\nthe large logarithmic terms proportional to $\\ln 1\/x$.\nThis contrasts with the usual \nDokshitzer-Gribov-Lipatov-Altarelli-Parisi \n(DGLAP)~[27] strategy where only the large \nlogarithmic terms proportional to $\\ln \\mu^2$ are taken into account.\nThe basic dynamical quantity of the $k_T$-factorization approach is \nthe unintegrated (i.e., ${\\mathbf k}_T$-dependent) parton distribution \n$f_a(x,{\\mathbf k}_T^2,\\mu^2)$ which determines the probability to find a \ntype $a$ parton carrying the longitudinal momentum fraction $x$ and the \ntransverse momentum ${\\mathbf k}_T$ at the probing scale $\\mu^2$.\nIn this approach, since each incoming parton carries its own nonzero\ntransverse momentum, the Born-level subprocess \n$q + \\bar q^\\prime \\to W^\\pm\/Z^0$ already generate the $p_T$ distribution\nof produced vector boson.\nSimilar to DGLAP, to calculate the cross sections of any \nphysical process the unintegrated parton density \n$f_a(x,{\\mathbf k}_T^2,\\mu^2)$ \nhas to be convoluted~[23, 24] with the relevant partonic cross section \nwhich has to be taken off mass shell (${\\mathbf k}_T$-dependent). \nThe soft gluon resummation formulas \nare the result of the approximate treatment of \nthe solutions of the CCFM evolution equation~[28].\nOther important properties of the \n$k_T$-factorization formalism are the additional contribution to the cross \nsections due to the integration over the ${\\mathbf k}_T^2$ region above $\\mu^2$\nand the broadening of the transverse momentum distributions due to extra \ntransverse momentum of the colliding partons\\footnote{For an introduction\nto $k_T$-factorization, see, for example, review~[29].}.\n\nThe $k_T$-factorization formalism has been already \napplied~[30] to calculate transverse momentum distribution of the \ninclusive $W^\\pm$ and $Z^0$ production at Tevatron. \nThe calculations~[30] were based on the usual \n(on-mass shell) matrix elements of the quark-antiquark \nannihilation subprocess $q + \\bar q^\\prime \\to W^\\pm\/Z^0$\nwhich embedded in precise off-shell kinematics.\nHowever, an important component of the calculations~[30] is the \nunintegrated quark distribution in a proton. At present these distributions\nare only available in the framework of the Kimber-Martin-Ryskin (KMR) approach~[31] \nsince there are some theoretical difficulties \nin obtaining the quark densities immediately from CCFM or BFKL \nequations\\footnote{Unintegrated quark density was considered recently in~[32].}\n(see, for example, review~[29] for more details).\nAs a result the dependence of the calculated cross sections \non the non-collinear evolution scheme has not been investigated.\nThis dependence in general can be significant and it is a special \nsubject of study in the $k_T$-factorization formalism. \nTherefore in the present paper we will try a different and more systematic way.\nInstead of using the unintegrated quark distributions and \nthe corresponding quark-antiquark annihilation cross\nsection we calculate off-shell matrix element of \nthe $g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime$ subprocess\nand then operate in terms of the unintegrated gluon \ndensities only. In this scenario, at the price of considering the \n$2 \\to 3$ rather than $2 \\to 1$ matrix \nelements, the problem of unknown unintegrated quark\ndistributions will reduced to the problem of gluon distributions.\nHowever, since the gluons are only responsible for the appearance of the sea but \nnot valence quarks, the contribution from the valence quarks should be \ncalculated separately. Having in mind that the valence quarks are only \nimportant at large $x$, where the traditional DGLAP evolution is accurate and \nreliable, this contribution can be taken into account within the usual collinear \nscheme based on the $q + g^* \\to W^\\pm\/Z^0 + q^\\prime$ and \n$q + \\bar q^\\prime \\to W^\\pm\/Z^0$ matrix elements\nconvoluted with the on-shell valence quark and\/or off-shell gluon \ndensities\\footnote{To avoid the double counting we have not considered here\n$q + \\bar q^\\prime \\to W^\\pm\/Z^0 + g$ subprocess.}.\nThus, the proposed way enables us with making comparisons \nbetween the different parton \nevolution schemes and parametrizations of parton \ndensities\\footnote{The similar scenario has been applied recently\nto the prompt photon hadroproduction at Tevatron~[33].}.\n\nWe should mention, of course, that this idea can only\nwork well if the sea quarks appear from the last step\nof the gluon evolution --- then we can absorb this\nlast step of the gluon ladder into hard matrix element.\nHowever, this method does not apply to the quarks\ncoming from the earlier steps of the evolution\n(i.e., from the second-to-last, third-to-last and other gluon\nsplittings). But it is not evident in advance, whether\nthe last gluon splitting dominates or not. The goal of our\nstudy is to clarify this point.\n\nThe outline of our paper is following. In Section~2 we \nrecall shortly the basic formulas of the $k_T$-factorization approach with a brief \nreview of calculation steps and the unintegrated \nparton densities used. We will concentrate mainly\non the $g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime$\nsubprocess. The evaluation of $q + g^* \\to W^\\pm\/Z^0 + q^\\prime$ and \n$q + \\bar q^\\prime \\to W^\\pm\/Z^0$ contributions is\nstraightforward and, for the reader's convenience, we only collect \nthe main relevant formulas in Appendix.\nIn Section~3 we present the numerical results\nof our calculations. \nThe central point is discussing the role\nof each contribution mentioned above to the cross section.\nSpecial attention is put on the transverse momentum\ndistributions of the $W^\\pm$ and $Z^0$ boson \nmeasured by the D$\\oslash$~[5, 8, 9] and CDF~[4] collaborations.\nSection~4 contains our conclusions.\n\n\\section{Theoretical framework} \\indent\n\nAs the off-shell gluon-gluon fusion $g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime$\nis calculated for the first time in the literature,\nwe find it reasonable to explain it in more detail.\n\n\\subsection{Kinematics} \\indent \n\nWe start from the kinematics (see Fig.~1). \nLet $p^{(1)}$ and $p^{(2)}$ be the four-momenta of the incoming protons and \n$p$ the four-momentum of the produced $W^\\pm$\/$Z^0$ boson.\nThe initial off-shell gluons have the four-momenta\n$k_1$ and $k_2$ and the final quark $q$ and antiquark $\\bar q^\\prime$ have the \nfour-momenta $p_1$ and $p_2$ and the masses $m_1$ and $m_2$, respectively.\nIn the $p \\bar p$ center-of-mass frame we can write\n$$\n  p^{(1)} = {\\sqrt s\\over 2} (1,0,0,1),\\quad p^{(2)} = {\\sqrt s\\over 2} (1,0,0,-1), \\eqno(1)\n$$\n\n\\noindent\nwhere $\\sqrt s$ is the total energy of the process \nunder consideration and we neglect the masses of the incoming protons.\nThe initial gluon four-momenta in the high energy limit can be written as\n$$\n  k_1 = x_1 p^{(1)} + k_{1T},\\quad k_2 = x_2 p^{(2)} + k_{2T}, \\eqno(2)\n$$\n\n\\noindent \nwhere $k_{1T}$ and $k_{2T}$ are the transverse four-momenta.\nIt is important that ${\\mathbf k}_{1T}^2 = - k_{1T}^2 \\neq 0$ and\n${\\mathbf k}_{2T}^2 = - k_{2T}^2 \\neq 0$. From the conservation laws \nwe can obtain the following relations:\n$$\n  {\\mathbf k}_{1T} + {\\mathbf k}_{2T} = {\\mathbf p}_{1T} + {\\mathbf p}_{2T} + {\\mathbf p}_{T},\n$$\n$$\n  x_1 \\sqrt s = m_{1T} e^{y_1} + m_{2T} e^{y_2} + m_T e^y, \\eqno(3)\n$$\n$$\n  x_2 \\sqrt s = m_{1T} e^{-y_1} + m_{2T} e^{-y_2} + m_T e^{-y},\n$$\n\n\\noindent \nwhere $p_T$, $m_T$ and $y$ are the transverse momentum, \ntransverse mass and center-of-mass rapidity of produced $W^\\pm$\/$Z^0$ boson, \n$p_{1T}$ and $p_{2T}$ are the transverse momenta of final quark\n$q$ and antiquark $\\bar q^\\prime$, $y_1$, $y_2$, $m_{1T}$ and $m_{2T}$ \nare their rapidities and \ntransverse masses, i.e. $m_{iT}^2 = m_i^2 + {\\mathbf p}_{iT}^2$.\n\n\\subsection{Off-shell amplitude of the $g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime$ subprocess} \\indent \n\nThere are eight Feynman diagrams (see Fig.~2) which describe the partonic\nsubprocess $g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime$ at $\\alpha \\alpha_s^2$ order.\nLet $\\epsilon_1$, $\\epsilon_2$ and $\\epsilon$ be the initial gluon and produced gauge boson \npolarization vectors, respectively, and $a$ and $b$ the eight-fold color indices of the off-shell\ngluons.