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+{"text":"\\section{Introduction}\n\\label{intro}\n\nInterest rate markets are a large and important part of global financial markets.\nThe figures published by the Bank for International Settlements (BIS) show that\ninterest rate derivatives represent more than 60\\% of the over-the-counter\nmarkets over the years, in terms of notional amount; cf. Table \\ref{BIS}. Hence,\nit is important to have models that can adequately describe the dynamics and\nmechanics of interest rates.\n\nThere is a notable difference between interest rate markets and stock or foreign\nexchange (FX) markets. While in the latter there is a single underlying to be\nmodeled, the stock price or the FX rate, in interest rate markets there is a\nwhole \\emph{family} of underlyings to be modeled, indexed by the time of maturity.\nThis poses unique challenges for researchers in mathematical finance and has led\nto some fascinating developments.\n\nThe initial approaches to interest rate modeling considered short rates or\ninstantaneous forward rates as modeling objects, and then deduced from them\ntradable rates. More recently, \\textit{effective rates}, i.e. tradable market\nrates such as the LIBOR or swap rate, were modeled directly. Models for\neffective rates consider only a discrete set of maturity dates, the so-called\n\\textit{tenor structure}, which consists of the dates when these rates are\nfixed. A review of the different approaches to modeling interest rates is beyond\nthe scope of the present article. There are many excellent books available,\nfocusing on the theoretical and practical aspects of interest rate theory. We\nrefer the reader e.g. to \\citeN{Bjoerk04}, \\citeN{MusielaRutkowski97},\n\\citeN{Filipovic09}, or \\citeN{BrigoMercurio06}.\n\nThe aim of this article is to review the construction and basic properties of\nmodels for LIBOR\\xspace rates. We consider the following popular approaches: LIBOR market models\\xspace,\nforward price models\\xspace and Markov-functional models\\xspace, as well as the recently developed class of affine LIBOR models\\xspace. In section\n\\ref{axioms} we will present and discuss some basic requirements that models\nfor LIBOR rates should satisfy. These are briefly: \\textit{positivity} of LIBOR\\xspace\nrates, \\textit{arbitrage freeness} and \\textit{analytical tractability}.\n\nThere are two natural starting points for modeling LIBOR rates: the rate itself\nand the forward price. Although they differ only by an additive and a\nmultiplicative constant, cf. \\eqref{basic}, the model dynamics are noticeably\ndifferent, depending on whether the model is based on the LIBOR or the forward\nprice. In addition, the consequences from the point of view of econometrics are\nalso significant.\n\nModeling LIBOR rates directly, leads to positive rates and arbitrage-free\ndynamics, but the model is not analytically tractable. On the other hand, models\nfor the forward price are analytically tractable, but LIBOR\\xspace rates can become\nnegative. The only models that can respect all properties simultaneously are\nMarkov-functional models\\xspace and affine LIBOR models\\xspace.\n\nThe article is organized as follows: in section \\ref{markets} we introduce some\nbasic notation for interest rates and in section \\ref{axioms} we describe the\nbasic requirements for LIBOR\\xspace models. In section \\ref{lmm} we review the\nconstruction of LIBOR market models\\xspace, describe its shortcomings and discuss some approximation\nmethods developed to overcome them. In section \\ref{fpm} we review forward price models\\xspace and in\nsection \\ref{mfm} we discuss Markov-functional models\\xspace. Finally, in section \\ref{alm} we present\naffine LIBOR models\\xspace and in section \\ref{extend} we outline the extensions of LIBOR models to\nthe multi-currency and default risk settings.\n\n\\begin{table}\n \\begin{center}\n  {\\renewcommand{\\arraystretch}{1.05}\n  \\begin{tabular}{lrrrr}\n                       & Dec 2006  & Dec 2007 & Jun 2008 & Dec 2008\\\\\n\\hline\n  Foreign exchange     &   40,271  &  56,238  &  62,983  &  49,753\\\\\n  Interest rate        &  291,581  & 393,138  & 458,304  & 418,678\\\\\n  Equity-linked        &    7,488  &   8,469  &  10,177  &   6,494\\\\\n  Commodity            &    7,115  &   8,455  &  13,229  &   4,427\\\\\n  Credit default swaps &   28,650  &  57,894  &  57,325  &  41,868\\\\\n  Unallocated          &   43,026  &  71,146  &  81,708  &  70,742\\\\\n\\hline\n  Total                &  418,131  & 595,341  & 683,726 & 591,963\\\\\n   \\end{tabular}}~\\\\[1ex]\n  \\caption{Amounts outstanding of over-the-counter (OTC) derivatives\n           by risk category and instrument (in billions of US dollars).\n           Source: BIS Quarterly Review, September 2009.}\n  \\label{BIS}\n  \\end{center}\n\\end{table}\n\n\n\n\\section{Interest rate markets -- notation}\n\\label{markets}\n\nLet us consider a discrete tenor structure $0=T_0<T_1<\\dots<T_N$, with constant\ntenor length $\\delta=\\tk[k+1]-T_k$. The following notation is introduced for\nconvenience; $K:=\\{1,\\dots,N-1\\}$ and $\\overline{K}:=\\{1,\\dots,N\\}$. Let us denote\nby $B(t,T)$ the time-$t$ price of a \\textit{zero coupon bond} maturing at time\n$T$; by $L(t,T)$ the time-$t$ \\textit{forward LIBOR\\xspace rate} settled at time $T$\nand received at time $T+\\delta$; and by $F(t,T,U)$ the time-$t$ \\textit{forward\nprice} associated to the dates $T$ and $U$. These fundamental quantities are\nrelated by the following basic equation:\n\\begin{align}\\label{basic}\n  1+\\delta L(t,T) = \\frac{B(t,T)}{B(t,T+\\delta)} = F(t,T,T+\\delta).\n\\end{align}\n\nThroughout this work, $\\mathscr{B}=(\\Omega,\\ensuremath{\\mathcal{F}},\\mathbf{F},\\ensuremath{\\mathrm{I\\kern-.2em P}})$ denotes a complete\nstochastic basis, where $\\ensuremath{\\mathcal{F}}=\\ensuremath{\\mathcal{F}}_T$, $\\mathbf{F}=(\\ensuremath{\\mathcal{F}}_t)_{t\\in[0,T]}$, and\n$T_N\\le T<\\infty$. We denote by $\\mathcal{M}(\\ensuremath{\\mathrm{I\\kern-.2em P}})$ the class of martingales on\n$\\mathscr{B}$ with respect to the measure $\\ensuremath{\\mathrm{I\\kern-.2em P}}$.\n\nWe associate to each date $T_k$ in the tenor structure a \\emph{forward martingale\nmeasure}, denoted by $\\pt[k]$, $k\\in\\overline{K}$. By the definition of forward measures,\ncf. \\citeN[Def. 14.1.1]{MusielaRutkowski97}, the bond price with maturity $T_k$\nis the numeraire for the forward measure $\\pt[k]$. Thus, we have that forward\nmeasures are related to each other via\n\\begin{align}\\label{Pk-to-next}\n\\frac{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k]}}{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k+1]}} \\Big|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\frac{F(t,\\tk[k],\\tk[k+1])}{F(0,\\tk[k],\\tk[k+1])}\n  = \\frac{B(0,\\tk[k+1])}{B(0,\\tk[k])}\n    \\times \\frac{B(t,\\tk[k])}{B(t,\\tk[k+1])},\n\\end{align}\nwhile they are related to the terminal forward measure via\n\\begin{align}\\label{Pk-to-final}\n\\frac{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k]}}{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[N]}} \\Big|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\frac{F(t,\\tk[k],\\tk[N])}{F(0,\\tk[k],\\tk[N])}\n  = \\frac{B(0,\\tk[N])}{B(0,\\tk[k])}\n    \\times \\frac{B(t,\\tk[k])}{B(t,\\tk[N])}.\n\\end{align}\nAll forward measures are assumed to be equivalent to the measure $\\ensuremath{\\mathrm{I\\kern-.2em P}}$.\n\n\n\n\\section{Axioms for LIBOR\\xspace models}\n\\label{axioms}\n\nIn this section, we present and discuss certain requirements that a model for\nLIBOR\\xspace rates should satisfy. These requirements are motivated both by the economic\nand financial aspects of LIBOR\\xspace rates, as well as by the practical demands\nfor implementing and using a model in practice. The aim here is to unify the\nline of thought in \\shortciteN{HuntKennedyPelsser00} and in \n\\shortciteANP{KellerResselPapapantoleonTeichmann09}\n\\citeyear{KellerResselPapapantoleonTeichmann09}.\n\nA model for LIBOR\\xspace rates should satisfy the following \\emph{requirements}:\n\\begin{itemize}\n\\item[(\\textbf{A1})] LIBOR\\xspace rates should be \\emph{non-negative}:\n            $L(t,T)\\ge0$ for all $t\\in[0,T]$.\n\\item[(\\textbf{A2})] The model should be \\emph{arbitrage-free}:\n            $L(\\cdot,T)\\in\\mathcal{M}(\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T+\\delta})$.\n\\item[(\\textbf{A3})] The model should be \\emph{analytically tractable},\n            easy to implement and quick to calibrate to market data.\n\\item[(\\textbf{A4})] The model should provide a \\emph{good calibration} to\n            market data of liquid derivatives, i.e. caps and swaptions.\n\\end{itemize}\n\nRequirements (A1) and (A2) are logical conditions originating from economics and\nmathematical finance. Below, we briefly elaborate on (A3) and (A4); they stem\nfrom practical demands and are more difficult to quantify precisely. In order to\nclarify their difference, we point out that e.g. the Black--Scholes model\nobviously satisfies (A3) but not (A4).\n\nRequirement (A3) means that we can price liquid derivatives, e.g. caps and\nswaptions in ``closed form'' in the model, so that the model can be calibrated\nto market data in a fast and easy manner. Ideally, of course we would like to\nbe able to price as many derivatives as possible in closed form. Here, ``closed\nform'' is understood in a broad sense meaning e.g. Fourier transform methods;\nreally closed form solutions \\'a la Black--Scholes are typically hard to achieve.\n\\shortciteN{HuntKennedyPelsser00} say that the model is analytically tractable\nif it is driven by a \\emph{low-dimensional} Markov process. In \\shortciteN{KellerResselPapapantoleonTeichmann09}, as well as\nin the present article, we say that a model is analytically tractable if the\nstructure of the driving process is \\emph{preserved} under the different forward\nmeasures.\n\nFinally, requirement (A4) means that the model is able to describe the\nobserved data accurately, without overfitting them. We will not examine this\nrequirement further in this article. On an intuitive level, since the models we\nwill describe in the sequel are driven by general Markov processes or general\nsemimartingales, we can always find a driving process that provides a good\ncalibration to market data. However, an empirical analysis should be performed\nin order to identify such a driving process (cf. e.g. \\shortciteNP{JarrowLiZhao07}\nand \\citeNP[Ch. III]{Skovmand08}).\n\n\n\n\\section{LIBOR market models}\n\\label{lmm}\n\nLIBOR market models\\xspace were introduced in the seminal papers of\nMiltersen et al. \\citeyear{MiltersenSandmannSondermann97},\n\\shortciteN{BraceGatarekMusiela97}\nand \\citeN{Jamshidian97}. In this framework, LIBOR\\xspace rates are modeled as the exponential\nof a Brownian motion under their corresponding forward measure, hence they are\nlog-normally distributed. This is the so-called \\emph{log-normal LIBOR market model}.\nCaplets are then priced by Black's formula (cf. \\citeNP{Black76}), which is in\naccordance with standard market practice. Later, LIBOR market models\\xspace were extended to\naccommodate more general driving processes such as L\\'{e}vy\\xspace processes, stochastic volatility\nprocesses and general semimartingales, in order to describe more accurately the market\ndata; cf. \\citeN{Jamshidian99}, Glasserman and Kou \\citeyear{GlassermanKou03},\n\\citeN{EberleinOezkan05}, \\citeANP{AndersenBrothertonRatcliffe05}\n\\citeyear{AndersenBrothertonRatcliffe05}, \\citeN{BelomestnySchoenmakers06} and\n\\shortciteN{BelomestnyMathewSchoenmakers06}, to mention just a fraction of the existing\nliterature.\n\nConsider an initial tenor structure of non-negative LIBOR\\xspace rates $L(0,T_k)$, $k\\in\\overline{K}$,\nand let $\\lambda(\\cdot,T_k):[0,T_k]\\to\\ensuremath{\\mathbb{R}}$ denote the volatility of the forward LIBOR\\xspace\nrate $L(\\cdot,T_k)$, $k\\in K$; the volatilities are assumed deterministic, for\nsimplicity. Let $H$ denote a semimartingale on $(\\Omega,\\ensuremath{\\mathcal{F}},\\mathbf{F},\\pt)$ with triplet\nof semimartingale characteristics $\\mathbb{T}(H|\\pt)=(B,C,\\nu)$ and $H_0=0$ a.s.;\n$H$ satisfies certain integrability assumptions which are suppressed for brevity\n(e.g. finite exponential moments, absolutely continuous characteristics). The\nprocess $H$ is driving the dynamics of LIBOR\\xspace rates and is chosen to have a\n\\emph{tractable} structure under $\\pt$ (e.g. $H$ is a L\\'{e}vy\\xspace or an affine process).\n\nIn LIBOR market models\\xspace, forward LIBOR\\xspace rates are modeled as follows:\n\\begin{align}\\label{LMM-1}\nL(t,T_k) &= L(0,T_k)\n \\exp\\left( \\int_0^t \\beta(s,T_k)\\ud s + \\int_0^t \\lambda(s,T_k)\\ensuremath{\\mathrm{d}} H_s \\right),\n\\end{align}\nwhere $\\beta(\\cdot,T_k)$ is the drift term that makes\n$L(\\cdot,T_k)\\in\\mathcal{M}(\\pt[k+1])$, for all $k\\in K$. Therefore, the model\nclearly satisfies requirements (A1) and (A2).\n\nNow, using Theorem 2.18 in \\citeN{KallsenShiryaev02}, we have that\n\\begin{align}\\label{LMM-2}\n\\beta(s,T_k)\n &= -\\lambda(s,T_k)b_s^{\\tk[k+1]}\n    -\\frac12\\lambda^2(s,T_k)c_s \\nonumber\\\\\n &\\quad\n    -\\int_{\\ensuremath{\\mathbb{R}}}\\big( \\mathrm{e}^{\\lambda(s,T_k)x}\n                     - 1 - \\lambda(s,T_k)x \\big) F_s^{\\tk[k+1]}(\\ud x),\n\\end{align}\nsuch that indeed $L(\\cdot,T_k)\\in\\mathcal{M}(\\pt[k+1])$. Here\n$(b_s^{\\tk[k+1]}, c_s, F_s^{\\tk[k+1]})$ denote the differential characteristics\nof $H$ under $\\pt[k+1]$.\nTherefore, in order to completely understand the dynamics of the model we have\nto calculate the characteristics $(b_s^{\\tk[k+1]}, c_s, F_s^{\\tk[k+1]})$.\n\nThese characteristics follow readily from Girsanov's theorem for semimartingales\n(cf. \\citeNP[III.3.24]{JacodShiryaev03}) once we have the density between the\nmeasure changes at hand. It is convenient to express this density as a stochastic\nexponential. Keeping \\eqref{Pk-to-final} in mind, and denoting \\eqref{LMM-1} as\nfollows\n\\begin{align}\\label{LMM-3}\n\\ensuremath{\\mathrm{d}} L(t,T_k) = L(t-,T_k)\\ensuremath{\\mathrm{d}} \\widetilde{H}_t^k,\n\\end{align}\ni.e. $\\widetilde{H}^k$ is the exponential transform of the exponent in\n\\eqref{LMM-1}, we get from \\eqref{basic} that\n\\begin{align}\\label{LIBOR-3}\n\\ensuremath{\\mathrm{d}} F(t,T_k,T_{k+1})\n &= \\delta\\, \\ensuremath{\\mathrm{d}} L(t,T_k) \\nonumber\\\\\n &= \\delta L(t-,T_k)\\ensuremath{\\mathrm{d}} \\widetilde{H}_t^k \\nonumber\\\\\n &= F(t-,T_k,T_{k+1}) \\frac{\\delta L(t-,T_k)}{1+\\delta L(t-,T_k)}\n    \\ensuremath{\\mathrm{d}} \\widetilde{H}_t^k \\nonumber\\\\\n \\Rightarrow\nF(t,T_k,\\tk[k+1])\n &= F(0,T_k,\\tk[k+1])\\,\n    \\mathcal{E}\\!\\left( \\int_0^\\cdot\\frac{\\delta L(s-,T_k)}{1+\\delta L(s-,T_k)}\n                                  \\ensuremath{\\mathrm{d}} \\widetilde{H}_s^k \\right)_{\\!t}.\n\\end{align}\nTherefore, the density between the measure changes takes the form\n\\begin{align}\\label{LMM-4}\n\\frac{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k+1]}}{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[N]}} \\Big|_{\\ensuremath{\\mathcal{F}}_t}\n  = \\frac{B(0,\\tk[N])}{B(0,\\tk[k+1])}\n    \\times \\prod_{l=k+1}^{N-1}\n     \\mathcal{E}\\!\\left( \\int_0^\\cdot\n       \\frac{\\delta L(s-,T_l)}{1+\\delta L(s-,T_l)}\\ensuremath{\\mathrm{d}} \\widetilde{H}_s^l \\right)_{\\!t}.\n\\end{align}\n\nThis calculation reveals the problem of LIBOR market models\\xspace: the density process between the\nmeasure changes -- and thus the characteristics of $H$ under the forward measures\n-- does not depend \\emph{only} on the dynamics of $\\widetilde{H}^k$, or\nequivalently on the dynamics of $\\int\\lambda(s,T_k)\\ensuremath{\\mathrm{d}} H_s$, as is the case in e.g.\nHJM models. It also crucially depends on \\emph{all subsequent} LIBOR\\xspace rates, as the\nproduct and the terms $\\frac{\\delta L(\\cdot,T_l)}{1+\\delta L(\\cdot,T_l)}$ in\n\\eqref{LMM-4} clearly indicate. This means, in particular, that the structure of\nthe model is \\emph{not preserved} under the different forward measures; e.g. if\n$H$ is a L\\'{e}vy\\xspace or an affine process under the terminal measure $\\pt$, then $H$ is\nneither a L\\'{e}vy\\xspace nor an affine process under any other forward measure -- not even\na time-inhomogeneous version of those. Therefore, LIBOR market models\\xspace do not satisfy\nrequirement (A3).\n\nThe semimartingale $H$, that drives the dynamics of LIBOR rates, has the following\ncanonical decomposition under the terminal martingale measure $\\pt$\n\\begin{align}\\label{LMM-4.5}\n H_t = B_t + \\int_0^t \\sqrt{c_s}\\ensuremath{\\mathrm{d}} W_s + \\int_0^t\\int_\\ensuremath{\\mathbb{R}} x(\\mu^H-\\nu)\\ensuremath{(\\ud s, \\ud x)},\n\\end{align}\n(cf. \\citeNP[II.2.38]{JacodShiryaev03} and \\citeNP[Theorem 3.4.2]{KaratzasShreve91})\nwhere $W$ denotes the $\\pt$-Brownian motion and $\\mu^H$ denotes the random measure\nof the jumps of $H$. The $\\pt$-compensator of $\\mu^H$ is $\\nu$ and\n$C=\\int_0^\\cdot c_s\\ud s$. Straightforward calculations using the density in\n\\eqref{LMM-4} \\linebreak (cf. e.g. \\citeNP{Kluge05} or \\citeNP{PapapantoleonSiopacha09})\nyield that the $\\pt[k+1]$-Brownian motion $W^{T_{k+1}}$ is related to the\n$\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_N}$-Brownian motion via\n\\begin{align}\\label{LMM-5}\nW_t^{T_{k+1}}\n & = W_t - \\int_0^t \\left(\\sum_{l=k+1}^{N-1}\n            \\frac{\\delta L(t-,T_l)}\n                 {1+\\delta L(t-,T_l)}\\lambda(t,T_l)\\right)\\sqrt{c_s} \\ensuremath{\\mathrm{d}} s,\n\\end{align}\nwhile the $\\pt[k+1]$-compensator of $\\mu^H$, $\\nu^{T_{k+1}}$, is related to the\n$\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_N}$-compensator of $\\mu^H$ via\n\\begin{align}\\label{LMM-6}\n\\nu^{T_{k+1}}\\ensuremath{(\\ud s, \\ud x)}\n &= \\left(\\prod_{l=k+1}^{N-1}\\gamma(s,x,T_l)\\right)\\nu\\ensuremath{(\\ud s, \\ud x)},\n\\end{align}\nwhere\n\\begin{align}\\label{LMM-7}\n\\gamma(s,x,T_l,)\n = \\frac{\\delta L(s-,T_l)}{1+\\delta L(s-,T_l)}\\Big(\\mathrm{e}^{\\lambda(s,T_l)x}-1\\Big) +1.\n\\end{align}\nIn addition, the drift term of the LIBOR\\xspace rate $L(\\cdot,T_k)$ relative to the\n$\\pt$ differential characteristics of $H$, i.e. $(b,c,F)$, is\n\\begin{align}\\label{LMM-8}\n\\widehat{\\beta}(s,\\ts[k])\n & = -\\frac12 \\lambda^2(s,T_k) c_s\n     - c_s \\lambda(s,T_k)\n        \\sum_{l=k+1}^{N-1}\\frac{\\delta L(s-,T_l)}{1+\\delta L(s-,T_l)}\\lambda(s,T_l)\n  \\nonumber\\\\ &\\,\\,\\,\\,\n     - \\int_{\\ensuremath{\\mathbb{R}}}\\left(\\Big(\\mathrm{e}^{\\lambda(s,T_k) x}-1\\Big)\n          \\prod_{l=k+1}^{N-1}\\gamma(s,x,T_l) - \\lambda(s,T_k) x\\right)F_s(\\ensuremath{\\mathrm{d}} x).\n\\end{align}\n\n\nThe consequences of the intractability of LIBOR market models\\xspace are the following. When the\ndriving process is a \\emph{continuous} semimartingale, then\n\\begin{itemize}\n\\item caplets can be priced in closed form;\n\\item swaptions and other multi-LIBOR products cannot be priced in closed form;\n\\item Monte-Carlo simulations are particularly time consuming, since one is\n      dealing with coupled high dimensional stochastic processes.\n\\end{itemize}\nWhen the driving process is a \\emph{general} semimartingale, then\n\\begin{itemize}\n\\item even caplets cannot be priced in closed form, let alone swaptions\n      or other multi-LIBOR\\xspace derivatives;\n\\item the Monte-Carlo simulations are equally time consuming.\n\\end{itemize}\n\nSeveral approximation methods have been developed in the literature in order to\novercome these problems. We briefly review three of the proposed methods below;\nfor other methods and empirical comparison we refer the interested reader to\nthe review paper by \\citeN{JoshiStacey08}.\n\n\n\\subsection{``Frozen drift'' approximation}\nThe first and easiest solution to the problem is the so-called ``frozen drift''\napproximation, where one replaces the random terms in \\eqref{LMM-8} or\n\\eqref{LMM-4} by their deterministic initial values, i.e.\n\\begin{align}\\label{LMM-FD1}\n\\frac{\\delta L(s-,T_l)}{1+\\delta L(s-,T_l)}\n \\approx \\frac{\\delta L(0,T_l)}{1+\\delta L(0,T_l)}.\n\\end{align}\nThis approximation was first proposed by \\shortciteN{BraceGatarekMusiela97}, and\nhas been used by many authors ever since. Under this approximation, the\nmeasure change becomes a structure preserving one -- observe that the density in\n\\eqref{LMM-4} depends now only on the driving process and the volatility\nstructure -- and the resulting \\emph{approximate} LIBOR market model\\xspace is analytically tractable;\ne.g. caplets and swaptions can now be priced in closed form even in models\ndriven by semimartingales with jumps.\n\nHowever, this method yields poor empirical results. Comparing the prices of\neither liquid options, or long-dated options, using the frozen drift approximation\nto the prices obtained by the simulation of the actual dynamics for the LIBOR rates,\nwe can observe notable differences both in terms of prices and in terms of\nimplied volatilities. See Figure \\ref{diffs} for an example. We refer to\n\\shortciteN{KurbanmuradovSabelfeldSchoenmakers}, \\citeN{SiopachaTeichmann07},\nand \\citeN{PapapantoleonSiopacha09} for further numerical illustrations.\n\n\\begin{figure}\n \\centering\n  \\includegraphics[width=177pt]{IV-SDE-FD}\n  \\includegraphics[width=177pt]{IV-SDE-ST}\n \\caption{Difference in implied volatilities between the actual LIBOR and the\n          frozen drift prices (left), and between the actual LIBOR and the\n          Taylor approximation prices (right). Source: Papapantoleon\n          and Siopacha (2009).}\n \\label{diffs}\n\\end{figure}\n\n\n\\subsection{Log-normal approximations}\nThe following approximation schemes for the log-normal LIBOR market model\\xspace were developed by\nKurbanmuradov et al.\n\\citeyear{KurbanmuradovSabelfeldSchoenmakers}. Consider the log-normal LIBOR market model\\xspace\ndriven by a one-dimensional Brownian motion, for simplicity. The dynamics of\nLIBOR rates (expressed under the terminal measure) take the form\n\\begin{align}\\label{LMM-LG-1}\n\\ensuremath{\\mathrm{d}} L(t,T_k)\n = L(t,T_k)\\big( \\lambda_t(T_k)\\ensuremath{\\mathrm{d}} W_t + \\beta_t(T_k)\\ud t \\big),\n\\end{align}\nwhere the drift term equals\n\\begin{align}\\label{LMM-LG-1+1\/2}\n\\beta_t(T_k)\n &= -\\lambda_t(T_k) \\sum_{l=k+1}^{N-1}\n     \\frac{\\delta L(t-,T_l)}{1+\\delta L(t-,T_l)}\\lambda_t(T_l),\n\\end{align}\ncf. \\eqref{LMM-8}; w.l.o.g. we can set $c\\equiv1$. A very crude log-normal\napproximation is to ``neglect'' the non-Gaussian terms in the SDE, i.e. to set\n$\\beta_t(\\ts[k])\\equiv0$. Of course, this approximation does not yield\nsatisfactory results -- in principle, results are even worse than the frozen\ndrift approximation.\n\nOne can develop more refined log-normal approximations as follows: let\n$f(x)=\\frac{\\delta x}{1+\\delta x}$, and define the process $Z$, which equals the\nterms that need to be approximated; i.e.\n\\begin{align}\\label{LMM-LG-2}\nZ(t,T_k)\n = \\frac{\\delta L(t,T_k) }{1+\\delta L(t,T_k)}\n =f\\big( L(t,T_k) \\big).\n\\end{align}\nApplying It\\^o's formula, we derive the SDE that $Z(\\cdot,T_k)$ satisfies\n\\begin{align}\\label{LMM-LG-3}\n\\ensuremath{\\mathrm{d}} Z(t,T_k)\n &= f'(L(t,T_k))L(t,T_k)\\lambda_t(T_k)\\ensuremath{\\mathrm{d}} W_t \\nonumber\\\\\n &\\,\\,+ \\big\\{f'(L(t,T_k))L(t,T_k)\\beta_t(T_k)\n  + \\frac12 f''(L(t,T_k))L^2(t,T_k)\\lambda^2_t(T_k)\\big\\}\\ud t \\nonumber\\\\\n &=: A_k(t,Z)\\ud t + B_k(t,Z)\\ensuremath{\\mathrm{d}} W_t,\n\\end{align}\nwith the initial condition $Z(0,T_k)=f(L(0,T_k))$. Note that the coefficients\n$A_k$ and $B_k$ can be calculated explicitly, by solving \\eqref{LMM-LG-2} for\n$L$ and substituting into \\eqref{LMM-LG-3}. Moreover, $Z$ in $A_k$ and $B_k$ denotes\nthe dependence on the whole vector $Z=[Z(\\cdot,T_1),\\dots,Z(\\cdot,T_N)]$. The\nfirst and second \\emph{Picard iterations} for the solution of this SDE are\n\\begin{align}\\label{LMM-LG-4}\nZ^0(t,T_k)\n &= Z(0,T_k)\n  = \\frac{\\delta L(0,T_k) }{1+\\delta L(0,T_k)},\n\\end{align}\nand\n\\begin{align}\\label{LMM-LG-5}\nZ^1(t,T_k)\n &= Z(0,T_k) + \\int_0^t A_k(s,Z^0)\\ud s + \\int_0^t B_k(s,Z^0)\\ensuremath{\\mathrm{d}} W_s.\n\\end{align}\nNote that $Z^0$ is constant, while $Z^1(t,T_k)$ has a Gaussian distribution\nsince the coefficients $A_k(\\cdot,Z^0)$ and $B_k(\\cdot,Z^0)$ are deterministic.\n\nNow, replacing the random terms $Z(\\cdot,T_k)$ in $\\beta(T_k)$ with the Picard iterates\n$Z^0(\\cdot,T_k)$ and $Z^1(\\cdot,T_k)$ leads to two different \\emph{log-normal\napproximations} to the dynamics of LIBOR\\xspace rates. Obviously, the approximation by\n$Z^0$ is nothing else than the frozen drift approximation. The approximation by\n$Z^1$ is again log-normal, cf. \\eqref{LMM-LG-1+1\/2} and \\eqref{LMM-LG-2}, and\nyields very good empirical results. This latter approximation has the advantage\nthat the law of the approximate LIBOR\\xspace rate is known at any time $t$, hence the\ntime-consuming Monte Carlo simulations can be avoided. For the empirical and\nnumerical analysis of these approximations we refer to\n\\shortciteN{KurbanmuradovSabelfeldSchoenmakers} and \\citeN[Ch. 6]{Schoenmakers05}.\n\n\n\\subsection{Strong Taylor approximation}\nAnother approximation method has been recently developed by\n\\citeN{SiopachaTeichmann07} and Papapantoleon and Siopacha\n\\citeyear{PapapantoleonSiopacha09}. The main idea\nbehind this method is to replace the random terms in the drift \\eqref{LMM-8} by\ntheir first-order strong Taylor approximations. The Taylor approximation is developed\nby a perturbation of the SDE for the LIBOR\\xspace rates and a subsequent Taylor expansion.\n\nLet us denote the log-LIBOR\\xspace rates by $G(\\cdot,T_k):=\\log L(\\cdot,T_k)$. Then, they\nsatisfy the linear SDE (under the terminal measure)\n\\begin{align}\\label{LMM-ST-1}\n\\ensuremath{\\mathrm{d}} G(t,T_k) = \\widehat{\\beta}(t,T_k)\\ud t + \\lambda(t,T_k)\\ensuremath{\\mathrm{d}} H_t,\n\\end{align}\nwith initial condition $G(0,T_k)=\\log L(0,T_k)$; cf. \\eqref{LMM-1} and \\eqref{LMM-8}.\nWe perturb this SDE by a real parameter $\\epsilon$, i.e.\n\\begin{align}\\label{LMM-ST-2}\n\\ensuremath{\\mathrm{d}} G^\\epsilon(t,T_k) = \\epsilon\\big(\\widehat{\\beta}^\\epsilon(t,T_k)\\ud t + \\lambda(t,T_k)\\ensuremath{\\mathrm{d}} H_t\\big),\n\\end{align}\nwhere $G^\\epsilon(0,T_k)=G(0,T_k)$ for all $\\epsilon$. The superscript $\\epsilon$ in the\ndrift term $\\widehat{\\beta}^\\epsilon(\\cdot,T_k)$ is a reminder that this term is\nalso perturbed by $\\epsilon$, since it contains all subsequent LIBOR\\xspace rates; see\n\\eqref{LMM-8} again. Now, the \\emph{first order strong Taylor approximation}\nof $G^\\epsilon$, denoted by $\\mathbf{T}G^\\epsilon$, is\n\\begin{align}\\label{LMM-ST-3}\n\\mathbf{T}G^\\epsilon(t,T_k)\n = G(0,T_k)\n  + \\epsilon \\frac{\\partial}{\\partial\\epsilon}\\big|_{\\epsilon=0} G^\\epsilon(t,T_k).\n\\end{align}\nWe denote the ``first variation'' process\n$\\frac{\\partial}{\\partial\\epsilon}\\big|_{\\epsilon=0} G^\\epsilon(\\cdot,T_k)$ by $Y(\\cdot,T_k)$,\nand then we can deduce, after some calculations, that it has the decomposition\n\\begin{align}\\label{LMM-ST-4}\nY(t,T_k)\n &= \\int_0^t \\widehat{\\beta}^0(s,T_k)\\ud s + \\int_0^t \\lambda(s,T_k)\\ensuremath{\\mathrm{d}} H_s,\n\\end{align}\nwhere $\\widehat{\\beta}^0(s,T_k):=\\widehat{\\beta}^\\epsilon(s,T_k)|_{\\epsilon=0}$. Hence,\nthis is a \\emph{deterministic} drift term, obtained by replacing the random\nterms in \\eqref{LMM-8} by their deterministic initial values. In particular, we\ncan easily deduce from \\eqref{LMM-ST-4} that, for example, if $H$ is a L\\'{e}vy\\xspace\nprocess then $Y(\\cdot,T_k)$ is a \\emph{time-inhomogeneous} L\\'{e}vy\\xspace process.\n\nConcluding, we have developed the following approximation scheme:\n\\begin{align}\\label{LMM-ST-5}\n\\log L(t,T_k) \\approx \\log L(0,T_k) + Y(t,T_k),\n\\end{align}\nwhere $Y(\\cdot,T_k)$ has the decomposition \\eqref{LMM-ST-4}; compare with\n\\eqref{LMM-1}.\n\nThe advantage of this method is threefold: (a) it is universal, and can be applied\nto LIBOR\\xspace models with stochastic volatility and\/or jumps, (b) it is faster and easier\nto simulate than the actual SDE for the LIBOR\\xspace rates, and (c) the empirical performance\nis very satisfactory; cf. Figure \\ref{diffs} and the aforementioned articles for\nfurther numerical illustrations. The drawback is that it is based on Monte Carlo\nsimulations, hence computational times can become long.\n\n\n\n\\section{Forward price models}\n\\label{fpm}\n\nForward price models were developed by \\citeN{EberleinOezkan05}, and further\ninvestigated by \\citeN{Kluge05}; see also \\citeN{EberleinKluge06}.\\!\nWe consider a setting similar to LIBOR market models\\xspace, i.e. an\ninitial tenor structure of non-negative LIBOR\\xspace rates, $\\lambda(\\cdot,T_k)$\ndenotes the volatility of the forward LIBOR\\xspace rate $L(\\cdot,T_k)$, and $H$ denotes\na semimartingale on $(\\Omega,\\ensuremath{\\mathcal{F}},\\mathbf{F},\\pt)$ with triplet of characteristics\n$(B,C,\\nu)$; again some assumptions on $H$ are suppressed. The process $H$ is\ndriving the dynamics of LIBOR\\xspace rates and has a \\emph{tractable} structure under\n$\\pt$ (e.g. $H$ is a L\\'{e}vy\\xspace or an affine process).\n\nInstead of modeling the forward LIBOR\\xspace rate directly, one now models the forward\nprice in a similar fashion; that is\n\\begin{align}\\label{fp-model-1}\n1\\!+\\!\\delta L(t,T_k)\n &= (1\\!+\\!\\delta L(0,T_k))\n    \\exp\\left( \\int_0^t \\beta(s,T_k)\\ud s + \\int_0^t\\lambda(s,T_k)\\ensuremath{\\mathrm{d}} H_s \\right),\n\\end{align}\nwhere again the drift term is such that $L(\\cdot,T_k)\\in\\mathcal{M}(\\pt[k+1])$,\nfor all $k\\in K$; i.e. $\\beta(\\cdot,T_k)$ has similar form to \\eqref{LMM-2}.\nTherefore the model obviously satisfies requirement (A2).\n\nNow, from \\eqref{Pk-to-final} and \\eqref{fp-model-1}, we get that the density\nbetween the forward measures is\n\\begin{align}\\label{fp-model-2}\n\\frac{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k+1]}}{\\dd \\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[N]}} \\Big|_{\\ensuremath{\\mathcal{F}}_t}\n &= \\frac{B(0,\\tk[N])}{B(0,\\tk[k+1])}\n    \\times \\prod_{l=k+1}^{N-1}\\big(1+\\delta L(t,T_l)\\big)\\\\\n &= \\frac{B(0,\\tk[N])}{B(0,\\tk[k+1])}\n    \\times \\exp\\left( \\int_0^t\\!\\sum_{l=k+1}^{N-1} \\beta(s,T_l)\\ud s +\n     \\int_0^t\\!\\sum_{l=k+1}^{N-1} \\lambda(s,T_l)\\ensuremath{\\mathrm{d}} H_s \\right). \\nonumber\n\\end{align}\nObserve that this density only depends on the driving process $H$ and the volatility\nstructures, hence we can deduce that the measure changes between forward\nmeasures are \\emph{Esscher transformations}; cf. \\citeANP{KallsenShiryaev02} \n\\linebreak\\citeyear{KallsenShiryaev02} for\nthe Esscher transform. Analogously to eqs.\n\\eqref{LMM-4.5}--\\eqref{LMM-6}, we have now that the $\\pt[k+1]$-Brownian motion\nis related to the $\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_N}$-Brownian motion via\n\\begin{align}\\label{fp-model-3}\nW_t^{T_{k+1}}\n & = W_t\n   - \\int_0^t \\left(\\sum_{l=k+1}^{N-1}\\lambda(t,T_l)\\right)\\sqrt{c_s} \\ensuremath{\\mathrm{d}} s,\n\\end{align}\nwhile the $\\pt[k+1]$-compensator of $\\mu^H$, is related to the\n$\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_N}$-compensator of $\\mu^H$ via\n\\begin{align}\\label{fp-model-4}\n\\nu^{T_{k+1}}\\ensuremath{(\\ud s, \\ud x)}\n &= \\exp\\left(x\\sum_{l=k+1}^{N-1}\\lambda(s,T_l)\\right)\\nu\\ensuremath{(\\ud s, \\ud x)}.\n\\end{align}\nThus the structure of the driving process is preserved. For example, if $H$\nis a L\\'{e}vy\\xspace or an affine process under $\\pt$, then it becomes a\n\\emph{time-inhomogeneous} L\\'{e}vy\\xspace or affine process respectively under any forward\nmeasure $\\pt[k+1]$. This implies that requirement (A3) is satisfied, so that\ncaplets and swaptions can be priced in closed form. In this class of models, we\nhave the additional benefit that we can even price some exotic path-dependent\noptions easily using Fourier transform methods; see \\citeN{KlugePapapantoleon06}\nfor an example.\n\nThe main shortcoming of forward price models\\xspace is that \\emph{negative} LIBOR\\xspace rates can occur,\nsimilarly to HJM models, since\n\\begin{align*}\n1+\\delta L(t,T_k)>0 \\,\\,\\nRightarrow\\,\\, L(t,T_k)>0\n\\qquad\\text{for all }\\, t\\in[0,T_k].\n\\end{align*}\nTherefore, this model can violate requirement (A1).\n\n\n\n\\section{Markov-functional models}\n\\label{mfm}\n\nMarkov-functional models\\xspace were introduced in the seminal paper of Hunt, Kennedy, and Pelsser\n\\citeyear{HuntKennedyPelsser00}. In\ncontrast to the other approaches described in this review, the aim of Markov-functional models\\xspace is\nnot to model some fundamental quantity, e.g. LIBOR\\xspace or swap rates, directly. Instead,\nMarkov-functional models\\xspace are constructed by inferring the model dynamics, as well as their functional\nforms, through matching the model prices to the market prices of certain liquid\nderivatives. That is, they are \\emph{implied interest rate models}, and should be\nthought of in a fashion similar to local volatility models and implied trees in\nequity markets.\n\nThe main idea behind Markov-functional models\\xspace is that bond prices and the numeraire are, at any point\nin time, a function of a \\emph{low-dimensional} Markov process under some martingale\nmeasure. The functional form for the bond prices is selected such that the model\naccurately calibrates to the relevant market prices, while the freedom to choose the\nMarkov process makes the model realistic and tractable. Moreover, the functional form\nfor the numeraire can be used to reproduce the marginal laws of swap rates or other\nrelevant instruments for the calibration.\n\nMore specifically, let $(M,\\mathbf{M})$ denote a numeraire pair, and consider a\n(time-inhomogeneous) Markov process \\prozess[X] under the measure $\\mathbf{M}$.\nIn the framework of Markov-functional models, one assumes that bond prices have\nthe functional form\n\\begin{align}\\label{MFM-1}\nB(t,S) = \\mathrm{B}(t,S;X_t),\n \\quad 0\\le t\\le \\partial_S\\le S,\n\\end{align}\nwhere $\\partial_S$ denotes some ``boundary curve''. In applications, the boundary\ncurve typically  has the form\n\\begin{align}\\label{MFM-2}\n\\partial_S\n = \\left\\{\n     \\begin{array}{ll}\n       S,   & S\\le T_{*}, \\\\\n       T_*, & S>T_{*},\n     \\end{array}\n   \\right.\n\\end{align}\nwhere $T_*$ is a common time of maturity. One further assumes that the numeraire\n$M$ is also a function of the driving Markov process $X$, i.e.\n\\begin{align}\\label{MFM-3}\nM_t = \\mathrm{M}(t;X_t),\n \\quad 0\\le t\\le T.\n\\end{align}\nTherefore, in order to specify a Markov-functional model\\xspace, the following quantities are required:\n\\begin{itemize\n\\item[(P1)] the law of $X$ under the measure $\\mathbf{M}$;\n\\item[(P2)] the functional form $\\mathrm{B}(\\partial_S,S;\\cdot)$ for $S\\in[0,T]$;\n\\item[(P3)] the functional form $\\mathrm{M}(t;\\cdot)$ for $0\\le t\\le T$.\n\\end{itemize}\n\nIn applications, the Markov process is specified first and is typically a diffusion\nprocess with time-dependent volatility. Then, the functional forms for the bond\nprices and the numeraire are implied by calibrating the model to market prices of\nliquid options. The choice of the calibrating instruments depends on the exotic\nderivative that should be priced or hedged with the model. If the exotic depends\non LIBOR rates, e.g. the flexible cap, then the model is calibrated to digital caplets,\nwhich leads to the \\emph{Markov-functional LIBOR model}. If the exotic depends on\nswap rates, e.g. the Bermudan swaption, then the model is calibrated to digital swaptions,\nwhich leads to the \\emph{Markov-functional swap rate model}. Let us point out that the\nfunctional forms are typically not known in closed form, and should be computed\nnumerically. These models typically satisfy requirements (A1), (A2) and (A3).\nFor further details and concrete applications we refer the reader to the books\nby \\citeN{HuntKennedy04} and \\citeN{Fries07}, and the references therein.\n\n\\begin{remark}\nLet us point out that forward price models\\xspace and affine LIBOR models\\xspace, that will be introduced in section \\ref{alm},\nbelong to the class of Markov-functional models\\xspace, while LIBOR market models\\xspace do not. In LIBOR market models\\xspace the LIBOR\\xspace rates are\nfunctions of a \\textit{high-dimensional} Markov process. \n\\end{remark}\n\n\n\\subsection{Markov-functional LIBOR model}\nIn order to gain a better understanding of the construction of Markov-functional models\\xspace, we will\nbriefly describe a Markov-functional model\\xspace calibrated to LIBOR\\xspace rates. This model is called the\n\\textit{Markov-functional LIBOR model}.\n\nThe set of relevant market rates are LIBOR\\xspace rates $L(\\cdot,T_k), k\\in K$. We will\nconsider the numeraire pair $(M,\\mathbf{M})=(B(\\cdot,T_N),\\pt)$.\n\nIn order to be consistent with Black's formula for caplets, we assume that\n$L(\\cdot,T_{N-1})$ is a log-normal martingale under \\pt, i.e.\n\\begin{align}\\label{MFL-1}\n\\ensuremath{\\mathrm{d}} L(t,T_{N-1})\n= \\sigma(t,T_{N-1}) L(t,T_{N-1}) \\ensuremath{\\mathrm{d}} W_t,\n\\end{align}\nwhere $W$ denotes a standard Brownian motion under $\\pt$ and $\\sigma(\\cdot,T_{N-1})$\nis a deterministic, time-dependent volatility function. Hence, we have that\n\\begin{align}\\label{MFL-2}\nL(t,T_{N-1})\n= L(0,T_{N-1}) \\exp\\Big( -\\frac12\\Sigma_t + X_t\\Big),\n\\end{align}\nwhere $\\Sigma=\\int_0^\\cdot \\sigma^2(s,T_{N-1})\\ud s$, and $X$ is a deterministic\ntime-change of the Brownian motion, that satisfies\n\\begin{align}\\label{MFL-3}\n\\ensuremath{\\mathrm{d}} X_t = \\sigma(t,T_{N-1})\\ensuremath{\\mathrm{d}} W_t.\n\\end{align}\nWe will use $X$ as the driving process of the model, which specifies (P1).\n\nRegarding (P2), the boundary curve is exactly of the form \\eqref{MFM-2} with\n$T_*=T_{N-1}$, hence we need to specify $\\mathrm{B}(T_i,T_i;X_{T_i})$ for\n$i\\in K$, which is trivially the unit map. We also need to specify\n$\\mathrm{B}(T_{N-1},T_N;X_{T_{N-1}})$; using \\eqref{basic} and\n\\eqref{MFL-2} we get that\n\\begin{align}\\label{MFL-4}\n\\mathrm{B}(T_{N-1},T_N;X_{T_{N-1}})\n= \\frac{1}{1 + \\delta L(0,T_{N-1}) \\exp\\big( -\\frac12\\Sigma_{T_{N-1}}+ X_{T_{N-1}}\\big)}.\n\\end{align}\nThen, we can recover bond prices in the interior of the region bounded by\n$\\partial_S$ using the martingale property:\n\\begin{align}\\label{MFL-5}\n\\mathrm{B}(t,S;X_t)\n= \\mathrm{B}(t,T_N;X_t)\n  \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\bigg[ \\frac{\\mathrm{B}(\\partial_S,S;X_{\\partial_S})}\n                      {\\mathrm{B}(\\partial_S,T_N;X_{\\partial_S})} \\Big|\\ensuremath{\\mathcal{F}}_t \\bigg].\n\\end{align}\n\nNow, it remains to specify the functional form $\\mathrm{B}(T_i,T_N;X_{T_i})$,\n$i\\in K$, for the numeraire, cf. (P3). In the framework of the Markov-functional\nLIBOR\\xspace model, this is done by deriving the numeraire from LIBOR\\xspace rates and inferring\nthe functional forms of the LIBOR\\xspace rates via calibration to market prices. Equation\n\\eqref{basic} combined with \\eqref{MFM-2} and the fact that $B(T_i,T_{i+1})$ is\na function of $X_{T_i}$, cf. \\eqref{MFL-5}, yield that $L(T_i,T_i)$ is also a\nfunction of $X_{T_i}$. The functional form is\n\\begin{align}\\label{MFL-6}\n1+\\delta\\mathrm{L}(T_i,T_i;X_{T_i})\n&= \\frac{1}{\\mathrm{B}(T_i,T_{i+1};X_{T_{i}})} \\nonumber\\\\\n&= \\frac{1}{\\mathrm{B}(T_i,T_N;X_{T_i})\n  \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\Big[ \\frac{1}{\\mathrm{B}(T_{i+1},T_N;X_{T_{i+1}})} \\Big|\\ensuremath{\\mathcal{F}}_{T_i} \\Big]}.\n\\end{align}\nRearranging, we get the following functional form for the numeraire\n\\begin{align}\\label{MFL-7}\n\\mathrm{B}(T_i,T_N;X_{T_i})\n&= \\frac{1}{(1+\\delta\\mathrm{L}(T_i,T_i;X_{T_i}))\n   \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\Big[ \\frac{1}{\\mathrm{B}(T_{i+1},T_N;X_{T_{i+1}})} \\Big|\\ensuremath{\\mathcal{F}}_{T_i} \\Big]}.\n\\end{align}\nThis formula provides a backward induction scheme to calculate\n$\\mathrm{B}(T_i,T_N;\\cdot)$ from $\\mathrm{B}(T_{i+1},T_N;\\cdot)$ for any value\nof the Markov process; the induction starts from $B(T_N,T_N)=1$.\n\nThe calibrating instruments are digital caplets with payoff\n$1_{\\{L(T_i,T_i)>\\mathcal{K}\\}}$, $i\\in K$, and their market values are provided\nby Black's formula; we denote them by $\\mathbb{V}_0(T_i,\\mathcal{K})$.\nAssuming that the map $\\xi\\mapsto\\mathrm{L}(T_i,T_i;\\xi)$ is strictly increasing,\nthere exists a unique strike $\\mathcal{K}(T_i,x^*)$ such that the set equality\n\\begin{align}\\label{MFL-10}\n\\big\\{X_{T_i}>x^*\\big\\}\n = \\big\\{\\mathrm{L}(T_i,T_i;X_{T_i})>\\mathcal{K}(T_i,x^*)\\big\\}\n\\end{align}\nholds almost surely. Define the model prices\n\\begin{align}\\label{MFL-11}\n\\mathbb{U}_0(T_i,x^*)\n = B(0,T_{N})\n   \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\bigg[\\frac{\\mathrm{B}(T_i,T_{i+1};X_{T_{i}})}\n                     {\\mathrm{B}(T_i,T_N;X_{T_{i}})} 1_{\\{X_{T_i}>x^*\\}}\\bigg],\n\\end{align}\nwhich have to be calculated numerically. Therefore, we can equate market and\nmodel prices\n\\begin{align}\\label{MFL-12}\n\\mathbb{V}_0(T_i,\\mathcal{K}(T_i,x^*)) = \\mathbb{U}_0(T_i,x^*),\n\\end{align}\nwhere the strike $\\mathcal{K}(T_i,x^*)$ is determined by Black's formula using\nsome numerical algorithm.\n\nHence, we have specified, numerically at least, the functional form for the\nLIBOR rates, which yields also the functional form for the numeraire via \\eqref{MFL-7}.\nThis completes the specification of the Markov-functional LIBOR model. This\nmodel satisfies requirement (A3), in the sense of \\shortciteANP{HuntKennedyPelsser00}\n\\citeyear{HuntKennedyPelsser00},\nsince all bond prices are functions of a one-dimensional diffusion.\n\n\n\n\\section{Affine LIBOR models}\n\\label{alm}\n\nAffine LIBOR\\xspace models were recently developed by Keller-Ressel, Papapantoleon, and Teichmann\n\\citeyear{KellerResselPapapantoleonTeichmann09} with the aim of combining the\nadvantages of LIBOR market models\\xspace and forward price models\\xspace, while circumventing their drawbacks. We provide\nhere a more general outline of this framework, which is based on two key\ningredients: martingales \\emph{greater than 1}, which are \\emph{increasing} in\nsome parameter.\n\nThe construction of martingales greater than 1 is done as follows: let $Y_T^u$\nbe an $\\ensuremath{\\mathcal{F}}_T$-measurable, integrable random variable, taking values in $[1,\\infty)$,\nand set\n\\begin{align}\\label{Mge1}\nM_t^u = \\ensuremath{\\mathrm{I\\kern-.2em E}}[Y_T^u|\\ensuremath{\\mathcal{F}}_t],\n\\qquad {0\\leq t\\leq T}.\n\\end{align}\nThen, using the tower property of conditional expectations, it easily follows that\n\\prozess[M^u] is a martingale. Moreover, it obviously holds that $M_t^u\\ge1$ for\nall $t\\in[0,T]$.\n\nIn addition, assume that the map $u\\mapsto Y_T^u$ is \\emph{increasing}; then, we\nimmediately get that the map\n\\begin{align}\\label{Mincrease}\nu\\mapsto M_t^u\n\\end{align}\nis also increasing, for all $t\\in[0,T]$. \n\nNow, using the family of martingales $M^u$ we can model quotients of bond prices\nas follows. Consider a \\emph{decreasing} sequence $(u_k)_{k\\in\\overline{K}}$ and set\n\\begin{align}\\label{bond-quots}\n\\frac{B(t,T_k)}{B(t,T_N)}\n &= M_t^{u_k},\n \\qquad t\\in\\ttk,\\, k\\in\\overline{K},\n\\end{align}\nrequiring that the initial values of the martingales fit today's observed market\nprices, i.e. $\\frac{B(0,T_k)}{B(0,T_N)}= M_0^{u_k}$. Since $M^u$ is increasing\nin $u$, we have that\n\\begin{align}\\label{order-M}\nM_t^{u_k} \\ge M_t^{u_l}\n \\quad\\text{ for }\\quad\nk\\le l \\Leftrightarrow u_k\\ge u_l.\n\\end{align}\nHence, we can deduce that bond prices are decreasing as functions of time of\nmaturity, i.e. $B(t,T_k)\\ge B(t,T_l)$ for $k\\le l$.\n\nTurning our attention to LIBOR rates, we get that\n\\begin{align}\\label{LIBOR-1}\n1 + \\delta L(t,T_k)\n &= \\frac{B(t,T_k)}{B(t,\\tk[k+1])}\n  = \\frac{M_t^{u_k}}{M_t^{u_{k+1}}}\n \\ge 1,\n\\end{align}\nfor all $t\\in\\ttk$ and all $k\\in K$; this is a trivial consequence of \\eqref{order-M}.\nMoreover, the martingale property of the LIBOR\\xspace rate under its corresponding forward\nmeasure follows easily from the structure of the measure changes \\eqref{Pk-to-final},\nand the structure of the martingales. Indeed, we have that\n\\begin{multline}\\label{LIBOR-MP}\n1+\\delta L(\\cdot,T_k)\n = \\frac{M^{u_k}}{M^{u_{k+1}}} \\in \\mathcal{M}(\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_{k+1}})\\\\\n \\quad\\text{ since }\\quad\n\\frac{M^{u_{k}}}{M^{u_{k+1}}}\n \\cdot \\frac{\\ensuremath{\\mathrm{d}}\\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[k+1]}}{\\ensuremath{\\mathrm{d}}\\ensuremath{\\mathrm{I\\kern-.2em P}}_{\\tk[N]}\n = \\frac{M^{u_{k}}}{M^{u_{k+1}}}\n  \\cdot \\frac{M^{u_{k+1}}}{M_0^{u_{k+1}}}\n \\in \\mathcal{M}(\\ensuremath{\\mathrm{I\\kern-.2em P}}_{T_{N}}).\n\\end{multline}\n\nTherefore, we have just described a broad framework for modeling LIBOR\\xspace rates, in\nwhich requirements (A1) and (A2) are satisfied. The next step is to show that\nrequirement (A3) is also satisfied. We will not pursue this in full generality,\ninstead we will consider a specific form for the variable $Y_T^u$, and thus for\nthe martingales $M^u$. In addition, the model is driven by an affine process,\nand is henceforth called the \\emph{affine LIBOR model}.\n\n\n\\subsection{Affine processes}\nLet \\prozess[X] be a stochastically continuous, time-homogeneous Markov process\nwith state space $D=\\mathbb{R}_{\\geqslant0}^d$, starting from $x\\in D$. The process $X$ is called\n\\emph{affine} if the moment generating function satisfies\n\\begin{align}\\label{affine}\n\\ensuremath{\\mathrm{I\\kern-.2em E}}_x \\big[\\mathrm{e}^{\\scal{u}{X_t}}\\big]\n = \\exp\\big( \\phi_t(u) + \\scal{\\psi_t(u)}{x} \\big),\n\\end{align}\nfor some functions $\\phi:[0,T]\\times\\mathcal{I}_T\\to\\ensuremath{\\mathbb{R}}$ and $\\psi:[0,T]\\times\\mathcal{I}_T\\to\\ensuremath{\\mathbb{R}}^d$,\nand all $(t,u,x)\\in[0,T]\\times\\mathcal{I}_T\\times D$, where\n\\begin{align}\n\\mathcal{I}_T\n:= \\set{u\\in\\ensuremath{\\mathbb{R}}^d: \\ensuremath{\\mathrm{I\\kern-.2em E}}_x\\big[\\mathrm{e}^{\\scal{u}{X_T}}\\big] < \\infty,\n        \\,\\,\\text{for all}\\; x \\in D}.\n\\end{align}\nWe will assume in the sequel that $0\\in\\mathcal{I}_T^\\circ$. The functions $\\phi$ and\n$\\psi$ satisfy the semi-flow property\n\\begin{equation}\\label{flow}\n\\begin{split}\n\\phi_{t+s}(u) &= \\phi_{t}(u)+\\phi_{s}(\\psi_t(u))\\\\\n\\psi_{t+s}(u) &= \\psi_{s}(\\psi_{t}(u)),\n\\end{split}\n\\end{equation}\nwith initial condition\n\\begin{align}\\label{phi-psi-0}\n \\phi_0(u)=0\n  \\quad\\text{ and }\\quad\n \\psi_0(u)=u,\n\\end{align}\nfor all $(t+s,u)\\in[0,T]\\times\\mathcal{I}_T$. Equivalently, $\\phi$ and $\\psi$ satisfy\ngeneralized Riccati differential equations. For comprehensive expositions of\naffine processes we refer the reader to \\shortciteN{DuffieFilipoviSchachermayer03}\nand \\citeN{KellerRessel08}.\n\n\n\\subsection{Affine LIBOR\\xspace model}\nIn the affine LIBOR model\\xspace, the random variable $Y_T^u$ has the following form:\n\\begin{align\nY_T^u = \\mathrm{e}^{\\scal{u}{X_T}},\n\\end{align}\nwhere $u\\in\\mathbb{R}_{\\geqslant0}^d$ and $X_T$ is a random variable from an $\\mathbb{R}_{\\geqslant0}^d$-valued affine\nprocess $X$. Hence, $Y_T^u\\ge1$, while the map $u\\mapsto Y_T^u$ is obviously\nincreasing; note that inequalities involving vectors are understood componentwise.\n\nUsing the Markov property of affine processes, we can deduce that the martingales\n$M^u$ have the form\n\\begin{align}\\label{alm-Mu}\nM_t^u\n &= \\ensuremath{\\mathrm{I\\kern-.2em E}}\\big[\\mathrm{e}^{\\scal{u}{X_T}}|\\ensuremath{\\mathcal{F}}_t\\big] \\nonumber\\\\\n &= \\exp\\big( \\phi_{T-t}(u) + \\scal{\\psi_{T-t}(u)}{X_t} \\big).\n\\end{align}\nTherefore, LIBOR\\xspace rates have the following evolution:\n\\begin{align}\n1+\\delta L(t,T_k)\n &= \\frac{M_t^{u_k}}{M_t^{u_{k+1}}}\n  = \\exp\\big(A_{k,t} + \\scal{B_{k,t}}{X_t}\\big),\n\\end{align}\nwhere\n\\begin{equation}\\label{Ak-Bk}\n\\begin{split}\n A_{k,t} &:= \\phi_{T_N-t}(u_k) - \\phi_{T_N-t}(u_{k+1})\\\\\n B_{k,t} &:= \\psi_{T_N-t}(u_k) - \\psi_{T_N-t}(u_{k+1}).\n\\end{split}\n\\end{equation}\nLet us point that, under reasonable assumptions on the driving affine process,\nwe can prove that the affine LIBOR model\\xspace can fit \\emph{any} term structure of\ninitial LIBOR\\xspace rates; cf. Proposition 6.1 in \\shortciteN{KellerResselPapapantoleonTeichmann09}.\n\nNow, regarding requirement (A3), let us turn our attention to the structure of\nthe driving process under the different forward measures. Using the connections\nbetween forward measures \\eqref{Pk-to-final}, the Markov property of affine\nprocesses, and the flow equations \\eqref{flow}, we can show that\n\\begin{align}\n\\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_k}\\Big[\\mathrm{e}^{\\langle v,X_r\\rangle}\\big|\\ensuremath{\\mathcal{F}}_s\\Big]\n &= \\ensuremath{\\mathrm{I\\kern-.2em E}}_{T_N}\\Big[ \\frac{M_r^{u_k}}{M_s^{u_k}} \\mathrm{e}^{\\langle v,X_r\\rangle}\\big|\\ensuremath{\\mathcal{F}}_s\\Big] \\\\\n &= \\exp\\Big(\\phi_{r-s}(\\psi_{T_N-r}(u_{k})+ v) - \\phi_{r-s}(\\psi_{T_N-r}(u_k))\\nonumber\\\\\n &\\qquad\n  + \\scal{\\psi_{r-s}(\\psi_{T_N-r}(u_{k})+ v) - \\psi_{r-s}(\\psi_{T_N-r}(u_k))}{X_s}\\Big);\n  \\nonumber\n\\end{align}\ncf. \\shortciteN[eq. (6.15)]{KellerResselPapapantoleonTeichmann09}. This means\nthat $X$ becomes a \\emph{time-inhomogeneous affine} process under \\emph{any}\nforward measure. Note that the measure changes are again Esscher\ntransformations, similarly to forward price models\\xspace. Consequently the affine LIBOR model\\xspace satisfies requirements\n(A1), (A2) \\emph{and} (A3).\n\nThe pricing of caplets and swaptions in the affine LIBOR model\\xspace using Fourier transform methods\nis described in \\shortciteN{KellerResselPapapantoleonTeichmann09}. In addition, closed-form valuation formulas -- in terms\nof the $\\chi^2$-distribution function -- are derived when the driving affine\nprocess is the Cox--Ingersoll--\\linebreak Ross (CIR) process.\n\n\n\n\\section{Extensions}\n\\label{extend}\n\nThe different approaches for modeling LIBOR\\xspace rates have been extended in two\ndifferent directions: (i) to model simultaneously LIBOR rates for different\ncurrencies and the corresponding foreign exchange rates, and (ii) to jointly\nmodel default-free and defaultable LIBOR\\xspace rates.\n\n\n\\subsection{Multiple currencies}\nThe log-normal LIBOR market model\\xspace has been extended to a multi-currency setting by \\citeN{Schloegl02}\nand by \\citeANP{Mikkelsen02} \\citeyear{Mikkelsen02}. The L\\'{e}vy\\xspace LIBOR\\xspace model and the L\\'{e}vy\\xspace forward price model\\xspace have\nbeen extended to model multiple currencies and foreign exchange rates by\n\\citeN{EberleinKoval06}. A multi-factor approach to multiple currency LIBOR\\xspace models\nhas been presented in \\shortciteN{BennerZyapkovJortzik09}. Markov-functional models\\xspace have been extended\nto the multi-currency setting by \\citeANP{FriesRott04} \\citeyear{FriesRott04} and\n\\citeN{FriesEcksteadt08}.\n\n\n\\subsection{Default risk}\nThe log-normal LIBOR market model\\xspace has been first extended to model default risk by\n\\citeN{LotzSchloegl00}, who used a deterministic hazard rate to model the time\nof default. \\shortciteN{EberleinKlugeSchoenbucher06}, \\linebreak borrowing also ideas from\n\\citeN{Schoenbucher00}, constructed a model for default-free and defaultable\nrates where they use time-inhomogeneous L\\'{e}vy\\xspace processes as the driving motion and\nthe ``Cox construction'' to model the time of default (cf. e.g.\n\\citeN{BieleckiRutkowski02} for the Cox construction). This has been extended\nto a model where defaultable bonds can have rating migrations by \\citeN{Grbac09}.\n\n\n\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section[]{The Diagrammatic Identities}\n\\label{app:D-ID}\n\n\\subsection*{The Primary Diagrammatic Identities}\n\n\\subsubsection*{The First Type}\n\n\\begin{equation}\n\\label{eq:WID}\n\t\\cd{WID-contract-b} = \\cd{WID-PF} + \\cd{WID-PFb} - \\cd{WID-PB} - \\cd{WID-PBb} + \\cdots\n\\end{equation}\n\n\\begin{eqnarray}\n\\label{eq:LdL-GRk-Pert}\n\t\\hspace{0.8em} \\stackrel{\\bullet}{\\rhd} & =  & 0\n\\\\\n\\label{eq:dalpha-GRk}\n\t\\begin{array}{c}\n\t\t\\Dal\n\t\\\\[-1.5ex]\n\t\t\\rhd\n\t\\end{array} & = & 0\n\\end{eqnarray}\n\n\\begin{equation}\n\\label{eq:D-ID-GR-TP}\n\t\\cd{GR-TP} = 0\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:Taylor}\n\t\\cd{Taylor-Parent-b} = \\cdeps{Taylor-PFa} + \\cdeps{Taylor-PFb} - \\cdeps{Taylor-PBa} - \\cdeps{Taylor-PBb} +\\cdots\n\\end{equation}\n\n\n\n\\subsubsection*{The Second Type}\n\n\\begin{equation}\n\t\\cd{EffPropReln-b}\t= \\cd{K-Delta-b} - \\cd{FullGaugeRemainder-b}\n\t\t\t\t\t\t\t= \\cd{K-Delta-b} - \\cd{DecomposedGR-b}\n\t\\label{eq:EPReln}\n\\end{equation}\n\n\n\\subsubsection*{The Third Type}\n\n\\begin{equation}\n\t\\cd{GR-TLTP} = 0\n\\label{eq:GR-TLTP}\n\\end{equation}\n\n\\begin{equation}\n\t\\cd{GR-relation} = 1\n\\label{eq:GR-relation}\n\\end{equation}\n\n\n\\begin{equation}\n\t\\cd{EP-GRpr} \\equiv \\cd{GR-PEP}\n\\label{eq:PseudoEP}\n\\end{equation}\n\n\n\\subsection*{The Secondary Diagrammatic Identities}\n\n\\subsubsection*{The First Type}\n\n\\begin{equation}\n\\label{D-ID-Trivial}\n\t\\cdeps{GR-hook} = 0\n\\end{equation}\n\n\n\\begin{equation}\n\\label{eq:D-ID-Bitten-hook}\n\t\\cdeps{Bitten-hook} - \\cdeps{Bitten-hook-R} \\equiv 0\n\\end{equation}\n\n\\begin{eqnarray}\n\t\\nonumber\n\t\t\\sum_{m'=1}^{m-1} \\frac{1}{m'!} \\cd{TLTP-EP-ArbGRs} \\hspace{2em}\n\t\t& = &  \\sum_{m'=1}^{m-1} \\frac{1}{(m'+1)!} \\cd{GR-ArbGRs} \\hspace{2em}\n\t\\\\ \n\t\\label{eq:D-ID-G}\n\t\t&& - \\sum_{m'=2}^{m-1} \\sum_{m''=0}^{m'-2} \\frac{1}{(m'-m'')!m''!} \n\t\t\\cd{ArbGRs-TLTP-EP-ArbGRs}\n\\end{eqnarray}\n\n\\clearpage\n\n\\subsubsection*{The Second Type}\n\n\\begin{equation}\n\\label{eq:D-ID-dGRk-GT-ring}\n\t2m(m-1) \\cd{GRs-dGRk} \\hspace{3em} = - \\decGR{ \\ }{\\ensuremath{\\sqcup} >^m}.\n\\end{equation}\n\n\\begin{eqnarray}\n\t\\lefteqn{\n\t\t\\sum_{m'=0}^m \n\t\t\\left[\n\t\t\t\\cd{CTP-E-GRs-Combo-GRk-GR} \n\t\t\t-\\cd{CTP-E-GRs-Combo-GR}\n\t\t\\right]\t\n\t}\n\t\\nonumber\n\t\\\\ & &\n\t\t- \\sum_{m'=0}^m \\!\\! \\nCr{m'}{m''} \\!\\! \\sum_{m''=0}^{m'} \n\t\t\\left[\n\t\t\t\\cd{CTP-E-GRs-Combo-CTP-EP-GR}\n\t\t\t\\hspace{1.8em}\n\t\t\t-\\cd{CTP-E-GRs-Combo-GRs-CTP-EP}\n\t\t\t\\hspace{1.8em}\n\t\t\\right]\n\t\t= 0\n\t\\label{eq:D-ID-Op2-G}\n\\end{eqnarray}\n\n\\clearpage\n\n\\section[]{Diagrammatic Expressions}\n\n\\subsection*{Decoration of diagram~\\ref{j+2-DEP}}\n\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2.5ex}\n\t\t\\left[\n\t\t\\begin{array}{c}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s} \\sum_{j=-1}^{n+s-m-3}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\\\\\n\t\t\t\\displaystyle\n\t\t\n\t\t\t\\dec{\n\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-DEP-vkR} \\!\\! \\Vertex{n_s,j}}{D-vj_+kR-DEP-vkR}\n\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\\end{array}\n\t\t\\right]\n\t\t-\n\t\t\\left[\n\t\t\\begin{array}{c}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t\\\\\n\t\t\t\\dec{\n\t\t\t\\sco[1]{\n\t\t\t\t\\LDi{vj_+R-DEP}{vj_+R-DEP}\n\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t\\end{array}\n\t\t\\right]\n\t\\\\\n\t\\vspace{2.5ex}\n\t\t-2\n\t\t\\left[\n\t\t\\begin{array}{c}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-1}^{n+s-m-2}\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t\t\\sum_{m'=1}^{m} \\nCr{m}{m'}\n\t\t\\\\\n\t\t\t\\displaystyle\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi[1]{v_j+R-DEP-GRs}{v_j+R-DEP-GRs}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t\\end{array}\n\t\t\\right]\n\t\\\\\n\t\\vspace{2.5ex}\n\t\t+\n\t\t\\left[\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2}\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\t\\sum_{m'=1}^{m} \\nCr{m}{m'} \n\t\t\t\\\\\n\t\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle \\nCr{m'}{m''}\\sum_{m''=1}^{m'-1} \n\t\t\t\t\t\\LDi{GRs-DEP-GRs}{GRs-DEP-GRs-MV} \\hspace{2em}\n\t\t\t\t\t+ \\LDi{Ubend-DEP}{Ubend-MV} \\hspace{0.5em}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\end{array}\n\t\\]\n\\caption{Decoration of diagram~\\ref{j+2-DEP} with\nthe differentiated effective propagator.}\n\\label{fig:app:DEP}\n\\end{figure}\\end{center}\n\n\\clearpage\n\n\\subsection*{Trapped gauge remainders}\n\\label{app:Trapped}\n\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2}  \\sum_{j=-1}^{n+s-m-4} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!} \\sum_{m'=0}^{m}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\nCr{m}{m'} \\!\\!\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{RV-RW-GR-B-RW}{RV-RW-GR-B-RW} \\ \n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s}>^{m-m'}}\n\t\t}\t\n\t\\\\\n\t\\vspace{2ex}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-4} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}  \\sum_{m'=0}^{m} \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{TLTP-RW-GR-B-RW}{TLTP-RW-GR-B-RW}}\n\t\t\t}{11\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-4}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}  \\sum_{m'=0}^{m} \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{Trapped-KBK}{Trapped-KBK}}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Terms spawned by diagrams~\\ref{WGRx2} \nand~\\ref{P-S_vj_+R-DW-GRx2}\nwhich possess a trapped gauge remainder.}\n\\label{fig:GRsx2-P}\n\\end{figure}\\end{center}\n\n\\subsection*{Diagrams spawned by~\\eq{eq:LdL-b-ba} in which a kernel\nbites its own tail}\n\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-2} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=0}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-WBT-GR-GRs-dGR-GRs}{CTP-E-WBT-GR-GRs-dGR-GRs} \n\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t-\\LDi{CTP-E-WBT-GR-GRs-GR-dGR-GRs}{CTP-E-WBT-GR-GRs-GR-dGR-GRs}\n\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r}  >^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-WBT-dGR-GRs}{CTP-E-WBT-dGR-GRs}\n\t\t\t\t\t-\\LDi{CTP-E-WBT-GR-dGR-GRs}{CTP-E-WBT-GR-dGR-GRs}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r}  >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=0}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-WBT-GRs-CTP-EP}{CTP-E-GRs-Combo-WBT-GRs-CTP-EP}\n\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r}  >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Diagrams spawned by~\\eq{eq:LdL-b-ba} in which a kernel\nbites its own tail, part~1.}\n\\label{fig:nLDlb-KBT}\n\\end{figure}\\end{center}\n\n\\clearpage\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-WBT}{CTP-E-GRs-Combo-WBT-b}\n\t\t\t\t\t-\\LDi{CTP-E-GRs-Combo-WBT-GR}{CTP-E-GRs-Combo-WBT-GR}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r}  >^{m-m'}}\n\t\t}\t\n\t\\end{array}\n\t\\]\n\\caption{Diagrams spawned by~\\eq{eq:LdL-b-ba} in which a kernel\nbites its own tail, part~2.}\n\\label{fig:nLDlb-KBT-2}\n\\end{figure}\\end{center}\n\n\n\\subsection*{Subtractions and Additions}\n\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{0ex}\n\t\t\\mp 4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{dvh_j-DEP-GRs-Skt-CTP-E}{dvh_j-DEP-GRs-Skt-CTP-E-s}{dvh_j-DEP-GRs-Skt-CTP-E-a}\n\t\t\t\t\t-\\LLDi{dPi_j+R-DW-GRs-Skt-CTP-E}{dPi_j+R-DW-GRs-Skt-CTP-E-s}{dPi_j+R-DW-GRs-Skt-CTP-E-a}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\t\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm 4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=0}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\left[\n\t\t\t\t\t\t\\LLDi{vh_k+R-DEP-GRs-Skt-CTP-E}{vh_kR-dvj+kh-s}{vh_kR-dvj+kh-a}\n\t\t\t\t\t\t-\\LLDi{Pi_kR-DW-GRs-Skt-CTP-E}{Pi_j+R-dvj+kh-s}{Pi_j+R-dvj+kh-a}\n\t\t\t\t\t\\right] \\cd{dvj+k}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+2}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for diagrams~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E}\nand~\\ref{Pi_j+R-DW-GRs-Skt-CTP-E}, part~1.}\n\\label{fig:vh_j+R-DEP-GRs-Skt-CTP-E-Add}\n\\end{figure}\\end{center}\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\mp 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{vh_j-DEP-TEGRs-Skt-CTP-E}{vh_j-DEP-TEGRs-Skt-CTP-E-s}{vh_j-DEP-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t-\\LLDi{Pi_j+R-DEP-TEGRs-Skt-CTP-E}{Pi_j+R-DEP-TEGRs-Skt-CTP-E-s}{Pi_j+R-DEP-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t\\hspace{1em}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t\\mp 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\\LLDi{vh_j+R-DEP-GRs-Skt-CTP-E-b}{vh_j+R-DEP-GRs-Skt-CTP-E-b-s}{vh_j+R-DEP-GRs-Skt-CTP-E-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\t-\\LLDi{Pi_j+R-DW-GRs-Skt-CTP-E-b}{Pi_j+R-DW-GRs-Skt-CTP-E-b-s}{Pi_j+R-DW-GRs-Skt-CTP-E-b-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m''}}\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm 4\n\t\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\t\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{Pi_j+R-dK-GRs-Skt-CTP-E}{Pi_j+R-dK-GRs-Skt-CTP-E-s}{Pi_j+R-dK-GRs-Skt-CTP-E-a}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for diagrams~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E}\nand~\\ref{Pi_j+R-DW-GRs-Skt-CTP-E}, part~2.}\n\\label{fig:vh_j+R-DEP-GRs-Skt-CTP-E-Add-II}\n\\end{figure}\\end{center}\n\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\mp 2 \n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\t\\LLDi{DW-GR-GRs-Skt-CTP-E}{DW-GR-GRs-Skt-CTP-E-dv-s}{DW-GR-GRs-Skt-CTP-E-dv-a}\n\t\t\t\t\t\t\t-2\\LLDi{DW-GRs-Skt-CTP-E}{DW-GRs-Skt-CTP-E-dv-s}{DW-GRs-Skt-CTP-E-dv-a}\n\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\cd{dvj+}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{}\n\t\t\t\t\\\\\n\t\t\t\t\t\\SumVertex\t\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm 2 \n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\t\\LLDi{dK-GR-GRs-Skt-CTP-E}{dK-GR-GRs-Skt-CTP-E-dv-s}{dK-GR-GRs-Skt-CTP-E-dv-a}\n\t\t\t\t\t\t-2\\LLDi{dK-GRs-Skt-CTP-E}{dK-GRs-Skt-CTP-E-dv-s}{dK-GRs-Skt-CTP-E-dv-a}\n\t\t\t\t\t}{}\n\t\t\t\t\\\\\n\t\t\t\t\t\\TopVertex \\SumVertex\t\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{DW-GR-TEGRs-Skt-CTP-E}{DW-GR-TEGRs-Skt-CTP-E-s}{DW-GR-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t-2\\LLDi{DW-TEGRs-Skt-CTP-E}{DW-TEGRs-Skt-CTP-E-s}{DW-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t}{}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for \ndiagrams~\\ref{DW-GR-GRs-Skt-CTP-E} and~\\ref{DW-GRs-Skt-CTP-E}, \npart~1.}\n\\label{fig:DW-GR-GRs-Skt-CTP-E-Add}\n\\end{figure}\\end{center}\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\pm\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\\LLDi{DW-GR-GRs-Skt-CTP-E-b}{DW-GR-GRs-Skt-CTP-E-TEGR-s}{DW-GR-GRs-Skt-CTP-E-TEGR-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\t-2\\LLDi{DW-GRs-Skt-CTP-E-b}{DW-GRs-Skt-CTP-E-TEGR-s}{DW-GRs-Skt-CTP-E-TEGR-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m''}}\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\]\n\\caption{Subtractions and additions for \ndiagrams~\\ref{DW-GR-GRs-Skt-CTP-E}\nand~\\ref{DW-GRs-Skt-CTP-E}, \npart~2.}\n\\label{fig:DW-GR-GRs-Skt-CTP-E-Add-II}\n\\end{figure}\\end{center}\n\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\mp 4 \n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\t\\LLDi{GRs-dEP-GR-Skt-CTP-E}{GRs-dEP-GR-Skt-CTP-E-dv-s}{GRs-dEP-GR-Skt-CTP-E-dv-a}\n\t\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\cd{dvj+}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{}\n\t\t\t\t\\\\\n\t\t\t\t\t\\SumVertex\t\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\t\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for \ndiagram~\\ref{GRs-dEP-GR-Skt-CTP-E}, part~1.}\n\\label{fig:GRs-dEP-GR-Skt-CTP-E-Add}\n\\end{figure}\\end{center}\n\n\\begin{center}\\begin{figure}[!tbp]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\pm 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\t\\LLDi{TEGRs-dEP-GR-Skt-CTP-E}{TEGRs-dEP-GR-Skt-CTP-E-s}{TEGRs-dEP-GR-Skt-CTP-E-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t\t+\\LLDi{GRs-dEP-GR-TEGRs-Skt-CTP-E}{GRs-dEP-GR-TEGRs-Skt-CTP-E-s}{GRs-dEP-GR-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t\t\\hspace{1.8em}\n\t\t\t\t\t}{}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\pm 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\! \\sum_{m'''=2}^{m''} \\!\\! \\nCr{m''}{m'''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\decp{\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\\LLDi{GRs-dEP-GR-Skt-CTP-E-b}{GRs-dEP-GR-Skt-CTP-E-TEGRs-s}{GRs-dEP-GR-Skt-CTP-E-TEGRs-a}\n\t\t\t\t\t\t\\hspace{2em}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m'''}}\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for \ndiagram~\\ref{GRs-dEP-GR-Skt-CTP-E}, part~2.}\n\\label{fig:GRs-dEP-GR-Skt-CTP-E-Add-II}\n\\end{figure}\\end{center}\n\n\n\n\\section{An Expression for $\\beta_n$}\n\\label{sec:beta_n}\n\n\n\n\\subsection{Initial Manipulations}\n\nIn this section,\nwe will derive a compact expression\nfor $\\beta_n$ which has no explicit\ndependence on either the seed action\nor the details of the covariantization \nof the cutoff. Our starting point is\nto consider \n\\[\n\t\\dec{\n\t\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) + \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)\n\t}{\\bullet}.\n\\]\n(We work with these auxiliary function\nbecause it is technically easier to do\nso than to work directly with\n${\\cal D}^{a,b}$. At the end of the calculation,\nwill be able to conveniently trade\n$\\mathcal{E}$s for ${\\cal D}$s.)\n\nAs a precursor to computing the flow of\nthese functions,\nit is clear that we will need to understand\nhow to compute the flow of a reduced vertex.\nRecall that a reduced vertex lacks a classical,\ntwo-point component. Consequently, the flow of\na reduced vertex lacks a component given\nby the flow of a classical, two-point vertex,\nwhich we can read off from \\fig{fig:TLTPs}.\nNow, the classical flow equation involves only\nthe $a_0$ term; there are no $\\beta$, $\\alpha$\nor $a_1$ terms. Hence, we need only take care\nwith the dumbbell term generated by the flow\nof a reduced vertex, which we will tag with an\n$R$. It is straightforward to check that a dumbbell \nterm tagged in this way must either possess\nat least one reduced vertex or must have a decorated\nkernel.\n\nWe begin by focusing on the terms\ngenerated by $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$\nand adjust the ranges the sums\nover $s$, $m$ and $j$ to remove diagrams which vanish\neven after the action of $-\\Lambda \\partial_\\Lambda|_\\alpha$.\nThis is trivial in the case where\n$-\\Lambda \\partial_\\Lambda|_\\alpha$ strikes an effective propagator: the limits of the\nsums just reduce to those of $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$.\nIf $-\\Lambda \\partial_\\Lambda|_\\alpha$ strikes a $>$ things are much\nthe same only now \nthe sum over $m$ must, of course, start from\none and not zero.\n\n\n\n\nHowever, things are more subtle in the case\nthat $-\\Lambda \\partial_\\Lambda|_\\alpha$ strikes a vertex. \nImmediately, since we must have at least one vertex,\nthe lower limit of the sum over $j$ increases from $-2$\nto $-1$. Consider\nnow the resulting $\\alpha$ and $\\beta$ terms.\nCompared to the parent vertex, the loop order\nof the daughter vertex is changed, but the number\nof legs remains the same. Thus, if\nthe parent\ndiagram vanishes at $\\Op{2}$, it is not necessarily\nthe case that the daughter $\\alpha$ and $\\beta$-terms also \nvanish\\footnote{Of course, it must be the case that the sum over\nall daughters, including those generated\nby $a_0$ and $a_1$, does vanish at $\\Op{2}$.}:\nit is clear from~\\eqs{eq:S_0-11}{eq:S_>0-11}\nthat a change to the loop order of a two-point\nvertex carrying momentum $p$ can\nchange the order of its leading contribution in $p$\n.\nHowever, if the parent diagram vanishes because it\npossesses a one-point vertex, then the daughter $\\alpha$ and $\\beta$-terms\nwill vanish for this reason also. Consequently, \nfor the $\\alpha$ and $\\beta$-terms\nthe minimum\nvalue  of $s$ stays the same, but the maximum values of both\n$m$ and $j$ are reduced by one. \n\nWith these points in mind, it is worth considering\nthe $\\alpha$ and $\\beta$-terms in more detail and\nso, to this end, we explicitly give them in \\fig{fig:beta+alpha-Prelim}.\nThe combinatoric factor follows from recognizing\nthat the $-\\Lambda \\partial_\\Lambda|_\\alpha$ can strike any of the $j+2$ vertices\nand using~\\eq{eq:norm-c}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t-2\n\t\\dec{\n\t\t\\begin{array}{c}\n\t\t\\vspace{1ex}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{u}\n\t\t\t\\sum_{s=0}^{n-1}   \\sum_{m=0}^{2s} \\sum_{j=-1}^{n+s-m-1-u}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\\sum_{v^{k}=1}^{v^{j_+}} \\delta(v^k - u)\n\t\t\\\\\n\t\t\t\\left[\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\t2 \\left( v^{j_+,k}-1 \\right) \\beta_{v^{k}}\n\t\t\t\t\t\t\t\t+ \\gamma_{v^{k}} \\pder{}{\\alpha}\n\t\t\t\t\t\\right]\n\t\t\t\t\t\\Vertex{v^{j_+\\!,k}}\n\t\t\t\t}{\\SumVertex}\n\t\t\t\\right]\n\t\t\\end{array}\n\t}{11\\Delta^{j+s+1} >^m}\n\t\\]\n\\caption{The $\\alpha$ and $\\beta$ terms produced by \n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$.}\n\\label{fig:beta+alpha-Prelim}\n\\end{figure}\\end{center}\n\nThere are a number of important points to make.