\nThen the relevant matrix element can be presented as follows:\n$$\n  {\\cal M}_1 = g^2 \\, \\bar u (p_1) \\, t^a \\gamma^\\mu \\epsilon_\\mu {\\hat p_1 - \\hat k_1 + m_1\\over m_1^2 - (p_1 - k_1)^2} T_{W,Z}^\\lambda \\, \\epsilon_\\lambda {\\hat k_2 - \\hat p_2 + m_2\\over m_2^2 - (k_2 - p_2)^2} t^b \\gamma^\\nu \\epsilon_\\nu \\, u(p_2), \\eqno(4)\n$$\n$$\n  {\\cal M}_2 = g^2 \\, \\bar u (p_1) \\, t^b \\gamma^\\nu \\epsilon_\\nu {\\hat p_1 - \\hat k_2 + m_1\\over m_1^2 - (p_1 - k_2)^2} T_{W,Z}^\\lambda \\, \\epsilon_\\lambda {\\hat k_1 - \\hat p_2 + m_2\\over m_2^2 - (k_1 - p_2)^2} t^a \\gamma^\\mu \\epsilon_\\mu \\, u(p_2), \\eqno(5)\n$$\n$$\n  {\\cal M}_3 = g^2 \\, \\bar u (p_1) \\, t^a \\gamma^\\mu \\epsilon_\\mu {\\hat p_1 - \\hat k_1 + m_1\\over m_1^2 - (p_1 - k_1)^2}\\, t^b \\gamma^\\nu \\epsilon_\\nu { - \\hat p_2 - \\hat p + m_1\\over m_1^2 - ( - p_2 - p)^2} T_{W,Z}^\\lambda \\, \\epsilon_\\lambda \\, u(p_2), \\eqno(6)\n$$\n$$\n  {\\cal M}_4 = g^2 \\, \\bar u (p_1) \\, t^b \\gamma^\\nu \\epsilon_\\nu {\\hat p_1 - \\hat k_2 + m_1\\over m_1^2 - (p_1 - k_2)^2}\\, t^a \\gamma^\\mu \\epsilon_\\mu { - \\hat p_2 - \\hat p + m_1\\over m_1^2 - ( - p_2 - p)^2} T_{W,Z}^\\lambda \\, \\epsilon_\\lambda \\, u(p_2), \\eqno(7)\n$$\n$$\n  {\\cal M}_5 = g^2 \\, \\bar u (p_1) \\, T_{W,Z}^\\lambda \\, \\epsilon_\\lambda {\\hat p_1 + \\hat p + m_2\\over m_2^2 - (p_1 + p)^2}\\, t^b \\gamma^\\nu \\epsilon_\\nu { \\hat k_1 - \\hat p_2 + m_2\\over m_2^2 - (k_1 - p_2)^2} t^a \\gamma^\\mu \\epsilon_\\mu \\, u(p_2), \\eqno(8)\n$$\n$$\n  {\\cal M}_6 = g^2 \\, \\bar u (p_1) \\, T_{W,Z}^\\lambda \\, \\epsilon_\\lambda {\\hat p_1 + \\hat p + m_2\\over m_2^2 - (p_1 + p)^2}\\, t^a \\gamma^\\mu \\epsilon_\\mu { \\hat k_2 - \\hat p_2 + m_2\\over m_2^2 - (k_2 - p_2)^2} t^b \\gamma^\\nu \\epsilon_\\nu \\, u(p_2), \\eqno(9)\n$$\n$$\n  \\displaystyle {\\cal M}_7 = g^2 \\, \\bar u (p_1) \\, \\gamma^\\rho C^{\\mu \\nu \\rho}(k_1,k_2,- k_1 - k_2){\\epsilon_\\mu \\epsilon_\\nu \\over (k_1 + k_2)^2} f^{abc} t^c \\times \\atop \n  \\displaystyle \\times { - \\hat p_2 - \\hat p + m_1\\over m_1^2 - ( - p_2 - p)^2}\\, T_{W,Z}^\\lambda \\, \\epsilon_\\lambda \\, u(p_2), \\eqno(10)\n$$\n$$\n  \\displaystyle {\\cal M}_8 = g^2 \\, \\bar u (p_1) \\, T_{W,Z}^\\lambda \\, \\epsilon_\\lambda {\\hat p_1 + \\hat p + m_2\\over m_2^2 - (p_1 + p)^2} \\times \\atop \n  \\displaystyle \\times \\gamma^\\rho C^{\\mu \\nu \\rho}(k_1,k_2,- k_1 - k_2) {\\epsilon_\\mu \\epsilon_\\nu \\over (k_1 + k_2)^2} f^{abc} t^c \\, u(p_2). \\eqno(11)\n$$\n\n\\vspace{0.2cm}\n\n\\noindent\nIn the above expressions $C^{\\mu \\nu \\rho}(k,p,q)$ and $T_{W,Z}^\\lambda$ are related to the standard QCD\nthree-gluon coupling and the $W^\\pm$\/$Z^0$-fermion vertexes:\n$$\n  C^{\\mu \\nu \\rho}(k,p,q) = g^{\\mu \\nu} (p - k)^\\rho + g^{\\nu \\rho} (q - p)^\\mu + g^{\\rho \\mu} (k - q)^\\nu, \\eqno(12)\n$$\n$$\n  T^\\lambda_W = {e\\over 2 \\sqrt 2 \\sin \\theta_W} \\gamma^\\lambda (1 - \\gamma^5) V_{qq^\\prime}, \\eqno(13)\n$$\n$$\n  T^\\lambda_Z = {e\\over \\sin 2 \\theta_W} \\gamma^\\lambda \\left[I_3^{(q)}(1 - \\gamma^5) - 2 e_q \\sin^2 \\theta_W\\right], \\eqno(14)\n$$\n\n\\noindent\nwhere $I_3^{(q)}$ and $e_q$ are the weak isospin and the fractional electric charge \n(in the positron charge $e$ units) of final-state quark $q$,  \n$\\theta_W$ is the Weinberg mixing angle and $V_{qq^\\prime}$ is the\nCabibbo-Kobayashi-Maskawa (CKM) matrix element. Of course, in\nthe case of $Z^0$ production $m_1$ equals $m_2$.\nThe summation on the $W^\\pm$\/$Z^0$ polarization is carried out by the\ncovariant formula\n$$\n  \\sum \\epsilon^\\mu (p) \\, \\epsilon^{* \\, \\nu} (p) = - g^{\\mu \\nu} + {p^\\mu p^\\nu\\over m^2}. \\eqno(15)\n$$\n\n\\noindent\nIn the case of the initial off-shell gluon we use the BFKL prescription~[23, 24]:\n$$\n  \\sum \\epsilon^\\mu (k_i) \\, \\epsilon^{* \\, \\nu} (k_i) = {k_{iT}^\\mu k_{iT}^\\nu \\over {\\mathbf k}_{iT}^2}. \\eqno(16)\n$$\n\n\\noindent\nThis formula converges to the usual expression \n$\\sum \\epsilon^\\mu \\epsilon^{* \\, \\nu} = -g^{\\mu \\nu}$ \nafter azimuthal angle averaging\nin the $k_T \\to 0$ limit. \nThe evaluation of the traces in~(4) --- (11) was done using the algebraic \nmanipulation system \\textsc{Form}~[34]. \nWe would like to mention here that the usual method \nof squaring of~(4) --- (11) results in enormously long\noutput. This technical problem was solved by applying the\nmethod of orthogonal amplitudes~[35].\n\nThe gauge invariance of the matrix element is a\nsubject of special attention in the $k_T$-factorization approach. Strictly speaking,\nthe diagrams shown in Fig.~2 are insufficient and have to be accompanied\nwith the graphs involving direct gluon exchange between the protons\n(these protons are not shown in Fig.~2). These graphs are \nnecessary to maintain the gauge invariance.\nHowever, they violate the factorization since they cannot be represented\nas a convolution of the gluon-gluon fusion matrix element with unintegrated gluon density.\nThe solution pointed out in~[24] refers to the fact that, within the \nparticular gauge~(16), the contribution from these unfactorizable diagrams\nvanish, and one has to only take into account the graphs depicted in Fig.~2.\nWe have successfully tested the gauge invariance of the matrix \nelement~(4) --- (11) numerically\\footnote{At the\npreliminary stage of the work we have made a cross-check \nof the matrix elements which have been \ncalculated independently by M.~Deak and F.~Schwennsen.}.\n\n\\subsection{Cross section for the inclusive $W^\\pm$\/$Z^0$ production} \\indent \n\nAccording to the $k_T$-factorization theorem, the \ninclusive $W^\\pm$\/$Z^0$ production cross section \nvia two off-shell gluon fusion \ncan be written as a convolution\n$$\n  \\displaystyle \\sigma (p + \\bar p \\to W^\\pm\/Z^0 + X) = \\sum_{q} \\int {dx_1\\over x_1} f_g(x_1,{\\mathbf k}_{1 T}^2,\\mu^2) d{\\mathbf k}_{1 T}^2 {d\\phi_1\\over 2\\pi} \\times \\atop \n  \\displaystyle \\times \\int {dx_2\\over x_2} f_g(x_2,{\\mathbf k}_{2 T}^2,\\mu^2) d{\\mathbf k}_{2 T}^2 {d\\phi_2\\over 2\\pi} d{\\hat \\sigma} (g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime), \\eqno(17)\n$$\n\n\\noindent \nwhere $\\hat \\sigma(g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime)$ is the partonic cross section, \n$f_g(x,{\\mathbf k}_{T}^2,\\mu^2)$ is the unintegrated gluon distribution in a proton \nand $\\phi_1$ and $\\phi_2$ are the azimuthal angles of the incoming gluons.\nThe multiparticle phase space $\\Pi d^3 p_i \/ 2 E_i \\delta^{(4)} (\\sum p^{\\rm in} - \\sum p^{\\rm out} )$\nis parametrized in terms of transverse momenta, rapidities and azimuthal angles:\n$$\n  { d^3 p_i \\over 2 E_i} = {\\pi \\over 2} \\, d {\\mathbf p}_{iT}^2 \\, dy_i \\, { d \\phi_i \\over 2 \\pi}. \\eqno(18)\n$$\n\n\\noindent\nUsing the expressions~(17) and~(18) we obtain the master formula:\n$$\n  \\displaystyle \\sigma(p + \\bar p \\to W^\\pm\/Z^0 + X) = \\sum_{q} \\int {1\\over 256\\pi^3 (x_1 x_2 s)^2} |\\bar {\\cal M}(g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime)|^2 \\times \\atop \n  \\displaystyle \\times f_g(x_1,{\\mathbf k}_{1 T}^2,\\mu^2) f_g(x_2,{\\mathbf k}_{2 T}^2,\\mu^2) d{\\mathbf k}_{1 T}^2 d{\\mathbf k}_{2 T}^2 d{\\mathbf p}_{1 T}^2 {\\mathbf p}_{2 T}^2 dy dy_1 dy_2 {d\\phi_1\\over 2\\pi} {d\\phi_2\\over 2\\pi} {d\\psi_1\\over 2\\pi} {d\\psi_2\\over 2\\pi}, \\eqno(19)\n$$\n\n\\noindent\nwhere $|\\bar {\\cal M}(g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime)|^2$ is the off-mass shell \nmatrix element squared and averaged over initial gluon \npolarizations and colors, $\\psi_1$ and $\\psi_2$ are the \nazimuthal angles of the final state quark and antiquark, respectively.\nWe would like to point out again that $|\\bar {\\cal M}(g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime)|^2$\nstrongly depends on the nonzero \ntransverse momenta ${\\mathbf k}_{1 T}^2$ and ${\\mathbf k}_{2 T}^2$.\nIf we average the expression~(19) over $\\phi_{1}$ and $\\phi_{2}$ \nand take the limit ${\\mathbf k}_{1 T}^2 \\to 0$ and ${\\mathbf k}_{2 T}^2 \\to 0$,\nthen we recover the expression for the $W^\\pm$\/$Z^0$ production cross section in the  \ncollinear $\\alpha \\alpha_s^2$ approximation.\n\nThe multidimensional integration in~(19) has been performed\nby means of the Monte Carlo technique, using the routine \n\\textsc{Vegas}~[36]. The full C$++$ code is available from the \nauthors upon request\\footnote{lipatov@theory.sinp.msu.ru}.\n\n\\subsection{The KMR unintegrated parton distributions} \\indent \n\nIn the present paper we have tried two different sets of \nunintegrated parton densities in a proton. First of them\nis the Kimber-Martin-Ryskin set.\n\nThe KMR approach~[31] is the formalism to construct\nparton distributions $f_a(x,{\\mathbf k}_T^2,\\mu^2)$ unintegrated over the parton \ntransverse momenta ${\\mathbf k}_T^2$ from the known conventional parton\ndistributions $xa(x,\\mu^2)$, where $a = g$ or $a = q$. This formalism \nis valid for a proton as well as a photon and\ncan embody both DGLAP and BFKL contributions. It also accounts for \nthe angular ordering which comes from coherence effects in gluon emission.\nThe key observation here is that the $\\mu$ dependence of the unintegrated \nparton distributions $f_a(x,{\\mathbf k}_T^2,\\mu^2)$ enters at the last step\nof the evolution, and therefore single scale evolution equations (pure DGLAP)\ncan be used up to this step. In this approximation, the unintegrated quark and \ngluon distributions are given by the expressions~[31]\n$$\n  \\displaystyle f_q(x,{\\mathbf k}_T^2,\\mu^2) = T_q({\\mathbf k}_T^2,\\mu^2) {\\alpha_s({\\mathbf k}_T^2)\\over 2\\pi} \\times \\atop {\n  \\displaystyle \\times \\int\\limits_x^1 dz \\left[P_{qq}(z) {x\\over z} q\\left({x\\over z},{\\mathbf k}_T^2\\right) \\Theta\\left(\\Delta - z\\right) + P_{qg}(z) {x\\over z} g\\left({x\\over z},{\\mathbf k}_T^2\\right) \\right],} \\eqno (20)\n$$\n$$\n  \\displaystyle f_g(x,{\\mathbf k}_T^2,\\mu^2) = T_g({\\mathbf k}_T^2,\\mu^2) {\\alpha_s({\\mathbf k}_T^2)\\over 2\\pi} \\times \\atop {\n  \\displaystyle \\times \\int\\limits_x^1 dz \\left[\\sum_q P_{gq}(z) {x\\over z} q\\left({x\\over z},{\\mathbf k}_T^2\\right) + P_{gg}(z) {x\\over z} g\\left({x\\over z},{\\mathbf k}_T^2\\right)\\Theta\\left(\\Delta - z\\right) \\right],} \\eqno (21)\n$$\n\n\\noindent\nwhere $P_{ab}(z)$ are the usual unregulated leading order DGLAP splitting \nfunctions, and $q(x,\\mu^2)$ and $g(x,\\mu^2)$ are the conventional quark \nand gluon densities\\footnote{Numerically, we have used the \nstandard GRV~(LO) parametrizations~[37].}. The theta functions which appear \nin~(20) and~(21) imply \nthe angular-ordering constraint $\\Delta = \\mu\/(\\mu + |{\\mathbf k}_T|)$ \nspecifically to the last evolution step to regulate the soft gluon\nsingularities. For other evolution steps, the strong ordering in \ntransverse momentum within the DGLAP equations automatically \nensures angular ordering. It is important that the parton \ndistributions $f_a(x,{\\mathbf k}_T^2,\\mu^2)$ extended now into \nthe ${\\mathbf k}_T^2 > \\mu^2$ region. This fact is in the clear contrast with the \nusual DGLAP evolution\\footnote{We would like to note that \ncut-off $\\Delta$ can be taken $\\Delta = |{\\mathbf k}_T|\/\\mu$ also~[31]. \nIn this case the unintegrated parton distributions given by (20) --- (21) \nvanish for ${\\mathbf k}_T^2 > \\mu^2$ in accordance with \nthe DGLAP strong ordering in ${\\mathbf k}_T^2$.}.\n\nThe virtual (loop) contributions may be resummed \nto all orders by the quark and gluon Sudakov form factors\n$$\n  \\ln T_q({\\mathbf k}_T^2,\\mu^2) = - \\int\\limits_{{\\mathbf k}_T^2}^{\\mu^2} {d {\\mathbf p}_T^2\\over {\\mathbf p}_T^2} {\\alpha_s({\\mathbf p}_T^2)\\over 2\\pi} \\int\\limits_0^{z_{\\rm max}} dz P_{qq}(z), \\eqno (22)\n$$\n$$\n  \\ln T_g({\\mathbf k}_T^2,\\mu^2) = - \\int\\limits_{{\\mathbf k}_T^2}^{\\mu^2} {d {\\mathbf p}_T^2\\over {\\mathbf p}_T^2} {\\alpha_s({\\mathbf p}_T^2)\\over 2\\pi} \\left[ n_f \\int\\limits_0^1 dz P_{qg}(z) + \\int\\limits_{z_{\\rm min}}^{z_{\\rm max}} dz z P_{gg}(z) \\right], \\eqno (23)\n$$\n\n\\noindent\nwhere $z_{\\rm max} = 1 - z_{\\rm min} = {\\mu\/({\\mu + |{\\mathbf p}_T|}})$.\nThe form factors $T_a({\\mathbf k}_T^2,\\mu^2)$ give the probability of \nevolving from a scale ${\\mathbf k}_T^2$ to a scale $\\mu^2$ without \nparton emission. In according with~(22) and~(23)\n$T_a({\\mathbf k}_T^2,\\mu^2) = 1$ in the ${\\mathbf k}_T^2 > \\mu^2$ region.\n\nNote that such definition of the $f_a(x,{\\mathbf k}_T^2,\\mu^2)$ is \ncorrect for ${\\mathbf k}_T^2 > \\mu_0^2$ only, where \n$\\mu_0 \\sim 1$ GeV is the minimum scale for which DGLAP evolution of \nthe collinear parton densities is valid. Everywhere in our numerical \ncalculations we set the starting scale $\\mu_0$ to be equal $\\mu_0 = 1$ GeV.\nSince the starting point of this derivation is the leading order \nDGLAP equations, the unintegrated parton distributions must satisfy\nthe normalisation condition\n$$\n  a(x,\\mu^2) = \\int\\limits_0^{\\mu^2} f_a(x,{\\mathbf k}_T^2,\\mu^2) d{\\mathbf k}_T^2. \\eqno(24)\n$$\n\n\\noindent\nThis relation will be exactly satisfied if one define~[31]\n$$\n  f_a(x,{\\mathbf k}_T^2,\\mu^2)\\vert_{{\\mathbf k}_T^2 < \\mu_0^2} = a(x,\\mu_0^2) T_a(\\mu_0^2,\\mu^2). \\eqno(25)\n$$\n\n\\subsection{The CCFM unintegrated gluon distribution} \\indent \n\nThe CCFM gluon density has been obtained~[38] \nfrom the numerical solution of the CCFM equation. \nThe function $f_g(x,{\\mathbf k}_T^2,\\mu^2)$ is determined\nby a convolution of the non-perturbative starting\ndistribution $f_g^{(0)}(x)$ and the CCFM evolution kernel\ndenoted by $\\tilde {\\cal A}(x,{\\mathbf k}_T^2,\\mu^2)$:\n$$\n  f_g(x,{\\mathbf k}_T^2,\\mu^2) = \\int {d x'\\over x'} f_g^{(0)}(x') \\tilde {\\cal A}\\left({x\\over x'},{\\mathbf k}_T^2,\\mu^2\\right). \\eqno(26)\n$$\n\n\\noindent\nIn the perturbative evolution the gluon splitting function\n$P_{gg}(z)$ including nonsingular terms\nis applied, as it was described in~[39]. The input parameters in $f_g^{(0)}(x)$\nwere fitted to reproduce the proton structure functions $F_2(x,Q^2)$.\nAn acceptable fit to the measured $F_2$ values was obtained~[38] with\n$\\chi^2\/ndf = 1.83$ using statistical and uncorrelated systematic\nuncertainties (compare to $\\chi^2\/ndf \\sim 1.5$ in the collinear approach\nat NLO).\n\n\\section{Numerical results} \\indent\n\nWe are now in a position to present our numerical results.\nFirst we describe the theoretical uncertainties of\nour consideration.\n\nExcept the unintegrated parton distributions in a \nproton $f_q(x,{\\mathbf k}_T^2,\\mu^2)$,\nthere are several parameters which determined the overall \nnormalization factor of the calculated $W^\\pm\/Z^0$ cross sections: \nthe quark masses $m_1$ and $m_2$ and the factorization and \nrenormalization scales $\\mu_F$ and $\\mu_R$\n(the first of them is related to the evolution of the parton distributions, \nthe other is responsible for the strong coupling constant).\nIn the numerical calculations the masses of light \nquarks were set to be equal to $m_u = 4.5$~MeV, $m_d = 8.5$~MeV, \n$m_s = 155$~MeV and the \ncharmed quark mass was set to $m_c = 1.5$~GeV. \nWe have checked that uncertainties which come \nfrom these quantities are negligible in comparison to the uncertainties\nconnected with the scale and\/or the unintegrated parton densities.\nAs it is often done, we choose the \nrenormalization and factorization scales to be equal: \n$\\mu_R = \\mu_F = \\mu = \\xi m_T$ (transverse mass of the\nproduced vector boson).\nIn order to investigate the scale dependence of our \nresults we vary the scale parameter\n$\\xi$ between $1\/2$ and~2 about the default value $\\xi = 1$.\nFor completeness, we set $m_W = 80.403$~GeV, $m_Z = 91.1876$~GeV,\n$\\sin^2 \\theta_W = 0.23122$ and use the LO formula for the strong \ncoupling constant $\\alpha_s(\\mu^2)$ with $n_f = 4$ \nactive quark flavors at \n$\\Lambda_{\\rm QCD} = 200$~MeV (so that $\\alpha_s(M_Z^2) = 0.1232$).\nNote that we use a special choice $\\Lambda_{\\rm QCD} = 130$~MeV \nin the case of CCFM gluon ($\\alpha_s(M_Z^2) = 0.1187$), \nas it was originally proposed in~[38].\n\nBefore we proceed to the numerical results,\nwe would like to comment on the effect of the\nhigher order QCD contributions~[30]. It is well-known\nthat the leading-order $k_T$-factorization approach\nnaturally includes a large part of them\\footnote{See, for example,\nreview~[29] for more details.}.\nIt is a corrections which are kinematic in nature \narising from the real parton emission during the\nevolution cascade. Another part of high-order contributions comes from\nthe logarithmic loop corrections which have\nalready been included in the Sudakov form factors~(22) and~(23).