\nFirst, it is understood \nthat the sum over the vertex argument $v^k$---which is identified with\n$v^{j+2}$---is\nperformed \\emph{after}\nthe sums over $v^1 \\cdots v^{j_+}$ buried in the diagrammatics \n({\\it cf.}\\  \\eq{eq:VertexTower}). Secondly, we notice a\nsum over the new variable, $u$, and a $\\delta$-function\n$\\delta(v^k - u)$. This is to ensure that the upper limit\non the sum over $j$ is given correctly, given that\nthe sum over\nvertex arguments is now $n-s-v^k$.\nFinally, the upper limit on the sum over $s$ has been reduced\nby unity. This follows from the minimum value of $v^k$\nbeing unity ($\\beta_0$ does not exist) which, in turn, forces $n-s$\nto take a minimum value of unity. The expression for the $\\beta$\nand $\\alpha$ terms is quite ugly in its current form,\nbut it can be neatened up by a change of variables:\n\\[\n v^{i>0} \\rightarrow v^i + v^k,\n\\]\nto give the diagrams of \\fig{fig:beta+alpha-Prelim-b}. Notice that\nwe have changed the dummy variable $k$ to $n'$ which\nnow appears as the outer sum and also as the label for\nboth $\\alpha$ and $\\beta$.\n\\begin{center}\\begin{figure}[h]\n\\[\n\t\t-2\\dec{\n\t\t\t\\begin{array}{c}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{n'=1}^{n}\n\t\t\t\t\\sum_{s=0}^{n-n'}   \\sum_{m=0}^{2s} \\sum_{j=-1}^{n-n'+s-m-1}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\\\\\n\t\t\t\t\\left[\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\t2 \\left( v^{j_+}-1 \\right) \\beta_{n'} \t+  \\gamma_{n'} \\pder{}{\\alpha}\n\t\t\t\t\t\t\\right]\n\t\t\t\t\t\t\\DisplacedVertex{v^{j_+}}\n\t\t\t\t\t}{\\Vertex{n_{n'+s}, j}}\n\t\t\t\t\\right]\n\t\t\t\\end{array}\n\t\t}{11\\Delta^{j+s+1} >^m}\n\\]\n\\caption{A re-expression of the $\\alpha$ and $\\beta$-terms.}\n\\label{fig:beta+alpha-Prelim-b}\n\\end{figure}\\end{center}\n\nWe now focus on the $s=0$ components of these diagrams.\nDemanding that the resulting terms are connected,\nwe discard all diagrams possessing one-point (Wilsonian\neffective action) vertices. Utilizing~\\eqs{eq:S_0-11}{eq:S_>0-11}\nit is straightforward to demonstrate that the sole\ncontribution which survives at $\\Op{2}$ is:\n\\begin{eqnarray*}\n\t&\n\t-2 \\ensuremath{\\Upsilon}_{0,0}\n\t\\left(\n\t\t-2\\beta_n + \\gamma_n \\pder{}{\\alpha}\n\t\\right)\n\t\\cd{CTP-EE}\n\\\\\n\t= & -4\\beta_n \\Box_{\\mu \\nu}(p) + \\Op{4},\n\\end{eqnarray*}\nwhere we have used~\\eq{eq:norm}.\n\nReturning to our consideration of \n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$,\nwe note that\nterms generated by the action of $a_0$ are particularly\nsimple to deal with: the ranges of the sums over \n$s$, $m$ and $j$ just stay the same. This follows \nbecause $a_0$ converts a Wilsonian effective action\nvertex into a dumbbell structure, components of\nwhich possess seed action vertices; seed action\nvertices, unlike their Wilsonian effective\naction counterparts, exist at the one-point level.\nSuppose\nthat the parent diagram vanishes at $\\Op{2}$,\non account of an $\\nLV{S}{n'}{\\mu\\nu}{}(p)$\nvertex. This can be converted\ninto (amongst other terms) a dumbbell possessing\na one-point, $n'$-loop seed action vertex,\njoined to a classical, three-point vertex.\nThis term no longer vanishes at $\\Op{2}$.\nIf the parent diagram vanishes on account\nof a one-point Wilsonian effective action\nvertex, the daughter\ncan survive \nsince the vanishing vertex of the parent can\nbe converted into a dumbbell structure\npossessing a one-point, seed action vertex\njoined to a two-point vertex.\n\nFinally, consider the effect of the action\nof $a_1$. Due to the equivalence of each of the vertices\nin a given term, we can take the $a_1$ to strike\nthe vertex with argument $v^{j_+;R}$ (so long as we multiply by $j+2$), causing\nthe argument to become\n$\\Sigma_{v^{j_+}-1}$.\nThe $R$ has been dropped,\nsince a quantum term is necessarily formed from a \nvertex whose argument is greater than zero. The\nlower limit on the sum over $v^{j_+}$ should\nbe now changed from zero to one, in recognition of the\nfact that\n$a_1$ does not act on tree level terms. Furthermore,\nthe sum over all vertex arguments is now $n+s-1$,\nrather than $n+s$, and so we should reduce the\nupper limit of the sum over $s$ by one. It is\nconvenient to change variables:\n\\[\n\tv^{i>0} \\rightarrow v^i + 1\n\\]\nand then to let $s \\rightarrow s-1$. This affects\nthe ranges of all three sums.\n\nThe diagrams formed by  $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$\nare shown in \\fig{fig:bn-b-P}. It is understood\nhere, and in all that follows, that the\nvertex argument $v^k$ is to be identified with $v^{j+2}$.\nIf this vertex argument appears in more than one vertex\n(as will always be the case) it is understood to be\nsummed over, in  the usual manner.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2.5ex}\n\t\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet} + \\Op{4} = -4 \\beta_n \\Box_{\\mu\\nu}(p) \n\t\\\\\n\t\\vspace{3ex} \n\t\t-2\\dec{\n\t\t\t\\begin{array}{c}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{n'=1}^{n-1}\n\t\t\t\t\\sum_{s=1}^{n-n'}   \\sum_{m=0}^{2s} \\sum_{j=-1}^{n-n'+s-m-1}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\\\\\n\t\t\t\t\\left[\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t\t\t\\vspace{1ex}\t\n\t\t\t\t\t\t\t\t\\LD{beta-terms}\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t& \\LD{alpha-terms}\t\t\t\n\t\t\t\t\t\t\t\\\\\t\n\t\t\t\t\t\t\t\t2 \\left( v^{j_+}-1 \\right) \\beta_{n'} \t& \\hspace{-1em} + \\hspace{-1em}\t& \\gamma_{n'} \\pder{}{\\alpha}\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\\right]\n\t\t\t\t\t\t\\DisplacedVertex{v^{j_+}}\n\t\t\t\t\t}{\\Vertex{n_{n'+s}, j}}\n\t\t\t\t\\right]\n\t\t\t\\end{array}\n\t\t}{11\\Delta^{j+s+1} >^m}\n\t\\\\\n\t\\vspace{3ex}\n\t\t\\displaystyle\n\t\t-\\sum_{s=0}^n   \\sum_{m=0}^{2s+1} \\sum_{j=-1}^{n+s-m-1}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\\dec{\n\t\t\t\\sco[1]{\n\t\t\t\t\\displaystyle\n\t\t\t\n\t\t\t\t\\left[\n\t\t\t\t\t\\LDi{Dumbbell-vj_+kb-vkb}{D-vj_+kb-vkb}\n\t\t\t\t\\right]_R\n\t\t\t}{\\Vertex{ n_s, j}}\n\t\t}{11\\Delta^{j+s+1} >^m}\n\t\\\\\n\t\\vspace{3ex}\n\t\t\\displaystyle\n\t\t+\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t\\dec{\n\t\t\t\\sco[1]{\n\t\t\t\t\\LDi{Padlock-Sig_vj_+}{P-Sig_vj_+}\n\t\t\t\t+\\LDi{WBT-Sig_vj_+}{WBT-Sig_vj_+}\n\t\t\t}{\\Vertex{ n_{s}, j}}\n\t\t}{11\\Delta^{j+s} >^m}\n\t\\\\\n\t\t\\displaystyle\n\t\t+\n\t\t\\left[\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s} \\sum_{j=-2}^{n+s-m-2}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\\\\n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\\LO{\\Vertex{v^{j_+\\!;R}}}{j+2-DEP}}{\\Vertex{n_s, j}}\n\t\t\t\t}{11\\Delta^{j+s} \\stackrel{\\odot}{\\Delta} >^m}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\t+\n\t\t\\left[\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t2 \\sum_{s=1}^n   \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{(m-1)!}\n\t\t\t\\\\\n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\\LO{\\Vertex{v^{j_+\\!;R}}}{j+2-DGR}}{\\Vertex{n_s, j}}\n\t\t\t\t}{11\\Delta^{j+s+1} >^{m-1} \\stackrel{\\bullet}{>}}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\end{array}\n\t\\]\n\\caption{An expression for $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$.}\n\\label{fig:bn-b-P}\n\\end{figure}\\end{center}\n\n\n\nIt is worth considering the diagrams\narising from explicitly decorating~\\ref{j+2-DEP}\nwith the differentiated effective propagator\nin more detail. For convenience, these diagrams\nare collected together in \\fig{fig:app:DEP}.\nExplicit decoration with the differentiated effective\npropagator has the capacity to further change\nthe limits on the various sums. For example,\nif we attach both ends to the same vertex,\nthen this will reduce the maximum value of $m$\nby one since, in order to form a connected\ndiagram, the vertex decorated by the\ndifferentiated effective propagator must be at least\nthree-point. This effectively uses up a decoration\nwhich could otherwise have filled a $>$.\n\n\n In the case that each end of the differentiated\neffective propagator attaches to a different vertex,\nwe note that such a diagram can only exist for $j>-1$.\nOur strategy here is to shift $j \\rightarrow j+1$,\nso that the sum over $j$ starts from $-1$,\nidentifying $v^{j+2}$ with $v^k$ as usual.\n\n\nWe now isolate all classical, two-point vertices\nin diagrams~\\ref{D-vj_+kb-vkb}--\\ref{WBT-Sig_vj_+}. \nThis step is crucial to\nthe entire diagrammatic procedure: if a classical,\ntwo-point vertex is attached to an effective propagator,\nwe can employ the effective propagator relation, whereas\nif it is attached to an external field, we can perform\nmanipulations at $\\Op{2}$. Although\nthe classical, two-point vertices of diagram~\\ref{P-Sig_vj_+}\nare fully decorated, it will nonetheless prove useful\nto isolate them, as this will help elucidate\nthe structure of the forthcoming cancellations.\n\nLet us consider isolating the classical, two-point\nvertices of diagram~\\ref{D-vj_+kb-vkb} in more detail.\nFirst, let us take both vertices\nof the dumbbell structure to be reduced vertices.\nThis immediately allows us to reduce the maximum \nvalue of the sum over $j$ by one,\nsince the\nformation of the dumbbell term causes the\ntotal number of vertices to increase from\n$j+2$ to $j+3$ (and the maximum number of reduced\nvertices we can decorate in the formation of\na connected diagram remains the same).\nOf the terms which remain, consider\nthose possessing exclusively \nWilsonian \neffective action vertices. Since we discard\none-point, Wilsonian effective action vertices\nin all diagrams in which $\\Lambda \\partial_\\Lambda|_\\alpha$ has acted,\nwe can reduce the maximum values of \n$j$ and $m$ and increase\nthe minimum value of $s$ by one.\nThe component of the resulting diagram in which\nthe kernel is undecorated is exactly cancelled\nby the component of diagram~\\ref{j+2-DEP}\nin which the differentiated effective propagator\njoins two different vertices (see diagram~\\ref{D-vj_+kR-DEP-vkR}). \nSince the surviving\ncomponent has a decorated kernel,\nthe maximum values of both $m$ and $j$ are reduced\nby one, again.\n\nIsolating a single classical, two-point vertex\nin diagram~\\ref{D-vj_+kb-vkb} is straightforward:\ntaking the argument of the top or bottom vertex\nof the dumbbell structure to be a classical, two-point\nvertex amounts to the same thing; hence we will choose\nto isolate the classical, two-point part of $\\overline{v}^k$ \nand multiply by two. When taking the classical\npart of $\\overline{v}^k$,\nthe other vertex argument,\n$\\overline{v}^{j_+,k;R} \\equiv \\overline{v}^{j_+;R} - \\overline{v}^{k;R}$, reduces to \nsimply $\\overline{v}^{j_+;R}$. \nWe now expanded out the barred vertices,\naccording to~\\eq{eq:bar}, noting that certain terms cancel\non account of us having\nset the classical, two-point seed action vertices\nequal to their Wilsonian effective action counterparts.\nThere is no need to change the \nlimits on any of the sums: compared to the\nparent diagram, we have an extra two-point vertex,\nbut also an extra two decorative fields, corresponding\nto the two ends of the kernel. \n\n\nTaking two classical, two-point vertices in\ndiagram~\\ref{D-vj_+kb-vkb} requires some thought.\nFirst, we note that the resulting dumbbell\nstructure cannot have been formed by\nthe flow of a one-point vertex. Since we are\ninterested only in one-point Wilsonian effective\naction vertices if\nthey have been processed,\nwe can reduce the maximum values of $m$\nand $j$ and increase the\nminimum value of $s$ to remove any unwanted diagrams.\nIf $j=-1$, then there are only two vertices\nin total, and so this case should be treated differently\nfrom $j \\geq 0$. In the latter case, it will prove\nuseful to shift variables $j \\rightarrow j+1$, so\nthat the sum over $j$ starts, once again, from $-1$.\nWe can then recombine terms into a diagram\nwith a sum that starts from $j=-2$.\nThe isolation of the\nclassical, two-point vertices of \ndiagram~\\ref{D-vj_+kb-vkb} is shown in \n\\fig{fig:bn-Ia}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\t\\begin{array}{l}\n\t\t\\vspace{2ex}\n\t\t\t-\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-1}^{n+s-m-4} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-RW-vkR} \\Vertex{n_s,j}}{D-vj_+kR-vkR}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\\\\\n\t\t\\vspace{2ex}\n\t\t\t+2\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s+1}  \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-vkhR} \\Vertex{n_s,j}}{D-vj_+kR-vkhR}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\\\\\n\t\t\t+\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\\LDi{Dumbbell-02x2-R}{D-02x2-R-VS}\n\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\\Vertex{v^{j_+\\!;R}}\n\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\Vertex{n_s, j}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t}{11\\Delta^{j+s+2} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\t+2\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s+1}  \\sum_{j=-1}^{n+s-m-1} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\t\\displaystyle\t\t\t\t\n\t\t\t\t\t\t\t\\LDi{Dumbbell-vj_+h-02}{D-vj_+R-02}\n\t\t\t\t\t\t}{\\Vertex{n_s, j}}\t\t\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\t\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\\end{array}\n\t\\]\n\\caption{Isolation of the classical, two-point vertices\nof diagram~\\ref{D-vj_+kb-vkb} plus diagram~\\ref{D-vj_+kR-DEP-vkR}.}\n\\label{fig:bn-Ia}\n\\end{figure}\\end{center}\n\nIt is now easy to isolate the classical, two-point\nvertices of diagrams~\\ref{P-Sig_vj_+} and~\\ref{WBT-Sig_vj_+},\nthough we note the following. First, on account\nof the equality of the Wilsonian effective action,\nclassical, two-point vertices and their seed action \ncounterparts, $\\Sigma_{0 RS}^{\\ XX}(k) = -S_{0 RS}^{\\ XX}(k)$;\nin other words,\nthe classical, two-point component of $\\Sigma$ has a minus\nsign, relative to the reduced part. Secondly, the\nkernel attaching to the classical, two-point\nvertex contained in diagram~\\ref{P-Sig_vj_+} must be\ndecorated, in order that a connected diagram\ncan be formed.\nFinally, the component of diagram~\\ref{P-Sig_vj_+}\nin which all vertices are Wilsonian effective\naction vertices and the kernel is undecorated\nis cancelled by\n the component of diagram~\\ref{j+2-DEP}\nin which the differentiated effective propagator\nattaches at both ends to the same vertex (see diagram~\\ref{vj_+R-DEP}).\nThe isolation of the\nclassical, two-point vertices of \ndiagrams~\\ref{P-Sig_vj_+} and~\\ref{WBT-Sig_vj_+} is shown in \n\\fig{fig:bn-Ib}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\n\t\t\t\t\t-2 \\LDi{Padlock-hS_vj_+R}{P-hS_vj_+R}\n\t\t\t\t}{\\Vertex{ n_{s}, j}}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\LDi{Padlock-02}{P-02-MV}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\Vertex{v^{j_+\\!;R}}\n\t\t\t\t\\\\\n\t\t\t\t\t\\Vertex{n_s, j}\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\t\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\displaystyle\t\n\t\t\t\t\t\\LDi{Padlock-S_vj_+R}{P-S_vj_+R}\t\t\t\n\t\t\t\t\t+\\LDi{WBT-vj_+R}{WBT-vj_+R}\n\t\t\t\t\t-2\\LDi{WBT-vj_+hR}{WBT-vj_+hR}\n\t\t\t\t}{\\Vertex{n_s, j}}\t\t\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\t\t\n\t\t}\n\t\t\t{\t\t\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\LDi{WBT-02}{WBT-02-VS}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\Vertex{v^{j_+\\!;R}}\n\t\t\t\t\\\\\n\t\t\t\t\t\\Vertex{n_s, j}\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Isolation of the classical, two-point vertices\nof diagrams~\\ref{P-Sig_vj_+} and~\\ref{WBT-Sig_vj_+} plus\ndiagram~\\ref{vj_+R-DEP}.}\n\\label{fig:bn-Ib}\n\\end{figure}\\end{center}\n\nThe next step of the diagrammatic procedure\nis to decorate the classical, two-point\nvertices of diagrams~\\ref{D-02x2-R-VS}, \\ref{D-vj_+R-02}\nand~\\ref{WBT-02-VS} with either\nan external field\nor an end of an effective propagator. \nIn the latter case we must then attach\nthe loose end of the effective propagator\nto an available structure. We refer to the \nprimary part of \na diagram as the component left over after\napplying the effective propagator relation\nas many times as possible but, each time,\nretaining only the \nKronecker-$\\delta$ contribution.\n\nAssuming that\nthe necessary structures exist, we can do the\nfollowing with a diagram possessing a single\nclassical, two-point vertex:\n\\begin{enumerate}\n\t\\item\tattach an external field;\n\n\t\\item\tattach one end of an effective propagator,\n\t\t\twith the other end attaching to:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tthe seed action vertex to which the kernel\n\t\t\t\t\t\tattaches;\n\t\t\t\t\\label{EP-SV}\n\n\t\t\t\t\\item\tone of the Wilsonian  effective action vertices;\n\t\t\t\t\\label{EP-DV}\n\n\t\t\t\t\\item\tthe kernel;\n\t\t\t\t\\label{EP-k}\n\n\t\t\t\t\\item\ta gauge remainder.\n\t\t\t\t\\label{EP-GR}\n\t\t\t\\end{enumerate}\n\\end{enumerate}\n\nIn each of~\\ref{EP-SV}--\\ref{EP-GR}\nthe effective propagator relation\ncan be applied. For the time\nbeing, we will concern ourselves\nwith just the primary\nparts of~\\ref{EP-SV}--\\ref{EP-k},\nwhich yield a series of cancellations,\nreturning to both the gauge remainder\ncontributions and other diagrams later.\nFirst, we analyse the result of\ndecorating the classical, two-point\nvertex of diagram~\\ref{D-vj_+R-02}.\n\n\\begin{cancel}[Diagram~\\ref{D-vj_+kR-vkhR}]\n\\label{Cancel:D-vj_+kR-vkhR}\nConsider the primary part of\ndiagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-DV}, above, which we note\nexists only for $j>-1$. \nFor comparison with diagram~\\ref{D-vj_+kR-vkhR},\nit is convenient\nto change variables $j \\rightarrow j+1$,\nso that the sum over $j$ once again starts\nfrom $-1$ and\nto identify $\\hat{v}^{j+2}$ with $\\hat{v}^{k}$.\nThus, \nthe two-point, tree level vertex of diagram\\ref{D-vj_+R-02}\ncan be joined to any of $j+2$ identical Wilsonian\neffective action vertices, using\nany of $j+s+2$ effective propagators.\nNoting that this effective propagator can\nattach either way round,  the\ncombinatoric factor is \n\\[\n\t2 (j+s+2)(j+2),\n\\]\nwhich, from \\eqs{eq:norm-b}{eq:norm-c}, \ncombines with $\\ensuremath{\\Upsilon}_{j+s+2,j+2}$\nto give $-\\ensuremath{\\Upsilon}_{j+s+1,j+1}$. Thus,\nthe primary part of\ndiagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-DV}, above,\nprecisely cancels\ndiagram~\\ref{D-vj_+kR-vkhR}.\n\\end{cancel}\n\n\n\\begin{cancel}[Diagram~\\ref{P-hS_vj_+R}]\n\\label{Cancel:P-Sig_vj_+R-hS}\n\nConsider the primary\npart of diagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-SV}, above, which we might\nhope cancels diagram~\\ref{P-hS_vj_+R}.\nAt first sight, though, it does not\nlook like the cancellation will quite work\nsince, although the combinatorics\nare fine, the limits on the sums over $s, \\ m$\nand $j$ differ between diagrams~\\ref{D-vj_+R-02}\nand~\\ref{P-hS_vj_+R}. However, the formation\nof a loop in diagram~\\ref{D-vj_+R-02}\nallows us to restrict the ranges\nover these sums (just as we did for diagram~\\ref{vj_+R-DEP}),\nand so the two diagrams do exactly cancel.\n\\end{cancel}\n\n\n\\begin{cancel}[Diagram~\\ref{WBT-vj_+hR}]\n\nConsider the primary\npart of diagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-k}, above. Following through\nsimilar arguments to those invoked in \ncancellation~\\ref{Cancel:P-Sig_vj_+R-hS}, it can\nbe shown that the primary\npart of diagram~\\ref{D-vj_+R-02} corresponding\nto~\\ref{EP-k} exactly cancels diagram~\\ref{WBT-vj_+hR}.\n\n\\end{cancel}\n\nThis concludes our initial discussion of\ndiagram~\\ref{D-vj_+R-02}.\nRather than immediately analysing\ndiagram~\\ref{WBT-02-VS}, which also possesses\na single classical, two-point vertex,\nit is useful to first consider\nthe decoration of diagram~\\ref{D-02x2-R-VS}\nwhich\npossesses two such vertices.\nAt $\\Op{2}$, we can discard the case in which\nan external field attaches to each of the classical,\ntwo-point vertices. If one of the vertices\nis decorated by an external field, then\nwe can attach an effective propagator to the\nother according to~\\ref{EP-DV}--\\ref{EP-GR}\nabove. If both vertices are decorated by an\nend of an effective propagator, then we can tie up\nthe associated loose ends in the following\nindependent ways: \n\\begin{enumerate}\n\t\\item\tJoin one classical, two-point vertex \n\t\t\tto a Wilsonian effective action vertex \n\t\t\tand the other to:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tthe same vertex;\n\t\t\t\t\\label{EPx2-SV}\n\n\t\t\t\t\\item\ta different vertex;\n\t\t\t\t\\label{EPx2-DV}\n\n\t\t\t\t\\item\tthe kernel;\n\t\t\t\t\\label{EPx2-V-k}\n\n\t\t\t\t\\item\ta gauge remainder;\n\t\t\t\\end{enumerate}\n\n\t\\item\tJoin one classical, two-point vertex  to the kernel and the other to:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tthe kernel;\n\t\t\t\t\\label{EPx2-k-k}\n\t\n\t\t\t\t\\item\ta gauge remainder;\n\t\t\t\t\\label{EPx2-k-GR}\n\t\t\t\\end{enumerate}\n\n\t\\item\tJoin one classical, two-point vertex  \n\t\t\tto a gauge remainder and the other to a gauge remainder\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tin the same gauge remainder structure;\n\n\t\t\t\t\\item\tin a different gauge remainder structure;\n\t\t\t\\end{enumerate}\n\n\t\\item\tJoin the classical, two-point vertices together.\n\t\t\t\\label{EP-02x2}\n\\end{enumerate}\n\n\nWe now consider the results of decorating\ndiagram~\\ref{D-02x2-R-VS} according\nto~\\ref{EPx2-SV}--\\ref{EP-02x2}.\n\n\\begin{cancel}[Diagram~\\ref{P-S_vj_+R}]\n\nThe primary part of~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-SV}, above, \nexactly cancels diagram~\\ref{P-S_vj_+R}.\n\\end{cancel}\n\n\\begin{cancel}[Diagram~\\ref{P-02-MV}]\nDiagram~\\ref{P-02-MV}\nis exactly cancelled by the primary part\nof diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EP-02x2},\nabove.\n\\end{cancel}\n\n\\begin{cancel}[Diagram~\\ref{D-vj_+kR-vkR}]\nThe primary part of diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-DV}, above,\nexactly cancels diagram~\\ref{D-vj_+kR-vkR}.\n\\end{cancel}\n\n\n\n\n\\begin{cancel}[Components of diagram~\\ref{WBT-02-VS}]\nThe primary part of diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-k-k}, above,\nexactly cancels the primary part of diagram~\\ref{WBT-02-VS}\nin which the classical, two-point vertex attaches\nto the kernel.\n\\label{Cancel:WBT-02}\n\\end{cancel}\n\n\nLooking at cancellation~\\ref{Cancel:WBT-02},\nit is tempting to say that the primary\nparts of diagram~\\ref{WBT-02-VS}\nare all\nexactly cancelled by a subset of the\nprimary parts\nof diagram~\\ref{D-02x2-R-VS}.\nHowever, this is not true: consider\nthe primary part of diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-k-GR}, above.\nIn this case, because each of the classical, two-point\nvertices attaches to something different,\nwe pick up a factor of two, in recognition\nof the fact that either of the vertices\ncould attach to the kernel \/ gauge remainder.\nHowever, the primary part of diagram~\\ref{WBT-02-VS}\nin which the classical, two-point vertex attaches\nto a gauge remainder can only be formed in one way;\nhence the cancellation is only partially realized.\nThe key to cancellation~\\ref{Cancel:WBT-02}\nis that both classical, two-point vertices\nof diagram~\\ref{D-02x2-R-VS} attach to the same structure,\nand so there is no factor of two picked up.\n\nA\ndiagram generated by the partial cancellation of\na component of\ndiagram~\\ref{WBT-02-VS}\nagainst a component of diagram~\\ref{D-02x2-R-VS}\nis immediately involved in a cancellation.\n\n\\begin{cancel}[Diagram~\\ref{WBT-vj_+R}]\n\\label{Cancel:WBT-vj_+R}\nThe primary part of diagram~\\ref{D-02x2-R-VS}\ncorresponding to~\\ref{EPx2-V-k} above and\nthe primary part of diagram~\\ref{WBT-02-VS}\nin which the classical, two-point attaches\nto a vertex combine to exactly cancel\ndiagram~\\ref{WBT-vj_+R}.\n\\end{cancel}\n\nUsing cancellations~\\ref{Cancel:D-vj_+kR-vkhR}--\\ref{Cancel:WBT-vj_+R} \nand partial cancellations of the type just discussed,\nwe can rewrite $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ as shown in \\fig{fig:bn-b-Pb}.\n$\\nLV{{\\cal G}}{n}{\\mu\\nu}{a}(p)$  and $\\nLV{{\\cal H}}{n}{\\mu\\nu}{a}(p)$\nare disjoint sets whose elements are\nall spawned by $a_0$ and $a_1$: all such diagrams which \npossess an $\\Op{2}$ stub belong to\n$\\nLV{{\\cal H}}{n}{\\mu\\nu}{a}(p)$;\nall elements\nof  $\\nLV{{\\cal G}}{n}{\\mu\\nu}{a}(p)$ possess a\nfull gauge remainder\n(arising from the application of the effective propagator relation).\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{3ex}\n\t\t\\displaystyle \\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet} + \\Op{4} = \n\t\t-4 \\beta_n \\Box_{\\mu\\nu}(p)  +\n\t\t\\mbox{\\ref{beta-terms}} + \\mbox{\\ref{alpha-terms}} +\n\t\t\\nLV{{\\cal G}}{n}{\\mu\\nu}{a}(p) + \\nLV{{\\cal H}}{n}{\\mu\\nu}{a}(p)\n\t\\\\\n\t\\vspace{3ex}\n\t\t+\n\t\t\\left[\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2}\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\t\\sum_{m'=1}^{m} \\!\\! \\nCr{m}{m'} \n\t\t\t\\\\\n\t\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle \\sum_{m''=1}^{m'-1}  \\!\\! \\nCr{m'}{m''} \\!\\! \n\t\t\t\t\t\\LDi{GRs-W-GRs}{GRs-W-GRs-MV} \\hspace{1.5em}\n\t\t\t\t\t+ \\LDi{Ubend-W}{Ubend-W-MV} \\hspace{1em}\n\t\t\t\t\t+ \\LDi{WBT-GRs}{WBT-GRs-MV}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\\\\n\t\\vspace{3ex}\n\t\t\\displaystyle\n\t\t+2 \\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{(m-1)!}\n\t\t\\dec{\n\t\t\t\\sco[1]{\\LO{\\Vertex{v^{j_+\\!;R}}}{j+2-DGR-b}}{\\Vertex{n_s, j}}\n\t\t}{11\\Delta^{j+s+1} >^{m-1} \\stackrel{\\bullet}{>}}\n\t\\\\\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=1}^{2s+1} \\sum_{m'=1}^m \\sum_{j=-1}^{n+s-m-1}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\nCr{m}{m'}\t\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\\LO{\\cd{v_j+hR-W-GRs}}{v_j+hR-W-GRs}}{\\Vertex{n_s, j}}\n\t\t\t\t-\\sco[1]{\\LO{\\cd{v_j+R-W-GRs}}{v_j+R-W-GRs}}{\\Vertex{n_s, j}}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{A re-expression of $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$.}\n\\label{fig:bn-b-Pb}\n\\end{figure}\\end{center}\n\nThe formation of diagram~\\ref{GRs-W-GRs-MV}\ndeserves further comment. This diagram\nis a combination of the following terms:\n\\begin{enumerate}\n\t\\item\tthe primary part\n\t\t\tof~\\ref{D-02x2-R-VS} in which both classical,\n\t\t\ttwo-point vertices attaches to a different\n\t\t\tgauge remainder structure\n\n\t\\item\tthe component of diagram~\\ref{j+2-DEP} in which both\n\t\t\tends of the differentiated effective propagator\n\t\t\tattaches to a different\n\t\t\tgauge remainder structure (see diagram~\\ref{GRs-DEP-GRs-MV}).\n\\end{enumerate}\n\nNotice that the limits on the sums over $m$ and $j$,\nin diagram~\\ref{GRs-W-GRs-MV} are the same as in\ndiagram~\\ref{j+2-DEP} and not~\\ref{D-02x2-R-VS}.\nDiagram~\\ref{D-02x2-R-VS} supplements diagram~\\ref{j+2-DEP},\nconverting a differentiated effective propagator\ninto a full kernel in all cases where there\nare sufficiently few gauge remainders and vertices\nto permit decoration of the kernel. \nThe values of $m$ and $j$ for which diagram~\\ref{j+2-DEP}\nexists but diagram~\\ref{D-02x2-R-VS} does not are\nthose for which the differentiated effective\npropagator of~\\ref{j+2-DEP} can be trivially\nreplaced by a full kernel, since this kernel\ncan effectively never be decorated.\n\nSimilar considerations apply to diagrams~\\ref{Ubend-W-MV}\nand~\\ref{v_j+R-W-GRs}. In the\nlatter case, to save space, we have put this diagram\nunder the same summations as diagram~\\ref{v_j+hR-W-GRs}.\nWhereas diagram~\\ref{v_j+hR-W-GRs} exists for\nthe maximum values of $m$ and $j$, diagram~\\ref{v_j+R-W-GRs}\npossesses a Wilsonian effective action, one-point\nvertex in these cases and so has no support: the\nmaximum values of $m$ and $j$ for which diagram~\\ref{v_j+R-W-GRs}\nexists are $2s$ and $n+s-m-2$, respectively.\nIn the same fashion, diagram~\\ref{WBT-GRs-MV} does \nnot in fact exist for the\nmaximum values of $m$ indicated. Having taken great\ncare, up until now, to restrict the ranges on\nthe sums as much as possible for each diagram,\nwe will now often put diagrams under the same summation\nsign to save space, mindful that not all diagrams\nnecessarily exist for all values of $s$, $m$ and $j$.\n\n\n\\subsection{Gauge Remainders}\n\\label{sec:bn-GRs}\n\nThe terms of $\\nLV{{\\cal G}}{n}{\\mu\\nu}{a}(p)$ can be usefully\ndecomposed into five sets comprising:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\Roman{enumi}}\n\t\\item\tdiagrams possessing a single full gauge remainder, \n\t\t\tthe $>$ part of which attaches to a kernel,\n\t\t\tthe other end of which attaches to:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\titself;\n\t\t\t\t\\label{GR-WBT}\n\t\n\t\t\t\t\\item\ta reduced seed action vertex;\n\t\t\t\t\\label{GR-S}\n\n\t\t\t\t\\item\ta reduced Wilsonian effective action vertex;\n\t\t\t\t\\label{GR-W}\n\n\t\t\t\t\\item\ta gauge remainder structure;\n\t\t\t\t\\label{GR-GRs}\n\t\t\t\\end{enumerate}\n\n\t\\item\tdiagrams possessing two full gauge remainders.\n\t\\label{GRsx2}\n\\end{enumerate}\n\nThe strategy for dealing with the gauge remainders\nis by now familiar:\nallowing the full gauge remainder\nto act, wherever possible, we isolate any\nclassical, two-point vertices and proceed as usual. \nWe start with the \ngauge remainder terms of type~\\ref{GR-WBT},\nwhich we collect together in \\fig{fig:bn:GR-WBT}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{WBT-GR-vj_+R}{WBT-GR-vj_+R}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{WBT-GR-B}{WBT-GR-B-MV}}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\t\n\t\\\\\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!} \\sum_{m'=2}^m \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{WBT-GR-GRs}{WBT-GR-GRs-MV}}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GR-WBT}.}\n\\label{fig:bn:GR-WBT}\n\\end{figure}\\end{center}\n\nThe pattern of cancellations we will find\nis very similar to that observed in the\ndemonstration of the transversality\nof $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$. Thus, although\ndiagram~\\ref{WBT-GR-GRs-MV}\npossesses a trapped gauge remainder, and so cannot\nbe processed, it has been collected with\nthe other terms in anticipation of a\ncancellation via diagrammatic identity~\\eq{eq:D-ID-G}.\n\n\nDiagram~\\ref{WBT-GR-B-MV} \ncan be processed by allowing the gauge remainder\nto act. This gauge remainder can strike one of three\nlocations:\n\\begin{enumerate}\n\t\\item\tthe top end of the kernel (which itself attaches\n\t\t\tto the middle of the kernel);\n\n\t\\item\tthe bottom end of the kernel;\n\n\t\\item\ta socket, which can be filled by one of the\n\t\t\tdecorations. \n\\end{enumerate}\n\nSumming over diagrams of the first item\nyields zero:\nbecause the kernel\nbites itself, its top end can be struck in two\nindependent ways. First, the active gauge remainder can \npush forward (pull back) onto the field on the\nkernel which the kernel bites; this field\nis identified with the top end of the kernel.\nSecondly, the active gauge remainder\ncan pull back (push forward)  unhindered to the top.\nThese two independent cases exactly cancel\neach other (see also the discussion around\ndiagrams~\\ref{Diags-WGRx2-Trap}--\\ref{Diags-WGRx2-Bottom-B}).\n\nNow we consider processing the active\ngauge remainder of diagram~\\ref{WBT-GR-vj_+R}\nand splitting off the classical, two-point\ncomponent from the reduced component.\nCompared to diagram~\\ref{CTP-EGR-Socket},\nthe classical, two-point vertex can,\nin addition to being attached  \nto a reduced Wilsonian\neffective action vertex or gauge remainder\nstructure, be attached  \nto the kernel or decorated with an \nexternal field. If it attaches\nto the kernel, this just cancels\nthe component of diagram~\\ref{WBT-GR-B-MV} \nwhere the gauge remainder bites a socket\non the kernel, up to a gauge remainder\ncontribution. This cancellation,\ntogether with the analogue of\nillustrative cancellation~\\ref{Icancel:Tansverse}\noccurs at each level of nesting.\nOf those diagrams which survive, two\ncancel against diagram~\\ref{WBT-GR-GRs-MV}\nvia diagrammatic identity~\\eq{eq:D-ID-G}.\n\nUp to those diagrams with an $\\Op{2}$\nstub, the only diagrams which survive\nare those spawned by diagram~\\ref{WBT-GR-B-MV} \nand its nested partners in which the\ngauge remainder strikes the bottom end\nof the kernel, as shown in \\fig{fig:bn-GR-WBT-Surv}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m'=1}^{2s-1} \\sum_{m=0}^{2s-1-m'} \\sum_{j=-2}^{n+s-m-m'-3} \t\n\t\t}\n\t\t\t{\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!m'!}\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{WBT-GRs}{WBT-ring}}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\]\n\\caption{Surviving terms generated by the diagrams of \\fig{fig:bn:GR-WBT}, \nup to terms with an $\\Op{2}$ stub.}\n\\label{fig:bn-GR-WBT-Surv}\n\\end{figure}\\end{center}\nThe sign and combinatoric factor of diagram~\\ref{WBT-ring} \nfollow from the discussion\naround \\fig{fig:nestedhook}.\n\n\n\\begin{cancel}[Diagram~\\ref{WBT-GRs-MV}]\n\tDiagram~\\ref{WBT-ring} exactly cancels\n\tdiagram~\\ref{WBT-GRs-MV}. This follows\n\tfrom first letting $m \\rightarrow m-m'$ in~\\ref{WBT-ring}\n\tand recognizing that \n\t\\[\n\t\\sum_{m'=1}^{2s-1} \\sum_{m=m'}^{2s-1} = \\sum_{m=1}^{2s-1} \\sum_{m'=1}^m,\n\t\\]\n\tand secondly recalling that the upper limits on the sums over\n\t$m$ and $j$ in diagram~\\ref{WBT-GRs-MV} can be reduced by\n\tone apiece.\n\\end{cancel}\n\n\nWe have thus demonstrated how all diagrams\nspawned by the type~\\ref{GR-WBT}\ngauge remainders, up to those with an $\\Op{2}$\nstub, cancel. \nIn preparation for our treatment of the other \ngauge remainders, it is worth reviewing this\nhow this happens.\nIf an active gauge remainder\nstrikes a socket on a reduced vertex\nor on a kernel, this diagram will be\ncancelled by a diagram in which the gauge\nremainder instead strikes a socket on\na classical, two-point vertex, which\nis subsequently attached to either\na reduced vertex or a kernel with an \neffective propagator. Such cancellations\nleave over diagrams of exactly the same\nbasic form as the parent diagrams,\nbut where the gauge remainder is nested.\nAllowing the nested gauge remainder\nto act then just repeats the cancellations\njust described. We are left with following:\n\\begin{enumerate}\n\t\\item\tDiagrams possessing a \n\t\t\tclassical, two-point vertex attached to\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tan $\\Op{2}$ stub;\n\n\t\t\t\t\\item  a gauge remainder nested with respect to\\ to\n\t\t\t\t\t\tthe original, active gauge remainder;\n\t\t\t\t\\label{it:nested-wrt-active}\n\n\t\t\t\t\\item\ta gauge remainder structure.\n\t\t\t\t\\label{it:GR-structure}\n\t\t\t\\end{enumerate}\n\n\t\\item\tThe original type~\\ref{GR-WBT} diagram possessing a trapped\n\t\t\tgauge remainder;\n\t\\label{it:trapped}\n\n\t\\item\tDiagrams in which a kernel is bitten at its end by\n\t\t\ta (nested) gauge remainder.\n\t\\label{it:BittenKernel}\n\\end{enumerate}\nDiagrams of type~\\ref{it:nested-wrt-active}, \\ref{it:GR-structure}\nand~\\ref{it:trapped} cancel, courtesy of diagrammatic\nidentity~\\ref{eq:D-ID-G}.\nDiagrams of type~\\ref{it:BittenKernel} cancel\nagainst terms generated elsewhere in the calculation.\n\n\nWe thus see that most of the diagrams spawned by the\ntype~\\ref{GR-WBT} gauge remainders cancel amongst themselves.\nThe wonderful thing is that this basic pattern of cancellations\nis repeated for all the other types of gauge remainders. \nThe differences come with the types of terms generated\nwhich cancel against terms generated elsewhere in\nthe calculation. With this in mind, we now treat\nthe gauge remainders of types~\\ref{GR-S} and~\\ref{GR-W}, which\nare collected together in \\figs{fig:bn-GR-S}{fig:bn-GR-W},\nrespectively.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{4ex}\n\t\t+2\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s}  \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-vkhR-GR} \\SumVertex}{D-vj_+kR-vkhR-GR}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\\\\\n\t\\vspace{4ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{Padlock-hS_vj_+R-GR}{P-hS_vj_+R-GR}\n\t\t\t\t\t+\\LDi{WBT-vj_+hR-GR}{WBT-vj_+hR-GR}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n \\sum_{m=2}^{2s} \\sum_{m'=2}^m \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\nCr{m}{m'}\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{vj_+h-W-GR-GRs}{vj_+h-W-GR-GRs}\n\t\t\t\t}{\\SumVertex}\t\t\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GR-S}.}\n\\label{fig:bn-GR-S}\n\\end{figure}\\end{center}\n\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{4ex}\n\t\t-2\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-1}^{n+s-m-4} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-vkR-DW-GR} \\SumVertex}{D-vj_+kR-vkR-DW-GR}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\\\\\n\t\\vspace{4ex}\n\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{Padlock-S_vj_+R-DW-GR}{P-S_vj_+R-DW-GR}\n\t\t\t\t\t+\\LDi{WBT-vj_+R-DW-GR}{WBT-vj_+R-DW-GR}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n \\sum_{m=2}^{2s-1} \\sum_{m'=2}^m \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\nCr{m}{m'}\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{vj_+R-DW-GR-GRs}{vj_+R-DW-GR-GRs}\n\t\t\t\t}{\\SumVertex}\t\t\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GR-W}.}\n\\label{fig:bn-GR-W}\n\\end{figure}\\end{center}\n\nIt is apparent that the gauge remainders\nof types~\\ref{GR-S} and~\\ref{GR-W} are\nalmost exactly the same, which is why\nwe choose to treat them together.\nThe differences are first that the gauge\nremainders of type~\\ref{GR-S} possess\na seed action vertex and secondly that the kernels\nof all type~\\ref{GR-W} diagrams\nmust be decorated. In diagrams~\\ref{D-vj_+kR-vkhR-GR}\nand~\\ref{vj_+h-W-GR-GRs}, we note\nthat we can reduce the maximum values\nof the sums over $m$ and $j$ by one more\nthan expected: if the seed action vertex is\none-point, then the kernel must be decorated,\nelse the full gauge remainder is forced\nto be in the $C^i$ sector---where of course it\nis null---by charge conjugation\\ invariance.\n\nProcessing the gauge remainders, it is\nobvious what will happen. \nDiagram~\\ref{D-vj_+kR-vkhR-GR} (\\ref{D-vj_+kR-vkR-DW-GR})\nhas two components: one in which the\ngauge remainder strikes a socket\non a reduced vertex and one in which\nthe gauge remainder strikes a socket\non a classical, two-point vertex.\nThe former case is cancelled, up to a nested \ngauge remainder by the contribution in which\nthe classical, two-point\nvertex is joined to one of the Wilsonian\neffective action vertices. If, instead,\nthe classical, two-point vertex is attached\nto a socket on either the seed action vertex or\nkernel then, up to a nested gauge remainder, \nthis cancels the contributions\nfrom diagrams~\\ref{P-hS_vj_+R-GR} \n(\\ref{P-S_vj_+R-DW-GR}) and~\\ref{WBT-vj_+hR-GR} (\\ref{WBT-vj_+R-DW-GR})\nin which, respectively, the gauge strikes a socket\non the reduced component of the seed action\nvertex and a socket on the kernel.\nUp to terms with an $\\Op{2}$ stub, the final type of diagram \nspawned by~\\ref{D-vj_+kR-vkhR-GR} (\\ref{D-vj_+kR-vkR-DW-GR})\nis one in which the classical, two-point vertex\nformed by the action of the gauge remainder\nis attached to a gauge remainder structure.\nPutting this to one side for the moment,\nwe note that  processing the nested gauge remainders\nrepeats this pattern of cancellations. \n\nAt the first level of nesting we get, as usual, a\nnew type of diagram: one in which the classical,\ntwo-point vertex generated by~\\ref{D-vj_+kR-vkhR-GR} \n(\\ref{D-vj_+kR-vkR-DW-GR}) attaches to the nested\ngauge remainder. Iterating the\ndiagrammatic procedure, diagrams of this type, together\nwith diagrams of the type temporarily put aside,\nexactly cancel diagram~\\ref{vj_+h-W-GR-GRs} (\\ref{vj_+R-DW-GR-GRs}),\ncourtesy of diagrammatic identity~\\eq{eq:D-ID-G}.\n\nOf the diagrams that remain, two are immediately involved\nin cancellations.\n\n\\begin{cancel}[Diagrams~\\ref{v_j+hR-W-GRs} and~\\ref{v_j+R-W-GRs}]\n\nDiagram~\\ref{v_j+hR-W-GRs} (\\ref{v_j+R-W-GRs}) is exactly\ncancelled by the component of diagrams~\\ref{WBT-vj_+hR-GR}\n(\\ref{WBT-vj_+R-DW-GR}) and its nested partners in which the\ngauge remainder strikes the top end of the kernel.\n\n\\end{cancel}\n\nUp to $\\Op{2}$ terms, the diagrams which survive are the following terms and\ntheir nested partners.\n\\begin{enumerate}\n\t\\item\tThe component of diagram~\\ref{P-hS_vj_+R-GR}\n\t\t\tin which the gauge remainder strikes\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\ta socket on a classical, two-point vertex;\n\t\t\t\t\t\t\\label{GR-S-CTP}\n\n\t\t\t\t\\item\tthe field which attaches to the\n\t\t\t\t\t\tbase of the kernel, and for which the\n\t\t\t\t\t\tseed action vertex is:\n\t\t\t\t\t\t\\begin{enumerate}\n\t\t\t\t\t\t\t\\item a reduced vertex;\n\t\t\t\t\t\t\t\\label{GR-S-BoK-R}\t\t\t\n\t\t\t\t\t\t\t\\item a classical, two-point vertex;\n\t\t\t\t\t\t\t\\label{GR-S-BoK-CTP}\n\t\t\t\t\t\t\\end{enumerate}\n\t\t\t\\end{enumerate}\n\n\t\\item\tThe component of diagram~\\ref{WBT-vj_+hR-GR} in which \n\t\t\tthe gauge remainder strikes the bottom end of the\n\t\t\tkernel;\n\t\t\t\\label{GR-S-K}\n\t\n\t\\item\tThe component of diagram~\\ref{P-S_vj_+R-DW-GR}\n\t\t\tin which the gauge remainder strikes\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\ta socket on a classical, two-point vertex;\n\t\t\t\t\t\t\\label{GR-W-CTP}\n\n\t\t\t\t\\item\tthe field which attaches to the\n\t\t\t\t\t\tbase of the kernel, and for which the\n\t\t\t\t\t\tseed action vertex is:\n\t\t\t\t\t\t\\begin{enumerate}\n\t\t\t\t\t\t\t\\item a reduced vertex;\n\t\t\t\t\t\t\t\\label{GR-W-BoK-R}\t\t\n\t\t\t\t\t\t\t\\item a classical, two-point vertex;\n\t\t\t\t\t\t\t\\label{GR-W-BoK-CTP}\n\t\t\t\t\t\t\\end{enumerate}\n\t\t\t\\end{enumerate}\n\n\t\\item\tThe component of diagram~\\ref{WBT-vj_+R-DW-GR} in which \n\t\t\tthe gauge remainder strikes the bottom end of the\n\t\t\tkernel;\n\t\t\t\\label{GR-W-K}\n\\end{enumerate}\n\nNow, \\ref{GR-S-BoK-R} and~\\ref{GR-S-K} exactly cancel\n({\\it cf.}\\  illustrative cancellation~\\ref{Icancel:v-W-GR-ai}).\nHowever, \\ref{GR-W-BoK-R} and~\\ref{GR-W-K} do not:\nin the former case, the kernel must be decorated\nwhereas, in the latter case, there is no such restriction.\nThus, a diagram possessing a differentiated effective\npropagator is left behind.\nThe component of diagram~\\ref{GR-S-CTP} (\\ref{GR-S-BoK-CTP})\nin which the kernel is decorated is exactly\ncancelled by diagram~\\ref{GR-W-CTP} (\\ref{GR-W-BoK-CTP}).\nThus, we are left with a whole set of diagrams,\ncollected together in \\fig{fig:TLTP-Survivors},\npossessing differentiated effective propagators.\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!} \\sum_{m'=0}^{m}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\nCr{m}{m'} \\!\\!\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{RV-DEP-B-DEP}{RV-DEP-B-DEP} \\ \n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s}>^{m-m'}}\n\t\t}\t\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}  \\sum_{m'=0}^{m} \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LDi{TLTP-DEP-B-DEP}{TLTP-DEP-B-DEP-MV} \\ \n\t\t\t\t\t+ \\LDi{TLTP-DEP-B-Soc}{TLTP-DEP-B-Soc-MV}}\n\t\t\t}{11\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Surviving terms from the type~\\ref{GR-S} and~\\ref{GR-W}\ngauge remainders, up to those with an $\\Op{2}$ stub.}\n\\label{fig:TLTP-Survivors}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{TLTP-DEP-B-Soc-MV} can be simplified\nby utilizing the primary diagrammatic identities\nand the classical flow equation. We illustrate this\nby considering the un-nested version; identical \nmanipulations can be performed in the nested case.\n\\begin{eqnarray}\n\t\\cd{TLTP-DEP-GR} \\\t& =\t& \\dec{\\cd{TLTP-EP-GR-B}}{\\bullet} - \\cd{TLTP-LdL-EP-GR} - \\cd{TLTP-EP-GR-LdL}\t\\nonumber\n\\\\[1ex]\n\t\t\t\t\t\t& =\t& \\dec{\\cd{TLTP-GR-PEP}}{\\bullet} - \\cd{TLTP-LdL-GR-PEP} - \\cd{GR-LdL} + \\cd{GR-GR-LdL} \\nonumber\n\\\\[1ex]\n\t\t\t\t\t\t& =\t& - \\cd{GR-LdL}\n\\label{eq:GR-LdL}\n\\end{eqnarray}\n\nTo go from the first line to the second, we have employed diagrammatic \nidentity~\\eq{eq:PseudoEP} and the effective\npropagator relation. On the second line, the\nfirst term vanishes courtesy of diagrammatic\nidentity~\\eq{eq:GR-TLTP}; similarly, the second\nterm, if we employ~\\eq{eq:LdL-GRk-Pert}. \nThe final term on the second line\nvanishes on account of diagrammatic identities~\\eq{eq:GR-relation}\nand~\\eq{eq:LdL-GRk-Pert}:\n\\[\n\t\\dec{ \\rhd \\!\\! >}{\\bullet} = 0 = \\stackrel{\\bullet}{\\rhd} \\! > + \\rhd \\! \\! \\stackrel{\\bullet}{>} = \\rhd \\!\\!  \\stackrel{\\bullet}{>}.\n\\]\nThus, the structure in diagram~\\ref{TLTP-DEP-B-Soc-MV}\ncomprising a classical, two-point vertex,\ndifferentiated effective propagator\nand $>$ reduces to just\na differentiated $>$. We now promote\nthe $\\stackrel{\\bullet}{>}$ to an implicit\ndecoration noting that, according to the rules\nfor converting implicit decorations to explicit\ndecorations, this comes with a minus sign\n(see the discussion around \\fig{fig:nestedhook}).\nFinally, the decorations $\\decGR{\\ }{>^{m'} \\stackrel{\\bullet}{>}}$\ncan be combined with the overall decorations $>^m$.\nUpon shifting $m \\rightarrow m' -1$ we find that:\n\\begin{cancel}[Diagram~\\ref{j+2-DGR-b}]\n\tDiagram~\\ref{TLTP-DEP-B-Soc-MV} exactly\n\tcancels diagram~\\ref{j+2-DGR-b}. \n\\end{cancel}\n\n\n\nWe now proceed to decorate the classical, two-point\nvertex of diagram~\\ref{TLTP-DEP-B-DEP-MV}. In\nthe case that we join this vertex to a Wilsonian \neffective action vertex, we just cancel diagram~\\ref{RV-DEP-B-DEP},\nup to a gauge remainder. Processing this\ngauge remainder, we can think of\nthe resultant diagram as versions \nof~\\ref{RV-DEP-B-DEP} and~\\ref{TLTP-DEP-B-DEP-MV} but where the end\nof the differentiated effective propagator\nwhich attaches to the two-point vertex\nnow attaches to one of the nested gauge\nremainders, instead. \nThus, this version\nof diagram~\\ref{RV-DEP-B-DEP} is cancelled also.\nUp to diagrams with an $\\Op{2}$ stub, the only terms\nthat remain are those of the \nform~\\ref{TLTP-DEP-B-DEP-MV} \n(but where the differentiated effective propagator\ncan attach to nested gauge remainders, in addition to\nthe classical, two-point vertex)\nwhere \nthe classical, two-point vertex is attached to\none of the nested gauge remainders or to a gauge remainder\nstructure. These terms can\nbe combined, courtesy of diagrammatic\nidentity~\\eq{eq:D-ID-G}, to yield the diagrams of\n\\fig{fig:TLTP-Survivors-b}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\t\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \\sum_{m'=1}^{m}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{(m-m')!(m'+1)!}  \n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{GR-ArbGRs-DEP}{GR-ArbGRs-DEP}} \\hspace{2em}\n\t\t\t}{11\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3}  \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!} \\sum_{m'=2}^{m} \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\sum_{m''=0}^{m'-2} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{GRs-TLTP-GRs-DEP}{GRs-TLTP-GRs-DEP}} \\hspace{2em}\n\t\t\t}{11\\Delta^{j+s}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The surviving \ndiagrams spawned by the gauge remainders\nof types~\\ref{GR-S} and~\\ref{GR-W}, up to those\nwith an $\\Op{2}$ stub.}\n\\label{fig:TLTP-Survivors-b}\n\\end{figure}\\end{center}\n\n\nIt is worth expanding on the origin \nof diagrams~\\ref{GR-ArbGRs-DEP} \nand~\\ref{GRs-TLTP-GRs-DEP} a little further,\nwhich is most readily done by returning to~\\eq{eq:D-ID-G}.\nEssentially, we have combined the following terms:\n\\begin{enumerate}\n\t\\item\ta version of the first term in which a differentiated\n\t\t\teffective propagator attaches at one end to the\n\t\t\texplicitly drawn $>$ and at the other end\n\t\t\tto either the socket decorating the classical, two-point\n\t\t\tvertex or to one of the implicit decorations;\n\n\t\\item\ta version of the final term in which a differentiated\n\t\t\teffective propagator attaches at one end to the explicitly\n\t\t\tdrawn $>$ and at the other end\n\t\t\tto either the socket decorating the classical, two-point\n\t\t\tvertex or to one of the implicit decorations which\n\t\t\tforms the string of gauge remainders biting this socket.\n\\end{enumerate}\nClearly, we can re-express these diagrams in terms of:\n\\begin{enumerate}\n\t\\item\tthe second diagram of~\\eq{eq:D-ID-G} in which\n\t\t\ta differentiated\n\t\t\teffective propagator attaches at one end to the\n\t\t\tfull gauge remainder and at the other end\n\t\t\tto one of the implicit decorations;\n\n\t\\item\ta version of the final term in which a differentiated\n\t\t\teffective propagator attaches at one end to the explicitly\n\t\t\tdrawn $>$ and at the other end to the gauge remainder\n\t\t\tcomponents which decorate the top of the diagram.\n\\end{enumerate}\n\n\nHaving completed the analysis\nof the type~\\ref{GR-S} and~\\ref{GR-W} gauge remainders,  \nwe now move on to gauge remainders of\ntype~\\ref{GR-GRs}, which we collect together\nin \\fig{fig:GR-GRs}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-1}^{n+s-m-3} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!} \\sum_{m'=1}^m \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{GRs-RW-GR-RWV}{GRs-RW-GR-RWV}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \\hspace{-1em}\n\t\t\t\t\t\\LDi{GRs-DW-GR-GRs}{GRs-DW-GR-GRs} \\hspace{1.5em}\n\t\t\t\t\t+\\LDi{Ubend-DW-GR}{Ubend-DW-GR} \\hspace{1em}\n\t\t\t\t\t+\\LDi{GRBT-GRs}{GRBT-GRs} \\hspace{0.5em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GR-GRs}.}\n\\label{fig:GR-GRs}\n\\end{figure}\\end{center}\n\nAllowing the gauge remainder to act in\ndiagrams~\\ref{GRs-RW-GR-RWV} and~\\ref{GRBT-GRs},\nwe uncover the by now familiar pattern of\ncancellations. Up to terms with an $\\Op{2}$\nstub, we are left with:\n\\begin{enumerate}\n\t\\item\tdiagram~\\ref{Ubend-DW-GR}, which we notice combines\n\t\t\twith diagram~\\ref{GR-ArbGRs-DEP};\n\n\t\\item\tdiagrams in which the (nested) gauge remainder of (the nested version of)\n\t\t\t\\ref{GRBT-GRs} strikes:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\tthe top end of the kernel;\n\n\t\t\t\t\\item\tthe bottom end of the kernel;\n\t\t\t\\end{enumerate}\n\n\t\\item\tdiagrams in which the classical, two-point vertex of~\\ref{GRs-RW-GR-RWV}\n\t\t\tand its nested partners\n\t\t\tis attached to one of the $m'$ gauge remainders,\n\t\t\tat the top of the diagram.\n\\end{enumerate}\n\nWith a little thought,\nwe see that diagrams of the last item are\nof the same structure as diagram~\\ref{GRs-TLTP-GRs-DEP}. \nIndeed, these diagrams\ncombine, to yield a single term with a full kernel.\nThe resultant term is shown in \\fig{fig:GR-GRs-P},\ntogether with the other surviving diagrams\nspawned by the gauge remainders of type~\\ref{GR-GRs}, up\nto those with an $\\Op{2}$ stub.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s} \\sum_{j=-2}^{n+s-m-2} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t\t\t\t\\LDi{GRs-W-GRs}{GRs-W-GRs} \\hspace{1.5em} \n\t\t\t\t\t+\\LDi{Ubend-W-GR}{Ubend-W-GR} \\hspace{1em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \n\t\t}\n\t\t\t{\n\t\t\t\\sum_{m''=0}^{m'-2} \\!\\! \\nCr{m'}{m''} \\!\\! \n\t\t\t\\dec{\n\t\t\t\t\\Tower{\t\t\t\t\t\n\t\t\t\t\t\\LDi{K-GRs-B-K-GRs}{K-GRs-B-K-GRs} \\hspace{2em}\t\n\t\t\t\t\t+\\LDi{GRs-TLTP-GRs-K}{GRs-TLTP-GRs-K} \\hspace{2em}\n\t\t\t\t} \n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Surviving diagrams spawned by the \ngauge remainders of type~\\ref{GR-GRs} plus \ndiagrams~\\ref{GR-ArbGRs-DEP} and~\\ref{GRs-TLTP-GRs-DEP}.}\n\\label{fig:GR-GRs-P}\n\\end{figure}\\end{center}\n\nWe note the following about the diagram of \\fig{fig:GR-GRs-P}.\nFirst, we should explain the sign of diagram~\\ref{K-GRs-B-K-GRs},\nsince the parent diagram comes with a minus sign.\nConsider the string of $m''+1$ gauge remainders (the $+1$ corresponding\nto the explicitly drawn gauge remainder). The\nlast of these bites the end of the kernel, which\nis then attached to one of the $m'-m''$ gauge remainders.\nIn other words, we can think of the first \nof the $m'-m''$ gauge remainders not as being at the beginning of a string\nof $m'-m''$ gauge remainders, but in the middle (or end if $m'-m'' =1$) \nof a string of $m'+1$ gauge remainders. Upon explicit decoration\nof diagram~\\ref{K-GRs-B-K-GRs} with the $m'$ gauge remainders,\nour rule for determining the sign is to associate\nassociate bites to the left (right)\nwith pushes forward (pulls back), yielding $(-1)^R$.\nHowever, if the gauge remainder at the end\nof the kernel is bitten on the left (right),\nit is in fact pulled back (pushed forward)\nonto ({\\it cf.}\\  the discussion around \\figs{fig:GRstring}{fig:nestedhook}).\nHence, given our rule for decoration, we must give\nthe diagram an overall minus sign.\n\nSecondly, notice that diagram~\\ref{GRs-W-GRs} is\nexactly the same as diagram~\\ref{GRs-W-GRs-MV}\n(the difference between the lower limits on\nthe sums over $m$ and $m'$ is just an artefact\nof putting diagram~\\ref{GRs-W-GRs} under the same\nsummation signs as diagram~\\ref{WBT-GRs-MV}---which does\nexist for $m,\\ m'=1$),\nup to a relative factor of $-2$.\nThus, these two diagrams do not exactly cancel;\nto find the missing term that completes the \ncancellation---and also to understand the\nroles of diagrams~\\ref{K-GRs-B-K-GRs}--\\ref{GRs-TLTP-GRs-K}---we \nmust conclude  our discussion of\nthe gauge remainders by analysing the\ngauge remainders of type~\\ref{GRsx2},\nwhich are collected together in \\figs{fig:GRsx2-a}{fig:GRsx2-b}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{s=0}^n \\sum_{m=0}^{2s-1}  \\sum_{j=-1}^{n+s-m-4} \n\t\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\t\\\\\n\t\t\t\t\t\\displaystyle\t\t\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\LO{\\cd{Dumbbell-vj_+kR-vkR-DW-GRx2} \\SumVertex}{D-vj_+kR-vkR-DW-GRx2}\n\t\t\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-3} \\sum_{j=-2}^{n+s-m-5}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\t\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LDi{Struc-WGRx2-B}{WGRx2}}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\\\\n\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n   \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{Padlock-S_vj_+R-DW-GRx2}{P-S_vj_+R-DW-GRx2}\n\t\t\t\t\t+2\\LDi{WBT-vj_+R-DW-GRx2}{WBT-vj_+R-DW-GRx2}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GRsx2}, part~1.}\n\\label{fig:GRsx2-a}\n\\end{figure}\\end{center}\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n \\sum_{m=2}^{2s-1} \\sum_{m'=2}^m \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\nCr{m}{m'}\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{vj_+R-DW-GRx2-GRs}{vj_+R-DW-GRx2-GRs}\n\t\t\t\t}{\\SumVertex}\t\t\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\n\t\t}\n\t\\\\\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \t\\hspace{-1em} \n\t\t\t\t\t\\LDi{GRs-DW-GRx2-GRs}{GRs-DW-GRx2-GRs} \t\\hspace{1.5em}\n\t\t\t\t\t+\\LDi{Ubend-GRsx2}{Ubend-GRsx2}\t\t\t\\hspace{1em}\n\t\t\t\t\t+2\\LDi{WBT-GRx2-GRs}{WBT-GRx2-GRs}\t\t\\hspace{0.8em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Gauge remainders of type~\\ref{GRsx2}, part~2.}\n\\label{fig:GRsx2-b}\n\\end{figure}\\end{center}\n\n\nThe first thing to note about the\ngauge remainders of type~\\ref{GRsx2} is\nthat we are guaranteed to generate\ntrapped gauge remainders in \ndiagrams~\\ref{WGRx2}\nand~\\ref{P-S_vj_+R-DW-GRx2} \n({\\it cf.}\\  \\figs{fig:DoubleGR}{fig:DoubleGR-A}). \nIn each of these\ncases, we can choose to act with\neither gauge remainder first, and one of\nthe things it will do is bite the field\non the structure to which the tip of\nthe other gauge remainder is attached. If,\non the other hand, the first action of \none of the gauge remainders is to bite a socket\nor the end of a kernel, then \nwe are free to perform the \nother gauge remainder as well. However,\nit is inefficient to allow the second\ngauge remainder to act immediately, \nas we can identify\ncancellations prior to this; we employ this\nstrategy with diagrams~\\ref{D-vj_+kR-vkR-DW-GRx2}\nand~\\ref{WBT-vj_+R-DW-GRx2}, also.\nNote, though, that diagram~\\ref{WBT-vj_+R-DW-GRx2} is not\nsymmetrical with respect to\\ the action of its two\ngauge remainders; the most efficient way to proceed\nis to be diplomatic:\nwe take one instance of this diagram \nwhere one gauge remainder\nacts first and one instance where the other \nacts first, dividing by two to avoid over-counting.\n\nAmongst the diagrams generated by processing\none of the gauge remainders of \ndiagrams~\\ref{D-vj_+kR-vkR-DW-GRx2}--\\ref{WBT-vj_+R-DW-GRx2}\nare versions of the parents, nested with respect to\\ the gauge\nremainder which acted. The strategy for dealing\nwith these terms is to process only the nested\ngauge remainder, for the time being.\n\nWe delay processing diagrams~\\ref{vj_+R-DW-GRx2-GRs}\nand~\\ref{WBT-GRx2-GRs}, which have only\na single active gauge remainder,\nuntil after one of the gauge remainders\nin each of \ndiagrams~\\ref{D-vj_+kR-vkR-DW-GRx2} \nand~\\ref{WBT-vj_+R-DW-GRx2}, and their nested partners,\nhas acted. It is easy to see why we do\nthis. Consider the component of diagram~\\ref{D-vj_+kR-vkR-DW-GRx2},\nand its nested partners, in which \na classical, two-point vertex, created\nby the action of one of the gauge remainders,\nis joined to either a nested gauge\nremainder or a gauge remainder structure. This\ndiagram partially cancels~\\ref{vj_+R-DW-GRx2-GRs},\ncourtesy of diagrammatic identity~\\eq{eq:D-ID-G} and without\nthe need to process any further gauge remainders.\nSimilarly, we process the gauge remainders\nof diagram~\\ref{WBT-vj_+R-DW-GRx2} according to the\nstrategy outlined above, and diagram~\\ref{WBT-GRx2-GRs}\nis also partially cancelled. Treating the partially\ncancelled diagrams together (and using also \ndiagram~\\ref{GRs-DW-GRx2-GRs}), it is evident that\ndiagrams in which the active gauge remainder\nstrikes a socket cancel in the usual way. Up to $\\Op{2}$ terms,\nthe only\nsurvivors from this chain of cancellations\nare the partially cancelled components of \n\\begin{enumerate}\n \t\\item\tdiagram~\\ref{vj_+R-DW-GRx2-GRs} and it nested partners\n\t\t\tin which a classical, two-point vertex, created\n\t\t\tby the action of the gauge remainder, is attached\n\t\t\tto the explicitly specified gauge remainder structure;\n\t\\label{it:CTP-ExplicitGR}\n\n\t\\item\tdiagram~\\ref{WBT-GRx2-GRs}  and it nested partners\n\t\t\tin which the active gauge\n\t\t\tremainder strikes either end of the kernel.\n\t\\label{it:GR-B-EOK}\n\\end{enumerate}\n\nReturning to diagrams~\\ref{WGRx2} \nand~\\ref{WBT-vj_+R-DW-GRx2} and their nested partners,\nif the first gauge remainder strikes the end\nof a kernel, we then have no choice but to\ngo ahead and process the second gauge remainder.\nDiagrams in which this second\ngauge remainder strikes a socket, combined\nwith diagrams of item~\\ref{it:GR-B-EOK}, above,\ncancel, in the\nusual manner.\n\nThus, up to diagrams with an $\\Op{2}$ stub, all\nsurviving terms possess either a trapped gauge\nremainder, or a kernel which is bitten at\none or both of its ends or a classical,\ntwo-point vertex attached to an explicitly\nspecified gauge remainder structure \n({\\it i.e.}\\  \\ref{it:CTP-ExplicitGR}, above).\n\nThe terms with a trapped gauge remainder,\nwhich are collected together in \n\\fig{fig:GRsx2-P}, can be simplified.\nNotice that the trapped gauge remainder\ndiagrams spawned by~\\ref{P-S_vj_+R-DW-GRx2}\n(see diagrams~\\ref{RV-RW-GR-B-RW} and~\\ref{TLTP-RW-GR-B-RW})\nare very similar to diagrams~\\ref{RV-DEP-B-DEP}\nand~\\ref{TLTP-DEP-B-DEP-MV}.\n(There is no analogue\nof diagram~\\ref{TLTP-DEP-B-Soc-MV}: if\none of the\ngauge remainders of\ndiagram~\\ref{P-S_vj_+R-DW-GRx2}\nwere to strike a socket on a classical,\ntwo-point vertex then this vertex would be\nkilled by the other gauge\nremainder, courtesy\nof diagrammatic identity~\\eq{eq:GR-TLTP}.)\nThe essential difference\nis the decoration of the kernel and the trapped\ngauge remainder. Now, we know\nthat diagrams~\\ref{RV-DEP-B-DEP}\nand~\\ref{TLTP-DEP-B-DEP-MV} can be processed\nto yield diagrams~\\ref{GR-ArbGRs-DEP} \nand~\\ref{GRs-TLTP-GRs-DEP}. Clearly, processing\nthe diagrams with trapped gauge remainders\nwill generate analogues of these diagrams, but\nwith the differentiated effective propagator\nreplaced by a decorated kernel, ending in a\ntrapped gauge remainder. Noting that the overall\nfactor of these diagrams is half of \ndiagrams~\\ref{GR-ArbGRs-DEP} \nand~\\ref{GRs-TLTP-GRs-DEP} we see that both the\ndiagrams corresponding to item~\\ref{it:CTP-ExplicitGR},\nabove, and diagram~\\ref{Ubend-GRsx2}\nare exactly cancelled.\n\nHowever, there are some new diagrams left over\nas a consequence of the kernel admitting\ndecorations. First, the\nclassical, two-point vertex of diagram~\\ref{TLTP-RW-GR-B-RW},\ncan be attached to the kernel. The corresponding\nprimary part is simply cancelled by\ndiagram~\\ref{Trapped-KBK}. The secondary\npart yields a version of diagram~\\ref{Trapped-KBK} \nwith an active gauge remainder inserted between\nthe end of the vertical line emanating from the\ntrapped gauge remainder and the kernel. If this\ngauge remainder strikes a socket, then\nwe can think of the resulting diagram as\na version of diagram~\\ref{Trapped-KBK} but\nwhere the trapped gauge remainder is contracted\ninto the gauge remainder which bites the kernel,\nrather than the kernel itself.\nThis diagram is cancelled by a term generated in the following way.\nAttach the classical, two-point vertex of diagram~\\ref{TLTP-RW-GR-B-RW}\nto a vertex and apply the effective propagator relation. \nFocusing on the gauge remainder part, allow\nthe gauge remainder to act and attach the resulting\nclassical, two-point vertex to the kernel. The primary part\nyields the term we require for our cancellation.\nIterating the diagrammatic\nprocedure, the only terms that survive are those\nin which an active gauge remainder bites one of\nthe ends of the kernel (or which possess an $\\Op{2}$ stub).\n\n\nWe collect together all surviving diagrams\nspawned by the gauge remainders of type~\\ref{GR-GRs},\nup to those with an $\\Op{2}$ stub,\nin \\fig{fig:GRsx2-S}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\t\n\t\\vspace{0.8ex}\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s} \\sum_{j=-2}^{n+s-m-2} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\displaystyle\t\n\t\t\t\t\t\\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t\t\t\t\\LDi{GRs-W-GRs}{GRs-W-GRs-b} \\hspace{1.5em}\n\t\t\t\t\t-\\LDi{Ubend-W}{Ubend-W-MV-b} \\hspace{1em} \n\t\t\t\t\t+ \\LDi{Ubend-W-GR}{Ubend-W-GR-b} \\hspace{1em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{0.8ex}\n\t\t-\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=1}^{m'-1} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LDi{K-GRs-B-K-GRs}{K-GRs-B-K-GRs-b} \\hspace{2em}\n\t\t\t\t\t+\\LDi{K-GRs-B-K-GR-GRs}{K-GRs-B-K-GR-GRs} \\hspace{2em}\n\t\t\t\t\t+\\LDi{GRs-TLTP-GRs-K}{GRs-TLTP-GRs-K-b} \\hspace{1em}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LDi{K-B-K-TLTP-EP}{K-B-K-TLTP-EP}\n\t\t\t\t}\n\t\t\t}{11\\Delta^{j+s} >^{m-m'}}\n\t\t}\t\n\t\\end{array}\n\t\\]\n\\caption{Surviving diagrams\nspawned by the gauge remainders of type~\\ref{GRsx2},\nup to those with an $\\Op{2}$ stub.}\n\\label{fig:GRsx2-S}\n\\end{figure}\\end{center}\n\n\n\nThe sign of diagram~\\ref{Ubend-W-MV-b}\nrequires comment. \nThis diagram is formed\nfrom diagram~\\ref{WGRx2} and its nested partners;\nhowever,\nwe can understand the sign from the un-nested term, alone.\nIn this case, the gauge remainder structure is formed\nas follows. The first gauge remainder bites the end\nof the kernel, just above the second (active)\ngauge remainder. This active gauge remainder\nthen bites the kernel at the end to which\nthe $>$ corresponding to the first gauge\nremainder attaches. The point is that each\nof the gauge remainders bites the end of a kernel\nand so, for this pair of gauge remainders---which\nform part of the same gauge remainder structure---bites \nto the left (right) must be interpreted as\npulls back (pushes forward). Thus, the sign of\nthe diagram is given by  $(-1)^{R-2}$.\nPromoting these gauge remainders to an explicitly\nspecified structure, $\\decGR{ \\ }{>^2}$,\nwe see from the discussion around \\fig{fig:nestedhook} that we must compensate with \nminus sign. For {\\it e.g.}\\  diagram~\\ref{GRs-W-GRs-b}\nthere is no such sign, since the gauge\nremainders of diagram~\\ref{WGRx2}\nand its nested partners have \nformed separate structures.\n\nFinally, we comment on diagram~\\ref{K-B-K-TLTP-EP}.\nThe loose end of the kernel can attach to any\nof the $m'$ $>$s or to the socket on the classical,\ntwo-point vertex. The primary part of the component\nof the diagram in which the loose end of the kernel attaches to the\nsocket\nis spawned by diagram~\\ref{WGRx2} and its\nnested partners. Indeed, the $m'=1$ case\ncorresponds to the illustrative \ndiagram~\\ref{Diags-WGR-N1-Bottom-PF} of \\fig{fig:DoubleGR-C}.\nThe corresponding secondary part \nis spawned by a version of diagram~\\ref{Trapped-KBK}\nwith a (nested) gauge remainder inserted between\nthe end of the vertical line emanating from the\ntrapped gauge remainder and the kernel. Both\nthe primary and secondary parts of the component\nof diagram~\\ref{K-B-K-TLTP-EP} in which the loose end of the \nkernel attaches to a gauge remainder are spawned\nby diagram~\\ref{WBT-vj_+R-DW-GRx2} and its nested\npartners.\n\n\n\\begin{cancel}[Diagram~\\ref{Ubend-W-MV}]\n\nDiagram~\\ref{Ubend-W-MV-b} exactly cancels\ndiagram~\\ref{Ubend-W-MV}.\n\n\\end{cancel}\n\n\n\\begin{cancel}[Diagram~\\ref{GRs-W-GRs-MV}]\n\nDiagrams~\\ref{GRs-W-GRs-b} and \\ref{GRs-W-GRs}\nexactly cancel diagram~\\ref{GRs-W-GRs-MV}.\n\n\\end{cancel}\n\n\nFinally, we conclude our treatment of\nthe gauge remainders with the\nfollowing elaborate cancellation.\n\n\\begin{cancel}[Diagram~\\ref{Ubend-W-GR}]\n\tDiagrams~\\ref{K-GRs-B-K-GRs} and~\\ref{GRs-TLTP-GRs-K}\n\tcombine with diagrams~\\ref{K-GRs-B-K-GRs-b} \n\tand~\\ref{GRs-TLTP-GRs-K-b}, reversing their\n\tsigns. The resultant diagrams, when combined with\n\tdiagrams~\\ref{K-GRs-B-K-GR-GRs} and~\\ref{K-B-K-TLTP-EP},\n\tcan be redrawn, courtesy of diagrammatic identity~\\eq{eq:D-ID-G},\n\tto yield an exact copy of diagram~\\ref{Ubend-W-GR-b}.\n\tThus the overall factor of diagram~\\ref{Ubend-W-GR-b}\n\tis doubled, providing precisely the necessary\n\tcontribution to exactly cancel diagram~\\ref{Ubend-W-GR}.\n\\end{cancel}\n\n\nReferring back to \\fig{fig:bn-b-P},\nwe have therefore demonstrated\nthat\n\\[\n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet} +\\Op{4} = -4 \\beta_n \\Box_{\\mu\\nu}(p) +\\mbox{\\ref{beta-terms}} + \\mbox{\\ref{alpha-terms}} + \\cdots,\n\\]\nwhere the ellipsis stands for terms with an $\\Op{2}$ stub, to which we\nnow turn.\n\n\n\n\\subsection{Terms with an $\\Op{2}$ Stub}\n\\label{sec:bn-Op2}\n\n\nEvery time a classical, two-point vertex was\ngenerated in the computation of $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$,\nwe had the option of decorating it with an external\nfield. In \\figs{fig:bn-Op2-a}{fig:bn-Op2-b} \nwe collect together the set of\nterms spawned by $a_0$ and $a_1$\nwhich remain after all active gauge remainders have\nbeen processed, all cancellations have been identified\nand all diagrams which manifestly vanish at $\\Op{2}$\nhave been discarded.\nThe treatment of gauge remainders in diagrams with an $\\Op{2}$\nstub is identical to the analysis just performed, and the\nsame patterns of cancellations are observed. \nWherever possible, we have combined terms possessing\na seed action vertex with their counterparts\npossessing just Wilsonian effective action vertices by\nintroducing the notation\n\\[\n\t\\Pi = S - \\hat{S}.\n\\]\nNotice, though, that since the terms possessing\nexclusively Wilsonian effective action vertices\nnecessarily possess a decorated kernel, we are\nleft over with terms possessing a seed action\nvertex attached to a differentiated effective propagator.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\t2\n\t\t\t\\bca{\n\t\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s} \\sum_{j=-1}^{n+s-m-2}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t}\n\t\t\t\t{\n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{Dumbbell-CTP-E-DEP-vh_j+R}{D-CTP-E-DEP-vh_j+R}\n\t\t\t\t\t\t-\\LDi{Dumbbell-CTP-E-DW-Pi_j+R}{D-CTP-E-DW-Pi_j+R}\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{1\\Delta^{j+s+1}>^m}\n\t\t\t}\n\t\\\\\n\t\t+\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-4}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT}\n\t\t\t\t}\n\t\t\t}{1\\Delta^{j+s+1}>^m}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{$\\Op{2}$ terms generated by $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$, part~1.}\n\\label{fig:bn-Op2-a}\n\\end{figure}\\end{center}\n\n\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\t\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{vh_j+R-DEP-GRs-Skt-CTP-E}{vh_j+R-DEP-GRs-Skt-CTP-E}\n\t\t\t\t\t-\\LDi{Pi_j+R-DW-GRs-Skt-CTP-E}{Pi_j+R-DW-GRs-Skt-CTP-E}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \n\t\t}\n\t\t\t{\n\t\t\t\\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{WBT-GRs-Skt-CTP-E}{WBT-GRs-Skt-CTP-E}\n\t\t\t\t\t-\\LDi{DW-GR-GRs-Skt-CTP-E}{DW-GR-GRs-Skt-CTP-E}\n\t\t\t\t\t+2\\LDi{DW-GRs-Skt-CTP-E}{DW-GRs-Skt-CTP-E}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-2}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \n\t\t}\n\t\t\t{\n\t\t\t\\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\! \\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LDi{GRs-dEP-GR-Skt-CTP-E}{GRs-dEP-GR-Skt-CTP-E}\n\t\t\t\t}{\\TopVertex \\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{$\\Op{2}$ terms generated by $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$, part~2.