\nHowever, there are also the non-logarithmic loop corrections,\narising, for example, from the gluon vertex corrections to Fig.~2.\nTo take into account these contributions we will\nuse the approach proposed in~[30]. It was demonstrated\nthat main part of the non-logarithmic loop corrections can be\nabsorbed in the so-called K-factor given by the expression\n$$\n  K(q + \\bar q^\\prime \\to W^\\pm\/Z^0) \\simeq \\exp \\left[C_F {\\alpha_s(\\mu^2)\\over 2 \\pi} \\pi^2 \\right], \\eqno(27)\n$$\n\n\\noindent\nwhere color factor $C_F = 4\/3$. A particular choice\n$\\mu^2 = {\\mathbf p}_T^{4\/3} m^{2\/3}$ has been proposed~[22, 30]\nto eliminate sub-leading logarithmic terms.\nWe choose this scale to evaluate the strong coupling constant \n$\\alpha_s(\\mu^2)$ in~(27).\n\nWe begin the discussion by presenting a comparison\nbetween the different contributions to the $W^\\pm\/Z^0$\ntotal cross section. The solid, dashed and\ndotted histograms in Figs.~3 --- 6 represent\nthe $g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime$,\n$q + q^* \\to W^\\pm\/Z^0 + q^\\prime$ and \n$q + \\bar q^\\prime \\to W^\\pm\/Z^0$ contributions\nto the rapidity distributions of gauge boson calculated\nat the Tevatron (Figs.~3 and~4) and LHC conditions\n(Figs.~5 and~6). It is important\nthat in the last two subprocesses we take into account only\nthe valence quarks within the usual collinear \napproximation. For illustration, we used here\nthe KMR unintegrated gluon density.\nWe found that the role of the gluon-gluon fusion\nsubprocess is greatly increased at the LHC energy:\nit contributes only about 2 or 3\\% of the valence quark \ncomponent at the Tevatron and more than 40\\% at \nthe LHC. Moreover, in the last case it \ndominates over the valence contributions at the \ncentral rapidities. The contribution of the valence quark-antiquark \nannihilation subprocess is important at the Tevatron and \ngives only a few percents at the LHC energy.\nAs expected, the contribution of the $q + g^* \\to W^\\pm\/Z^0 + q^\\prime$\nsubprocess is significant in the forward rapidity region, $|y| > 2$.\nAt this point, we can conclude that the gluon-gluon\nfusion becomes an important production mechanism \nat high energies and therefore should be taken into\naccount in the calculations. However, we would like to note that \nthere is an additional contribution which is not included\nin the simple decomposition scheme proposed above.\nAs it was mentioned above, in this scheme \nit was assumed that sea quarks appear only at last \ngluon splitting and there is no contribution from the \nquarks coming from the earlier steps of the evolution\n(and we absorb last step of the \ngluon ladder into hard matrix element \n$g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime$). \nIt is not clear in advance, whether\nthe last gluon splitting dominates or not.\nIn order to model this additional component, we have\nrepeated the calculations using the KMR unintegrated quark \ndensities~(20) and the quark-antiquark annihilation \n$q + \\bar q^\\prime \\to W^\\pm\/Z^0$ matrix element.\nBut in these evaluations we omited the last term and keep only sea quark \nin first term of~(20). Thus, we switch\noff the pure gluon component of the sea quark distributions\nand remove the valence quarks from the evolution ladder.\nIn this way only the contributions to the $f_q(x,{\\mathbf k}_T^2,\\mu^2)$ \noriginating from the earlier (involving quarks) evolution steps\nare taken into account. \nSo, the dash-dotted histograms in Figs.~3 ---~6 represent\nthe results of our calculations. We have found the\nsignificant (by about of 50\\%) enhancement of the cross sections\nat both the Tevatron and LHC conditions.\nTherefore in all calculations below we will consider this\nmechanism as an additional production one.\nFinally, taking into account all described above components,\nwe can conclude that the gluon-gluon fusion contributes about\n$\\sim 1$\\% to the total cross section at Tevatron and up to $\\sim 25$\\%\nat LHC energies. \n\nNow we turn to the transverse momentum distributions of\nthe $W^\\pm$ and $Z^0$ bosons.\nThe experimental data for the transverse momentum\ndistributions come from both the D$\\oslash$~[8, 9] and CDF~[4] \ncollaborations at Tevatron. These data were obtained at center-of-mass\nenergy $\\sqrt s = 1800$~GeV. Measurements were made for $W \\to l\\nu$\nand $Z \\to l^+l^-$ decays; so that we should multiply our\ntheoretical predictions by the relevant branching fractions \n$f(W \\to l\\nu)$ and $f(Z \\to l^+l^-)$. \nThese branching fractions \nwere set to $f(W \\to l\\nu) = 0.1075$ and $f(Z \\to l^+l^-) = 0.03366$~[40].\nIn Figs.~7 --- 9 we display a comparison of the calculated \ndifferential cross sections $d\\sigma\/dp_T$ of the\n$W^\\pm$ and $Z^0$ boson production\nwith the experimental data~[4, 8, 9] in the low $p_T$ region, namely $p_T < 20$~GeV.\nNext, in Figs.~10 --- 12,\nwe demonstrate the $W^\\pm$ and $Z^0$ transverse momentum\ndistributions in the intermediate and high $p_T$ regions.\nAdditionally, in Figs.~13 and 14, we plot the normalized \ndifferential cross section\n$(1\/\\sigma)\\,d\\sigma\/dp_T$ of the $W^\\pm$ boson production.\nThe solid and dashed histograms correspond to the \nresults obtained with the CCFM and KMR unintegrated gluon densities, \nrespectively. All contributions discussed above are taken\ninto account. The dotted histograms were obtained using\nthe quark-antiquark annihilation matrix element \nconvoluted with the KMR unintegrated quark distributions\nin a proton (in this case the transverse mometum of the \nproduced vector boson is defined by the transverse momenta\nof the incoming quarks).\nWe found an increase in the cross section calculated\nin the proposed decomposition scheme (where only the unintegrated \ngluon densities used).\nIn this scheme, we obtain that both the CCFM and KMR gluon distributions\nreproduce well the Tevatron data within the uncertainties,\nalthough the KMR gluon tends to slightly underestimate\nthe data in the low $p_T$ region. \nThe difference between the solid and dashed histograms in Figs.~7 --- 14\nis due to different behaviour of the CCFM and KMR gluon densities.\nThe predictions based of the \nquark-antiquark annihilation subprocess lie\nbelow the experimental data but agree with them in shape.\nThis observation coincides with the one from~[30] where\nan additional factor of about 1.2 was introduced\nto eliminate the visible disagreement between the data and \ntheory\\footnote{In Ref.~[30] authors have explained the origin of \nthis extra factor by the fact that the input\nparton densities (used to determine the unintegrated ones)\nshould themselves be determined from data using\nthe appropriate non-collinear formalism.}. \n\nAn additional possibility to distinguish\nthe two calculation schemes comes from the\nstudying of the ratio of the $W^\\pm$ and $Z^0$ boson \ncross sections. In fact, since $W^\\pm$ and $Z^0$ production\nproperties are very similar, as\nthe transverse momentum of the vector boson becomes\nsmaller, the radiative corrections affecting the individual distributions and \nthe cross sections of hard process are\nfactorized and canceled in this ratio.\nTherefore the results of calculation of this ratio \nin the decomposition scheme (where\nthe ${\\cal O}(\\alpha \\alpha_s)$ and ${\\cal O}(\\alpha \\alpha_s^2)$ \nsubprocesses are taken into account) and the predictions \nbased on the ${\\cal O}(\\alpha)$\nquark-antiquark annihilation should\ndiffer from each other at moderate and high $p_T$ values.\nThis fact is clearly illustrated in Fig.~15 where the ratio of\n$W^\\pm$ and $Z^0$ cross sections as a function of the\ntransverse momentum is displayed.\nAs it was expected, there is practically no difference\nbetween all plotted histograms in the low $p_T$ region.\n\nAs a final point of our study, we discuss the scale dependence of\nour results. In Figs.~16 and~17 we show the total\ncross section of the $W^\\pm$ and $Z^0$ boson \nproduction as a function of the total center-of-mass \nenergy $\\sqrt s$. Here, the solid and dotted histograms \ncorrespond to the results obtained with the CCFM \nand KMR unintegrated gluon densities, respectively. The\nthe upper and lower dashed histograms \ncorrespond to the scale variations in the CCFM gluon density \nas it was described above. We find that the\nscale uncertainties are the same order approximately \nas the uncertainties coming from the unintegrated\ngluon distributions. This fact is the typical one for the\nleading-order $k_T$-factorization calculations. \nOur predictions for the $W^\\pm$ and $Z^0$ boson \ntotal cross section agree well with the data\nin a wide $\\sqrt s$ range.\n\n\\section{Conclusions} \\indent \n\nWe have studied the production of \nelectroweak gauge bosons in hadronic collisions at high energies\nin the $k_T$-factorization approach of QCD.\n Our consideration is based on the scheme which provides solid \ntheoretical grounds \nfor adequately taking into account the effects of initial parton \nmomentum. The central part of our derivation is the off-shell gluon-gluon \nfusion subprocess $g^* + g^* \\to W^\\pm\/Z^0 + q \\bar q^\\prime$. \nAt the price of considering the corresponding matrix element \nrather than $q + \\bar q^\\prime \\to W^\\pm\/Z^0$ one, \nwe have reduced the problem of unknown unintegrated quark distributions to the \nproblem of gluon distributions. \nThis way enables us with making comparisons between the different parton \nevolution schemes and parametrizations of parton densities. \nSince the gluons are only responsible for the appearance of sea, but \nnot valence quarks, the contribution from the valence quarks has been \ncalculated separately. Having in mind that the valence quarks are only \nimportant at large $x$, where the traditional DGLAP evolution is accurate and \nreliable, we have calculated this contribution within the usual collinear \nscheme based on $q + g^* \\to W^\\pm\/Z^0 + q^\\prime$ and \n$q + \\bar q^\\prime \\to W^\\pm\/Z^0$ partonic subprocesses \nand on-shell parton densities. \n\nWe have studied in detail the different production \nmechanisms of $W^\\pm$ and $Z^0$ bosons. \nWe find that the off-shell gluon-gluon fusion\ngives $\\sim 1$\\% and $\\sim 25$\\% contributions\nto the inclusive $W^\\pm\/Z^0$ production cross sections\nat the Tevatron and LHC.\nSpecially we simulate the contribution from\nthe quarks involved into the earlier steps of the \nevolution cascade (i.e., into the second-to-last, third-to-last and \nother gluon splittings) and find that these quarks\nplay an important role at both the Tevatron and LHC energies. \nIt was demonstrated that corresponding corrections\nshould be taken into accout in the numerical calculations \nwithin the $k_T$-factorization approach.\n\nWe have calculated the total and differential $W^\\pm$ and $Z^0$ \nproduction cross sections and have made comparisons with the Tevatron \ndata. In the numerical analysis we have used \nthe unintegrated gluon densities obtained from the  \nCCFM evolution equation and from the KMR prescription.\nOur numerical results agree well with\nthe experimental data.\n\nWhen the present paper was ready for publication,\nwe have learned about the results obtained by\nM.~Deak and F.~Schwennsen~[41], who used the same theoretical approach,\nbut focused attention on slightly different aspects of the problem.\nThese authors concentrate on the associated $W^\\pm\/Z^0$ production\nwith heavy quark pairs (mainly on the $Z b\\bar b$ final \nstate), where the gluon-gluon fusion subprocess dominates.\nIn additional to that, we consider quark subprocesses, which \nare important for inclusive $W^\\pm\/Z^0$ production. We show that the \nexperimental data can be described with taking quark contributions into account.\n\n\\section{Acknowledgements} \\indent \n\nWe thank H.~Jung for his encouraging interest, very helpful discussions\nand for providing the CCFM code for \nunintegrated gluon distributions. We\nthank M.~Deak and F.~Schwennsen for letting us\nknow about the results of their work before\npublication and useful remarks.\nThe authors are very grateful to \nDESY Directorate for the support in the \nframework of Moscow --- DESY project on Monte-Carlo\nimplementation for HERA --- LHC.\nA.V.L. was supported in part by the grants of the president of \nRussian Federation (MK-438.2008.2) and Helmholtz --- Russia\nJoint Research Group.\nAlso this research was supported by the \nFASI of Russian Federation (grant NS-8122.2006.2)\nand the RFBR fundation (grant 08-02-00896-a).\n\n\\section{Appendix A} \\indent \n\nHere we present the compact analytic expressions for the \ncross section of the $W^\\pm\/Z^0$ production\nvia $q + \\bar q^\\prime \\to W^\\pm\/Z^0$ subprocess\nin the $k_T$-factorization approach. \nLet us define the transverse momenta and azimuthal\nangles of incoming quark $q$ and antiquark $\\bar q^\\prime$ as \n${\\mathbf k}_{1T}$ and ${\\mathbf k}_{2T}$ and\n$\\phi_1$ and $\\phi_2$, respectively. \nThe produced vector boson has the transverse momentum\n${\\mathbf p}_{T}$ (${\\mathbf p}_{T} = {\\mathbf k}_{1T} + {\\mathbf k}_{2T}$) \nand center-of-mass rapidity $y$.\nThe $W^\\pm\/Z^0$ production cross section can be written as\n$$\n  \\displaystyle \\sigma(p + \\bar p \\to W^\\pm\/Z^0 + X) = \\sum_{q} \\int {2 \\pi \\over (x_1 x_2 s)^2} |\\bar {\\cal M}(q + \\bar q^\\prime \\to W^\\pm\/Z^0)|^2 \\times \\atop \n  \\displaystyle \\times f_q(x_1,{\\mathbf k}_{1 T}^2,\\mu^2) f_q(x_2,{\\mathbf k}_{2 T}^2,\\mu^2) d{\\mathbf k}_{1 T}^2 d{\\mathbf k}_{2 T}^2 dy {d\\phi_1\\over 2\\pi} {d\\phi_2\\over 2\\pi}, \\eqno(A.1)\n$$\n\n\\noindent \nwhere $f_q(x,{\\mathbf k}_{T}^2,\\mu^2)$ is the unintegrated quark\ndistributions given by~(20). In the high-energy limit the \nfractions $x_1$ and $x_2$ of the\ninitial proton's longitudinal momenta are given by\n$$\n  x_1 \\sqrt s = m_T\\,e^y, \\quad x_2 \\sqrt s = m_T \\, e^{-y}. \\eqno(A.2)\n$$\n\n\\noindent \nwhere $m_T$ is the transverse mass of the vector boson.\nThe squared matrix element $|\\bar {\\cal M}(q + \\bar q^\\prime \\to W^\\pm)|^2$ \nsummed over final polarization states and averaged over initial ones is\n$$\n  |\\bar {\\cal M}(q + \\bar q^\\prime \\to W^\\pm)|^2 = - {e^2\\over 72 m_W^2 \\sin^2 \\theta_W} \\left[(m_1^2 - m_2^2)^2 + m_W^2 (m_1^2 + m_2^2) - 2 m_W^4\\right], \\eqno(A.3)\n$$\n\n\\noindent \nwhere $m_1$ and $m_2$ are the masses of incoming quarks.\nIn the case of $Z^0$ boson production,\nthe squared matrix element $|\\bar {\\cal M}(q + \\bar q \\to Z^0)|^2$ \nsummed over final polarization states and averaged over initial ones is\n$$\n  \\displaystyle |\\bar {\\cal M}(q + \\bar q \\to Z^0)|^2 = {2 e^2\\over 9 \\sin^2 2 \\theta_W} \\times \\atop\n  \\displaystyle \\times \\left[ (m_Z^2 - m^2)\\left[I_3^{(q)}\\right]^2 + 2 e_q (2 m^2 + m_Z^2) \\sin^2 \\theta_W \\left(e_q \\sin^2\\theta_W - I_3^{(q)}\\right) \\right], \\eqno(A.4)\n$$\n\n\\noindent \nwhere $m$, $e_q$ and $I_3^{(q)}$ is the mass, fractional electric charge and\nweak isospin of incoming quark. Note that there is no obvious \ndependence on the transverse momenta of the initial quark and antiquark.\nHowever, this dependence is present because \nthe true off-shell kinematics is used. \nIn particular, the incident parton momentum fractions\n$x_1$ and $x_2$ have some ${\\mathbf k}_{T}$ \ndependence. If we take the limit ${\\mathbf k}_{1 T}^2 \\to 0$\nand ${\\mathbf k}_{2 T}^2 \\to 0$,\nthen we recover the relevant expression in the standard \ncollinear approximation of QCD.\n\n\\section{Appendix B} \\indent \n\nHere we present the analytic expressions for the \ncross section of the $W^\\pm\/Z^0$ production\nvia $q + g^* \\to W^\\pm\/Z^0 + q^\\prime$ subprocess\nin the $k_T$-factorization approach. \nLet us define the transverse momenta and azimuthal\nangles of incoming quark and off-shell gluon as \n${\\mathbf k}_{1T}$ and ${\\mathbf k}_{2T}$ and\n$\\phi_1$ and $\\phi_2$, respectively. \nIn the following, $\\hat s$, $\\hat t$ and $\\hat u$ are usual Mandelstam \nvariables for $2 \\to 2$ subprocess.\nThe $W^\\pm\/Z^0$ production cross section can be written as follows:\n$$\n  \\displaystyle \\sigma(p + \\bar p \\to W^\\pm\/Z^0 + X) = \\sum_{q} \\int {1\\over 16\\pi (x_1 x_2 s)^2} |\\bar {\\cal M}(q + g^* \\to W^\\pm\/Z^0 + q^\\prime)|^2 \\times \\atop \n  \\displaystyle \\times f_q(x_1,{\\mathbf k}_{1 T}^2,\\mu^2) f_g(x_2,{\\mathbf k}_{2 T}^2,\\mu^2) d{\\mathbf k}_{1 T}^2 d{\\mathbf k}_{2 T}^2 d{\\mathbf p}_{T}^2 dy dy^\\prime {d\\phi_1\\over 2\\pi} {d\\phi_2\\over 2\\pi}, \\eqno(B.1)\n$$\n\n\\noindent\nwhere $y^\\prime$ is the rapidity of the final quark $q^\\prime$. The\nfractions $x_1$ and $x_2$ of the\ninitial proton's longitudinal momenta are given by\n$$\n  x_1 \\sqrt s = m_{T}\\,e^y + m_{T}^\\prime \\,e^{y^\\prime}, \\quad x_2 \\sqrt s = m_{T}\\, e^{-y} + m_{T}^\\prime \\,e^{-y^\\prime}. \\eqno(B.2)\n$$\n\n\\noindent \nwhere $m_T$ and $m_T^\\prime$ are the transverse masses of the \nvector boson and final quark $q^\\prime$.