}\n\\label{fig:bn-Op2-b}\n\\end{figure}\\end{center}\n\nBefore processing diagrams~\\ref{D-CTP-E-DEP-vh_j+R}--\\ref{GRs-dEP-GR-Skt-CTP-E},\nwe emphasise a very important observation.\nConsider the set of diagrams obtained by promoting\nthe explicitly drawn external field to an implicit\ndecoration. This set of diagrams is, of course,\nnothing other than\nthe complete set of terms generated by\n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$ which possess a single,\nclassical, \ntwo-point vertex (with a free socket)\n\\emph{after} all gauge remainders\nhave acted. We can use this to immediately deduce\nthe set of diagrams left over after \ntaking $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}$\n(see \\fig{fig:nL-b})\nand iterating the diagrammatic procedure until exhaustion.\nFirst, there will be the $\\alpha$ and $\\beta$ terms.\nSecondly, there will be diagrams in which $-\\Lambda \\partial_\\Lambda|_\\alpha$\nstrikes either one of the $\\ensuremath{\\star}$s or one of the elements\nof a string of $>$s. Finally, there are those diagrams\narising from attaching all classical, two-point vertices\ngenerated at each stage of the calculation either to a\n$\\ensuremath{\\star}$ or to one of the elements of a string of\n$>$s. In the case that there is just a single,\nclassical, two-point vertex to play with, \nthe resulting set of diagrams can be directly\ndeduced from diagrams~\\ref{D-CTP-E-DEP-vh_j+R}--\\ref{GRs-dEP-GR-Skt-CTP-E}.\nIn the case that there are two classical, two-point \nvertices to play with, the resultant diagrams\nwill arise from an analogue of diagram~\\ref{D-02x2-R-VS}.\n(Note that the two classical, two-point \nvertices must be attached to elements of\nthe same \nstructure comprising a single\n$\\ensuremath{\\star}$ plus string\nof $>$s plus vertices plus $>$s, \nas follows from  \\sec{sec:Not:GRs}.)\n\n\nDiagram~\\ref{D-CTP-E-DEP-vh_j+R} is simple\nto deal with. First, notice that the differentiated\neffective propagator must be in the $A^1$ sector\n(which, incidentally, means that the seed action vertex, to which\nit attaches at one end, cannot be a one-point vertex).\nNow we redraw this diagram by detaching \nthe differentiated effective\npropagator from the seed action vertex\nand\ndecorating the seed action vertex with an $A^1$ which\ncarries the same index as the loose end of the\neffective propagator, say $\\alpha$. Next, we\ndecorate the diagram with the remaining external\nfield, which we suppose carries index $\\nu$.\nPutting the $\\Op{2}$ stub and differentiated\neffective propagator to one side, consider\ncontracting the remaining part of\nthe diagram with $(-p)_\\nu p_\\alpha$.\nIt is straightforward to demonstrate\nthat the final outcome is zero. Therefore,\nby Lorentz invariance, the diagram to which\nthe $\\Op{2}$ stub attaches via the \ndifferentiated effective propagator \nis transverse in $p$ and, just like\n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$, in fact\ngoes as\n$\\Box_{\\mu \\nu}(p) + \\Op{4}$. \nConsequently,\ndiagram~\\ref{D-CTP-E-DEP-vh_j+R} as a whole\nis $\\Op{4}$\nand so does not contribute to \n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$ at $\\Op{2}$.\n\n \nTo process diagrams~\\ref{D-CTP-E-DW-Pi_j+R}--\\ref{DW-GRs-Skt-CTP-E}\nwe construct subtractions. We begin with diagram~\\ref{CTP-E-WBT}\nfor which, as we know from \\sec{sec:Subtractions-G},\nthe subtractions completely kill the parents. This leaves\nbehind only the additions, which are collected together in\n\\fig{fig:CTP-E-WBT-Add}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-3} \\sum_{j=-2}^{n+s-m-5}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\\LDi{CTP-E-WBT-E}{CTP-E-WBT-E}}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-3} \\sum_{j=-1}^{n+s-m-5}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT-dV}\n\t\t\t\t\t}{\\cd{dvj+hR}}\n\t\t\t\t}{} \\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-3} \\sum_{j=-2}^{n+s-m-5}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\t\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT-dCTP}\n\t\t\t\t\t}{\\cd{dCTP}}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2}>^m}\n\t\t}\n\t\\\\\n\t\t+\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=3}^{2s-2} \\sum_{j=-2}^{n+s-m-4}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=3}^m\t\\nCr{m}{m'} \n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT-GRs}\n\t\t\t\t\t}{\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m'}}}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Additions for diagram~\\ref{CTP-E-WBT}.}\n\\label{fig:CTP-E-WBT-Add}\n\\end{figure}\\end{center}\n\nThe first thing to notice about the\ndiagrams of \\fig{fig:CTP-E-WBT-Add}\nis that they are very similar\nto the diagrams of \\fig{fig:nLV-c-Properties-a}\n(which we recall arose from considering \n$\\decp{\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)}{}$). Consequently,\nthey will be very easy to manipulate.\nThe differences are essentially three-fold.\nFirst, we must be careful to \nproperly take account of the\neffects of $\\decp{\\ }{}$, since it does not\nnecessarily apply to all elements of a fully\nfleshed out diagram.\nConsider first the \nnon-factorizable components of\ndiagrams~\\ref{CTP-E-WBT-E}--\\ref{CTP-E-WBT-dCTP}.\nIn this case, the socket of each diagram is\nfilled\nby the external field; there can\nbe no effective propagators carrying just $p$\nand all diagrammatic\nelements can be brought under the influence\nof $\\decp{\\ }{}$. In other words, \neach of the additions can be thought of\nas arising from Taylor expanding the corresponding components\nof the parent\ndiagram to $\\Op{2}$.\nIn the factorizable case, however,\n$p$ dependence comes not only from the $\\Op{2}$\nstub, but also from the effective propagators\ncarrying just $p$\nand the sub-diagrams to which they\nattach. These sub-diagrams may \npossess components which are \nnot Taylor expandable in $p$ and so we leave them alone.\nOnly those diagrammatic components which are not\npart of a $p$ dependent sub-diagrams can be brought\nunder the influence of $\\decp{\\ }{}$, in this case.\n\nThe second difference between the diagrams\nof \\figs{fig:nLV-c-Properties-a}{fig:CTP-E-WBT-Add}\nis that the latter diagrams do not possess\na string of gauge remainders decorating their\nclassical, two-point vertex decorated by\nthe external field. This makes life easier\nand means that we will never generate the analogue\nof the diagrams of \\fig{fig:nLV-c-Properties-e}.\nThe third difference, however, makes our life slightly\nharder: the presence of a kernel in the diagrams\nof \\fig{fig:CTP-E-WBT-Add} means there is an additional\nstructure to which effective propagators can attach.\nAs usual, this ultimately generates a set of\nsurviving diagrams in which gauge remainders\nwhich bite the kernel slide down to its base.\nImmediately, then, we can deduce the single diagram\nleft over from iterating the diagrammatic procedure\nuntil exhaustion, which is shown in \\fig{fig:CTP-E-WBT-Final}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!r!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-WBT}{CTP-E-WBT-Final}\n\t\t\t\t}{\\ensuremath{\\star}^r }\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r} >^m}\n\t\t}\n\t\\]\n\\caption{The only diagram left from the full treatment\nof the additions of \\fig{fig:CTP-E-WBT-Add}.}\n\\label{fig:CTP-E-WBT-Final}\n\\end{figure}\\end{center}\n\nWhat is to be the fate of diagram~\\ref{CTP-E-WBT-Final}?\nAs we will demonstrate shortly, it will be exactly\ncancelled by terms arising from processing\n\\begin{equation}\n\\label{eq:b-derivs}\n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}  -\n\t\\sum_{n'=1}^{n-1} \\dec{\\nLV{\\mathcal{E}}{n-n'}{\\mu\\rho}{b}(p)}{\\bullet}\n\t\\Delta^{11}_{\\rho \\sigma}(p) \\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p)\n\\end{equation}\n(see \\fig{fig:nL-b} and equation~\\eq{eq:b'-decomp}).\nBefore seeing this explicitly, we will deal with\nthe remaining diagrams of \\figs{fig:bn-Op2-a}{fig:bn-Op2-b}.\nFirst, we deal with diagram~\\ref{D-CTP-E-DW-Pi_j+R},\nthe only term other than~\\ref{CTP-E-WBT} \nthat is completely killed\nby its subtractions. Inspired by the analysis\nof~\\ref{CTP-E-WBT}, it is obvious that the\nfinal result of processing diagram~\\ref{D-CTP-E-DW-Pi_j+R}\nis the\ndiagram of \\fig{fig:CTP-E-DW-Pi_j+R-Add}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t4\n\t\\bca{\n\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+1}}{m!r!}\n\t}\n\t\t{\n\t\t\\dec{\n\t\t\t\\decp{\n\t\t\t\t\\LDi{Dumbbell-CTP-E-DW-Pi_j+R}{D-CTP-E-DW-Pi_j+R-Final}\n\t\t\t}{\\ensuremath{\\star}^r}\n\t\t\t\\SumVertex\n\t\t}{1\\Delta^{j+s+1-r}  >^m}\n\t}\n\t\\]\n\\caption{The final result of processing diagram~\\ref{D-CTP-E-DW-Pi_j+R}.}\n\\label{fig:CTP-E-DW-Pi_j+R-Add}\n\\end{figure}\\end{center}\n\n\nWe must now bite the bullet and deal with those diagrams\nwhich are not completely cancelled by their subtractions,\nstarting with diagram~\\ref{WBT-GRs-Skt-CTP-E}---whose\nsubtractions and additions are collected together in \n\\fig{fig:WBT-GRs-Skt-CTP-E-Add}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{0.5ex}\n\t\t\\mp 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-2} \\sum_{j=-2}^{n+s-m-4}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{dWBT-GRs-Skt-CTP-E}{dWBT-GRs-Skt-CTP-E-s}{dWBT-GRs-Skt-CTP-E-a}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{0.5ex}\n\t\t\\pm 2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LOLO{\n\t\t\t\t\t\t\\cd{WBT-GRs-Skt-CTP-E}  \\hspace{0.5em} \\cd{dvj+}\n\t\t\t\t\t}{WBT-GRs-Skt-CTP-E-dvj+-s}{WBT-GRs-Skt-CTP-E-dvj+-a}\n\t\t\t\t}{}\n\t\t\t\t\\SumVertex\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{0.5ex}\n\t\t\\mp\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=2}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\sum_{m''=2}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LOLO{\n\t\t\t\t\t\t\\cd{WBT-GRs-Skt-CTP-E-b}\n\t\t\t\t\t\t\\hspace{2em}\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m''}}\n\t\t\t\t\t}{WBT-GRs-Skt-CTP-E--TEGRs-s}{WBT-GRs-Skt-CTP-E--TEGRs-a}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\\\\n\t\t\\mp\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{m=1}^{2s-1} \\sum_{j=-1}^{n+s-m-3}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=1}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LLDi{WBT-TEGRs-Skt-CTP-E}{WBT-TEGRs-Skt-CTP-E-s}{WBT-TEGRs-Skt-CTP-E-a}\n\t\t\t\t\t\\hspace{1em}\n\t\t\t\t}{}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Subtractions and additions for diagram~\\ref{WBT-GRs-Skt-CTP-E}.}\n\\label{fig:WBT-GRs-Skt-CTP-E-Add}\n\\end{figure}\\end{center}\n\n\nWe proceed as before by\nisolating the classical, two-point component\nof the differentiated vertex of \ndiagram~\\ref{WBT-GRs-Skt-CTP-E-dvj+-a}, decorating it with\ntwo effective propagators and utilizing~\\eq{eq:EP-dCTP-EP}.\nHowever, compared to the treatment of diagram~\\ref{CTP-E-WBT},\nwe will encounter four novelties, all associated with\nthe $m'$ $>$s which bite\nthe classical, two-point vertex decorated by an external field.\nFirst, the active gauge \nremainders associated with~\\eq{eq:EP-dCTP-EP}\ncan \nbe contracted into one of these $m'$  $>$s or the\nsocket which they bite.\nSecondly, classical two-point vertices\ngenerated by the action of (arbitrarily)\nnested gauge remainders\ncan be similarly attached.\nThirdly, we can no longer exactly combine terms into total\nmomentum derivatives since, at this\nstage of the calculation, those terms involving\nmomentum derivatives of any of the $m'$ $>$s\nor the $>$ which bites them are missing.\nHowever, it proves\nconvenient to express the sum of momentum derivative terms\nthat we do have as a total momentum derivative (which we\nthrow away) minus the missing terms.\nLastly, we have not yet encountered\nan analogue of diagram~\\ref{WBT-TEGRs-Skt-CTP-E-a},\nand so we put this to one side for a moment.\n\nIterating the diagrammatic procedure until exhaustion,\nwe can re-express the additions~\\ref{dWBT-GRs-Skt-CTP-E-a}, \n\\ref{WBT-GRs-Skt-CTP-E-dvj+-a} and~\\ref{WBT-GRs-Skt-CTP-E--TEGRs-a}\nas shown in \\figs{fig:WBT-GRs-Skt-CTP-E-+Subs}{fig:WBT-GRs-Skt-CTP-E-+Subs-II},\nwhere we have absorbed diagram~\\ref{CTP-E-WBT-Final}\ninto diagram~\\ref{CTP-E-GRs-Combo-WBT}, in preparation for what is to follow.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-dGRs-WBT-GR}{CTP-E-dGRs-WBT-GR}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r}  >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-WBT}{CTP-E-GRs-Combo-WBT}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r}  >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The final result of processing\ndiagrams~\\ref{dWBT-GRs-Skt-CTP-E-a}, \n\\ref{WBT-GRs-Skt-CTP-E-dvj+-a} and~\\ref{WBT-GRs-Skt-CTP-E--TEGRs-a}\nplus\ndiagram~\\ref{CTP-E-WBT-Final}, part~1.}\n\\label{fig:WBT-GRs-Skt-CTP-E-+Subs}\n\\end{figure}\\end{center}\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1-r,j+2}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-GRk-WBT-GR}{CTP-E-GRs-Combo-GRk-WBT-GR}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+1-r}  >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=2}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)-1} \\sum_{j=-2}^{n+s-2r-m-3}\n\t\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s-r,j+1}}{m!(r-1)!} \\sum_{m'=0}^m \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\t\\sum_{m''=0}^{m'} \\!\\! \\nCr{m'}{m''} \\!\\!\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\decp{\n\t\t\t\t\t\\LDi{CTP-E-GRs-Combo-CTP-EP-WBT-GR}{CTP-E-GRs-Combo-CTP-EP-WBT-GR}\n\t\t\t\t\t\\hspace{1em}\n\t\t\t\t}{\\ensuremath{\\star}^{r-1}}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s-r}  >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The final result of processing\ndiagrams~\\ref{dWBT-GRs-Skt-CTP-E-a}, \n\\ref{WBT-GRs-Skt-CTP-E-dvj+-a} and~\\ref{WBT-GRs-Skt-CTP-E--TEGRs-a}\nplus\ndiagram~\\ref{CTP-E-WBT-Final}, part~2.}\n\\label{fig:WBT-GRs-Skt-CTP-E-+Subs-II}\n\\end{figure}\\end{center}\n\n\nThe cancellation of diagrams~\\ref{WBT-TEGRs-Skt-CTP-E-a} \nand~\\ref{CTP-E-dGRs-WBT-GR}--\\ref{CTP-E-GRs-Combo-CTP-EP-WBT-GR}\nfollows from processing~\\eq{eq:b-derivs}\nand\nfocusing on all surviving terms in which a kernel\nbites its own tail. As commented under \\figs{fig:bn-Op2-a}{fig:bn-Op2-b},\nthis set of terms arises from the versions\nof diagrams~\\ref{CTP-E-WBT} and~\\ref{WBT-GRs-Skt-CTP-E}\nappropriate to~\\eq{eq:b-derivs}  {\\it i.e.}\\  diagrams\nfor which the classical, two-point vertices of\n\\ref{CTP-E-WBT} and~\\ref{WBT-GRs-Skt-CTP-E}, rather than being decorated\nby an external field, are instead attached  either to a\n$\\ensuremath{\\star}$ or to one of the elements of a string of\n$>$s. \nUsing~\\eq{eq:TLTP-Combo}, we collect together\nall terms spawned by \n\\begin{equation}\n\\label{eq:LdL-b-ba}\n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}\n\t-\n\t\\sum_{n'=1}^{n-1} \\dec{\\nLV{\\mathcal{E}}{n-n'}{\\mu\\rho}{b}(p)}{\\bullet}\n\t\\Delta^{11}_{\\rho \\sigma}(p) \\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p)\n\\end{equation}\nin which a kernel\nbites its own tail\nin \\figs{fig:nLDlb-KBT}{fig:nLDlb-KBT-2}.\n\n\\begin{cancel}[Diagram~\\ref{CTP-E-dGRs-WBT-GR}]\nDiagrams~\\ref{CTP-E-WBT-GR-GRs-dGR-GRs} \nand~\\ref{CTP-E-WBT-dGR-GRs} exactly\ncancel diagram~\\ref{CTP-E-dGRs-WBT-GR}.\n\n\\end{cancel}\n\n\\begin{cancel}[Diagram~\\ref{CTP-E-GRs-Combo-WBT}]\n\nDiagram~\\ref{CTP-E-GRs-Combo-WBT-b}\nexactly cancels diagram~\\ref{CTP-E-GRs-Combo-WBT}.\n\n\\end{cancel}\n\n\n\\begin{cancel}[Diagram~\\ref{WBT-TEGRs-Skt-CTP-E-a}]\n\nDiagrams~\\ref{CTP-E-WBT-GR-GRs-GR-dGR-GRs}\nand~\\ref{CTP-E-WBT-GR-dGR-GRs}\nexactly cancel diagram~\\ref{WBT-TEGRs-Skt-CTP-E-a},\ncourtesy of diagrammatic identity~\\eq{eq:D-ID-dGRk-GT-ring}.\n\n\\end{cancel}\n\n\\begin{cancel}[Diagrams~\\ref{CTP-E-GRs-Combo-GRk-WBT-GR} and~\\ref{CTP-E-GRs-Combo-CTP-EP-WBT-GR}]\n\nDiagrams~\\ref{CTP-E-GRs-Combo-WBT-GR} and~\\ref{CTP-E-GRs-Combo-WBT-GRs-CTP-EP}\nexactly cancel \ndiagrams~\\ref{CTP-E-GRs-Combo-GRk-WBT-GR} \nand~\\ref{CTP-E-GRs-Combo-CTP-EP-WBT-GR},\ncourtesy of diagrammatic identity~\\eq{eq:D-ID-Op2-G}.\n\n\\end{cancel}\n\nThis completes the analysis of\nthe additions of diagram~\\ref{WBT-GRs-Skt-CTP-E}.\nNote that the only diagrams remaining\nin which a kernel bites its own tail\nare diagram~\\ref{WBT-GRs-Skt-CTP-E}, itself,\nand its subtractions. It should therefore\nnot come as a surprise that we can\nnow reduce \n\\begin{equation}\n\\label{eq:al+be+subs}\n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)+ \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}\n\t-\n\t\\sum_{n'=1}^{n-1} \\dec{\\nLV{\\mathcal{E}}{n-n'}{\\mu\\rho}{b}(p)}{\\bullet}\n\t\\Delta^{11}_{\\rho \\sigma}(p) \\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p)\n\\end{equation}\nto just $\\alpha$-terms, $\\beta$-terms and\nthe diagrams of \\fig{fig:bn-Op2-b} and their \nsubtractions. To see this,\nwe must explicitly construct subtractions for\ndiagrams~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E},\n\\ref{Pi_j+R-DW-GRs-Skt-CTP-E},\n\\ref{DW-GR-GRs-Skt-CTP-E},\n\\ref{DW-GRs-Skt-CTP-E}\nand~\\ref{GRs-dEP-GR-Skt-CTP-E},\nwhich are shown in \n\\figrange{fig:vh_j+R-DEP-GRs-Skt-CTP-E-Add}{fig:GRs-dEP-GR-Skt-CTP-E-Add-II}.\n\nWhen processing the additions, the only novelty is\nassociated with diagram~\\ref{dvh_j-DEP-GRs-Skt-CTP-E-a}:\nthis diagram has a component which\npossesses a classical, two-point vertex,\ndifferentiated with respect to\\ momentum, that cannot\nbe manipulated in the usual way, on\naccount of the \\emph{differentiated} effective\npropagator, already attached to one of\nits sockets. We can proceed by utilizing the\nvarious primary diagrammatic identities to\nre-express\n\\begin{eqnarray}\n\t\\cd{GR-DEP-dCTP-EP}\t& =\t& \\cd{DGR-dEP} - \\cd{GR-DEP-dFGR}\n\\nonumber\n\\\\\n\t\t\t\t\t\t& =\t& -\\cd{Dcombo} - \\cd{DdEP-GR} + \\displaystyle \\frac{1}{2} \\cd{GR-DdEP-FGR}\n\\nonumber\n\\\\\n\t\t\t\t\t\t&\t& -\\frac{1}{2} \\cd{GR-DEP-dGR-GRk} + \\frac{1}{2} \\cd{dGR-DEP-FGR}\n\\label{eq:GR-DEP-dCTP-EP}\n\\end{eqnarray}\nwhere,\nfor all elements for which decoration is not defined, we\nhave traded $\\odot$ for $\\bullet$.\n\nIncluding~\\eq{eq:GR-DEP-dCTP-EP} amongst our tricks,\nwe can now iterate the diagrammatic procedure\nuntil exhaustion. There are no further novelties\nand it is straightforward to crank the handle and\ndemonstrate that~\\eq{eq:al+be+subs}\ndoes indeed reduce \nto just $\\alpha$-terms, $\\beta$-terms and\nthe diagrams of \\fig{fig:bn-Op2-b} and their \nsubtractions. This latter collection of\nterms is remarkably easy to treat,\nand we can immediately write down\nthe result, by using the techniques of\n\\sec{sec:Subtractions-G}. As an illustration,\nwe shown the result of combining\ndiagram~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E} with its subtractions, \n\\ref{dvh_j-DEP-GRs-Skt-CTP-E-s}, \\ref{vh_kR-dvj+kh-s}, \n\\ref{vh_j-DEP-TEGRs-Skt-CTP-E-s} and~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E-b-s}\nin \\fig{fig:Subs-P}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{1ex}\n\t\t\\displaystyle\n\t\t-2 \\sum_{s=1}^n \\sum_{m=0}^{2s-1} \\sum_{m'=1}^{2s-1-m} \\sum_{j=-1}^{n+s-m-m'-3}\n\t\t\\sum_{j'=-1}^{n+s-m-m'-j-5} \\sum_{s'=-j'+1}^{j+s+1}\n\t\\\\\n\t\\vspace{2ex}\n\t\t\\displaystyle\n\t\t\\Delta^{1\\, 1}_{\\rho \\sigma}(p) \\frac{\\ensuremath{\\Upsilon}_{j'+s'+2,j'+2}}{m'!} \\frac{\\ensuremath{\\Upsilon}_{j+s+1-s',j+1}}{m!} \\sum_{m''=0}^{m}\n\t\\\\\n\t\t\\decNTEs{\n\t\t\t\\begin{array}{c}\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\cd{vj+jpr+3-R}\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\Vertex{v^{j+2},j'}\n\t\t\t\\\\\n\t\t\t\t\\cd{CTP-E}\n\t\t\t\\end{array}\n\t\t}{1_\\rho \\Delta^{j'+s'+2} >^{m'}}\n\t\t\n\t\t\\dec{\n\t\t\t\\decp{\n\t\t\t\t\\cd{vh_j+R-DEP-GRs-Skt-CTP-E-c} \\hspace{0.5em}\n\t\t\t}{}\n\t\t\t\\SumVertex\n\t\t}{1_\\sigma \\Delta^{j+s+1-s'} >^{m-m''}}\n\t\\end{array}\n\t\\]\n\\caption{The result of combining\ndiagram~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E} with its subtractions.}\n\\label{fig:Subs-P}\n\\end{figure}\\end{center}\n\nAs expected, the contributions factorize into two sub-diagrams\njoined by an effective propagator, $\\Delta^{1\\, 1}_{\\rho \\sigma}(p)$.\nThe first sub-diagram is under the influence of $\\not{\\mathrm{T}}_{M} (p)$.\nThe second sub-diagram has exactly the same structure\nas the additions for diagram~\\ref{vh_j+R-DEP-GRs-Skt-CTP-E}.\nIf we now include those terms obtained by combining\ndiagrams~\\ref{Pi_j+R-DW-GRs-Skt-CTP-E}--\\ref{GRs-dEP-GR-Skt-CTP-E}\nwith their subtractions we will generate the additions\nfor these diagrams as factorizable sub-diagrams also. \nThe crucial point to recognize is that we already know exactly\nwhat happens when we manipulate this complete set of\nadditions! We can therefore write:\n\\begin{eqnarray}\n\t-4\\beta_n \\Box_{\\mu\\nu}(p) + \\Op{4} = \n\t\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) + \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}\n\\nonumber\n\\\\\n\t\t-\\sum_{n'=1}^{n-1} \n\t\t\\dec{\n\t\t\t\\mathcal{E}_{n-n' \\mu \\rho}^{b \\hspace{1.2em} 1 \\, 1}(p)\n\t\t}{\\bullet}\n\t\t\\Delta^{1\\,1}_{\\rho \\sigma}(p) \n\t\t\\left(\n\t\t\t\\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p) +\n\t\t\t\\decNTEs{ \\nLV{\\overline{\\mathcal{E}}}{n'}{\\sigma\\nu}{c}(p)}{}\n\t\t\\right)\n\\nonumber\n\\\\\n\t\t+\n\t\t\\sum_{n'=1}^{n-1} \\sum_{n''=1}^{n'-1}\n\t\t\\dec{\n\t\t\t\\nLV{\\mathcal{E}}{n-n'-n''}{\\mu\\rho}{b}(p) \n\t\t}{\\bullet}\n\t\t\\Delta^{11}_{\\rho \\kappa}(p)\n\t\t\\nLV{\\mathcal{E}}{n''}{\\kappa\\tau}{a}(p)\n\t\t\\Delta^{11}_{\\tau \\sigma}(p)\n\t\t\\decNTEs{\\nLV{\\overline{\\mathcal{E}}}{n'}{\\sigma \\nu}{c}(p)}{}\n\\nonumber\n\\\\\n\t\t+ \\cdots\n\\label{eq:bn-nearly}\n\\end{eqnarray}\nwhere the ellipsis represents the $\\alpha$ and $\\beta$ terms.\nThere is no need to explicitly draw all of\nthese, since the complete set can be built up out\nof just the\n$\\alpha$ and $\\beta$ terms formed by\n$\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$\nand $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}$.\nThese diagrams are collected together in\n\\fig{fig:bn-al+be}; the overall signs of\nthe diagrams are appropriate to the right-hand side\\ \nof~\\eq{eq:bn-nearly}.\n\\bc\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\displaystyle\n\t\t\t\\sum_{n'=1}^{n-1}\n\t\t\t\\sum_{s=1}^{n-n'}   \\sum_{m=0}^{2s} \\sum_{j=-1}^{n-n'+s-m-1}\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\left[\n\t\t\t\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t\t\\vspace{1ex}\t\n\t\t\t\t\t\t\t\\LD{beta-terms-a}\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t& \\LD{alpha-terms-a}\t\t\t\n\t\t\t\t\t\t\\\\\t\n\t\t\t\t\t\t\t2 \\left( v^{j_+}-1 \\right) \\beta_{n'} \t& \\hspace{-1em} + \\hspace{-1em}\t& \\gamma_{n'} \\pder{}{\\alpha}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\right]\n\t\t\t\t\t\\DisplacedVertex{v^{j_+}}\n\t\t\t\t}{\\Vertex{n_{n'+s}, j}}\n\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t}\n\t\\\\\n\t\t4\n\t\t\\bca{\n\t\t\t\\sum_{n'=1}^{n-1} \\sum_{s=1}^{n-n'}  \\sum_{r=1}^s \n\t\t\t\\sum_{m=0}^{2(s-r)} \\sum_{j=-1}^{n-n'+s-2r-m-1}\n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+1}}{m!r!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\left[\n\t\t\t\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t\t\\vspace{1ex}\t\n\t\t\t\t\t\t\t\\LD{beta-terms-b}\t\t\t\t\t\t\t&\t\t\t\t\t\t\t\t& \\LD{alpha-terms-b}\t\t\t\n\t\t\t\t\t\t\\\\\t\n\t\t\t\t\t\t\t2 \\left( v^{j_+}-1 \\right) \\beta_{n'} \t& \\hspace{-1em} + \\hspace{-1em}\t& \\gamma_{n'} \\pder{}{\\alpha}\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\right]\n\t\t\t\t\t\\DisplacedVertex{v^{j_+}}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\Vertex{n_{n^{'} +s, j}}\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1 \\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^m }\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The $\\alpha$ and $\\beta$ terms contributing \nto $-4 \\beta_n \\Box_{\\mu\\nu}(p)$\nformed by $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$\nand $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)}{\\bullet}$.}\n\\label{fig:bn-al+be}\n\\end{figure}\\end{center}\n\nExamining these diagrams, and recalling the expressions\nfor the $\\mathcal{E}^{i}$, we see that we have succeeded\nin deriving an expression for $\\beta_n$\nwhich has no explicit dependence on either\nthe seed action or the details of the covariantization!\n\n\\subsection{The $\\alpha$ and $\\beta$ Terms}\n\\label{sec:al+be}\n\nWe can simplify the diagrams of \\fig{fig:bn-al+be}\nand do so by starting with the\n$\\alpha$\nterms.\nThis is done by using the diagrammatic relationship:\n\\begin{equation}\n\\label{eq:EP-dalDTP-EP}\n\t\\cd{A-EP-dalCTP-EP-B} = -\\cd{A-dalEP-B} - \\frac{1}{2} \\cd{A-GR-dalCombo-B} -\\frac{1}{2} \\cd{A-dalCombo-GR-B},\n\\end{equation}\nwhere\n\\[\n\t\\cd{dalCombo-GR} \\hspace{1em} \\equiv \\cd{EP-dalGR-GR} \\hspace{1em} -\\frac{1}{2} \\cd{GR-dalPEP-GR}\n\\]\n(and we recall that $\\Dal \\equiv \\partial \/ \\partial \\alpha$).\nComparing~\\eq{eq:EP-dalDTP-EP} with~\\eq{eq:EP-dCTP-EP}\nit is clear that the analysis of the $\\alpha$\nterms is very similar to the analysis\nof diagrams with momentum derivatives.\nIndeed, it is straightforward\nto show that the $\\alpha$ terms\nyield a contribution to\n$-4\\beta_n \\Box_{\\mu \\nu}(p)$ \nsuch that for every appearance\nof $\\dec{X_n}{\\bullet}$, there\nis a partner term\nwhich\ncan be generated by making the substitution\n\\[\n\\dec{X_n}{\\bullet} \\rightarrow - \\sum_{n'=1}^{n-1} \\gamma_{n'} \n\t\\pder{}{\\alpha} X_{n-n'}.\n\\]\nEquivalently, we can explicitly include the\n$\\alpha$-terms in~\\eq{eq:bn-nearly}\nby inserting $\\sum_n g^{2n+1}$\nat the beginning of both sides of the equation\nand then\nmaking the replacement\n\\begin{equation}\n\\label{eq:replace}\n\t-\\Lambda \\partial_\\Lambda|_\\alpha \\rightarrow -\\Lambda \\partial_\\Lambda.\n\\end{equation}\n\nThe $\\beta$ terms are more subtle\nthan one might expect. Let us begin with\ndiagram~\\ref{beta-terms-a}. Clearly, the way to proceed\nis to isolate any diagrams which possess\na classical, two-point vertex and then decorate\nthis vertex. There are two possible decorations:\neither we can attach one external field and one\neffective propagator, or we can attach two\neffective propagators; either way, we can apply\nthe effective propagator relation. In the former\ncase, the primary part simply yields\n\\begin{equation}\n\\label{eq:beta-a-nLV-a}\n\t-4 \\sum_{n'=1}^{n-1} \\beta_{n'} \n\t\\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{a}(p).\n\\end{equation}\nThe secondary part vanishes, since the\nassociated gauge remainder strikes $\\mathcal{E}^a$, which \nwe know to be transverse.\nWith this in mind,\nwe re-express\ndiagram~\\ref{beta-terms-a}  in \\fig{fig:bn-beta}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\displaystyle\n\t\t-4 \\sum_{n'=1}^{n-1} \\beta_{n'} \\mathcal{E}_{n-n' \\mu \\nu}^{a \\hspace{1.2em} 1 \\, 1}(p)\n\t\\\\\n\t\\vspace{2ex}\n\t\t+4\n\t\t\\bca{\n\t\t\t\\sum_{n'=1}^{n-1} \\sum_{s=1}^{n-n'}   \\sum_{m=0}^{2s} \n\t\t\t\\sum_{j=-1}^{n-n'+s-m-2} \\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t\t\\beta_{n'}\n\t\t}\n\t\t\t{\n\t\t\t\\left(\n\t\t\t\t\\begin{array}{ccc}\n\t\t\t\t\t\\LD{beta-vj+}\t&\t& \\LD{beta-novj+}\n\t\t\t\t\\\\[1ex]\n\t\t\t\t\tv^{j+}\t\t\t& - & 1\n\t\t\t\t\\end{array}\n\t\t\t\\right)\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\Vertex{n_{n'+s},j}}\n\t\t\t}{11\\Delta^{j+s+1} >^m}\n\t\t}\n\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{n'=1}^{n-1} \\sum_{s=1}^{n-n'}   \\sum_{m=0}^{2s} \n\t\t\t\\sum_{j=-2}^{n-n'+s-m-2} \\frac{\\ensuremath{\\Upsilon}_{j+s,j+2}}{m!}\n\t\t\t\\beta_{n'}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\begin{array}{ccc}\t\t\t\t\t\n\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\\LD{beta-EP}\t& \t& \\LD{beta-EP-GR}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\cd{Vert-EP}\t& -\t& \\cd{Vert-EP-GR}\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\Vertex{n_{n'+s},j}\n\t\t\t\t\\end{array}\n\t\t\t}{11\\Delta^{j+s} >^m}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{A re-expression of \ndiagram~\\ref{beta-terms-a}.}\n\\label{fig:bn-beta}\n\\end{figure}\\end{center}\n\nWe can now see that there are some\nnovelties associated with the\n$\\beta$ terms.\nFirst, diagram~\\ref{beta-novj+}\ndoes not have a canonical normalization factor:\nthere are $j+2$ identical vertices, but the\nnormalization factor $\\sim 1\/(j+1)!$. \nSecondly, the presence of the multiplicative\nfactor $v^{j_+}$ in diagram~\\ref{beta-vj+}\nmakes this term different from any we have \nencountered so far. Note, though, that\nthe combinatoric factor is effectively\ncanonical, since the factor of $v^{j_+}$\nsingles out one of the vertices, leaving\nbehind only $j+1$ identical vertices.\nThirdly, just as diagram~\\ref{beta-vj+} has a special\nvertex, so diagram~\\ref{beta-EP} has a special\neffective propagator. This effective\npropagator cannot be promoted to an implicit\ndecoration in the usual way, since then\nthe combinatorics will be wrong. \nTo put\nit another way, if we decorate a vertex with\na total of $q+1$ effective propagators,\ncomprising the special one and $q$\nothers, the combinatoric factor\nassociated with choosing the effective propagators\nin this way \nwill be $\\nCr{j+s}{q}$ and \\emph{not} $\\nCr{j+s+1}{q}$.\n\nNevertheless, diagrams~\\ref{beta-vj+}, \\ref{beta-novj+}\nand~\\ref{beta-EP} combine in a miraculous way.\nStarting with diagram~\\ref{beta-vj+}, we exploit\nthe indistinguishability of the $j+2$ vertices by\nreplacing $v^{j_+}$ with\n\\[\n\t\\frac{1}{j+2} \\left[ \\sum_{i=0}^j v^{i,i_+} + v^{j_+} \\right] = \\frac{n_{n'}-s}{j+2},\n\\]\nwhere we have used~\\eq{eq:VertexSum} (but with $v^0 = n_{n'} -s$).\n\nNext, consider creating some fully fleshed out\ndiagram from~\\ref{beta-EP}. The total of $j+s+1$\neffective propagators are to be divided into $q$\nsets, each containing $L_i$ effective propagators. \nSince the special effective propagator can reside\nin any of these sets, there are $q$ different ways\nto make the sets. The overall combinatoric factor\nassociated with this partitioning\nis, therefore,\n\\[\n\t\\frac{(j+s)!}{\\prod_i L_i!} \\sum_i L_i = \\frac{(j+s+1)!}{\\prod_i L_i!},\n\\]\nwhich is just the combinatoric factor expected\nfrom partitioning $j+s+1$\neffective propagators into $q$ sets.\nTherefore, we can now promote the special effective propagator\nof diagram~\\ref{beta-EP}\nto an implicit decorations but, counterintuitively, \nthe combinatoric\nfactor of the diagram, $\\ensuremath{\\Upsilon}_{j+s,j+2}$, \\emph{stays the same}!\n\nWith these points in mind, it is a trivial matter to show that \n\\begin{equation}\n\\label{eq:nLV-d}\n\t\\mbox{\\ref{beta-vj+}} + \\mbox{\\ref{beta-novj+}} \n\t\t+ \\mbox{\\ref{beta-EP}}\n\t= -2 \\sum_{n'=1}^{n-1} \\beta_{n'} (n_{n'}-1) \\nLV{\\mathcal{E}}{n-n'}{\\mu \\nu}{a}(p)\n\\end{equation}\n\n\n\n\n\n\nDiagram~\\ref{beta-EP-GR} can be processed, yielding:\n\\begin{equation}\n\\label{eq:beta-a-nLV-c}\n\t-2 \\sum_{n'=1}^{n-1} \\beta_{n'} \n\t\\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{c}(p)\n\\end{equation}\nPutting together~\\eq{eq:beta-a-nLV-a}, \\eq{eq:nLV-d} and~\\eq{eq:beta-a-nLV-c},\ndiagram~\\ref{beta-terms-a} ultimately reduces to \n\\begin{equation}\n\\label{beta-terms-a-Final}\n\t-2\\sum_{n'=1}^{n-1} \\beta_{n'} \n\t\\left[\n\t\t(n_{n'}+1) \t\\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{a}(p) + \\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{c}(p)\n\t\\right].\n\\end{equation}\n\nThe treatment of diagram~\\ref{beta-terms-b} follows in \nsimilar fashion. Now, however, our options for\ndecorating the classical, two-point vertex\nare to attach either two effective propagators or\none effective propagator and one $\\ensuremath{\\star}$\n(it is illegal to attach two $\\ensuremath{\\star}$s---see\n\\sec{sec:Not:GRs}).\nIn the latter case we proceed by expressing\n\\[\n\t\\cd{EP-CTP-Combo} = \n\t\\frac{1}{2}\n\t\\left[\n\t\t\\cdeps{Combo-b} - \\cdeps{fGR-Combo} + \\cdeps{EP-dGR-b} - \\cdeps{EP-fGR-dGR-b}\n\t\\right].\n\\]\nIn this way, we can reduce diagram~\\ref{beta-terms-b} \nto \n\\begin{equation}\n\\label{beta-terms-b-Final}\n\t\\sum_{n'=1}^{n-1} \\beta_{n'} \n\t\\left[\n\t\t2 \\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{c'}(p) - (2n_{n'}+1) \\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{b}(p) + 2\\nLV{\\tilde{\\mathcal{E}}}{n-n'}{\\mu\\nu}{b}(p)\n\t\\right],\n\\end{equation}\nwhere\n\\[\n\t\\nLV{\\tilde{\\mathcal{E}}}{n}{\\mu\\nu}{b}(p)  \\equiv\n\t\\bca{ \n\t\t4 \\sum_{s=1}^n  \\sum_{r=1}^s \\sum_{m=0}^{2(s-r)+1} \n\t\t\\sum_{j=-2}^{n+s-2r-m-1} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1)!}\n\t}\n\t\t{\n\t\t\\qquad\n\t\t\\decp{\n\t\t\t\t\\Tower{\\cd{CTP-E}}\n\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^m }{}\t\t\n\t}.\n\\]\nNotice that the $r=1$ contribution is the same as\nfor $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)$.\n\nBy combining~\\eq{eq:bn-nearly} with~\\eqs{beta-terms-a-Final}{beta-terms-b-Final}\nand utilizing~\\eq{eq:replace} we can write down the\ncomplete\nexpression for $\\beta_n$:\n\\begin{equation}\n\\label{eq:bn-Complete}\n\\fl\n\t\\begin{array}{c}\n\t\\vspace{1ex}\n\t\t\\displaystyle\n\t\t-4 \\sum_{n} g^{2n+1} \\beta_n \\Box_{\\mu\\nu}(p) + \\Op{4} = \n\t\\\\\n\t\\vspace{1ex}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{n} g^{2n+1}\n\t\\\\\n\t\t\\displaystyle\n\t\t\\left[\n\t\t\t\\begin{array}{l}\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t-\\Lambda \\partial_\\Lambda\\dec{\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p) + \\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)}{}\n\t\t\t\t\n\t\t\t\\\\\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle\n\t\t\t  \t-2\\sum_{n'=1}^{n-1} \\beta_{n'}\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\qquad\n\t\t\t\t\\times\n\t\t\t\t\\left\\{\n\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t(n_{n'} +1) \\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{a}(p)\n\t\t\t\t\t\t+\\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{c}(p)\n\t\t\t\t\t\t-\\decpb{\\nLV{\\overline{{\\cal D}}}{n-n'}{\\mu\\nu}{c}(p)}{}\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t+(n_{n'}+1) \\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{b}(p) -  \\nLV{\\tilde{{\\cal D}}}{n-n'}{\\mu\\nu}{b}(p)\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\right\\}\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t+\n\t\t\t\t\\sum_{n'=1}^{n-1}\n\t\t\t\t\\left[\n\t\t\t\t\t\\nLV{{\\cal D}}{n'}{\\mu\\rho}{a}(p) +\n\t\t\t\t\t\\decNTEsb{\n\t\t\t\t\t\t\\nLV{\\overline{{\\cal D}}}{n'}{\\mu\\rho}{c}(p)\n\t\t\t\t\t}{}\n\t\t\t\t\\right]\n\t\t\t\t\\Delta^{1\\,1}_{\\rho \\sigma}(p) \n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\qquad\n\t\t\t\t\\times\n\t\t\t\t\\left\\{\n\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\\Lambda \\partial_\\Lambda\n\t\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\t\\nLV{{\\cal D}}{n-n'}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t}{}\t\t\t\t\t\t\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t-2\\sum_{n''=1}^{n'-1} \\beta_{n''}\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\t\\decpb{\\nLV{\\overline{{\\cal D}}}{n-n'-n''}{\\sigma\\nu}{c}(p)}{} \n\t\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\t- (n_{n'+n''}+1)\\nLV{{\\cal D}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\t+ \\nLV{\\tilde{{\\cal D}}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\\right]\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\right\\}\n\t\t\t\\\\\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t-\\sum_{n'=1}^{n-1} \\sum_{n''=1}^{n'-1}\n\t\t\t\t\n\t\t\t\t\\decNTEsb{\n\t\t\t\t\t\t\\nLV{\\overline{{\\cal D}}}{n'}{\\mu\\rho}{c}(p)\n\t\t\t\t\t}{}\n\t\t\t\t\\Delta^{11}_{\\rho\\tau}(p)\n\t\t\t\t\\nLV{{\\cal D}}{n''}{\\tau\\kappa}{a}(p)\n\t\t\t\t\\Delta^{11}_{\\kappa \\sigma}(p)\n\t\t\t\\\\\n\t\t\t\t\\qquad\n\t\t\t\t\\times\n\t\t\t\t\\left\\{\n\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\\Lambda \\partial_\\Lambda\n\t\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\t\\nLV{{\\cal D}}{n-n'-n''}{\\sigma\\nu}{b}(p) \n\t\t\t\t\t\t}{}\t\t\t\t\t\t\n\t\t\t\t\t\\\\\n\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t-2\\sum_{n'''=1}^{n''-1} \\beta_{n'''}\n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\t\\decpb{\\nLV{\\overline{{\\cal D}}}{n-n'-n''-n'''}{\\sigma\\nu}{c}(p)}{} \n\t\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\t\t\t- (n_{n'+n''+n'''}+1)\\nLV{{\\cal D}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t\t\\\\\n\t\t\t\t\t\t\t\t+ \\nLV{\\tilde{{\\cal D}}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\\right]\t\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\right\\}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\end{array}\n\\end{equation}\n\nNotice that we have substituted for $\\mathcal{E}^{c'}$\nvia~\\eq{eq:nLV-c-Op2-b} and have traded\nthe $\\mathcal{E}^i$ for the ${\\cal D}^i$, as this is the\ncorrect time to discard all diagrams which\nvanish at $\\Op{2}$ and \/ or contain a Wilsonian\neffective action one-point vertex.\n\nUp until now, we have been very careful not to\nTaylor expand individual diagrams to $\\Op{2}$ unless\nwe are certain that this step is valid. However,\nwe know that the \\emph{sum} \nof diagrams contributing to $\\beta_n$ is\nTaylor expandable to $\\Op{2}$. It must therefore\nbe the case that all contributions which include\nnon-polynomial dependence on $p$, at $\\Op{2}$,\ncancel out of~\\eq{eq:bn-Complete}. At two loops\nit has already been demonstrated that a subset of\nsuch contributions cancel out diagrammatically~\\cite{Thesis,mgierg2}.\nIt is very likely that this analysis can be extended\nto any number of loops, but we leave the investigation\nof this for the future. In the meantime,\nwe note that we can considerably simplify~\\eq{eq:bn-Complete}\nif we focus just on \nthe $\\Op{2}$ component of each diagram, \nwhich yields\nthe following equation (we have used~\\eq{nLDc-factorize}). \n\\begin{equation}\n\\fl\n\t\\begin{array}{c}\n\t\\vspace{1ex}\n\t\t\\displaystyle\n\t\t-4 \\sum_{n} g^{2n+1} \\beta_n \\Box_{\\mu\\nu}(p) + \\Op{4} = \n\t\\\\\n\t\\vspace{1ex}\n\t\t\t\\displaystyle\n\t\t\t\\sum_{n} g^{2n+1}\n\t\\\\\n\t\t\\displaystyle\n\t\t\\left[\n\t\t\t\\begin{array}{l}\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t-\\Lambda \\partial_\\Lambda\\dec{\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p) + \\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)}{}\n\t\t\t\t+\\sum_{n'=1}^{n-1}\n\t\t\t\t\t\\nLV{{\\cal D}}{n'}{\\mu \\rho}{a}(p) \\Delta^{1\\,1}_{\\rho \\sigma}(p) \n\t\t\t\t\t\\Lambda \\partial_\\Lambda\n\t\t\t\t\t\\dec{\n\t\t\t\t\t\t\\nLV{{\\cal D}}{n-n'}{\\sigma\\nu}{b}(p)\n\t\t\t\t\t}{}\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\left.\n\t\t\t\t\\rule{0em}{4ex}\n\t\t\t  \t-2\\sum_{n'=1}^{n-1} \\beta_{n'}\n\t\t\t\t\\right\\{\n\t\t\t\t\t\t(n_{n'} +1) \n\t\t\t\t\t\t\\left[\n\t\t\t\t\t\t\t\\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{a}(p)\n\t\t\t\t\t\t\t+\\nLV{{\\cal D}}{n-n'}{\\mu\\nu}{b}(p)\n\t\t\t\t\t\t\\right]\n\t\t\t\t\t\t- \\nLV{\\tilde{{\\cal D}}}{n-n'}{\\mu\\nu}{b}(p)\n\t\t\t\\\\\n\t\t\t\\vspace{2ex}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\qquad +\n\t\t\t\t\\left.\n\t\t\t\t\\sum_{n''=1}^{n'-1} \n\t\t\t\t\t\\nLV{{\\cal D}}{n''}{\\mu \\rho}{a}(p) \\Delta^{1\\,1}_{\\rho \\sigma}(p) \n\t\t\t\t\\left[\n\t\t\t\t\t\\nLV{\\tilde{{\\cal D}}}{n-n'-n''}{\\sigma\\nu}{b}(p) - \n\t\t\t\t\t(n_{n'+n''}+1)\\nLV{{\\cal D}}{n-n'-n''}{\\sigma\\nu}{b}(p)\n\t\t\t\t\\right]\n\t\t\t\t\\rule{0em}{4ex}\n\t\t\t\t\\right\\}\n\t\t\t\\end{array}\n\t\t\\right]_{p^2}\n\t\\end{array}\n\\label{eq:bn-Final}\n\\end{equation}\n\nEquation~\\eq{eq:bn-Final} reduces to particularly simple\nforms at one and two loops:\n\\begin{eqnarray}\n\\label{eq:beta1}\n\\vspace{2ex}\n\t-4 \\beta_1 \\Box_{\\mu\\nu}(p) & = & \n\t\\decp{\n\t\t\\nLV{\\overline{{\\cal D}}}{1}{\\mu\\nu}{a}(p) + \\nLV{\\overline{{\\cal D}}}{1}{\\mu\\nu}{b}(p)\n\t}{\\bullet}\n\\\\\n\\nonumber\n\t-4 \\beta_2 \\Box_{\\mu\\nu}(p) & = &\n\t\\decp{\n\t\t\\nLV{\\overline{{\\cal D}}}{2}{\\mu\\nu}{a}(p)\n\t\t+ \\nLV{\\overline{{\\cal D}}}{1}{\\mu\\rho}{a}(p)\n\t\t\\Delta^{1\\,1}_{\\rho \\sigma}(p) \\nLV{\\overline{{\\cal D}}}{1}{\\sigma\\nu}{b}(p)\t\n\t\t+ \\nLV{\\overline{{\\cal D}}}{2}{\\mu\\nu}{b}(p)\n\t}{\\bullet}\n\\\\\n\\label{eq:beta2}\n\t& & \\qquad + \\gamma_1 \\pder{}{\\alpha} \n\t\t\\decp{\n\t\t\t\\nLV{\\overline{{\\cal D}}}{1}{\\mu\\nu}{a}(p) + \\nLV{\\overline{{\\cal D}}}{1}{\\mu\\nu}{b}(p)\n\t\t}{},\n\\end{eqnarray}\nwhere we have used~\\eqs{eq:EP-leading}{nLDa-factorize} and\nhave defined\n\\[\n\t\\nLV{\\overline{{\\cal D}}}{n}{\\mu\\nu}{b}(p) \\equiv\n\t\\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)\n\t+\n\t\\frac{1}{2}\n\t\\sum_{n'=1}^{n-1} \\nLV{{\\cal D}}{n-n'}{\\mu\\rho}{b}(p)\n\t\\Delta^{11}_{\\rho\\sigma}\n\t\\nLV{{\\cal D}}{n'}{\\sigma\\nu}{b}(p).\n\\]\nEquations~\\eqs{eq:beta1}{eq:beta2} reproduce the\nresults of~\\cite{mgierg1,mgierg2,Thesis}, but written\nin a remarkably compact form.\n\n\n\\section{Conclusion}\n\\label{sec:Conc}\n\n\nWe have constructed a manifestly gauge invariant\ncalculus for $SU(N)$ Yang-Mills, which can\nbe applied at any number of loops. The calculus, which\nis entirely diagrammatic in nature,\nhas been comprehensively illustrated by demonstrating\nthat the $\\beta$-function is independent of both the seed\naction and the details of the covariantization of the cutoff,\nto all orders in perturbation theory.\n\nThe inspiration for this methodology, the first\nelements of which were developed\nin~\\cite{aprop}, is the immense freedom in the\nconstruction of ERGs. Given\nthat we specialize not only to those ERGS that are\nmanifestly gauge invariant but to those which also allow \nconvenient renormalization to any loop order,\nthere are still an infinite number with which\nwe can work. The differences between these\nERGs amount to non-universal details which\nneed never be exactly specified, instead satisfying\ngeneral constraints. In the computation of\nuniversal quantities, these non-universal details\nmust cancel out; by leaving them unspecified,\nwe are thus guided towards a very constrained calculational\nprocedure. However, as recognized in~\\cite{Thesis,Primer,RG2005},\nsuch cancellations are embedded within the formalism\nat a particularly deep level: at least\nsome of them occur in the calculation of quantities which\nare not universal, such as $\\beta$-function coefficients\nbeyond two loops. This observation formed the basis for developing\nthe diagrammatic calculus to the level\ndescribed in this paper.\n\nGiven a diagrammatic representation of the flow equation,\nthe calculus comprises an operator which implements \nthe flow {\\it i.e.}\\  $-\\Lambda \\partial_\\Lambda$,\na rule for implementing\nthe effects of charge conjugation\\ invariance, a set of primary\nand secondary diagrammatic identities and the subtraction\ntechniques.\n\nThere are three types of primary diagrammatic identities\n(see \\sec{sec:Review}).\nThose of the first type make no reference to\nperturbation theory and reflect general\nproperties of the exact flow equation\nand underlying theory. The single identity\nof the second type is the effective propagator\nrelation, which arises as a solution to\nthe classical flow equation, given a \nconvenient choice of\nseed action we are free to make. \nThe effective propagator relation states\nthat for each classical two-point vertex\n(which cannot be consistently set to zero)\nthere exists an effective propagator, which\nis the inverse of this vertex, up to a gauge remainder.\nThe primary\ndiagrammatic identities of the third type\nfollow directly from those of the first\nand second types; they are stated,\nnonetheless, due to the central role they play\nin perturbative calculations.\n\n\nThe secondary diagrammatic identities, of which\nthere are two types, encode the equivalence\nof distinct diagrammatic representations\nof structures comprising\nparticular arrangements of components of\ngauge remainders. Those of the second\ntype (see \\sec{sec:D-ID-Secondary-II}) are applied only when we focus on\nthe component of a diagram which has been Taylor\nexpanded to zeroth order in its external momentum\n(in this paper we have dealt only with diagrams\ncarrying a single external momentum);\nthose of the first type (see \\sec{sec:D-ID-Secondary-I}) \nare more generally applicable.\n\nThe final element of the calculus is the subtraction\ntechniques. Working to some order in the external momentum\nof a diagram, these techniques are used to isolate\nthose components which are of precisely the desired\norder from those\nwhich have an additional, non-polynomial contribution.\n\n\nA crucial step which facilitates the practical\napplication of the calculus is the notation\nintroduced in \\sec{sec:Notation}. As with\nperturbative methods in general, the number\nof diagrams contributing to some function grows\nvery rapidly with loop order, $n$. However, rather\nthan explicitly drawing each of these\ndiagrams, it was realized that we can construct\na function of both $n$ and the various diagrammatic\n\\emph{components} which can be used to generate the\ncorrect set of diagrams, at any loop order.\nSuch expressions are very compact, with a beautifully\nintuitive structure.\n\nFor the treatment of $\\beta_n$,\nwe constructed\na set of diagrammatic functions \n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$, $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)$\nand $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c}(p)$\n(see \\sec{sec:Prelim-DiagFns})\nwhich each include two external $A^1_\\mu$s, carrying momenta $p_\\mu$\nand $-p_\\nu$.  We proceeded by\nfocusing on the $\\Op{2}$ \ncomponents of these diagrammatic functions\n(up to additional non-polynomial dependence on $p$)\n and\ncomputed the flow of $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$. Amongst the terms generated\nis a single instance of $\\beta_n$, multiplied by\na universal coefficient. Of the remaining terms,\nthose involving either lower order $\\beta$-function\ncoefficients or $\\alpha$-derivatives were put to one side.\nThe remaining diagrams\nwere manipulated, wherever possible, using the \nfollowing scheme.\n\nThe first stage\nwas to isolate any classical, two-point vertices\nformed by the flow equation and, wherever\npossible, to attach\neither an external field or an effective propagator.\nIn the former case, the diagram possesses a manifestly $\\Op{2}$\nstub; these $\\Op{2}$ terms were put to one side. \nIn the latter case, the effective propagator\nrelation was applied. This\nrelation collapses all classical, two-point vertices\nwhich are contracted into an effective propagator\ndown into a Kronecker~$\\delta$ and a gauge remainder.\nThe gauge remainders were either be processed using\nthe primary diagrammatic identities or re-expressed\nusing the secondary diagrammatic identities. In\nthe former case, the whole procedure\nwas repeated: classical, two-point vertices were\nisolated and decorated and the effective propagator\nrelation was applied {\\it etc}. The\nset of remaining diagrams now cancelled amongst\nthemselves, up to the $\\Op{2}$ terms.\n\nTo process the $\\Op{2}$ terms, the subtraction\ntechniques were employed, allowing the $\\Op{2}$ terms\nto be\neffectively split into\nthose which have additional non-polynomial\ndependence on $p$ and those which do not.\nThe latter terms were manipulated\nusing the primary and secondary diagrammatic\nidentities, yielding a set\nof diagrams which, up to\nfurther $\\alpha$ and $\\beta$-terms,\nwere precisely\ncancelled by~\\eq{eq:LdL-b-ba}.\nThe $\\Op{2}$ diagrams containing additional,\nnon-polynomial, dependence on $p$ each factorized\ninto an un-manipulable component, $\\decNTEs{\\nLV{\\mathcal{E}}{n-n'}{\\mu\\nu}{c}(p)}{}$,\nand an $n'$-loop component of exactly the same form\nas~\\eq{eq:LdL-b-ba}. \nThis allowed us to immediately write down an expression for\n$\\beta_n$  which is manifestly independent of\nthe seed action and details of the covariantization.\nFinally, we neatened the expression up by processing\nthe $\\alpha$ and $\\beta$ terms, to yield~\\eq{eq:bn-Complete}.\n\n\nThis expression was derived with the utmost rigour. In particular,\nall components carrying non-polynomial dependence on $p$\nhave been retained, even though it is know that\nthey must cancel out. Indeed, it may well be\nthat these cancellations can be demonstrated diagrammatically\n(this has already been done to some extent at two loops~\\cite{mgierg2,Thesis}),\nwhich would doubtless serve to considerably illuminate\nunderlying features of~\\eq{eq:bn-Complete}. If one is\nprepared to simply accept that these contributions do\nindeed cancel, then~\\eq{eq:bn-Complete} can be simplified\nby focusing on the strictly $\\Op{2}$ components \nof each diagram. This yields the considerably simpler\nexpression~\\eq{eq:bn-Final}. \n\nFrom~\\eq{eq:bn-Final}, the\ndiagrammatic expressions for $\\beta_1$ and\n$\\beta_2$, \\eqs{eq:beta1}{eq:beta2}, follow\ndirectly. In the context of assessing the\npractical usability of the calculus, the importance\nof this can hardly be overstated. Recall\nthat the first time $\\beta_2$ was computed in our\nERG framework~\\cite{Thesis} the derivation of~\\eq{eq:beta2}\ninvolved the generation of $\\order{10^4}$ diagrams (!)\nat intermediate\nstages of the calculation,\nalmost all of which cancelled to yield the final diagrammatic\nexpression.\nAlthough the number of intermediate diagrams can\nbe hugely reduced using the refinements of~\\cite{Primer},\nthe benefits of an expression from which the\ndiagrammatic expression can be read off at any loop\norder is obvious.\n\n\nThe derivation of the new formula for $\\beta_n$\nprovides insight into two subtle issues, both\nof which relate to the regularization. First, although the\nphysical $SU(N)$ gauge theory is properly \nregularized, care must be taken with\nthe flow equation. In particular, the flow\nequation generates (non-universal) diagrams in which\nthe kernel `bites its own tail' which \nare not properly regularized~\\cite{ym,ymi,ymii,aprop,mgierg1}\n(see the discussion around \\fig{fig:Flow}).\nUp until~\\cite{Primer}, these diagrams were excluded,\nvia an imposed constraint on the covariantization.\nHowever, as speculated in~\\cite{Primer,Thesis},\nall explicit instances\nsuch non-universal objects should cancel out\nin the reduction of $\\beta$-function\ncoefficients to a form with no explicit dependence\non either the seed action or details of the\ncovariantization. This has been borne out here,\nthough it should be emphasised that implicit\ndependence on these objects remains. It\nthus seems very likely that, ultimately,\na suitable constraint on the covariantization\nis required. \n\nNonetheless, it is very interesting that\nthe basic structure of the flow equation\nis sufficient to formally\nremove certain non-universal objects\nto $\\beta_n$, even if these objects are\nnot strictly well defined. This very much\nstrengthens the case that\nthe diagrammatics is driving us towards a\nnew framework formulated directly\nin terms of functions such as the $\\mathcal{E}^i$,\nwith the seed action\nand details of the covariantization of the\ncutoff operating in the background. \n\n\nThe second issue pertaining to the regularization\nthat the new formula for $\\beta_n$\nsheds some light on is that of pre-regularization.\nRecall that, in order the regularization\nbe properly defined, a pre-regulator must\nbe used in $D=4$~\\cite{SU(N|N)}\nto unambiguously define contributions\nwhich are finite only by virtue of the \nPauli-Villars regularization provided by\nthe massive regularizing fields. The\npre-regulator amounts to a prescription\nfor discarding otherwise non-vanishing\nsurface terms which can be generated by\nshifting loop momenta. In this paper,\ndimensional regularization was used.\nHowever, the only place that the\npre-regulator is explicitly used in practise is to throw away diagrams\nsuch as~\\ref{nLV-c-MomDer} which possess a sum of momentum\nderivative with respect to\\ the loop momenta. Thus,\nfor the purposes of the calculation performed\nin this paper,\nwe could have adopted the prescription\nthat we simply throw away all diagrams\nof this form. Of course, the diagrammatic\nfunctions contributing to $\\beta_n$ require\npre-regularization themselves, as they\ncontain diagrams which are finite only\nby virtue of the PV regularization. It is\nnot immediately obvious that the new prescription\nsuffices in this case but it will\nbe interesting to see in the future if\nit can, in fact, be adopted \nin complete generality, thereby allowing\nus to work directly in $D=4$.\n\nWhilst the application of the calculus to\n$\\beta$-function coefficients was vital\nin the development of the formalism,\nit is very important to ask whether the techniques\nof this paper can be applied in the computation of\nother quantities. The most obvious\nextension is to try to\ncalculate the expectation values\nof gauge invariant variables in perturbation\ntheory; efforts in this direction are \nunderway~\\cite{FutureWork}.\nAdditionally, we should not lose\nsight of the fact that the current\nformulation was originally developed\nwith a view to non-perturbative \napplications~\\cite{ymi,aprop}. This is\nhardly surprising, since the ERG has proven\nitself to be a flexible and powerful tool\nfor addressing such problems in a range of \nQFTs~\\cite{Wilson:1974sk,Hasenfratz:1985dm,Wetterich:1992yh,Bergerhoff:1995zm,Bergerhoff:1995zq,Aoki:1996fh,Fisher:1998kv,Morris:1998da,Berges:2000ew,Bagnuls:2000ae,Polonyi:2001se,Salmhofer:2001tr,Pawlowski:2003hq,Pawlowski:2005xe,Reuter:1997gx,Bergerhoff:1997cv,Gies:2002af,Ellwanger:1996wy,Pawlowski:2003hq,Fischer:2004uk,Ellwanger:1997wv,Ellwanger:1999vc,Ellwanger:2002xa};\nfurthermore, an interesting link\nhas been recently made between\nelements of the formalism presented\nhere and the\nAdS \/ CFT correspondence~\\cite{EMR}.\n\nA very interesting question to ask from the\nperspective of this paper is whether it is\npossible to repeat the type of cancellations\nseen in the treatment of $\\beta_n$ in a non-perturbative context.\nAn immediate obstacle to this is that non-perturbative\nERG  treatments typically rely on truncations (in our case\nof the Wilsonian effective action), after which\nuniversality is lost. However, as stated already,\nit seems that the diagrammatics are driving us\ntowards a more direct framework for\nperforming calculations in QFTs, whilst retaining\nthe advantages of an ERG approach. If this turns\nout to be the case, then we can hope that the new\nframework is defined non-perturbatively. Indeed, \nconsider again the expression~\\eq{eq:bn-Complete} for\n$\\beta_n$. We could imagine formally lifting\nthe diagrammatic expression on the right-hand side\\ to a\nnon-perturbative expression, such that its weak\ncoupling expansion just reproduces~\\eq{eq:bn-Complete}.\n\nOf course, even if this action is possible (and we\nnote that the presence of reduced vertices in~\\eq{eq:bn-Complete}\ncomplicates matters) we stress that\nit would\nrepresent only one step towards extracting non-perturbative\ninformation. After all, our hypothetical non-perturbative\nexpression for the $\\beta$-function would presumably\nstill require truncation before any numbers\ncould be computed. Nonetheless, the starting point\nwould surely be far more appealing with \nthe explicit dependence on the seed action and\ndetails of the covariantization removed from\nthe start. \n\n\n\n\n\n\n\\ack \nI would like to thank Daniel Litim for encouragement\nand useful discussions.\nI acknowledge financial support from PPARC.\n\n\\section{Further Diagrammatics}\n\\label{sec:Further}\n\n\\subsection{Gauge Remainders, Again}\n\\label{sec:GRs-2}\n\nThe various additional properties of gauge remainders\nwhich we require are most readily introduced by example.\nTo this end, consider the three diagrams shown\nin \\fig{fig:GR-Ex}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\dec{\n\t\t\\begin{array}{ccc}\n\t\t\t\\vspace{1ex}\n\t\t\t\\LID{v-W-GR}&\t\t& \\LID{v-WGR}\n\t\t\\\\\n\t\t\t\\cd{v-W-GR} & \\displaystyle+ \t& \\cd{v-WGR} \n\t\t\\end{array}\n\t}{\\{f\\}}\n\t-\\frac{1}{2}\n\t\\dec{\n\t\t\\LIDi[1]{v-W-GR-w}{v-W-GR-w}\n\t}{\\{f\\}\\Delta}\n\t\\]\n\\caption{Examples of diagrams possessing a gauge remainder.}\n\\label{fig:GR-Ex}\n\\end{figure}\\end{center}\n\nThere are a number of things to note. First, since these\ndiagrams are used for illustrative purposes, they\nare labelled \\textbf{I.\\#}; for the $\\beta_n$ diagrammatics,\nthe labels will be \\textbf{D.\\#}. Secondly, $v$ and $w$ are just\nvertex arguments, and so take the values of non-negative\nintegers. Since these arguments do not carry a hat or a bar\n(see~\\ref{eq:bar}), the vertices belong to the Wilsonian effective action.\nThirdly, the decorations of diagram~\\ref{v-W-GR-w}\ninclude not just the set of\nfields, $\\{f\\}$, but also an effective propagator, $\\Delta$, which\ncan be in any sector. The rules for decorating \ndiagram~\\ref{v-W-GR-w}\nwith the effective propagator are simple: if each end\nattaches to a different object, then the combinatoric factor\nof the attachment is two, recognizing that the effective propagator\ncan attach either way around. If the two ends attach to the same\nobject, then the combinatoric factor is unity, recognizing that\neach vertex represents a sum over all permutations of the fields arranged\ninto all possible supertrace structures. Whilst the effective\npropagator and fields, $\\{f\\}$, are not explicitly drawn on the\ndiagrams, they will be referred to as implicit, or unrealized decorations.\n\nLet us now consider what happens when\nthe gauge remainders in each\nof the diagrams~\\ref{v-W-GR}--\\ref{v-W-GR-w} act. \nSince no other gauge remainders have acted and since\nno Taylor expansions have been performed, we can\nuse charge conjugation\\ to collect together the push forward and\nthe pull back  (see \\sec{sec:CC}).\nIn diagram~\\ref{v-W-GR}, the gauge remainder can strike one\nof two things: either the field to which the kernel\nattaches or one of the fields, $\\{f\\}$.\nIt proves to be technically\nvery convenient not to specify precisely which field\nthe gauge remainder\nhits in the latter case, only that it hits something.\nConsequently, we take the gauge remainder to strike a `socket',\nwhich we suppose can be filled by any of the fields, $\\{f\\}$.\nIn diagram~\\ref{v-WGR}, the gauge remainder can strike\na socket, the top end of the kernel or the bottom end \nof the kernel. In diagram~\\ref{v-W-GR-w}, the gauge remainder\ncan only strike a socket, though we note\nthat this socket can be filled not only by $\\{f\\}$ but\nalso by an end of the effective propagator. \\Fig{fig:GR-Ex-P} shows the result\nof processing the gauge remainders of \ndiagrams~\\ref{v-W-GR}--\\ref{v-W-GR-w}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\t\n\t\\begin{array}{rcl}\n\t\\vspace{2ex}\n\t\t\\dec{\n\t\t\t\\cd{v-W-GR}\t \n\t\t}{\\{f\\}}\n\t\t& =2 & \n\t\t\\dec{\n\t\t\t\\begin{array}{ccc}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\CID{v-W-GR-a}{v-W-GR-ai}\t&\t& \\CID{v-W-GR-b}{v-W-GR-b2}\n\t\t\t\\\\\n\t\t\t\t\\cd{v-W-GR-a}\t\t\t\t&-\t& \\cd{v-W-GR-b}\n\t\t\t\\end{array}\n\t\t}{\\{f\\}}\n\t\\\\\n\t\\vspace{2ex}\n\t\t\\dec{\n\t\t\t\\cd{v-WGR}\t \n\t\t}{\\{f\\}}\n\t\t& =-2 &\n\t\t\\dec{\n\t\t\t\\begin{array}{ccccc}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\CID{v-W-GR-ai}{v-W-GR-a}\t&\t& \\CID{v-WGR-b}{v-WGR-b2}\t&\t& \\LID{v-WGR-c}\n\t\t\t\t\\\\\n\t\t\t\t\\cd{v-W-GR-a}\t\t\t\t&+\t& \\cd{v-WGR-b}\t\t\t\t&-\t& \\cd{v-WGR-c}\n\t\t\t\\end{array} \n\t\t}{\\{f\\}}\n\t\\\\\n\t\t-\\displaystyle \\frac{1}{2}\n\t\t\\dec{\n\t\t\t\\cd{v-W-GR-w}\n\t\t}{\\{f\\}\\Delta}\n\t\t& = &\n\t\t\\dec{\n\t\t\t\\LIDi[1]{v-W-GR-w-b}{v-W-GR-w-b}\n\t\t}{\\{f\\}\\Delta}\n\t\\end{array}\n\t\\]\n\\caption{The result of allowing the gauge remainders\nof diagrams~\\ref{v-W-GR}--\\ref{v-W-GR-w} to act. In all\ndiagrams with the same (opposite) sign to the parent, the\npush forward and pull back have been collected into\ntwice the push forward (pull back).}\n\\label{fig:GR-Ex-P}\n\\end{figure}\\end{center}\n\nNotice that, rather than terminating the pushed forward \/ \npulled back field-line with a half arrow ({\\it cf.}\\  \\eq{eq:WID-A}), \nwe just utilize the fact\nthat the corresponding field line already ends \nin a $>$ and use this to indicate the field hit.\\footnote{\nNote that we will not always have a $>$ at our disposal.\nIn particular, we will encounter gauge remainders involving\npseudo effective propagators terminating in just $\\rhd$. In\nthis case, the notation of \\sec{sec:WIDs} must be used.\n}\nImmediately, we find a cancellation, which we indicate by\nenclosing the reference number of the cancelled diagram\nin curly brackets, together with the reference number\ndiagram against\nwhich it cancels.\n\n\\ICancelCom{v-W-GR-ai}{v-W-GR-a}{. Although these diagrams\nlook exactly the same, one might worry that they are\ndifferent: in diagram~\\ref{v-W-GR-ai}, the gauge remainder\npulls back along the kernel, whereas in diagram~\\ref{v-W-GR-a}\nthe gauge remainder has nothing to do with the kernel, instead\npushing forward around the vertex. However, gauge invariance\nensures that these two diagrams are indeed \nequivalent~\\cite{Thesis,mgierg1}, and so they cancel.}\n\nReturning to \\fig{fig:GR-Ex-P}, consider next\ndiagram~\\ref{v-WGR-c}. The line segment\nwhich joins the top\nof the kernel to the $>$---thereby forming a `hook'---performs \nno role other \nthan to make this join. In other words, it is neither a section of \nthe kernel nor an effective propagator. We could imagine\ndeforming this line segment so that the hook becomes arbitrarily large. \nDespite\nappearances, we must always remember \nthat this line segment simply performs the role\nof a Kronecker delta. When part of a complete diagram, this line\nsegment can always be distinguished from an effective propagator,\nto which it can be made to look identical, by the context. This\nfollows because hooks in which the line segment is a Kronecker~$\\delta$\nonly ever attach to effective propagators or kernels, whereas \nhook-like structures\nmade out of an effective propagator only ever attach to vertices\n(this will be particularly clear from the perspective\nof \\sec{sec:beta_n}).\nWhen viewed in isolation, we will always take the hook structure to\ncomprise just a line segment and so will draw the hook as tightly as\npossible.\n\nTo conclude our discussion of the hook, we give its algebraic\nform. First, we note from table~\\ref{tab:NFE:k,k'} that\nthe gauge\nremainder that forms the hook must be in the $D$ $(\\bar{D})$ sector,\nelse the loop integral over its momenta is odd and vanishes. \nConsequently, \nthe field to which the hook attaches must be\nin the  $C^1$ ($C^2$) sector; a conclusion which could\nalso have been drawn from consideration of charge conjugation\\ invariance.\nThus, in the former (latter)\ncase, the gauge remainder of the parent, which is in the\n$F$ ($\\bar{F}$) sector, strikes the $F$ ($\\bar{F}$) at\nthe top of the kernel, turning it into a $C^1$ ($-C^2$).\nThe sign in the latter case essentially arises from\nthe minus sign in fifth component of $\\bar{F}_N$ (see\n\\eqs{eq:F}{eq:Fbar}) (see~\\cite{Thesis,mgierg1} for \npainstaking detail). Our prescription is to absorb\nany signs associated with the end of the kernel\ninto the definition of the hook.\n\n\n\nThe final ingredient we need to obtain the algebraic form for the hook is to\nrealize that the inside of the hook constitutes an empty loop and so\ngives a group theory factor of $\\pm N$, depending on flavour. Thus we have:\n\\numparts\n\\begin{eqnarray}\n\\label{eq:hook-R-1}\n\t\\cd{hook-R-1} & = & +(-N) \\int_l g_l \n\\\\\n\\label{eq:hook-R-2}\n\t\\cd{hook-R-2} & = & - (N) \\int_l g_l.\n\\end{eqnarray}\n\\endnumparts\nThe numbers at the base of the hook in the above equations indicate the sector\nof the $C^i$ field to which the hooks attach.\n\nWe now see a further advantage to having absorbed signs into the definition of\nthe hook: we can trivially extend the diagrammatic effect of charge conjugation\\ to cover\nits action on a disconnected hook.\nQuite simply, we have that\n\\begin{equation}\n\t\\cd{hook-R} = - \\cd{hook-L},\n\\label{eq:hookCC}\n\\end{equation}\nconsistent with our previous definition that we take the mirror image, picking\nup a sign for every performed gauge remainder. \n\n\nNow we focus on diagram~\\ref{v-W-GR-w-b}. This diagram,\nlike~\\ref{v-W-GR-b} and~\\ref{v-WGR-b}, possesses a socket;\nwe will now see why it is so useful to leave this socket\nempty. To process this diagram further, we suppose that\nthe vertex with argument $w$ possesses a classical, two-point\ncomponent. Of course, in this case, this means that\n$w$ must be a classical vertex. However, in the computation\nof $\\beta_n$, we will generally be summing over vertex arguments,\nsuch that $w$ could take all values between zero and some\npositive integer. In preparation for this, we  leave the\nvertex argument as $w$ rather than explicitly writing it\nas zero. To separate off the classical, two-point component\nof $w$, we define the reduction of a vertex to be\nthe full vertex minus its classical, two-point component\n(should this component exist). The reduction is denoted\nby appending the appropriate vertex argument with a superscript $R$\n{\\it viz.}\\  $w^R$. This decomposition of diagram~\\ref{v-W-GR-w-b}\nis shown in \\fig{fig:Decompose}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\dec{\n\t\t\\cd{v-W-GR-w-b}\n\t}{\\{f\\}\\Delta}\n\t=\n\t\\dec{\n\t\t\\begin{array}{ccc}\n\t\t\\vspace{1ex}\n\t\t\t\\LID{v-W-GR-w-bR}\t&\t& \\LID{v-W-GR-w-b02}\n\t\t\\\\\n\t\t\t\\cd{v-W-GR-w-bR}\t& + & \\cd{v-W-GR-w-b02}\n\t\t\\end{array}\n\t}{\\{f\\}\\Delta}\n\t\\]\n\\caption{Decomposing the vertex of diagram~\\ref{v-W-GR-w-b}\n(which in this case is implicitly a classical vertex)\ninto a reduced part and a classical, two-point part.}\n\\label{fig:Decompose}\n\\end{figure}\\end{center}\n\nThe top vertex of diagram~\\ref{v-W-GR-w-bR}\nis reduced. The vertex argument $0^2$ of \ndiagram~\\ref{v-W-GR-w-b02} tells us that this\nvertex must not only be classical but must also\npossess precisely two decorations. Thus in addition\nto the socket, this vertex must have one and only\none additional decoration. Clearly, this additional\ndecoration can either be one of the fields $\\{f\\}$\nor one of the ends of the effective propagator. \nThe\nformer case will not concern us here, so we focus\non the latter case. Given that one end of the effective\npropagator decorates the classical, two-point vertex,\nthere are three locations to which the loose end can attach:\n\\begin{enumerate}\n\t\\item\tthe socket;\n\t\\label{it:socket}\n\n\t\\item\tthe vertex with argument $v$.\n\t\\label{it:vertex}\n\n\t\\item\tthe kernel;\n\t\\label{it:kernel}\n\\end{enumerate}\n\nThe first case vanishes by charge conjugation\\ invariance. \nTo see this,\nconsider the parent of diagram~\\ref{v-W-GR-w-b},\ndiagram~\\ref{v-W-GR-w}. To generate the diagram\ncorresponding to~\\ref{it:socket}, we should\nconsider the component of diagram~\\ref{v-W-GR-w}\nin which\nthe top vertex is a three-point vertex, \nwith one of the fields corresponding to the gauge\nremainder and the other two corresponding to the\ntwo ends of the effective propagator. Now,\nsince net fermionic vertices vanish, the gauge\nremainder must be bosonic. Furthermore, by charge conjugation\\ invariance,\nthe gauge remainder is forced to be in the $C^{1,2}$\nsector, in which it is null (see \\sec{sec:CC}).\n\nIn both cases~\\ref{it:vertex} and~\\ref{it:kernel},\nwe can employ the effective propagator relation.\nThis is shown in \\fig{fig:EP-Ex}. We note that\nfollowing. First,\nusing the effective propagator to join the\nclassical, two-point vertex to some other object\nyields a combinatoric factor of two. Secondly,\nwhen we henceforth talk\nof decorating a classical, two-point\nvertex we always mean that we are explicitly\ndrawing on the second decoration of the vertex,\nthe socket being the first decoration. If we\nmean to fill the socket, this will be explicitly\nstated.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray*}\n\t\\dec{\n\t\t\\cd{v-W-GR-w-b02}\n\t}{\\{f\\} \\Delta}\n\t& = &\n\t2\n\t\\dec{\n\t\t\\begin{array}{ccc}\n\t\t\\vspace{1ex}\n\t\t\t\\LID{v-W-GR-w-b02-v}&\t& \\LID{v-W-GR-w-b02-k}\n\t\t\\\\\n\t\t\t\\cd{v-W-GR-w-b02-v}\t&+\t& \\cd{v-W-GR-w-b02-k}\n\t\t\\end{array}\n\t}{\\{f\\}} + \\cdots\n\t\\\\\n\t& = &\n\t2\n\t\\dec{\n\t\t\\begin{array}{ccc}\n\t\t\\vspace{1ex}\n\t\t\t\\CID{v-W-GR-b2}{v-W-GR-b}\t&\t& \\LID{v-W-nGR}\n\t\t\\\\\n\t\t\t\\cd{v-W-GR-b}\t\t\t\t& - & \\cd{v-W-nGR}\n\t\t\\\\\n\t\t\\vspace{1ex}\n\t\t\t\\CID{v-WGR-b2}{v-WGR-b}\t\t&\t& \\LID{v-WnGR}\n\t\t\\\\\n\t\t\t\\cd{v-WGR-b}\t\t\t\t& - & \\cd{v-WnGR}\n\t\t\\end{array}\n\t}{\\{f\\}} + \\cdots\n\t\\end{eqnarray*}\n\\caption{The decoration of the classical, two-point vertex\nof diagram~\\ref{v-W-GR-w-b02} by the effective propagator,\ntogether with the subsequent application of the\neffective propagator relation. The ellipses\ndenote diagrams in which the classical, two-point\nvertex is decorated by one of the fields, $\\{f\\}$, rather\nthan by the effective propagator.