\nIf we take the limit ${\\mathbf k}_{1 T}^2 \\to 0$ and ${\\mathbf k}_{2 T}^2 \\to 0$,\nthen we recover the relevant expression in the usual \ncollinear approximation.\nThe squared matrix elements $|\\bar {\\cal M}(q + g^* \\to W^\\pm + q^\\prime)|^2$ \nand $|\\bar {\\cal M}(q + g^* \\to Z^0 + q)|^2$ summed \nover final polarization states and averaged over initial ones are\n$$\n  |\\bar {\\cal M}(q + g^* \\to W^\\pm + q^\\prime)|^2 = {e^2 g^2 \\over 192 \\sin^2 \\theta_W} {F_W \\over (m_1^2 - \\hat s)^2 (m_2^2 - \\hat t)^2 m_W^2}, \\eqno(B.3)\n$$\n$$\n  |\\bar {\\cal M}(q + g^* \\to Z^0 + q)|^2 = {2 e^2 g^2 \\over 3 \\sin^2 2 \\theta_W} {F_Z \\over (m^2 - \\hat s)^2 (m^2 - \\hat t)^2 m_Z^2}, \\eqno(B.4)\n$$\n\n\\noindent where\n$$\n  F_W = -8 (m_1^8 (3 m_2^2 - \\hat t) + m_1^6 (m_2^4 + m_2^2 (2 m_W^2 - 5 \\hat s - 7 \\hat t) + \\hat t (\\hat s + 2 \\hat t)) +\n$$\n$$ \n  m_1^4 (m_2^6 + m_2^4 (8 m_W^2 - 3 (\\hat s + \\hat t)) + m_2^2 (-6 m_W^4 + 3 \\hat s^2 + 13 \\hat s \\hat t + 5 \\hat t^2 - 6 m_W^2 (\\hat s + \\hat t)) - \n$$\n$$\n  \\hat t (-2 m_W^4 - 2 m_W^2 \\hat s + (\\hat s + \\hat t)^2)) + m_1^2 (3 m_2^8 + m_2^6 (2 m_W^2 - 7 \\hat s - 5 \\hat t) + \n$$\n$$\n  m_2^4 (-6 m_W^4 + 5 \\hat s^2 + 13 \\hat s \\hat t + 3 \\hat t^2 - 6 m_W^2 (\\hat s + \\hat t)) + m_2^2 (4 m_W^6 - \\hat s^3 - 11 \\hat s^2 \\hat t - 11 \\hat s \\hat t^2 - \\hat t^3 + \n$$\n$$\n  6 m_W^4 (\\hat s + \\hat t) + 6 m_W^2 (\\hat s^2 + \\hat t^2)) + \\hat t (-4 m_W^6 + 2 m_W^4 \\hat s + \\hat s (\\hat s + \\hat t)^2 - \n$$\n$$\n  2 m_W^2 (2 \\hat s^2 - \\hat s \\hat t + \\hat t^2))) + \\hat s (-m_2^8 + m_2^6 (2 \\hat s + \\hat t) + 2 m_W^2 \\hat t (2 m_W^4 + \\hat s^2 + \\hat t^2 - 2 m_W^2 (\\hat s + \\hat t)) + \n$$\n$$\n  m_2^4 (2 m_W^4 + 2 m_W^2 \\hat t - (\\hat s + \\hat t)^2) + m_2^2 (-4 m_W^6 + 2 m_W^4 \\hat t + \\hat t (\\hat s + \\hat t)^2 - \n$$\n$$\n  2 m_W^2 (\\hat s^2 - \\hat s \\hat t + 2 \\hat t^2))) + (m_1^8 + m_2^8 + m_1^6 (m_W^2 - 2 (\\hat s + \\hat t)) + m_2^6 (m_W^2 - 2 (\\hat s + \\hat t)) + \n$$\n$$\n  m_2^4 (-2 m_W^4 - 2 m_W^2 \\hat t + (\\hat s + \\hat t)^2) + m_2^2 m_W^2 (5 \\hat s^2 + 4 \\hat s \\hat t + \\hat t^2 + m_W^2 (-8 \\hat s + 4 \\hat t)) - \n$$\n$$\n  2 m_W^2 (2 \\hat s \\hat t (\\hat s + \\hat t) + m_W^2 (\\hat s^2 - 4 \\hat s \\hat t + \\hat t^2)) + m_1^4 (-2 m_2^4 - 2 m_W^4 - 2 m_W^2 \\hat s + (\\hat s + \\hat t)^2 + \n$$\n$$\n  m_2^2 (m_W^2 + 2 (\\hat s + \\hat t))) + m_1^2 (m_W^2 (\\hat s^2 + 4 m_W^2 (\\hat s - 2 \\hat t) + 4 \\hat s \\hat t + 5 \\hat t^2) + m_2^4 (m_W^2 + 2 (\\hat s + \\hat t)) + \n$$\n$$\n  m_2^2 (8 m_W^4 - 6 m_W^2 (\\hat s + \\hat t) - 2 (\\hat s + \\hat t)^2))) ( - {\\mathbf k}_{2T}^2 ) + 4 m_W^2 (m_1^2 - \\hat s) (m_2^2 - \\hat t) {\\mathbf k}_{2T}^4), \\eqno(B.5)\n$$\n$$\n  F_Z = -2 e_q I_3^{(q)} m_Z^2 \\sin^2 \\theta_W (6 m^8 - \\hat s \\hat t (2 m_Z^4 + \\hat s^2 + \\hat t^2 - 2 m_Z^2 (\\hat s + \\hat t)) - \n$$\n$$\n  m^4 (2 m_Z^4 + 3 \\hat s^2 + 14 \\hat s \\hat t + 3 \\hat t^2 - 2 m_Z^2 (\\hat s + \\hat t)) + m^2 (\\hat s^3 - 8 m_Z^2 \\hat s \\hat t + 7 \\hat s^2 \\hat t + \n$$\n$$\n  7 \\hat s \\hat t^2 + \\hat t^3 + 2 m_Z^4 (\\hat s + \\hat t))) + 2 e_q^2 m_Z^2 \\sin^4 \\theta_W (6 m^8 - \\hat s \\hat t (2 m_Z^4 + \\hat s^2 + \\hat t^2 - 2 m_Z^2 (\\hat s + \\hat t)) - \n$$\n$$\n  m^4 (2 m_Z^4 + 3 \\hat s^2 + 14 \\hat s \\hat t + 3 \\hat t^2 - 2 m_Z^2 (\\hat s + \\hat t)) + m^2 (\\hat s^3 - 8 m_Z^2 \\hat s \\hat t + 7 \\hat s^2 \\hat t + 7 \\hat s \\hat t^2 + \\hat t^3 + \n$$\n$$\n  2 m_Z^4 (\\hat s + \\hat t))) + \\left[I_3^{(q)}\\right]^2 (-4 m^{10} + m^8 (-6 m_Z^2 + 8 (\\hat s + \\hat t)) - m_Z^2 \\hat s \\hat t (2 m_Z^4 + \\hat s^2 + \\hat t^2 - \n$$\n$$\n  2 m_Z^2 (\\hat s + \\hat t)) + m^6 (6 m_Z^4 - 5 \\hat s^2 - 14 \\hat s \\hat t - 5 \\hat t^2 + 6 m_Z^2 (\\hat s + \\hat t)) + \n$$\n$$\n  m^4 (-2 m_Z^6 + \\hat s^3 + 7 \\hat s^2 \\hat t + 7 \\hat s \\hat t^2 + \\hat t^3 - 4 m_Z^4 (\\hat s + \\hat t) - m_Z^2 (3 \\hat s^2 + 2 \\hat s \\hat t + 3 \\hat t^2)) + \n$$\n$$\n  m^2 (-2 m_Z^4 \\hat s \\hat t + 2 m_Z^6 (\\hat s + \\hat t) - \\hat s \\hat t (\\hat s + \\hat t)^2 + m_Z^2 (\\hat s^3 + \\hat s^2 \\hat t + \\hat s \\hat t^2 + \\hat t^3))) +\n$$\n$$ \n  m_Z^2 (2 e_q I_3^{(q)} \\sin^2 \\theta_W (2 m^4 (m_Z^2 - \\hat s - \\hat t) - 2 \\hat s \\hat t (\\hat s + \\hat t) - m_Z^2 (\\hat s^2 - 4 \\hat s \\hat t + \\hat t^2) - \n$$\n$$\n  2 m^2 (-4 \\hat s \\hat t + m_Z^2 (\\hat s + \\hat t))) + 2 e_q^2 \\sin^4 \\theta_W (2 \\hat s \\hat t (\\hat s + \\hat t) + 2 m^4 (-m_Z^2 + \\hat s + \\hat t) + \n$$\n$$\n  m_Z^2 (\\hat s^2 - 4 \\hat s \\hat t + \\hat t^2) + 2 m^2 (-4 \\hat s \\hat t + m_Z^2 (\\hat s + \\hat t))) + \\left[I_3^{(q)}\\right]^2 (-2 m^6 + 2 \\hat s \\hat t (\\hat s + \\hat t) + \n$$\n$$\nm_Z^2 (\\hat s^2 - 4 \\hat s \\hat t + \\hat t^2) + m^4 (-2 m_Z^2 + 4 (\\hat s + \\hat t)) + m^2 (-3 \\hat s^2 - 4 \\hat s \\hat t - 3 \\hat t^2 + \n$$\n$$\n  2 m_Z^2 (\\hat s + \\hat t)))) ( - {\\mathbf k}_{2T}^2 ) - 2 m_Z^2 (m^2 - \\hat s) (\\left[I_3^{(q)}\\right]^2 - \n$$\n$$\n  2 e_q I_3^{(q)} \\sin^2 \\theta_W + 2 e_q^2 \\sin^4 \\theta_W) (m^2 - \\hat t) {\\mathbf k}_{2T}^4. \\eqno(B.6)\n$$\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently the Alpha Magnetic Spectrometer (AMS-02) on the International\nSpace Station has reported that the antiproton to proton ratio stays\nconstant from 20 GeV to 450 GeV kinetic energy~\\cite{AMS-02DATA}.\nThis behavior cannot be explained by the secondary antiprotons from\ncollisions of ordinary cosmic rays with interstellar\nmedium~\\cite[e.g.,][]{Kachelriess:2015wpa}.  This suggests a new\nsource such as astrophysical accelerators and annihilating or decaying\ndark matter, although there are still uncertainties in the background\nmodeling~\\cite{Giesen:2015ufa}.\n\nThe excess of antiprotons looks surprisingly similar to what we\npredicted~\\cite{Fujita:2009wk} when the PAMELA experiment detected the\npositron excess~\\cite{Adriani:2008zr} and the Fermi, HESS, and\nATIC\/PPB-BETS experiments observed the electron\nanomaly~\\cite{cha08,Abdo:2009zk,Aharonian:2009ah}.  We considered\nrecent supernova explosions in a dense gas cloud (DC) near the Earth.\nThe antiprotons and positrons are produced as secondaries by the $pp$\ncollisions between cosmic-ray protons accelerated by the supernova\nremnant (SNR) and target protons in a DC which surrounds the\nSNRs~\\cite{Fujita:2009wk}.  Since the fundamental process determines\nthe branching fraction, the positron excess should accompany the\nantiproton excess.\n\nThere are several variants of such a hadronic model, e.g., the\nreacceleration of secondaries by SNR shocks~\\cite{Blasi:2009bd} or the\nnon-standard propagation that increases secondaries from ordinary\ncosmic-ray collisions with interstellar\nmatter~\\cite{Blum:2013zsa,Cowsik:2013woa,Guo:2014laa}.  \nSince the element ratio is the same as the\nordinary cosmic rays in these models, \nthe ratio of secondaries (e.g., Li, Be, B) to\nprimaries (C, N, O) must also rise with energy beyond $\\sim 100$ GeV,\nwhich is not\nobserved yet~\\cite{Mertsch:2009ph,Mertsch:2014poa,Cholis:2013lwa}.  In\ncontrast to these models, our model can accommodate the observations\nas shown below.\n\nBefore AMS-02, the antiproton observations were consistent with the\nsecondary background~\\cite{Adriani:2008zq}.  Thus the leading models\nfor the positron excess are leptonic, such as pulsars and leptophillic\ndark\nmatter~\\cite[e.g.,][]{Serpico:2011wg,Fan:2010yq,Ioka:2008cv,Kashiyama:2010ui}.\nThe pulsars cannot usually explain the antiproton excess.  On the\nother hand, the dark matter for the positron excess is now severely\nconstrained by other messengers such as gamma-rays and cosmic\nmicrowave background~\\cite[e.g.,][]{Ackermann:2015zua,Ade:2015xua},\nand hence we may need fine tunings in dark matter models to reproduce\nboth the antiproton and positron excesses~\\cite{Giesen:2015ufa,DM2015,Evoli:2015vaa,Kohri:2013sva}.\n\nFollowing Occam's razor, we reexamine our nearby SNR model \nand simultaneously fit the antiproton fraction, positron fraction, \nand total electron and positron flux in light of new AMS-02 data.\nIn particular the $pp$ collisions give the correct branching fraction\nfor the observed positron to antiproton ratio.\nThroughout this paper we adopt units of $c = \\hbar = k_B = 1$.\n\n\\section{Supernova explosions in a Dense Cloud}\n\nHere we consider supernova explosions which occurred around $\\sim\n10^{5} - 10^{6} $ years ago in a  DC. We assume the\nDC is located at around $\\sim$~100--200~pc away from the Earth \nlike the progenitor DCs that produced the Local Bubble (LB) or\nLoop~I. In general a massive star tends to be born in a giant\nDC~\\cite{lar82} which explodes as a supernova. In this paper, we\nassume that the Giant DC is ionized and the temperature is\napproximately $\\sim 10^4$~K at the time of its\nexplosion~\\cite{whi79}. The shock of an SNR accelerates protons, which\nproduce copious energetic mesons (pions and kaons, etc.) and baryons\n(antiproton, proton, antineutron, neutron, etc.)  through the $pp$\ncollisions in the surrounding DC. The mesons further decay into\nenergetic positrons, electrons, gamma-rays, and neutrinos.  In total\nthose local secondary particles can be observed at the Earth as\ncosmic-rays in addition to the standard background components.\n\nThe energy spectrum of the accelerated protons is parametrized by\n\\begin{equation}\n\\label{eq:NE}\n \\frac{dn_p}{dE_p} \\propto E_p^{-s}  e^{- \\frac{E_p}{E_{\\rm max, p}}},\n\\end{equation}\nwhere $s$ is the spectral index. \nThe age of SNR, $t_{\\rm age}$, approximately determines the maximum\nenergy~\\cite{yam06},\n\\begin{equation}\n E_{\\rm max,p} \\sim 2 \\times10^2  v_{s,8}^2\n\\left(\\frac{B_{\\rm d}}{10~\\rm\\mu G}\\right)\n\\left(\\frac{t_{\\rm age}}{10^5{\\rm yr}}\\right)~{\\rm TeV}~,\n\\label{eq:Emax_p}\n\\end{equation}\nwhere the shock velocity, $v_s$, is $v_{s,8}=v_s\/10^8$cm~s$^{-1} \\sim\nO(1)$. $B_{\\rm d}$ is the downstream magnetic field. We take\nthe minimum energy of the protons to be its rest mass.  We assume that\nthe supernova explodes at the center of a DC for simplicity.  In\naddition, we also assume that the acceleration stops when the Mach\nnumber of the shock decreases to 7~\\cite{Fujita:2009wk}. We define\nthis time as the acceleration time\n$t_{\\rm acc}=t_{\\rm age}$, \nand the energy spectrum at\nthis time is given by $s \\sim 2$ and $E_{\\rm max,p} \\sim 120$\nTeV~\\cite{Fujita:2009wk}. The SNR continues to expand even at $t_{\\rm\n  age} > t_{\\rm acc}$.\n\nThe radius is $50$~pc at\n$t_{\\rm age}=5\\times 10^5 $~yr. Since it is comparable to the size of\na giant DC, $R_{\\rm DC}$,~\\cite{mck07} and the initial energy of the\nejecta from the supernovae is larger than the binding energy of a DC,\nthe cloud would be destroyed around this time. Until it is destroyed,\nthe DC is illuminated by the accelerated protons from the inside with\nthe spectrum of Eq. (\\ref{eq:NE}) given at $t_{\\rm age}\\sim t_{\\rm\n  acc}$. The duration of the exposure, $t_{pp}$, could be approximated\nby the time elapsing from the explosion of the supernovae to the\ndestruction of the DC  because the timescale $t_{\\rm acc}$ is\nshorter than $5\\times 10^5 $~yr. \n\n\nAfter the destruction of the DC, the produced charged particles such\nas $\\bar{p}$, $p$, $e^{+}$, or $e^{-}$ propagate through diffusion\nprocesses and reach to the Earth. Since we assume that the DC has\nalready been destroyed well before the present epoch, there are some\ndifferences in arrival times between those charged particles and\nmassless neutral particles such as photons. It should be a reasonable\nassumption that we would not detect any photon and neutrino signals\nfrom the DC $\\sim 10^{5-6}$ years after the destruction of the DC.\n\nWe have calculated spectra of those daughter particles through the\n$pp$ collisions by performing the PYTHIA Monte-Carlo event\ngenerator~\\cite{Sjostrand:2006za} (See~\\cite{yam06} for the\ndetails). Then we solve the diffusion equation of the charged particle\n``$i$'' ($i$ runs $\\bar{p}$, $p$, $e^{+}$, and $e^{-}$, ),\n\\begin{equation}\n    \\label{eq:diff_eq}\n    \\frac{\\partial f_{i}}{\\partial t} \n= K(\\varepsilon_{i}) \\Delta f_{i} +\n    \\frac{\\partial}{ \\partial \\varepsilon_{i}} \\left[\n    B(\\varepsilon_{i}) \nf_{i}\\right] + Q(\\varepsilon_{i})\n\\end{equation}\nwhere $f_{i}(t,{\\boldmath{x},\\varepsilon_{i}})$ is the distribution\nfunction of an $i$ particle, and $\\varepsilon_{i} = E_{i}\/{\\rm GeV}$\nwith $E_{i}$ being the energy of the $i$ particle.  The flux is given\nby \n\\begin{eqnarray}\n  \\label{eq:totalflux0}\n  \\Phi_{i} (t,{\\boldmath{x},\\varepsilon_{i}}) = \\frac{1}{4\\pi} f_{i}.\n\\end{eqnarray}\nWe adopt a diffusion model 08-005 given in~\\cite{Moskalenko:1997gh}\nwith the diffusion coefficient,\n\\begin{equation}\n K(\\varepsilon_{e}) = K_{0}\n \\left(1 +\n\\frac{\\varepsilon_{e}}{3{\\rm GeV} }\n\\right)^{\\delta},\n\\end{equation}\nwith $K_{0} = 2 \\times 10^{28} {\\rm cm}^{2}{\\rm s}^{-1}$ and $\\delta =\n0.42$~\\cite{AMS-02:2013conf,Genolini:2015cta,Evoli:2015vaa}.  The cooling rate through the synchrotron emission and the\ninverse Compton scattering is collectively parametrized to be~\\cite{Baltz:1998xv}\n\\begin{equation}\n B(\\varepsilon_{e}) \\sim  10^{-16} {\\rm s}^{-1} \\varepsilon_{e}^{2 }\n\\left[ 0.2\n\\left(\n\\frac{B_{\\rm diff} }  {3 \\mu {\\rm G}}\n\\right)^{2} + 0.9 \n\\right],\n\\end{equation}\nwhere $B_{\\rm diff}$ is the magnetic field outside the DC. This set of\nthe parameters approximately corresponds to the MED model of the\ncosmic-ray propagation~\\cite{Bottino:2005xy}.\n\n\nIf we assume that the timescale of the production is shorter than that\nof the diffusion, $\\sim d^{2}\/(Kc)$ with $d$ being the distance to the\nsource, and the source of the daughter particles is spatially\nlocalized sufficiently, we can use the known analytical solution\nin~\\cite{Atoian:1995ux}.  When the shape of the source spectrum is a\npower-law with an index $\\alpha$ to be\n\\begin{eqnarray}\n  \\label{eq:Qdetail}\n  Q =Q_{0}\\varepsilon^{-\\alpha}\\delta(\\boldmath{x})\\delta(t),\n\\end{eqnarray}\nthen the solution is given by\n\\begin{equation}\n    \\label{eq:f_diff}\nf_{e}=\\frac{Q_0 }{\\pi^{3\/2} d_{\\rm diff}^3}\n\\varepsilon_e^{-\\alpha}\n\\left(1-\\frac{\\varepsilon_e}{\\varepsilon_{\\rm cut}}\\right)^{\\alpha-2}\ne^{-(\\frac{\\bar{d}} {d_{\\rm diff}})^2},\n\\end{equation}\nwhere $\\varepsilon_{\\rm cut} = \\varepsilon_e^2 \/ B t_{\\rm diff}$, and the\ndiffusion length is represented by\n\\begin{equation}\n d_{\\rm diff} = 2 \\sqrt{K t_{\\rm diff} \\frac{1 - (1 -\n\\frac{\\varepsilon_e} {\\varepsilon_{\\rm cut}})^{1 - \\delta}}{\n(1-\\delta) \n\\frac {\\varepsilon_e } {\\varepsilon_{\\rm cut}} }}\\:.\n\\end{equation}\n$\\bar{d}$ means the effective distance to the source by spatially\naveraging the distance to the volume element of the source, and we\nassume $\\alpha \\simeq s$. We approximately have\n\\begin{eqnarray}\n  \\label{eq:Q0emalpha}\n  Q_0\\varepsilon_{i}^{-\\alpha} \\sim  V_{s}t_{pp} \\frac{d^{2}n_{i}}{dtdE_{i}} \n\\end{eqnarray}\nwith $V_{s}$ the  source volume  where\n\\begin{equation}\n \\frac{d^{2}n_{i}}{dtdE_{i}} = \\int d E_{p} n_{0} \\frac{dn_p}{dE_p}  \\sum_{j}\ng_{j}  \\frac{v_pd\\sigma_{j}}{dE_{i}} \\:.\n\\end{equation}\nThe differential cross section of the ``$j$''-mode for the production\nof the $i$ particle is represented to be\n$d\\sigma_{j}(E_{p},E_{i})\/dE_{i}$ with the multiplicity into the\n$j$-mode, $g_{j} = g_{j}(E_{p},E_{i})$. $v_p=v_p(E_p)$ is the velocity of the\nprimary proton.  We also consider the free neutron (antineutron)  decay\nfor the electron (positron) production process, which is not included\nin the original version of PYTHIA. The initial proton spectrum\n$\\frac{dn_p}{dE_p}$ can be obtained by a normalization to satisfy\n\\begin{equation}\n V_{s}\\int dE_{p} \\frac{dn_p}{dE_p}= E_{\\rm tot, p} .\n\\end{equation}\nFor the local propagation of protons and antiprotons, their cooling is\nnegligible unlike electrons and positrons.  Additionally we can omit\nannihilations of antiprotons through scattering off the background\nprotons because the scattering rate is small. We can also omit\nconvection by interstellar turbulence\nwithin the galaxy. An analytical\nsolution for the proton and the antiproton is also given by the same\nequation as Eq.