}\n\\label{fig:EP-Ex}\n\\end{figure}\\end{center}\n\n\\ICancel{v-W-GR-b2}{v-W-GR-b}\n\n\\ICancel{v-WGR-b2}{v-WGR-b}\t\n\nThere are some comments worth making about\ndiagrams~\\ref{v-W-nGR} and~\\ref{v-WnGR}.\nFirst, \nthe $>$\npart of the full gauge remainder plays the role of\nthe socket in the parent diagrams. \nSecondly, these diagrams\nhave a very similar\nstructures to diagrams~\\ref{v-W-GR} and~\\ref{v-WGR}.\nThe only difference is that the full gauge remainder\nin the new diagrams is nested~\\cite{Thesis}, meaning\nthat it does not attach directly to the kernel, but\ninstead is hit by the gauge remainder component which\nends the kernel.\nProcessing nested gauge remainders\nis much the same as processing un-nested gauge\nremainders, but charge conjugation\\ invariance cannot generally be used\nagain to collect together the \nnested push forward and pull back~\\cite{Thesis,Primer}:\nwe must count them separately. However, everything\nelse goes through as before and so we will find that\ncancellation~\\ref{Icancel:v-W-GR-ai} is repeated\ntwice (twice because we count the nested push forward\nseparately from the nested pull back).\n\nIf there were more vertices available, we could imagine\niterating the above procedure and thus generating\narbitrarily nested gauge remainders.\nIt would then be useful for us to know if there is \nany simple way of\nkeeping track of which gauge remainders have pushed \nforward and which have pulled back.\nWith sufficient thought, it is always \npossible to stare at a complicated\ndiagram and deduce the pattern of pushes \nforward and pulls back\n(up to the ambiguity associated with illustrative \ncancellation~\\ref{Icancel:v-W-GR-ai}). However, there is\nan easier way. Note in diagrams~\\ref{v-W-nGR} and~\\ref{v-WnGR}\nthat the $>$ part of the full gauge remainder \nis bitten on its right-hand edge by the $>$ which \nterminates the kernel. (We define the right-hand edge\nto be the edge on our right as we traverse an imaginary\nline running into the socket at the apex {\\it viz.}\\  \n$\\left.^{\\scriptscriptstyle \\mathrm{L}}_{\\scriptscriptstyle \\mathrm{R}} \\cd{EP-GR}\\right.$;\nlikewise for sockets on vertices \/ kernels.)\nHad we pushed forward, rather than pulled \nback with the initial\ngauge remainder, then this bite would have been to the left. \nWith this in mind, consider a string\nof gauge remainders which bite a socket \ndecorating either a kernel or a vertex; the latter\ncase is shown in \\fig{fig:GRstring}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\LIDi{v-GRstring}{v-GRstring}\n\t\\]\n\\caption{An arbitrarily nested gauge remainder bites a socket on a vertex.}\n\\label{fig:GRstring}\n\\end{figure}\\end{center}\n\n\n\nWe know from the discussion in illustrative \ncancellation~\\ref{Icancel:v-W-GR-ai} that,\nwere we to fill the socket of diagram~\\ref{v-GRstring}\nwith the end of a kernel (the other end of which\nattaches to the gauge remainder at the beginning\nof the string), then the sense in which the socket\nis bitten can be interpreted in two ways:\neither as a push forward around the vertex or\nas a pull back along the kernel. Given a diagram like~\\ref{v-GRstring},\nwe will always\ninterpret the gauge remainder at the end of the string of\ngauge remainders as having\nbitten the vertex, using this to determine the sense\nin which the gauge remainder acts.\n\n\n\n\nTo determine\nin which sense the nested gauge remainders are bitten\nwe can equate bites\non the left with pushes forward and bites on the right with pulls back,\nthe only exception being when a gauge remainder bites\nthe end of kernel to form a hook: the gauge remainder\nthat pushed forward to form\nthe un-nested hook of diagram~\\ref{v-WGR-c}\ncan be thought of as biting itself on the right.\nThus, moving on to consider\nthe arbitrarily nested hook shown in \\fig{fig:nestedhook},\nthe number of pushes forward is equal to the number of \nbites on the left, plus one, and the number of pulls\nback is equal to the number of bites on the right, minus one.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\cd{Struc-GR-ring}\n\t\\]\n\\caption{An arbitrarily nested version of the hook.}\n\\label{fig:nestedhook}\n\\end{figure}\\end{center}\n\nStructures like those in \\fig{fig:GRstring}\nhave a compact representation, shown below:\n\\[\n\t\\frac{1}{m!}\\cd{compact} \\ .\n\\]\n\nThe notation $>^m$ stands for $m$ instances of $>$.\nThese gauge remainder\ncomponents form a string which bites the socket. We sum over\nall possible ways in which each gauge remainder can bite,\nand so diagram~\\ref{v-GRstring} is just one component included\nour compact diagrammatic representation. When going from\nthe compact diagrammatic representation to its explicit \ncomponents, we note that any of the $m$ gauge remainders\ncan bite the socket, and this gauge remainder can, in turn,\nbe bitten by any of the remaining $m-1$ gauge remainders.\nThis explains the normalization factor of $1\/m!$.\nDenoting the number of bites to the left \/ right by L \/ R,\nthe sign of each explicitly drawn diagram is just $(-1)^R$.\n\nSimilarly, there is a compact diagrammatic representation\t\nof the nested hook---or ring---of \\fig{fig:nestedhook}, \nwhich is simply\n\\[\n\t\\frac{1}{(m-1)!} \\decGR{ \\ }{>^m}.\n\\] \nThe rule to go from this compact\nrepresentation to explicit diagrams is as follows:\nwe draw the set of rings corresponding to\nall independent sequences bites to the left \/ right.\nThe normalization factor follows \nfrom noting that a ring is invariant\nunder a cyclic permutations of the gauge remainders.\nThe sign of each explicitly drawn ring is $(-1)^{R-1}$.\n\n\n\n\nWe conclude this discussion of the gauge \nremainders by examining diagrams possessing\ntwo active gauge remainders. The techniques we use to process such terms are\nexactly the same as  the ones detailed already in this section. There is,\nhowever, a qualitatively new type of diagram that arises, \ntogether with a source\nof possible confusion. To investigate both of these issues, we focus on an\nexample\nin which two full gauge remainders bite a kernel\\footnote{\nDouble gauge remainder diagrams are not restricted to those in which both gauge\nremainders bite a kernel; one or both of the gauge remainders can bite a vertex.},\nas shown in \\fig{fig:DoubleGR}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\frac{1}{2}\n\t\\LIDi[1]{Struc-WGRx2-B}{Diags-WGRx2-B}\n\t\\]\n\\caption{An example of a double gauge remainder diagram.}\n\\label{fig:DoubleGR}\n\\end{figure}\\end{center}\n\nWe proceed by allowing first one gauge remainder to act and then, if possible,\nallowing the second gauge remainder to act. The qualitatively new type of diagram\narises because one of the\neffects of the first gauge remainder can be to `trap' the second gauge remainder,\nby biting the field on the kernel to which the second gauge remainder attaches.\nThe remaining full gauge remainder now does \nnot have the same momentum flowing through\nit as the field it is trying to bite, and so it cannot act.\nThe other diagrams generated are those in\nwhich the processed gauge remainder bites one of the\nends of the kernel. The result of allowing the first gauge remainder to\nact is shown in \\fig{fig:DoubleGR-A}, where we have collected pulls\nback and pushes forward, as usual.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\t\\LID{Diags-WGRx2-Top}\t&\t& \\LID{Diags-WGRx2-Trap}\t&\t& \\LID{Diags-WGRx2-Bottom}\n\t\\\\[1ex]\n\t\t\\cd{Struc-WGRx2-Top}\t& -\t& \\cd{Struc-WGRx2-Trap}\t& +\t& \\cd{Struc-WGRx2-Bottom}\n\t\\end{array}\n\t\\]\n\\caption{Result of allowing one gauge remainder in diagram~\\ref{Diags-WGRx2-B} to act.}\n\\label{fig:DoubleGR-A}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{Diags-WGRx2-Trap} possesses the \ntrapped gauge remainder. In diagram~\\ref{Diags-WGRx2-Bottom}\nwe have recognized that the kernel ends where it attaches \nto the active gauge remainder\nand so this is where the processed gauge remainder bites. \nNote that\nwe can trivially redraw this diagram, as shown in \\fig{fig:DoubleGR-B}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\LIDi[1]{Struc-WGRx2-Bottom-B}{Diags-WGRx2-Bottom-B}\n\t\\]\n\\caption{A trivial redrawing of diagram~\\ref{Diags-WGRx2-Bottom}.}\n\\label{fig:DoubleGR-B}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{Diags-WGRx2-Bottom-B} highlights how one must be \nvery careful when drawing\nwhich end of an active gauge remainder a \nprocessed gauge remainder bites; diagrams~\\ref{Diags-WGRx2-Bottom-B}\nand~\\ref{Diags-WGRx2-Trap} are clearly different. Notice, however, that if\nthe active gauge remainder were \\emph{absent}, then\ndiagram~\\ref{Diags-WGRx2-Bottom}\nwould cancel diagram~\\ref{Diags-WGRx2-Trap}; we will exploit this\nlater.\n\n\nFinally, consider allowing the active gauge remainder to \nact in diagrams~\\ref{Diags-WGRx2-Top}\nand~\\ref{Diags-WGRx2-Bottom-B}. We are not \ninterested in all the contributions. Rather,\nwe just want to focus on\n\\begin{enumerate}\n\t\\item\tthe term produced by diagram~\\ref{Diags-WGRx2-Top} where the\n\t\t\tgauge remainder pulls back along the kernel, to the same end as the hook;\n\n\t\\item\tthe term produced by diagram~\\ref{Diags-WGRx2-Bottom-B} in which the\n\t\t\tgauge remainder pulls back to the bottom of the kernel.\n\\end{enumerate}\nReflecting the former diagram about a horizontal line, we arrive at the two diagrams\nof \\fig{fig:DoubleGR-C}. (The reflection does not yield a net sign as\nwe pick up one for each of the processed gauge remainders.) \n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{cccc}\n\t\t\t& \\LID{Diags-WGRx2-L-Bottom}&\t& \\LID{Diags-WGR-N1-Bottom-PF}\n\t\\\\[1ex]\n\t\t-\t& \\cd{Struc-WGRx2-L-Bottom}\t& -\t& \\cd{Struc-WGR-N1-Bottom-PF}\n\t\\end{array}\n\t\\]\n\\caption{Two of the terms produced by processing diagrams~\\ref{Diags-WGRx2-Top}\nand~\\ref{Diags-WGRx2-Bottom-B}.}\n\\label{fig:DoubleGR-C}\n\\end{figure}\\end{center}\n\nFrom the way in which these two diagrams have been drawn, it is clear that\nthey are distinct and that they must be treated as such. We can, however, redraw\ndiagram~\\ref{Diags-WGR-N1-Bottom-PF} by \nsliding the outer gauge remainder round the hook\nto where the inner gauge remainder bites the kernel. This is shown in \n\\fig{fig:DoubleGR-D}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t-\\LIDi[1]{Struc-WGR-N1-Bottom-PF-B}{Diags-WGR-N1-Bottom-PF-B}\n\t\\]\n\\caption{A trivial redrawing of diagram~\\ref{Diags-WGR-N1-Bottom-PF}.}\n\\label{fig:DoubleGR-D}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{Diags-WGR-N1-Bottom-PF-B} is, of course, still the same as\ndiagram~\\ref{Diags-WGR-N1-Bottom-PF} but it is starting to look very similar\nto diagram~\\ref{Diags-WGRx2-L-Bottom}. Indeed, we must make sure that we never\nslide the gauge remainder so far round the hook that it appears to bite the kernel\nat the same point as the gauge remainder which forms the hook. If this\nwere to happen, then such a diagram would be ambiguous.\n\n\n\n\n\n\\subsection{The Secondary Diagrammatic Identities}\n\\label{sec:Secondary}\n\nThere are two families of secondary diagrammatic identities.\nMembers of the first family are\ngenerally applicable, whereas\n members of\nthe second family are\napplicable only to diagrams which have\nbeen manipulated at $\\Op{2}$.\n\n\n\n\n\n\n\\subsubsection{The First Family}\n\\label{sec:D-ID-Secondary-I}\n\nThe first diagrammatic identity in this\nfamily is trivial:\n\\begin{equation}\n\\label{D-ID-Trivial-A}\n\t\\cdeps{GR-hook} = 0.\n\\end{equation}\nThis follows directly from charge conjugation\\ invariance:\nthe field attaching to the hook must\nbe in the $C^i$ sector, but in this sector\nthe gauge remainder contracted into the\nhook is null. Before moving on, notice\nthat the diagram contains one empty socket,\ncorresponding to the $>$ part of the\nfull gauge remainder.\n\nThere is a second trivial identity\ninvolving a diagram possessing a single\nsocket, which we have seen already. We\nre-draw it here in what will become a\nsuggestive form:\n\\begin{equation}\n\\label{eq:D-ID-Trivial-PEP}\n\t\\cd{TLTP-Soc-EP-GR} = 0,\n\\end{equation}\nrecalling that this identity follows as\na consequence of the effective propagator\nrelation and~\\eq{eq:GR-relation}. We\nexplicitly indicate that the vertex is\na two-point vertex, since such structures\nwill generically \noccur as sub-diagrams in diagrams\nwith unrealized decorations.\n\n\nTo uncover the next diagrammatic identity,\nwhich is the first we will encounter involving two\nsockets, \nconsider the pair of diagrams in\n\\fig{fig:D-ID-Trivial-b}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\\vspace{1ex}\n\t\t\\LID{Trivial-b-A}\t&\t& \\LID{Trivial-b-A2}\t\n\t\\\\\n\t\t\\cd{Trivial-b-A} \t& -\t& \\cd{Trivial-b-A2} \t\n\t\\end{array}\n\t\\]\n\\caption{A pair of diagrams whose sum vanishes.}\n\\label{fig:D-ID-Trivial-b}\n\\end{figure}\\end{center}\nThese\ndiagrams naturally arise from \na common parent: the differences between\nthe two diagrams come from the sense in\nwhich the full gauge remainder of\nthe parent, of which \nthe component labelled $X$ is left behind, acts.\nThe point now is that diagrams~\\ref{Trivial-b-A}\nand~\\ref{Trivial-b-A2} cancel. By construction,\nthe unspecified structure at the bottom of both\ndiagrams is common. Furthermore, the gauge\nremainder labelled $X$ and its attachment to\nthe common structure is the same between the\ntwo diagrams. From~\\eqs{eq:hook-R-1}{eq:hook-R-2},\nthe two hooks yield the same algebraic contribution.\nThe only difference between the two diagrams is\nthe sense in which the gauge remainder labelled $X$\nacts. However, effectively\nall that this gauge remainder does is determine\nwhether the hook is of type~\\eq{eq:hook-R-1} \nor~\\eq{eq:hook-R-2}; and we know that the hook is\njust a common algebraic factor between the two diagrams.\nTherefore, diagrams~\\ref{Trivial-b-A}\nand~\\ref{Trivial-b-A2} are in fact identical up to a\nsign and so cancel.\n\nThis leads us to the following diagrammatic\nidentity:\n\\begin{equation}\n\\label{eq:D-ID-Bitten-hook-A}\n\t\\cdeps{Bitten-hook} - \\cdeps{Bitten-hook-R} \\equiv 0,\n\\end{equation}\nwhere we understand that this diagram should be viewed\nas part of some larger diagram, attached via internal\nfields. \nNote that in this diagrammatic\nidentity, and all that follow, we can strip off\nthe common $>$ which attaches to some\nother part of a larger diagram, replacing it with\nthe half arrow notation of~\\eq{eq:WID-A}.\n\nThe non-trivial diagrammatic identities which now follow can\nbe thought of as arising from versions \nof~\\eq{D-ID-Trivial-A} and~\\eq{eq:D-ID-Bitten-hook-A}\nin which the hook is nested and versions of~\\eq{eq:D-ID-Trivial-PEP}\nin which the gauge remainder is nested. To be\nspecific, we will find relationships between\nthe nested versions of~\\eq{D-ID-Trivial-A}--\\eq{eq:D-ID-Bitten-hook-A}\nfor which the numbers of sockets are equal; equivalently,\nthese relationships are between diagrams with an\nequal number of \\emph{performed} gauge remainders.\n\nThe key to proving the non-trivial diagrammatic\nidentities resides in the following relationship\nbetween gauge remainder components. Using the notation\ndescribed around~\\eq{GR:flavour}, and taking $j_2 = j_1 \\ \\xor j_3$\nwe have:\n\\begin{equation}\n\\label{eq:Template}\n\t1 - \\rhd^{j_1}_{l-m_1-k} >^{j_2}_{l-m_1} - \\rhd^{j_3}_k >^{j_2}_{l-m_1} = 0.\n\\end{equation}\t\nTo demonstrate this, we simply substitute for the\ngauge remainder components using \\tab{tab:NFE:k,k'}\nand employ~\\eq{eq:xf+2g}, as shown in \\tab{tab:template}.\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\\begin{table}[h]\n\t\\[\n\t\\begin{array}{cc|c|rcl}\n\t\tj_1\t& j_3\t& j_2\t& \t\n\t\\\\ \\hline\n\t\t0\t& 0\t\t& 0\t\t& 1 - \\frac{(l-m_1-k)\\cdot(l-m_1)}{(l-m_1)^2} - \\frac{k \\cdot (l-m_1)}{(l-m_1)^2} & = & 0\n\t\\\\\n\t\t0\t& 1\t\t& 1\t\t& 1 - f_{l-m_1} \\frac{(l-m_1-k)\\cdot(l-m_1)}{\\Lambda^2} - f_{l-m_1} \\frac{k\\cdot(l-m_1)}{\\Lambda^2} - 2g_{l-m_1} & = & 0\n\t\\\\\n\t\t1\t& 0\t\t& 1\t\t& 1 - f_{l-m_1} \\frac{(l-m_1-k)\\cdot(l-m_1)}{\\Lambda^2} - 2g_{l-m_1} - f_{l-m_1} \\frac{k\\cdot(l-m_1)}{\\Lambda^2} & = & 0\n\t\\\\\n\t\t1\t& 1\t\t& 0\t\t& 1 - \\frac{(l-m_1-k)\\cdot(l-m_1)}{(l-m_1)^2} - \\frac{k \\cdot (l-m_1)}{(l-m_1)^2} & = & 0\n\t\\end{array}\n\t\\]\n\\label{tab:template}\n\\caption{Break down of~\\eq{eq:Template} into the different\nsectors.}\n\\end{table}\\end{center}\n\\renewcommand{\\arraystretch}{1}\n\n\nThe first non-trivial relationship involves versions \nof~\\eq{D-ID-Trivial-A}--\\eq{eq:D-ID-Bitten-hook-A} with\ntwo sockets. This identity is, in some sense, still a\nspecial case, since the two-socket version \nof~\\eq{eq:D-ID-Bitten-hook-A} is of course trivial.\nConsequently, we treat the two-socket case---which is\nshown in \\fig{fig-D-ID-2}---separately from the rest. \n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\\label{eq:D-ID-2}\n\t\t\\begin{array}{ccccccccc}\n\t\t\\vspace{1ex}\n\t\t\t\t& \\LID{TLTP-EP-LL}\t&\t& \\LID{GR-ring-RR}\t&\t& \\LID{TLTP-EP-RR}\t&\t& \\LID{GR-ring-LL}\t&\n\t\t\\\\\n\t\t\\vspace{1ex}\n\t\t\t\t& \\cd{TLTP-EP-LL} \t& -\t& \\cd{GR-ring-RR}\t& +\t& \\cd{TLTP-EP-RR}\t& -\t& \\cd{GR-ring-LL}\t&\n\t\t\\\\\n\t\t\\vspace{1ex}\n\t\t\t\t&\\LID{TLTP-EP-RL}\t&\t& \\LID{GR-ring-RL}\t&\t& \\LID{TLTP-EP-LR}\t&\t& \\LID{GR-ring-LR}\t&\n\t\t\\\\\n\t\t\t-\t&\\cd{TLTP-EP-RL}\t& + &\\cd{GR-ring-RL} \t& -\t& \\cd{TLTP-EP-LR}\t& +\t& \\cd{GR-ring-LR} \t& = 0\n\t\t\\end{array}\n\t\\end{equation}\n\\caption{Two sets of diagrams, each of which will be shown\nto sum individually to zero.}\n\\label{fig-D-ID-2}\n\\end{figure}\\end{center}\n\nDiagrams~\\ref{TLTP-EP-RR}, \\ref{TLTP-EP-RL}\nand~\\ref{TLTP-EP-LR} can each be obtained\nfrom diagram~\\ref{TLTP-EP-LL} by changing\nthe senses in which the gauge remainders\nbite (the remaining four diagrams are similarly\nrelated).\nThere are actually some stronger relationships than\nthe one given above. For example, the first four diagrams\nonly ever cancel amongst themselves. However,\nfor our analysis of $\\beta_n$, only the\nstatement~\\eq{eq:D-ID-2} is required.\n\n\n\nTo prove \\eq{eq:D-ID-2},\nwe begin by focusing on the first\npair of diagrams, which we redraw in\n\\fig{fig-D-ID-2b}. Notice that we\nhave used \nthe effective propagator relation in\ndiagram~\\ref{TLTP-EP-LL},\nhave filled the sockets \nwith fields of a specific flavour \nand have chosen a convenient momentum\nrouting for each of the diagrams.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\\vspace{1ex}\n\t\t\\LID{TLTP-EP-LL-Kd2}\t&\t&\\LID{TLTP-EP-LL-GR}\t&\t&\\LID{GR-ring-RR-2}\n\t\\\\\n\t\t\\cd{TLTP-EP-LL-Kd2}\t\t& -\t& \\cd{TLTP-EP-LL-GR}\t& -\t& \\cd{GR-ring-RR-2}\n\t\\end{array}\n\t\\]\n\\caption{Re-expression of diagrams~\\ref{TLTP-EP-LL} and~\\ref{GR-ring-RR}.}\n\\label{fig-D-ID-2b}\n\\end{figure}\\end{center}\n\nThe letters $U$--$Z$ denote field flavour, \nand the subscripts\n$T$ and $R$ represent indices. It is straightforward to\nverify, and intuitively obvious, that $W_R$ and $X_T$\nmust be either both bosonic or both fermionic. In the former\ncase, the internal fields $U, \\ V,\\ Y$ and $Z$ are all\nof the same flavour, being either all bosonic or\nall fermionic. In the case that the internal\nfields are all bosonic, diagrams~\\ref{TLTP-EP-LL-Kd2}--\\ref{GR-ring-RR-2}\nreduce to a common factor multiplied by\n\\begin{equation}\n\\label{eq:D-ID-2-Alg-a}\n\t1 - \\rhd_l^0 >_{l+k}^0 - \\rhd_k^0 >^0_{l+k},\n\\end{equation}\nwhereas, in the case that the internal fields\nare all fermionic, they reduce to some\nother common factor multiplied by\n\\begin{equation}\n\\label{eq:D-ID-2-Alg-b}\n\t1 - \\rhd_l^1 >_{l+k}^1 - \\rhd_k^0 >^1_{l+k}.\n\\end{equation}\nIt is immediately\napparent from~\\eq{eq:Template}\nthat both~\\eq{eq:D-ID-2-Alg-a} and~\\eq{eq:D-ID-2-Alg-b}\nyield zero, upon identifying $-m_1$ with $k$.\nIn exactly the same way, diagrams~\\ref{TLTP-EP-RR} and\n\\ref{GR-ring-LL}, \\ref{TLTP-EP-RL} and~\\ref{GR-ring-RL},\n\\ref{TLTP-EP-LR} and~\\ref{GR-ring-LR} cancel\nwhen the external fields are bosonic.\n\nLet us now consider filling the\nsockets of diagrams~\\ref{TLTP-EP-LL}--\\ref{GR-ring-LR}\nwith fermionic fields. The first thing\nto realize is that the latter four diagrams\ndo not exist with these decorations, since\nsuch decoration is incompatible with the group\ntheory structure~\\cite{Thesis}. In \\fig{fig:D-ID-2-F2},\nwe collect together all surviving terms\nfor which $W_R = \\bar{F}$ and $X_T = F$\n(and identical analysis can be done for \n$F \\leftrightarrow \\bar{F}$), and sum\nover all possible flavours of the internal fields.\n\\begin{center}\\begin{figure}[h]\n\\numparts\n\t\\begin{eqnarray}\n\t\t\\begin{array}{ccccccc}\n\t\t\\vspace{1ex}\t\t\t\n\t\t\t\\LID{TLTP-EP-LL-F2a}&\t& \\LID{GR-ring-RR-F2a}\t&\t& \\LID{TLTP-EP-RR-F2a}\t&\t& \\LID{GR-ring-LL-F2a}\n\t\t\\\\\n\t\t\t\\cd{TLTP-EP-LL-F2a}\t& -\t& \\cd{GR-ring-RR-F2a}\t& +\t& \\cd{TLTP-EP-RR-F2a}\t& -\t& \\cd{GR-ring-LL-F2a}\t\t\n\t\t\\end{array}\n\t\\\\\n\t\t\\begin{array}{ccccccc}\n\t\t\\vspace{1ex}\n\t\t\t\\LID{TLTP-EP-LL-F2b}&\t& \\LID{GR-ring-RR-F2b}\t&\t& \\LID{TLTP-EP-RR-F2b}\t&\t& \\LID{GR-ring-LL-F2b}\n\t\t\\\\\n\t\t\t\\cd{TLTP-EP-LL-F2b}\t& -\t& \\cd{GR-ring-RR-F2b}\t& +\t& \\cd{TLTP-EP-RR-F2b}\t& -\t& \\cd{GR-ring-LL-F2b}\t\t\n\t\t\\end{array}\n\t\\end{eqnarray}\n\\endnumparts\n\\caption{Field decomposition of diagrams~\\ref{TLTP-EP-LL}--\\ref{GR-ring-LL}\nin the case that\nthe external fields are fermionic.}\n\\label{fig:D-ID-2-F2}\n\\end{figure}\\end{center}\n\nOn the basis of~\\eqs{eq:D-ID-2-Alg-a}{eq:D-ID-2-Alg-b},\nwe might expect diagram~\\ref{TLTP-EP-LL-F2a}\nto cancel diagram~\\ref{GR-ring-RR-F2a} {\\it etc.}\\  However,\nthis expectation is not borne out; rather we find\nthat the Kronecker~$\\delta$ part of~\\ref{TLTP-EP-LL-F2a} (\\ref{TLTP-EP-RR-F2a})\ncombines with the gauge remainder part of~\\ref{TLTP-EP-RR-F2a} (\\ref{TLTP-EP-LL-F2a})\nand~\\ref{GR-ring-LL-F2a} (\\ref{GR-ring-RR-F2a}), to yield a common factor\nmultiplied by a term of the form~\\eq{eq:Template}.\nIn this way, we thus find that each row of diagrams\nin \\fig{fig:D-ID-2-F2} vanishes.\n\nThis completes the discussion of diagrammatic\nidentity~\\eq{eq:D-ID-2}\nand so now we move on to the general case,\nwhich relates versions \nof~\\eq{D-ID-Trivial-A}--\\eq{eq:D-ID-Bitten-hook-A}\nwith any number of sockets~$\\geq 3$ to each other. To begin, we consider\nby looking at those diagrams with the specific\npatterns of bites on the left \/ right shown in\n\\fig{fig:D-ID-G-a}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray}\n\t\t\\begin{array}{ccccc}\n\t\t\\vspace{1ex}\n\t\t\t\\LID{TLTP-EP-G-Kd}\t&\t\t\t\t\t& \\LID{TLTP-EP-G-GR}\t\t\t\t&\t& \\LID{GR-ring-G}\n\t\t\\\\\n\t\t\t\\cd{TLTP-EP-G-Kd} \t& \\hspace{-2em} - \t& \\hspace{-1em} \\cd{TLTP-EP-G-GR} \t& -\t& \\cd{GR-ring-G}\n\t\t\\end{array}\n\t\\nonumber \\\\\n\t\\label{eq:D-ID-G-R}\n\t\t\\begin{array}{ccccc}\n\t\t\\vspace{1ex}\n\t\t\t\t& \\LID{GR-ring-G-c}\t\t&\t& \\LID{GR-ring-G-b}\t&\n\t\t\\\\\n\t\t\t- \t& \\cd{GR-ring-G-c}\t\t& +\t& \\cd{GR-ring-G-b} \t& = 0 \n\t\t\\end{array}\n\t\\end{eqnarray}\n\\caption{A generalized diagrammatic identity.}\n\\label{fig:D-ID-G-a}\n\\end{figure}\\end{center}\n\nThere are a number of things to note about \ndiagrams~\\ref{TLTP-EP-G-Kd}--\\ref{GR-ring-G-b}.\nFirst, each of the diagrams possesses the fields\n$W$, $X$ and $Z^1, \\cdots, Z^r$ with indices\n$R$, $T$ and $S_1, \\cdots S_r$, respectively.\nThe cyclic ordering of these fields is the same\nin each diagram.\nIn the case that each diagram possesses only\nthree external fields, we should identify\n$Z^1$ with $Z^r$. \nSecondly, \nfor each of the fields $X$,\n$Z^1, \\cdots Z^r$ there is, in each\ndiagram, an associated $>$. The \nflavour of this $>$, which may\nof course differ from that of the \nassociated external\nfield, is denoted by  $X'$, ${Z^1}', \\cdots {Z^r}'$,\nas appropriate. Notice that, in diagram~\\ref{TLTP-EP-G-GR},\nthe field line $X$ is associated with not one\nbut two internal objects, the flavours of which\nare denoted by $X'$ and $X''$, as indicated. Diagram~\\ref{GR-ring-G}\nis the only one in which the field line $W$\nis associated not only with an external $>$\n(the flavour of which, $W'$, is just the same as\nthat of $W$) but also an internal $>$, the flavour\nof which is denoted by $W''$.\nFinally, we comment on the momentum routing, which is\ntaken to be the same each diagram. The external field\n$W$ is taken to carry momentum $k$. The external\nfields ${Z^1}, \\cdots, {Z^r}$ carry momenta $m_1, \\cdots m_r$,\nrespectively; momentum conservation now fixes the momentum\ncarried by $X$. The loop momentum is $l$, and is routed\nsuch that it is carried by ${Z^1}'$. This now uniquely\ndetermines the momenta carried by all other internal fields.\n\nWith these points in mind, we now wish to show that the sum\nof the five diagrams vanishes. Immediately, we notice\nthat, whatever the flavours of the internal and external\nfields, diagrams~\\ref{TLTP-EP-G-Kd} and~\\ref{GR-ring-G-c}\ncancel. The only difference between these two diagrams\nis the order in which $W$ and $Z^r$ bite the field\nline of $X$, which affects neither the flavours\nof the internal fields or the group theory factors\n(this conclusion will change when we consider\nversions of these diagrams with different patterns\nof bites to the left \/ right).\n\nNow let us consider diagrams~\\ref{TLTP-EP-G-GR},\n\\ref{GR-ring-G} and~\\ref{GR-ring-G-b}. The first\nthing that we must do to show that these diagrams\ncancel is demonstrate that the flavours of\nthe external fields and all common internal fields\ncan be arranged\nto be the same in all three cases. Clearly, we \ncan always choose the external fields to be\nthe same. \nAssigning a value of 0 (1) to the bosonic (fermionic) \nfields, as usual, and denoting \nthe statistics of the field $V$ by $\\stats{V}$,\nwe see that\n\\begin{eqnarray*}\n\t\\stats{X'} & = & \\stats{X} \\ \\xor \\stats{Z^{r'}},\n\\\\\n\t\\stats{{Z^1}'} & = & \\stats{Z^1} \\ \\xor \\stats{X'} \\ \\xor \\stats{W'},\n\\\\\n\t\\stats{{Z^{(r-i)}}'} & = & \\stats{Z^{r-i}} \\ \\xor \\stats{Z^{r-i-1}} \\ \\xor \\cdots \\ \\xor \\stats{{Z^1}'},\n\\end{eqnarray*}\nin all three diagrams, where $i$ is in the range $\\{0,r-2\\}$.\nThus, if in addition to choosing all \nexternal fields to be the same in\ndiagrams~\\ref{TLTP-EP-G-GR},\n\\ref{GR-ring-G} and~\\ref{GR-ring-G-b}\nwe also choose $Z^{r'}$ to be the same,\nit follows that all flavours \n$W', \\ X'$ and ${Z^1}', \\cdots {Z^r}'$ agree.\nConsequently, we can write the sum\nof the three diagrams as:\n\\[\n\t\\int_l F(k,m_i,l,j_1,j_2,j_3)\n\t\\left[\n\t\t- \\rhd^{j_1}_{l-m_1-k} >^{j_2}_{l-m_1} - \\rhd^{j_3}_k >^{j_2}_{l-m_1} + 1\n\t\\right],\n\\]\nwhere $j_2 = j_1 \\ \\xor j_3$ and $F(k,m_i,l,j_1,j_2,j_3)$ is common to all\nthree terms. This term vanishes, courtesy of~\\eq{eq:Template}.\n\n\nOur next task is to show that the versions of~\\eq{eq:D-ID-G-R}\nwith different patterns of bites to the left \/ right\nalso vanish. The first thing to notice is that\nchanging the senses in which the gauge remainder\ncomponents carrying the labels ${Z^2}', \\cdots {Z^{r-1}}'$\nare bitten induces a change which is common to all\nfive diagrams, and so the cancellations detailed above\ngo through, just the same. In the case that there\nare at least four external fields---in which case\nwe do not identify $Z^r$ with $Z^1$---the sense in\nwhich \n${Z^r}'$ is bitten factors out as well. \n\n\nLet us now consider alternatively nested versions\nof diagrams~\\ref{TLTP-EP-G-Kd} and~\\ref{GR-ring-G-c},\nin which we consider changing the sense in which\none and only one gauge reminder is bitten \/ bites.\nChanging the sense in which ${Z^1}'$ is bitten\ninduces the same change in both diagrams, and so they\nstill cancel. Next, consider changing the sense in which ${Z^r}'$\n\\emph{bites}. \nThis does not induce the same change\nin each diagram, on account of the differing orders\nin which ${Z^r}'$ and $W'$ act in each of the diagrams.\nMoreover, in diagram~\\ref{TLTP-EP-G-Kd}\nwe can change the sense in which $W'$ bites. We cannot\ndo this in diagram~\\ref{GR-ring-G-c} since it has,\nin a sense, been done already: diagrams~\\ref{GR-ring-G-c}\nand~\\ref{GR-ring-G-b} form a pair of diagrams, essentially\ndiffering only in the sense in which $W'$ acts.\n(This is particularly clear when we consider these\ndiagrams as part of a larger diagram, in which case\nall indices simply become dummy indices which are summed\nover).\n\nSimilarly, consider alternatively nested versions\nof diagrams~\\ref{TLTP-EP-G-GR}, \\ref{GR-ring-G}\nand~\\ref{GR-ring-G-b}. Changing the sense in which\n$X'$ is bitten induces the same change in all three\ndiagrams, and so they still cancel. However,\nchanging the sense in which ${Z^1}'$ is bitten\ndoes not induce the same change  in the\nfinal diagram as in the first two. \nMoreover, in diagram~\\ref{TLTP-EP-G-GR} we can\nchange the sense in which $W'$ bites, and in \ndiagram~\\ref{GR-ring-G} we can change the sense\nin which $W''$ is bitten. Again, neither of\nthese terms is obviously cancelled.\n\nWe collect together the diagrams which no longer\ncancel after changing the sense in which one\nof the gauge remainders acts\nin \\fig{fig:D-ID-G-b}. \n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{1ex}\n\t\t\\begin{array}{cccccc}\n\t\t\\vspace{1ex}\t\n\t\t\t\t& \\LID{TLTP-EP-G-Kd-B}\t&\t& \\LID{TLTP-EP-G-GR-B}\t&\t& \\LID{GR-ring-G-B}\n\t\t\\\\\n\t\t\t-\t& \\cd{TLTP-EP-G-Kd-B}\t& +\t& \\cd{TLTP-EP-G-GR-B}\t& +\t& \\cd{GR-ring-G-B-Mac}\n\t\t\\end{array}\n\t\\\\\n\t\\vspace{1ex}\n\t\t\\begin{array}{cccccc}\n\t\t\\vspace{1ex}\t\n\t\t\t\t& \\LID{TLTP-EP-G-Kd-C}\t&\t& \\LID{TLTP-EP-G-GR-C}\t&\t& \\LID{GR-ring-G-C}\n\t\t\\\\\n\t\t\t-\t& \\cd{TLTP-EP-G-Kd-C}\t& +\t& \\cd{TLTP-EP-G-GR-C}\t& +\t& \\cd{GR-ring-G-C-Mac}\n\t\t\\end{array}\n\t\\\\\n\t\t\\begin{array}{cccc}\n\t\t\\vspace{1ex}\n\t\t\t\t& \\LID{GR-ring-G-c-B}\t\t&\t& \\LID{GR-ring-G-b-B}\n\t\t\\\\\n\t\t\t+\t& \\cd{GR-ring-G-c-B}\t\t& -\t& \\cd{GR-ring-G-b-B}\n\t\t\\end{array}\n\t\\end{array}\n\t\\]\n\\caption{The set of diagrams, obtained from \nchanging the sense in which precisely\none of the gauge remainders of\ndiagrams~\\ref{TLTP-EP-G-Kd}--\\ref{GR-ring-G-b}\nbites, which do not cancel\nin the same way as the diagrams of \\fig{fig:D-ID-G-a}.}\n\\label{fig:D-ID-G-b}\n\\end{figure}\\end{center}\n\nNotice that, in diagram~\\ref{TLTP-EP-G-Kd-B},\nwe have changed the labelling of\nthe fields, compared to diagram~\\ref{TLTP-EP-G-Kd}.\nWith the fields labelled in this way,\nwe can repeat the analysis of \ndiagrams~\\ref{TLTP-EP-G-GR},\n\\ref{GR-ring-G} and~\\ref{GR-ring-G-b}\nand thus prove that, given a set of common external\nfields, all internal\nfields can be chosen to be the same in \ndiagrams~\\ref{TLTP-EP-G-Kd-B}--\\ref{GR-ring-G-B}.\nFurthermore, we see that the group theory structure\nof these diagrams is identical: in each case\nthe field $Z^1_{S_1}$ is inside the loop,\nwith the remaining field lines having a\ncyclic order, common to the three diagrams, on the outside.\nConsequently, we can write the sum\nof the three diagrams as:\n\\[\n\t\\int_l F'(k,m_i,l,j_1,j_2,j_3)\n\t\\left[\n\t\t1 - \\rhd^{j_1}_{l-m_1-k} >^{j_2}_{l-m_1} - \\rhd^{j_3}_k >^{j_2}_{l-m_1}\n\t\\right],\n\\]\nwhere once again $j_2 = j_1 \\ \\xor j_3$ and $F'(k,m_i,l,j_1,j_2,j_3)$ is \ncommon to all\nthree terms. Thus, in accord with~\\eq{eq:Template},\n the sum of diagrams~\\ref{TLTP-EP-G-Kd-B}--\\ref{GR-ring-G-B}\nvanishes. In exactly the same fashion, we can\nshow that diagrams~\\ref{TLTP-EP-G-Kd-C}--\\ref{GR-ring-G-C}\nalso sum to zero.\n\nThis still leaves us with diagrams~\\ref{GR-ring-G-c-B}\nand~\\ref{GR-ring-G-b-B}. Let us put these\nto one side for a moment and consider instead\ndifferent patterns of bites to the left \/ right\nin diagrams~\\ref{TLTP-EP-G-Kd-B}--\\ref{GR-ring-G-C}.\nLet us start with the first three diagrams.\nThe only changes we can make which do not\nlead to diagrams which cancel in exactly the\nsame way as the parents\n(or take us back to the diagrams of \\fig{fig:D-ID-G-a})\nare as follows:\n\\begin{enumerate}\n\t\\item\tchanging the sense in which $W'$ bites\n\t\t\tin diagrams~\\ref{TLTP-EP-G-Kd-B} and~\\ref{TLTP-EP-G-GR-B};\n\n\t\\item\tchanging the sense in which $X'$ bites in\n\t\t\tdiagram~\\ref{GR-ring-G-B}.\n\\end{enumerate}\nThe resulting diagrams are shown in \\fig{fig:D-ID-G-c}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\\vspace{1ex}\n\t\t\\LID{TLTP-EP-G-Kd-B2}\t&\t& \\LID{TLTP-EP-G-GR-C2}\t&\t& \\LID{GR-ring-G-C2}\n\t\\\\\n\t\t\\cd{TLTP-EP-G-Kd-B2}\t& -\t& \\cd{TLTP-EP-G-GR-C2}\t& -\t& \\cd{GR-ring-G-C2}\n\t\\end{array}\n\t\\]\n\\caption{The set of diagrams, obtained from \nchanging the sense in which precisely\none of the gauge remainders of\ndiagrams~\\ref{TLTP-EP-G-Kd-B}--\\ref{GR-ring-G-C}\nacts, which do not cancel in exactly\nthe same way as the parent diagrams or reproduce already\ndrawn diagrams.}\n\\label{fig:D-ID-G-c}\n\\end{figure}\\end{center}\n\nNow consider making similar changes to \ndiagrams~\\ref{TLTP-EP-G-Kd-C}--\\ref{GR-ring-G-C}.\nThe point is, that the set of diagrams\nwe would draw simply reproduces the\npattern of bites to the\nleft \/ right that we see in \ndiagrams~\\ref{TLTP-EP-G-Kd-B2}--\\ref{GR-ring-G-C2}.\nIn other words, we have taken account of\nthese terms already and so we have no further\ndiagrams to draw. Now, diagram~\\ref{TLTP-EP-G-Kd-B2}\ncancels diagram~\\ref{GR-ring-G-b-B} and\ndiagrams~\\ref{TLTP-EP-G-GR-C2} and~\\ref{GR-ring-G-C2}\ncancel diagram~\\ref{GR-ring-G-c-B}. Thus everything closes.\nAny further changes to the pattern of bites to\nthe left \/ right will simply replicate cancellations\nwe have seen already.\n\nWe have therefore demonstrated that, in addition\nto the diagrams of \\fig{fig:D-ID-G-a} summing to\nzero, so too do appropriate combinations\nof their alternatively nested partners.\nWe can use this fact, together with\nidentity~\\eq{eq:D-ID-2},\nto derive a relationship between all\ndiagrams of the basic structure just discussed,\npossessing between two and $m$ empty sockets.\nThe relationship, which is crucial in our treatment\nof $\\beta_n$, is shown in \\fig{fig:D-ID-App}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\\label{eq:D-ID-G-Pre}\n\t\\begin{array}{ll}\n\t\\vspace{1ex}\n\t\t&\n\t\t\\AGRO{\n\t\t\t\\LIDi[1]{Struc-LXL}{D-ID-App-LXL} \\hspace{0.225em}\n\t\t\t+\n\t\t\t\\left[\n\t\t\t\t\\begin{array}{llll}\n\t\t\t\t\t& \\LIDi[1]{Struc-LLXL}{D-ID-App-LLXL} \t& \\hspace{0.5em} + & \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t- & \\LIDi[1]{Struc-LXRL}{D-ID-App-LXLL} \t& \\hspace{0.5em} + & \\cdots\n\t\t\t\t\\\\ \n\t\t\t\t\t& & & \\vdots\n\t\t\t\t\\end{array}\n\t\t\t\\hspace{5em}\n\t\t\t\\right]\n\t\t}\n\t\\\\\n\t\t= &\n\t\t\\AGRO{\n\t\t\t\\begin{array}{lll}\n\t\t\t\\vspace{0.2in}\n\t\t\t\t\\LIDi[1]{Struc-GR-RR}{D-ID-App-GR-LL} & - & \\hspace{2.3em} \\LIDi[1]{Struc-GR-RRR}{D-ID-App-GR-LLL} \\hspace{1.6em} +\\cdots\n\t\t\t\\\\\n\t\t\t\t& - &\n\t\t\t\t\\left[\n\t\t\t\t\t\\begin{array}{lll}\n\t\t\t\t\t\t\\hspace{1em} \\LIDi[1]{Struc-PF-TLTP-EP-RR}{D-ID-App-PB-TLTP-EP-LL} \t\n\t\t\t\t\t\t& \\hspace{0.5em} -  \\LIDi[1]{Struc-PF-TLTP-EP-RRR}{D-ID-App-PB-TLTP-EP-LLL}  \n\t\t\t\t\t\t& +\\cdots \n\t\t\t\t\t\\\\\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t& \n\t\t\t\t\t\t\\hspace{0.5em} - \\LIDi[1]{Struc-R-PF-TLTP-EP-RR}{D-ID-App-L-PB-TLTP-EP-LL}\n\t\t\t\t\t\t& +\\cdots\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\right]\n\t\t\t\\end{array}\n\t\t}\n\t\\end{array}\n\t\\end{equation}\n\\caption{A relationship between certain diagrams\npossessing between two and $m$ empty sockets.}\n\\label{fig:D-ID-App}\n\\end{figure}\\end{center}\n\nThe notation $\\AGRO{\\ }$ tells us to sum over Every Sense\nin which all the gauge remainders can bite. Thus, \ndiagrams~\\ref{D-ID-App-LXL} and~\\ref{D-ID-App-GR-LL}\nhave thee partners, diagrams~\\ref{D-ID-App-LLXL}, \n\\ref{D-ID-App-LXLL}, \\ref{D-ID-App-GR-LLL} and~\\ref{D-ID-App-PB-TLTP-EP-LL} \t\nseven partners {\\it etc.}\\  \n\nIt is worth clarifying the diagrams represented by the various ellipses.\nThe ellipsis following diagram~\\ref{D-ID-App-LLXL} denotes additionally\nnested diagrams, where the classical, two-point, vertex is still\njoined to the innermost gauge remainder. The ellipsis after\ndiagram~\\ref{D-ID-App-LXLL} denotes additionally nested diagrams,\nwhere the classical, two-point vertex joins to the innermost\nbut one gauge  remainder. The vertical dots on the next line represent additionally\nnested diagrams in which the classical, two-point vertex joins to\ngauge remainders successively further away from the innermost one.\nNote, though, that we never make the join to the outermost gauge remainder.\nThus, we have arranged things such that all diagram in a given\ncolumn possess the same number of sockets.\nThe relationship between the top set of diagrams and\nthe bottom set holds for any number of columns, so \nlong as this number is the same for both sets of\ndiagrams.\n\nWe now expound on how the diagrammatic identities of this\nsection combine to yield~\\eq{eq:D-ID-G-Pre}.\nDiagram~\\ref{D-ID-App-LXL} and its partners\nare clearly\nequal to~\\ref{D-ID-App-GR-LL} and its partners\nby diagrammatic identity~\\eq{eq:D-ID-2}.\nApplying the effective propagator relationship to\ndiagram~\\ref{D-ID-App-PB-TLTP-EP-LL} and its partners, \nthe Kronecker~$\\delta$\npart combines with diagram~\\ref{D-ID-App-GR-LLL}\nto give diagram~\\ref{D-ID-App-LXLL}, via\ndiagrammatic identity~\\eq{eq:D-ID-G-R}\n(and its alternatively nested versions). On the other\nhand, the gauge remainder term and its partners\nare just equal to\ndiagram~\\ref{D-ID-App-LLXL} \nand its partners courtesy of \ndiagrammatic identity~\\eq{eq:D-ID-2}.\n\nUsing  the\nnotation described under \\fig{fig:nestedhook}, we can represent~\\eq{eq:D-ID-G-Pre}\nin the compact form shown in \\fig{fig:D-ID-G-Compact}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray}\n\t\\nonumber\n\t\t\\sum_{m'=1}^{m-1} \\frac{1}{m'!} \\cd{TLTP-EP-ArbGRs} \\hspace{2em}\n\t\t& = &  \\sum_{m'=1}^{m-1} \\frac{1}{(m'+1)!} \\cd{GR-ArbGRs} \\hspace{2em}\n\t\\\\ \n\t\\label{eq:D-ID-G-A}\n\t\t&& - \\sum_{m'=2}^{m-1} \\sum_{m''=0}^{m'-2} \\frac{1}{(m'-m'')!m''!} \n\t\t\\cd{ArbGRs-TLTP-EP-ArbGRs}\n\t\\end{eqnarray}\n\\caption{A compact form of~\\eq{eq:D-ID-G-Pre}.}\n\\label{fig:D-ID-G-Compact}\n\\end{figure}\\end{center}\n\n\nThe loose end of the effective propagator\nin the first diagram can attach to any\nof the $m'$ gauge remainder components. We need not\nstrictly prohibit it\nfrom attaching to the socket on the vertex since\nsuch a diagram vanishes by charge conjugation\\ invariance, anyway.\nThe relationship~\\eq{eq:D-ID-G-A} is, of course, valid separately\nfor every integer value of $m$ from one to infinity.\nThe explicitly drawn  $>$ in the first and third diagrams\ncan bite in either sense. Notice that,\nsince the explicitly drawn $>$ is common\nto each diagram (in the second diagram it just sits\nat the bottom end of the full gauge remainder),\nwe could strip it off completely, \nreplacing it where necessary \nwith the \nhalf arrow notation of~\\eq{eq:WID-A}\n({\\it i.e.}\\  we could remove the $>$ completely from the second\ndiagram, where it does not push forward or pull back\non to another field, but would have to replace it\nwith a half arrow in the first and third diagrams).\n\n\n\\subsubsection{The Second Family}\n\\label{sec:D-ID-Secondary-II}\n\nThe first diagrammatic identity\nin this family\nfollows from the two equivalent\nway of processing the diagram on\nthe left-hand side\\\nof \\fig{fig:D-ID-dGR}: our first operation can\neither be to process the gauge remainder\nor to Taylor expand the vertex (we assume that\nthe socket is decorated by an $A^i$ sector field).\nIn the former case, although we do not explicitly\nindicate it at this stage, it is understood that\nthe socket in the daughter diagrams is filled\nby an $A^i$ sector field carrying zero momentum,\njust as in the parent.\nIn the latter case, we can move the momentum derivative\nfrom the vertex to the $\\rhd$ which bites the vertex,\nat the expense of a minus sign, courtesy of diagrammatic\nidentity~\\eq{eq:GR-TLTP}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{lcccrr}\n\t\t\\cd{D-ID-dGR-A}\t& =\t& \t\t\t\t\t\t\t\t \t& =\t& \\displaystyle \\frac{1}{m} \\cd{D-ID-dGR-B} \t& \\displaystyle - \\frac{1}{m} \\cd{D-ID-dGR-C}\n\t\\\\[1ex]\n\t\t\t\t\t\t& = & - 2 \\cd{D-ID-dGR-D} \\hspace{1em} \t& =\t& -2 \\cd{D-ID-dGR-E} \t\t\t\t& + 2 \\cd{D-ID-dGR-F}\n\t\\end{array}\n\t\\]\n\\caption{Deducing a secondary diagrammatic identity; \nthe socket must be decorated by an $A^i$ sector\nfield which carries zero momentum.}\n\\label{fig:D-ID-dGR}\n\\end{figure}\\end{center}\n\nThe factor of two on the second line arises from using charge conjugation\\ invariance\nto collect together momentum derivative terms; we have\nnot used charge conjugation\\ to collect together pushes forward and\npulls back on the first line. By equating the two expressions\nwith each other, we are lead to the following diagrammatic\nidentity:\n\\begin{equation}\n\\label{eq:D-ID-dGRk-GT}\n\t\\frac{1}{m} \\cd{D-ID-dGR-Bi} \\hspace{1em} = -2 \\cd{D-ID-dGR-Ei}\n\\end{equation}\nwhere, in both diagrams, the socket must be decorated by\nan $A^i$ sector carrying zero momentum. In the\nspecial case that the socket of the\nfirst diagram can be identified with the sockets of one\nof the $m$ gauge remainders~\\eq{eq:D-ID-dGRk-GT} becomes:\n\\begin{equation}\n\\label{eq:D-ID-dGRk-GT-ring-A}\n\t2m(m-1) \\cd{GRs-dGRk} \\hspace{3em} = - \\decGR{ \\ }{\\ensuremath{\\sqcup} >^m}.\n\\end{equation}\nNotice that, in the second diagram, \neven though the socket is identified with one of\nthe gauge remainders, we have kept it\nto remind us that the relationship~\\eq{eq:D-ID-dGRk-GT-ring-A}\nholds only if the sockets on each side of the equation\nare filled with an $A^i$ field carrying zero momentum.\n\n\n\n\nThe second diagrammatic identity in\nthis family follows as a consequence of\ndiagrammatic identities~\\eq{eq:WID} \nand~\\eq{eq:GR-TLTP}. Defining\n\\begin{equation}\n\\label{eq:Combo}\n\t\\ensuremath{\\star} \\equiv \\cd{Combo} \\equiv \\cd{EP-dGR} -\\frac{1}{2} \\cd{dPEP-GR}\n\\end{equation}\nwe have\n\\begin{equation}\n\\label{eq:D-ID-Trapped}\n\t\\cd{TLTP-Trapped-PF} = \\cd{TLThP-GRx2} = - \\cd{TLTP-Trapped-PB}.\n\\end{equation}\n\n\nA number of comments are in order.\nStrictly speaking, the presence of $\\ensuremath{\\star}$ (as opposed\nto some other internal line involving a momentum derivative) is not necessary\nfor either this diagrammatic identity or those that follow. However,\nin practise, it is a $\\ensuremath{\\star}$ that most commonly occurs in such\nscenarios.\nNote\nthat each diagram effectively possesses two external $A^1$s (after\ndecoration of the socket associated with the $\\ensuremath{\\star}$), which \nwe suppose to be the only external fields. \nWe understand that the socket\npossessed by the $>$ and the loose end of the $\\ensuremath{\\star}$\nare internal fields, which are somehow tied up into a complete\ndiagram. \nIn the second diagram, if the right-most\ngauge remainder pushes forward onto the internal\nfield, we recover the parent. If this gauge remainder\nwere instead to pull back onto the internal field,\nthen the supertrace structure of the diagram\nwould be uniquely determined to be \n$\\mathrm{str} A^1_\\mu \\mathrm{str} A^1_\\nu = 0$.\nIf either gauge remainder strikes the external field, then the \nother gauge remainder kills the diagram,\ncourtesy \nof~\\eq{eq:GR-TLTP}. To go from the second diagram to the\nthird, we allow the left-most gauge remainder to act;\nthe only surviving contribution is the pull back onto\nthe internal field.\n\n\nThe next diagrammatic identity\nis the first of this family which necessitates\nconsideration of the algebraic form of gauge\nremainders, and is shown in \\fig{fig:D-ID-Op2-C}.\nNotice that, in the first diagram, both the $\\ensuremath{\\star}$\nand the $>$ bite the socket.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\\label{eq:D-ID-Op2-C}\n\t\t\\begin{array}{cccccc}\n\t\t\t\\LID{D-ID-CTP-E-Cb}\t&\t& \\LID{D-ID-CTP-E-Ab}\t&\t& \\LID{D-ID-CTP-E-Bb}\t&\n\t\t\\\\[1ex]\n\t\t\t\\cd{D-ID-CTP-E-Cb} \t& -\t& \\cd{D-ID-CTP-E-Ab}\t& -\t& \\cd{D-ID-CTP-E-Bb} \t& = 0\n\t\t\\end{array}\n\t\\end{equation}\n\\caption{The first diagrammatic identity of the second\nfamily which relies on~\\eq{eq:Template}.}\n\\label{fig:D-ID-Op2-C}\n\\end{figure}\\end{center}\n\nThis diagrammatic identity follows directly\nfrom~\\eq{eq:Template}.\nAlthough we have given \nboth~\\eqs{eq:D-ID-Trapped}{eq:D-ID-Op2-C}\nin un-nested form, it is clear that\nthey both hold in appropriately nested\ncases. Indeed, we now combine\nthe nested versions of these\ndiagrammatic identities to give\nthe final diagrammatic\nidentity. This is shown in \\fig{fig:D-ID-Op2-D}\nand is, in some sense, the second\nfamily analogue of diagrammatic\nidentity~\\eq{eq:D-ID-G-A}. Wherever a $\\ensuremath{\\star}$\nand $>$ bite the same thing, the $\\ensuremath{\\star}$\nis taken to act first, as in diagram~\\ref{D-ID-CTP-E-Cb}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray}\n\t\\lefteqn{\n\t\t\\sum_{m'=0}^m \n\t\t\\left[\n\t\t\t\\cd{CTP-E-GRs-Combo-GRk-GR} \n\t\t\t-\\cd{CTP-E-GRs-Combo-GR}\n\t\t\\right]\t\n\t}\n\t\\nonumber\n\t\\\\ & &\n\t\t- \\sum_{m'=0}^m \\!\\! \\nCr{m'}{m''} \\!\\! \\sum_{m''=0}^{m'} \n\t\t\\left[\n\t\t\t\\cd{CTP-E-GRs-Combo-CTP-EP-GR}\n\t\t\t\\hspace{1.8em}\n\t\t\t-\\cd{CTP-E-GRs-Combo-GRs-CTP-EP}\n\t\t\t\\hspace{1.8em}\n\t\t\\right]\n\t\t= 0\n\t\\label{eq:D-ID-Op2-G-A}\n\t\\end{eqnarray}\n\\caption{The final diagrammatic identity belonging to the second\nfamily.}\n\\label{fig:D-ID-Op2-D}\n\\end{figure}\\end{center}\n\n\n\\subsection{Subtraction Techniques}\n\\label{sec:Subtraction}\n\n\\subsubsection{Introduction}\n\\label{sec:Subtractions-I}\n\nAs discussed in the introduction, and as\nwe will see shortly, the $n$-loop $\\beta$ function\nis computed by evaluating a set of diagrams at\n$\\Op{2}$:\n\\[\n\t\\beta_n \\Box_{\\mu \\nu}(p) = \\left.\\mathrm{Diagrams}\\right|_{p^2}.\n\\]\nA subset of the diagrams contributing to $\\beta_n$ possess\na stub which is\nmanifestly $\\Op{2}$ and so we would like to Taylor expand\nthe remaining components of these diagrams to zeroth order in $p$.\nThe problem is that individual diagrams may\npossess components which are not Taylor expandable to\n$\\Op{2}$.\nWe could proceed simply by ignoring these\ncomponents since we know that\nthe sum of all diagrams contributing to $\\beta_n$\nmust be Taylor expandable to $\\Op{2}$. However, for\nthe purposes of this paper we adopt a higher level of\nrigour and so keep them. \nAs an example,\nconsider the first diagram of \\fig{fig:TLM-M:TE-Ex-B}\nwhere, in anticipation of what follows, we have added and subtracted\na term.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t\\begin{array}{ccc}\n\t\t\t\\LID{d:111b.11}\t\t&\t\t& \\LIDLID{d:111b.11-s}{d:111b.11-a}\n\t\t\\\\[1ex]\n\t\t\t\\cd{Diagram111b.11} & \\mp \t& \\left[\\cd{Diagram111b.11-s}\\right]_{p^2}\n\t\t\\end{array}\n\t\\]\n\\caption{A two-loop diagram with an $\\Op{2}$ stub, components\nof which cannot be Taylor\nexpanded to $\\Op{2}$, and a term constructed to isolate\nthese components.}\n\\label{fig:TLM-M:TE-Ex-B}\t\n\\end{figure}\\end{center}\n\nWe begin by focusing on diagram~\\ref{d:111b.11}.\nSince we can always Taylor expand vertices in momenta, let us suppose that\nwe take a power of $l$ from the top-most vertex \n(we cannot take any powers of $p$, at $\\Op{2}$) and\nlet us choose to take a power of $k$ from the other vertex. \nUsing \\tab{tab:NFE:k,k'} and~\\eq{eq:EP-leading}, we see that \nthe leading IR behaviour\nof the $l$-integral is given by\n\\[\n\t\\int_l \\frac{1}{l^2 (l-p)^2},\n\\]\nwhich is not Taylor expandable to $\\Op{2}$. Note that had we taken a power of $l$ from\nthe right-hand vertex, rather than a power of $k$, then the extra power of $l$\nin the integrand would render the diagram Taylor expandable in $p$ (to the\nrequired order).\n\nNext, let us consider the pair of \ndiagrams~\\ref{d:111b.11-s} and~\\ref{d:111b.11-a},\nwhere the first of these---the subtraction---comes\nwith the minus sign and the corresponding\naddition comes with the plus sign. The tag $p^2$\ntells us that only the stub carries momentum\n$p$ {\\it i.e.}\\  these diagrams are constructed\nby setting $p=0$ in diagram~\\ref{d:111b.11}\neverywhere but the stub.\n\nThe effect of the subtraction on the parent is to cancel\nall those components which are Taylor expandable to $\\Op{2}$. This immediately tells us\nthe following about any surviving contributions to diagram~\\ref{d:111b.11}:\n\\begin{enumerate}\n\t\\item\tall fields carrying momentum $l$ must be in the $A^1$-sector;\n\n\t\\item\twe must take $\\mathcal{O}(l^0)$ from the $k$-integral (note that\n\t\t\tthe $k$-integral is Taylor expandable in $l$);\n\n\t\\item\twe must discard any remaining contributions to the $l$\n\t\t\tintegral which are Taylor expandable to $\\Op{0}$.\n\\end{enumerate}\nThe contributions to diagram~\\ref{d:111b.11} not removed by\n diagram~\\ref{d:111b.11-s} are shown on the left-hand side\\ of \\fig{fig:TLM-M:TE-Ex-C}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t\\left[\n\t\t\t\\LIDi{Diagram111b.11-Rem}{d:111b.11-Rem}\n\t\t\\right]_{\\NTE{p}}\n\t\t\\rightarrow\n\t\t\\LIO{\n\t\t\t\\left[\n\t\t\t\t\\cd{CTP-E-EP-Thpt-EP-GR}\n\t\t\t\\right]_{\\NTE{p}}\n\t\t\t\\Delta^{11}_{\\alpha \\beta}(p)\n\t\t\t\\decp{\n\t\t\t\t\\cd{CTP-E-EP-Thpt-K-GR}\n\t\t\t}{}\n\t\t}{d:111b.11-Rem-b}\n\t\\]\n\\caption{The contribution to diagram~\\ref{d:111b.11} not removed by its subtraction.}\n\\label{fig:TLM-M:TE-Ex-C}\n\\end{figure}\\end{center}\n\nFocusing first on diagram~\\ref{d:111b.11-Rem},\nwe see that,\nas required, we have taken the $\\mathcal{O}(l^0)$ from the $k$-integral. \nThe tag $\\BigNTE{p}$\ndemands that we retain only the component of the diagram\nwhich, whilst possessing an overall $\\Op{2}$ contribution,\nis not Taylor expandable in $p$ (in dimensional\nregularization, this contribution would go\nas $\\Op{2} p^{-2\\epsilon}$, for this diagram).\nThe bar sitting just before the\n$>$ which bites the classical,\ntwo-point vertex indicates a discontinuity in\nmomenta: the $>$ which bites\nthe classical, two-point vertex carries momentum $l$,\nbut the loop to its right carries only momentum\n$k$ along all lines.\nWe can neaten this diagram up (getting rid of\nthe momentum discontinuity) \nto obtain diagram~\\ref{d:111b.11-Rem-b}.\nTo do this, we first \nnotice \nthat the sub-diagram of~\\ref{d:111b.11-Rem}\nwhich is independent of $l$\ncarries two indices, say $\\rho$ and $\\tau$.\nBy Lorentz invariance, this sub-diagram must go\nas $\\delta_{\\rho \\tau}$, which allows us to tie\nup the effective propagator attaching to the\nmomentum derivative to the gauge remainder\nwhich follows the bar. This procedure yields\nthe $\\BigNTE{p}$ component of diagram~\\ref{d:111b.11-Rem-b},\nbut with indices $\\mu$ and $\\nu$, rather than\n$\\mu$ and $\\alpha$. We now re-express\n\\begin{eqnarray*}\n\t\\Box_{\\mu \\nu}(p)\t& =\t& \\Box_{\\mu \\alpha}(p) \\frac{\\delta_{\\alpha \\beta}}{2 p^2} 2 \\Box_{\\beta \\nu}(p)\n\\end{eqnarray*}\nwhich allows us to re-write diagram~\\ref{d:111b.11-Rem}\nin the form~\\ref{d:111b.11-Rem-b}, up to $\\Op{4}$ corrections\n({\\it cf.}\\ ~\\eq{eq:EP-leading}).\n\n\nBefore discussing the addition, we note that we\ndefine the subtraction to be the diagram which removes\nthe components of the parent diagram which are Taylor\nexpandable to $\\Op{2}$, not by its sign.\nWe could always move  \nthe momentum derivatives in diagrams~\\ref{d:111b.11-s}\nfrom one side of the vertex they strike to\nthe other, at the expense of a minus sign\n(this\nwould amount to replacing $\\partial_\\nu^{-l}$\nwith $\\partial_\\nu^{l}$); were we to do so,\nthe subtraction would now come with a plus sign\nand the addition with a minus. \n \nAdditions like our example~\\ref{d:111b.11-a} can\nbe manipulated by using the effective\npropagator relation and diagrammatic\nidentity~\\eq{eq:PseudoEP}. For the two structures,\n$A$ and $B$ we have:\n\\begin{equation}\n\\label{eq:EP-dCTP-EP}\n\t\\cd{A-EP-dCTP-EP-B} = \n\t\\begin{array}{cccccc}\n\t\t\t& \\LID{A-dEP-B}\t&\t& \\LID{A-GR-combo-B}\t&\t& \\LID{A-combo-GR-B}\n\t\\\\[1ex]\n\t\t-\t& \\cd{A-dEP-B}\t& -\t& \\cd{A-GR-combo-B}\t\t& -\t& \\cd{A-combo-GR-B},\n\t\\end{array}\n\\end{equation}\nwhere\n\\[\n\t\\cd{Combo-GR-b} \\equiv \\cd{EP-dGR-GR-b} -\\frac{1}{2} \\cd{dPEP-GR-GR} \n\t\t\\qquad \\mathrm{and} \\qquad\n\t\\cd{Combo-GR} \\equiv \\cd{EP-dGR-GR} -\\frac{1}{2} \\cd{dPEP-GR-GR}.\n\\]\nIt is now apparent why the definition~\\eq{eq:Combo} is\nuseful.\n\n\nAs a consequence of the decomposition~\\eq{eq:Combo},\nit is clear that diagrams~\\ref{A-GR-combo-B} and~\\ref{A-combo-GR-B}\nhave components where an active gauge\nremainder strikes both structures $A$ and\n$B$. However, in practical calculations,\nthis turns out to be a red-herring: it\nis most efficient never to perform\nthis decomposition and so we only ever \nprocess the explicitly drawn gauge remainders\nin~\\eq{eq:EP-dCTP-EP}.\n\nIt is useful to consider the special case\nwhere the realizations of $A$ and $B$ \nare summed over, as we can \nuse charge conjugation\\ invariance\nto combine terms, in this case. \nSpecifically, can we identify the structure $A$ $(B)$\nof diagram~\\ref{A-combo-GR-B} with the structure\n$B$ $(A)$ of diagram~\\ref{A-GR-combo-B}.\nThis is shown\nin \\fig{fig:Op2-CC} where, for the sake\nof generality, we suppose that the\nactive gauge remainders in~\\eq{eq:EP-dCTP-EP}\ncan be arbitrarily nested.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\t\\LID{A-GR-combo-A}\t&\t\t\t\t& \\LID{A-combo-GR-A}\n\t\\\\[1ex]\n\t\t\\cd{A-GR-combo-A}\t& \\hspace{1em} +& \\cd{A-combo-GR-A}\n\t\\end{array}\n\t\\]\n\\caption{Generalized versions of the second and third\ndiagrams of~\\eq{eq:EP-dCTP-EP}, where $A$ and $B$\nhave been identified with each other.}\n\\label{fig:Op2-CC}\n\\end{figure}\\end{center}\n\nLet us now consider a specific arrangement of\nbites to the left \/ right in diagrams~\\ref{A-GR-combo-A} and~\\ref{A-combo-GR-A}.\nTo be precise, we suppose that there are a total of $G$ such operations,\nwhere $G = L + R = L' + R'$, the (un)primed variable\ncorresponding to diagram (\\ref{A-GR-combo-A}) \\ref{A-combo-GR-A}.\nThe signs of diagrams~\\ref{A-GR-combo-A} and~\\ref{A-combo-GR-A} \nare, according to the rules of \\sec{sec:GRs-2}, $(-1)^R$ and $(-1)^{R'}$,\nrespectively.\n\nWe now choose to focus on pairs of terms for which\n$R' = L$ and consider taking the charge conjugate\nof diagram~\\ref{A-GR-combo-A} \nwhich, we recall, amounts to reflecting it, picking up a sign\nfor each of the $G$ performed gauge remainders\nand the momentum derivative. We remove this\nlatter sign by reversing\nthe direction of the momentum derivative symbol \n so that its sense is once again\ncounterclockwise. By doing this, we have\narranged for the reflected version of diagram~\\ref{A-GR-combo-A}\nto look exactly the same as diagram~\\ref{A-combo-GR-A}.\nThe associated sign of the\nreflected diagram\nis $(-1)^{G+R} = (-1)^{R'}$\nwhich is, of course, just the sign associated with\ndiagram~\\ref{A-combo-GR-A}. Therefore \ndiagrams~\\ref{A-GR-combo-A}\nand~\\ref{A-combo-GR-A} can be combined.\n\n\nAt the beginning of this section,\nwe stated that the effect of\nsubtractions is to isolate the\ncomponents of the parent which\nare not Taylor expandable to $\\Op{2}$\nand that, by construction, the\nadditions are Taylor expandable to the desired\norder. Strictly speaking, this latter\nstatement requires a caveat. \nThe point is that we can encounter\ndiagrams like~\\ref{d:111b.11}\nas a sub-diagram of some larger,\nfactorizable diagram. By factorizable,\nwe mean that the complete diagram \nhas (at least one) internal line which\ncarries just $p$. Thus, we could imagine\nthe top-most external field of\ndiagram~\\ref{d:111b.11} instead\nbeing an effective propagator, carrying $p$,\nattaching to some other sub-diagram\n(which would be decorated by the spare\nexternal field).\nThis latter sub-diagram, which we note\ncannot possess a kernel (only one is allowed\nper complete diagram),\nmay contain\ncomponents which are not Taylor expandable\nto $\\Op{2}$. The strategy is to simply leave \nsuch sub-diagrams alone, but proceed\nas usual with the other part of\nthe complete diagram {\\it i.e.}\\  the sub-diagram\nof the form~\\ref{d:111b.11}. The rule that\nwe will effectively obey is that we only ever\nconstruct subtractions for (sub) diagrams\nwhich contain a kernel. The corresponding\nadditions, whilst always possessing a sub-diagram\nwhich is Taylor expandable in $p$, may also\npossess a sub-diagram which is not. This does\nnot worry us: we simply manipulate the\nTaylor expandable sub-diagram and effectively\nignore the other one. \n\n\n\n\n\\subsubsection{General Analysis}\n\\label{sec:Subtractions-G}\n\nIn this section, we analyse the construction\nand effects of subtractions in complete generality.\nIn \\sec{sec:bn-Op2}, we will discover\nthat we only have to construct subtractions for\ndiagrams with two basic templates, which\nare shown in \\fig{fig:Subs-Templates}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\t\\LID{CTP-E-DK}\t&\t\t\t& \\LID{CTP-E-GRs-K}\n\t\\\\[1ex]\n\t\t\\cd{CTP-E-DK}, \t& \\qquad\t& \\cd{CTP-E-GRs-K}\n\t\\end{array}\n\t\\]\n\\caption{The template for diagrams for which we will construct\nsubtractions.}\n\\label{fig:Subs-Templates}\n\\end{figure}\\end{center}\n\nClearly, we have suppressed all other diagrammatic\ningredients which can include vertices, effective\npropagators and $>$s. Our aim now is to start\nfleshing out the sub-diagrams~\\ref{CTP-E-DK}\nand~\\ref{CTP-E-GRs-K} to see if we can find\nany components which are not Taylor expandable to\n$\\Op{2}$. To stand any chance of doing so,\nwe must find effective propagators or gauge remainders\ncarrying $l-p$\n(where $l$ is a loop momentum) since we recall\nthat, in the $A^i$ sector, these objects go as\n$ 1\/ (l-p)^2$ and $(l-p)_\\alpha \/ (l-p)^2$, respectively.\nTo this end, consider decorating diagram~\\ref{CTP-E-DK}\nwith a single socket, and consider the complete set\nof momentum routings, as shown in \\fig{fig:Subs-MomRoute}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\t\\LID{CTP-E-DK-Soc-A}\t&\t\t\t& \\LID{CTP-E-DK-Soc-B}\t&\t\t\t& \\LID{CTP-E-DK-Soc-C\n\t\\\\[1ex]\n\t\t\\cd{CTP-E-DK-Soc-A}, \t& \\qquad\t& \\cd{CTP-E-DK-Soc-B}, \t& \\qquad\t& \\cd{CTP-E-DK-Soc-C\n\t\\end{array}\n\t\\]\n\\caption{Momentum routings for diagram~\\ref{CTP-E-DK}.}\n\\label{fig:Subs-MomRoute}\n\\end{figure}\\end{center}\n\nFirst, consider diagram~\\ref{CTP-E-DK-Soc-A}.\nThere is no possibility of having an effective\npropagator carrying $l-p$ and so the (sub) diagram\nderived from fully fleshing out \\ref{CTP-E-DK-Soc-A}\nis Taylor expandable to $\\Op{2}$ (we re-iterate\nthat the field carrying momentum $p$ could attach\nto a factorizable sub-diagram which is not Taylor\nexpandable to the desired order in $p$).\n\nNow we move on to diagram~\\ref{CTP-E-DK-Soc-B}.\nNotice that there is no need to consider also\na diagram with the momentum arguments\ninterchanged because such a diagram identical,\nby momentum re-routing invariance. Since $l-p$\nis carried by the kernel (rather than by an\neffective propagator), there is no chance\nof generating a factor of $1\/(l-p)^2$ from\nthe structures drawn. However, the $p$\ncould, in principle, flow into an effective\npropagator or gauge remainder such that their\narguments are {\\it e.g.}\\  $k-p$, as illustrated\nin \\fig{fig:Subs-MomRoute-b}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccccc}\n\t\t\\LID{CTP-E-DK-Soc-Ba}\t&\t\t\t& \\LID{CTP-E-DK-Soc-Bc}\t\t&\t\t\t& \\LID{CTP-E-DK-Soc-Bb}\n\t\\\\[1ex]\n\t\t\\cd{CTP-E-DK-Soc-Ba}, \t& \\qquad\t& \\cd{CTP-E-DK-Soc-Bc}, \t& \\qquad\t& \\cd{CTP-E-DK-Soc-Bb}\n\t\\end{array}\n\t\\]\n\\caption{Fleshings out of diagram~\\ref{CTP-E-DK-Soc-B} such that\neither an effective propagator or gauge remainder\ncarries $k-p$.}\n\\label{fig:Subs-MomRoute-b}\n\\end{figure}\\end{center}\n\n\nAt first sight, diagram~\\ref{CTP-E-DK-Soc-Ba} seems to possess\nthe momentum dependence we are looking for, so long as we put\nthe $k$-dependent effective propagators in the $A^i$ sector, which\nwe do for all that follows. (\nGroup theory considerations in fact force them to be in\nthe $A^1$ sector.)\nIn this case,\nthe $k$-dependent effective propagators\n\\[\n\t\\sim \\frac{1}{k^2 (k-p)^2},\n\\]\nwhich is certainly not Taylor expandable to zeroth order in $p$.\nHowever, we must also consider the vertices. Specifically,\nby charge conjugation\\ invariance, the field filling the socket\nof the topmost vertex must be in the $A^1$ sector. \nTaylor expanding this vertex to zeroth order in $p$\n(which we can always do) must yield at least one power of $k$ by\nLorentz invariance. Furthermore, Lorentz invariance\nand the requirement that (up to powers of $p^{-2 \\epsilon}$)\nthe diagram is $\\Op{2}$\nforces us to pick up a second power of $k$ from the\nfour-point vertex. It is thus quite clear, despite\nfirst appearances, that the diagram is Taylor expandable\nto $\\Op{2}$.\n\nA similar analysis of diagrams~\\ref{CTP-E-DK-Soc-Bc}\nand~\\ref{CTP-E-DK-Soc-Bb}\nleads to the conclusion that the worst momentum\ndependence is\n\\[\n\t\\frac{\\order{l^2, l.k}}{(k-p)^2 (l-k)^2 l^2},\n\\]\nin both cases. Consequently, both diagrams are Taylor expandable to\n$\\Op{2}$.\n\n\nWe now attempt to worsen the behaviour of \ndiagram~\\ref{CTP-E-DK-Soc-Ba} by adding\nadditional diagrammatic elements. Clearly, \nsimply adding more effective propagators\nwill do nothing. What we can try, however,\nis splitting open one of the existing\neffective propagators, as shown in \n\\fig{fig:Subs-MomRoute-b2}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\LIDi{CTP-E-DK-Soc-Ba-i}{CTP-E-DK-Soc-Ba-i}\n\t\\]\n\\caption{Attempting to worsen the IR behaviour of diagram~\\ref{CTP-E-DK-Soc-Ba}.}\n\\label{fig:Subs-MomRoute-b2}\n\\end{figure}\\end{center}\n\nThis seems to have done the trick: rather than having\na single effective propagator giving a factor\nof $1\/k^2$, we now have two (attached to an \narbitrary structure), and the diagram\nappears not to be Taylor expandable to $\\Op{2}$.\nHowever, in all the cases we encounter, we will\nfind that when we sum over all\npossible diagrams which can be represented by\nthe arbitrary structure, the arbitrary\nstructure turns out to be transverse in its\nexternal momentum (when the legs are in the $A^i$\nsector). Once again we are thwarted: diagrams\nof the type~\\ref{CTP-E-DK-Soc-Ba-i} are, in practise\nTaylor expandable to $\\Op{2}$.\nSimilarly, we cannot modify diagrams~\\ref{CTP-E-DK-Soc-Bc}\t\nand~\\ref{CTP-E-DK-Soc-Bb} such that they are\nno longer Taylor expandable to $\\Op{2}$.\n\nWe can repeat precisely the same arguments with \ndiagram~\\ref{CTP-E-DK-Soc-C} \nand are thus lead to the conclusion\nthat all diagrams with~\\ref{CTP-E-DK} as\na template are Taylor expandable to $\\Op{2}$.\nWe can of course still construct subtractions,\nbut they exactly cancel the parent (leaving\nover the additions).\n\nDiagrams built up from the template~\\ref{CTP-E-GRs-K}\nare, however, a different kettle of fish.\nIndeed, the example with\nwhich we began \\sec{sec:Subtractions-I} is of\nthis type. From diagram~\\ref{CTP-E-GRs-K},\nthere are two ways in which we can construct\ndiagrams which are not Taylor expandable to $\\Op{2}$,\nas shown in \\fig{fig:Subs-NTE}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\t\\LID{CTP-E-NTE-a}\t&\t\t\t& \\LID{CTP-E-NTE-b}\n\t\\\\[1ex]\n\t\t\\cd{CTP-E-NTE-a},\t& \\qquad\t& \\cd{CTP-E-NTE-b}\n\t\\end{array}\n\t\\]\n\\caption{The two types of diagram derived from~\\ref{CTP-E-GRs-K}\nwhich are not Taylor expandable to $\\Op{2}$.}\n\\label{fig:Subs-NTE}\n\\end{figure}\\end{center}\n\nExcluding those terms in which the loose ends\nof the effective propagator and kernel of\ndiagram~\\ref{CTP-E-NTE-a} attach to something\ntransverse in $l$ (in the $A^i$ sector),\nboth diagrams have a component which goes as\n\\[\n\t\\frac{1}{l^2 (l-p)^2},\n\\]\nand so both diagrams possess\na component which is not Taylor expandable to\norder $p^2$. We know from the example\nat the beginning of \\sec{sec:Subtractions-I}\nthat further progress can be made by combining\nsuch diagrams with their subtractions. \nTo isolate the components which survive, \nthe strategy\nis as follows:\n\\begin{enumerate}\n\t\\item\tidentify all loop integrals over momenta other than\n\t\t\t$l$ but which carry dependence on $l$;\n\n\t\\item\tdetermine which, if any, of these loop integrals\n\t\t\tare factorizable from the others;\n\n\t\\item\tFocus on the loop integral \/ non-factorizable \n\t\t\tloop integrals for which the kernel is in the\n\t\t\tintegrand and either\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item \tTaylor expand to zeroth order in $l$,\n\t\t\t\t\t\tdirectly;\n\n\t\t\t\t\\item\tconstruct subtractions if appropriate\n\t\t\t\t\t\tand iterate the procedure.\n\t\t\t\\end{enumerate}\n\\end{enumerate}\n\nAn example of this is shown in \\fig{fig:Op2-NTE-Ex},\nwhere we recognize that the first diagram\nis a generalization of~\\ref{d:111b.11}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{ccc}\n\t\t\t\\LID{CTP-E-NTE-EX}\t&\t\t& \\LIDLID{CTP-E-NTE-EX-s}{CTP-E-NTE-EX-a}\n\t\t\\\\[1ex]\n\t\t\t\\cd{CTP-E-NTE-EX} \t& \\mp \t& \\decp{\\cd{CTP-E-NTE-EX-s}}{}\n\t\t\\end{array}\n\t\\]\n\\caption{A diagram possessing a component which is not Taylor\nexpandable to $\\Op{2}$, together with its subtraction and addition.}\n\\label{fig:Op2-NTE-Ex}\n\\end{figure}\\end{center}\n\nFollowing the above recipe, we recognize that\nthe integrals over $k, \\ m_1$ and $m_2$ all contain\ndependence on $l$. Given that the $m_2$~integral contains\nthe kernel and that the $m_1$~integral has dependence\non $m_2$ but the $k$~integral does not, we can\nfactorize out the integral over $k$. \nThe parent\nand subtraction combine to yield the first diagram\nof \\fig{fig:Op2-NTE-Ex-b}, for which\nwe have constructed a subtraction designed to isolate\nthe part of the $m_1, \\ m_2$ dependent sub-diagram\nwhich is not Taylor expandable in $l$.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\begin{array}{ccc}\n\t\t\t\t\\LID{CTP-E-NTE-EX-b}\t\t\t\t\t\t&\t\t& \\LIDLID{CTP-E-NTE-EX-ss}{CTP-E-NTE-EX-sa}\n\t\t\t\\\\[1ex]\n\t\t\t\t\\left[\\cd{CTP-E-NTE-EX}\\right]_{\\NTE{p}} \t& \\mp \t& \\left[\\cd{CTP-E-NTE-EX-ss}\\right]_{\\NTE{p}}\n\t\t\t\\end{array}\n\t\\\\\n\t\\vspace{2ex}\n\t\t\\rightarrow\n\t\t\\dec{\n\t\t\t\\LIO{\n\t\t\t\t\\left[\n\t\t\t\t\t\\cd{CTP-E-NTE-EX-Rem} \\hspace{1em}\n\t\t\t\t\\right]_{\\NTE{p}}\n\t\t\t\t\\Delta^{11}_{\\alpha \\beta}(p)\n\t\t\t\t\\decp{\n\t\t\t\t\t\\cd{CTP-E-EP-Thpt-K-GR}\n\t\t\t\t}{}\n\t\t\t}{CTP-E-NTE-EX-Rem}\n\t\t}{}\n\t\\\\\n\t\t\\qquad\n\t\t+\n\t\t\\dec{\n\t\t\t\\LIO{\n\t\t\t\t\\left[\n\t\t\t\t\t\\cd{CTP-E-NTE-EX-a}\n\t\t\t\t\\right]_{\\NTE{p}}\n\t\t\t\t\\Delta^{11}_{\\alpha \\beta}(p)\n\t\t\t\t\\decp{\n\t\t\t\t\t\\cd{Diagram111b.11-s-b}\n\t\t\t\t}{}\n\t\t\t}{CTP-E-NTE-EX-Rem-a}\t\n\t\t}{}\n\t\\end{array}\n\t\\]\n\\caption{The result of combining diagrams~\\ref{CTP-E-NTE-EX} and~\\ref{CTP-E-NTE-EX-s}.}\n\\label{fig:Op2-NTE-Ex-b}\n\\end{figure}\\end{center}\n\n\nDiagram~\\ref{CTP-E-NTE-EX-b} combines\nwith diagram~\\ref{CTP-E-NTE-EX-ss} \nto yield diagram~\\ref{CTP-E-NTE-EX-Rem}.\nJust as we have utilized Lorentz\ninvariance to draw\ndiagram~\\ref{CTP-E-NTE-EX-Rem} in an\nappealing form,\nso too do\nwe recognize that a similar thing can be\ndone with the addition, \\ref{CTP-E-NTE-EX-sa},\nyielding diagram~\\ref{CTP-E-NTE-EX-Rem-a}.\n\nReturning to diagram~\\ref{CTP-E-NTE-EX-Rem},\nnotice that we must take the component of the\n$m_1$ integral which, up to the non-polynomial\ndependence on $l$ goes as $l^0$. Were we to\ntake the $l^2$ component, instead,\nthen we could not take the $\\BigNTE{p}$ component of\nthe diagram as a whole. The appearance of the\ntag $\\BigNTE{l}$ is awkward, since it requires\nspecification of a loop momentum, which is unnatural\nfrom our diagrammatic view point. However, with a little\nthought, we can replace this tag with something\nwhich refers only to the external momentum, $p$.\nDiagram~\\ref{CTP-E-NTE-EX-Rem} obviously possesses\nthree loop integrals. However, only two of them (the ones over $l$ and $m_1$)\nare tagged such that we must take the $\\BigNTE{\\mathrm{mom}}$ component,\nwhere `$\\mathrm{mom}$' refers to the momentum external with respect to\\\nthe appropriate sub-diagram. There is a simple rule to\ndetermine the loops from which we must take the\n$\\BigNTE{\\mathrm{mom}}$ component (a similar\nrule can be derived for diagrams of type~\\ref{CTP-E-NTE-a}). In all that\nfollows, if there are any sub-diagrams which\n attach to\njust two effective propagators carrying equal momenta ({\\it cf.}\\ \\\n\\fig{fig:Subs-MomRoute-b2}), we can\nignore each sub-diagram and one of the\neffective propagators to which it attaches.\nStarting with a diagram of arbitrary loop order, the\nrecipe is as follows:\n\\begin{enumerate}\n\t\\item\tIdentify the loop momentum associated with the internal\n\t\t\tfield which attaches to the classical, two-point vertex\n\t\t\t(in the above examples, this momentum has always been $l$)\n\t\t\tand route momenta such that \n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item the external field attaching to the classical,\n\t\t\t\t\t\ttwo-point vertex carries momentum $+p$;\n\n\t\t\t\t\\item  momentum $l+p$ flows \n\t\t\t\t\t\tinto the vertex decorated by the other external field;\n\t\t\t\\end{enumerate}\n\n\n\t\\item\tCut the diagram in every place that there is a gauge remainder\n\t\t\tcarrying $l$ and cut the effective propagator carrying $l$\n\t\t\tat the end out of which $+l$ flows.\n\n\t\\item\tDiscard the sub-diagrams for which there is\n\t\t\ta gauge remainder which carries just $l$\n\t\t\t(this includes the sub-diagram carrying $p$\n\t\t\tand $l$, which is supposed to have already been tagged with $\\BigNTE{p}$);\n\t\\label{it:discard}\n\n\t\\item\tIdentify the remaining sub-diagram as the one that must be\n\t\t\ttagged $\\BigNTE{l}$;\n\n\t\\item\tIterate the procedure, if necessary, taking\n\t\t\tthe sub-diagram tagged with $\\BigNTE{l}$ as the starting point.\n\\end{enumerate}\n\nIn \\fig{fig:Op2-NTE-Ex-c} we apply this procedure to\ndiagram~\\ref{CTP-E-NTE-EX-Rem}, with the red lines\nindicating the cuts. The above recipe tells us to\ntag the sub-diagram with loop momentum $m_1$, but\nnot the one with loop momentum $k$, since the latter sub-diagram\nincludes a $>_l$ and so is excluded by~\\ref{it:discard}, above.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\cd{CTP-E-NTE-EX-Rem-Cut}\n\t\\]\n\\caption{Algorithmic determination of which loop integrals\nare tagged with $\\NTE{}$.}\n\\label{fig:Op2-NTE-Ex-c}\n\\end{figure}\\end{center}\n\nSince this recipe works at any number of loops,\nthere is no need to explicitly tag the appropriate\nloop integrals of some diagram\nalready tagged by $\\BigNTE{p}$; rather, we simply tag the diagram as \na whole by $\\not{\\! \\mathrm{T}}_{M} (p)$.\n\n\n\\section{Introduction}\n\\label{sec:Intro}\n\n\n\nOf all the problems in theoretical physics,\nsurely one of the most pressing is a better understanding\nof Yang-Mills theories, particularly in the\nnon-perturbative domain. A promising framework\nfor addressing this issue\nis the Exact Renormalization Group \n(ERG)~\\cite{Wil,W&H,Pol}, the continuum version of Wilson's RG.\n\nThe essential physical idea behind this approach is\nthat of integrating out degrees of freedom\nbetween the bare scale of the quantum theory and\nsome effective scale, $\\Lambda$. The effects\nof these modes are encoded in the Wilsonian effective action, \n$S_\\Lambda$, which describes the physics of the theory\nin terms of parameters relevant to the effective scale.\nCentral to this methodology is the ERG equation\nwhich determines how the Wilsonian effective action\nchanges under infinitesimal changes of the scale.\nBy relating physics at different\nscales, this equation provides access to the\nlow energy dynamics of Yang-Mills theories.\nIndeed, more generally, \nthe ERG has proven itself to be a flexible\nand powerful tool for studying \nboth perturbative and non-perturbative problems\nin a range of field theories \n(see~\\cite{Fisher:1998kv,Morris:1998da,Aoki:2000wm,Litim:1998nf,Berges:2000ew,Bagnuls:2000ae,Polonyi:2001se,Salmhofer:2001tr,Delamotte:2003dw} \nfor reviews). A particular advantage conferred by the ERG is\nthat renormalization is built in:\nsolutions to the flow equation (in pretty much any approximation scheme),\nfrom which physics can be extracted,\ncan be naturally phrased directly in terms of renormalized parameters.\n\n\nGiven that the notion of a momentum cutoff\nis fundamental to the entire ERG construction,\nit is clear that a regulator\nis required which incorporates this feature. \nImmediately,\nthis presents a problem for Yang-Mills theories,\nsince the direct implementation of a momentum \ncutoff breaks non-Abelian gauge invariance. \nTraditional (gauge fixed) approaches,  which can be broadly divided\ninto those which employ the background field \nmethod~\\cite{Reuter:1993kw,Reuter:1997gx,Bergerhoff:1997cv,Litim:2002ce,Bonini:2001,Freire:2000bq,Gies:2002af,Litim:2002hj}\nand those which do \nnot~\\cite{Becchi:1993,Bonini:1993sj,Bonini:1994kp,Ellwanger:1994iz,Ellwanger:1995qf,Ellwanger:1996wy,Ellwanger:1997tp,Ellwanger:1997wv,Ellwanger:1999vc,Ellwanger:2002xa,D'Attanasio:1996jd,Litim:1998qi,Simionato:2000,Simionato:2000-B,Panza:2000,Pawlowski:2003hq,Fischer:2004uk}, \naccept this breaking,\nrecovering the physical symmetry in \nthe limit that all quantum fluctuations\nhave been integrated out.\\footnote{\nFor an approach based on the geometric \neffective action see~\\cite{Veneziano,Pawlowski-VDW} and for\na summary of the various approaches see~\\cite{Pawlowski:2005xe}.}\n\nNevertheless, in~\\cite{SU(N|N)}, a regulator was constructed\nfor $SU(N)$ Yang-Mills based on a real, gauge invariant cutoff.\nThis is achieved by embedding the physical gauge theory\ninto a spontaneously broken $SU(N|N)$ supergauge theory\nwhich is itself regularized by covariant higher \nderivatives. Combining this with earlier work~\\cite{ym,ymi,ymii}\nallowed the construction of\nan ERG for Yang-Mills which respects \ngauge invariance at all scales~\\cite{aprop}.  In addition to\nthe obvious advantages this has over the alternative approaches,\nthere is a major additional benefit: the gauge invariance\nis in fact \\emph{manifest}, no gauge fixing being required\nat any stage in the computation of the Wilsonian effective\naction. Whilst having considerably novelty value, manifest\ngauge invariance also provides powerful technical simplifications:\nthe gauge field is protected from field strength renormalization and\nthe Ward identities take a particularly simple form~\\cite{ymi},\nsince the Wilsonian effective action is built only from\ngauge invariant combinations of the covariant derivative,\neven at the quantum level. In the non-perturbative domain, not only are\nGribov copies~\\cite{Gribov} entirely avoided, but it should\npossible to make statements\nabout {\\it e.g.}\\  confinement in\na completely gauge independent way. Furthermore, \nsuch a framework\nhas the ability to underlie gauge-fixed ERG formalisms\nas it should, in principle, be possible to derive all\nresults obtained by the latter by gauge fixing\nat an appropriate stage.\n\n\n\n\nThe key to constructing the manifestly gauge invariant \nscheme of this paper\nresides in the immense\nfreedom inherent in  the  ERG~\\cite{jose1,jose2,mgierg1}: of the infinite\nnumber suitable for $SU(N)$ Yang-Mills, an infinite subset\nallow the Wilsonian effective action to be computed without\nfixing the gauge. \nOf these manifestly gauge invariant\nERGs, we further specialize to those\nwhich allow convenient renormalization\nto any loop order~\\cite{mgierg1,mgierg2,Thesis}. (By this we mean\nthat the flow equation is of a suitably general form to\ntreat the physical gauge field and an unphysical copy, \nwhich\nis part of the regularizing structure, asymmetrically. Since\neach of these fields comes with their own coupling, which\nrenormalize separately, allowing the flow equation to distinguish\nbetween them facilitates convenient\ncomputation.) \n\nDespite these restrictions, there are still an infinite\nnumber of admissible ERGs, the differences between them\namounting to the following non-universal details. The first two \nrelate to the\nimplementation of a gauge invariant cutoff:\nthe exact forms of both the cutoff\nfunctions and their covariantizations amount to non-universal\nchoices. The final source of non-universal differences between\nthe ERGs with which we work is\nthe `seed action', $\\hat{S}$~\\cite{Thesis,mgierg1,mgierg2,scalar1,scalar2,aprop,giqed}:\na functional which respects the same symmetries as the Wilsonian effective \naction, $S$, and has the same structure. However, whereas our\naim is to solve the flow for $S$, $\\hat{S}$ acts as an input.\nPhysically, the seed action can be thought of as (partially)\nparameterizing a general Kadanoff blocking~\\cite{Kadanoff} in the \ncontinuum~\\cite{jose1,jose2,mgierg1}. Crucially, these non-universal\ndetails\nneed never\nbe explicitly specified, instead just satisfying general\nconstraints to ensure that the flow equation is well defined.\nWe turn this residual freedom\nin the construction to our advantage\nby recognizing that since all non-universal\ndetails must cancel out in the computation of a universal\nquantity, they can be efficiently absorbed into diagrammatic\nrules. This observation formed the basis for the manifestly gauge invariant\nand universal calculus proposed in~\\cite{aprop}, in which\na scheme was developed whereby these non-universal contributions\ncan be iteratively cancelled out, in perturbative calculations.\n\nIn the original work~\\cite{aprop},\na small subset of the diagrammatic rules now known\nto exist were uncovered and were used in the initial\nstages of a manifestly gauge invariant\ncomputation of the one-loop $\\beta$\nfunction. In a series of \nworks since then~\\cite{scalar2,mgierg1,mgierg2,Thesis,giqed},\nthese diagrammatic rules were extended,\nallowing both the one and two-loop $\\beta$ functions\n(in a variety of Quantum Field Theories (QFTs)) to be reduced\nto manifestly universal diagrammatic expressions,\nfrom which the correct numerical coefficients \nwere directly extracted. In this paper, \nworking in $SU(N)$ Yang-Mills,\nwe bring together and complete\nthe overlapping sets of ideas from this collection of works,\ndeveloping the diagrammatic calculus to a stage where it\nis applicable at any number of loops.  The calculus\nis then comprehensively illustrated by \nderiving an expression for the $n$-loop $\\beta$ function, $\\beta_n$,\nwhich has no explicit dependence on\neither the seed action or the details of the\ncovariantization of the cutoff.\n\n\nThis result is, in itself, interesting for several \nreasons. First, let us recall the standard argument as to why\nthe coefficients $\\beta_1$ and $\\beta_2$, but not $\\beta_{\\geq 3}$,\nare guaranteed to\nagree between certain renormalization schemes~\\cite{Weinberg}.\nIt is important to recognize that each choice of the non-universal\ndetails within our ERG corresponds, in general, to a different renormalization\nscheme. Focusing on one such scheme, we take the coupling\nof the physical $SU(N)$ gauge field to be $g(\\Lambda)$.\\footnote{Beyond this\ndiscussion, we will take $g(\\Lambda)$ to represent the coupling\nfor all renormalization schemes implicitly defined by our approach.}\nNow consider a second scheme---either corresponding to some other\nchoice of non-universal details within our ERG or to an entirely\nindependent scheme such as $\\overline{MS}$---with coupling $\\tilde{g}(\\mu \\mapsto \\Lambda)$.\nGiven the dimensionless\ncoefficient, $\\eta$, we can perturbatively\nmatch the two couplings:\n\\begin{equation}\n\\label{eq:match}\n\t\\frac{1}{\\tilde{g}^2} = \\frac{1}{g^2} + \\eta + \\Or (g^2).\n\\end{equation}\nUsing the usual definition for the $\\beta$ function of $g$\n\\[\n\t\\beta \\equiv \\Lambda \\partial_\\Lambda g = \\sum_{i=1}^{\\infty} g^{2i+1} \\beta_i,\n\\]\nand a similar definition for $\\tilde{\\beta} (\\tilde{g})$, we can differentiate~\\eq{eq:match}\nwith respect to\\ $\\Lambda$ to obtain\n\\begin{equation}\n\t\\tilde{\\beta}_1 + g^2 \\tilde{\\beta}_2 = \\beta_1 + g^2 \\beta_2 + \\Lambda \\partial_\\Lambda \\eta + \\Or (g^4).\n\\label{eq:Universal-beta}\n\\end{equation}\nTherefore, \nif $\\Lambda \\partial_\\Lambda \\eta = 0$ (or, at any rate, does not \ncontribute until $\\mathcal{O}(g^4)$), \nwe will obtain $\\tilde{\\beta}_1 = \\beta_1$\nand $\\tilde{\\beta}_2 = \\beta_2$. We can expect this \nagreement to be spoilt, however, if there exist\nrunning, dimensionless couplings, besides $g$\n(which is equivalent to the introduction of\nadditional mass scales).\n\n\nIndeed, within our setup there \ngenerically exist dimensionless couplings\nwhich run even at tree level\nlevel~\\cite{aprop}, spoiling\nagreement between $\\beta$-function coefficients\nat one loop. Of course, \nthis is not a sign\nof a sick formalism, just a sign that\n$\\beta_1$ and $\\beta_2$ are not physically\nobservable and can be scheme dependent. Nonetheless, it is possible\nto recover the universal values of\n$\\beta_1$ and $\\beta_2$ by suitably tailoring the\nsetup. The running of\n$\\eta$ has two sources.\nThe first is $\\hat{S}$, which contributes\nto the running of $\\eta$ at all orders, including\ntree level. However, through an implicit choice\nof $\\hat{S}$, this running can be removed, not just\nat one-loop, but actually to all orders~\\cite{Thesis,mgierg2}.\nIndeed, we now assume that this has been done, and promote\nthis choice to a requirement, necessary in the specification\nof the subset of manifestly gauge invariant\nERGs with which we choose to work.\n\nThe second contribution to $\\Lambda \\partial_\\Lambda \\eta$ comes from\nan unphysical, dimensionless coupling, $g_2$,\nassociated with the regularizing \nstructure~\\cite{SU(N|N),Thesis,mgierg1,mgierg2}.\nThe running of this coupling,\nwhich occurs from the one-loop level onwards,\ncannot be removed through\na choice of $\\hat{S}$; the solution is to tune it to\nzero at the end of a calculation. In this manner,\n$\\beta_2$ can be arranged to coincide with its\nstandard value~\\cite{Thesis,mgierg2}. \nFor convenience, we work not with $g_2$ directly\nbut with\n\\begin{equation}\n\t\\alpha := g^2_2\/g^2\n\\label{eq:alpha-defn}\n\\end{equation}\nand so it is $\\alpha$, in practise, which is tuned to zero.\n\nBeyond two loops, it is apparent from~\\eq{eq:Universal-beta}\nthat there is no reason to expect any agreement\nbetween $\\beta$ function coefficients. In light of\nthis, it is surprising not only that all contributions\nto $\\Lambda \\partial_\\Lambda \\eta$ coming from $\\hat{S}$ can be removed to all\norders but also that all dependence of $\\beta_n$ on\nthe seed action and details of the covariantization\nof the cutoff cancels out, to all orders. (For speculations\non whether it may be possible to push the removal\nof non-universal details further still, see~\\cite{RG2005}.)\nThat we do find such\ncancellations is perhaps indicative that there\nis a more direct framework for performing calculations\nin QFTs which retains the advantages of the ERG approach\nwhilst removing some or all of the redundancy inherent in\nthe approach. In particular, these observations\ncould inspire a manifestly \ngauge invariant formalism where the seed action and\ndetails of the covariantization of the cutoff\nare relegated to a background role. \n\nThe second point to make about the\nderivation of an expression for $\\beta_n$\nwhich has no explicit dependence on\neither the seed action or the details of the\ncovariantization of the cutoff is that this is\na huge step forward in\nturning this formalism into a practical computational\nscheme.\nThe calculation of $\\beta_1$ and $\\beta_2$\nin this ERG approach was, up until now, an arduous\ntask; just getting to the expression from which the universal\nvalue can be extracted was extremely difficult. Now, however,\nthis step is trivial as we simply specialize the new\nformula for $\\beta_n$ to the appropriate loop order!\n\n\nTo appreciate the various elements of the diagrammatic\ncalculus, it is worth reviewing the \nprocedure for computing $\\beta$ function coefficients employed\nin~\\cite{Primer}.\nTo compute $\\beta_{1,2}$, we start by using the flow\nequation to compute the flow of the two-point vertex\ncorresponding to the physical $SU(N)$ gauge field,\n$A^1_\\mu$, which we suppose carries momentum, $p$. Next, we specialize to the\nappropriate loop order and work at $\\Op{2}$; this\nlatter step constrains the equation by allowing the renormalization\ncondition for the physical coupling to feed in. At this\npoint, the equation for $\\beta_{1,2}$ contains a small\nnumber of diagrams, each of which contains explicit\ndependence on both the seed action and the details\nof the covariantization of the cutoff.\n\nCentral to the diagrammatic calculus\nis the `effective propagator \nrelation'~\\cite{Thesis,mgierg1,mgierg2,Primer,scalar1,scalar2,aprop,giqed}.\nThis convenient computational device\narises as a consequence of a choice\nof seed action we are free to make: \nwe choose the classical, two-point seed action vertices equal to \ntheir Wilsonian effective action counterparts.\nIn turn,\nthis ensures that \nfor each independent classical two-point vertex (that cannot be consistently\nset to zero~\\cite{Thesis}) there exists\nan `effective propagator', denoted by $\\Delta$, \nwhich is the inverse of the\ngiven vertex, up to a `gauge remainder'. \nDenoting the classical two-point vertex corresponding to\nthe fields $X$ and $Y$, which carry indices $R$ and $S$ and\nmomenta $p$ and $-p$, respectively, \nby $S_{0 R S}^{\\ X Y}(p)$\nthe effective propagator relationship in the $A^1$ sector reads\n\\[\n\tS^{\\ A^1 A^1}_{0 \\mu \\ \\; \\alpha} (p) \\Delta^{A^1 A^1}_{\\alpha \\ \\, \\nu} (p) = \\delta_{\\mu\\nu} - \\frac{p_\\mu p_\\nu}{p^2}.\n\\]\nThus, $p_\\mu p_\\nu \/ p^2$ is a gauge remainder\nand it appears as \na consequence of the manifest gauge invariance: the effective\npropagators are inverses of the classical, two-point vertices\nonly in the transverse space.\nAs we will see later, it is convenient to split this gauge remainder \ninto two components,\n$p_\\nu$ and $p_\\mu\/p^2$. \n\n\nTo proceed, we recognize that certain diagrams generated\nby the flow comprise exclusively Wilsonian\neffective action vertices joined together by $\\bigdot{\\Delta}$,\nwhere we define\n\\begin{equation}\n\\label{eq:dot}\n\t\\bigdot{X} \\equiv -\\Lambda \\partial_\\Lambda|_\\alpha X.\n\\end{equation}\nThese terms are processed by moving the $-\\Lambda \\partial_\\Lambda|_\\alpha$\nfrom the effective propagator to the diagram as a whole,\nminus correction terms in which $-\\Lambda \\partial_\\Lambda|_\\alpha$ strikes\nthe vertices. The former diagrams are called $\\Lambda$-derivative\nterms; the latter can be processed using the flow equation\nand the resulting set of diagrams simplified, using\na set of primary diagrammatic identities, of which the\neffective propagator relation is one.\nAt this point, we are able to identify\ncancellations of non-universal contributions, at the\ndiagrammatic level.\nThere is, however, a complication to this diagrammatic procedure:\nparticular classes of sub-diagrams can have two distinct diagrammatic\nrepresentations. The equivalence of these representations\nis encoded in the secondary diagrammatic identities.\n\nIterating the diagrammatic procedure,\nthe expressions for $\\beta_{1,2}$ ultimately\nreduce to the following sets of diagrams:\n\\begin{enumerate}\n\t\\item \t$\\Lambda$-derivative terms, which are built out\n\t\t\tof Wilsonian effective action vertices, effective\n\t\t\tpropagators and (components of) gauge remainders;\n\n\t\\item \t`$\\alpha$-terms', consisting of diagrams\n\t\t\tbuilt out of the same elements as the \n\t\t\t$\\Lambda$-derivative terms but\n\t\t\tstruck by $\\partial \/ \\partial \\alpha$\n\t\t\t(there are no $\\alpha$-terms at one loop~\\cite{Thesis,mgierg1});\n\n\t\\item \t`$\\Op{2}$-terms', which contain an $\\Op{2}$ stub\n\t\t\t{\\it i.e.}\\  a diagrammatic component which is manifestly\n\t\t\t$\\Op{2}$.\n\\end{enumerate}\n\n\nThe $\\Op{2}$-terms can be manipulated. In the calculation\nof $\\beta_1$, at any rate, the structure attaching to the stub\ncan be directly Taylor expanded to zeroth order in $p$---which \ncan once again be done diagrammatically. The above diagrammatic\nprocedure is then iterated.\nAt two loops (and beyond), \nthis procedure is not so straightforward, since\nna\\\"ive Taylor expansion can generate spurious infra-red (IR)\ndivergences~\\cite{Thesis,mgierg1,mgierg2}. The\nsolution is to isolate the components which\ncannot be Taylor expanded using the \n`subtraction techniques' of~\\cite{Thesis,mgierg2}.\nNow the $\\Op{2}$ can\nbe processed, and so $\\beta_{1,2}$ can be reduced to\njust $\\Lambda$-derivative and $\\alpha$-terms.\nFrom these terms, the universal values of\n$\\beta_{1,2}$ can be extracted~\\cite{aprop,mgierg2,Thesis}.\n\nGiven the diagrammatic representation of the flow equation, \nthe diagrammatic calculus comprises the following elements:\n\\begin{enumerate}\n\t\\item\tAn operator which implements the flow {\\it i.e.}\\  $-\\Lambda \\partial_\\Lambda$;\n\n\t\\item\tA diagrammatic rule reflecting the charge conjugation\\ invariance\n\t\t\tof the theory;\n\n\t\\item\tThe primary diagrammatic identities which allow\n\t\t\tthe direct manipulation of diagrams possessing particular\n\t\t\telements;\n\n\t\\item\tThe secondary diagrammatic identities which encode\n\t\t\tthe equivalence of distinct diagrammatic representations\n\t\t\tof particular classes of sub-diagrams;\n\n\t\\item\tThe subtraction techniques.\n\\end{enumerate}\n\nThe primary diagrammatic identities further decompose into three sets. \nThose of the first type are defined\nwithout any reference to perturbation theory, and follow from\ngeneral constraints such as gauge invariance. The single primary\ndiagrammatic identity of the second type is the effective propagator\nrelation, which we recall follows from the classical flow equation,\ngiven a choice we impose on the seed action. Those of the third type\nfollow directly from those of the first and second types. However, it\nis convenient to give them in their own right, as they are not\nnecessarily obvious and are heavily used in this paper.\n\nThe secondary diagrammatic identities decomposes into two families,\none of which is applicable only to those diagrams which possess\nan $\\Op{2}$ stub and the other of which is more generally\napplicable.\n\nIt is apparent that, \nof all the elements of the diagrammatic calculus, \nonly one is reliant on perturbation\ntheory.  An obvious\nquestion, to which we return in the conclusion,\nis whether the calculus has a non-perturbative\nextension. We emphasise\nthat, irrespective of whether or not\nthis is the case, the framework admits \nstandard ERG analyses in the non-perturbative domain, \nbut with the benefits of manifest gauge invariance.\n\nTo derive an expression for arbitrary $\\beta_n$,\nwe could pursue the above strategy (see \\cite{Thesis} for an attempt\nin this direction). However, there is a much more\nefficient way to proceed, motivated by\nthe form of the $\\Lambda$-derivative terms\nat one and two loops. We start by constructing\na set of $n$-loop functions which depend only on\nWilsonian effective action vertices, effective\npropagators and (components of) gauge remainders.\nConsidering the $\\Op{2}$ parts of these diagrams\n(strictly, up to functions not polynomial  in $p$),\nwe compute their flow. Amongst the generated\nterms is $\\beta_n$ multiplied by a\ncoefficient\nwhich is universal as a consequence of specializing\nto $\\Op{2}$. Using the various diagrammatic identities,\nwe then demonstrate that all dependence on\nthe seed action and details of the covariantization\ncancels out between the remaining terms. \nBy taking the $\\Lambda$-derivative terms\nas the starting point for the calculation, we\navoid having to iteratively construct them, which\nreaps better and better dividends with each loop order.\n\n\nThis paper is organized as follows. In \\sec{sec:Review},\nwe review the aspects of the manifestly gauge\ninvariant ERG we require for this paper. Following\na brief exposition of $SU(N|N)$ gauge theory, the flow\nequation is introduced via its diagrammatic representation.\nSome properties of the various diagrammatic\nelements of the flow equation are discussed\nand the primary diagrammatic identities of the first type\nare stated. The section concludes with the examination\nof the\nform taken by the flow equation in the perturbative regime;\nwe describe the effective propagator relation\nand derive the primary diagrammatic identities of the third type.\n\n\n\\Sec{sec:Further}, the basis of which comes from~\\cite{Thesis}, \nis devoted to developing\nthe diagrammatic techniques to the level\nsufficient for computing $\\beta_n$. First,\nwe revisit the gauge remainders, discussing\nthe type of diagrams they can generate\nin the perturbative domain. Next, we\nstate and prove the secondary diagrammatic\nidentities and finally we describe\nthe subtraction techniques necessary for\nmanipulating the $\\Op{2}$ terms. This completes\nthe description of the calculus and concludes\nthe first part of the paper.\n\nThe second part of the paper is devoted to\na detailed illustration of the calculus.\n\\Sec{sec:Preliminary}\nbegins with a description of some further notation\nwhich facilitates the practical\napplication of the calculus, by allowing us to\nrepresent sets (with a potentially huge\nnumber) of $n$-loop diagrams in an extremely compact\nmanner. Using this notation,\nwe introduce a set of diagrammatic functions\nwhich will play a central role in the treatment\nof $\\beta_n$ to follow. Various important properties of\nthese functions are analysed, which allows an\nillustration of many of the diagrammatic techniques\nin their most natural setting.\nIn \\sec{sec:beta_n} we derive an expression for $\\beta_n$\nwhich is independent of the seed action and the\ndetails of the covariantization of the cutoff\nand we conclude in \\sec{sec:Conc}.\nThe primary and secondary diagrammatic identities are\ncollected together in \\app{app:D-ID} for easy reference.\n\n\n\n\n\n\n\\section{Preliminary Diagrammatics}\n\\label{sec:Preliminary}\n\n\\subsection{Additional Notation}\n\\label{sec:Notation}\n\n\\subsubsection{(Pseudo) Effective Propagators}\n\nIn \\sec{sec:GRs-2}, we encountered a single effective\npropagator amongst the implicit decorations \n(see {\\it e.g.}\\  diagram~\\ref{v-W-GR-w}). We now\ngeneralize to consider an arbitrary number of such\neffective propagators. The rule for explicit decoration\nis defined as follows. If we wish to join two objects\n(say two vertices) together with $j'$ out of a total\nof $j$ effective propagators, then there are\n$\\nCr{j}{j'} 2^{j'}$ ways to do this. \nIntuitively, the first factor captures the \nnotion that,\nso long as they are implicit decorations, the effective\npropagators are indistinguishable. The factor of\n$2^{j'}$ allows for the fact that we can interchange\nthe two ends of an effective propagator. If these\neffective propagators were instead used to form $j'$\nloops on a single vertex, then the factor of\n$2^{j'}$ would disappear, since the vertices\nare defined such that all cyclically independent\narrangement of their decorative fields are summed\nover.\n\n\nGiven that we have seen how\nfields and effective propagators can usefully appear\nas implicit decorations, it is not surprising that,\nin our calculation of $\\beta_n$, \ngauge remainder components and pseudo effective\npropagators also appear in this fashion.\nThe combinatorics for pseudo effective propagators\nis exactly the same as for effective\npropagators. When we come to decorate\nwith pseudo effective propagators, we can make life easy\nfor ourselves by noting that, in practise, pseudo\neffective propagators can be arranged to appear\nonly in one way and always in combination with another\nterm, precisely as in~\\eq{eq:Combo}.\nThe combinatoric factor\nassociated with partitioning $j$ $\\ensuremath{\\star}$s\ninto two sets of $j'$ and $j-j'$ is just\n$\\nCr{j}{j'}$. However, unlike effective\npropagators, the ends of $\\ensuremath{\\star}$ are not\ninterchangeable.\n\n\n\\subsubsection{Gauge Remainders}\n\\label{sec:Not:GRs}\n\nWe begin\nby giving the rules for converting a set of $m$\ngauge remainder components of type $>$ from implicit\nto explicit decorations. Thus, we are considering\ndecorations of the form\n\\begin{equation}\n\\label{eq:Implict-GR}\n\t\\dec{ \\\n\t}{>^m \\cdots},\n\\end{equation}\nwhere the square brackets could enclose some\ndiagrammatic structure, but need not. The ellipsis\nrepresents any additional implicit decorations,\nso long as they are not further instances of $>$.\nThe superscript notation $>^m$ simply tells\nus, as before, that there are $m$ instances of $>$.\n\nThere are only two structures we are allowed to form from\nimplicit decorations containing $>$. The first\nof these is a ring, as \nin~\\fig{fig:nestedhook}. Noting that, in some\ncompletely fleshed out diagram, the sockets of\nall the $>$ will be filled, this means that\nwe can generate a diagram like~\\ref{Diags-WGR-N1-Bottom-PF}\nfrom implicit decorations, but\nnever\na diagram like~\\ref{Diags-WGRx2-L-Bottom}.\nHowever, there is no restriction that we\ncreate a single ring structure from our $m$\ngauge remainders: we can create up to $m$\nindependent structures. Furthermore,\nwe need not promote all of the $>$\nto explicit decorations at the same time.\nThus, we can imagine\npartitioning the $m$ gauge remainders into\n$q+1$ sets, with the first $q$ of these sets forming\nan independent ring structure and the final\nset remaining as implicit decorations\n{\\it viz.}\\ \n\\begin{equation}\n\\label{eq:Implict-GR-b}\n\tM \\dec{\n\t\t\\decGR{ \\\n\t\t}{>^{m^1}} \n\t\t\\decGR{ \\\n\t\t}{>^{m^2}}\n\t\t\\cdots\n\t\t\\decGR{ \\\n\t\t}{>^{m^q}}\n\t}{>^{m^{q+1}} \\cdots},\n\\end{equation}\nwhere $M$ a combinatoric factor and the notation,\n$\\decGR{ \\ }{>^{m^1}}$, tells us that\nthese decorative gauge remainders must form a\nsingle ring {\\it i.e.}\\  they cannot be further partitioned,\nnor can this ring be added to by the remaining\nimplicit decorations.\nThe $m^i$ are positive integers which sum up to $m$.\n\nThe combinatoric factor\nis easy to compute: if we partition $m$ gauge\nremainders into two sets of $m'$ and $m-m'$\ngauge remainders, then the combinatoric factor\nis defined to be\n\\[\n\t\\nCr{m}{m'}.\n\\]\nTherefore, we\ncan rewrite~\\eq{eq:Implict-GR-b} as:\n\\[\n\t\\frac{m!}{m^{q+1}!} \n\t\\dec{\n\t\t\\prod_{i=1}^q \\frac{1}{m^i!}\n\t\t\\decGR{ \\ }{>^{m^i}}\n\t}{>^{m^{q+1}}\\cdots}.\n\\]\n\n\nThe final gauge remainder structure\nwe can construct out of implicit decorations\ncontaining $>$s occurs only in diagrams\npossessing an $\\Op{2}$ stub. The\ncontexts in which this structure occurs are\nillustrated in \\fig{fig:GR-Op2String}. \n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\left[\n\t\t\\LIDi{Op2-GR-String-Ex}{Op2-GR-String-Ex}\n\t\\right]_{p^2}, \\qquad\n\t\\LIDi{Op2-GR-String-Ex-B-Mac}{Op2-GR-String-Ex-B}\n\t\\]\n\\caption{A gauge remainder structure which only occurs in diagrams\npossessing an $\\Op{2}$ stub.}\n\\label{fig:GR-Op2String}\n\\end{figure}\\end{center}\n\nDiagrams~\\ref{Op2-GR-String-Ex} and~\\ref{Op2-GR-String-Ex-B}\nboth possess a string (not a ring) of $m'$ gauge remainders\n({\\it cf.}\\  \\fig{fig:GRstring}),\nthe last of which bites the socket on the classical,\ntwo-point vertex (in either sense).\nIn diagram~\\ref{Op2-GR-String-Ex}, the\nfirst of the $m'$ gauge remainders is \nbitten by a \\ensuremath{\\star}\\ whereas, in diagram~\\ref{Op2-GR-String-Ex-B}\nit is bitten by a $>$ attached to an effective\npropagator. The loose end of the \\ensuremath{\\star}\\ \/ effective propagator \ncan attach to any of the $>$s or to the\nsocket on the classical, two-point vertex or\nto any (un-drawn) structures (the \\ensuremath{\\star}\\ cannot\nattach to its own socket, though).\n\nAll diagrams possessing a string of gauge remainders\nbitten by a \\ensuremath{\\star}\\ are tagged $p^2$. This\ntells\nus that everything attaching to\nthe $\\Op{2}$ stub \nis independent of\n$p$ (as in \\sec{sec:Subtraction}). \nDiagrams tagged with $p^2$ can possess\nmore than one instance of the structure\nwhich attaches to the stub in \ndiagram~\\ref{Op2-GR-String-Ex}: although there\nis only one stub, structures involving a $\\ensuremath{\\star}$\ncan attach to the momentum derivative symbol\nof another such structure, as illustrated in\n\\fig{fig:GR-Op2String-b}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\left[\n\t\t\\cd{Op2-GR-String-Ex-b} \\hspace{1em}\n\t\\right]_{p^2}\n\t\\]\n\\caption{An example of a diagram possessing multiple $\\ensuremath{\\star}$s.}\n\\label{fig:GR-Op2String-b}\n\\end{figure}\\end{center}\n\nWe note the following. First,\nit is not legal to join any of the $m'$\ngauge remainders to any of the $m''$ gauge remainders.\nIn other words, no two $\\ensuremath{\\star}$s are permitted to \ncarry a common loop momentum. Secondly, the $\\ensuremath{\\star}$s are always\ntied up such that the momentum derivative symbols\nare on the `outside' of the diagram. In other words, we can draw\na line from any momentum derivative symbol \/ external field to any other\nwithout crossing an internal line. Diagrams in which this\nis not the case vanish by a combination of group theory\nand charge conjugation\\ invariance considerations~\\cite{Thesis,mgierg1}.\n\n\n\n\\subsubsection{Vertices}\n\nWhen deriving an expression for $\\beta_n$,\nwe will find it invaluable to compactly\nrepresent all vertices in a diagram\nwhich are yet to be explicitly decorated.\nTo this end, we introduce a set of vertex arguments, $v^j$,\nwhere the upper roman index acts as a label. Thus, the $v^j$\nare integers, denoting the loop orders of some set of vertices.\nWe denote the reduction of these vertices by $v^{j;R}$, where\nwe recall that a reduced vertex lacks a classical, two-point \ncomponent.\n\nNext, we introduce the compact notation\n\\begin{eqnarray*}\n\tv^{j,j_+} & \\equiv & v^j - v^{j+1},\n\\\\\n\tv^{j,j_+;R} & \\equiv & v^{j;R} - v^{j+1;R}.\n\\end{eqnarray*}\nWe use this notation to define\n\\begin{equation}\n\\label{eq:CompactVertices}\n\\SumVertex \\equiv \\prod_{i=0}^j \\sum_{v^{i_+} = 0}^{v^i} \\Vertex{v^{i,i_+;R}},\n\\end{equation}\nwhere the first argument of the structure on the left-hand side, $n_s$, gives\nthe value of $v^0$. Notice that all other vertex arguments are summed over.\nThe interpretation of the product symbol is as a generator\nof  $j+1$ vertices.\n\nThe structure shown in~\\eq{eq:CompactVertices} always appears as a part of\ndiagrams which\npossess an additional vertex, which carries the argument $v^{j_+}$ (this\nargument need not appear on its own---it could be part of something more complicated\n{\\it e.g.}\\  $v^{j_+,k}$). An example, which will play an important role\\ later,\nis\n\\begin{equation}\n\\label{eq:VertexTower}\n\t\\left[\n\t\t\\sco[2]{\\TopVertex}{\\SumVertex}\n\t\\right] \\equiv\n\t\\prod_{i=0}^j \\sum_{v^{i_+} = 0}^{v^i}\n\t\\left[\n\t\\sco[2]{\\TopVertex}{\\Vertex{v^{i,i_+;R}}}\n\t\\right].\n\\end{equation}\nNotice that the sum over all vertex arguments is trivially $n_s$:\n\\begin{equation}\n\\label{eq:VertexSum}\n\t\\sum_{i=0}^j v^{i,i_+} + v^{j_+} = \\sum_{i=0}^j \\left( v^i - v^{i+1} \\right) + v^{j+1} = v^0 = n_s.\n\\end{equation}\n\nThe interpretation that the structure defined by~\\eq{eq:CompactVertices}\npossesses $j+1$ vertices allows us to usefully define~\\eq{eq:VertexTower}\nfor $j=-1$:\n\\begin{equation}\n\\label{eq:Tower-1}\n\t\\left[\n\t\t\\sco[2]{\\TopVertex}{\\SumVertex}\n\t\\right]_{j=-1} \\equiv\n\t\\Vertex{n^R_s \\hspace{0.25em}},\n\\end{equation}\nwhere $n_s^R$ is, of course, just an $n-s$ loop, reduced vertex. (Note that\nthis example illustrates the rule that $v^{j_+}$ is replaced by $v^0$;\nthis holds irrespective of whether or not $v^{j_+}$ occurs only as part of some\nmore complicated vertex argument.) We can\neven usefully define what we mean by~\\eq{eq:VertexTower}\nfor $j=-2$:\n\\begin{equation}\n\\label{eq:Tower-2}\n\t\\left[\n\t\t\\sco[2]{\\TopVertex}{\\SumVertex}\n\t\\right]_{j=-2} \\equiv\n\t\\delta(n-s).\n\\end{equation}\nNotice that~\\eq{eq:Tower-1} still makes\nsense if one and only one of the vertices\nis decorated (in practise, we will\nalways take this to be the top vertex):\nif more than one vertex is decorated then this\nimplies that the number of vertices is at least two,\nwhich leads to a contradiction.\nOn the other hand, \\eq{eq:Tower-2} makes sense\nonly as is, and not if any of the vertices are\ndecorated. In the computation of $\\beta_n$,\nwe will find structures like~\\eq{eq:VertexTower},\nwhere we sum over $j$. The lower value of this\nsum will start out at $-2$. However, as we perform\nexplicit decorations of the vertices, so we will\nneed to raise the lower limit on $j$, such\nthat the diagrams still make sense.\n\n\n\n\n\n\\subsubsection{Decoration of Kernels {\\it etc.}\\ }\n\\label{sec:Not-K}\n\nIt will prove useful to isolate the decorated\ncomponents of a kernel from the undecorated\nparts.  To facilitate this separation, we introduce the \nsymbol $\\odot$ to indicate an undecorated kernel and $\\circ$ \nto indicate a decorated kernel,\nsuch that\n\\begin{equation}\n\\label{eq:Kernel-Notation}\n\t\\ensuremath{\\stackrel{\\bullet}{\\mbox{\\rule{1cm}{.2mm}}}} \\ = \\ \\ensuremath{\\stackrel{\\odot}{\\mbox{\\rule{1cm}{.2mm}}}} + \\ensuremath{\\stackrel{\\circ}{\\mbox{\\rule{1cm}{.2mm}}}}.\n\\end{equation}\nNoting that decoration is not defined for\neither $\\ensuremath{\\star}$s or pseudo effective propagators\n(or gauge remainder components)\nwe define\n\\begin{equation}\n\\label{eq:Combo-Decoration}\n\t\\cd{LdL-Combo} \\equiv \\cd{DEP-dGR} + \\cd{EP-DdGR} -\\frac{1}{2} \\cd{DdPEP-GR},\n\\end{equation}\nwhere we have used~\\eqs{eq:LdL-GRk-Pert}{eq:Combo}.\n\n\n\n\\subsection{Diagrammatic Functions}\n\\label{sec:Prelim-DiagFns}\n\n\nIn this section we introduce a set of\ndiagrammatic functions,\n$\\nLV{{\\cal D}}{n}{\\mu\\nu}{i}(p)$, which will\nprove crucial in our treatment of $\\beta_n$\n(recall that the $1$s are shorthand for $A^1$s).\nBy analysing their various\nproperties, we will be able to illustrate\nthe use of many of the diagrammatic techniques,\nin their natural setting. We will ultimately\nbe interested in the $\\Op{2}$ parts of these\ndiagrams (up to functions not polynomial in $p$).\n\n\n\\subsubsection{The Function $\\protect \\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$}\n\nWe introduce the function\n\\begin{equation}\n\\label{eq:nL-a}\n\t\\nLV{{\\cal D}}{n}{\\mu \\nu}{a}(p) \\equiv\n\t2 \\sum_{s=1}^n  \\sum_{m=0}^{2s} \\sum_{j=-2}^{n+s-m-2} \n\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\n\t\\dec{\n\t\t\\sco[1]{\\TopVertex}{\\SumVertex}\n\t}{11\\Delta^{j+s+1}>^m},\n\\end{equation}\nwhere, for the non-negative integers $a$ and $b$,\nwe define\n\\begin{equation}\n\\label{eq:norm}\n\t\\ensuremath{\\Upsilon}_{a,b} = \\frac{(-1)^{b+1}}{a!b!} \\left(\\frac{1}{2}\\right)^{a+1},\n\\end{equation}\nwhich we see obeys the following relationships:\n\\numparts\n\\begin{eqnarray}\n\\label{eq:norm-b}\n\t2a \\ensuremath{\\Upsilon}_{a,b} \t\t& = & \\ensuremath{\\Upsilon}_{a-1,b},\n\\\\\n\\label{eq:norm-c}\n\tb\\ensuremath{\\Upsilon}_{a,b} \t\t& = & -\\ensuremath{\\Upsilon}_{a,b-1}.\n\\end{eqnarray}\n\\endnumparts\nIn the case that either $a$ or $b$ are negative, $\\ensuremath{\\Upsilon}_{a,b}$\nis null. The overall combinatoric factor, $\\ensuremath{\\Upsilon}_{j+s+1,j+2}\/m!$,\nis said to be canonical, since the three arguments, $j+s+1$, $j+2$\nand $m$ match, respectively, the number of effective propagators,\nthe number of vertices and the number of $>$s.\n\nThere is a simple, intuitive explanation for the relationship\nbetween the number of vertices, the number of effective propagators\nand the sum over the vertex arguments.\nConsider the components of $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$\nfor which there is at least one vertex {\\it i.e.}\\  for which $j>-2$ and for which\n$m=0$. We\nknow from~\\eq{eq:VertexSum} that the sum of the vertex arguments is $n-s$.\nNow, given\n$j+2$ vertices, exactly $j+1$ effective propagators\nare required to create a connected diagram. This leaves\nover $s$ effective propagators, each of which must create\na loop. Therefore, the loop order of the diagram is $n-s+s = n$,\nas must be the case.\nNow we relax the restriction that $m=0$ and suppose that\nwe create $m'$ gauge remainder rings ({\\it cf.}\\  \\fig{fig:nestedhook}). \nTo ensure that each of these rings is connected to\nsome other part of the diagram, we must use up \n $m'$ of our\n$s$ `spare' effective propagators. However, this is\nprecisely counteracted by the contribution of\neach of the rings to the loop order of the diagram.\nNext, consider the component of the diagram\nfor which $j=-2$:\n\\begin{equation}\n\\label{eq:Dn-j=-2}\n\t2 \\sum_{m=0}^{2n} \\frac{\\ensuremath{\\Upsilon}_{n-1,0}}{m!} \n\t\t\\dec{\n\t\t\t\\hspace{1em}\n\t\t}{11\\Delta^{n-1} >^{m}}.\n\\end{equation}\nTo ensure that the diagram is fully connected, $m$ must\ntake the value $2n$ since,\nfor any smaller value, there are not enough $>$s\nfor the $2(n-1)+2$ fields which require a socket in\nwhich to reside. The loop order of all diagrams\ngenerated by~\\eq{eq:Dn-j=-2} is clearly $n$.\n\n\nHaving discussed how the various diagrammatic elements \nconspire to yield a diagram of loop order $n$, we now\ndiscuss the ranges on the various sums. The maximum\nvalue of $s$ clearly follows from the requirement\nthat the loop order of the diagram is $\\geq 0$. \nThe maximum values of the sums over $m$ and $j$\nand the minimum value of $s$\nfollow from\nthe constraint that all fully fleshed out diagrams\nmust be connected and must have an $\\Op{2}$ contribution. \nSuppose that there were a term contributing to $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$\nwith $s=0$.\nIn this case, we would no longer have any\ndecorative gauge remainder components, being left\nwith just $j+2$ reduced, Wilsonian effective\naction vertices, $j+1$ effective propagators\nand two external fields. Recalling that effective\npropagators are two-ended objects, we see that\nthere are $2j+4$ available decorations.\nNow, from \n\\sec{sec:elements} we know that it is imposed as a constraint that all\none-point, Wilsonian effective action vertices vanish\n(though one-point, \\emph{seed} action vertices do exist, beyond tree level),\nin order that the vacuum expectation value of the\nsuperscalar which breaks the $SU(N|N)$ symmetry\nis not shifted by quantum corrections.\nWe can thus insist that all \nWilsonian effective action vertices are at \nleast two-point. This requires $2(j+2)$\ndecorations, which is precisely equal to\nthe number of available decorations\nfor the diagram\nas a whole to be connected. Consequently, \neach vertex\nmust be precisely two-point. Immediately,\nthis precludes the existence of any classical\nvertices, since reduced vertices\ndo not possess such a component.\nThus we have a line of vertices with a loop\norder of at least one apiece. At each end\nof the line is a vertex decorated by one external\nfield and by one end of an effective propagator;\nall other vertices are joined to two other vertices,\nin each case by a single effective propagator.\nNow, since none of these vertices is classical,\nthey must each, by~\\eq{eq:S_>0-11}, contribute $\\Op{4}$.\nAlthough the effective propagators each contain\na $1\/p^2$ component this is insufficient to prevent\nthe diagram as a whole vanishing at $\\Op{2}$.\n\nIn similar fashion, we can understand the\nmaximum values of the sums over $m$ and $j$.\nOf the $j+2$ vertices, suppose\nthat $T$ are classical vertices. Since the\nvertices are reduced,\na minimum of $3T$ decorations are required for\nthe classical vertices and so, if there are\n$m$ gauge remainders, a minimum of  \n\\begin{equation}\n\\label{eq:required}\n\t2(j+2-T) + 3T + m = 2j+T+m+4\n\\end{equation}\ndecorations are required. \nNoting that the \nnumber of\navailable decorations is\n$2(j+s+1)+2$, it is clear\nthat \n\\begin{equation}\n\\label{eq:T+m}\n\tT+m \\leq 2s.\n\\end{equation}\nIt therefore follows that\n\\begin{equation}\n\\label{eq:Maxm}\n\tm \\leq 2s.\n\\end{equation}\n\nNext, let us deduce the maximum number of vertices {\\it i.e.}\\ \nthe maximum number taken by $j+2$ for some values of\n$s$ and $m$. From~\\eq{eq:VertexSum},\nwe know that\nthe sum over vertex arguments is $n-s$.\nTherefore, we can have at most $n-s$ vertices which\nare not tree level and so a total of at most $n-s+T$\nvertices. Hence,\n\\[\n\tj+2 \\leq n+s-m,\n\\]\nwhere we have used~\\eq{eq:T+m}.\nBefore moving on, let us return to the conclusion~\\eq{eq:Maxm}.\nIn fact, there may be a tighter constraint than this\nsince, if there are more than $n-s$ vertices,\nthen $T$ is compelled to be greater than zero.\nHowever, we stick with the limit given by~\\eq{eq:Maxm},\ndiscarding any diagrams which\nturn out to be disconnected and adopting\nthe prescription that terms for which the\nmaximum value of the sum over $j$ is less\nthan its minimum value do not exist.\n\nThis completes our discussion of the form of \n$\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$ and we will now\ndiscuss some of its properties. However,\nrather than working with $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$\ndirectly, we introduce the auxiliary function\n\\begin{equation}\n\\label{eq:nLaux-a}\n\t\\nLV{\\mathcal{E}}{n}{\\mu \\nu}{a}(p) \\equiv\n\t2 \\sum_{s=0}^n  \\sum_{m=0}^{2s+1} \\sum_{j=-2}^{n+s-m-1} \n\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\n\t\\dec{\n\t\t\\sco[1]{\\TopVertex}{\\SumVertex}\n\t}{11\\Delta^{j+s+1}>^m}.\n\\end{equation}\nComparing $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ with $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$,\nwe notice \nthat the ranges on the sums over $s, \\ m$ and $j$\ndiffer between the two expressions.\nThus, we know from our previous arguments that\nthere are diagrams contained in $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ \nwhich vanish\nat $\\Op{2}$ and \/ or contain one-point, Wilsonian effective\naction vertices. The point is that, in our computation of\n$\\beta_n$, we will consider $\\dec{\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$.\nTo process this expression, we will use the weak coupling\nflow equations. However, when we do so, there are\ngenerically  elements\nof the resulting set of terms which,\nthough not generated by $\\dec{\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)}{\\bullet}$,\ndo not \\emph{individually}\nvanish.  We will find that some of these terms cancel against\nterms generated by the flow of other diagrams and so\nwe can view $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ as being \nthe same as $\\nLV{{\\cal D}}{n}{\\mu\\nu}{a}(p)$, at $\\Op{2}$,\nup to a useful re-expression of zero.\n\nWe now demonstrate that \n\\begin{equation}\n\\label{eq:Transverse+TE}\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) \\sim \\Box_{\\mu \\nu}(p) + \\Op{4}.\n\\end{equation}\nTo do this, we first show that $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$\nis transverse in $p$. This is done by contracting\n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$\nwith the momenta\n$p_\\mu$ and $(-p)_\\nu$ of the external \nfields.\\footnote{\nTo prove the transversality of  $\\nLDl{a}(p)$ \nit is necessary only to contract\n$\\nLDl{a}(p)$ with one of the external momenta,\nbut it technically easier to do it with both.}\nSince we are not computing the flow of any of\nthe vertices, we can discard all one-point\nvertices, and so decrease the upper limits\non the sums over $m$ and $j$ by one each.\nExplicitly decorating with one of\nthe external fields but leaving the other\nas an implicit decoration \nyields the diagrams of \\fig{fig:bn-Transverse}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray*}\n\t\tp_\\mu (-p)_\\nu \\nLDl{a}(p) & = &\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m=0}^{2s} \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LIDi{vj+R-EGR}{vj+R-EGR}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+1}>^m}\n\t\t}\n\t\\\\[2ex]\n\t\t& & +2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m=2}^{2s} \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^m\n\t\t}\n\t\t\t{\n\t\t\t \\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LIDi{GRs-EGR}{GRs-EGR}\n\t\t\t\t}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+2}>^{m-m'}}\n\t\t}\n\t\\end{eqnarray*}\n\\caption{Contraction of $\\nLDl{a}(p)$ with its external\nmomenta.}\n\\label{fig:bn-Transverse}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{vj+R-EGR} comes with a factor of $j-2$,\ncompared to the parent (which have have absorbed into\n$\\ensuremath{\\Upsilon}$, using~\\eq{eq:norm-c}), recognizing\nthe indistinguishability of the $j+2$ vertices,\nprior to explicit decoration. \nFurthermore, the lower limit on the sum over $j$ is\n$-1$ and not $-2$, since diagram~\\ref{vj+R-EGR}\nmust possess at least one vertex.\nThe combinatorics\nfor diagram~\\ref{GRs-EGR} follows from~\\sec{sec:Not:GRs};\nwe have recognized that for the diagram as a whole to\nbe connected (or, alternatively, on account of diagrammatic\nidentity~\\eq{D-ID-Trivial}), the explicit gauge remainder\nstructure must possess a minimum of two $>$s.\n\nThe reduced vertex of \ndiagram~\\ref{vj+R-EGR} \nwhich is struck by the gauge remainder\ncan\nbe promoted to a full vertex, since a classical,\ntwo-point vertices contracted into a $\\rhd$\ndies, courtesy of diagrammatic \nidentity~\\eq{eq:GR-TLTP}.\nAllowing the gauge remainder\nto act---whereupon it necessarily strikes \na socket---we split the resulting vertex\ninto a reduced part and a classical, two-point\npart. In the latter case, we shift $j \\rightarrow j+1$.\nSince no other gauge remainders have acted (and no Taylor\nexpansions have been performed), we could use\ncharge conjugation\\ invariance to collect together\npushes forward and pulls back. However, we refrain\nfrom doing this, for reasons that will become\napparent.\nThe result of processing diagram~\\ref{vj+R-EGR} \nis shown in \\fig{fig:bn-Transverse-b} where,\nfor each term,\nthe explicitly drawn external field can bite the\nsocket in either sense.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m=0}^{2s-1} \\sum_{j=-1}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\LIDi{vj+R-EGR-Socket}{vj+R-EGR-Socket}\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+1}>^m}\n\t\t}\n\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m=0}^{2s-1} \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LIDi{CTP-EGR-Socket}{CTP-EGR-Socket}\n\t\t\t\t}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+2}>^m}\n\t\t}\n\t\\]\n\\caption{The result of processing diagram~\\ref{vj+R-EGR}.}\n\\label{fig:bn-Transverse-b}\n\\end{figure}\\end{center}\n\nNotice that the limits on the sums over $m$ and \n$j$ differ from those of the parent. For example, \nin diagram~\\ref{vj+R-EGR-Socket}, where the gauge\nremainder strikes a socket on a reduced vertex,\nwe have `used up' a decoration and so can \ndecrease the limits on the sums over\n$m$ and $j$ by one each.\nIn diagram~\\ref{CTP-EGR-Socket}\nsimilar consideration apply, but here \nnotice that the lower limit on the sum over $j$ is now\n$-2$, as a consequence of having\nshifted $j \\rightarrow j+1$.\n\nThe next step is to decorate the classical,\ntwo-point vertex of diagram~\\ref{CTP-EGR-Socket},\njust as we did with diagram~\\ref{v-W-GR-w-b02}\n(see \\figs{fig:Decompose}{fig:EP-Ex}). If we\nattach the remaining \\ensuremath{\\cdeps{ExtGR}}, the diagram\ndies, courtesy of diagrammatic identity~\\eq{eq:GR-TLTP}.\nThus, all we  can do is\nattach an effective propagator,\nthe loose end of which joins\neither to a reduced Wilsonian effective\naction vertex or to a gauge remainder\nstructure (recall that if the both\nends of the effective propagator attach\nto the classical, two-point vertex then the\ndiagram dies as a consequence of charge conjugation\\ invariance). \nIn each case, the combinatoric\nfactor associated with choosing one effective\npropagator, either end of which can attach to\nthe classical, two-point vertex, is\n$2(j+s+2)$. In the case that the other end attaches\nto one of the $j+2$ reduced vertex, there is an additional\nfactor of $j+2$.\n\n\\begin{Icancel}\n\\label{Icancel:Tansverse}\n\tThe component of diagram~\\ref{CTP-EGR-Socket}\n\tin which the classical, two-point vertex\n\tis attached to a reduced, Wilsonian effective\n\taction vertex cancels diagram~\\ref{vj+R-EGR-Socket},\n\tup to a gauge remainder contribution.\n\\end{Icancel}\n\nCrucially, the surviving gauge remainder contribution\nis a nested version of diagram~\\ref{vj+R-EGR}.\nThus, processing the nested gauge remainder\njust repeats the above cancellations,\ngenerating successively more nested diagrams.\nThe process terminates when there are insufficient\ndecorations for the gauge remainder to strike\na reduced vertex. What, then, remains? At each level of\nnesting (including the un-nested case, as discussed\nalready), we can\nattach the classical, two-point vertex to a\ngauge remainder structure. Additionally,\nfrom the first level of nesting onwards,\nwe can attach the classical, two-point vertex\nto one of the gauge remainders which is nested\nwith respect to\\ the gauge remainder which bites the vertex. The\ncomplete set of surviving diagrams is collected\ntogether in \\fig{fig:bn-Transverse-c}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m'=1}^{2s-1} \\sum_{m=0}^{2s-1-m'} \\sum_{j=-1}^{n+s-m-m'-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LIDi{EGR-GRs-EP-TLTP}{EGR-GRs-EP-TLTP}\n\t\t\t\t}\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+1}>^m}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=0}^n  \\sum_{m''=0}^{2s-3}  \\sum_{m=2}^{2s-1-m''} \\sum_{j=-1}^{n+s-m-m''-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!m''!} \\sum_{m'=2}^{m}\n\t\t}\n\t\t\t{\n\t\t\t\\!\\! \\nCr{m}{m'} \\!\\!\n\t\t\t\\dec{\n\t\t\t\t\\Tower{\\LIDi{EGR-Skt-CTP-EP-GRs}{EGR-Skt-CTP-EP-GRs}}\t\n\t\t\t}{\\ensuremath{\\cdeps{ExtGR}} \\!\\! \\Delta^{j+s+1}>^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{All surviving terms generated by the diagrams\nof \\fig{fig:bn-Transverse-b}.}\n\\label{fig:bn-Transverse-c}\n\\end{figure}\\end{center}\n\nNotice that, in diagram~\\ref{EGR-Skt-CTP-EP-GRs},\nthe sum over $m'$ starts from two, as the $m'=1$\ncase vanishes on account of diagrammatic \nidentity~\\eq{eq:D-ID-Bitten-hook}.\nThere is a very important point to make about\ndiagrams~\\ref{EGR-GRs-EP-TLTP} \nand~\\ref{EGR-Skt-CTP-EP-GRs}. Recall that, in\n\\sec{sec:Not:GRs}, we stated strings of gauge remainders\nformed by implicit decorations can only occur\nin diagrams with $\\Op{2}$ stubs, yet here we\nappear to have a counter-example. \nHowever, the strings of gauge remainders\nin diagrams~\\ref{EGR-GRs-EP-TLTP} \nand~\\ref{EGR-Skt-CTP-EP-GRs} are\nformed by the action of nested gauge remainders\nand \\emph{not} by the original set of implicit decorations.\nWith these structures formed, it is useful\nto sum over nestings, in the usual way,\nusing the notation $\\decGR{ \\ }{>^{q}}$, as indicated\n(remembering to compensate with a\ncombinatoric factor of $1\/q!$).\nHowever, we can not further promote these gauge\nremainders to join the implicit decorations of the diagram\nas a whole.\n\n\nWe now find a wonderful cancellation.\n\n\\begin{Icancel}\n\tDiagrams~\\ref{GRs-EGR}, \\ref{EGR-GRs-EP-TLTP}\n\tand~\\ref{EGR-Skt-CTP-EP-GRs} cancel,\n\tcourtesy of diagrammatic identity~\\eq{eq:D-ID-G}.\n\tThis is seen most clearly my making the following\n\tchanges of variables:\n\t\\begin{description}\n\t\t\\item[Diagram~\\ref{GRs-EGR}] \n\t\t\t$m \\rightarrow m+1$, $m' \\rightarrow m'+1$;\n\t\n\t\t\\item[Diagram~\\ref{EGR-GRs-EP-TLTP}] \n\t\t\t$m \\rightarrow m-m'$,\n\t\t\tand recognising that\n\t\t\t\\[\n\t\t\t\\sum_{m'=1}^{2s-1} \\sum_{m=m'}^{2s-1} = \\sum_{m=1}^{2s-1} \\sum_{m'=1}^m;\n\t\t\t\\]\n\n\t\t\\item[Diagram~\\ref{EGR-Skt-CTP-EP-GRs}]\n\t\t\t$m \\rightarrow m-m''$, $m' \\rightarrow m'-m''$\n\t\t\tand recognising that\n\t\t\t\\[\n\t\t\t\\sum_{m''=0}^{2s-3} \\sum_{m=m''+2}^{2s-1} \\sum_{m'=2+m''}^m =\n\t\t\t\\sum_{m=2}^{2s-1} \\sum_{m'=2}^m \\sum_{m''=0}^{m'-2}.\n\t\t\t\\]\t\n\t\\end{description}\n\\end{Icancel}\n\nWe have thus demonstrated that\n\\begin{equation}\n\\label{eq:Transverse}\n\tp_\\mu (-p)_\\nu \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) = 0,\n\\end{equation}\nas we set out to. The patterns of cancellations\ninvolved are the template for all cancellations\nwe will encounter when we deal with gauge\nremainders henceforth.\n\nAt first sight, \\eq{eq:Transverse+TE} follows\ndirectly from~\\eq{eq:Transverse}.\nHowever, consider the components of $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$\nwhich are factorizable. The worry is that such diagrams\nnecessarily contain at least one\neffective propagator carrying just\nthe external momentum, which contributes\na factor of $1\/p^2 + \\Op{0}$. The resolution to this\nproblem comes from breaking all such contributions\ndown into non-factorizable sub-diagrams and\nrecognizing that these sub-diagrams are\ntransverse, thereby providing\nthe necessary powers of $p$ to cancel\nthose coming from the effective propagators. \n\n\nTo\nsee this explicitly, we consider the component of\n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$ which possesses two non-factorizable\nsub-diagrams. To this end, we split up \n\\begin{enumerate}\n\t\\item\tthe two external fields;\n\n\t\\item\tthe $j+s+1$ effective\n\t\t\tpropagators into three sets of $j+s-s'$, $s'$ and 1\n\t\t\teffective propagator(s) with the single effective\n\t\t\tpropagator required to join together the two factorizable\n\t\t\tsub-diagrams;\n\n\t\\item\tthe $j+2$ vertices into two sets of $j-j'$\n\t\t\tand $j'+2$ vertices; \n\n\t\\item\tthe $m$ gauge remainders\n\t\t\tinto two sets of $m-m'$ and $m'$ gauge remainders;\n\n\t\n\\end{enumerate}\n\nNoting that we sum over $j'$, $m'$ and $s'$, the combinatoric\nfactor is\n\\begin{equation}\n\\label{eq:Factorize-combins}\n\t2 (j+s+1) \\nCr{j+s}{j+s-s'} \\nCr{j+2}{j'+2} \\nCr{m}{m'}.\n\\end{equation}\nIncluding this\nwith~\\eq{eq:Factorize-combins} and combining the\nresult\nwith $\\ensuremath{\\Upsilon}_{j+s+1,j+2} \/ m!$ yields\n\\[\n\t-2 \\frac{\\ensuremath{\\Upsilon}_{j+s-s', j-j'}}{(m-m')!} \\frac{\\ensuremath{\\Upsilon}_{s',j'+2}}{m'!}.\n\\]\nIt is now convenient to shift $j \\rightarrow j+j'+2$,\n$s' \\rightarrow s' + j' +1$ and $m \\rightarrow m+m'$,\nwhich yields the contribution shown in \\fig{fig:Factorize}.\nIt is understood that the repeated vertex argument $v^{j+2}$\nis summed over.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\\begin{array}{c}\n\t\\vspace{2ex}\n\t\t\\displaystyle\n\t\t-4 \t\\sum_{s=0}^n  \\sum_{m=0}^{2s+1} \\sum_{m'=0}^{2s+1-m} \\sum_{j=-2}^{n+s-m-m'-1} \n\t\t\t\\sum_{j'=-2}^{n+s-m-m'-j-3} \\sum_{s'=-j'-1}^{j+s+1} \n\t\\\\\n\t\t\\left[\n\t\t\t\\begin{array}{cc}\n\t\t\t\\vspace{1ex}\n\t\t\t\t\\displaystyle \\frac{\\ensuremath{\\Upsilon}_{j'+s'+1,j'+2}}{m'!} \t& \n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\cd{vj+jpr+3-R}\n\t\t\t\t\t}{\\Vertex{v^{j+2},j'}}\n\t\t\t\t}{1 \\Delta^{j'+s'+1} >^{m'}}\n\t\t\t\\\\\n\t\t\t\\vspace{1ex}\n\t\t\t\t& \\hspace{-4.5em} \\mbox{\\rule{.2mm}{1cm}}\n\t\t\t\\\\\n\t\t\t\t\\displaystyle \\frac{\\ensuremath{\\Upsilon}_{j+s+1-s',j+2}}{m!} &\n\t\t\t\t\\dec{\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\cd{vj_+j+2-R}\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{1 \\Delta^{j+s+1-s'} >^{m}}\n\t\t\t\\end{array}\n\t\t\\right]\n\t\\end{array}\n\t\\]\n\\caption{A factorizable contribution to $\\nLDl{a}(p)$. The two sub-diagrams are understood to be non-factorizable.}\n\\label{fig:Factorize}\n\\end{figure}\\end{center}\n\nThe crucial point to recognize is that the\ncombinatoric factor for each of the sub-diagrams\nis the canonical one. Immediately we see\nthat each of these sub-diagrams is transverse.\nIt is trivial to generalize this analysis to the case\nwhere there are an arbitrary number of non-factorizable\nsub-diagrams, from which~\\eq{eq:Transverse+TE} follows.\nIn view of this, it is hardly surprising\nthat we can split  $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p)$\ninto Non-Factorizable (NF) components\nin an enlightening way.\nIndeed, shifting $s \\rightarrow s+s'$\nand $v^{i\\leq j+2} \\rightarrow v^i - s'$ it is apparent that\n\\begin{equation}\n\\label{nLDa-factorize}\n\t\\nLV{{\\mathcal{E}}}{n}{\\mu\\nu}{a}(p) \n\n\t=\n\t\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{a}(p)\n\t\t\t- \\sum_{n'=1}^{n-1}\n\t\t\t\\nLV{\\overline{\\mathcal{E}}}{n-n'}{\\mu \\rho}{a}(p) \n\t\t\t\\Delta^{1\\,1}_{\\rho \\sigma}(p)\n\t\t\t\\nLV{\\overline{\\mathcal{E}}}{n'}{\\sigma \\nu}{a}(p)\n\t+\\ldots,\n\\end{equation}\nwhere \n$\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{a}(p)\n\\equiv \\left. \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{a}(p) \\right|_{\\mathrm{NF}}$,\nand the ellipsis builds up a full geometric series.\n\n\n\\subsubsection{The Function $\\protect \\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)$}\n\nWe define the function $\\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)$\nand two auxiliary functions as shown in \\fig{fig:nL-b}.\nNotice that the range for $s$ is the same for all three\nfunctions; only the upper limits on $m$ and $j$ differ.\n\\begin{center}\\begin{figure}\n\\begin{eqnarray*}\n\t\\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p) \\equiv\n\t\\bca{\n\t\t4 \\sum_{s=1}^n  \\sum_{r=1}^s \\sum_{m=0}^{2(s-r)} \n\t\t\\sum_{j=-2}^{n+s-2r-m-2} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!r!}\n\t}\n\t\t{\n\t\t\\decp{\n\t\t\t\t\\Tower{\\cd{CTP-E}}\n\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^m }{}\n\t}\n\\\\[2ex]\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p) \\equiv\n\t\\bca{ \n\t\t4 \\sum_{s=1}^n  \\sum_{r=1}^s \\sum_{m=0}^{2(s-r)+1} \n\t\t\\sum_{j=-2}^{n+s-2r-m-1} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!r!}\n\t}\n\t\t{\n\t\t\\qquad \\times\n\t\t\\decp{\n\t\t\t\t\\Tower{\\cd{CTP-E}}\n\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^m }{}\t\t\n\t}\n\\\\[2ex]\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b'}(p) \\equiv\n\t\\bca{ \n\t\t4 \\sum_{s=1}^n  \\sum_{r=1}^s \\sum_{m=0}^{2(s-r)+1} \n\t\t\\sum_{j=-2}^{n+s-2r-m-1} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!r!}\n\t}\n\t\t{\n\t\t\\dec{\n\t\t\t\t\\Tower{\\decp{\\cd{CTP-E}}{\\ensuremath{\\star}^r}}\n\t\t\t}{1\\Delta^{j+s+2-r}  >^m }\n\t}\n\\end{eqnarray*}\n\\caption{The definition of $\\nLV{{\\cal D}}{n}{\\mu\\nu}{b}(p)$\nand two auxiliary functions.}\n\\label{fig:nL-b}\n\\end{figure}\\end{center}\n\nIn the case of $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b'}(p)$,\nwe must be careful to define precisely\nwhat it is we mean by $\\decp{\\ }{}$, in this context.\nThe basic notion is that for all diagrammatic \ncomponents enclosed by $\\decp{\\ }{}$,\nall $p$ dependence comes from the $\\Op{2}$ stub, alone;\nthose components not explicitly enclosed\nby $\\decp{\\ }{}$ can either be brought under\nits influence or not, as we begin to flesh\nout the diagrams. Thus, for the\nnon-factorizable components of $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b'}(p)$,\nall diagrammatic elements can be brought under\nthe influence of $\\decp{\\ }{}$, whereas this is\nnot the case for\nthe factorizable components. Notice, though, that all $\\ensuremath{\\star}$s\nmust decorate the part of the diagram under the influence of\n$\\decp{\\ }{}$.\nWith these points in mind, we can\nwrite\n\\begin{equation}\n\\label{eq:b'-decomp}\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b'}(p) = \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)\n\t- \\sum_{n'=1}^{n-1} \\nLV{\\mathcal{E}}{n-n'}{\\mu\\rho}{b}(p)\n\t\\Delta^{11}_{\\rho \\sigma}(p) \\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p).\n\\end{equation}\n\n\n\n\\subsubsection{The Function $\\protect \\nLV{{\\cal D}}{n}{\\mu\\nu}{c}(p)$}\n\nWe define the function $\\nLV{{\\cal D}}{n}{\\mu\\nu}{c}(p)$,\ntogether with an auxiliary function, as follows:\n\\begin{eqnarray}\n\\label{eq:nL-c}\n\t\\nLV{{\\cal D}}{n}{\\mu\\nu}{c}(p) \\equiv\n\t\\sum_{s=1}^n \\sum_{m=1}^{2s} \\sum_{j=-2}^{n+s-m-2} \n\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\\dec{\n\t\t\\Tower{\\cd{CTP-E}}\n\t}{1 \\Delta^{j+s+2}>^m}\n\\\\\n\\label{eq:nLaux-c}\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c}(p) \\equiv\n\t\\sum_{s=1}^n \\sum_{m=1}^{2s+1} \\sum_{j=-2}^{n+s-m-1} \n\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\\dec{\n\t\t\\Tower{\\cd{CTP-E}}\n\t}{1 \\Delta^{j+s+2}>^m}.\n\\end{eqnarray}\n\nAs with $\\mathcal{E}^a$, it is useful to isolate the non-factorizable components\nof $\\mathcal{E}^c$:\n\\begin{equation}\n\\label{nLDc-factorize}\n\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c}(p) = \\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)\n\t- \\sum_{n'=1}^{n-1}\n\t\\nLV{\\overline{\\mathcal{E}}}{n-n'}{\\mu\\rho}{c}(p)\n\t\\Delta^{1\\,1}_{\\rho \\sigma}(p)\n\t\\nLV{\\mathcal{E}}{n'}{\\sigma\\nu}{a}(p).\n\\end{equation}\n\nWe now demonstrate that\n\\begin{equation}\n\\label{eq:nLV-c-Op2}\n\t2 \\decp{\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)}{} = \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p) + \\mathrm{remainders}.\n\\end{equation}\n\nThe presence of the $\\decp{\\ }{}$ allows us to Taylor\nexpand all diagrammatic components of $\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)$, besides\nthe $\\Op{2}$ stub, to $\\Op{0}$. The result\nof this procedure is shown in \\fig{fig:nLV-c-Properties-a}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIDi{dvj+R}{nLV-c-dvj+R}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+2} >^m}\n\t\t}\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+3,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIDi{dCTP}{nLV-c-dCTP}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1 \\Delta^{j+s+3} >^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=2}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=2}^m \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIO{\\TopVertex}{nLV-c-CTP-GRs-TE}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E-TEGRs}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+2} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=4}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=3}^{m-1} \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIO{\n\t\t\t\t\t\t\\decGR{\\ }{\\ensuremath{\\sqcup} >^{m'}}\n\t\t\t\t\t}{nLV-c-TE-GRs}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\TopVertex\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+2} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The diagrams contributing to $2 \\protect \\decp{\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)}{}$.}\n\\label{fig:nLV-c-Properties-a}\n\\end{figure}\\end{center}\n\nThere are a number of comments to make about the\ndiagrams of \\fig{fig:nLV-c-Properties-a}. First,\neach diagram\nhas a socket that we demand is filled by the\nexternal field (we will see shortly why\nwe do not fill the socket).  \nIn diagram~\\ref{nLV-c-CTP-GRs-TE} \nit is understood that, given that the final\ngauge remainder in the string is the one\nthat bites the vertex,\nthe first gauge remainder in the string\ncannot be filled by the external field,\nas this generates an illegal diagram\n(see \\sec{sec:Not:GRs}).\n\nSecondly, the factor of two\nthat diagrams~\\ref{nLV-c-dvj+R}\nand~\\ref{nLV-c-dCTP}\nhave acquired relative to the parent diagram\ncomes from using charge conjugation\\ to combine the pull back like\nmomentum derivative with the push forward like derivative\nmomentum.\nThere is no such factor of two in  diagrams~\\ref{nLV-c-CTP-GRs-TE}\nand~\\ref{nLV-c-TE-GRs} since are no momentum\nderivatives.\n\nThirdly, the fact that the diagrams are under the influence\nof $\\decp{\\ }{}$ leads to some novelties. Notice\nthat in diagram~\\ref{nLV-c-TE-GRs},\nthe explicitly drawn gauge remainder structure\nis required to\nhave at least three gauge remainders. If there\nwere only a single gauge remainder, whose socket\nis necessarily filled by an  $A^1$ sector field, then\nthe diagram would vanish as a consequence of charge conjugation\\ \ninvariance---nothing\nunusual there. But why is it not possible for there\nto be only two gauge remainders? The reason\nis that such a structure which, crucially, is\ndecorated by a $\\ensuremath{\\sqcup}$, would be compelled\nto be attached to the rest of the diagram by an \neffective propagator carrying just $p$. \nSuch a term\nis forbidden by\nthe influence of $\\decp{\\ }{}$.\n\nTo make progress, we attach two effective propagators\nto the classical, two-point vertex of diagram~\\ref{nLV-c-dCTP}.\nAs a consequence of the  $\\decp{\\ }{}$, these\ntwo effective propagators must carry the same\n(loop) momentum and so we can utilise~\\eq{eq:EP-dCTP-EP}.\nProcessing\nthe gauge remainders, we arrive at the diagrams\nof \\fig{fig:nLV-c-Properties-b}, where we have \nused charge conjugation\\ invariance to collect\nterms together, as described under~\\eq{eq:EP-dCTP-EP}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIO{\\cdeps{dEP}}{nLV-c-dEP}\n\t\t\t\t\\\\\n\t\t\t\t\t\\InvTower{\\cd{CTP-E}}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1} >^m}\n\t\t}\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s-1}  \\sum_{j=-1}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+1}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\LIDi{Combo-Bite-Vertex-nn}{nLV-c-Combo-Bite-Vertex-nn}\n\t\t\t\t\\\\\n\t\t\t\t\\vspace{1ex}\n\t\t\t\t\t\\SumVertex\n\t\t\t\t\\\\\n\t\t\t\t\t\\cd{CTP-E}\n\t\t\t\t\\end{array}\n\t\t\t}{1\\Delta^{j+s+1} >^m}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=3}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=2}^{m-1} \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIO{\n\t\t\t\t\t\\cd{CTP-E} \\cd{Combo-GR-GRs} \\hspace{1em}\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{nLV-c-Combo-GR-GRs}\n\t\t\t}{1\\Delta^{j+s+1} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t+4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+1,j+2}}{m!} \\sum_{m'=1}^{m} \\nCr{m}{m'} m'\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\Tower{\n\t\t\t\t\t\\LIDi{CTP-E-GRs-Combo-b}{nLV-c-CTP-E-GRs-Combo-b}\n\t\t\t\t\t\\hspace{1.5em}\n\t\t\t\t\t+\n\t\t\t\t\t\\LIDi{CTP-E-GRs-Combo-c}{nLV-c-CTP-E-GRs-Combo-c}\n\t\t\t\t}\n\t\t\t}{1\\Delta^{j+s+1} >^{m-m'}}\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{Diagrams spawned by \\ref{nLV-c-dCTP}.}\n\\label{fig:nLV-c-Properties-b}\n\\end{figure}\\end{center}\n\nDiagrams~\\ref{nLV-c-CTP-E-GRs-Combo-b} and~\\ref{nLV-c-CTP-E-GRs-Combo-c}\nare related, arising from\nisolating particular components of a single diagram. In the\nparent diagram from which~\\ref{nLV-c-CTP-E-GRs-Combo-b} and~\\ref{nLV-c-CTP-E-GRs-Combo-c}\nare derived, the classical, two-point vertex is decorated\nby $m'$ $>$s. The $\\rhd$ at the end of the $\\ensuremath{\\star}$\ncan be contracted into any of these $>$s or the\nsocket on the classical, two-point vertex. However,\nit is useful to isolate the component in which\nthe $\\rhd$ is contracted into the $>$\nat the top of the string {\\it i.e.