~(\\ref{eq:f_diff}) with a limit of\n$\\varepsilon_p\/\\varepsilon_{\\rm cut} = 0$.\n\n\\section{Antiproton and positron fittings}\n\n\n\n\\begin{figure}[t]\n  \\begin{center}\n    \\vspace{-0. cm}\n    \\includegraphics[width=100mm]{pbar20151204.eps}\n    \\vspace{-1. cm}\n    \\caption{Antiproton fraction fitted to the data. The data points\n      are taken from~\\cite{AMS-02DATA} for AMS-02, and\n      from~\\cite{Adriani:2008zq} for PAMELA. The dotted line is plotted\n      only by using the background flux~\\cite{Nezri:2009jd}. The\n      shadow region represents the uncertainties of the background\n      flux among the propagation models shown\n      in~\\cite{AMS-02DATA}. Cosmic rays below an energy $\\lesssim$\n      10GeV are affected by the solar modulation. \n        We choose the background line and its uncertainty band only\n        for a demonstration purpose. This choice is not\n        essential for our conclusion (See the text \n        about Fig.~\\ref{fig:posipbar}).} \n  \\end{center}\n\\label{fig:anti_p}\n\\end{figure}\n\n \n\nIn Fig.~1, we plot the antiproton fraction at the Earth\nin our model (See also a similar model named ``model B'' given in\nRef.~\\cite{Fujita:2009wk}). For the background flux, we adopted the\n20$\\%$ smaller value of the mean value shown\nin~\\cite{Nezri:2009jd}.  Here, the radius of a spherical DC,\n$R_{\\rm DC}=40$~pc is adopted. The target proton density is set to be\n$n_0 = 50\\rm\\: cm^{-3}$. The spectral index $s=2.15$ \nand the maximum\nenergy, $E_{\\rm max}=100$~TeV, are assumed. We take the duration of the\n$pp$ collision to be $t_{pp}=2\\times 10^5$~yr.  The total energy of the\naccelerated protons is assumed to be\n$E_{\\rm tot,p}=2.6 \\times 10^{50}$~erg.  The distance to the front of\nthe DC is set to be $d=200$~pc. About the diffusion time of $e^-$ and\n$e^+$, $t_{\\rm diff}=2\\times 10^5$~yr is adopted. We take the magnetic\nfield outside the DC to be $B_{\\rm diff}=3\\rm\\: \\mu G$\n(See~\\cite{Fujita:2009wk} for the further details).\n\n\n\\begin{figure}[t]\n  \\begin{center}\n    \\includegraphics[width=100mm]{combC20151204.eps} \n    \\vspace{-1.3 cm}\n    \\caption{(a) Positron fraction (solid line), which includes the\n      electrons and positrons coming from the DC and background\n      electrons (dotted line, for example see\n      Refs.~\\cite{Moskalenko:1997gh,Baltz:1998xv}). Filled circles\n      correspond to the AMS-02 data~\\cite{AMS-02DATA,AMSe+e-14,AMSv1}\n      and PAMELA data~\\cite{Adriani:2008zr} (b) Total electron and\n      positron flux (solid line). The flux of the electrons and\n      positrons created only in the DC (background) is plotted by the\n      dashed (dotted) line. Observational data by AMS-02, Fermi, HESS,\n      BETS, PPB-BETS, and\n      ATIC2~\\cite{cha08,Abdo:2009zk,Aharonian:2009ah,AMSe++e-14} are\n      also plotted. The shadow region represents the uncertainty of\n      the HESS data.}\n  \\end{center}\n \\label{fig:posifra}\n\\end{figure}\n\n\n\nIn Fig.~2, we also plot the positron fraction and the total\n$e^-$+$e^+$ flux.  It is remarkable that we can automatically fit the\nobservational data of both the positron fraction and the total $e^-$ +\n$e^+$ flux by using the same set of the\nparameters~\\cite{Fujita:2009wk}. Here the cooling cutoff energy is\napproximately given by $\\varepsilon_{\\rm cut} = \\epsilon_e^2  \/ B t_{\\rm diff}\n\\sim$~1~TeV ($t_{\\rm diff} \/ 2 \\times 10^5$ yrs)$^{-1}$.\n\nThe positron fraction rises at higher energies than that of the\nantiproton fraction (Fig.~\\ref{fig:posifra}), because the spectral\nindex of the background antiproton is harder than that of the\nbackground positron.  This comes from a difference between their\ncooling processes.  Only for background positrons and electrons\nthe cooling is effective in the current situation.\n \n\nIn Fig.~\\ref{fig:posipbar}, we plot the positron to antiproton ratio\nas a function of the rigidity. Here the local components represent the\ncontribution of the nearby SNRs produced only by the $pp$ collisions.\nFrom this figure, we find that both of the positron and the\nantiproton can be consistently fitted only by adding astrophysical\nlocal contributions produced from the same $pp$ collision sources.\n\n\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=100mm]{posipbar20151204.eps} \n    \\vspace{-1.1 cm}\n    \\caption{Positron to antiproton ratio as a function of the\n      rigidity with adding the local components produced by the $pp$\n      collisions occurred at SNRs near the Earth. The thick solid line\n      represents the case of the total flux. From the upper right to\n      the lower left, we plot the flux ratios of 1) the one at the\n      source (without cooling), 2) only the local components, 3) the\n      total of the local and the background components, and 4) only\n      the background components. The observational data reported by\n      AMS-02 are also plotted. }\n  \\end{center}\n \\label{fig:posipbar}\n\\end{figure}\n\n\\section{Conclusion}\nWe have discussed the anomaly of the antiproton fraction\nrecently-reported by the AMS-02 experiment. By considering the same\norigin of the $pp$ collisions between cosmic-ray protons accelerated\nby SNRs and a dense cloud which surrounds the SNRs, we can fit the\ndata of the observed antiproton and positron simultaneously in the\nnatural model parameters. The observed fluxes of both antiprotons and\npositrons are consistent with our predictions shown in\nRef.~\\cite{Fujita:2009wk}.\n\nRegardless of the model details, the ratio of antiproton to positron\nis essentially determined by the fundamental branching fraction into\neach mode of the $pp$ collisions.  Thus the observed antiproton excess\nshould entail the positron excess, and vice versa.  This does not\ndepend on the propagation model since both antiparticles propagate in\na similar way below the cooling cutoff energy $\\sim$ TeV.\n\n\nThe cutoff energy of $e^-$ cooling marks the supernova age of\n$\\sim 10^{5}$ years~\\cite{Ioka:2008cv,Kawanaka:2009dk}, while we also\nexpect a $e^{+}$ cutoff.  The trans-TeV energy will be probed by the\nfuture CALET, DAMPE and CTA experiments\n\\cite{Kobayashi:2003kp,Kawanaka:2010uj}.  An anisotropy of the arrival\ndirection is also a unique signature, e.g., \\cite{Linden:2013mqa}. We\nmay estimate the amplitude of anisotropy as\n$\\delta_e \\sim 3d\/2ct_{\\rm diff} \\sim 0.5\\%$, which is below the upper\nlimits by Fermi observations~\\cite{Ackermann:2010}.\n\nThe boron to carbon ratio as well as the Li to carbon ratio have no\nclear excesses~\\cite{AMS-02DATA}.  This suggests that the carbon\nfraction of the excess-making cosmic rays is smaller than that of the\nordinary cosmic rays.  In general the supernovae in the DC would not\nbe the main channel of cosmic-ray production.  Most of cosmic rays \nabove $\\sim 30$ GeV may\nbe produced in chemically enriched regions, such as superbubbles, as\nimplied by the hard spectrum of cosmic-ray\nhelium~\\cite{Ohira:2010eq}. Or the carbon abundance of the destroyed\nDC might happen to be lower than the Galactic\naverage~\\cite{Fujita:2009wk}.\n\nWe should be careful about the background systematics.  In particular\nthe propagation uncertainties yield the largest\nerrors~\\cite{Yuan:2014pka,Giesen:2015ufa}.  However, in the energy\nregion above $\\sim 100$ GeV where the background contribution is\nsmall, the observed positron to antiproton ratio is very close to the\nbranching fraction of the $pp$ collisions (source components in\nFig.~\\ref{fig:posipbar}).\n This fact is free from the background choice and partially supports our model.\n\n\\section*{ACKNOWLEDGMENTS}\nThis work was supported in part by Grant-in-Aid for Scientific\nresearch from the Ministry of Education, Science, Sports, and Culture\n(MEXT), Japan, Nos. 26105520, 15H05889 (K.K.), 26247042 (K.K. and K.I.),\n26287051, 24103006, 24000004 (K.I.), 15K05080 (Y.F.), and 15K05088\n(R.Y.).  The work of K.K. and K.I. was also supported by the Center\nfor the Promotion of Integrated Science (CPIS) of Sokendai\n(1HB5804100). \n\n\n\\section*{Note added}\nWhile finalizing this manuscript, Ref.~\\cite{Kachelriess:2015oua}\nappeared which has some overlaps with this work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}