}\\  diagram~\\ref{nLV-c-CTP-E-GRs-Combo-c}. \nIn this case alone,\nwe can use diagrammatic identity~\\eq{eq:GR-relation}.\nWe recognize the resulting term as the $r=1$ component\nof $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)$, demonstrating\nthat we are well on the way to proving~\\eq{eq:nLV-c-Op2}.\n\nIn anticipation of what is to follow,\nwe note that diagrams~\\ref{nLV-c-dvj+R}\nand~\\ref{nLV-c-dEP} naturally form part of a set\nin which all diagrammatic elements\nare struck by a momentum derivative.\nTo find a further member of the\nset, we process diagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\nby isolating the component possessing a classical,\ntwo-point vertex and proceeding as usual.\nThe reason we have held back with this step,\nrather than using our experience of\ndealing with such terms to jump straight\nto the result of iterating the diagrammatic\nprocedure until exhaustion, is because\nof the presence of $\\decp{\\ }{}$.\n\nThe loose end of the $\\ensuremath{\\star}$\ncan attach to any available socket in the\ndiagram, besides its own. Suppose that it attaches to the\nvertex with argument $v^{j_+}$, such that\nthe $\\ensuremath{\\star}$ bites its own base. In this\ncase, the vertex does not possess a (classical)\ntwo-point component due to the influence of\n$\\decp{\\ }{}$. Given that the vertex\nwith argument $v^{j_+}$ must be at least\nthree point, how do we proceed?\nThe point is that, since the $\\ensuremath{\\star}$ bites\nitself, the field on the vertex to \nwhich the $\\ensuremath{\\star}$ attaches carries zero momentum.\nFurthermore, charge conjugation\\ invariance forces\nthis field to be in the $A^i$\nsector (group theory considerations then\nforce it to be in the $A^1$ sector). \nThus, we \\emph{effectively} have a vertex\nwhich has been Taylor expanded to zeroth\norder in the momentum of a decorative $A^1$\nfield, and so we can repeat the above analysis!\nThe difference is that, rather than the socket\nassociated with the Taylor expansion being filled\nby an external field, it is now filled by a $\\ensuremath{\\star}$\nwhich bites its own base. \n\nPutting this diagram to one side for the moment,\nconsider those components of diagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\nthat we \\emph{can} process in the usual way.\nCancelling terms where possible, we\nare left with the following diagrams:\n\\begin{enumerate}\n\t\\item\tthe component of the nested version of\n\t\t\tdiagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\n\t\t\tin which the $\\ensuremath{\\star}$ bites a socket on a classical,\n\t\t\ttwo-point vertex, the \\emph{other} socket of which\n\t\t\tis filled by the other end of the $\\ensuremath{\\star}$;\n\t\\label{it:nLV-c-Combo-Bite-Vertex}\n\n\t\\item\tnested versions of\n\t\t\tdiagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\n\t\t\twhere the bitten field on the\n\t\t\tvertex with argument $v^{j_+}$ carries\n\t\t\tzero momentum;\n\t\\label{it:nLV-c-combo-kv}\n\n\t\\item\tremainders from cancellations via diagrammatic identity~\\eq{eq:D-ID-G};\n\t\t\tthe cancellation no longer being exact, as some of the diagrams necessary for\n\t\t\tthe cancellation are forbidden when under the influence of\n\t\t\t$\\decp{\\ }{}$;\n\t\\label{it:nLV-c-combo-GR}\n\n\t\\item\tthe component of the nested version of\n\t\t\tdiagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\n\t\t\tin which the classical, two-point vertex\n\t\t\tbitten by the $\\ensuremath{\\star}$ is attached\n\t\t\tto the gauge remainder structure or socket\n\t\t\tdecorating the $\\Op{2}$ stub.\n\t\\label{it:nLV-c-TLTP-EP-GRs}\n\\end{enumerate}\n\nWe begin by analysing the diagrams of item~\\ref{it:nLV-c-Combo-Bite-Vertex},\nwhich we have drawn in \\fig{fig:nLV-c-Properties-c}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t-4\n\t\\bca{\n\t\t\\sum_{s=1}^n  \\sum_{m=2}^{2s-1}  \\sum_{j=-2}^{n+s-m-3} \n\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=1}^{m-1} \\nCr{m}{m'}\n\t}\n\t\t{\n\t\t\\decp{\n\t\t\t\\LIO{\n\t\t\t\t\\cd{CTP-E} \\cd{TLTP-Combo-GR}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{nLV-c-TLTP-Combo-GR}\n\t\t}{1\\Delta^{j+s+2} >^{m-m'}}\n\t}\n\t\\]\n\\caption{One of the surviving terms produced by processing\ndiagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}.}\n\\label{fig:nLV-c-Properties-c}\n\\end{figure}\\end{center}\n\nTo proceed, we decompose the $\\ensuremath{\\star}$,  according to~\\eq{eq:Combo}.\n The first\nterm arising from this decomposition can be processed\nusing the effective propagator relation, whereas the\nsecond term vanishes, courtesy of diagrammatic \nidentity~\\eq{eq:GR-TLTP}. The result of\ncombining\nthe surviving terms with diagrams~\\ref{nLV-c-dvj+R}\nand~\\ref{nLV-c-dEP} is shown in \\fig{fig:nLV-c-Properties-d}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\left[\n\t\t\t\t\t\\Tower{\\LIDi{CTP-E}{nLV-c-MomDer}}\n\t\t\t\t\t\\cdeps{MomDer} \\hspace{-1.8em}\n\t\t\t\t\\right]\n\t\t\t\t\\hspace{1em}\n\t\t\t}{1 \\Delta^{j+s+2} >^{m}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=1}^{2s}  \\sum_{j=-2}^{n+s-m-2} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=1}^{m} \\nCr{m}{m'}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIO{\n\t\t\t\t\t\\cd{CTP-E-dGRs}\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{nLV-c-dGRs}\n\t\t\t}{1 \\Delta^{j+s+2} >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-4\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n  \\sum_{m=4}^{2s-1}  \\sum_{j=-2}^{n+s-m-3} \n\t\t\t\\frac{\\ensuremath{\\Upsilon}_{j+s+2,j+2}}{m!} \\sum_{m'=3}^{m-1} \\nCr{m}{m'} m'(m'-1)\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIO{\n\t\t\t\t\t\\cd{CTP-E} \\cd{GRs-Full-dGR} \\hspace{2ex}\n\t\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t\t}{\\SumVertex}\n\t\t\t\t}{nLV-c-GRs-Full-dGR}\n\t\t\t}{1 \\Delta^{j+s+2} >^{m-m'}}\n\t\t}\t\n\t\\end{array}\n\t\\]\n\\caption{The result of\ncombining diagrams~\\ref{nLV-c-dvj+R}\nand~\\ref{nLV-c-dEP} with the surviving components\nof diagram~\\ref{nLV-c-TLTP-Combo-GR}.}\n\\label{fig:nLV-c-Properties-d}\n\\end{figure}\\end{center}\n\nDiagram~\\ref{nLV-c-MomDer} contains a sum  of\ntotal momentum derivatives with respect to\\\nthe various loop momenta carried by the fully fleshed\nout constituent diagrams~\\cite{mgierg2}. Thus, diagram~\\ref{nLV-c-MomDer}\nvanishes as a consequence of the \ndimensional pre-regularisation. We comment on this further\nin the conclusions.\n\n\\begin{Icancel\n\nDiagram~\\ref{nLV-c-GRs-Full-dGR} exactly cancels\ndiagram~\\ref{nLV-c-TE-GRs}. \nThis follows from~\\eq{eq:D-ID-dGRk-GT-ring}\nupon recognizing\nthat diagrammatic \nidentity~\\eq{eq:GR-relation}\nallows\nthe momentum derivative\nof diagram~\\ref{nLV-c-GRs-Full-dGR} to be transfered from the $>$ to the\n$\\rhd$, at the expense of a minus sign. \n\n\\end{Icancel}\n\nNext, consider the diagrams of item~\\ref{it:nLV-c-combo-kv}, above.\nFrom the earlier comments about the\nun-nested diagram~\\ref{nLV-c-Combo-Bite-Vertex-nn}\nit is clear that the diagrams of item~\\ref{it:nLV-c-combo-kv}\nsimply yield versions of\ndiagram~\\ref{nLV-c-dvj+R} and~\\ref{nLV-c-dCTP}\nbut where the socket associated with the momentum\nderivative is filled by an external field of a sub-diagram\ncontaining a $\\ensuremath{\\star}$ (this is why we left the socket\nunfilled in the first place, rather than decorating\nit with the external field). The similar generalization\nof diagram~\\ref{nLV-c-TE-GRs} arises from the\ndiagrams of item~\\ref{it:nLV-c-combo-GR}, above.\nWe can manipulate the new version of\ndiagram~\\ref{nLV-c-dCTP} just as we manipulated\nthe original version and precisely the same pattern\nof terms and cancellations will be generated. In particular,\nwe will generate versions of diagram~\\ref{nLV-c-CTP-E-GRs-Combo-c}.\nWe recognise these as the $r>1$ components of\n$\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p)$. Therefore,\nwe have succeeded in demonstrating~\\eq{eq:nLV-c-Op2};\nnow we find an explicit expression for the remainders.\n\n\n\nThe remainders can be thought of as a set of primitive\ndiagrams, together with their analogues possessing\nimplicit $\\ensuremath{\\star}$s. The primitive diagrams are:\nthe diagrams of item~\\ref{it:nLV-c-TLTP-EP-GRs}\nand diagrams~\\ref{nLV-c-CTP-GRs-TE},\n\\ref{nLV-c-CTP-E-GRs-Combo-b} and~\\ref{nLV-c-dGRs}.\n\n \nDiagrams of item~\\ref{it:nLV-c-TLTP-EP-GRs}\ncan be redrawn  via diagrammatic identities~\\eq{eq:D-ID-Op2-G},\nupon which\none of the resultant diagrams\nexactly cancels \ndiagram~\\ref{nLV-c-CTP-E-GRs-Combo-b}.\nDiagram~\\ref{nLV-c-CTP-GRs-TE} can be\nredrawn using diagrammatic identity~\\eq{eq:D-ID-dGRk-GT-ring}.\nThe resultant diagram naturally combines\nwith diagram~\\ref{nLV-c-dGRs} upon recognizing\nthat\n\\begin{equation}\n\\label{eq:TLTP-Combo}\n\t\\cd{CTP-Combo} = \\cd{CTP-EP-dGR} = \\cd{dGR} + \\cd{GR-dGRk-GR}.\n\\end{equation}\nWe thus find that\n\\begin{equation}\n\\label{eq:nLV-c-Op2-b}\n\t2 \\decp{\\nLV{\\overline{\\mathcal{E}}}{n}{\\mu\\nu}{c}(p)}{} = \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{b}(p) \n\t+ 2 \\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c'}(p),\n\\end{equation}\nwhere $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c'}(p)$\nis given in \\fig{fig:nLV-c-Properties-e}.\n\\begin{center}\\begin{figure}\n\t\\[\n\t\\begin{array}{l}\n\t\\vspace{2ex}\n\t\t\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c'}(p) \\equiv\n\t\\\\\n\t\\vspace{2ex}\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{r=1}^{s-1} \\sum_{m=1}^{2(s-r)+1}\n\t\t\t\\sum_{j=-2}^{n+s-m-2r-1} \\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1!)}\n\t\t\t\\sum_{m'=1}^{m} \\nCr{m}{m'} \\sum_{m''=1}^{m'} \\nCr{m'}{m''}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIDi{CTP-E-GRs-Combo-GRs-CTP-EP-b}{nLV-c-CTP-E-GRs-Combo-GRs-CTP-EP}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^{m-m'}}\n\t\t}\n\t\\\\\n\t\\vspace{2ex}\n\t\t+2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)+1}\n\t\t\t\\sum_{j=-2}^{n+s-m-2r-1} \\frac{\\ensuremath{\\Upsilon}_{j+s+2-r,j+2}}{m!(r-1!)}\n\t\t\t\\sum_{m'=0}^{m} \\nCr{m}{m'} \n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIDi{CTP-E-GRs-Combo-GR}{nLV-c-CTP-E-GRs-Combo-GR}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+2-r} \\ensuremath{\\star}^r >^{m-m'}}\n\t\t}\n\t\\\\\n\t\t-2\n\t\t\\bca{\n\t\t\t\\sum_{s=1}^n \\sum_{r=1}^{s-1} \\sum_{m=0}^{2(s-r)+1}\n\t\t\t\\sum_{j=-2}^{n+s-m-2r-1} \\frac{\\ensuremath{\\Upsilon}_{j+s+3-r,j+2}}{m!(r-1!)}\n\t\t\t\\sum_{m'=0}^{m} \\nCr{m}{m'} \\sum_{m''=0}^{m'} \\nCr{m'}{m''}\n\t\t}\n\t\t\t{\n\t\t\t\\decp{\n\t\t\t\t\\LIDi{CTP-E-GRs-Combo-CTP-GRs}{nLV-c-CTP-E-GRs-Combo-CTP-GRs}\n\t\t\t\t\\hspace{1.5em}\n\t\t\t\t\\sco[1]{\n\t\t\t\t\t\t\\TopVertex\n\t\t\t\t}{\\SumVertex}\n\t\t\t}{1\\Delta^{j+s+3-r} \\ensuremath{\\star}^r >^{m-m'}}\t\n\t\t}\n\t\\end{array}\n\t\\]\n\\caption{The expression for $\\nLV{\\mathcal{E}}{n}{\\mu\\nu}{c'}(p)$.}\n\\label{fig:nLV-c-Properties-e}\n\\end{figure}\\end{center}\n\n\n\n\n\\section{Review}\n\\label{sec:Review}\n\n\\subsection{Elements of $SU(N|N)$ Gauge Theory}\n\\label{sec:elements}\n\nThroughout this paper, we work in Euclidean dimension, D.\nWe regularize $SU(N)$ Yang-Mills by embedding\nit in spontaneously broken $SU(N|N)$ Yang-Mills,\nwhich is itself regularized by covariant higher derivatives~\\cite{SU(N|N)}. \nThe supergauge field, ${\\cal A}_\\mu$, is valued in\nthe Lie superalgebra and, using the defining representation,\ncan be written as a Hermitian supertraceless supermatrix:\n\\[\n\t{\\cal A}_\\mu = \n\t\\left(\n\t\t\\begin{array}{cc}\n\t\t\tA_\\mu^1 \t& B_\\mu \n\t\t\\\\\n\t\t\t\\bar{B}_\\mu & A_\\mu^2\n\t\t\\end{array} \n\t\\right) + {\\cal A}_\\mu^0 \\hbox{1\\kern-.8mm l}.\n\\]\nHere, $A^1_\\mu(x)\\equiv A^1_{a\\mu}\\tau^a_1$ is the\nphysical $SU(N)$ gauge field, $\\tau^a_1$ being the $SU(N)$\ngenerators orthonormalized to\n${\\mathrm{tr}}(\\tau^a_1\\tau^b_1)=\\delta^{ab}\/2$, while $A^2_\\mu(x)\\equiv\nA^2_{a\\mu}\\tau^a_2$ is a second unphysical $SU(N)$ gauge field.\nWhen labelling {\\it e.g.}\\  vertex coefficient functions,\nwe often abbreviate $A^{1,2}$ to just 1,2.\nThe $B$ fields are fermionic gauge fields which will gain a mass\nof order $\\Lambda$ from the spontaneous symmetry breaking; they play the\nrole of gauge invariant Pauli-Villars (PV) fields, furnishing the\nnecessary extra regularization to supplement the covariant\nhigher derivatives. In order\nto unambiguously define contributions which are finite\nonly by virtue of the PV regularization, a pre-regulator must\nbe used in $D=4$~\\cite{SU(N|N)}. For the time being, we will use\ndimensional regularization, emphasising that\nthis makes sense\nnon-perturbatively, since \nit is not being used to renormalize the theory, but rather\nas a prescription for discarding surface terms in loop\nintegrals~\\cite{SU(N|N)}.\n\n\n\nThe theory is\nsubject to the local invariance:\n\\begin{equation}\n\\label{Agauged}\n\\delta{\\cal A}_\\mu = [\\nabla_\\mu,\\Omega(x)] +\\lambda_\\mu(x) \\hbox{1\\kern-.8mm l}.\n\\end{equation}\nThe first term, in which $\\nabla_\\mu = \\partial_\\mu -i{\\cal A}_\\mu$, \ngenerates supergauge transformations. Note that the coupling, $g$,\nwhich we now take to represent the coupling for all renormalization\nschemes implictly defined by our approach,\nhas been scaled out of this definition. It is worth doing\nthis: since we do not gauge fix, the exact preservation of~\\eq{Agauged}\nmeans that none of the fields suffer wavefunction\nrenormalization, even in the broken phase~\\cite{aprop}.\n\nThe second term in~\\eq{Agauged} divides out the centre of the algebra.\nThis `no ${\\cal A}^0$ shift symmetry' ensures that nothing depends on ${\\cal A}^0$\nand that ${\\cal A}^0$ has no degrees of freedom. We adopt a\nprescription whereby we can effectively ignore the field ${\\cal A}^0$,  altogether, \nusing it  to map us into a particular diagrammatic\npicture~\\cite{Thesis,mgierg1}.\n\nThe spontaneous symmetry breaking is carried by a superscalar\nfield\n\\[\n{\\cal C} =\n\t\\left(\n\t\t\\begin{array}{cc}\n\t\t\tC^1\t\t& D\n\t\t\\\\\n\t\t\t\\bar{D}\t& C^2\n\t\t\\end{array}\n\t\\right),\n\\]\nwhich transforms homogeneously:\n\\begin{equation}\n\\label{Cgauged}\n\\delta{\\cal C} = -i\\,[{\\cal C},\\Omega].\n\\end{equation}\n\n\n\nIt can be shown that, at the classical level, the spontaneous\nsymmetry\nbreaking scale (effectively the mass of $B$) tracks the covariant\nhigher derivative effective cutoff scale, $\\Lambda$, if ${\\cal C}$ is\nmade dimensionless (by using powers of $\\Lambda$) and  $\\hat{S}$ has\nthe minimum of its effective potential at:\n\\begin{equation}\n\\label{sigma}\n\\sigma \\equiv \\pmatrix{\\hbox{1\\kern-.8mm l} & 0\\cr 0 & -\\hbox{1\\kern-.8mm l}}.\n\\end{equation}\nIn this case the classical action $S_0$ also has a minimum \nat~\\eq{sigma}. At the quantum level this can be imposed as a\nconstraint on $S$ by taking $\\expectation{C} = \\sigma $ as a renormalization\ncondition. This ensures that the Wilsonian effective action\ndoes not possess any one-point vertices, which can be\ntranslated into a constraint on\n$\\hat{S}$~\\cite{aprop,Thesis}. In the broken phase, $D$ is\na super-Goldstone mode (eaten by $B$ in the unitary gauge) whilst the\n$C^i$ are Higgs bosons and can be given a running mass of order\n$\\Lambda$~\\cite{ym,SU(N|N),aprop}. Working in our manifestly\ngauge invariant formalism, $B$ and $D$ gauge transform into each\nother; in recognition of this, we define the fields\n\\numparts\n\\begin{eqnarray}\n\\label{eq:F}\n\tF_M & = & (B_\\mu, D),\n\\\\\n\\label{eq:Fbar}\n\t\\bar{F}_N & = & (\\bar{B}_\\nu, -\\bar{D}),\n\\end{eqnarray}\n\\endnumparts\nwhere $M$, $N$ are\nfive-indices~\\cite{Thesis,mgierg1}.\\footnote{The summation\nconvention for these indices is that we take each product of\ncomponents to contribute with unit weight.} \n\nThe couplings $g$ and $\\alpha$ (recall~\\eq{eq:alpha-defn})\nare defined through their\nrenormalization conditions:\n\\begin{eqnarray}\n\\label{defg}\n\tS[{\\cal A}=A^1, {\\cal C}=\\sigma]\t& =\t& {1\\over2g^2}\\,\\mathrm{str}\\!\\int\\!\\!d^D\\!x\\,\n\t\t\t\t\t\t\t\t\t\\left(F^1_{\\mu\\nu}\\right)^2+\\cdots,\n\\\\\t\n\\label{defg2}\n\tS[{\\cal A}=A^2, {\\cal C}=\\sigma] \t& =\t& {1\\over2 \\alpha g^2}\\,\\mathrm{str}\\!\\int\\!\\!d^D\\!x\\,\n\t\t\t\t\t\t\t\t\t\\left(F^2_{\\mu\\nu}\\right)^2+\\cdots,\n\\end{eqnarray}\nwhere the ellipses stand for higher dimension operators and the\nignored vacuum energy. The field strength tensors in the $A^1$ and\n$A^2$ sectors, $F^1_{\\mu\\nu}$ and $F^2_{\\mu\\nu}$, should really\nbe embedded in the top left \/ bottom right entries of a supermatrix,\nin order for the supertraces in~\\eqs{defg}{defg2} \nto make sense. We will frequently employ\nthis minor abuse of notation, for convenience.\n\n\n\n\\subsection{Diagrammatics for the Flow Equation}\n\n\\subsubsection{The Exact Flow Equation}\n\\label{sec:Flow}\n\nThe diagrammatic representation of the flow\nequation is shown in \\fig{fig:Flow}~\\cite{Thesis,mgierg1}.\n(For a comprehensive description of the diagrammatics see~\\cite{Thesis,mgierg1}.)\n\\begin{center}\\begin{figure}[h]\n\t\\begin{eqnarray*}\n\t\\displaystyle\n\t-\\Lambda \\partial_\\Lambda \n\t\\dec{\n\t\t\\cd{Vertex-S}\n\t}{\\{f\\}}\n\t& = & a_0[S,\\Sigma_g]^{\\{f\\}} - a_1[\\Sigma_g]^{\\{f\\}}\n\\\\\n\t& = &\n\t\\displaystyle\n\t\\frac{1}{2}\n\t\\dec{\n\t\t\\cd{Dumbbell-S-Sigma_g} - \\cd{Padlock-Sigma_g} - \\cd{WBT-Sigma_g}\n\t}{\\{f\\}}\n\t\\end{eqnarray*}\n\\caption{The diagrammatic form of the flow equation.}\n\\label{fig:Flow}\n\\end{figure}\\end{center}\n\nThe left-hand side\\ depicts the flow of all independent Wilsonian effective action\nvertex \\emph{coefficient functions}, \nwhich correspond to the set of broken phase\nfields, $\\{f\\}$. Each coefficient function has associated\nwith it an implied supertrace structure (and symmetry factor which,\nas one would want, does not appear in the diagrammatics).\nFor example,\n\\begin{equation}\n\\label{eq:Vertex-C1C1}\n\t\\dec{\n\t\t\\cd{Vertex-S}\n\t}{C^1C^1}\n\\end{equation}\nrepresents both the coefficient functions $S^{C^1 C^1}$ and\n$S^{C^1,C^1}$ which, respectively, are associated with the\nsupertrace structures $\\mathrm{str} C^1 C^1$ and $\\mathrm{str} C^1 \\mathrm{str} C^1$.\n\nThe objects on the right-hand side\\ of \\fig{fig:Flow}\nhave two different types of component. The lobes\nrepresent vertices of action functionals,\nwhere $\\Sigma_g \\equiv g^2S - 2 \\hat{S}$ (recall that $\\hat{S}$ is the seed action). \nThe object attaching\nto the various lobes, \\ensuremath{\\stackrel{\\bullet}{\\mbox{\\rule{1cm}{.2mm}}}},  is\nthe sum over vertices of the covariantized ERG kernels~\\cite{ymi,aprop}\nand, like the action vertices, can be decorated by fields belonging to $\\{f\\}$.\nThe appearance of the symbol $\\scriptstyle \\bullet$ is not accidental, \nmeaning $-\\Lambda \\partial_\\Lambda|_\\alpha$, as in~\\eq{eq:dot}.\nThe dumbbell-like\nterm, which corresponds to the bilinear functional $a_0$, is referred\nto as the classical term. The next two terms, which are both\ngenerated by $a_1$, are referred to as quantum terms. The second\nof these contains a kernel which `bites its own tail'. This\ndiagram is not properly \nUV regularized by the $SU(N|N)$ regularization \nand, in the past, it has been argued that\nit can and should be discarded, through an appropriate\nconstraint on the covariantization~\\cite{mgierg1,ym,ymi,aprop}.\\footnote{\nThese diagrams are artefacts of the flow equation. The $SU(N)$\ngauge theory \\emph{is} fully regularized by the $SU(N|N)$ scheme. However,\nregularization of the flow equation does not trivially follow from\nthe regularization of the underlying theory.}\nHere, though, we will keep these diagrams for as\nlong as possible: as recognized in~\\cite{Thesis},\nin any calculation $\\beta$ function coefficients, \nthe explicit ultraviolet divergences in kernel-biting-their-tail\ndiagrams, which can be dimensionally regularized,\nshould be cancelled by divergences hidden in other\nterms. In this paper, we demonstrate that this\nis indeed the case: all explicit dependence\nof $\\beta_n$ on kernel-biting-their-tail\ndiagrams cancels out. Nonetheless, there is an implicit\ndependence left behind and it seems inevitable that, to further\nproceed, the covariantization \nmust be suitably constrained,\nafter all.\n\n\n\nIt is worth drawing attention to the fact that dimensional\nregularization is thus being used for two entirely independent\npurposes. On the one hand, it is used as a temporary\nmeasure, at intermediate stages of our calculations, \nto properly define\ndiagrams possessing a kernel which bites its own tail. \nOn the\nother hand, it is being employed as a pre-regulator. \nAs we will argue in the conclusions, there is evidence\nto suggest the existence of a purely diagrammatic\npre-regulator, which would make sense in $D=4$.\n\n\nEmbedded within the diagrammatic rules is a prescription for evaluating the\ngroup theory factors. \nSuppose that we wish to focus on the flow of a particular\nvertex coefficient function, which necessarily has a unique\nsupertrace structure. \nFor example, we might be interested in just\nthe $S^{C^1 C^1}$ component of~\\eq{eq:Vertex-C1C1}.\n\nOn the right-hand side\\ of the flow equation, we must\nfocus on the components of each diagram\nwith precisely the same (implied) \nsupertrace structure as the left-hand side,\nnoting that the kernel, like the vertices,\nhas multi-supertrace contributions (for more\ndetails see~\\cite{Thesis,mgierg1}).\nIn this more explicit diagrammatic picture,\nthe kernel is to be considered a double\nsided object.\nThus, whilst the dumbbell like term of \\fig{fig:Flow}\nhas at least one associated supertrace, the next diagram\nhas  at least two, on a account of the loop\n(this is strictly true only in the\ncase that kernel attaches to fields on the same\nsupertrace). If a closed\ncircuit formed by a kernel is devoid\nof fields then\nit contributes \na factor of $\\pm N$, depending on\nthe flavours of the fields to which the kernel forming\nthe loop attaches. This is most easily appreciated by\ndefining the projectors\n\\begin{equation}\n\\label{eq:Projectors}\n\t\\sigma_{+} \\equiv\n\t\\left(\n\t\t\\begin{array}{cc}\n\t\t\t\\hbox{1\\kern-.8mm l}\t&\t0\n\t\t\\\\\n\t\t\t\t0\t&\t0\n\t\t\\end{array}\n\t\\right), \\ \\ \n\t\t\\sigma_{-} \\equiv\n\t\\left(\n\t\t\\begin{array}{cc}\n\t\t\t0\t&\t0\n\t\t\\\\\n\t\t\t0\t&\t\\hbox{1\\kern-.8mm l}\n\t\t\\end{array}\n\t\\right)\n\\end{equation}\nand noting that $\\mathrm{str} \\sigma_\\pm = \\pm N$. \nIn the counterclockwise sense, a $\\sigma_+$\ncan always be inserted for free after an $A^1$, $C^1$ or $\\bar{F}$,\nwhereas a $\\sigma_-$\ncan always be inserted for free after an $A^2$, $C^2$ or $F$. \n\n\nThe rules thus described receive $1\/N$ corrections in \nthe $A^1$ and $A^2$ sectors. If a kernel\nattaches to an $A^1$ or $A^2$, it comprises a direct\nattachment and an indirect attachment, as shown \nin \\fig{fig:Attach} (see~\\cite{Thesis,mgierg1} for more detail).\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t\\cd{Direct} \\rightarrow \\left. \\cd{Direct} \\right|_{\\mbox{direct}} + \\frac{1}{N} \\left[ \\cd{Indirect-2} - \\cd{Indirect-1} \\right]\n\t\\]\n\\caption{The $1\/N$ corrections to the group theory factors.}\n\\label{fig:Attach}\n\\end{figure}\\end{center} \n\nWe can thus consider the diagram on the left-hand side\\ as having been unpackaged,\nto give the terms on the right-hand side. The dotted lines in the diagrams with indirect\nattachments serve to remind us where the loose end of the kernel attaches\nin the parent diagram.\n\n\n\\subsubsection{The Ward Identities}\n\\label{sec:WIDs}\n\nAll vertices, whether they belong to either\nof the actions or to the covariantized kernels\nare subject to Ward identities which, due to the\nmanifest gauge invariance, take a \nparticularly simple form:\n\\begin{equation}\n\\label{eq:WID-A}\n\t\\cd{WID-contract} = \\cd{WID-PF} + \\cd{WID-PFb} - \\cd{WID-PB} - \\cd{WID-PBb} + \\cdots\n\\end{equation}\n\n\nEquation~\\eq{eq:WID-A} is the first primary diagrammatic identity\nof the first type. On the left-hand side, we contract a vertex with the momentum of\nthe field which carries $p$. This field---which we will\ncall the active field---can be either\n$A^1_\\rho$, $A^2_\\rho$, $F_R$ or $\\bar{F}_R$.\nIn the first two cases, the triangle $\\rhd$ represents\n$p_\\rho$ whereas, in the latter two cases, it represents\n$p_R \\equiv (p_\\rho,2)$. (Given that we often sum over\nall possible fields, we can take the Feynman rule for\n$\\rhd$ in the $C$-sector to be null.)\nIn all cases,  $\\rhd$ is independent of $\\Lambda$ and $\\alpha$, which is\nencoded in the  second of the primary diagrammatic identities of the \nfirst type:\n\\numparts\n\\begin{eqnarray}\n\\label{eq:LdL-GRk-Pert-a}\n\t\\hspace{0.8em} \\stackrel{\\bullet}{\\rhd} & =  & 0,\n\\\\\n\\label{eq:dalpha-GRk-a}\n\t\\begin{array}{c}\n\t\t\\Dal\n\t\\\\[-1.5ex]\n\t\t\\rhd\n\t\\end{array} & = & 0,\n\\end{eqnarray}\n\\endnumparts\nwhere $ \\Dal\\ \\equiv \\partial \/ \\partial{\\alpha}$.\n\nOn the right-hand side\\ of~\\eq{eq:WID-A}, we push the contracted momentum forward onto \nthe field which directly follows the active field, in the counterclockwise\nsense, and pull back (with a minus sign) onto\nthe field which directly precedes the active field. \nSince our diagrammatics is permutation symmetric, the struck field---which\nwe will call the target field---can\nbe either $X$, $Y$ or any of the un-drawn fields represented\nby the ellipsis.\nAny field(s) besides the active field and the\ntarget field will be called spectators.\nNote that we can take\n$X$ and \/ or $Y$ to represent the end of a kernel.\nIn this case, the struck field is determined to be unambiguously\non one side of the (double sided) kernel; \nthe contributions\nin which the struck field is on the other side are included\nin the ellipsis. \nThis highlights the point that allowing the active\nfield to strike another field necessarily involves a partial\nspecification of the supertrace structure: it must be the case that\nthe struck field either directly followed or preceded the active\nfield. In turn, this means that the Feynman rule for particular\nchoices of the active and target fields can be zero. For example,\nan $F$ can follow, but never precede an $A^1_\\mu$, and so the \npull back of an $A^1_\\mu$ onto an $F$ should be assigned a value\nof zero. \nThe momentum routing follows in an obvious manner: for example,\nin the first diagram on the right-hand side, momenta $q+p$ and $r$ now flow into\nthe vertex. In the case that the active field is fermionic,\nthe field pushed forward \/ pulled back onto is transformed\ninto its opposite statistic partner. \nThere are some signs \nassociated with this in the $C$ and $D$-sectors, \nfor which we refer the reader to~\\cite{mgierg1,Thesis}. \n\nThe half arrow which terminates the pushed forward \/ pulled back\nactive field is of no significance and can go on either side\nof the active field line. It is necessary to\nkeep the active field line---even though the active field\nis no longer part of the vertex---in order that\nwe can unambiguously deduce flavour changes \nand momentum routing, without reference to the parent diagram.\n\n\nWe illustrate~\\eq{eq:WID-A} by considering contracting\n$\\rhd$ into the Wilsonian the effective action\ntwo-point vertex:\n\\begin{equation}\n\\label{eq:GR-TP}\n\t\\cd{GR-TP} =  \\cd{GR-TP-PF} - \\cd{GR-TP-PB}.\n\\end{equation}\nGiven that $\\rhd$ is null in the $C^i$ sector,\nthe fields decorating the two-point\nvertex on the right-hand side\\ can be either both $A^i$s\nor both fermionic. In the former case, \n\\eq{eq:GR-TP} reads:\n\\[\n\tp_\\mu S^{A^i A^i}_{\\mu \\ \\, \\nu}(p) = S^{A^i}_{\\nu}(0) - S^{A^i}_{\\nu}(0) = 0\n\\]\nwhere we note that $S^{A^i}_{\\nu}$ is in fact zero by itself,\nas follows by both Lorentz invariance and gauge invariance.\nIn the latter case, \\eq{eq:GR-TP} reads:\n\\[\n\tp_M S^{\\bar{F} \\; F}_{M N}(p) = \\left[S^{C^2}(0) - S^{C^1}(0)\\right] \\delta_{5N},\n\\]\nwhere we have used~\\eq{eq:F} and have discarded\ncontributions which go like $S^{A^i}_{\\nu}(0)$.\nHowever, the $S^{C^i}(0)$ must vanish. This\nfollows from demanding\nthat the minimum of the superhiggs potential is\nnot shifted by quantum corrections~\\cite{aprop}.\nTherefore, we arrive at the diagrammatic identity\n\\begin{equation}\n\\label{eq:D-ID-GR-TP-A}\n\t\\cd{GR-TP} = 0.\n\\end{equation}\n\n\n\n\\subsubsection{Taylor Expansion of Vertices}\n\\label{sec:Taylor}\n\nFor the formalism to be properly defined,\nit must be the case that all vertices\nare Taylor expandable to all orders\nin momenta~\\cite{ym,ymi,ymii}.\nConsider a vertex which is part\nof a complete diagram, decorated by some set of internal\nfields and by a single external $A^1$ (or $A^2$).\nThe diagrammatic representation for the zeroth order expansion\nin the momentum of the external field is all that is required\nfor this paper~\\cite{Thesis,mgierg1}:\n\\begin{equation}\n\\label{eq:Taylor-A}\n\t\\cd{Taylor-Parent} = \\cdeps{Taylor-PFa} + \\cdeps{Taylor-PFb} - \\cdeps{Taylor-PBa} - \\cdeps{Taylor-PBb} +\\cdots\n\\end{equation}\nThis is the final primary diagrammatic identity of the first type;\nnote the similarity to~\\eq{eq:WID-A}.\n\n\nThe interpretation of the diagrammatics is as follows. In the first diagram\non the right-hand side, the vertex is differentiated with respect to\\ the momentum carried\nby the field $X$, whilst holding the momentum of the preceding field fixed\n(we assume for the time being that both $X$ and the\npreceding field carry non-zero momentum).\nOf course, using our current diagrammatic notation,\nthis latter field can be any of those\nwhich decorate the vertex, and so we sum over all possibilities. Thus,\neach cyclically ordered push forward like term has a partner,\ncyclically ordered pull back like term, such that\nthe pair can be interpreted as\n\\begin{equation}\n\t\\left( \\left. \\partial^r_\\mu \\right|_s - \\left. \\partial^s_\\mu \\right|_r \\right) \\mathrm{Vertex},\n\\label{eq:Momderivs}\n\\end{equation}\nwhere $r$ and $s$ are momenta entering the vertex. \nIn the case that $r=-s$, we can and will\ndrop either the push forward like term or pull back like term, since\nthe combination can be expressed as $\\partial^r_\\mu$; we\ninterpret the diagrammatic notation appropriately.\nIf any of the fields decorating the vertex carry\nzero momentum (besides the explicitly drawn $A^i$),\nthen they are transparent to this entire procedure.\nThus, they are never differentiated and, if they precede\na field which is, we must look to the first field carrying\nnon-zero momentum to figure\nout which of the vertex's momenta is held constant.\nJust as in~\\eq{eq:WID-A}, \nthe fields $X$ and or $Y$ can be interpreted as the end\nof a kernel. In this case, we introduce some new\nnotation, since it proves confusing in complete diagrams\nto actually locate the derivative\nsymbol at the end of such an object. The notation for the\nderivative with respect to\\ the momentum entering the end of a kernel \nis introduced in figure~\\ref{fig:Taylor-EndofKernel}.\n\\begin{center}\\begin{figure}[h]\n\t\\[\n\t\t\\cd{DifferentiatedKernel-A}\n\t\\]\n\\caption{Notation for the derivative with respect to\\ the momentum entering an\nundecorated kernel.}\n\\label{fig:Taylor-EndofKernel}\n\\end{figure}\\end{center}\n\nRecalling that a \nkernel, whose fields are explicitly\ncyclically ordered, is a two-sided object, we note\nthat the field whose momentum we have expanded in is sat on the\ntop-side of the vertex.\nThe derivative is taken to be with respect to\\\nthe momentum which flows \\emph{into} the end of the vertex which\nfollows the derivative, in the sense indicated by the\narrow on the derivative symbol.\nIt is clear that the direction of the\narrow on the derivative symbol can be reversed \nat the expense of a minus sign.\n\n\n\n\\subsubsection{Charge Conjugation Invariance}\n\\label{sec:CC}\n\nCharge conjugation invariance can be used to simplify the\ndiagrammatics, by allowing us to discard certain terms\nand to combine others. The diagrammatic prescription for\nreplacing a diagram which possesses exclusively\nbosonic external fields with its charge conjugate is~\\cite{Thesis,mgierg1}\nto reflect the diagram, picking up a sign for each\n\\begin{enumerate}\n\t\\item external $A^i$,\n\n\t\\item \\emph{performed} gauge remainder,\n\n\t\\item momentum derivative symbol (note\n\t\t\tthat the direction of the arrow accompanying such\n\t\t\tsymbols is reversed by the reflection of the diagram).\n\\end{enumerate}\n\n\\subsection{The Weak Coupling Expansion}\n\\label{sec:Weak}\n\n\\subsubsection{Perturbative Diagrammatics}\n\\label{sec:WeakDiag}\n\nIn the perturbative domain, we have the following\nweak coupling expansions~\\cite{ymi,aprop,Thesis,mgierg1}.\nThe Wilsonian effective action is given by\n\\begin{equation}\n\tS = \\sum_{i=0}^\\infty \\left( g^2 \\right)^{i-1} S_i = \\frac{1}{g^2}S_0 + S_1 + \\cdots,\n\\label{eq:Weak-S}\n\\end{equation}\nwhere $S_0$ is the classical effective action and the $S_{i>0}$\nthe $i$th-loop corrections. The seed action has a similar expansion:\n\\begin{equation}\n\t\\hat{S} = \\sum_{i=0}^\\infty  g^{2i}\\hat{S}_i.\n\\label{eq:Weak-hS}\n\\end{equation}\nRecalling~\\eq{eq:alpha-defn} we have:\n\\begin{eqnarray}\n\t\\beta \\equiv \\Lambda \\partial_\\Lambda g \t\t& = & \\sum_{i=1}^\\infty  g^{2i+1} \\beta_i(\\alpha)\n\\label{eq:beta}\n\\\\[1ex]\n\t\\gamma \\equiv \\Lambda \\partial_\\Lambda \\alpha \t& = & \\sum_{i=1}^{\\infty}  g^{2i} \\gamma_i(\\alpha).\n\\label{eq:gamma}\n\\end{eqnarray}\n\nDefining $\\Sigma_i = S_i - 2\\hat{S}_i$, the weak coupling flow equations\nfollow from substituting~\\eq{eq:Weak-S}--\\eq{eq:gamma}\ninto the flow equation, as shown in \n\\fig{fig:WeakCouplingFE}~\\cite{Thesis,mgierg1}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\t\\dec{\n\t\t\t\\cd{Vertex-n-LdL} \n\t\t}{\\{f\\}}\n\t\t= \n\t\t\\dec{\n\t\t\t\\begin{array}{c}\n\t\t\t\t\\displaystyle\n\t\t\t\t\\sum_{r=1}^n \\left[2\\left(n_r -1 \\right) \\beta_r +\\gamma_r \\pder{}{\\alpha} \\right]\\cd{Vertex-n_r-B} \n\t\t\t\\\\[4ex]\n\t\t\t\t\\displaystyle\n\t\t\t\t+ \\frac{1}{2} \n\t\t\t\t\\left( \n\t\t\t\t\t\\sum_{r=0}^n \\cd{Dumbbell-n_r-r} - \\cd{Vertex-Sigma_n_-B} - \\cd{WBT-Sigma_n-}\n\t\t\t\t\\right)\n\t\t\t\\end{array}\n\t\t}{\\{f\\}}\n\t\\label{eq:WeakFlow}\n\t\\end{equation}\n\\caption{The weak coupling flow equations.}\n\\label{fig:WeakCouplingFE}\n\\end{figure}\\end{center}\n\nWe refer to the first two terms on the right-hand side\\ of~\\eq{eq:WeakFlow} as\n$\\beta$ and $\\alpha$-terms, respectively.\nThe symbol $\\bullet$, as in equation~\\eq{eq:dot}, means\n$-\\Lambda \\partial_\\Lambda|_\\alpha$. A vertex whose argument is an unadorned letter, say $n$,\nrepresents $S_n$. We define $n_r \\equiv n-r$ and $n_\\pm \\equiv n \\pm 1$. The\nbar notation of the dumbbell term is defined as follows:\n\\begin{equation}\n\\label{eq:bar}\n\ta_0[\\bar{S}_{n-r}, \\bar{S}_r] \t\\equiv \ta_0[S_{n-r}, S_r] - a_0[S_{n-r}, \\hat{S}_r] - a_0[\\hat{S}_{n-r}, S_r].\n\\end{equation}\n\nThe renormalization condition\nfor $g$, equation~\\eq{defg}, constrains the two-point  vertex of\nthe physical field $S_{\\mu \\ \\; \\nu}^{A^1 A^1}(p)$ as follows:\n\\begin{eqnarray}\n\\label{eq:S_0-11}\n\tS_{0 \\mu  \\nu}^{\\ 1 \\, 1}(p) & = & 2 (p^2 \\delta_{\\mu\\nu} - p_\\mu p_\\mu) + \\Op{4}\n\t\\equiv 2\\Box_{\\mu \\nu}(p) + \\Op{4}\n\\\\\n\\label{eq:S_>0-11}\n\tS_{n>0 \\mu  \\nu}^{\\ \\ \\ \\ 1 \\, 1}(p) & = & \\Op{4},\n\\end{eqnarray}\nwhere we have abbreviated $A^1$ by just `1'. \n\n\n\\subsubsection{The Effective Propagator Relation}\n\\label{sec:EP-Reln}\n\nThe effective propagator\nrelation (which we recall is the sole primary\ndiagrammatic identity of the second type) arises\nfrom examining the flow of all two-point, tree level vertices.\nThis is done by setting $n=0$ in~\\eq{eq:WeakFlow}\nand specializing $\\{f\\}$ to contain two fields, \nas shown in \\fig{fig:TLTPs}.\nWe note that we can and do choose\nall such vertices to be single supertrace terms~\\cite{Thesis,mgierg1}.\n\\begin{center}\\begin{figure}[h]\n\t\\begin{equation}\n\t\t\\cd{Vertex-TLTP-LdL} = \\cd{Dumbbell-S_0-S_0} - \\cd{Dumbbell-S_0-hS_0} - \\cd{Dumbbell-hS_0-S_0}\n\t\\label{eq:TLTP-flow}\n\t\\end{equation}\n\\caption{Flow of all possible two-point, tree level vertices.}\n\\label{fig:TLTPs}\n\\end{figure}\\end{center}\n\nFollowing~\\cite{ym,ymi,ymii,aprop,Thesis,mgierg1,scalar2} \nwe use the freedom inherent in $\\hat{S}$ by choosing the two-point, tree\nlevel seed action vertices equal to the corresponding Wilsonian effective\naction vertices. Equation~\\eq{eq:TLTP-flow} now simplifies.\nRearranging, integrating with respect to\\ $\\Lambda$ and choosing the appropriate\nintegration constants~\\cite{Thesis,mgierg1}, we arrive at the following\nrelationship between the integrated ERG kernels---{\\it a.k.a.}\\  the\neffective propagators---and the two-point,\ntree level vertices. \n\\begin{equation}\n\t\t\\cd{EffPropReln}\t= \\cd{K-Delta} - \\cd{FullGaugeRemainder}\n\t\t\t\t\t\t\t= \\cd{K-Delta} - \\cd{DecomposedGR}\n\t\\label{eq:EPReln-A}\n\\end{equation}\n\nNote\nthat we have attached the effective propagator, which only\never appears as an internal line, to an arbitrary structure.\nThe field labelled by $M$ can be any of the broken phase\nfields. The object $\\cdeps{GR} \\!\\! \\equiv \\; > \\!\\!\\! \\rhd$ is a gauge remainder.\nThe individual components of\n$> \\!\\! \\rhd$ will often be loosely\nreferred to as gauge remainders; where it is necessary to\n unambiguously refer to the composite structure, we will use\nthe terminology `full gauge remainder'.\n\nThe various components on the right-hand side\\ of~\\eq{eq:EPReln-A}\ncan be interpreted, in the different sectors,\naccording to table~\\ref{tab:NFE:k,k'}, where we take\nthe gauge remainder to carry momentum $p$ (as denoted\nby the subscripts carried by the gauge remainder components).\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{table}[h]\n\t\\[\n\t\\begin{array}{c|ccc}\n\t\t\t\t\t& \\delta_{MN}\t\t& >_p\t\t\t\t\t\t& \\rhd_p\n\t\\\\ \\hline \n\t\tF,\\bar{F}\t& \\delta_{MN}\t\t& (f_p p_\\mu \/ \\Lambda^2, g_p)\t& (p_\\nu, 2)\n\t\\\\\n\t\tA^i\t\t\t& \\delta_{\\mu \\nu}\t& p_\\mu \/ p^2\t\t\t\t\t& p_\\nu\n\t\\\\\n\t\tC^i\t\t\t& \\hbox{1\\kern-.8mm l}\t\t\t\t& \\mbox{---}\t\t\t\t\t& \\mbox{---}\n\t\\end{array}\n\t\\]\n\\caption{Prescription for interpreting~\\eq{eq:EPReln-A}.}\n\\label{tab:NFE:k,k'}\n\\end{table}\n\\end{center}\n\\renewcommand{\\arraystretch}{1}\n\n\nFrom~\\eq{eq:S_0-11} and \\tab{tab:NFE:k,k'}, it follows\nthat\n\\begin{equation}\n\\label{eq:EP-leading}\n\t\\Delta^{11}_{\\rho \\sigma}(p) = \\frac{\\delta_{\\rho \\sigma}}{2 p^2} + \\Op{0},\n\\end{equation}\nwhich we will need later. \nThe functions $f(p^2\/\\Lambda^2)$ and $g(p^2\/\\Lambda^2)$ \nneed never be exactly determined (though for a concrete\nrealization, see~\\cite{Thesis});\nrather, they must satisfy general constraints enforced\nby the requirements of\nproper UV regularization of the physical $SU(N)$ theory\nand gauge invariance. We will see the effects of the\nlatter shortly. \n\n\n\nGenerally speaking, when we encounter\ncomponents of gauge remainders in complete diagrams,\nwe will either sum over flavours, or the flavour\nwill be obvious, from the context. However, there\nis one place where we will find it useful to \nexplicitly indicate the flavour (and momenta) \nof both $\\rhd$ and $>$.\nThus we introduce the notation\n\\begin{equation}\n\\label{GR:flavour}\n\t\\rhd^{i}_{p}, \\qquad >^{j}_q.\n\\end{equation}\nThe superscript indices take the values 0 or 1, denoting\na (non-null) bosonic or fermionic gauge remainder, \nrespectively. Later, we will find\nit useful to form logical expressions with\nthese indices. \n\n\n\n\\subsubsection{Primary Diagrammatic Identities of the Third Type}\n\nWe recall that the primary diagrammatic identities\nof the third type follow directly from\nthose of the first and second types but are given\nexplicitly nonetheless due to the central role\nthey will play in this paper. The first of\nthese identities\nis simply the classical part of~\\eq{eq:D-ID-GR-TP-A}:\n\\begin{equation}\n\t\\cd{GR-TLTP} = 0.\n\\label{eq:GR-TLTP-A}\n\\end{equation}\n\n\n\nFrom the effective propagator relation and~\\eq{eq:GR-TLTP-A},\ntwo further diagrammatic identities follow.\nFirst, consider attaching\nan effective propagator to the right-hand field in~\\eq{eq:GR-TLTP-A}\nand applying\nthe effective propagator before $\\rhd$ has acted. Diagrammatically,\nthis gives\n\\[\n\t\\cd{GR-TLTP-EP} = 0 = \\cd{k} - \\cd{kkprk},\n\\]\nwhich implies the following diagrammatic identity:\n\\begin{equation}\n\t\\cd{GR-relation} = 1.\n\\label{eq:GR-relation-A}\n\\end{equation}\nNotice that this means that the $f$ and $g$ of\n\\tab{tab:NFE:k,k'} are related by\n\\begin{equation}\n\\label{eq:xf+2g}\n\tx_p f_p + 2g_p = 1,\n\\end{equation}\nwhere we have defined $x_p = p^2 \/ \\Lambda^2$.\n\nThe effective propagator relation, together\nwith~\\eq{eq:GR-relation-A}, implies that\n\\[\n\t\\cd{TLTP-EP-GR} = \\cd{kpr} - \\cd{kprkkpr} = 0.\n\\]\nIn other words, the (non-zero) structure $\\cd{EP-GR}$ kills\na classical, two-point vertex. But, by~\\eq{eq:GR-TLTP-A}, \nthis suggests that the structure $\\cd{EP-GR}$\nmust be equal, up to some factor, to $\\lhd$. Hence, \nthe last of\nthe primary diagrammatic identities is\n\\begin{equation}\n\t\\cd{EP-GRpr} \\equiv \\cd{GR-PEP},\n\\label{eq:PseudoEP-A}\n\\end{equation}\nwhere the dot-dash line represents the pseudo effective propagators \nof~\\cite{Thesis,mgierg1}, also denoted by $\\stackrel{\\sim}{\\Delta}$.\n\n\n\\section*{References}\n\\input{.\/Bibliography\/Biblio}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Main Results}\n\nWe set $ \\Delta = \\partial_{11} + \\partial_{22} $  on open analytic domain $ \\Omega $ of $ {\\mathbb R}^2 $.\n\n\\bigskip\n\nWe consider the following equation:\n\n$$ (P)   \\left \\{ \\begin {split} \n       -\\Delta u & = |x|^{2 \\beta} V e^{v} \\,\\, &\\text{in} \\,\\, & \\Omega  \\subset {\\mathbb R}^2, \\\\\n      - \\Delta v & = W e^{u} \\,\\, &\\text{in} \\,\\, & \\Omega  \\subset {\\mathbb R}^2, \\\\\n                    u & = 0  \\,\\,             & \\text{in} \\,\\,    &\\partial \\Omega,\\\\ \n                    v & = 0  \\,\\,             & \\text{in} \\,\\,    &\\partial \\Omega.                        \n   \\end {split}\\right.\n$$\n\nHere, we assume that:\n\n$$ 0 \\leq V \\leq b_1 < + \\infty, \\,\\, e^u \\in L^1({\\Omega})\\,\\, {\\rm and} \\,\\,  u \\in W_0^{1,1}(\\Omega), $$\n\n$$ 0 \\leq W \\leq b_2 < + \\infty, \\,\\, |x|^{2\\beta} e^v \\in L^1({\\Omega})\\,\\, {\\rm and} \\,\\,  v \\in W_0^{1,1}(\\Omega), $$\n\nand,\n\n$$  0 \\in \\partial \\Omega, \\,\\,\\, \\beta \\geq 0. $$\n \nWhen $ u=v $ and $ \\beta =0 $, the above system is reduced to an equation which was studied by many authors, with or without  the boundary condition, also for Riemann surfaces,  see [1-17],  one can find some existence and compactness results, also for a system.\n\nAmong other results, we  can see in [6] the following important Theorems ($ \\beta=0 $):,\n\n\\smallskip\n\n{\\bf Theorem A.}{\\it (Brezis-Merle [6])}.{\\it  Consider the case of one equation; if $ (u_i)_i=(v_i)_i $ and $ (V_i)_i=(W_i)_i $ are two sequences of functions relatively to the problem $ (P) $ with, $ 0 < a \\leq V_i \\leq b < + \\infty $, then, for all compact set $ K $ of $ \\Omega $,\n\n$$ \\sup_K u_i \\leq c = c(a, b, K, \\Omega). $$}\n\n\\smallskip\n\n{\\bf Theorem B} {\\it (Brezis-Merle [6])}.{\\it Consider the case of one equation and assume that $ (u_i)_i $ and $ (V_i)_i $ are two sequences of functions relatively to the previous problem $ (P) $ with, $ 0 \\leq V_i \\leq b < + \\infty $, and,\n\n$$ \\int_{\\Omega} e^{u_i} dy  \\leq C, $$\n\nthen, for all compact set $ K $ of $ \\Omega $,\n\n$$ \\sup_K u_i \\leq c = c(b, C, K, \\Omega). $$}\n\nNext, we call energy the following quantity:\n\n$$ E= \\int_{\\Omega} e^{u_i} dy. $$\n\nThe boundedness of the energy is a necessary condition to work on the problem $ (P) $ as showed in $ [6] $, by the following counterexample ($\\beta =0$):\n\n\\smallskip\n\n{\\bf Theorem C} {\\it (Brezis-Merle [6])}.{\\it  Consider the case of one equation, then there are two sequences $ (u_i)_i $ and $ (V_i)_i $ of the problem $ (P) $ with, $ 0 \\leq V_i \\leq b < + \\infty $, and,\n\n$$ \\int_{\\Omega} e^{u_i} dy  \\leq C, $$\n\nand\n\n$$ \\sup_{\\Omega}  u_i \\to + \\infty. $$}\n\nWhen $ \\beta = 0 $, the above system have many properties in the constant and the Lipschitzian cases. Indeed we have (when $ \\beta = 0 $):\n\nIn [12], Dupaigne-Farina-Sirakov proved (by an existence result of Montenegro, see [16]) that the solutions of the above system when $ V $ and $ W $ are constants can be extremal and this condition imply the boundedness of the energy and directly the compactness.\nNote that in [11], if we assume (in particular) that $ \\nabla \\log V $ and $ \\nabla \\log W $ and $ V >a >0 $ or $ W > a'>0 $ and $ V, W $ are nonegative and uniformly bounded then the energy is bounded and we have a compactness result.\n\nNote that in the case of one equation (and $ \\beta = 0 $), we can prove by using the Pohozaev identity that if $ + \\infty >b \\geq V \\geq a >0 $, $ \\nabla V $ is uniformely Lipschitzian that the energy is bounded when $ \\Omega $ is starshaped. In [15] Ma-Wei, using the moving-plane method showed that this fact is true for all domain $ \\Omega $ with the same assumptions on $ V $. In [11] De Figueiredo-do O-Ruf extend this fact to a system by using the moving-plane method for a system.\n\n\nTheorem C, shows that we have not a global compactness to the previous problem with one equation, perhaps we need more information on $ V $ to conclude  to the boundedness of the solutions. When $ \\nabla \\log V $ is  Lipschitz function and $ \\beta = 0 $, Chen-Li and Ma-Wei see [7] and [15], showed that we have a compactness on all the open set. The proof is via the moving plane-Method of Serrin and Gidas-Ni-Nirenberg. Note that in [11], we have the same result for this system when $ \\nabla \\log V $ and $ \\nabla \\log W $ are uniformly bounded. We will see below that for a system we also have a compactness result when $ V $ and $ W $ are Lipschitzian and $ \\beta \\geq 0 $.\n\nNow consider the case of one equation. In this case our equation have nice properties.\n\nIf we assume $ V $ with more regularity, we can have another type of estimates, a $ \\sup + \\inf $ type inequalities. It was proved by Shafrir see [17], that, if $ (u_i)_i, (V_i)_i $ are two sequences of functions solutions of the previous equation without assumption on the boundary and, $ 0 < a \\leq V_i \\leq b < + \\infty $, then we have the following interior estimate:\n\n$$ C\\left (\\dfrac{a}{b} \\right ) \\sup_K u_i + \\inf_{\\Omega} u_i \\leq c=c(a, b, K, \\Omega). $$\n\n\\bigskip\n\nNow, if we suppose $ (V_i)_i $ uniformly Lipschitzian with $ A $ the\nLipschitz constant, then, $ C(a\/b)=1 $ and $ c=c(a, b, A, K, \\Omega)\n$, see [5]. \n\\smallskip\n\nHere we are interested by the case of a system of this type of equation. First, we give the behavior of the blow-up points on the boundary, with weight and boundary singularity, and in the second time we have a proof of  compactness of the solutions to Gelfand-Liouville type system with weight and boundary singularity and Lipschitz condition.\n\n\\smallskip\n\nHere, we write an extention of Brezis-Merle Problem (see [6]) to a system:\n\n\\smallskip\n\n{\\bf Problem}. Suppose that $ V_i \\to  V $ and  \n$ W_i \\to  W $ in $ C^0( \\bar \\Omega ) $, with, $ 0 \\leq V_i $ and $ 0 \\leq W_i $. Also, we consider two sequences of solutions $ (u_i), (v_i) $ of $ (P) $ relatively to $ (V_i), (W_i) $ such that,\n\n$$ \\int_{\\Omega} e^{u_i} dx \\leq C_1,\\,\\,\\,  \\int_{\\Omega} |x|^{2\\beta} e^{v_i} dx \\leq C_2,  $$\n\nis it possible to have:\n\n$$ ||u_i||_{L^{\\infty}}\\leq C_3=C_3(\\beta, C_1, C_2, \\Omega) ? $$\n\nand,\n\n$$ ||v_i||_{L^{\\infty}}\\leq C_4=C_4(\\beta, C_1, C_2, \\Omega) ? $$\n\n\nIn this paper we give a caracterization of the behavior of the blow-up points on the boundary and also a proof of the compactness theorem when $ V_i $ and $ W_i $ are uniformly Lipschitzian and $ \\beta \\geq 0 $. For the behavior of the blow-up points on the boundary, the following condition are enough,\n\n$$ 0 \\leq  V_i \\leq b_1, \\,\\,\\, 0 \\leq  W_i \\leq b_2, $$\n\nThe conditions $ V_i \\to  V $ and $ W_i \\to  W $ in $ C^0(\\bar \\Omega) $ are not necessary.\n\n\\bigskip\n\nBut for the proof of the compactness  for the system, we assume that:\n\n$$ ||\\nabla V_i||_{L^{\\infty}}\\leq  A_1, \\,\\,\\, ||\\nabla W_i||_{L^{\\infty}}\\leq  A_2,\\,\\, \\beta \\geq 0. $$\n\nOur main result are:\n\n\\begin{Theorem}  Assume that $ \\max_{\\Omega} u_i \\to +\\infty $ and $ \\max_{\\Omega} v_i \\to +\\infty $ Where $ (u_i) $ and $ (v_i) $ are solutions of the probleme $ (P) $ with ($ \\beta \\geq 0 $), and:\n \n $$ 0 \\leq V_i \\leq b_1,\\,\\,\\, {\\rm and } \\,\\,\\, \\int_{\\Omega}  e^{u_i} dx \\leq C_1, \\,\\,\\, \\forall \\,\\, i, $$\n \nand,\n\n $$ 0 \\leq W_i \\leq b_2,\\,\\,\\, {\\rm and } \\,\\,\\, \\int_{\\Omega}  |x|^{2 \\beta} e^{v_i} dx \\leq C_2, \\,\\,\\, \\forall \\,\\, i, $$\n \n \n then;  after passing to a subsequence, there is a finction $ u $,  there is a number $ N \\in {\\mathbb N} $ and $ N  $ points $ x_1, x_2, \\ldots, x_N \\in  \\partial \\Omega $, such that, \n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} u_i  \\phi  \\to \\int_{\\partial \\Omega} \\partial_{\\nu} u  \\phi +\\sum_{j=1}^N \\alpha_j  \\phi(x_j), \\, \\alpha_j \\geq 4\\pi, $$\n$ \\text{for\\, any}\\,\\, \\phi\\in C^0(\\partial \\Omega) $, \nand,\n\n$$ u_i \\to u \\,\\,\\, {\\rm in }\\,\\,\\, C^1_{loc}(\\bar \\Omega-\\{x_1,\\ldots, x_N \\}). $$\n\n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} v_i  \\phi  \\to \\int_{\\partial \\Omega} \\partial_{\\nu} v  \\phi +\\sum_{j=1}^N \\beta_j  \\phi(x_j), \\, \\beta_j \\geq 4\\pi, $$\n$ \\text{for\\, any}\\,\\, \\phi\\in C^0(\\partial \\Omega) $,  and,\n\n$$ v_i \\to v \\,\\,\\, {\\rm in }\\,\\,\\, C^1_{loc}(\\bar \\Omega-\\{x_1,\\ldots, x_N \\}). $$\n\n\n\\end{Theorem} \n\n In the following theorem, we have a proof for the global a priori estimate which concern the problem $ (P) $.\n\n\\bigskip\n\n\\begin{Theorem}Assume that $ (u_i), (v_i) $ are solutions of $ (P) $ relatively to $ (V_i), (W_i) $ with the following conditions:\n\n$$ x_1=0 \\in \\partial \\Omega, \\,\\,\\beta \\geq 0, $$\n\nand,\n\n$$ 0 \\leq  V_i \\leq b_1, \\,\\,  ||\\nabla V_i||_{L^{\\infty}} \\leq A_1,\\,\\, {\\rm and } \\,\\,\\, \\int_{\\Omega} e^{u_i} \\leq C_1, $$\n\n\n$$ 0 \\leq  W_i \\leq b_2, \\,\\,  ||\\nabla W_i||_{L^{\\infty}} \\leq A_2,\\,\\, {\\rm and } \\,\\,\\, \\int_{\\Omega} |x|^{2\\beta} e^{v_i} \\leq C_2, $$\n\nWe have,\n\n$$  || u_i||_{L^{\\infty}} \\leq C_3(b_1, b_2, \\beta, A_1, A_2, C_1, C_2, \\Omega), $$\n\nand,\n\n$$  || v_i||_{L^{\\infty}} \\leq C_4(b_1, b_2, \\beta, A_1, A_2, C_1, C_2, \\Omega), $$\n\n\n\\end{Theorem} \n\n\\section{Proof of the theorems} \n\n\\bigskip\n\n\\underbar {\\it Proof of theorem 1.1:} \n\n\\bigskip\n\nWe have:\n\n$$ u_i, v_i \\in W_0^{1,1}(\\Omega). $$\n\nSince $ e^{u_i} \\in L^1(\\Omega) $ by the corollary 1 of Brezis-Merle's paper (see [6]) we have $ e^{v_i} \\in L^k(\\Omega) $ for all $ k  >2 $ and the elliptic estimates of Agmon and the Sobolev embedding (see [1]) imply that:\n\n$$ u_i \\in W^{2, k}(\\Omega)\\cap C^{1, \\epsilon}(\\bar \\Omega). $$ \n\nAnd,\n\nWe have:\n\n$$ v_i, u_i \\in W_0^{1,1}(\\Omega). $$\n\nSince $ |x|^{2\\beta} e^{v_i} \\in L^1(\\Omega) $ by the corollary 1 of Brezis-Merle's paper (see [6]) we have $ e^{u_i} \\in L^k(\\Omega) $ for all $ k  >2 $ and the elliptic estimates of Agmon and the Sobolev embedding (see [1]) imply that:\n\n$$ v_i \\in W^{2, k}(\\Omega)\\cap C^{1, \\epsilon}(\\bar \\Omega). $$ \n\n\n\nSince $ |x|^{2\\beta} V_ie^{v_i} $ and $ W_ie^{u_i} $ are bounded in $ L^1(\\Omega) $, we can extract from those two sequences two subsequences which converge to two nonegative measures $ \\mu_1 $ and $ \\mu_2 $. (This procedure is similar to the procedure of Brezis-Merle, we apply corollary 4 of Brezis-Merle paper, see [6]).\n\nIf $ \\mu_1(x_0) < 4 \\pi $, by a Brezis-Merle estimate for the first equation, we have $ e^{u_i} \\in L^{1+\\epsilon} $ around $ x_0 $, by the elliptic estimates, for the second equation, we have $ v_i \\in W^{2, 1+\\epsilon} \\subset L^{\\infty} $ around $ x_0 $, and , returning to the first equation, we have $ u_i \\in L^{\\infty} $ around $ x_0 $.\n\nIf $ \\mu_2(x_0) < 4 \\pi $, then $ u_i $ and $ v_i $ are also locally bounded around $ x_0 $.\n\nThus, we take a look to the case when, $ \\mu_1(x_0) \\geq 4 \\pi $ and $ \\mu_2(x_0) \\geq 4 \\pi $. By our hypothesis, those points $ x_0 $ are finite.\n\nWe will see that inside $ \\Omega $ no such points exist. By contradiction, assume that, we have $ \\mu_1(x_0) \\geq 4 \\pi $. Let us consider a ball $ B_R(x_0) $ which contain only $ x_0 $ as nonregular point. Thus, on $ \\partial B_R(x_0) $, the two sequence $ u_i $ and $ v_i $ are uniformly bounded. Let us consider:\n\n$$ \\left \\{ \\begin {split} \n       -\\Delta z_i & = |x|^{2\\beta}V_i e^{v_i} \\,\\, &\\text{in} \\,\\, & B_R(x_0)  \\subset {\\mathbb R}^2, \\\\\n                       z_i & = 0  \\,\\,             & \\text{in} \\,\\,    &\\partial B_R(x_0).                                            \n   \\end {split}\\right.\n$$\n\nBy the maximum principle we have:\n\n$$ z_i \\leq u_i $$ \n\nand $ z_i \\to z $ almost everywhere on this ball, and thus,\n\n$$ \\int e^{z_i} \\leq \\int e^{u_i} \\leq C, $$\n\nand,\n\n$$ \\int e^z \\leq C.$$\n\nbut, $ z  $ is  a solution in $ W_0^{1,q}(B_R(x_0)) $, $ 1\\leq q <2 $, of the following equation:\n\n\n$$ \\left \\{ \\begin {split} \n       -\\Delta z & = \\mu_1\\,\\, &\\text{in} \\,\\, & B_R(x_0) \\subset {\\mathbb R}^2, \\\\\n                     z & = 0  \\,\\,             & \\text{in} \\,\\,    &\\partial B_R(x_0).                                            \n   \\end {split}\\right.\n$$\n\nwith, $ \\mu_1 \\geq 4 \\pi $ and thus, $ \\mu_1 \\geq 4\\pi \\delta_{x_0} $ and then, by the maximum principle in $ W_0^{1,q}(B_R(x_0)) $:\n\n$$ z \\geq -2 \\log |x-x_0|+ C $$\n\nthus,\n\n$$ \\int e^z = + \\infty, $$\n\nwhich is a contradiction. Thus, there is no nonregular points inside  $ \\Omega $\n\nThus, we consider the case where we have nonregular points on the boundary, we use two estimates:\n\n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} u_i d\\sigma \\leq C_1,\\,\\,\\, \\int_{\\partial \\Omega} \\partial_{\\nu} v_i d\\sigma \\leq C_2, $$\n \n and,\n \n $$ ||\\nabla u_i||_{L^q} \\leq C_q, \\,\\,\\, ||\\nabla v_i||_{L^q} \\leq C'_q, \\,\\,\\forall \\,\\, i\\,\\, {\\rm and  }  \\,\\, 1< q < 2. $$\n\nWe have the same computations, as in the case of one equation.\n\nWe consider a points $ x_0 \\in \\partial \\Omega $ such that:\n\n$$ \\mu_1(x_0) < 4 \\pi. $$\n\nWe consider a test function on the boundary $ \\eta $ we extend $ \\eta $ by a harmonic function on $ \\Omega $, we write the equation:\n\n$$ -\\Delta ((u_i-u)\\eta) =|x|^{2\\beta}(V_i e^{v_i}-Ve^v)\\eta+ <\\nabla (u_i-u)|\\nabla \\eta> = f_i $$\n\nwith,\n\n$$ \\int |f_i| \\leq 4 \\pi-\\epsilon +o(1) < 4\\pi-2\\epsilon <4\\pi, $$\n\n$$ -\\Delta ((v_i-v)\\eta) =(W_i e^{u_i}-We^u)\\eta+ <\\nabla (v_i-v)|\\nabla \\eta> = g_i, $$\n\nwith,\n\n$$ \\int |g_i| \\leq 4 \\pi-\\epsilon +o(1) < 4\\pi-2\\epsilon <4\\pi, $$\n\nBy the Brezis-Merle estimate, we have uniformly, $ e^{u_i} \\in L^{1+\\epsilon} $ around $ x_0 $, by the elliptic estimates, for the second equation,  we have $ v_i \\in W^{2, 1+\\epsilon} \\subset L^{\\infty} $ around $ x_0 $, and , returning to the first equation, we have $ u_i \\in L^{\\infty} $ around $ x_0 $.\n\n\nWe have the same thing if we assume:\n\n$$ \\mu_2(x_0) < 4 \\pi. $$\n\nThus, if $ \\mu_1(x_0) < 4 \\pi $ or $ \\mu_2(x_0) < 4 \\pi $, we have for $ R >0 $ small enough:\n\n$$ (u_i,v_i) \\in L^{\\infty}(B_R(x_0)\\cap \\bar \\Omega). $$\n\nBy our hypothesis the set of the points such that:\n\n$$ \\mu_1(x_0)  \\geq  4 \\pi,  \\,\\,\\, \\mu_2(x_0)   \\geq 4 \\pi, $$\n\nis finite, and, outside this set $ u_i $ and $ v_i $ are locally uniformly bounded. By the elliptic estimates, we have the $ C^1 $ convergence to  $ u $ and $ v $  on each compact set of $ \\bar \\Omega- \\{x_1, \\ldots x_N\\} $.\n\nIndeed,\n\nBy the Stokes formula we have, \n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} u_i d\\sigma \\leq C, $$\n\nWe use the weak convergence in the space of Radon measures to have the existence of a nonnegative Radon measure $ \\mu_1 $ such that,\n\n$$ \\int_{\\partial \\Omega} \\partial_{\\nu} u_i \\phi  d\\sigma \\to \\mu_1(\\phi), \\,\\,\\, \\forall \\,\\,\\, \\phi \\in C^0(\\partial \\Omega). $$\n\nWe take an $ x_0 \\in \\partial \\Omega $ such that, $ \\mu_1({x_0}) < 4\\pi $. For $ \\epsilon >0 $ small enough set $ I_{\\epsilon}= B(x_0, \\epsilon)\\cap \\partial \\Omega $ on the unt disk or one can assume it as an interval. We choose a function $ \\eta_{\\epsilon} $ such that,\n\n$$ \\begin{cases}\n    \n\\eta_{\\epsilon} \\equiv 1,\\,\\,\\,  {\\rm on } \\,\\,\\,  I_{\\epsilon}, \\,\\,\\, 0 < \\epsilon < \\delta\/2,\\\\\n\n\\eta_{\\epsilon} \\equiv 0,\\,\\,\\, {\\rm outside} \\,\\,\\, I_{2\\epsilon }, \\\\\n\n0 \\leq \\eta_{\\epsilon} \\leq 1, \\\\\n\n||\\nabla \\eta_{\\epsilon}||_{L^{\\infty}(I_{2\\epsilon})} \\leq \\dfrac{C_0(\\Omega, x_0)}{\\epsilon}.\n\n\\end{cases} $$\n\nWe take a $\\tilde \\eta_{\\epsilon} $ such that,\n\n\\begin{displaymath}  \\left \\{ \\begin {split} \n      -\\Delta \\tilde \\eta_{\\epsilon}  & = 0              \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  \\tilde\\eta_{\\epsilon} & =  \\eta_{\\epsilon}   \\,\\,             && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\n{\\bf Remark:} We use the following steps in the construction of $ \\tilde \\eta_{\\epsilon} $:\n\n\nWe take a cutoff function $ \\eta_{0} $ in $ B(0, 2) $ or $ B(x_0, 2) $:\n\n1- We set $ \\eta_{\\epsilon}(x)= \\eta_0(|x-x_0|\/\\epsilon) $ in the case of the unit disk it is sufficient.\n\n2- Or, in the general case: we use a chart $ (f, \\tilde \\Omega) $ with $ f(0)=x_0 $ and we take $ \\mu_{\\epsilon}(x)= \\eta_0 ( f( |x|\/ \\epsilon)) $ to have  connected  sets $ I_{\\epsilon} $ and we take $ \\eta_{\\epsilon}(y)= \\mu_{\\epsilon}(f^{-1}(y))$. Because $ f, f^{-1} $ are Lipschitz, $ |f(x)-x_0| \\leq k_ 2|x|\\leq 1 $ for $ |x| \\leq 1\/k_2 $ and $ |f(x)-x_0| \\geq k_ 1|x|\\geq 2 $ for $ |x| \\geq 2\/k_1>1\/k_2 $, the support  of $ \\eta $ is in $ I_{(2\/k_1)\\epsilon} $.\n\n$$ \\begin{cases}\n    \n\\eta_{\\epsilon} \\equiv 1,\\,\\,\\,  {\\rm on } \\,\\,\\,  f(I_{(1\/k_2)\\epsilon}), \\,\\,\\, 0 < \\epsilon < \\delta\/2,\\\\\n\n\\eta_{\\epsilon} \\equiv 0,\\,\\,\\, {\\rm outside} \\,\\,\\, f(I_{(2\/k_1)\\epsilon }), \\\\\n\n0 \\leq \\eta_{\\epsilon} \\leq 1, \\\\\n\n||\\nabla \\eta_{\\epsilon}||_{L^{\\infty}(I_{(2\/k_1)\\epsilon})} \\leq \\dfrac{C_0(\\Omega, x_0)}{\\epsilon}.\n\n\\end{cases} $$\n\n3- Also, we can take: $ \\mu_{\\epsilon}(x)= \\eta_0(|x|\/\\epsilon) $ and $ \\eta_{\\epsilon}(y)= \\mu_{\\epsilon}(f^{-1}(y)) $, we extend it by $ 0 $ outside $ f(B_1(0)) $.  We have $ f(B_1(0)) = D_1(x_0) $, $ f (B_{\\epsilon}(0))= D_{\\epsilon}(x_0) $ and $ f(B_{\\epsilon}^+)= D_{\\epsilon}^+(x_0) $ with $ f $ and $ f^{-1} $ smooth diffeomorphism.\n\n\n\n$$ \\begin{cases}\n    \n\\eta_{\\epsilon} \\equiv 1,\\,\\,\\,  {\\rm on \\, a \\, the \\, connected \\, set } \\,\\,\\,  J_{\\epsilon} =f(I_{\\epsilon}), \\,\\,\\, 0 < \\epsilon < \\delta\/2,\\\\\n\n\\eta_{\\epsilon} \\equiv 0,\\,\\,\\, {\\rm outside} \\,\\,\\, J'_{\\epsilon}=f(I_{2\\epsilon }), \\\\\n\n0 \\leq \\eta_{\\epsilon} \\leq 1, \\\\\n\n||\\nabla \\eta_{\\epsilon}||_{L^{\\infty}(J'_{\\epsilon})} \\leq \\dfrac{C_0(\\Omega, x_0)}{\\epsilon}.\n\n\\end{cases} $$\n\nAnd, $ H_1(J'_{\\epsilon}) \\leq C_1 H_1(I_{2\\epsilon}) = C_1 4\\epsilon $, since $ f $ is Lipschitz. Here $ H_1 $ is the Hausdorff measure.\n\nWe solve the Dirichlet Problem:\n\n\\begin{displaymath}  \\left \\{ \\begin {split} \n      -\\Delta \\bar \\eta_{\\epsilon}  & = -\\Delta \\eta_{\\epsilon}              \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  \\bar \\eta_{\\epsilon} & = 0   \\,\\,             && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\nand finaly we set $ \\tilde \\eta_{\\epsilon} =-\\bar \\eta_{\\epsilon} + \\eta_{\\epsilon} $. Also, by the maximum principle and the elliptic estimates we have :\n\n$$ ||\\nabla \\tilde \\eta_{\\epsilon}||_{L^{\\infty}} \\leq C(|| \\eta_{\\epsilon}||_{L^{\\infty}} +||\\nabla \\eta_{\\epsilon}||_{L^{\\infty}} + ||\\Delta \\eta_{\\epsilon}||_{L^{\\infty}}) \\leq \\dfrac{C_1}{\\epsilon^2}, $$\n\nwith $ C_1 $ depends on $ \\Omega $.\n\n\nWe use the following estimate, see [8],\n\n$$ ||\\nabla v_i||_{L^q}\\leq C_q,\\,\\, ||\\nabla u_i||_q \\leq C_q, \\,\\,\\forall \\,\\, i\\,\\, {\\rm and  }  \\,\\, 1< q < 2. $$\n\nWe deduce from the last estimate that, $ (v_i) $ converge weakly in $ W_0^{1, q}(\\Omega) $, almost everywhere to a function $ v \\geq 0 $ and $ \\int_{\\Omega} |x|^{2\\beta} e^v < + \\infty $ (by Fatou lemma). Also, $ V_i $ weakly converge to a nonnegative function $ V $ in $ L^{\\infty} $. \n\nWe deduce from the last estimate that, $ (u_i) $ converge weakly in $ W_0^{1, q}(\\Omega) $, almost everywhere to a function $ u \\geq 0 $ and $ \\int_{\\Omega} e^u < + \\infty $ (by Fatou lemma). Also, $ W_i $ weakly converge to a nonnegative function $ W $ in $ L^{\\infty} $.\n\n\n\nThe function $ u, v $ are in $ W_0^{1, q}(\\Omega) $ solutions of :\n\n\\begin{displaymath} \\left \\{ \\begin {split} \n      -\\Delta u  & = |x|^{2 \\beta} V e^{v} \\in L^1(\\Omega)    \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  u  & = 0  \\,\\,                                     && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n \nAnd,\n\n\\begin{displaymath} \\left \\{ \\begin {split} \n      -\\Delta v  & = W e^{u} \\in L^1(\\Omega)    \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  v  & = 0  \\,\\,                                     && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\n According to the corollary 1 of Brezis-Merle's result, see [6],   we have $ e^{k u }\\in L^1(\\Omega), k >1 $. By the elliptic estimates, we have $ v \\in C^1(\\bar \\Omega) $.\n\nAccording to the corollary 1 of Brezis-Merle's result, see [6],   we have $ e^{k v }\\in L^1(\\Omega), k >1 $. By the elliptic estimates, we have $ u \\in C^1(\\bar \\Omega) $.\n\n\nFor two vectors $ f $ and $ g $ we denote by $  f \\cdot g $ the inner product of $ f $ and $ g $. \n\nWe can write:\n\n\\be -\\Delta ((u_i-u) \\tilde \\eta_{\\epsilon})= |x|^{2 \\beta}(V_i e^{v_i} -Ve^v)\\tilde \\eta_{\\epsilon} -2\\nabla (u_i- u)\\cdot \\nabla \\tilde \\eta_{\\epsilon}. \\label{(1)}\\ee\n\n\n$$ -\\Delta ((v_i-v) \\tilde \\eta_{\\epsilon})= (W_i e^{u_i} -We^u)\\tilde \\eta_{\\epsilon} -2\\nabla (v_i- v)\\cdot \\nabla \\tilde \\eta_{\\epsilon}. $$\n\nWe use the interior esimate of Brezis-Merle, see [6],\n\n\\bigskip\n\n\\underbar {\\it Step 1:} Estimate of the integral of the first term of the right hand side of $ (\\ref{(1)}) $.\n\n\\bigskip\n\nWe use the Green formula between $ \\tilde \\eta_{\\epsilon} $ and $ u $, we obtain,\n\n\\be  \\int_{\\Omega} |x|^{2 \\beta}Ve^v \\tilde \\eta_{\\epsilon} dx =\\int_{\\partial \\Omega} \\partial_{\\nu} u \\eta_{\\epsilon} \\leq C'\\epsilon ||\\partial_{\\nu}u||_{L^{\\infty}}= C \\epsilon \\label{(2)}\\ee\n\nWe have,\n\n\\begin{displaymath} \\left \\{ \\begin {split} \n      -\\Delta u_i  & = |x|^{2 \\beta} V_i e^{v_i}                      \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                  u_i  & = 0  \\,\\,                                     && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\nWe use the Green formula between $ u_i $ and $ \\tilde \\eta_{\\epsilon} $ to have:\n\n\\be \\int_{\\Omega} |x|^{2 \\beta} V_i e^{v_i} \\tilde \\eta_{\\epsilon} dx = \\int_{\\partial \\Omega} \\partial_{\\nu} u_i \\eta_{\\epsilon} d\\sigma \\to \\mu_1(\\eta_{\\epsilon}) \\leq \\mu_1(J'_{\\epsilon}) \\leq 4  \\pi - \\epsilon_0, \\,\\,\\, \\epsilon_0 >0 \\label{(3)}\\ee\n\nFrom $ (\\ref{(2)}) $ and $ (\\ref{(3)}) $ we have for all $ \\epsilon >0 $ there is $ i_0 =i_0(\\epsilon) $ such that, for $ i \\geq i_0 $,\n\n\n\\be \\int_{\\Omega} ||x|^{2 \\beta}(V_ie^{v_i}-Ve^v) \\tilde \\eta_{\\epsilon}| dx \\leq 4 \\pi -\\epsilon_0 + C \\epsilon \\label{(4)}\\ee\n\n\\bigskip\n\n\\underbar {\\it Step 2:} Estimate of integral of the second term of the right hand side of $ (\\ref{(1)}) $.\n\n\\bigskip\n\nLet $ \\Sigma_{\\epsilon} = \\{ x \\in \\Omega, d(x, \\partial \\Omega) = \\epsilon^3 \\} $ and $ \\Omega_{\\epsilon^3} = \\{ x \\in \\Omega, d(x, \\partial \\Omega) \\geq \\epsilon^3 \\} $, $ \\epsilon > 0 $. Then, for $ \\epsilon $ small enough, $ \\Sigma_{\\epsilon} $ is hypersurface.\n\nThe measure of $ \\Omega-\\Omega_{\\epsilon^3} $ is $ k_2\\epsilon^3 \\leq meas(\\Omega-\\Omega_{\\epsilon^3})= \\mu_L (\\Omega-\\Omega_{\\epsilon^3}) \\leq k_1 \\epsilon^3 $.\n\n{\\bf Remark}: for the unit ball $ {\\bar B(0,1)} $, our new manifold is $ {\\bar B(0, 1-\\epsilon^3)} $.\n\n( Proof of this fact; let's consider $ d(x, \\partial \\Omega) = d(x, z_0), z_0 \\in \\partial \\Omega $, this imply that $ (d(x,z_0))^2 \\leq (d(x, z))^2 $ for all $ z \\in \\partial \\Omega $ which it is equivalent to $ (z-z_0) \\cdot (2x-z-z_0) \\leq 0 $ for all $ z \\in \\partial \\Omega $, let's consider a chart around $ z_0 $ and $ \\gamma (t) $ a curve in $ \\partial \\Omega $, we have;\n\n$ (\\gamma (t)-\\gamma(t_0) \\cdot (2x-\\gamma(t)-\\gamma(t_0)) \\leq 0 $ if we divide by $ (t-t_0) $ (with the sign and tend $ t $ to $ t_0 $), we have $ \\gamma'(t_0) \\cdot (x-\\gamma(t_0)) = 0 $, this imply that $ x= z_0-s \\nu_0 $ where $ \\nu_0 $ is the outward normal of $ \\partial \\Omega $ at $ z_0 $))\n\nWith this fact, we can say that $ S= \\{ x, d(x, \\partial \\Omega) \\leq \\epsilon \\}= \\{ x= z_0- s \\nu_{z_0}, z_0 \\in \\partial \\Omega, \\,\\, -\\epsilon \\leq s \\leq \\epsilon \\} $. It  is sufficient to work on  $ \\partial \\Omega $. Let's consider a charts $ (z, D=B(z, 4 \\epsilon_z), \\gamma_z) $ with $ z \\in \\partial \\Omega $ such that $ \\cup_z B(z, \\epsilon_z) $ is  cover of $ \\partial \\Omega $ .  One can extract a finite cover $ (B(z_k, \\epsilon_k)), k =1, ..., m $, by the area formula the measure of $ S \\cap B(z_k, \\epsilon_k) $ is less than a $ k\\epsilon $ (a $ \\epsilon $-rectangle).  For the reverse inequality, it is sufficient to consider one chart around one point of the boundary.\n\n\nWe write,\n\n\\be \\int_{\\Omega} |\\nabla ( u_i -u) \\cdot \\nabla \\tilde \\eta_{\\epsilon} | dx =\n\\int_{\\Omega_{\\epsilon^3}} |\\nabla (u_i -u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx + \\int_{\\Omega - \\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx.  \\label{(5)}\\ee\n\n\\bigskip\n\n\\underbar {\\it Step 2.1:} Estimate of $ \\int_{\\Omega - \\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx $.\n\n\\bigskip\n\nFirst, we know from the elliptic estimates that  $ ||\\nabla \\tilde \\eta_{\\epsilon}||_{L^{\\infty}} \\leq C_1 \/\\epsilon^2 $, $ C_1 $ depends on $ \\Omega $\n\n\nWe know that $ (|\\nabla u_i|)_i $ is bounded in $ L^q, 1< q < 2 $, we can extract  from this sequence a subsequence which converge weakly to $ h \\in L^q $. But, we know that we have locally the uniform convergence to $ |\\nabla u| $ (by Brezis-Merle's theorem), then, $ h= |\\nabla u| $ a.e. Let $ q' $ be the conjugate of $ q $.\n\n\nWe have, $  \\forall f \\in L^{q'}(\\Omega)$\n\n$$ \\int_{\\Omega} |\\nabla u_i| f dx \\to \\int_{\\Omega} |\\nabla u| f dx $$\n\n\nIf we take $ f= 1_{\\Omega-\\Omega_{\\epsilon^3}} $, we have:\n\n$$ {\\rm for } \\,\\, \\epsilon >0 \\,\\, \\exists \\,\\, i_1 = i_1(\\epsilon) \\in {\\mathbb N}, \\,\\,\\, i \\geq  i_1,  \\,\\, \\int_{\\Omega-\\Omega_{\\epsilon^3}} |\\nabla u_i| \\leq \\int_{\\Omega-\\Omega_{\\epsilon^3}} |\\nabla u| + \\epsilon^3. $$\n\nThen, for $ i \\geq i_1(\\epsilon) $,\n\n$$ \\int_{\\Omega-\\Omega_{\\epsilon^3}} |\\nabla u_i| \\leq meas(\\Omega-\\Omega_{\\epsilon^3}) ||\\nabla u||_{L^{\\infty}} + \\epsilon^3 = \\epsilon^3(k_1 ||\\nabla u||_{L^{\\infty}} + 1). $$\n\nThus, we obtain,\n\n\\be \\int_{\\Omega - \\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx \\leq  \\epsilon C_1(2 k_1 ||\\nabla u||_{L^{\\infty}} + 1) \\label{(6)}\\ee\n\nThe constant $ C_1 $ does  not depend on $ \\epsilon $ but on $ \\Omega $.\n\n\\bigskip\n\n\\underbar {\\it Step 2.2:} Estimate of $ \\int_{\\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx $.\n\n\\bigskip\n\nWe know that, $ \\Omega_{\\epsilon} \\subset \\subset \\Omega $, and ( because of Brezis-Merle's interior estimates) $ u_i \\to u $ in $ C^1(\\Omega_{\\epsilon^3}) $. We have,\n\n$$ ||\\nabla (u_i-u)||_{L^{\\infty}(\\Omega_{\\epsilon^3})} \\leq \\epsilon^3,\\, {\\rm for } \\,\\, i \\geq i_3 = i_3(\\epsilon). $$\n\nWe write,\n \n$$ \\int_{\\Omega_{\\epsilon^3}} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx \\leq ||\\nabla (u_i-u)||_{L^{\\infty}(\\Omega_{\\epsilon^3})} ||\\nabla \\tilde \\eta_{ \\epsilon}||_{L^{\\infty}} \\leq C_1 \\epsilon \\,\\, {\\rm for } \\,\\, i \\geq i_3, $$\n\n\nFor $ \\epsilon >0 $, we have for $ i \\in {\\mathbb N} $, $ i \\geq \\max \\{i_1, i_2, i_3 \\} $,\n\n\\be \\int_{\\Omega} |\\nabla (u_i-u) \\cdot \\nabla \\tilde \\eta_{\\epsilon}| dx \\leq \\epsilon C_1(2 k_1 ||\\nabla u||_{L^{\\infty}} + 2) \\label{(7)}\\ee\n\nFrom $ (\\ref{(4)}) $ and $ (\\ref{(7)}) $, we have, for $ \\epsilon >0 $, there is $ i_3= i_3(\\epsilon) \\in {\\mathbb N}, i_3 = \\max \\{ i_0, i_1, i_2 \\} $ such that,\n\n\n\\be \\int_{\\Omega} |-\\Delta [(u_i-u)\\tilde \\eta_{\\epsilon}]|dx \\leq 4 \\pi-\\epsilon_0+  \\epsilon 2 C_1(2 k_1 ||\\nabla u||_{L^{\\infty}} + 2 + C) \\label{(8)}\\ee\n\nWe choose $ \\epsilon >0 $ small enough to have a good estimate of  $ (\\ref{(1)}) $.\n\nIndeed, we have:\n\n\\begin{displaymath} \\left \\{ \\begin {split} \n      -\\Delta [(u_i-u) \\tilde \\eta_{\\epsilon}]   & = g_{i,\\epsilon}                   \\,\\, &&\\text{in} \\!\\!&&\\Omega \\subset {\\mathbb R}^2, \\\\\n                 (u_i-u) \\tilde \\eta_{\\epsilon}    & = 0  \\,\\,                                     && \\text{in} \\!\\!&&\\partial \\Omega.               \n\\end {split}\\right.\n\\end{displaymath}\n\nwith $ ||g_{i, \\epsilon} ||_{L^1(\\Omega)} \\leq 4 \\pi -\\dfrac{\\epsilon_0}{2}. $\n\nWe can use Theorem 1 of [6] to conclude that there are $ q \\geq \\tilde q >1 $ such that:\n\n$$ \\int_{V_{\\epsilon}(x_0)} e^{\\tilde q |u_i-u|} dx \\leq \\int_{\\Omega} e^{q|u_i -u| \\tilde \\eta_{\\epsilon}} dx \\leq C(\\epsilon,\\Omega). $$\n \nwhere, $ V_{\\epsilon}(x_0) $ is a neighberhood of $ x_0 $ in $ \\bar \\Omega $. Here we have used that in a neighborhood of $ x_0 $  by the elliptic estimates, \n$ 1- C \\epsilon \\leq \\tilde \\eta_{\\epsilon} \\leq 1 $.\n\nThus, for each $ x_0 \\in \\partial \\Omega - \\{ \\bar x_1,\\ldots, \\bar x_m \\} $ there is $ \\epsilon_{x_0} >0, q_{x_0} > 1 $ such that:\n\n\\be \\int_{B(x_0, \\epsilon_{x_0})} e^{q_{x_0} u_i} dx \\leq C, \\,\\,\\, \\forall \\,\\,\\, i. \\label{(9)}\\ee\n\nNow, we consider a cutoff function $ \\eta \\in C^{\\infty}({\\mathbb R}^2) $ such that\n\n$$ \\eta \\equiv 1 \\,\\,\\, {\\rm on } \\,\\,\\, B(x_0, \\epsilon_{x_0}\/2) \\,\\,\\, {\\rm and } \\,\\,\\, \\eta \\equiv 0 \\,\\,\\, {\\rm on } \\,\\,\\, {\\mathbb R}^2 -B(x_0, 2\\epsilon_{x_0}\/3). $$\n\nWe write\n\n$$ -\\Delta (v_i \\eta) = W_i e^{u_i} \\eta - 2 \\nabla v_i \\cdot \\nabla \\eta  - v_i \\Delta \\eta. $$\n\nBecause, by Poincar\\'e and Gagliardo-Nirenberg-Sobolev inequalities:\n\n$$ ||v_i||_{q^*} \\leq c_q ||\\nabla v_i||_q \\leq  C_q, \\,\\, 1\\leq q <2, $$\n\nwith, $ q^*=2q\/(2-q) >2 >1 $.\n\n\\smallskip\n\nBy the elliptic estimates, $ (v_i)_i $ is uniformly bounded in $ L^{\\infty}(V_{\\epsilon}(x_0)) $. Finaly, we have, for some $ \\epsilon > 0 $ small enough,\n\n$$ || v_i||_{C^{0,\\theta}[B(x_0, \\epsilon)]} \\leq c_3 \\,\\,\\, \\forall \\,\\,\\, i. $$\n\nNow, we consider a cutoff function $ \\eta \\in C^{\\infty}({\\mathbb R}^2) $ such that\n\n$$ \\eta \\equiv 1 \\,\\,\\, {\\rm on } \\,\\,\\, B(x_0, \\epsilon_{x_0}\/2) \\,\\,\\, {\\rm and } \\,\\,\\, \\eta \\equiv 0 \\,\\,\\, {\\rm on } \\,\\,\\, {\\mathbb R}^2 -B(x_0, 2\\epsilon_{x_0}\/3). $$\n\nWe write\n\n$$ -\\Delta (u_i \\eta) = |x|^{2\\beta} V_i e^{v_i} \\eta - 2 \\nabla u_i \\cdot \\nabla \\eta  - u_i \\Delta \\eta. $$\n\nBy the elliptic estimates, $ (u_i)_i $ is uniformly bounded in $ L^{\\infty}(V_{\\epsilon}(x_0)) $ and also in $ C^{0,\\theta} $ norm.\n\nIf we repeat this procedure another time, we have a boundedness of $ (u_i)_i $ and $ (v_i)_i $ in the $ C^{1,\\theta} $ norm, because they are bounded in $ W^{2,q}\\subset W^{1,q^*} $ norms with $ 2q\/(2-q)=q^* >2 $.\n\nWe have the same computations and conclusion if we consider a regular point $ x_0 $ for the measure $ \\mu_2 $.\n\n\nWe have proved that, there is a finite number of points $ \\bar x_1, \\ldots, \\bar x_m $ such that the squence $ (u_i)_i  $ and $ (v_i)_i $ are locally uniformly bounded (in $ C^{1,\\theta}, \\theta >0 $) in $ \\bar \\Omega - \\{ \\bar x_1, \\ldots , \\bar x_m \\} $.\n\n\\bigskip\n\n\\underbar {\\it Proof of theorem 1.2:} \n\nWithout loss of generality, we can assume that $ 0=x_1 $ is a blow-up point. Since the boundary is an analytic curve $ \\gamma_1(t) $, there is a neighborhood of  $ 0 $ such that the curve $ \\gamma_1 $ can be extend to a holomorphic map such that $ \\gamma_1'(0) \\not = 0 $ (series) and by the inverse mapping one can assume that this map is univalent around $ 0 $. In the case when the boundary is a simple Jordan curve the domain is simply connected. In the case that the domains has a finite number of holes it is conformally equivalent to a disk with a finite number of disks removed. Here we consider a general domain. Without loss of generality one can assume that $ \\gamma_1 (B_1^+) \\subset \\Omega $ and also $ \\gamma_1 (B_1^-) \\subset (\\bar \\Omega)^c $ and $ \\gamma_1 (-1,1) \\subset \\partial \\Omega $ and $ \\gamma_1 $ is univalent. This means that $ (B_1, \\gamma_1) $ is a local chart around $ 0 $ for $ \\Omega $ and $ \\gamma_1 $ univalent. (This fact holds if we assume that we have an analytic domain, (below a graph of an analytic function), we have necessary the condition $ \\partial \\bar \\Omega = \\partial \\Omega $ and the graph is analytic, in this case $ \\gamma_1 (t)= (t, \\phi(t)) $ with $ \\phi $ real analytic and an example of this fact is the unit disk  around the point $ (0,1) $ for example).\n\nBy this conformal transformation, we can assume that $ \\Omega =B_1^+ $, the half ball, and $ \\partial^+ B_1^+ $ is the exterior part, a part which not contain $ 0 $ and on which  $ u_i $ converge in the $ C^1 $ norm to $ u $. Let us consider $ B_{\\epsilon}^+ $, the half ball with radius $ \\epsilon >0 $. Also, one can consider a $ C^1 $ domain (a rectangle between two half disks) and by charts its image is a $ C^1 $ domain)\nWe know that:\n\n$$ u_i \\in C^{2, \\epsilon}(\\bar \\Omega). $$ \n\nThus we can use integrations by parts (Stokes formula). The Pohozaev identity applied around the blow-up $ 0 $:\n\n\\be  \\int_{B_{\\epsilon}^+} \\Delta u_i < x |\\nabla v_i > dx = -  \\int_{B_{\\epsilon}^+} \\Delta v_i < x |\\nabla u_i > dx + \\int_{\\partial^+ B_{\\epsilon}^+}  g(\\nabla u_i, \\nabla v_i)d\\sigma, \\label{(10)}\\ee\n\nThus,\n\n\n\\be  \\int_{B_{\\epsilon}^+} |x|^{2 \\beta} V_i e^{v_i}< x |\\nabla v_i > dx = -  \\int_{B_{\\epsilon}^+} W_ie^{u_i} < x |\\nabla u_i > dx - \\int_{\\partial^+ B_{\\epsilon}^+}  g(\\nabla u_i, \\nabla v_i)d\\sigma, \\label{(11)}\\ee\n\nAfter integration by parts, we obtain:\n\n\n$$  \\int_{B_{\\epsilon}^+} 2V_i(1+\\beta) |x|^{2 \\beta} e^{v_i} dx +  \\int_{B_{\\epsilon}^+} < x |\\nabla V_i > |x|^{2 \\beta} e^{v_i} dx+  \\int_{\\partial B_{\\epsilon}^+} < \\nu |x >|x|^{2 \\beta} V_i e^{v_i} d\\sigma+ $$\n\n$$ + \\int_{B_{\\epsilon}^+} W_i e^{u_i} dx +  \\int_{B_{\\epsilon}^+} < x |\\nabla W_i > e^{u_i} dx+  \\int_{\\partial B_{\\epsilon}^+} < \\nu | x > W_i e^{u_i}  d\\sigma = $$\n\n$$ = - \\int_{\\partial^+ B_{\\epsilon}^+} g(\\nabla u_i, \\nabla v_i)d\\sigma, \\label{(10)} $$ \n\nAlso, for $ u $ and $ v $, we have:\n\n\n$$  \\int_{B_{\\epsilon}^+} 2V(1+ \\beta) |x|^{2 \\beta} e^{v} dx +  \\int_{B_{\\epsilon}^+} < x |\\nabla V > |x|^{2 \\beta} e^{v} dx+  \\int_{\\partial B_{\\epsilon}^+} < \\nu | x >|x|^{2 \\beta} V e^{v}  d\\sigma+ $$\n\n$$ + \\int_{B_{\\epsilon}^+} We^{u} dx +  \\int_{B_{\\epsilon}^+} < x |\\nabla W> e^{u} dx+  \\int_{\\partial B_{\\epsilon}^+} < \\nu | x > W e^{u} d\\sigma = $$\n\n$$ = - \\int_{\\partial^+ B_{\\epsilon}^+} g(\\nabla u, \\nabla v)d\\sigma, $$ \n\nIf, we take the difference, we obtain:\n\n$$ \\int_{\\gamma_1 (B_{\\epsilon}^+)} |x|^{2 \\beta} V_i e^{v_i} dx + \\int_{\\gamma_1 (B_{\\epsilon}^+)} W_i e^{u_i} dx  = o(\\epsilon)+o(1) $$\n\nBut,\n\n $$ \\int_{\\gamma_1 (B_{\\epsilon}^+)} |x|^{2 \\beta} V_i e^{v_i} dx + \\int_{\\gamma_1 (B_{\\epsilon}^+)} W_i e^{u_i} dx  = \\int_{\\partial \\gamma_1 (B_{\\epsilon}^+)} \\partial_{\\nu} u_i d\\sigma + \\int_{\\partial \\gamma_1 (B_{\\epsilon}^+)} \\partial_{\\nu} v_i d\\sigma $$\n\nand,\n\n\\be \\int_{\\partial \\gamma_1 (B_{\\epsilon}^+)} \\partial_{\\nu} u_i d\\sigma + \\int_{\\partial \\gamma_1 (B_{\\epsilon}^+)} \\partial_{\\nu} v_i d\\sigma \n\\to \\alpha_1+\\beta_1 >0 \\label{(12)}\\ee\n\na contradiction.\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{sec:Intro}}\n\n\nThe presence of additional light scalar degrees of freedom is a common feature of many \nextensions of the Standard Model (SM).  Examples include the \nQCD axion~\\cite{PecceiQuinn1,PecceiQuinn2,WeinbergAxion,WilczekAxion}, a more\ngeneral set of axions and axion-like particles~\\cite{LowEnergyFrontier}, \nmajorons~\\cite{ChikashigeMajorons,GelminiMajorons}, familons~\\cite{WilczekFamilon}, \nchameleons~\\cite{RatraChameleons,KhouryChameleons1,KhouryChameleons2,BraxChameleons},\nbranons~\\cite{DobadoBranon}, dilatons, and a variety of other string and geometric \nmoduli~\\cite{Polchinski,BBS,GSW}.  Scalars of this sort are typically \nlight due to symmetries of the high-scale theory which prevent them from acquiring masses.\nAs a result, masses for these particles must be generated by dynamical processes at lower scales \nwhich break these symmetries --- often in conjunction with a cosmological\nphase transition.  This dynamics also generically leads to a non-trivial time-dependence for \nthe masses of these fields, the details of which can have a significant impact on  \ntheir late-time energy densities.  Understanding these effects is crucial, since    \nsuch particles can potentially have a variety of phenomenological and cosmological \nconsequences, both helpful and harmful.  Indeed, scalars of this sort can \ncontribute to the total abundances of dark matter or dark energy; induce additional, \nlate periods of reheating after cosmic inflation, potentially resulting in substantially \nmodified cosmologies~\\cite{WatsonNonThermalHistories};  \nprematurely induce matter-domination and\/or overclose the universe; \nand disrupt the formation of light elements through late decays. \n\nWhile the detailed dynamics of mass generation can have important consequences\neven in scenarios involving only a single light scalar, the situation becomes far \nricher in scenarios involving multiple such fields.  \nIndeed, in cases in which these scalars have the same quantum numbers, the \ndynamics of mass generation can lead them to experience time-dependent mixing.\nThis mixing in turn leads to a \ncontinual redistribution of energy density among the scalars during the mass-generation\nepoch.  Such a redistribution can also have profound effects on the late-time energy densities\nof these fields, several of which were pointed out in Ref.~\\cite{TwoTimescales}.  Specifically,\neven in a theory involving only two scalars, such a redistribution can lead their late time \nabundances to experience either an enhancement or a suppression,  depending on\nthe choice of model parameters.  Moreover, in certain regimes, the system may experience\na parametric resonance which can have further dramatic effects on these abundances.\nFinally, the system may also experience a ``re-overdamping'' phenomenon \nin which its energy density reverts to behaving like vacuum energy \nat a time after its initial transition from the overdamped to underdamped regime.\n\nA variety of scenarios for new physics predict large numbers of scalar fields with \nsimilar quantum numbers.  Indeed, such collections of fields emerge naturally from a \nvariety of string constructions~\\cite{WittenStringAxion,SvrcekWitten}, from axiverse \nconsiderations~\\cite{Axiverse}, and more \ngenerally from theories with extra dimensions.  Of course, many features of such \ntheories are highly model-dependent, including the number of fields present, \nthe form of the mass-squared matrix for these fields (both at very early and at very \nlate times), and the time or temperature scales associated with the dynamical mechanisms \nfor mass generation.  However,\na given specific \nnew-physics context is likely to lead to certain theoretical structures in which\nsome of these features become fixed.\nFor example, theories in which a scalar propagates in extra spacetime dimensions  give \nrise to infinite numbers of scalar fields which are organized into Kaluza-Klein (KK) \ntowers in which the form of the mass matrix at early times --- even prior to the onset \nof any mass-generating phase transition --- is dictated by the geometry of the \ncompactification manifold.  Because such KK towers arise generically in scenarios \ninvolving compact extra dimensions (such as string theories, supergravity theories, and \nmany other high-scale unification models),\nthe features associated with the cosmological evolution of such KK towers --- including \nthe dynamical flow of energy amongst the different KK modes prior to, during, and after\ncosmological phase transitions --- are likely to play an important role in \nearly-universe cosmology.\n\nIn this paper we tackle this important question.  In particular, we study\nthe early-universe cosmology of a KK tower of scalar fields in the presence\nof a mass-generating phase transition, focusing on the time-development of the total \ntower energy density (which determines the relic abundance) as well as its distribution across the different \nKK modes.  On the one hand, this study is critical for understanding the cosmological \nissues associated with extra spacetime dimensions {\\it per se}\\\/.  On the other hand, \nthis study also represents an extension of the analysis presented in \nRef.~\\cite{TwoTimescales}.  In that analysis, only two scalar fields were considered, \nbut these fields were permitted to experience the most general possible\nmixings between them.  In this analysis, by contrast, the number of scalar fields is \ninfinite but the structure of the KK tower and the interplay between brane and bulk \nphysics imposes a strict mixing structure.  A goal of this paper is therefore to \nunderstand the consequences of this simultaneous extension in the number of\nfields and narrowing of the allowed mixing structure. \n\nAs we shall see, studying the dynamics of a complete KK tower can be a significant\nundertaking due to its infinite number of fields --- especially in the presence of \na non-trivial, time-dependent, mass-generating and mixing-inducing phase transition.\nTherefore, in this paper, we shall work towards this goal in distinct stages by\nstarting with only one KK mode and then introducing more and more KK excitations into \nthe system.  Ultimately, we shall build towards the full, infinite  KK tower.\nIn each case, our goal is to trace the flow of energy through the system and to examine \nthe behavior of such quantities as the total late-time energy density as well as its \ndistribution across the available KK modes.  Moreover, our goal is to retain as much \ngenerality as possible.  We shall therefore refrain from specifying the exact nature \nof our KK tower other than to specify that its constituents are scalars.   Our results \nwill therefore be broadly applicable to any of the particular scalar fields listed above.\n\nThis paper is organized as follows.  In Sect.~\\ref{sec:TheModel}, we begin by establishing \nthe context in which we shall be working and review the general properties of our setup.\nThen, in Sect.~\\ref{sec:FourDimensionalLimit}, we begin our study by considering the \ncase of only a single scalar field undergoing a mass-generating phase transition.\nIn particular,\nour analysis will extend beyond the usual ``adiabatic'' and ``abrupt'' regimes commonly \ndiscussed in the literature.\nIn Sect.~\\ref{sec:FiniteModeNumber}, we then broaden our perspective by\nconsidering the case of only the $N$ lowest-lying KK excitations, where $N$\nis arbitrary but finite.  As we shall see, situations with finite $N$ have a number of \nproperties which render their cosmological evolution somewhat distinct.\nIn Sect.~\\ref{Approaching}, we then examine how our system behaves as a function of \n$N$ for \\il{N\\gg 1}, \nand in Sect.~\\ref{sec:KKLimit} we present our results for the full, infinite KK tower.\nIn order to distinguish this case from the large-$N$ case\nin Sect.~\\ref{Approaching}, we shall refer to the KK tower as representing the case with \\il{N=\\infty}.\nUntil this point, our analysis is completely general and applicable to any scalar\nfield.  However, in Sect.~\\ref{sec:AxionintheBulk}, we then apply our machinery to the case\nin which our scalar is an axion or axion-like particle.  This allows us to interpret our \nresults within a particular phenomenologically relevant context, and illustrate the \nsignificant phenomenological implications that our results can have.  Finally, in \nSect.~\\ref{sec:Conclusions}, we summarize our main conclusions and discuss further \npotential consequences of our results.\n\nAs evident from this outline, our method for studying the properties of a full KK theory \ninvolves truncating the KK tower to its $N$ lowest-lying modes and then considering the \n\\il{N\\to\\infty} limit.  Indeed, this is a valid, standard approach.  However, as a by-product \nof our analysis, we shall discover that such a truncation is not unique.  Specifically, we \nshall develop an alternate, ``UV-based'' effective truncation of KK theories which has a \nnumber of interesting theoretical properties that distinguish it from the more traditional \n``IR-based'' truncation\ntypically used in the extra-dimension literature.  Both truncations lead to the same \nphysics as \\il{N\\to\\infty}, but differ significantly for finite $N$.\nThis alternate truncation will be discussed in Sect.~\\ref{alttrunc}.\n\n\n\\FloatBarrier\n\\section{The Framework\\label{sec:TheModel}}\n\n\n\nIn this section, we establish the framework in which our analysis\nwill take place.\nThis framework is similar to that considered in Refs.~\\cite{DDGAxions,DDM1,DDM2,DDMAxion}, \nand is essentially a KK-oriented extension\nof the framework considered in Ref.~\\cite{TwoTimescales}. \nSpecifically, we consider a flat, five-dimensional spacetime geometry\nof the form \\il{{\\cal M} \\times S^1\/\\IZ_2}, where\n${\\cal M}$ denotes our usual four-dimensional Minkowski spacetime and\nwhere $S^1\/\\IZ_2$ denotes an orbifolded circle ({\\it i.e.}\\\/, a line segment) of \nlength $2\\pi R$.\nOur spacetime coordinates shall be denoted \\il{(x^\\mu,x^5)}, \\il{\\mu=0,1,2,3}, \nwhere \\il{0\\leq x^5 \\leq 2\\pi R} and where the orbifold action is given by \\il{x^5\\to -x^5}.\nWe shall further imagine that\nour usual Standard-Model (SM) fields and interactions are confined to a four-dimensional\nbrane localized at the\norbifold fixed point at \\il{x^5=0}.\n\nWithin this geometry we consider \na five-dimensional scalar field $\\Phi(x^\\mu, x^5)$  and a\ncorresponding five-dimensional action ${\\cal S}$ of the form\n\\begin{equation}\\label{eq:5Daction}\n {\\cal S} ~=~ \\int d^4x\\,dx^5 \\left[\\mathcal{L}_{\\rm bulk}(\\Phi)\n+ \\delta(x^5)\\mathcal{L}_{\\rm brane}(\\psi_i,\\Phi)\\right] \n\\end{equation}\nwhere $\\psi_i$ are fields confined to the SM brane. \nIn general, our bulk action will contain\ngeneric kinetic and mass terms of the form \n\\begin{equation}\\label{eq:generalLbulk}\n\\mathcal{L}_{\\rm bulk}(\\Phi) ~=~ \\frac{1}{2}\\partial_M\\Phi^*\\partial^M\\Phi - \\frac{1}{2}M^2\\abs{\\Phi}^2~.\n\\end{equation}\nHowever, many well-motivated \nscalars such as moduli and axions have a primordial shift \nsymmetry under which \\il{\\Phi\\to \\Phi+c} for arbitrary constants $c$. \nWith this motivation in mind, we shall henceforth assume that \\il{M=0}.\n\nThe brane Lagrangian $\\mathcal{L}_{\\rm brane}$, on the other hand, \nincludes the SM Lagrangian in addition \nto interactions between $\\Phi$ and the SM fields. In general, these interactions can\nopen up decay channels from $\\Phi$ to SM states.   However, such interactions can also lead\nan {\\it effective}\\\/ four-dimensional mass for $\\Phi$ on the brane.\nSuch masses are typically generated as the result of some brane-localized dynamics \n(such as a brane-localized phase transition) \nwhich occurs at some time during the cosmological evolution on the brane\nand which explicitly breaks the shift symmetry. \nFor this reason, we shall allow our brane Lagrangian to include\na time-dependent effective brane-mass term of the form\n\\begin{equation}\\label{eq:Lbrane}\n{1\\over {\\cal V}}\\, \\mathcal{L}_{\\rm brane}(\\Phi) ~=~ -  \\frac{1}{2}m^2(t) \\abs{\\Phi}^2 + ...  \\ \n\\end{equation}\nwhere \\il{{\\cal V}\\equiv 2\\pi R} is the compactification volume.\nThe behavior of $m(t)$ is ultimately determined by the details of the non-perturbative dynamics on \nthe brane and thus highly model-dependent. \n\nThis information alone is sufficient to allow us to determine\nthe effective four-dimensional theory that emerges\nupon compactification.\nAssuming that $\\Phi$ has even parity under the orbifold action \\il{x^5\\to -x^5},\nwe may perform a Kaluza-Klein expansion of the form\n\\begin{equation}\n\\Phi(x^{\\mu},x^5) ~=~ \\frac{1}{\\sqrt{\\cal V}}\\sum_{k=0}^{\\infty}r_k\n\\phi_k(x^{\\mu})\\cos\\left(\\frac{kx^5}{R}\\right) \\ \n\\label{eq:orbifoldmodeexpansion}\n\\end{equation}\nwith normalization constants \\il{r_0=1} and \\il{r_{k} =\\sqrt{2}} for all \\il{k>0}, as appropriate\nfor a $S^1\/\\IZ_2$ compactification.\nInserting this into our action and integrating over $x^5$ we obtain\nan effective four-dimensional action of the form \n\\begin{align}\n\\mathcal{L}_{\\rm eff} ~=~ \\sum_{k=0}^{\\infty}\\left[\\frac{1}{2}(\\partial_{\\mu}\\phi_k)^2\n- {1\\over 2}\\sum_{\\ell=0}^{\\infty}\\phi_k\\mathcal{M}^2_{k\\ell}(t)\\phi_\\ell\\right] ~+~...~~~ \n\\label{eq:4Dlagrangian}\n\\end{align}\nwhere our time-dependent mass matrix ${\\cal M}_{k\\ell}^2$ is given by \n\\begin{eqnarray}\n     {\\cal M}_{k\\ell}^2 &=& k\\ell \\, \\delta_{k\\ell} \\, M_c^2 ~+~\n       r_k r_\\ell\\, m^2 (t)~\\nonumber\\\\\n&=& m^2(t)\n\\begin{bmatrix}\n1  & \\sqrt{2} & \\sqrt{2} & \\cdots \\\\\n\\sqrt{2} & ~~2 + \\frac{M_c^2}{m^2(t)} & 2 & \\cdots \\\\\n\\sqrt{2} & 2 & ~~2 + \\frac{4M_c^2}{m^2(t)}~ & \\cdots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots \n\\end{bmatrix} ~~~~~\n\\label{kmassmatrix}\n\\end{eqnarray}\nwith \\il{M_c\\equiv 1\/R}.\nThe first term in the top line of Eq.~(\\ref{kmassmatrix}) represents the expected KK contribution\nfrom the physics in the bulk,\nwhile the second term reflects the contribution from physics on the brane\nwhich breaks the translational symmetry in the $x^5$-direction and thereby\nintroduces a mixing between the different KK modes.\n\nGiven the Lagrangian in Eq.~(\\ref{eq:4Dlagrangian}), \nit follows that in a flat Friedman-Robertson-Walker (FRW) \nspacetime the zero-momentum modes of the KK scalars $\\phi_k$ obey the coupled equations of motion\n\\begin{equation}\n\\ddot{\\phi}_k + 3H(t)\\dot{\\phi}_k + \\sum_{\\ell = 0}^{\\infty}\\mathcal{M}^2_{k\\ell}(t)\\phi_\\ell ~=~ 0 ~\n\\label{eq:KKequationsofmotion}\n\\end{equation}\nwhere \\il{H(t)\\equiv \\kappa\/(3t)} is the Hubble parameter.\nNote that \\il{\\kappa=2} and \\il{\\kappa=3\/2} for matter- and radiation-dominated epochs, respectively.\nIf ${\\cal M}^2_{k\\ell}$ were constant as a function of time, it would be possible to \npass to a mass-eigenstate basis \\il{\\lbrace\\phi_\\lambda\\rbrace} \nin which the different differential equations in Eq.~(\\ref{eq:KKequationsofmotion})\nwould decouple.  Unfortunately, the time-dependence of $m(t)$ leads to a time-dependence\nfor ${\\cal M}^2_{k\\ell}$.    This time-dependence induces an unavoidable mixing between the different\nmodes, since even the mass-eigenstate basis is continually changing.\nIt is this feature which underpins the non-trivial dynamics we shall be studying in this paper.\nLikewise, at any moment in time, the total energy density associated \nwith our system of KK modes is given by\n\\begin{equation}\n\\rho ~=~ {1\\over 2}\\sum_k \\dot\\phi_k^2 + \n         {1\\over 2}\\sum_{k,\\ell} \\phi_k {\\cal M}_{k\\ell}^2 \\phi_k~.\n\\label{energydensity}\n\\end{equation}\nIf the mass matrix ${\\cal M}_{k\\ell}^2$ were time-independent, it would be\npossible to rewrite this energy density in the mass-eigenstate basis \nas the sum of individual contributions: \n\\begin{equation}\n            \\rho  ~=~ \\sum_\\lambda \\rho_\\lambda\n\\label{energydecomp}\n\\end{equation}\nwhere\n\\begin{equation} \n\t\\rho_\\lambda ~=~ {1\\over 2}\\left(\n\t\\dot\\phi_\\lambda^2 + \\lambda^2 \\phi_\\lambda^2 \\right)~.\n\\label{rholambda}\n\\end{equation}\nUnfortunately, once again, the time-dependence of the mass matrix renders such a decomposition impossible.\nThus, we may consider the total energy density in Eq.~(\\ref{energydensity}) as receiving contributions\nfrom individual mass eigenstates as in Eq.~(\\ref{energydecomp}) only during those periods of cosmological evolution\nfor which the mass matrix might not be changing with time. \n\nIn general, we shall assume that the mass $m(t)$ is generated through some sort of phase\ntransition on the brane whose occurrence is centered at some time $t_G$ in cosmological history  \nand which requires a duration $\\Delta_G$ over which to unfold.\nWe shall therefore assume that \\il{m(t)=0}\nlong before the phase transition ({\\it i.e.}\\\/, for \\il{t\\ll t_G}), \nwhile we shall assume that $m(t)$ is given by some fixed mass $\\overline{m}$ long after the phase\ntransition ({\\it i.e.}\\\/, for \\il{t\\gg t_G}).\nThe precise details of how $m(t)$ evolves from $0$ to $\\overline{m}$ clearly depend on the particular\nstructure of the phase transition in question, but we shall find that the most important feature of\nthis function will be its ``width'' $\\Delta_G$. \nFor concreteness, we shall therefore adopt the function~\\cite{TwoTimescales} \n\\begin{equation}\nm(t) ~=~\n\\frac{1}{2}\\,{\\overline{m}}\\, \\left\\{1 + \n\\text{erf}\\left[\\frac{1}{\\sqrt{2}\\delta_G}\\log\\left(\\frac{t}{t_G}\\right)\\right]\\right\\} \\ ,\n\\label{sigmoid}\n\\end{equation}\nwhere ${\\overline{m}}$ is the mass at asymptotically late times, where $t_G$ is the central time\ndefined by \\il{m(t_G)={\\overline{m}}\/2}, and where `erf' denotes the so-called ``error'' function. \nIn Eq.~(\\ref{sigmoid}), the quantity \\il{\\delta_G\\in\\left[0,1\\right]} is a \ndimensionless parameter describing the ``width'' of the phase transition,\nwith  \\il{\\delta_G=0} corresponding to an essentially\ninstantaneous phase transition and \\il{\\delta_G=1} describing a phase transition\nwhich proceeds as slowly as possible.\nGiven the dimensionless parameter $\\delta_G$, the physical timescale $\\Delta_G$ over which\nthe phase transition occurs is then given by~\\cite{TwoTimescales}\n\\begin{equation}\n\\Delta_G ~\\equiv~ \\sqrt{2\\pi}\\delta_G t_G \\ .\n\\label{bigwidth}\n\\end{equation}\nIndeed, given Eq.~(\\ref{sigmoid}), we may view the transition from \\il{m=0} to \\il{m={\\overline{m}}} as \noccurring over a ``width'' $\\Delta_G$ logarithmically centered at $t_G$. \n\nThis choice of function is discussed more fully in Ref.~\\cite{TwoTimescales}.  Indeed, it is \nshown in Ref.~\\cite{TwoTimescales} that this function models the time-dependence of a variety \nof phase transitions extremely well in the neighborhood of $t_G$, where the mass function is \nchanging most rapidly.   This includes the instanton-induced phase transition \nthat gives mass to the QCD axion.   Of course, as discussed in \nRefs.~\\cite{WantzShellard1,WantzShellard2,diCortonaQCDAxion,BorsanyiQCDAxion},\nthe actual QCD phase transition leads to a mass function $m(t)$ which exhibits a power-law \ndependence on $t$ at early times $t\\ll t_G$, and which is then often taken to be followed by \none or more slope discontinuities near \\il{t\\sim t_G} before becoming a constant for \\il{t\\gg t_G}.    \nBy contrast, our functional form for $m(t)$ is designed to interpolate smoothly between fixed \nearly- and late-time values in such a way that there is an adjustable and easily identifiable \n``width'' $\\Delta_G$.  Despite these differences, however, it is shown in Ref.~\\cite{TwoTimescales} \nthat our functional form models the QCD functional form extremely well near \\il{t\\sim t_G}.\n \nOf course, many other choices are possible for the particular functional form of \n$m(t)$, with different choices being more or less suitable in different contexts.\nHowever, none of the qualitative results of this paper will ultimately depend on \nthese specific details.\n\nIn general, we may choose to consider phase transitions with any \\il{\\delta_G\\leq 1}.\nIndeed, as discussed in Ref.~\\cite{TwoTimescales}, situations with \\il{\\delta_G > 1} suffer\nfrom an unphysical boundary artifact near \\il{t\\approx 0}, namely the emergence of a ``fake'' \nmass-generating phase transition whose width tends to decrease as $\\delta_G$ increases beyond \\il{\\delta_G=1}.\nHowever, when \\il{{\\overline{m}}\/M_c\\gg 1}, it turns out that an additional\nboundary effect emerges which restricts $\\delta_G$ even more strongly, ultimately imposing an upper limit on $\\delta_G$ which\nis less than $1$ and which depends on ${\\overline{m}}\/M_c$.   \n(Specifically, as $\\delta_G$ increases for \\il{{\\overline{m}}\/M_c\\gg 1}, the effective ``mixing angle'' $\\theta$ between our ground\nstate and the first excited state starts changing, evolving from \\il{\\theta=0} to \\il{\\theta=\\overline{\\theta}} \nmore and more rapidly and at earlier and earlier\ntimes.  Ultimately we reach a new critical value of $\\delta_G$ beyond which the $\\theta$-width $\\Delta_\\theta$ in\nEq.~(4.3) of Ref.~\\cite{TwoTimescales} actually starts to {\\it decrease}\\\/ when $\\delta_G$ is\nincreased, giving the appearance of a new ``fake'' \nphase transition at early times near $t\\approx 0$.)    \nAvoiding this unphysical behavior then \nimposes an additional upper bound on $\\delta_G$\nwhich we can evaluate numerically as a function of ${\\overline{m}}\/M_c$ and $N$.\nIn this paper we shall therefore restrict our attention to situations with \\il{\\delta_G\\leq 1} but also impose\nthis additional constraint as appropriate for large ${\\overline{m}}\/M_c$. \n\nIn this context, we note that\ntheories with extra spacetime dimensions must generally obey additional ``normalcy-temperature'' \nconstraints~\\cite{ADD,ADDPheno} \nwhich are more easily satisfied within cosmological low-temperature reheating (LTR) scenarios \nin which inflation occurs late and reheating occurs at temperatures as low as \n\\il{T_{\\rm RH}\\sim \\mathcal{O}(\\text{MeV})}~\\cite{KawasakiLTR}. \nIf we assume the dynamics that generate $m(t)$ are at higher scales than $T_{\\rm RH}$, then our \nmass-generating phase transition can be assumed to occur during \nan inflaton-dominated --- {\\it i.e.}\\\/, a matter-dominated --- epoch.\nHowever this assumption will not play a significant role in our analysis, and the primary\nresults of this paper will hold regardless of the specific cosmological timeline assumed.\n\nGiven the differential equations of motion in Eq.~(\\ref{eq:KKequationsofmotion}),\nthe only ingredients of our model remaining to be specified \nare the initial conditions on each field mode $\\phi_k$.   \nWhile in principle many possibilities \nexist, \nthe existence of the shift symmetry \\il{\\Phi \\rightarrow \\Phi + c} prior to mass\ngeneration on the brane suggests \nthat $\\Phi$ might have an arbitrary fixed displacement at early times \\il{t_i\\ll t_G},\nwith a fixed non-zero vacuum expectation value (VEV) \\il{\\langle \\Phi\\rangle\\neq 0}.\nUpon KK reduction, this corresponds to initial conditions given by\n\\begin{align}\\label{eq:initialconditions}\n\\phi_k(t_i) ~&=~ \\expt{\\phi_0} \\delta_{0k} \\nonumber \\\\\n\\dot{\\phi}_k(t_i) ~&=~ 0 \\ ,\n\\end{align}\nwhere \\il{\\expt{\\phi_0}\\equiv \\sqrt{2\\pi R}\\expt{\\Phi}}.\nWe shall therefore take these to be the initial \nconditions for our differential equations (\\ref{eq:KKequationsofmotion}) \nin what follows.\nIt is important to note that these initial conditions \nhave several additional advantages beyond their natural origins outlined above.\nFirst, as long as the initial time $t_i$ is sufficiently prior to $t_G$, our ensemble of KK modes\nwill have zero energy.   Thus, as we shall see, {\\it all energy accrued by our KK system is \nsolely the result of the phase transition on the brane}\\\/. \nMoreover, these initial conditions, being essentially static,\nfree our system and its subsequent dynamics\nfrom all details concerning\nthe generation of the initial VEV \\il{\\langle \\Phi\\rangle}.\nAs a result, the time at which the initial VEV \\il{\\langle \\Phi\\rangle} \nis generated is arbitrary and we need not concern ourselves with its origins. \nIndeed, as we shall see, the precise value of $t_i$ will not affect our analysis, or any of our conclusions.\nFinally, we remark that while $\\expt{\\phi_0}$ sets an overall energy scale, this scale\nwill be irrelevant for our purposes since the equations of motion are linear and\nour main interest will be on relative {\\it comparisons}\\\/ between energy scales \nrather than their absolute magnitudes.\n\nFinally, let us discuss the masses of the individual KK modes in this model.\nEven though our overall brane mass $m(t)$ is continually changing in time,\ndiagonalizing the matrix in Eq.~(\\ref{kmassmatrix}) at any moment in time\nleads directly to the corresponding instantaneous (mass)$^2$ eigenvalues $\\lambda_k^2$.\nWhile no closed-form analytical expressions for these eigenvalues\nexist, they may be readily approximated in the \\il{m\/M_c\\ll 1} and \\il{m\/M_c\\gg 1} limits.\nIn the \\il{m\/M_c\\ll 1} limit,\nwe find\n\\begin{equation}\n      \\lambda_k ~\\approx~ \\begin{cases}\n              m  &   k=0\\cr\n             k M_c & k>0~.\n           \\end{cases}\n\\label{eq:unmixedmassspectrum}\n\\end{equation}\nThere are two ways to interpret this result.\nIf we imagine holding $M_c$ fixed, this limit corresponds to taking $m$\nextremely small.\nThis renders the brane irrelevant in the KK mass decomposition, and\nindeed \nthe results in Eq.~(\\ref{eq:unmixedmassspectrum})\nare then the eigenvalues expected from a straightforward compactification on a circle.\nAlternatively, if we imagine holding $m$ fixed, we see that this limit corresponds\nto taking $M_c$ extremely large.   \nWe then have only a single light mode with mass \\il{\\lambda_0\\approx m}, which is nothing\nbut the four-dimensional limit.\nBy contrast, for \\il{m\/M_c\\gg 1},\n the mixing between the modes is maximized and our corresponding eigenvalues are given \nby~\\cite{DDGAxions,DDM1} \n\\begin{equation}\n      \\lambda_k ~\\approx~ (k+1\/2) \\,M_c~.\n\\label{kmixedlimit}\n\\end{equation}\nIt is remarkable that the effect of the brane mass in this limit is entirely $m$-independent, and merely amounts\nto shifting our eigenvalues by $M_c\/2$.   Indeed, these shifted eigenvalues are those that would have emerged from \nan {\\it anti}\\\/-periodic compactification on a circle.\nFinally, for intermediate values of $m\/M_c$, we find that our eigenvalues \n$\\lambda_k$\ntend to follow Eq.~(\\ref{kmixedlimit}) for \\il{k\\ll \\pi m^2\/M_c^2} and   \nEq.~(\\ref{eq:unmixedmassspectrum}) for \\il{k\\gg \\pi m^2\/M_c^2},\nwith $\\lambda_k$ taking values that smoothly evolve between these two extremes for other values\nof $k$.\n\nIt is important to stress that this behavior for the eigenvalues emerges\nonly in the limit in which we consider the full KK tower with its infinite complement\nof KK modes, \\il{k=0,1,...,\\infty}.\nBy contrast, if we truncate the mass matrix in Eq.~(\\ref{kmassmatrix}) to its first\n$N$ rows and columns so that \\il{0 \\leq \\lbrace k,\\ell\\rbrace \\leq N-1}, \nour general expectations described above continue to apply\nonly for the eigenvalues \\il{\\lambda_0, ..., \\lambda_{N-2}}.\nHowever, for the highest eigenvalue $\\lambda_{N-1}$, we find that the above\nexpectations continue to apply only for \\il{m^2\/M_c^2 \\lsim {\\cal O}(\\sqrt{N})}.\nBy contrast, for \\il{N\\gg 1} and \nfor \\il{m^2\/M_c^2 \\gsim {\\cal O}(\\sqrt{N})},\nwe instead find that\n\\begin{equation}\n     \\lambda_{N-1} ~\\approx~  \\sqrt{2 N} \\, m ~.\n\\label{artifact}\n\\end{equation}\nThe normalized eigenvalue $\\lambda_{N-1}\/M_c$ thus diverges as \\il{m\/M_c\\to \\infty}.\n\n\\begin{figure}\n\\includegraphics[keepaspectratio, width=0.5\\textwidth]{massspectrumlinear.pdf}\n\\caption{The mass spectrum $\\lambda_k$ for \\il{N=10}, plotted as a function of $m\/M_c$.\nWhen \\il{{\\overline{m}}\\ll M_c}, \nour mass eigenvalues follow the simple form given in Eq.~(\\ref{eq:unmixedmassspectrum}).\nBy contrast, when\n\\il{{\\overline{m}}\\gg M_c},\nall but the highest mass eigenvalue shift upwards\nby $M_c\/2$, while the highest mass eigenvalue diverges according\nto Eq.~(\\ref{artifact}) for \\il{m^2\/M_c^2\\gsim {\\cal O}(\\sqrt{N})}.\nAs discussed in the text, this anomalous behavior\nof the highest mass eigenvalue is an artifact \nof having truncated our KK tower to only a finite number of states, and \ndisappears as \\il{N\\to \\infty}.}\n\\label{fig:massspectrum}\n\\end{figure}\n \nThe behavior of the eigenvalues $\\lambda_k$ is illustrated in Fig.~\\ref{fig:massspectrum},\nwhere we have plotted the values of $\\lambda_{k=0,...,9}$\nas functions of $m\/M_c$ for \\il{N=10}.\nAs is clear from Fig.~\\ref{fig:massspectrum}, the somewhat anomalous behavior of \n$\\lambda_{N-1}$ is ultimately an artifact of truncating \nour KK tower to its first $N$ values.\nHowever, as $N$ increases, this anomalous behavior of $\\lambda_{N-1}$ \nsets in only at increasingly \nlarge values of $m\/M_c$.\nThus, in the \\il{N\\to\\infty} limit, this anomalous behavior disappears\naltogether, as expected.\n\nIn our model the quantity $m$ is  \ngenerally time-dependent, rising from zero to some final, fixed late-time value ${\\overline{m}}$\nas the result of our mass-generating phase transition on the brane.\nThe corresponding eigenvalues $\\lambda_k^2$ are thus set  \nby Eq.~(\\ref{eq:unmixedmassspectrum}) with \\il{m=0} at early times, but \nsubsequently evolve towards late-time results which are governed \nby the magnitude of the late-time ratio ${\\overline{m}}\/M_c$.\nMeanwhile, each of the corresponding KK fields $\\phi_k$ is evolving \naccording to Eq.~(\\ref{eq:KKequationsofmotion}), with the non-diagonal\nmass matrix ${\\cal M}^2_{k\\ell}$ inducing a time-dependent mixing between the KK modes\nand even between their instantaneous mass eigenstates.\nThis, coupled with the continuously decreasing Hubble friction term $H(t)$ in \nEq.~(\\ref{eq:KKequationsofmotion}), leads to a highly non-trivial dynamical system,\nwith energy slowly being introduced into the system as the result of the mass-generating\nphase transition while it is simultaneously and continually redistributed across the different\nmodes and dissipated from each mode.\nUnderstanding this energy flow is critical if we are to properly\nunderstand the cosmological implications of such KK scalar towers in the presence\nof mass-generating phase transitions in the early universe. \nThis is therefore the main task to which we now turn.\n\n\n\\FloatBarrier\n\\section{\\texorpdfstring{$N=1$}{N=1}:  ~Mapping out the 4D Limit \\label{sec:FourDimensionalLimit}}\n\n\nWe begin our analysis by focusing on the simplest case:  that of only one scalar field, {\\it i.e.}\\\/, the case\nwith \\il{N=1}.   This case lacks any dependence on $M_c$, and therefore can be \nconsidered equivalent to  a four-dimensional limit in which \\il{M_c\\gg m(t)} \nat all times (so that all KK modes with \\il{k\\geq 1} effectively decouple from the problem).\nThis case of a single scalar field $\\phi(t)$ undergoing a mass-generating phase transition\nhas been studied extensively in the literature (see, {\\it e.g.}\\\/, Refs.~\\cite{TurnerEffect,UpdatedAxionAbundanceCalc}), \nbut only in certain ``adiabatic'' or ``abrupt'' limits.\nIn this section, by contrast, we shall provide a complete mapping of the entire relevant \nparameter space.\n\nIn general, given the equation of motion in Eq.~(\\ref{eq:KKequationsofmotion}),\nwe see that the field $\\phi(t)$ simply follows the trajectory of a damped harmonic oscillator with time-dependent\ncritical damping coefficient \\il{\\zeta \\equiv 3H\/2m}.  Such an oscillator is overdamped if \\il{\\zeta>1} \nand only experiences oscillations \nin the underdamped regime with \\il{\\zeta<1}.\nSince \\il{H(t)\\sim \\kappa\/(3t)} \nis monotonically decreasing while $m(t)$ is monotonically increasing,  \nwe see that $\\zeta$ is constantly decreasing.\nAs a result,\nthe field $\\phi(t)$ is necessarily overdamped at times \\il{t \\ll t_G} \nand does not begin to undergo coherent oscillations until \\il{\\zeta = 1}.\nWe shall let $t_\\zeta$ denote this time at which such coherent oscillations begin.\nIn general, we know that $t_\\zeta$ cannot come too much earlier than $t_G$ \n({\\it i.e.}\\\/, we cannot have \\il{t_G-t_\\zeta \\gg \\Delta_G}) since our\nfield $\\phi$ is presumed massless prior to the onset of the phase transition.\nLikewise, if $t_{\\zeta}$ occurs long after the phase transition has \nalready occurred ({\\it i.e.}\\\/, if \\il{t_\\zeta - t_G\\gg \\Delta_G}), the specific details of\nthe phase transition such as its shape or width will have a negligible effect \non the resulting dynamics and on the corresponding energy \ndensity $\\rho$. However, if \\il{|t_{\\zeta}-t_G|\\lsim \\Delta_G},\nthe specific properties of the phase transition can be significant.\n\nTwo approximations are typically used in the literature in order to estimate the effects of\nthe phase transition  within this third regime.\nBoth rest upon considering extreme limits of the phase-transition width $\\Delta_G$. \nIf the width is large enough so that \\il{\\dot m\/m \\ll m} at all times during the  \nfield oscillations ({\\it i.e.}\\\/, at all times after $t_\\zeta$),\nthen the field undergoes many oscillations during the phase transition. \nThis is the so-called \\emph{adiabatic approximation}:  the fields remain virialized during the phase transition and we \ncan approximate the energy density by\n\\begin{equation}\n\\rho_{\\rm 4D}(t) ~\\approx~ \n\\frac{1}{2}\\, \\langle\\phi_0\\rangle^2 \\, m(t_{\\zeta}) \\, m(t)\\, \\left[\\frac{a(t_{\\zeta})}{a(t)}\\right]^3 \n\\label{eq:adiabaticapproximation}\n\\end{equation}\nwhere \\il{a(t)\\sim t^{\\kappa\/3}} is the cosmological scale factor.\nBy contrast, if the phase transition width $\\Delta_G$ is extremely small, we are in the so-called\n\\emph{abrupt approximation}:   our assumption that\n\\il{|t_{\\zeta}-t_G|\\lsim \\Delta_G}\nforces $t_\\zeta$ and $t_G$ to coincide, and the phase transition happens so rapidly \nthat the field $\\phi(t)$ retains its initial value \\il{\\langle\\phi_0\\rangle} \nuntil $t_G$ (or equivalently $t_\\zeta$), at which point it immediately begins oscillating coherently.\nThe evolution of the energy density is then easily obtained by solving \nEq.~\\eqref{eq:KKequationsofmotion} with \\il{N=1} analytically.   This yields the exact solution\n\\begin{equation}\n\\rho_{\\rm 4D}(t) ~=~ \\frac{\\pi^2}{8}\\, \\langle\\phi_0\\rangle^2\\, {\\overline{m}}^4\\, \\frac{t_G^{\\kappa+1}}{t^{\\kappa-1}}\\, \n\\left[B_1^2(t) +B_2^2(t)\\right] \\ ,\n\\label{eq:abruptapproximation}\n\\end{equation}\nwhere \n\\begin{eqnarray}\nB_1(t) &\\equiv& J_{\\kappa_+}({\\overline{m}} t_G)Y_{\\kappa_-}({\\overline{m}} t) - \nJ_{\\kappa_-}({\\overline{m}} t)Y_{\\kappa_+}({\\overline{m}} t_G) \\nonumber \\\\\nB_2(t) &\\equiv& J_{\\kappa_+}({\\overline{m}} t)Y_{\\kappa_+}({\\overline{m}} t_G) - \nJ_{\\kappa_+}({\\overline{m}} t_G)Y_{\\kappa_+}({\\overline{m}} t) \\nonumber\\\\\n\\end{eqnarray}\nand where we have defined \\il{\\kappa_{\\pm} \\equiv (\\kappa \\pm 1)\/2}. \nNote that in both Eq.~(\\ref{eq:adiabaticapproximation}) \nand Eq.~(\\ref{eq:abruptapproximation}),\nthe quantity $t$ is measured against  the same cosmological clock\nthat measures $t_G$. \n\nWhile the approximations above are applicable in two limits of parameter space, it is \nimportant to understand the behavior of the late-time energy density over the entire parameter space.\nIn particular, although the mass-generating phase transition generally pumps energy into the system,\nand although this energy density is ultimately dissipated by the Hubble friction that \nslowly damps the resulting field oscillations,\nit turns out that there can be an additional source of dissipation\n  and therefore an additional source of suppression of the\nlate-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ compared to our usual expectations based on the abrupt approximation.\nThis arises if the corresponding field is undergoing oscillations \n {\\it during}\\\/ the phase transition, while the mass of the field is changing appreciably.\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.48\\textwidth]{singlefielddensityplot.pdf}\n\\hskip 0.2 truein\n\\includegraphics[keepaspectratio, width=0.48\\textwidth]{singlefieldadiabatic.pdf}\n\\caption{The late-time energy density ${\\overline{\\rho}}_\\protect{\\rm 4D}$, plotted within the \\il{({\\overline{m}} t_G, \\delta_G)} plane\nand normalized to the value it would have had in the abrupt approximation (left panel)\nor adiabatic approximation (right panel).\nIn each case the colors and black lines indicate the contours of the normalized ${\\overline{\\rho}}_{\\rm 4D}$ \n({\\it i.e.}\\\/, the degrees to which the true late-time energy density is enhanced or suppressed\nrelative to its approximated value),\nwhile the blue lines indicate the corresponding contours of \\il{(t_\\zeta-t_G)\/\\Delta_G}.\nAs additional relevant guidelines,  \nthe dashed line in the left panel \nindicates the contour\nalong which \\il{\\Delta_G=2\\pi\/{\\overline{m}}},\nwhile in the right panel \nit indicates the contour along which \\il{\\dot{m}(t_{\\zeta}) = m^2(t_{\\zeta})}.\nIn general, the energy densities in regions above or to the left of these\nguidelines \nin the left (right) panel tend to obey the abrupt (adiabatic) approximation \nmore strongly than those below or to the right.  \nWe also find that neither approximation holds \nacross the majority of the parameter space shown, with the true\nvalues of the late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ often experiencing significant\nsuppressions (blue regions) or enhancements (red regions) as compared with the \napproximated expectations. }\n\\label{fig:singlefielddensityplot}\n\\end{figure*}\n\nIt is easy to identify those regions of parameter space for which this will be the case.\nIn general, there are two criteria that must be satisfied \nif the scalar field is to undergo at least one oscillation \nwhile the mass of the field is changing appreciably.\nFirst, the scalar field must indeed be undergoing oscillations at some point during the phase\ntransition --- {\\it i.e.}\\\/, our system must be in the underdamped regime before the mass of our field reaches ${\\overline{m}}$.\nWe thus must have \\il{3H \\lsim 2 {\\overline{m}}}, or equivalently \\il{{\\overline{m}} t_G\\gsim  \\kappa\/2},\nwhere we have taken $t_G$ as a rough benchmark time for the phase transition.  \nSecond, \nin order to ensure that at least one or more field oscillations\noccur during the interval over which the mass is changing,\nwe require that the timescale associated with the field oscillations \nbe shorter than the phase-transition timescale associated with\nthe changing mass.\nThe former timescale is generally given by $2\\pi\/m$ and thus decreases\nthroughout the phase transition, ultimately reaching a minimum value $2\\pi\/{\\overline{m}}$,\nwhile the latter timescale is nothing but $\\Delta_G$.   \nAs a rough benchmark, our phase transition will therefore include at least one oscillation\nso long as \\il{\\Delta_G \\gsim 2\\pi\/{\\overline{m}}}, or equivalently\n\\il{{\\overline{m}} t_G \\gsim \\sqrt{2\\pi}\/\\delta_G}.\nCombining our two conditions, we thus find that an additional suppression\nof the late-time energy density compared with what we might expect based on the abrupt approximation\nwill occur as long as our system satisfies \nthe rough benchmark criterion\n\\begin{equation}\n            {\\overline{m}} t_G ~\\gsim~ \n         {\\rm max} \\left\\lbrace  {\\sqrt{2\\pi}\\over \\delta_G}\\, ,\\, {\\kappa\\over 2} \\right\\rbrace~.\n\\end{equation}\nNote that for a matter- or radiation-dominated background cosmology, the underdamped condition will automatically\nbe satisfied whenever the oscillation condition is satisfied. \n\nIn the left panel of Fig.~\\ref{fig:singlefielddensityplot}, we \nplot the exact numerical value of the late-time\nenergy-density ratio ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)\/{\\overline{\\rho}}_{\\rm 4D}(0)$ within \nthe (${\\overline{m}} t_G$,$\\delta_G$) plane.\nFor reference, the benchmark contour with\n\\il{\\Delta_G=2\\pi\/{\\overline{m}}} is also plotted.\nIn general, \nthe regions above and to the right\nof this contour ({\\it i.e.}\\\/, regions with \\il{{\\overline{m}} t_G \\gg 1})\nindeed experience suppressions which grow increasingly\nsevere compared with what we might expect from the abrupt approximation.\nAs indicated in Fig.~\\ref{fig:singlefielddensityplot}, these are regions\nwith \\il{t_\\zeta > t_G}.\nSurprisingly, however, we note that there is also a region in which\nthe late-time energy density is {\\it enhanced}\\\/ compared with its abrupt expectation.\nThis region arises only for \\il{{\\overline{m}} t_G\\sim {\\cal O}(1)}, and typically has \\il{t_\\zeta-t_G \\lsim\n\\Delta_G}.\nLike the corresponding suppression,\nthis too is a feature of considering\na non-zero timescale for the phase transition, occurring only when \\il{\\delta_G >0}.\nFinally, for \\il{{\\overline{m}} t_G \\ll 1} or for \\il{\\delta_G \\ll 1},\nwe see that our late-time energy density is neither enhanced \nnor suppressed.  Indeed, these are the regions in which the abrupt approximation\napplies.\n \nIn the right panel of Fig.~\\ref{fig:singlefielddensityplot}, \nwe plot the same late-time energy density ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)$, only now normalized  \nto what we might expect from the {\\it adiabatic}\\\/ approximation.\nIn this case, the contour \nwith \\il{\\dot m\/m^2=1} at \\il{t=t_\\zeta} is shown\nas a relevant guideline.\nUnlike the comparison to the abrupt\napproximation, in this case\nwe see that the late-time energy density ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)$\nis {\\it enhanced}\\\/ throughout the entire region\nwith \\il{\\delta_G >0}.   As expected, this enhancement is relatively mild  \nin regions to the left of the\n\\il{\\dot m\/m^2=1} contour but grows increasingly severe below and to the right.\nOnce again, it is only for \\il{{\\overline{m}} t_G \\ll 1} that the adiabatic approximation\nappears to hold.\n\nWe see, then, that our usual expectations based on the abrupt or adiabatic approximations\napply only in relatively small regions of the full \\il{({\\overline{m}} t_G,\\delta_G)} parameter space, the\nformer consisting of the \nregions with \\il{{\\overline{m}} t_G \\ll 1} or \\il{\\delta_G \\ll 1} and the latter consisting of\nthe region with \\il{{\\overline{m}} t_G \\ll 1}.\nInterestingly, we see that {\\it either}\\\/ approximation yields the same \n(reliable) result for \\il{{\\overline{m}} t_G\\ll 1}, regardless of the value of $\\delta_G$.\nIn all other regions, however, we see that these standard approximations break down ---  often\nsignificantly ---  with the adiabatic approximation tending to underestimate the true value\nand the abrupt approximation either under- or over-estimating the true value, depending\non the particular values of ${\\overline{m}} t_G$ and $\\delta_G$.\n\nOne of the advantages of the abrupt and\/or adiabatic approximations is that they \nprovide relatively simple, {\\it analytical}\\\/ expressions for the late-time energy\ndensity ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)$.\nUnfortunately, as we have seen, these expressions are accurate\nonly within relatively narrow slices of the full parameter space.\nGiven this deficiency, we now offer  \ntwo approximate analytical expressions for ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)\/{\\overline{\\rho}}_{\\rm 4D}(0)$ which\ntogether describe its numerical values fairly accurately across the entire parameter space shown in\nFig.~\\ref{fig:singlefielddensityplot}. \nSubsequent use of the analytical expression in \nEq.~(\\ref{eq:adiabaticapproximation})\nthen allows us to isolate ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)$.\n\nWe begin by noting that within the \nsuppressed region \nin the left panel of \nFig.~\\ref{fig:singlefielddensityplot}, \nthe contours of ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G)\/{\\overline{\\rho}}_{\\rm 4D}(0)$ \nroughly follow the same slope as the \ncontour with \n\\il{\\Delta_G=2\\pi\/{\\overline{m}}}.  Indeed, for\n\\il{\\delta_G\\lsim 0.3}, \none finds the approximate power-law scaling behavior\n\\begin{equation}\n      {{\\overline{\\rho}}_{\\rm 4D}(\\delta_G) \\over {\\overline{\\rho}}_{\\rm 4D}(0)}  ~\\sim~ { 1\\over {\\overline{m}} t_G \\delta_G}~.\n\\label{fdscaling}\n\\end{equation}\nOf course, as we move closer to the \\il{\\delta_G=1} boundary, \nthe contours deviate from this scaling behavior and the suppression grows increasingly severe. \nNevertheless, we find an approximate expression \n\\begin{equation}\\label{eq:4Dsuppressedformula}\n\\frac{{\\overline{\\rho}}_{\\rm 4D}(\\delta_G)}{{\\overline{\\rho}}_{\\rm 4D}(0)} ~\\approx~ \n\\frac{({\\overline{m}} t_G \\delta_G)^{-0.92}}{1 + \\left[({\\overline{m}} t_G)^{1\/4}-3\/2\\right]\\delta_G^2}~ \n\\end{equation}\nwhich holds to within \\il{\\pm 20\\%} across\nthe entire suppressed region and which approximately reduces to the power-law result \nin Eq.~(\\ref{fdscaling}) for very small $\\delta_G$.\n\nBy contrast, turning to the enhanced region near \\il{{\\overline{m}} \\sim 1\/t_G},\nwe see the maximum of this enhancement always occurs at \\il{\\delta_G=1}. \nIndeed, we find that we can approximate this enhancement analytically in\nthe Gaussian form\n\\begin{equation}\\label{eq:4Denhancedformula}\n\\frac{{\\overline{\\rho}}_{\\rm 4D}(\\delta_G)}{{\\overline{\\rho}}_{\\rm 4D}(0)} ~\\approx~ \n1 + 0.42 \\exp\\left[-\\ve{v}^T\\!\\!\\begin{pmatrix}1.64 & 0.68 \\\\ 0.68 & 0.67\\end{pmatrix}\\!\\ve{v}\\right] \\ ,\n\\end{equation}\nwhere \\il{\\ve{v}\\equiv \\left[\\, \\log({\\overline{m}} t_G\/0.54) , \\, \\log(\\delta_G)\\,\\right]^T}. \nIn this case, the error associated with this approximation\nis bounded to lie between $-10\\%$ and $5\\%$ throughout\nthe non-suppressed portion of the parameter space shown.\nThus, with the analytical expressions in Eq.~\\eqref{eq:4Dsuppressedformula} and \nEq.~\\eqref{eq:4Denhancedformula},\nwe now cover the entire single-scalar parameter space.\nWe shall nevertheless continue to use our exact results throughout the rest of this paper.\n\nIn Fig.~\\ref{fig:singlefielddensityplot} \nwe have presented our results for the late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ \nas fractions of the values this quantity would have had in either the abrupt or adiabatic\napproximations.\nThese results therefore enabled us to understand the effects that can accrue beyond\nthe regions in which these approximations are valid.\nHowever, it is also important to consider the {\\it absolute}\\\/ magnitudes of ${\\overline{\\rho}}_{\\rm 4D}$\nthat are generated throughout our \\il{({\\overline{m}} t_G, \\delta_G)} parameter space.\nThese can be trivially obtained at each point in parameter space by multiplying the results in \nthe left panel of Fig.~\\ref{fig:singlefielddensityplot} \nby the corresponding values in \nEq.~(\\ref{eq:abruptapproximation})\n[or equivalently the results in the right panel of \nFig.~\\ref{fig:singlefielddensityplot} \nby the corresponding values in \nEq.~(\\ref{eq:adiabaticapproximation})].\n\n\\begin{figure}[b]\n\\includegraphics[keepaspectratio, width=0.5\\textwidth]{singlefieldabsolute.pdf}\n\\caption{The {\\it absolute}\\\/ late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$, plotted in units\nof \\il{{\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}} within the \\il{({\\overline{m}} t_G,\\delta_G)} plane.\nNote that our late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$\nis generally independent of $\\delta_G$ for \\il{\\lbrace {\\overline{m}} t_G\\lsim 10^3, \\delta_G\\lsim 0.5\\rbrace}\nand scales roughly as $({\\overline{m}} t_G)^2$ within this region.}\n\\label{fig:fdabs}\n\\end{figure}\n\nThe results are shown in Fig.~\\ref{fig:fdabs}, where we plot the absolute\nmagnitude of ${\\overline{\\rho}}_{\\rm 4D}$ in units of \\il{{\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}}.\nOnce again, $t$ refers to universal time, measured against the same cosmological clock as $t_G$.\nThus, for example, if our late-time mass is chosen to be \\il{{\\overline{m}} t_G=10} and our mass-generating phase transition width\nis chosen to be \\il{\\delta_G=0.4}, we find  \nfrom Fig.~\\ref{fig:fdabs} that our resulting late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ is given approximately as \n\\begin{equation}\n            {\\overline{\\rho}}_{\\rm 4D} (t) ~\\approx~ 50 \\, \\langle\\phi\\rangle^2 \\, t^{-2}~.\n\\end{equation} \nIn general, we see from Fig.~\\ref{fig:fdabs} that our late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$\nis independent of $\\delta_G$ for \\il{\\lbrace {\\overline{m}} t_G\\lsim 10^3, \\delta_G\\lsim 0.5\\rbrace}\nand moreover scales roughly as \n\\begin{equation}\n      {\\overline{\\rho}} ~\\sim~ ({\\overline{m}} t_G)^2\n\\label{4Dscaling}\n\\end{equation}\nfor \\il{{\\overline{m}} t_G \\gsim 1}.\nOf course, this scaling behavior extends to larger and larger values \nof ${\\overline{m}} t_G$ as \\il{\\delta_G\\to 0}, as we expect from the abrupt approximation.\n\n\n\\FloatBarrier\n\\section{\\texorpdfstring{$N>1$}{N>1}:  ~A General Study with Multiple Fields\\label{sec:FiniteModeNumber}}\n\n\nWe now consider how these results evolve as we move away from the \\il{N=1} limit\nand consider larger, arbitrary values of $N$.\nNote that although we will eventually be considering the \\il{N\\to\\infty} limit, the theories we analyze\nhere with finite \\il{N>1} are interesting in their own right, as will be discussed further in the Conclusions.\nAs discussed in Sect.~\\ref{sec:TheModel}, taking \\il{N>1} introduces not only additional $\\phi$ fields\n(each potentially with its own critical oscillation time $t_\\zeta$)\nbut also the non-trivial {\\it mixing}\\\/ between these fields that is a consequence of the mass-generating\nphase transition on the brane (resulting in a non-diagonal squared-mass matrix ${\\cal M}_{k\\ell}^2$).\nSystems with \\il{N>1} can therefore be expected to be considerably more complex than their simpler\n\\il{N=1} cousins.\n\nTowards this end, we shall concentrate not only on the behavior of the {\\it total}\\\/ late-time energy density ${\\overline{\\rho}}$\nbut also on its {\\it distribution}\\\/ across the different $\\phi$ fields.  As discussed in  \nSect.~\\ref{sec:TheModel}, one cannot resolve the individual contributions to the total energy density $\\rho$\nduring times when the mass matrix is appreciably changing, since the non-vanishing time-derivatives of \n${\\cal M}_{k\\ell}^2$ introduce new terms into $\\rho$ which prevent its simple decomposition into the form in\nEq.~(\\ref{energydecomp}).\nHowever, at late times $t$ for which \\il{(t-t_G)\/\\Delta_G \\gg 1} ---\n{\\it i.e.}\\\/, at times when the phase transition is largely completed --- \nthe corresponding mass matrix ${\\cal M}_{k\\ell}^2$ becomes essentially time-independent.\nThe mass-eigenstate fields $\\phi_\\lambda$ then decouple from each other and virialize.\nAt such late times, a decomposition\nsuch as that in Eq.~(\\ref{energydecomp})\nbecomes possible, with each such late-time energy density contribution $\\rho_\\lambda$ given in Eq.~(\\ref{rholambda}).\nThus, in order to characterize the late-time energy configuration of \nthe system with $N$ fields, we can calculate not only \n${\\overline{\\rho}}$ but also the individual contributions ${\\overline{\\rho}}_\\lambda$ corresponding to the\n$N$ individual mass eigenstates $\\phi_\\lambda$.\n\nFinally, following Ref.~\\cite{DDM1}, another useful quantity we may define\nis the so-called {\\it tower fraction} $\\eta$.   This quantity measures the \nfraction of the total energy density which is carried by all but the most abundant mode in the tower:\n\\begin{equation}\\label{eq:etadefinition}\n{\\overline{\\eta}} ~\\equiv~ 1 - \\max\\limits_{\\lambda}\\left\\{\\frac{{\\overline{\\rho}}_{\\lambda}}{{\\overline{\\rho}}}\\right\\} \\ .\n\\end{equation}\nFor large but finite $N$, the tower fraction $\\eta$ takes values within \\il{0 \\leq \\eta < 1}\nand can be viewed as quantifying the extent to which our system really has multiple components, with each carrying\nsome relevant portion of the total energy density.   If $\\eta$ is extremely close to zero, \nthen almost all of the total energy density is captured within a single field, rendering the other fields \nlargely irrelevant from a phenomenological point of view.    For this reason we shall be most interested in systems with larger values of $\\eta$. \n  \nGiven this, we shall analyze the general-$N$ system by tracing its energy flow, moment by moment through the mass-generating \nphase transition, ultimately evaluating the total late-time energy density\n${\\overline{\\rho}}$, its distribution across the individual contributions ${\\overline{\\rho}}_\\lambda$, and the corresponding late-time value of ${\\overline{\\eta}}$ --- \nall as functions of $m$, $M_c$, $t_G$, and $\\delta_G$.\n\n\\subsection{Instantaneous phase transition\\label{instantsection}}\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.9\\textwidth]{initialabruptabundance.pdf}\n\\caption{The abundance fractions $\\rho_\\lambda\/\\rho$ at time \\il{t=t_G}, evaluated within \nthe abrupt \\il{\\delta_G\\rightarrow 0}\nlimit and plotted as functions of ${\\overline{m}}\/M_c$ for \\il{N=10} (left panel) and \\il{N=100} (right panel).\nIn both panels we observe that the lightest mode carries the largest fractional abundance for small\n$m\/M_c$, but that this abundance is more fully distributed amongst the modes as\n$m\/M_c$ increases.   For finite $N$, this abundance then ultimately collects in the {\\it heaviest}\\\/\nmode as \\il{m\/M_c\\to\\infty}.    This last feature is ultimately an artifact of the truncation to only\nfinitely many modes, and disappears in the \\il{N\\to\\infty} limit.   In all cases, \nthe sum of the fractional abundances shown is equal to $1$ for all $m\/M_c$.} \n\\label{fig:initialabruptabundance}\n\\end{figure*}\n\nIt turns out that we can actually perform this calculation analytically in the special abrupt-transition \nlimit \\il{\\delta_G\\to 0}.\nIndeed, in this limit our mass matrix ${\\cal M}_{k\\ell}^2$ is time-independent both before and after $t_G$, and all that \nwe need calculate is the sudden change of \nbasis \nfrom the KK-eigenstate basis $\\phi_k$ which is appropriate before $t_G$\nto the mass-eigenstate basis $\\phi_\\lambda$ which is appropriate after.\nPrior to $t_G$, only $\\phi_{k=0}$ has an initial displacement \\il{\\langle \\phi_0\\rangle}: \nthis field is massless prior to $t_G$, and thus the system has no energy and none of the fields oscillate.\nHowever, for times after $t_G$, our mass eigenstates $\\phi_\\lambda$ become\n\\begin{equation}\n          \\phi_\\lambda ~=~ \\sum_{k=0}^{N-1} U_{\\lambda k}\\, \\phi_k~\n\\label{basischange}\n\\end{equation}\nwhere $U_{\\lambda k}$ is the time-independent but $N$-dependent \nunitary basis-change matrix that diagonalizes ${\\cal M}_{k\\ell}^2$.\nThe initial conditions in Eq.~(\\ref{eq:initialconditions})\nthen become\n\\begin{eqnarray}\n        && \\phi_\\lambda(t_G)  ~=~  U_{\\lambda 0} \\, \\langle \\phi_0\\rangle~ \\nonumber\\\\  \n        && \\dot \\phi_\\lambda(t_G) ~=~ 0~, \n\\end{eqnarray} \nwhereupon our total energy density at \\il{t=t_G} is given by  \n\\begin{eqnarray}\n      \\rho(t_G) &=&  \\sum_\\lambda \\rho_\\lambda (t_G) = {\\textstyle{1\\over 2}} \\sum_\\lambda \\lambda^2 \\phi_\\lambda^2 \n                 \\nonumber\\\\\n             &=& {\\textstyle{1\\over 2}} \\, \\sum_\\lambda \\, \\lambda^2 \\, U_{\\lambda 0}^2 \\, \\langle \\phi_0\\rangle^2 ~.\n\\end{eqnarray}\nHowever, for all $N$,  our basis-change matrix $U_{\\lambda k}$ satisfies the remarkable identity\n\\begin{equation}  \n              \\sum_\\lambda \\, \\lambda^2 \\, U_{\\lambda 0}^2 ~=~ m^2 ~\n\\label{magic}\n\\end{equation}\nwhich completely eliminates from $\\rho(t_G)$ what has otherwise been a highly non-trivial \ndependence on $M_c$ (and $N$).\nWe thus find that our phase transition at \\il{t=t_G} injects \ninto our $N$-mode system \na total energy density  \n\\begin{equation}\n            \\rho(t_G) ~=~ {\\textstyle{1\\over 2}} \\, m^2 \\, \\langle \\phi_0\\rangle^2~, \n\\label{rhotott}\n\\end{equation}\nwith individual fractional contributions\n\\begin{equation}\n           {\\rho_\\lambda(t_G)\\over \\rho(t_G)} ~=~ \\left( \\lambda\\over m\\right)^2 U_{\\lambda 0}^2~.\n\\label{rhoconts}\n\\end{equation}\nNote that because we are working in the abrupt \\il{\\delta_G\\to 0} limit,\nthe mass $m$, eigenvalues $\\lambda$, and basis-change matrix $U_{\\lambda k}$\nin Eqs.~(\\ref{basischange}) through (\\ref{rhoconts})\nare constant for all times after $t_G$ and thus equal to their late-time values:\n\\il{m={\\overline{m}}}, \\il{\\lambda={\\overline{\\lambda}}}, and\n\\il{U_{\\lambda k} = \\overline U_{\\lambda k}}.\n\nIn general, we see from Eq.~(\\ref{rhotott}) that the total energy density\ninjected into our system grows polynomially with $m$.\nHowever, the individual contributions to this total energy density \nvary non-trivially as a function of $m\/M_c$ --- even when expressed \nas a fraction relative to the total.\nThese individual fractional contributions to the total energy density \nat \\il{t=t_G}\nare plotted as functions of $m\/M_c$ \nfor \\il{N=10} in the left panel of Fig.~\\ref{fig:initialabruptabundance} \nand\nfor \\il{N=100} in the right panel of Fig.~\\ref{fig:initialabruptabundance}. \nIn each case, we see that these energy-density distributions\nare highly sensitive to $m\/M_c$,\nwith the majority of the energy density at \\il{t=t_G}\ncarried by the lightest mode for small \\il{m\/M_c}\n(as consistent with our expectation from the 4D limit).\nHowever, as $m\/M_c$ increases,\nthis picture changes, with more and more of the states beginning to carry\ngreater fractions of the total energy density.\nNote that for all values of $m\/M_c$, the sum of these fractional\nenergy densities is fixed at $1$.\n\nUltimately, for very large \\il{m^2\/M_c^2\\gsim {\\cal O}(\\sqrt{N})},\nthe majority of the energy density \nbegins to collect in the {\\it heaviest}\\\/ mode.\nHowever, we have already seen in Sect.~\\ref{sec:TheModel}\nthat this is precisely the regime in which the heaviest eigenvalue $\\lambda_{N-1}$ \nbegins to diverge.\nIndeed, as discussed in Sect.~\\ref{sec:TheModel}, \nthis feature is ultimately an artifact of truncating our system to a finite\nnumber of modes, with \\il{N<\\infty}.\nA similar situation persists for the corresponding energy densities.\nEven though the energy density associated with the heaviest mode tends to dominate\nin the region with \\il{m^2\/M_c^2\\gsim {\\cal O}(\\sqrt{N})},\nthis region becomes increasingly remote\nas \\il{N\\to \\infty} and effectively vanishes.\nThus, for \\il{m\/M_c\\gg 1} and \\il{N\\to\\infty}, all of the modes in our theory \ntend to share the total energy density\nessentially equally.\n\n\\begin{figure}[t]\n\\includegraphics[keepaspectratio, width=0.5\\textwidth]{etaRAPID_vsmBAR.pdf}\n\\caption{The tower fractions $\\eta(t_G)$ corresponding to the fractional\nabundances shown in Fig.~\\ref{fig:initialabruptabundance},\nplotted as functions of $m\/M_c$ for different values of $N$.\nWe see that the tower fraction begins near zero for small $m\/M_c$ but then begins to grow as\n$m\/M_c$ increases.    For finite $N$, the tower fraction ultimately hits a peak before returning\nto zero at large $m\/M_c$, while in \nthe \\il{N\\to\\infty}  limit the tower fraction continues monotonically to $1$.}\n\\label{fig:towerfraction}\n\\end{figure}\n\nIn Fig.~\\ref{fig:towerfraction}\nwe show the\ncorresponding values of the tower fraction $\\eta(t)$ at \\il{t=t_G}, plotted \nas functions of $m\/M_c$.   \nAs we see, $\\eta(t_G)$ is non-monotonic as a function of\n$m\/M_c$ for all finite values of $N$, starting near \\il{\\eta\\approx 0} for small\n$m\/M_c$ and reaching a maximum at a certain critical\n$N$-dependent value $(m\/M_c)_{\\rm peak}$  \nbefore declining back to \\il{\\eta\\approx 0} for large $m\/M_c$.\nThis non-monotonic behavior for finite $N$ is ultimately a result of the fact\nthat the identity of the mode carrying the maximum abundance is itself\na function of $m\/M_c$, with the lightest mode carrying the largest\nabundance for small $m\/M_c$ but the heaviest mode carrying the largest\nabundance for large $m\/M_c$.   However, as \\il{N\\to\\infty}, we find\nthat the magnitude of the peak tends towards unity while\nthe position of peak itself shifts towards increasingly large values of $m\/M_c$.\nIndeed, with excellent precision, we find that the position of the peak is given by\n\\begin{equation}\n    \\left( {m\\over M_c}\\right)_{\\rm peak} \\approx~ {4\\over 9} \\, N^{4\/9}~,\n\\end{equation}\nwhereupon it follows that \\il{(m\/M_c)_{\\rm peak}\\to\\infty} as \\il{N\\to \\infty}.\nWe thus find that $\\eta(t_G)$ becomes completely monotonic in the infinite-$N$ \nlimit, transitioning from \\il{\\eta\\approx 0} at small $m\/M_c$ to \\il{\\eta \\approx 1} at large $m\/M_c$.\n\nThe results illustrated in Fig.~\\ref{fig:initialabruptabundance} and \\ref{fig:towerfraction}\napply to the energy configuration \nof our system at \\il{t=t_G}.\nA natural question, however, concerns the extent to which these results apply for the {\\it late-time}\\\/\nenergy-density fractions ${\\overline{\\rho}}_\\lambda\/{\\overline{\\rho}}$ and late-time tower fraction ${\\overline{\\eta}}$. \nEven though the mass matrix $M_{k\\ell}^2$ and  associated eigenvalues \nare time-independent for all \\il{t> t_G} in the abrupt \\il{\\delta_G\\to 0} limit,   \nthe mass eigenstates $\\phi_\\lambda(t)$ \nand corresponding energy densities $\\rho_\\lambda(t)$ nevertheless \ncontinue have a non-trivial time-dependence.    In general, \na single field $\\phi_\\lambda$ with constant mass $\\lambda$\nremains essentially fixed for all \\il{t < t_\\zeta^{(\\lambda)} \\equiv\\kappa\/(2\\lambda)}\nand then transitions to damped-oscillatory behavior for \\il{t> t_\\zeta^{(\\lambda)}}.\nLikewise, the corresponding energy density is essentially fixed for  \n\\il{t < t_\\zeta^{(\\lambda)}}\nand then decays as $t^{-\\kappa}$ for \n\\il{t > t_\\zeta^{(\\lambda)}}.\n(Exact analytical results for the fields and energy densities can be found\nin Appendix~A and Fig.~1 of Ref.~\\cite{TwoTimescales}.)\nThus, in our present situation with an abrupt phase transition at \\il{t=t_G},\nall of our fields \nwill already be underdamped and commence oscillations \nsimultaneously \nat $t_G$\nas long as \\il{t_\\zeta^{(\\lambda)} \\leq t_G} for all $\\lambda$. \nIn such cases, the ratios $\\rho_\\lambda\/\\rho$ as well as the corresponding\ntower fraction $\\eta$ will be essentially time-independent, with the same values\nat late times as they have at $t_G$.\n\n\\subsection{Phase transition with arbitrary width \\texorpdfstring{$\\delta_G$}{delta_G}}\n\nThe results given above apply only in the abrupt \\il{\\delta_G\\to 0} limit.\nHowever, \nwe now wish to explore the full parameter space  and calculate our total and fractional\nlate-time energy  densities and tower fractions throughout the \n\\il{({\\overline{m}} t_G, \\delta_G)} plane.\nWe already did this for the \\il{N=1} limiting case in Sect.~\\ref{sec:FourDimensionalLimit},\nwhere we found that the effects of taking \\il{\\delta_G>0} tended to produce either\nsuppressions or enhancements in the resulting late-time energy \\il{{\\overline{\\rho}}_{\\rm 4D}(\\delta_G)} compared\nwith $\\rho_{\\rm 4D}(0)$  depending on the precise values of ${\\overline{m}} t_G$ and $\\delta_G$.\nThese results were shown in the left panel of Fig.~\\ref{fig:singlefielddensityplot}.\nWe therefore now seek to know what additional effects \nbeyond those in Fig.~\\ref{fig:singlefielddensityplot}\nemerge from considering arbitrary values of $N$ rather than merely \\il{N=1}.\n\n\\begin{figure*}[bht]\n\\includegraphics[keepaspectratio, width=0.9\\textwidth]{finitemodedensityplotsA.pdf}\n\\caption{The total late-time abundance ${\\overline{\\rho}}$ for $N$ modes, \nevaluated within the \\il{({\\overline{m}} t_G,\\delta_G)} plane\nfor \\il{N=2} (left panel) and \\il{N=10} (right panel)\nand expressed as a fraction of the \ncorresponding value ${\\overline{\\rho}}_{\\rm 4D}$ for the 4D \\il{N=1} special case.\nWe have chosen \\il{t_G=10^2\/M_c} as a reference value for both plots above.\nWe see that the introduction of additional modes relative to the 4D special case\neither {\\it preserves}\\\/ the 4D result nearly exactly for \\il{{\\overline{m}} t_G\\lsim 10}  or\n{\\it suppresses}\\\/ it for \\il{{\\overline{m}} t_G\\gsim 10}, with this suppression becoming\nincreasingly severe for larger ${\\overline{m}} t_G$ (or equivalently, larger ${\\overline{m}}\/M_c$)\nand \\il{\\delta_G\\lsim 1}.  Note that the gray regions in the upper right corners of\nthese panels (and in similar subsequent figures throughout this paper) \nare excluded for the reasons discussed in the paragraph below Eq.~(\\ref{bigwidth}).  }\n\\label{fig:finitemodedensityplotsgenN}\n\\end{figure*}\n\nOur results are shown in Fig.~\\ref{fig:finitemodedensityplotsgenN} for\n\\il{t_G=10^2\/M_c}.\nRemarkably, \nfor that portion of the \\il{({\\overline{m}} t_G,\\delta_G)} parameter space\n  with \\il{{\\overline{m}} t_G\\lsim 10}, \n{\\it we find that the total late-time energy density ${\\overline{\\rho}}$\nis virtually identical to what it would have been for only one mode!}\\\/ \nOf course, this behavior is consistent with what we already saw for the \nabrupt \\il{\\delta_G\\to 0} limit and very small $m\/M_c$, but what is remarkable\nis that this behavior extends even for non-zero $\\delta_G$ and larger $m\/M_c$\nas well.\nMoreover, for \\il{{\\overline{m}} t_G\\gsim 10}, \nwe find that our\nlate-time energy density is actually\nsuppressed rather than enhanced\nrelative to our 4D expectations --- \nall this despite the presence of extra modes whose masses are also lifted by \n(and which therefore also receive an additional energy-density insertion from)\nthe mass-generating phase transition.\n\nThese results may be understood as follows.   \nSince we have taken \\il{t_G= 10^2\/M_c} as a reference value,\nwe see that \\il{{\\overline{m}} t_G = 10^2 \\, {\\overline{m}}\/M_c}.\nThus values \\il{{\\overline{m}} t_G \\lsim 10^2} correspond to \n\\il{{\\overline{m}}\/M_c\\lsim 1}, and we have already seen that \nthe bulk of the energy density remains concentrated  \nin the lightest mode for such values of ${\\overline{m}}\/M_c$.\nAs a result, \nour system is functionally no different than the single-mode 4D system\nfor small values of ${\\overline{m}} t_G$. \nBy contrast, as we increase the value of ${\\overline{m}} t_G$, the total energy density \nof our system is more equally distributed across the different modes.\nThe oscillations of the heavier modes then dissipate the\nenergy density more rapidly than the lighter modes during the phase transition, resulting in a more\nrapid dissipation of the total energy density and thus an overall suppression \nof the late-time energy density as compared with 4D expectations.\nIt also is important to note that this latter effect requires \\il{\\delta_G>0}, even\nfor large ${\\overline{m}} t_G$.\nIndeed, if \\il{\\delta_G=0}, our energy density is partitioned across the different\nmodes precisely as described in Sect.~\\ref{instantsection},\nwhereupon the unitarity relation in Eq.~(\\ref{magic})\nremoves all dependence on $N$ and ensures the same results as we would\nhave had in 4D!\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.9\\textwidth]{finitemodedensityplotsB.pdf}\n\\caption{The total late-time tower fraction ${\\overline{\\eta}}$ corresponding to the panels\nshown in Fig.~\\protect\\ref{fig:finitemodedensityplotsgenN}.\nIn each case we see that the \\il{({\\overline{m}} t_G,\\delta_G)} plane is subdivided into\ntwo disjoint \\il{{\\overline{\\eta}}\\approx 0} regions (white) by a narrow \\il{{\\overline{\\eta}}>0} ribbon (green):    \nthe lightest mode carries the bulk of the energy density in the region to the left of the ribbon,\nwhile the heaviest mode carries the bulk of the energy density in the region to the right.\nInterestingly, the division between these two regions \nis non-monotonic as a function of not only ${\\overline{m}} t_G$ but also $\\delta_G$.}\n\\label{fig:finitemodedensityplotsgenN2}\n\\end{figure*}\n\nIn Fig.~\\ref{fig:finitemodedensityplotsgenN2} we plot the corresponding values of the\nlate-time tower fraction ${\\overline{\\eta}}$.\nAs evident from this figure, the \\il{({\\overline{m}} t_G, \\delta_G)} plane is subdivided\ninto two disjoint \\il{{\\overline{\\eta}} \\approx 0} regions by \na fairly narrow ``mountain range'' along which \\il{{\\overline{\\eta}}>0}.\nAs we already determined in Sect.~\\ref{instantsection} for the \\il{\\delta_G\\to 0} limit,\nthe region with smaller ${\\overline{m}} t_G$ (or equivalently smaller ${\\overline{m}}\/M_c$) \ncorresponds to the region in which the lightest mode carries all of the energy density.\nBy contrast, as we dial ${\\overline{m}} t_G$ towards greater values, the energy density\nis increasingly distributed  \nacross all the modes, resulting in a greater tower fraction. \nUltimately, however, as we dial \\il{{\\overline{m}} t_G \\to\\infty} in our finite-$N$ system,\nall of the energy density finds itself in the heaviest mode, and the tower fraction returns\nto zero.  This behavior is consistent with that shown in \nFig.~\\ref{fig:towerfraction} for the \\il{\\delta_G\\to 0} limit.\n\nThis explains the non-monotonicity of the late-time tower fraction ${\\overline{\\eta}}$ \nas a function of ${\\overline{m}} t_G$.\nHowever, we additionally learn from Fig.~\\ref{fig:finitemodedensityplotsgenN2} that \nthe late-time tower fraction is also non-monotonic as a function of $\\delta_G$.\nThis too is relatively straightforward to explain.\nFor ${\\overline{m}} t_G$ sufficiently large and for small $\\delta_G$, we have already seen that the bulk of the\nenergy density is carried by the heaviest mode. \nHowever, as we increase $\\delta_G$, we are increasing the timescale associated with our phase \ntransition --- {\\it i.e.}\\\/, the timescale over which energy is pumped into the system\nand each mode becomes populated.    \nEventually this timescale reaches the timescale associated with the oscillations of the heaviest mode.\nThis then suppresses the energy transfer into the (nevertheless dominant) heaviest mode, \nthereby increasing the tower fraction. \nIndeed, as we increase $\\delta_G$ still further, we begin to suppress the energy densities of lighter\nand lighter modes, and this too has the effect of increasing the tower fraction.\nEventually, however,\nas we continue to increase $\\delta_G$, \nwe reach a point\nwhere we have suppressed the majority of the modes and the bulk of the energy density\nbegins collecting in the lightest mode.   Subsequent increases in $\\delta_G$ then only\nreinforce the dominance of the lightest mode, thereby causing \nthe tower fraction to return to zero.\nIndeed, as shown in Fig.~\\ref{fig:finitemodedensityplotsgenN2}, we have crossed the green \n``mountain range'', leaving us\nin the region in which the lightest mode carries all of the energy density.\n\nThus far, our discussion has focused on the regime with \\il{t_G M_c \\gg 1}.\nIn this regime, all of our excited modes are underdamped and have begun oscillating at very early \ntimes prior to the phase transition.   This will be true even for the lightest mode\nif ${\\overline{m}} t_G$ is chosen sufficiently large.\nHowever, if \\il{t_G M_c \\ll 1} and \\il{N\\gsim 1\/(t_G M_c)},  \nonly an upper subset of the modes will have begun oscillating\nprior to the phase transition --- \nthe lightest modes will remain overdamped and fixed.\nThus, the introduction of the heavier modes relative to the 4D case has the \neffect of distributing some of the energy density into heavier modes\nwhich begin oscillating earlier than the lightest mode, \nthereby enhancing the dissipation of the total energy\ndensity relative to the 4D case. \nIt is important to stress that this source of suppression is\ncompletely distinct from that discussed above, resulting instead from\ndifferences in the times at which individual modes begin oscillating.\nAs such, this effect --- which was originally discussed in Ref.~\\cite{DDGAxions}\nwithin the context of the \nabrupt \\il{\\delta_G\\to 0} limit --- is largely independent of $\\delta_G$.\n\n\\FloatBarrier\n\\section{Approaching Asymptotia: ~\\texorpdfstring{\\il{N\\to \\infty}}{N->infty}\\label{Approaching}}\n\n\n\\begin{figure*}\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{Nscaling.pdf}\n\\caption{\nThe total late-time abundance ${\\overline{\\rho}}$ and late-time tower fraction ${\\overline{\\eta}}$ of our $N$-mode system,\nplotted as functions of $N$ for different values of $\\delta_G$. The left and center panels show \nthe total late-time abundance expressed as a fraction of the value it would have in the abrupt \n\\il{\\delta_G\\to 0} and 4D limits, respectively, while the right panel shows the late-time tower fraction.\nFor all three panels we have taken  \\il{t_G=10^2\/M_c} and \\il{{\\overline{m}}=10^2 M_c}, thereby fixing \n\\il{{\\overline{m}} t_G=10^4} as a benchmark value. The blue curves are calculated using the usual ``IR-based'' \ntruncation which has been employed thus far in Sects.~\\ref{sec:TheModel} through \\ref{sec:FiniteModeNumber},\nwhile the red curves are calculated using the alternative ``UV-based'' truncation to be discussed in \nSect.~\\protect\\ref{alttrunc}; both truncations have the same \\il{N\\to\\infty} asymptotic behavior\nbut the UV-based truncation reaches asymptotia far more rapidly and smoothly. The black-dashed line that \noccurs in each of the energy-density panels indicates the results of the multi-component adiabatic \napproximation of Eq.~\\eqref{eq:multiadiabaticapproximation} which serves as a lower bound for the energy \ndensity of the ensemble.}\n\\label{fig:Nscaling}\n\\end{figure*}\n\nWe now wish to study how our $N$-mode system evolves as a function of $N$, \nwith an eye towards understanding the asymptotic behavior of our system for large \\il{N\\gg 1}.\nThis will ultimately enable us to extract the behavior of the \nfull \\il{N=\\infty} Kaluza-Klein tower, as we shall do in Sect.~\\ref{sec:KKLimit}.~ \nThis will also enable us to understand the effects that come from truncating\nthe KK system to finite but large $N$. \n\n\\subsection{Large-\\texorpdfstring{$N$}{N} behavior:  The road to asymptotia\\label{road}}\n\nIn Fig.~\\ref{fig:Nscaling}, the blue curves indicate the $N$-dependence of our \ntwo quantities of interest, namely\nthe late-time energy density ${\\overline{\\rho}}$ of our $N$-mode system \nas well as its corresponding late-time tower fraction ${\\overline{\\eta}}$.\nWhile the latter quantity is shown in the right panel,\nthe former is plotted in the left panel \nas a fraction of the value it would\nhave in the abrupt \\il{\\delta_G\\to 0} limit \nand in the middle panel \nas a fraction of its corresponding 4D value.\nFor all three panels we have taken  \\il{t_G=10^2\/M_c}\nand \\il{{\\overline{m}}=10^2 M_c}, thereby fixing \\il{{\\overline{m}} t_G =10^4} as a benchmark value.\nWe have then \nplotted the resulting curves \n(blue) as functions of $N$ \nfor different values of $\\delta_G$.\n(By contrast, the red curves will be discussed in Sect.~\\ref{alttrunc}.)\n\nAs expected, each of the blue curves shown in Fig.~\\ref{fig:Nscaling} eventually\nheads towards a finite asymptote as \\il{N\\to\\infty}.\nOne immediate observation from the left and middle panels of \nFig.~\\ref{fig:Nscaling}\nis that increasing the number of modes\nin our system \ngenerally results in an {\\it increasing suppression}\nof the total late-time energy density when compared to the\nvalues this energy density would have in the abrupt \\il{\\delta_G\\to 0} or\n4D limits.\nThus, {\\it the larger the value of \n$N$, the more the abrupt and 4D approximations fail to accurately\nestimate the late-time energy density}\\\/.\nThis is an important result, given that most approaches to calculating\nthe energy densities of such multiple-component systems in the literature\nassume that the phase transition occurs when the fields are still overdamped.\nCompared with these static fields, any such phase transition is therefore essentially occurring\nwith an infinitely short timescale $\\delta_G$, and is thereby functionally equivalent to the abrupt\napproximation.    The results shown in Fig.~\\ref{fig:Nscaling} \nthus illustrate what happens as one moves away from these assumptions. \n\nIn this vein, it is perhaps also worthwhile to consider the adiabatic \napproximation which is traditionally applied in single-component\nscalar theories undergoing mass-generating phase transitions.  \nOf course, while an adiabatic approximation can be realized\nfor a single mode, we do not expect such an approximation to be appropriate\nfor a large collection of modes with vastly different masses and non-trivial mixings.\nDespite this, we can nevertheless imagine limiting cases in which\nall excited modes above the lightest mode are\ndrained of significant energy density.    \nIn such a regime we can then imagine applying the adiabatic approximation \nto the only surviving mode $\\phi_{\\lambda_0}$ in the ensemble, leading to a\ndefinition\n\\begin{equation}\n\\rho(t)|_{\\rm ad}  ~\\equiv ~ \\frac{1}{2}\\langle\\phi_0\\rangle^2 \\lambda_0(t^{(\\lambda_0)}_{\\zeta}) \\, \n\\lambda_0(t)\\left[\\frac{a(t^{(\\lambda_0)}_{\\zeta})}{a(t)}\\right]^3 \\ ,\n\\label{eq:multiadiabaticapproximation}\n\\end{equation}\nwhere $t^{(\\lambda)}_{\\zeta}$ is defined as the time at which \\il{3H(t) = 2\\lambda(t)}\nand where we implicitly assume \\il{\\delta_G=1} when evaluating $\\lambda(t)$.\nWe have already seen in the left panel of Fig.~\\ref{fig:singlefielddensityplot} that the\nadiabatic approximation sets a lower bound for the energy density associated with\na single mode --- indeed, in all other regions of parameter space the energy density \nis either the same or enhanced.\nLikewise, the approximation in Eq.~(\\ref{eq:multiadiabaticapproximation})\nconsists of disregarding whatever energy density might reside in the excited modes.\nThus, we expect the adiabatic approximation in \nEq.~(\\ref{eq:multiadiabaticapproximation})\nto provide us with a lower bound on the total energy density of our $N$-mode\nsystem throughout all regions of parameter space.\n\nThe dashed-black curves in the left and middle panels of Fig.~\\ref{fig:Nscaling} show the \nbehavior of Eq.~\\eqref{eq:multiadiabaticapproximation} as a function of $N$. \nIndeed, as expected, we see that \\emph{the adiabatic approximation serves \nas a lower bound on the total energy density of the tower}, just as it does in the single-field\nscenario.  Indeed, we see from Fig.~\\ref{fig:Nscaling} \nthat the adiabatic limit seems to be reached most directly for large $N$\n(such as for a full KK tower) and \\il{\\delta_G\\to 1}.\nBy contrast, the results in other regions of parameter space differ significantly from this limit.\n\n\\subsection{A new truncation of KK theory:  An alternate road to asymptotia\\label{alttrunc}}\n\nOur results in Fig.~\\ref{fig:Nscaling} clearly illustrate a successful road to asymptotia,\nas each of the blue curves heads towards a finite asymptotic value as \\il{N\\to \\infty}.\nThis is not a surprise, since\nwe began our analysis by truncating our infinite-dimensional KK mass matrix\n${\\cal M}_{k\\ell}^2$ in Eq.~(\\ref{kmassmatrix}) by \nretaining only its first $N$ rows and columns.\nThus it is natural that taking $N\\to\\infty$ restores the physics of the infinite-matrix limit.\nIndeed, truncating our mass matrix in\nEq.~(\\ref{kmassmatrix}) \nto only its first $N$ rows and columns\nrepresents one way of constructing a finite $N$-mode theory \nwhose \\il{N\\to\\infty} limit reproduces the physics of the full KK tower \nat any moment in time.  \n\nHowever, this is not a unique truncation to a finite $N$-mode theory.\nThere is, in fact, an alternative truncation \nof our full KK theory at any moment in time which also has only $N$ modes and which also reproduces the physics\nof the full KK tower as \\il{N\\to \\infty}, yet yields significantly different physical results for finite $N$.\nIndeed, as we shall see, this alternate truncation exhibits an even more rapid path\nto asymptotia, yielding what are essentially the full infinite-$N$ values of\nthe late-time energy densities and tower fraction for even smaller values of $N$\nthan are required using the standard truncation that we have employed thus far.\n\nRecall that our standard truncation began with the full, infinite-dimensional\nmass matrix ${\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{kmassmatrix}), expressed in the KK basis of the individual KK modes $\\phi_k$.\nThis matrix was then truncated to its first $N$ rows and columns.\nThe resulting \\il{N\\times N} matrix then defined our truncated KK theory,\nand all subsequent calculations proceeded from this truncated matrix.\nSpecifically, all calculations were performed directly from \nthe equations of motion~(\\ref{eq:KKequationsofmotion})\nin the KK basis (which has the advantage of representing a basis choice that does not change with time, even in the presence of the mass-generating phase transition), \nand the results at late times were \nconverted to the mass-eigenstate basis (such as for quoting late-time quantities such as ${\\overline{\\rho}}_\\lambda$ or ${\\overline{\\eta}}$ which\npertain to individual mass eigenstates) only at the final step.\n\nHowever, an alternative approach is {\\it not}\\\/ to truncate our infinite-dimensional mass matrix \nin the KK basis $\\phi_k$,  but rather to transform the infinite-dimensional matrix into its\nmass-eigenstate basis and then truncate the resulting {\\it mass-eigenstate}\\\/ matrix.\nIn general, this matrix will take the form\n\\begin{equation}\n\\mathcal{M}_{\\lambda\\lambda'}^2 ~=~ \n\\begin{pmatrix}[1.1]\n\\lambda_0^2 & 0 & 0 & \\cdots \\\\\n0 & \\lambda_1^2 & 0 & \\cdots  \\\\\n0 & 0 & \\lambda_2^2 & \\cdots  \\\\\n\\vdots & \\vdots & \\vdots & \\ddots\n\\end{pmatrix} \\ ,\n\\label{eq:massbasismassmatrix}\n\\end{equation}\nwhere the $\\lambda_k^2$ are the mass eigenvalues that are calculated in \nthe infinite-$N$ limit at any moment in time.   These eigenvalues will be discussed below.\n\nNote that truncating the mass matrix in the mass-eigenstate basis \nis mathematically different than truncating the mass matrix in the KK-eigenstate basis.\nOf course, these procedures are in some sense parallel in that they both\nbegin from equivalent infinite-dimensional matrices which describe the \nsame KK system and which are related to each other through a simple,\nunitary basis change.   \nHowever, their truncations to \\il{N\\times N} submatrices are mathematically different, and thus have\ndifferent physical effects.\n\nHaving truncated our mass-eigenstate matrix ${\\cal M}^2_{\\lambda \\lambda'}$\nin Eq.~(\\ref{eq:massbasismassmatrix}),\nour final step is to convert the resulting\nmatrix back to the KK basis.\nIn order to do this, we \nuse a similarly truncated version of the exact basis-change matrix $U_{\\lambda k}$ \nthat\nwould have related ${\\cal M}^2_{k\\ell}$ and ${\\cal M}^2_{\\lambda\\lambda'}$ in the full infinite-dimensional \nlimit.\nSpecifically, we define\n\\begin{equation}\n     \\widetilde {\\cal M}^2_{k\\ell} ~\\equiv ~ \\sum_{\\lambda,\\lambda'} \\, \n              (\\widehat U^\\dagger)_{k\\lambda} \\, \\widehat {\\cal M}^2_{\\lambda\\lambda'}  \\, \\widehat U_{\\lambda' \\ell} \n\\label{tildeM}\n\\end{equation}\nwhere $\\widehat{\\cal M}^2_{\\lambda\\lambda'}$ and $\\widehat U_{\\lambda'\\ell}$ here represent the first \n\\il{N\\times N} submatrices within the exact infinite-$N$ matrices \n${\\cal M}^2_{\\lambda\\lambda'}$ and $U_{\\lambda k}$ respectively.\nNote that since $\\widehat U$ is a truncated version of the unitary infinite-$N$ matrix $U$,\nthe truncated matrix $\\widehat U$ is not unitary by itself.   In particular, $\\widehat U$ is {\\it not}\\\/ the matrix \nthat diagonalizes the new matrix $\\widetilde {\\cal M}^2_{k\\ell}$ defined in Eq.~(\\ref{tildeM}), and thus, strictly speaking, \nwe should not regard $\\widetilde {\\cal M}^2_{k\\ell}$\ndefined in Eq.~(\\ref{tildeM})\nas representing the KK-basis version of $\\widehat {\\cal M}^2_{\\lambda\\lambda'}$.\nWe can nevertheless proceed to use $\\widetilde{\\cal M}^2_{k\\ell}$ directly in our equations of motion~(\\ref{eq:KKequationsofmotion}), \ntreating $\\widetilde{\\cal M}^2_{k\\ell}$ as we would any other mass matrix.\nConverting our final results \ninto statements about individual mass eigenstates \nat late times is then done in the usual way by \ndiagonalizing $\\widetilde{\\cal M}^2_{k\\ell}$ and calculating its mass eigenvalues and eigenvectors.\nNote that these eigenvalues will generally differ from the $\\lambda_i$ which\nappear in the \n$\\widetilde{\\cal M}^2_{k\\ell}$ matrix\nand only approach these $\\lambda_i$ as \\il{N\\to \\infty}.\n\nBecause this alternate truncation of our KK theory\nutilizes the exact eigenvalues $\\lambda_k$  and exact basis-change matrix elements $U_{\\lambda k}$ corresponding\nthe full KK theory, it may at first glance seem that this    \nalternate truncation cannot be realized in practice.\nHowever, it turns out to be relatively straightforward to solve the eigensystem corresponding\nto the full infinite-dimensional mass matrix ${\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{kmassmatrix}) --- not only \nnumerically, but even analytically.\nOne finds~\\cite{DDGAxions,DDM1} \nthat the mass eigenvalues $\\lambda_i$ at any moment in time are the (infinite) set\nof solutions to the transcendental equation\n\\begin{equation}\n         {\\pi \\lambda \\over M_c} \\, \\cot\\left( \\pi \\lambda\\over M_c\\right) ~=~ {\\lambda^2\\over m^2}~,\n\\label{eigeq}\n\\end{equation}\n and likewise the corresponding $U_{\\lambda k}$ matrix is given by\n\\begin{equation}\n            U_{\\lambda k} ~=~ \\left(  {r_k \\lambda^2 \\over \\lambda^2 - k^2 M_c^2} \\right) \\, A_\\lambda\n\\end{equation}\nwhere the $r_k$ are defined below Eq.~(\\ref{eq:orbifoldmodeexpansion}) and\nwhere \n\\begin{equation}\n           A_\\lambda~\\equiv  ~  {\\sqrt{2} \\, m^2 \\over \\lambda \\sqrt{ \\lambda^2 + m^2 + \\pi^2 m^4\/M_c^2 }}~.  \n\\label{Adefn}\n\\end{equation}\nFollowing the procedure outlined above, we then find that our alternate mass matrix\nis given by\n\\begin{equation}\n          \\widetilde {\\cal M}^2_{k\\ell} ~=~  \\sum_{\\lambda=\\lambda_0}^{\\lambda_{N-1}}  { r_k r_l  \\, A_\\lambda^2  \\, \\lambda^6 \\over\n                  (\\lambda^2 - k^2 M_c^2)\n                  (\\lambda^2 - \\ell^2 M_c^2)}~.\n\\label{massmatrixnew}\n\\end{equation}\nIn this connection, we remark \nthat although constructing $\\widetilde {\\cal M}^2_{k\\ell}$ requires  \nexplicit knowledge \nof the exact eigenvalues $\\lambda_i$ and matrix elements $U_{\\lambda k}$, \nthis in no way implies that we \nhave already solved the problems we originally set out to investigate.\nIndeed, our goals are far deeper than mere KK spectroscopy and instead pertain  \nto understanding the dynamical energy flow within our KK system in the presence of \nmode-mixing and mass-generating phase transitions.\n\nAlthough each element of the matrix $\\widetilde{\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{massmatrixnew}) depends on $N$ \nexplicitly through the upper limit of the $\\lambda$-summation, \nthis matrix smoothly reproduces our original ${\\cal M}^2_{k\\ell}$ mass matrix  \nas \\il{N\\to\\infty}.\nIndeed, through clever use of the eigenvalue equation~(\\ref{eigeq}), \ncotangent summation identities such as  \n\\begin{equation}\n      \\sum_{k=0}^\\infty {2 \\lambda^2 \\over \\lambda^2-k^2 M_c^2} ~=~ 1 \n         + {\\pi \\lambda \\over M_c} \\, \\cot\\left( \\pi \\lambda\\over M_c\\right)~, \n\\end{equation}\nand unitarity relations such as that in Eq.~(\\ref{magic}),\nit is possible to demonstrate explicitly that performing the summation in Eq.~(\\ref{massmatrixnew})\nover the {\\it full}\\\/ spectrum ({\\it i.e.}\\\/, taking \\il{N\\to\\infty}) \nreproduces our original mass matrix in Eq.~(\\ref{kmassmatrix}), as it must by construction.\nFor example, in what is perhaps the simplest case, we see from Eq.~(\\ref{massmatrixnew}) that\n\\begin{equation}\n         \\widetilde {\\cal M}^2_{00} ~=~   \\sum_{\\lambda=\\lambda_0}^{\\lambda_{N-1}}  A_\\lambda \\, \\lambda^2~.\n\\end{equation}\nIf the summation had proceeded over the entire infinite spectrum, the unitarity relation in\nEq.~(\\ref{magic}) would have given us the correct infinite-$N$ result \\il{{\\cal M}^2_{00} = m^2}.\nThus, we see in this case \nthat our UV-based truncation consists of gently draining away the contributions   \nto the unitarity sum that come from the most massive modes, thereby deforming this\nfirst matrix element in a gentle, $N$-dependent way.\nOther matrix elements are similar.\n\nThe new matrix $\\widetilde {\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{massmatrixnew}) \nthus defines an alternate truncation of the full KK theory. \nIndeed, both our original truncated mass matrix ${\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{kmassmatrix}) and our new mass matrix\n$\\widetilde {\\cal M}^2_{k\\ell}$ \nin Eq.~(\\ref{massmatrixnew})\ndescribe the same KK theory in their \\il{N\\to\\infty} limits.\nHowever, for any finite $N$ these mass matrices define distinct theories.   \nWe shall refer to our traditional truncation as being ``IR-based'', since it builds our finite-$N$ theory\ndirectly from the ground up, KK mode by KK mode, directly as they were in the full theory without regard for any \nof the UV physics.\nBy contrast, we shall refer to our alternative truncation as being ``UV-based'' in the sense that it utilizes\nthe full UV values of the mass eigenvalues and basis-changing matrices prior to truncation,\nand gently builds these quantities into our recipe for truncation. \nThis is thus more of a ``top-down'', UV-sensitive approach.\nOf course, it is only because of the mixing of KK states induced by the phase transition on the brane \nthat these two truncations are distinct.\n\nWhat makes this alternate UV-based truncation \nparticularly important for our purposes in this paper\nis that it provides a much more rapid road to asymptotia than does our usual truncation.\nIn other words, the asymptotic values of late-time quantities such as ${\\overline{\\rho}}_\\lambda$\nand ${\\overline{\\eta}}$ are approached more rapidly as functions of $N$\nvia our UV-based truncation than via the traditional IR-based truncation.\nWe can also see this from Fig.~\\ref{fig:Nscaling}.\nIn Fig.~\\ref{fig:Nscaling}, the blue curves illustrate \nthe IR-based approach to asymptotia for these quantities.\nHowever, in Fig.~\\ref{fig:Nscaling} we have also\nsuperimposed the red curves which represent the results of our  UV-based  \napproach to asymptotia for these same quantities.\nIn all cases, we see that the UV-based approach to asymptotia tends to differ\nsignificantly from the IR-based approach for small $N$.   \nIndeed, as $N$ increases, we see that the finite-$N$ energy densities associated\nwith the IR-based approach tend to approach their asymptotic limits {\\it from above}\\\/,\nwhile the finite-$N$ energy densities associated with the UV-based approach tend to approach\nthese same asymptotic limits {\\it from below}.\\\/ \nNevertheless, as \\il{N\\to\\infty},\nwe see that the UV-based truncation approaches \nasymptotia more rapidly (for smaller values of $N$) than does the IR-based truncation.\nWe also observe that the approach to asymptotia provided by the UV-based truncation\nis monotonic, particularly for the late-time tower fraction ${\\overline{\\eta}}$,\nwhereas the path provided by the IR-based approach is non-monotonic.    However, this too is \nstraightforward to understand.\nIn general, the non-monotonicity of \n${\\overline{\\eta}}(N)$ in the IR-based approach is due to the somewhat spurious effects \nof the highest mass eigenstate in the finite-$N$ system, \nas discussed in Sect.~\\ref{sec:TheModel} and sketched in \nFigs.~\\ref{fig:massspectrum} and \\ref{fig:initialabruptabundance}.  \nIn the UV-based approach, by contrast, the highest mode no longer behaves\nanomalously for finite $N$.\n\n\\begin{figure}[b]\n\\includegraphics[keepaspectratio, width=0.5\\textwidth]{lambdascaling}\n\\caption{Eigenvalues of the IR-based and UV-based mass matrices, plotted respectively in blue and red \nas functions of $N$ for \\il{m\/M_c=2}.  For visual clarity, only even eigenvalues are shown.\nIn each case \nthe blue and red curves asymptote to the exact eigenvalues $\\lambda_k$ as\n\\il{N\\to\\infty}, but the UV-based red curves generally approach this limit more rapidly and more smoothly \nthan do the IR-based blue curves.\nNote that each blue curve begins with an anomalously high eigenvalue;\nthis is nothing but the truncation artifact already sketched in Fig.~\\protect\\ref{fig:massspectrum}\nfor the highest mode $\\lambda_{N-1}$ in each case. \nBy contrast, such artifacts are entirely eliminated in the UV-based approach.  }\n\\label{fig:lambdascaling}\n\\end{figure}\n\nThese features are also readily apparent simply by comparing the eigenvalues \nof the IR-based truncated mass matrix ${\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{kmassmatrix}) with those\nof the UV-based truncated mass matrix $\\widetilde {\\cal M}^2_{k\\ell}$ in Eq.~(\\ref{massmatrixnew}) as functions of $N$.\nOur results are shown in Fig.~\\ref{fig:lambdascaling} for \\il{m\/M_c=2}. Once again, we see that the \nasymptotic \\il{N\\to\\infty} limit is approached more rapidly and more smoothly in the UV-based approach\n(red curves) than in the IR-based approach (blue curves).   Moreover, we see that the anomalous highest \neigenvalue which appears in the latter approach no longer exists in the former.\n\nAt the end of Sect.~\\ref{road}, we introduced a multi-component adiabatic \napproximation~(\\ref{eq:multiadiabaticapproximation}) and noted\nthat this quantity provides a lower bound on the total energy density of\nour $N$-mode system because it includes only the contributions from the lightest mode.\nLike other physical finite-$N$ quantities, however, the value of this quantity and its\ninterpretation as a lower limit depend \non the specific KK truncation chosen, since \nthe truncation in some sense determines what is meant by the lightest mode\nand whether its contributions are affected by the removal of the \nhigher modes.   \nThis is readily apparent in Fig.~\\ref{fig:Nscaling}, where the red ``UV-based''\nvalues of the total energy density are clearly \nsmaller for certain small values of $N$ than the ``IR-based''\nvalues of the adiabatic lower limit.\nThese red ``UV-based'' total energies nevertheless strictly exceed \nthe values of a corresponding ``UV-based'' adiabatic lower limit. \n\nWe conclude with a final comment.  Although we have presented our alternate UV-based \ntruncation of the full KK theory as providing a more rapid road to asymptotia, the \nexistence of such an alternate truncation is interesting in its own right. In any theory \ninvolving extra spacetime dimensions, one can never probe all energy scales and thereby \ndetect all KK modes.  Instead, we expect the physics of our full KK system to be \napproximately represented at low energies through some sort of truncation that focuses \non the lower modes. {\\it However, if the physics of our extra dimensions results in a \nmixing of KK modes (as must always arise in any theory which breaks translational invariance \nin the extra compactified dimension), we now see that there are multiple options for \nperforming such a truncation.}\\\/ Indeed, one could even argue that our UV-based truncation \nis more appropriate for certain calculations since it incorporates and thus is more sensitive \nto the actual masses of the physically propagating mass-eigenstates that we would expect \nto observe experimentally.  This last statement is of course subject to various \nrenormalization-group effects which could potentially deform the observed KK masses and \ncouplings, as discussed in Refs.~\\cite{sky1,sky2,sky3}, and which exist even for theories such as those \nconsidered in Refs.~\\cite{sky1,sky2,sky3} in which no KK-mixing is present. These observations nevertheless \npotentially give our UV-based truncation a theoretical importance in its own right that \nrenders it worthy of further study.\n\nOf course, there does exist a well-defined procedure through which one can unambiguously \ndescribe the physics of our full KK theory at low energies: one can use the methods of \neffective field theory (EFT).~ Specifically, one carefully integrates out the modes with energies \nabove a particular cutoff scale, thereby obtaining not only a truncated KK tower exhibiting \nrenormalized masses, but also a set of effective operators which reflect the underlying structure \nof the full theory.  In general, the structure of such an EFT can be quite complicated. \nThus, it is traditional in the literature to simply adopt the IR-truncated theory as an \napproximation to this EFT.~ Our point, then, is that our UV-based truncation might profitably \nserve as an alternative approximation to the complete EFT --- an approximation which, as we have \ndiscussed, may have certain advantages.  Needless to say, the UV and IR truncations, as well as \nthe complete EFT, all converge to the full KK theory as \\il{N\\rightarrow\\infty}.\n\n\n\\FloatBarrier\n\\section{KK Tower Limit: ~\\texorpdfstring{$N=\\infty$}{N=infty}\\label{sec:KKLimit}}\n\n\n\\begin{figure*}\n\\includegraphics[keepaspectratio, width=0.9\\textwidth]{infinitemodedensityplots.pdf}\n\\caption{The late-time energy density ${\\overline{\\rho}}$ \nof the full KK tower, plotted in the \\il{({\\overline{m}} t_G, \\delta_G)} plane for \\il{t_G= 10^2\/M_c}.\nThe left and right panels show contours of \n${\\overline{\\rho}}\/ {\\overline{\\rho}}(\\delta_G=0)$ and\n${\\overline{\\rho}}\/ {\\overline{\\rho}}_{\\rm 4D}$, respectively.}\n\\label{fig:infinitemodedensityplots}\n\\end{figure*}\n\nHaving studied our system as a function of $N$ for large \\il{N\\gg 1}, \nwe are now finally in a position to present our \nresults for the late-time energy density ${\\overline{\\rho}}$ and tower fraction ${\\overline{\\eta}}$\nfor the full, infinite KK tower. \nAs we shall see, certain features emerge in the full infinite-$N$ limit\nthat were not present for any finite $N$.\nTaken together, these results thus describe the late-time energy configuration\nacross our infinite KK tower when \nit has been subjected to a mass-generating phase transition \nof arbitrary width $\\delta_G$ and arbitrary magnitude (set by the late-time brane mass ${\\overline{m}}$)\nat time $t_G$ during its cosmological history. \n\nWe shall present our results in several different ways.\nFirst, in Fig.~\\ref{fig:infinitemodedensityplots},\nwe plot our results for the total late-time energy density ${\\overline{\\rho}}$ as a fraction of\n${\\overline{\\rho}}(\\delta_G=0)$ (left panel)\nor ${\\overline{\\rho}}_{\\rm 4D}$ (right panel).\nOur first observation \nfrom the results in Fig.~\\ref{fig:infinitemodedensityplots}\nis that while a quantity such as ${\\overline{\\rho}}(\\delta_G=0)$ \nmight indeed be a useful approximation for ${\\overline{\\rho}}$ which is \nvalid in certain regions of parameter space, \nsuch an approximation can fail badly in others.\nFor example, we see that the abrupt (\\il{\\delta_G=0}) approximation works\nwell for small \\il{{\\overline{m}} t_G\\lsim 0.1} even if $\\delta_G$ is sizable but \nis otherwise capable of either significantly underestimating or overestimating the \ntrue late-time energy density ${\\overline{\\rho}}$, the latter often by many orders\nof magnitude.\n\nAnother immediate observation is that the contours in\nFig.~\\ref{fig:infinitemodedensityplots}\nfollow a power-law type behavior in those\nregions of \\il{({\\overline{m}} t_G,\\delta_G)} parameter space in which we are eventually going to be the\nmost interested,\nnamely those regions with\n\\il{{\\overline{m}} t_G\\gsim 10} and \\il{\\delta_G\\lsim 0.3}\nin which multiple components contribute non-trivially to the total late-time abundance and \\il{{\\overline{\\eta}}>0}. \nThis power-law behavior enables us to extract approximate analytical expressions for\n${\\overline{\\rho}}\/ {\\overline{\\rho}}(\\delta_G=0)$ and\n${\\overline{\\rho}}\/ {\\overline{\\rho}}_{\\rm 4D}$ which are valid in these regions.\nSpecifically, given the results in Fig.~\\ref{fig:infinitemodedensityplots}\n(as well as analogous results calculated for different values of $t_G$),\nwe find\n\\begin{equation}\\label{eq:simplewidthnormapproximation}\n\\frac{{\\overline{\\rho}}(\\delta_G)}{{\\overline{\\rho}}(0)} ~\\approx~ \\frac{0.6}{\\delta_G}\\, \\frac{e^{-\\frac{1}{2} M_c\/{\\overline{m}}}}{{\\overline{m}} t_G}\n         \\left(\\frac{M_c}{{\\overline{m}}}\\right)^{0.9}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:mixingnormformula}\n  \\frac{{\\overline{\\rho}}}{{\\overline{\\rho}}_{\\rm 4D}} ~\\approx~ \\frac{1}{1 + 2({\\overline{m}}\/M_c) e^{-\\frac{1}{2}M_c\/{\\overline{m}} }} \\ .\n\\end{equation}\nWhile the expression in Eq.~(\\ref{eq:simplewidthnormapproximation})  holds \nto within $\\pm 25\\%$ across the relevant \\il{{\\overline{\\eta}}>0} region,\nthis result is actually somewhat sensitive to the value of $t_G$, with the upper error limit \ngrowing smaller with increasing $t_G$ and bigger with decreasing $t_G$.\nBy contrast, the expression in Eq.~(\\ref{eq:mixingnormformula}) holds to within $\\pm 5\\%$ across the relevant\nregion, making it one of the most accurate analytical approximations we have presented in this paper.\n\nWe can also deduce approximate scaling laws from these expressions.\nFor \\il{{\\overline{m}}\/M_c \\gg 1}, we find that\nEq.~(\\ref{eq:simplewidthnormapproximation}) approximately reduces to\n\\begin{equation}\n    \\frac{{\\overline{\\rho}}(\\delta_G)}{{\\overline{\\rho}}(0)} ~\\sim~ {1\\over {\\overline{m}} t_G \\delta_G}  \\left( {M_c\\over {\\overline{m}}}\\right)~\n\\label{scaling1}\n\\end{equation}\nwhich is a factor of $M_c\/{\\overline{m}}$\ngreater than the corresponding expression we observed in Eq.~(\\ref{fdscaling})\nfor the 4D case with \\il{N=1}.\nThus the presence of an entire of tower of KK states\nsuppresses what would otherwise have been\nthe late-time energy density of the zero mode alone\nby an additional factor of $M_c\/{\\overline{m}}$.\nLikewise, for \\il{{\\overline{m}}\/M_c\\gg 1},\nwe observe from\nEq.~(\\ref{eq:mixingnormformula})\nthat\n\\begin{equation}\n    \\frac{{\\overline{\\rho}}}{{\\overline{\\rho}}_{\\rm 4D}} ~\\sim~ {M_c\\over {\\overline{m}}}~.\n\\label{scaling2}\n\\end{equation}\nThus, combining these results for \\il{{\\overline{m}}\/M_c\\gg 1}, we find that\n\\begin{equation}\n         {{\\overline{\\rho}}(\\delta_G) \\over {\\overline{\\rho}}(0)} ~\\sim~\n         {{\\overline{\\rho}}(\\delta_G) \\over {\\overline{\\rho}}_{\\rm 4D}(\\delta_G)} \\cdot\n         {{\\overline{\\rho}}_{\\rm 4D}(\\delta_G) \\over {\\overline{\\rho}}_{\\rm 4D}(0)}~\n\\end{equation}\nfrom which we deduce that \n\\begin{equation}\n      {\\overline{\\rho}}(0) ~\\sim~ {\\overline{\\rho}}_{\\rm 4D}(0)~.\n\\label{same}\n\\end{equation}\nThis in turn requires that there exist a constant $c$ for which\n\\begin{equation}\n           \\lim_{\\delta_G\\to 0} \\left( {{\\overline{\\rho}}\\over {\\overline{\\rho}}_{\\rm 4D}} \\right) ~\\approx ~c~,\n\\label{limitratio}\n\\end{equation}\nand indeed this last relation is true for \\il{c=1}. \nAccording to the results shown in the right panel of Fig.~\\ref{fig:infinitemodedensityplots},\nthis relation with \\il{c=1} is manifestly true for \\il{{\\overline{m}}\/M_c \\lsim 1}.  \nHowever, as $\\delta_G$ approaches zero, this relation with \\il{c=1} becomes true\nfor larger and larger values of ${\\overline{m}}\/M_c$.\nThus, all of the scaling relations we have quoted here are self-consistent\nwithin the regions of validity claimed.\n\nIn this connection, we remark that it is not only the power-law scaling relations\nin Eqs.~(\\ref{scaling1}) and (\\ref{scaling2}) \nwhich must be consistent with each other;\nthe same must also be true of the more complete expressions such as those\nin Eqs.~(\\ref{eq:simplewidthnormapproximation})\nand (\\ref{eq:mixingnormformula}) from which these power-law relations are derived.\nHowever, although such self-consistent pairs of expressions exist,\nthe specific expressions provided in\nEqs.~(\\ref{eq:simplewidthnormapproximation})\nand (\\ref{eq:mixingnormformula}) \ndo not constitute such a pair.  Rather, these expressions are provided instead because they \nyield even greater numerical accuracy over their appropriate regions of\nparameter space.\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{infinitemodeabsolute.pdf}\n\\caption{The {\\it absolute}\\\/ late-time energy density ${\\overline{\\rho}}$ \nin units of ${\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}$, plotted within the $({\\overline{m}} t_G,\\delta_G)$ \nplane for three different values of $M_c t_G$.}\n\\label{fig:absoluteplots}\n\\end{figure*}\n\nAs we have seen, considering the late-time energy\ndensity ${\\overline{\\rho}}$ as a fraction of related benchmarks such as ${\\overline{\\rho}}(\\delta_G=0)$ or ${\\overline{\\rho}}_{\\rm 4D}$ \nhas been useful for understanding the {\\it relative}\\\/ effects of increasing $\\delta_G$ or \nincreasing the numbers of modes.\nHowever, we also wish to understand the late-time energy densities ${\\overline{\\rho}}$ on an {\\it absolute}\\\/ scale. \nOf course, \nthese absolute magnitudes can in principle be obtained through a sequence of multiplications of previous \nintermediate results.\nFor example, one possible path is given by\n\\begin{equation}\n           {\\overline{\\rho}} ~=~ \\left({{\\overline{\\rho}} \\over {\\overline{\\rho}}_{\\rm 4D}}\\right) \n        \\, \n            \\left({{\\overline{\\rho}}_{\\rm 4D} \\over {\\overline{\\rho}}_{\\rm 4D}(\\delta_G=0)}\\right) \n        \\,  \n            {\\overline{\\rho}}_{\\rm 4D} (\\delta_G=0)~\n\\end{equation}\nwhere these three different factors are indicated in the left panel of Fig.~\\ref{fig:infinitemodedensityplots},\nthe left panel of Fig.~\\ref{fig:singlefielddensityplot}, and \nthe large-$t$ behavior of Eq.~(\\ref{eq:abruptapproximation}) respectively.\nHowever, since each of these quantities is generally a complicated function of ${\\overline{m}} t_G$ and $\\delta_G$ \nthroughout the $({\\overline{m}} t_G,\\delta_G)$ parameter space,\nit is not readily apparent what composite behavior might emerge from these individual factors.\nIt is therefore useful to compile our intermediate results together in order to present the resulting \nvalues for ${\\overline{\\rho}}$ as full, absolute quantities.\n\nOur results are shown in Fig.~\\ref{fig:absoluteplots} for three different values of $t_G$.\nIn general, we see that taking larger values of ${\\overline{m}} t_G$ results in larger values of ${\\overline{\\rho}}$.\nHowever, we also see that taking larger values of $\\delta_G$ tends to suppress ${\\overline{\\rho}}$.\nTogether, these effects conspire to produce the curved contours shown.\nIndeed, increasing $t_G$ relative to $M_c$ then tends to push these contours to the right, thereby\nagain increasing the late-time energy density ${\\overline{\\rho}}$ still further.\n\n\\begin{figure}[t]\n\\includegraphics[keepaspectratio, width=0.49\\textwidth]{infinitemodeeta.pdf}\n\\caption{The late-time tower fraction ${\\overline{\\eta}}$ of the KK tower,\nplotted within the \\il{({\\overline{m}} t_G,\\delta_G)} plane for \\il{t_G=10^2\/M_c}.\nUnlike the tower fractions illustrated in Fig.~\\ref{fig:finitemodedensityplotsgenN2} for finite $N$,\nwe see that the tower fraction for \\il{N=\\infty} \nis now monotonic in both ${\\overline{m}} t_G$ and $\\delta_G$\nand approaches unity for large ${\\overline{m}} t_G$ and small $\\delta_G$.}\n\\label{fig:infinitemodeeta}\n\\end{figure}\n\nIn Fig.~\\ref{fig:absoluteplots} we indicated the \nfull, absolute magnitudes of ${\\overline{\\rho}}$ for entire our KK tower as functions of ${\\overline{m}} t_G$ and $\\delta_G$.\nHowever, we are also interested in \nthe distribution of this total energy density across the different modes of our KK tower.\nAs we have seen throughout this paper,\none measure of this distribution is the tower fraction $\\eta$ --- the \nfraction of the total abundance which is carried by all but the most abundant mode in the tower.\nThe late-time values of these tower fractions for the full KK tower are plotted \nin Fig.~\\ref{fig:infinitemodeeta}.\nAs we see upon comparison with its finite-$N$ \nequivalents in Fig.~\\ref{fig:finitemodedensityplotsgenN},\n {\\it passing to the infinite-$N$ limit has the effect of removing all non-monotonicities in ${\\overline{\\eta}}$}.\nIndeed, we see that ${\\overline{\\eta}}$ now increases monotonically as either ${\\overline{m}} t_G$ is increased or $\\delta_G$ is \ndecreased.   We stress that this is a feature which emerges only for the full KK tower with \\il{N=\\infty},\nbut which would not be true for any finite value  of $N$.   \nAs a result, {\\it it is the region with large ${\\overline{m}} t_G$ (or equivalently large ${\\overline{m}}\/M_c$) and relatively\nsmall $\\delta_G$ for which the total energy density ends up distributed most broadly across the different\nstates in the KK tower at late times.}\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.8\\textwidth]{rhofracpies.pdf}\n\\caption{A graphical representation of the distribution of the\nenergy densities ${\\overline{\\rho}}_\\lambda$ across the entire KK tower, calculated for a\nsquare ``lattice'' of locations\nin the \\il{({\\overline{m}} t_G,\\delta_G)} plane.\nEach pie chart \nindicates how the total ${\\overline{\\rho}}$ is distributed across the different KK modes,\nwith the blue slices indicating the contribution associated with the lightest mode\nand the remaining slices (colored pink through dark red) indicating\nthe contributions from successively heavier KK modes.\nThe results for each pie correspond to the parameters \\il{({\\overline{m}} t_G,\\delta_G)}\nassociated with the location of the center of the pie,\nand the yellow contours indicate the corresponding values of ${\\overline{\\eta}}$,\nas taken from Fig.~\\protect\\ref{fig:infinitemodeeta}.\nWe see that the total energy density is preferentially captured by the lightest\nmode for larger $\\delta_G$ or smaller ${\\overline{m}} t_G$, but that the total energy density\nis distributed more democratically across the KK tower for smaller $\\delta_G$ and larger ${\\overline{m}} t_G$.\nThus a wide variety of energy-density distributions across the KK tower can be\nrealized simply by adjusting the parameters associated with the mass-generating phase transition.}\n\\label{fig:rhofracpies}\n\\end{figure*}\n\nWhile ${\\overline{\\eta}}$ represents one measure of the degree to which the late-time total energy density of the KK tower\nis distributed across its different modes, \nthis quantity still does not tell us how many modes are actually\ncarrying a significant abundance.\nMore specifically, we would like to know   the shape the overall ``profile'' of the \nenergy-density distribution across the entire KK tower as a function of ${\\overline{m}} t_G$ and $\\delta_G$.\nTowards this end, can visualize the shape of a given energy-density distribution profile by means of \na pie chart whose different pie slices illustrate how ${\\overline{\\rho}}$ is distributed across the different KK modes.\nWe then seek to understand how the relative slices of such pies \nevolve as functions of ${\\overline{m}} t_G$ and $\\delta_G$.\n\nOur results are shown in Fig.~\\ref{fig:rhofracpies}  for \\il{t_G=10^2\/M_c}.\nIn Fig.~\\ref{fig:rhofracpies}\nwe have focused on the non-trivial green region of Fig.~\\ref{fig:infinitemodeeta} wherein \\il{{\\overline{\\eta}}>0},\nand then superimposed a set of pie charts illustrating how the energy-density profile varies\nacross this region. \nWe see from this figure that the total energy density tends to be preferentially captured by the lightest\nmode as $\\delta_G$ increases or ${\\overline{m}} t_G$ decreases, but that \nthe total energy density tends to be more democratically distributed across the KK modes of the tower\notherwise.  {\\it We see, then, that a wide variety of energy-density distributions across the KK tower are possible\nand can be realized simply by adjusting the parameters associated with the mass-generating phase transition.}\n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.8\\textwidth]{rhofracpiesgrand.pdf}\n\\caption{A grand summary of the results of this section, combining \nour information concerning the absolute {\\it magnitudes}\\\/ of the total late-time\nenergy density ${\\overline{\\rho}}$ from Fig.~\\protect\\ref{fig:absoluteplots} \nwith our information concerning the {\\it distribution}\\\/ of that energy density from\nFig.~\\protect\\ref{fig:rhofracpies}, all plotted \nwithin the \\il{({\\overline{m}} t_G, \\delta_G)} plane.\nThis figure is essentially the same as Fig.~\\protect\\ref{fig:rhofracpies} except that the overall\nareas of our pie charts have been rescaled in proportion to the magnitudes of the total late-time\nenergy densities ${\\overline{\\rho}}$ shown in Fig.~\\protect\\ref{fig:absoluteplots}.\nIn the background we have also indicated the contours of ${\\overline{\\rho}}$ from Fig.~\\protect\\ref{fig:absoluteplots}\n(dashed black lines)\nas well as the contours of ${\\overline{\\eta}}$ from Fig.~\\protect\\ref{fig:infinitemodeeta} (solid blue lines).\nWe see that there is a non-trivial {\\it correlation}\\\/ between the overall  magnitude of ${\\overline{\\rho}}$\nand the distribution of that energy density, with larger energy densities distributed more\ndemocratically across the KK tower and smaller energy densities captured more and more preferentially\nby the lightest KK mode.   Both features are nevertheless extremely sensitive to the parameters governing\nthe mass-generating phase transition, with the former behavior dominating for smaller $\\delta_G$ and larger\n${\\overline{m}} t_G$ and the latter \ndominating for larger $\\delta_G$ and smaller ${\\overline{m}} t_G$.} \n\\label{fig:rhofracpiesgrand}\n\\end{figure*}\n\nFinally, we can summarize the results of this section by combining our \ninformation concerning the absolute {\\it magnitudes}\\\/ of the late-time total energy density ${\\overline{\\rho}}$,\nas indicated in Fig.~\\ref{fig:absoluteplots},\nwith our information concerning the late-time {\\it distributions}\\\/ of that total energy density,\nas indicated in Fig.~\\ref{fig:rhofracpies}.\nTo do this, we can begin with the information in Fig.~\\ref{fig:rhofracpies} but then \n{\\it rescale the size of each pie chart so that \nthe area of each pie chart is proportional to the total energy density ${\\overline{\\rho}}$.}\nThe result is shown in Fig.~\\ref{fig:rhofracpiesgrand}.\n\nOne important lesson that emerges from \nFig.~\\ref{fig:rhofracpiesgrand} is that\nthere is a non-trivial correlation between the overall {\\it magnitude}\\\/ of ${\\overline{\\rho}}$\nand the {\\it distribution}\\\/ of that energy density, with larger energy densities distributed more\ndemocratically across the KK tower and smaller energy densities captured more and more preferentially\nby the lightest KK mode.   \nThe results in Fig.~\\ref{fig:rhofracpiesgrand} allow us to see rather dramatically\nthe effects of our mass-generating phase transition on the eventual late-time energy configuration\nof our KK tower.\nFor any fixed ${\\overline{m}} t_G$, we see that increasing the phase-transition timescale $\\delta_G$ \nsuppresses the energy density that remains in the massive KK modes, causing the lighter\nmodes to assume an increasing fractional share of the total energy density.\nYet, at the same time, increasing $\\delta_G$ suppresses the total energy\ndensity that is ultimately pumped into the KK tower by the phase transition.\nLikewise, for any $\\delta_G$, we see that increasing ${\\overline{m}} t_G$ \nincreases the total energy pumped into the system by the phase transition\nwhile simultaneously causing this energy density to be more democratically distributed.\n{\\it Thus, merely by choosing appropriate values of ${\\overline{m}} t_G$ and $\\delta_G$, it is possible to adjust the \ntotal absolute energy density remaining in the KK tower at late times to any value one might\nselect for phenomenological purposes while simultaneously retaining the ability to adjust\nthe distribution of that energy density across the different KK modes.}   This, then, is the power\nof the mass-generating phase transition and the influence of its associated timescale.\n \n\n\\FloatBarrier\n\\section{Example:~ Axion in the Bulk\\label{sec:AxionintheBulk}}\n\n\n\nUntil this point, we have maintained generality\nby considering a higher-dimensional field $\\Phi$\nand specifying little more about this field than that it is a scalar.\nLikewise, we have assumed little more about our phase transition\nthan that it generates masses in a time-dependent way.\nHowever, in order to explore one possible set of phenomenological implications of our results,\nwe shall now apply our machinery to the case in which $\\Phi$ is an axion-like\nparticle and in which our phase transition on the brane is one in which \ninstanton-like effects give mass to that axion.\n\nWe begin by considering the setup\ndescribed in Refs.~\\cite{DDM2,DDMAxion}, which is itself a generalization of\nan earlier framework considered in Ref.~\\cite{DDGAxions}.  \nSpecifically, \nwe consider the same five-dimensional geometry as discussed in previous sections\nand henceforth take \n\\begin{equation}\n    M_c ~=~ 4.49 \\,\\times\\, 10^{-12} ~\\text{GeV} \n\\end{equation}\nas our compactification scale.\nThis value of $M_c$ corresponds to \\il{R\\approx 44~\\mu\\text{m}}, \nwhich is the largest flat extra dimension \nallowed according to data from torsion-balance experiments~\\cite{KapnerEotvosExpt}. \nWithin the bulk of this extra dimension \nwe shall consider\na (pseudo-)scalar field $\\Phi$ which is\na straightforward generalization of the traditional QCD axion~\\cite{PecceiQuinn1,WeinbergAxion,WilczekAxion}.\nIn particular,\nwe shall take $\\Phi$ to be \nthe Nambu-Goldstone boson associated with a global chiral Peccei-Quinn-like $U(1)_X$ symmetry which is spontaneously \nbroken at some scale $f_X$.  \nThis $U(1)_X$ symmetry is assumed anomalous, and for a non-Abelian gauge group $G$ on the brane\nwith coupling $g$ and field strength $\\mathcal{G}_{\\mu\\nu}$, this anomaly therefore generates a topological \nbrane term of the form\n\\begin{equation}\n\\mathcal{L}_{\\text{brane}} ~\\rightarrow~  \n\\mathcal{L}_{\\text{brane}} +\n\\frac{\\mathcal{C} g^2}{32\\pi^2}\\frac{\\Phi}{f_X^{3\/2}}\\text{Tr }\\mathcal{G}_{\\mu\\nu}\\tilde{\\mathcal{G}}^{\\mu\\nu} \n\\end{equation}\nwhere $\\mathcal{C}$ is a model-dependent constant. \nWhile this term has no effect on the classical \nequations of motion, it affects the vacuum structure of the theory. \nAs the universe cools \nand reaches \\il{T\\sim \\Lambda_G},\nwhere $\\Lambda_G$ is the confinement scale associated with the group $G$, \ninstanton effects on the brane\nexplicitly break the $U(1)_X$ axion shift symmetry that prevents the axion from acquiring a mass.\nAs a result, a small temperature-dependent axion mass $m_X(T)$ is generated on the brane. \nIn general, we shall model the time dependence of this mass \nexactly as in previous sections, as resulting from a phase transition \noccurring at a time $t_G$ with a width $\\delta_G$.\nIn this connection we note that our adoption of an LTR cosmology,\nas discussed in Sect.~\\ref{sec:TheModel}, \nimplies that this phase transition takes place \nduring an inflaton-dominated (and thus matter-dominated) epoch.\nThe corresponding  time\/temperature relationship then yields\nthe result\n\\begin{equation}\nt_G ~=~ \\sqrt{\\frac{45 g_*(T_{\\rm RH})}{2\\pi^2}}\\frac{T_{\\rm RH}^2 M_p}{g_*(\\Lambda_G)\\Lambda_G^4} ~\n\\label{timetemp}\n\\end{equation}\nwhere \\il{T_{\\rm RH}\\sim\\mathcal{O}(\\text{MeV}) < \\Lambda_G} is the reheating temperature, \nwhere $g_*(T)$ is the effective number of relativistic degrees of freedom at temperature $T$, \nand where \\il{M_p\\equiv 1\/\\sqrt{8\\pi G}} is the reduced Planck mass. \nFor concreteness we shall take \\il{T_{\\rm RH}= 30}~MeV in what follows.\nFinally, at late times, our phase transition leaves our axion-like field with\na brane mass given by\n\\begin{equation}\n{\\overline{m}}_X^2 ~\\equiv~ \\frac{\\mathcal{C}^2 g^2}{32\\pi^2}\\frac{\\Lambda_G^4}{{\\hat{f}_X}^2}~ \n\\label{mbarXdefn}\n\\end{equation}\nwhere we have defined the effective four-dimensional $U(1)_X$-breaking \nscale \\il{{\\hat{f}_X}\\equiv \\sqrt{2\\pi R f_X^3}}. \n\nThe above setup defines our five-dimensional axion theory.\nCompactifying this theory via KK reduction  \nthen yields an effective four-dimensional theory consisting of a KK tower of axion modes $\\phi_k$ \nwhose mass matrix takes exactly the form given in Eq.~(\\ref{kmassmatrix}), with $m(t)$ now \nidentified as $m_X(t)$. \nThus, we see that our axion model is nothing but a special case of the model we have already considered\nthus far --- the special case in which we identify ${\\overline{m}}$ as ${\\overline{m}}_X$ and identify $t_G$ as the cosmological time \ncorresponding to the temperature \\il{T\\sim\\Lambda_G}, as in Eq.~(\\ref{timetemp}).\n\nIn previous sections, we studied the behavior of our general KK system \nas a function of the four parameters \\il{\\lbrace {\\overline{m}}, M_c, t_G, \\delta_G\\rbrace}.\nFor our axion theory, by contrast, we see that \n\\il{{\\overline{m}}\\to {\\overline{m}}_X}, and moreover we see that both ${\\overline{m}}_X$ and $t_G$ are themselves\nrelated to the more fundamental parameters $\\Lambda_G$ and $\\hat f_X$ through \nEqs.~(\\ref{mbarXdefn}) and (\\ref{timetemp})\nrespectively.\nThus, we shall henceforth consider our axion theory \nto be parametrized by\n\\il{\\lbrace \\Lambda_G, \\hat f_X, M_c, \\delta_G\\rbrace}.\n\nIn four dimensions, the methods used to estimate the late-time abundance of a given axion field typically fall into one\nof two classes examined in Sect.~\\ref{sec:FourDimensionalLimit}: the adiabatic approximation\nor the abrupt approximation.\nAs we have seen, the former only applies when the mass is generated sufficiently slowly during field oscillations,\nwith \\il{\\dot m \\ll m^2}, \nwhile the latter applies only for very small $\\delta_G$ (such that the phase-transition width $\\Delta_G$ is much\nsmaller than the timescale of field oscillations) or for situations in which \\il{t_\\zeta\\gg t_G} (so that the phase\ntransition occurs while the field is still overdamped and essentially frozen).\nHowever, as we have seen in previous sections, these approximations break down in\nrelatively large regions of parameter space.\nWe should also expect deviations from our standard expectations in the case of\nan infinite KK tower of axion modes.\nFor example, the virialization condition that underpins the adiabatic approximation\nnow becomes \n \\il{\\dot{\\lambda}_0 \\ll \\lambda_0^2}, and $\\lambda_0$ can be much smaller than\n$m_X$~\\cite{DDGAxions}.\nLikewise, there is always some subset of modes   \nin the KK tower for which \\il{t_\\zeta^{(\\lambda)}\\lesssim t_G}.\nSuch modes are therefore necessarily affected by the time-dependence of the phase transition.\n\n\\begin{figure*}\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{4Daxion.pdf}\n\\caption{ The late-time energy density ${\\overline{\\rho}}_{\\rm 4D}$ of our generalized axion field in the 4D limit, \nplotted within the \\il{(\\Lambda_G, \\hat f_X)} plane for \\il{\\delta_G = 0.1} (top row) and\n\\il{\\delta_G=1.0} (bottom row).   Note that the horizontal $\\Lambda_G$ axis (as shown along the bottom of each panel)\nis equivalently a $t_G$ axis (as shown along the top).\nThe panels in the left and middle columns plot\n${\\overline{\\rho}}$ as fractions of the abrupt and adiabatic approximations ${\\overline{\\rho}}_{\\rm 4D}(\\delta_G=0)$ \nand ${\\overline{\\rho}}_{\\rm 4D}|_{\\rm abs}$, respectively,\nwhile the panels in the right column \nplot the absolute magnitude of ${\\overline{\\rho}}_{\\rm 4D}$ in units of    \n${\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}$.   \nAlso shown in each panel is a purple dashed line indicating the contour\nalong which \\il{{\\overline{m}}_X t_G=1}.}   \n\\label{fig:4Daxion}\n\\end{figure*}\n\nIn this section,\nwe shall therefore present exact results for the late-time energy densities of our axion fields.  We shall do this both\nfor the four-dimensional \\il{N=1} case as well as the infinite-$N$ case of our full KK axion tower.\nAs discussed above, we shall take our axion parameter\nspace to be parametrized by\n\\il{\\lbrace \\Lambda_G, \\hat f_X, M_c, \\delta_G\\rbrace}.\nHowever, we stress that the plots to be presented in this section are not merely translations of our previous\nplots into these new variables.   First, we have extended our range of \ninterest within this parameter space into those regions of specific interest for axion physics.\nSecond, we shall now regard $\\Lambda_G$ and $\\hat f_X$ as the fundamental axion parameters\nrelative to which we wish to consider continuous variations. In other words,\nwe shall plot our the late-time energy densities ${\\overline{\\rho}}$ \nas functions of $\\Lambda_G$ and $\\hat f_X$ within the \\il{(\\Lambda_G,\\hat f_X)} plane, choosing only \ndiscrete representative choices for $\\delta_G$.    This too is different than what was done in previous sections,\nand thus represents a different, independent slice through our four-dimensional parameter space.  \n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=0.7\\textwidth]{KKaxion1.pdf}\n\\caption{ The total late-time energy density ${\\overline{\\rho}}$ of our generalized axion KK tower, \nplotted as fractions of ${\\overline{\\rho}}(\\delta_G=0)$ (left column) or ${\\overline{\\rho}}_{\\rm 4D}$ (right column)\nwithin the \\il{(\\Lambda_G, \\hat f_X)} plane for \\il{\\delta_G = 0.1} (top row) and \\il{\\delta_G=0.3} (bottom row).   \nAs in Fig.~\\protect\\ref{fig:4Daxion},\nthe $\\Lambda_G$ axis is shown along the bottom of each panel\nwhile the equivalent $t_G$ axis is shown along the top.\nAlso shown in each panel are three purple dashed lines indicating the contours along which \n\\il{{\\overline{m}}_X t_G=1}, \\il{{\\overline{m}}_X = M_c}, or \\il{M_c t_G=1}.\nAs with other figures in this paper, the gray regions \nare excluded for the reasons discussed in the paragraph below Eq.~(\\ref{bigwidth}).}\n\\label{fig:KKaxion1}\n\\end{figure*}\n\nWe begin, as in Sect.~\\ref{sec:FourDimensionalLimit}, by considering the four-dimensional \\il{N=1} limit.\nIn this case, we find the results shown in Fig.~\\ref{fig:4Daxion}.\nWe observe from the panels in the left and center columns of \nFig.~\\ref{fig:4Daxion} that there are distinct regions\nof parameter space in which the abrupt and adiabatic approximations fail\nto model the true late-time energy density, with the adiabatic approximation\nconsistently underestimating the true energy density and the abrupt approximation\neither under- or overestimating the true energy density, in some cases by many orders\nof magnitude.   We also see that increasing either the confinement scale $\\Lambda_G$   \nor the $U(1)_X$ symmetry-breaking scale $\\hat f_X$  \ngenerally decreases the late-time energy density of our axion field.  \nIncreasing the width $\\delta_G$ of our axion-induced phase transition also has the same effect.\nIndeed, we see from the relatively straight contours in Fig.~\\ref{fig:4Daxion} that\nthe absolute magnitude of the late-time energy density scales approximately as\n\\begin{equation} \n           {\\overline{\\rho}}_{\\rm 4D} ~\\sim~ 1\/(\\Lambda_G^4 \\hat f_X^2)~,\n\\end{equation}\nwhere the constant of proportionality is a non-trivial function of $\\delta_G$.\nThis scaling behavior is of course consistent with the analogous result in Eq.~(\\ref{4Dscaling}).  \nFinally, we note that all of the contour lines in Fig.~\\ref{fig:4Daxion}  \nexperience slight ``ripples'' at \\il{\\Lambda_G\\approx 2}~GeV and $5$~GeV.\nThese ripples are physical, and correspond to the energy scales $\\Lambda_G$ at which there\nare changes in the number $g_\\ast(\\Lambda_G)$  of relativistic degrees of freedom (the former\ncorresponding to the threshold for the charm quark and tau lepton, and the latter corresponding to the threshold for the bottom quark). \n\n\\begin{figure*}[t]\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{KKaxion3with0.pdf}\n\\caption{ The absolute magnitude of the total late-time energy density ${\\overline{\\rho}}$ of our generalized axion KK tower, \nplotted in units of ${\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}$\nwithin the \\il{(\\Lambda_G, \\hat f_X)} plane for \n\\il{\\delta_G=0} (left panel),\n\\il{\\delta_G = 0.1} (middle panel),\nand \\il{\\delta_G=0.3} (right panel).  }   \n\\label{fig:KKaxion2}\n\\bigskip\n\\bigskip\n\\includegraphics[keepaspectratio, width=1.0\\textwidth]{KKaxion2with0.pdf}\n\\caption{ The late-time tower fraction ${\\overline{\\eta}}$ of our generalized axion KK tower,\nplotted within the \\il{(\\Lambda_G, \\hat f_X)} plane for \n\\il{\\delta_G=0} (left panel),\n\\il{\\delta_G = 0.1} (middle panel),\nand \\il{\\delta_G=0.3} (right panel).  }   \n\\label{fig:KKaxion3}\n\\end{figure*}\n\nWe now turn to the case in which a full KK tower of axion modes experiences\nthe instanton-induced phase transition.\nIn this case, our results are plotted in Figs.~\\ref{fig:KKaxion1} through \\ref{fig:KKaxion3}. \nIn Fig.~\\ref{fig:KKaxion1} \nwe have plotted the values of the late-time total energy density ${\\overline{\\rho}}$ of the KK tower\nrelative to our usual benchmarks ${\\overline{\\rho}}(\\delta_G=0)$ and ${\\overline{\\rho}}_{\\rm 4D}$.\nOnce again, we see that the introduction of a non-zero width $\\delta_G$ for our\ninstanton-induced phase transition leads to either enhancements or suppressions\nin ${\\overline{\\rho}}$, depending on the specific region of parameter space.\nMoreover, we see that increasing $\\delta_G$ only makes these enhancements or suppressions\nmore severe.   By contrast, turning to ${\\overline{\\rho}}\/{\\overline{\\rho}}_{\\rm 4D}$, we see the introduction of the extra KK modes in the tower\nonly suppresses ${\\overline{\\rho}}$ by an amount that grows {\\it less}\\\/ severe for increasing $\\delta_G$.\nIn all cases, however, these suppressions tend to be more pronounced for \nsmaller $\\Lambda_G$ and smaller $\\hat f_X$ than they are in other regions of parameter space. \nIt is worth noting that for smaller $\\Lambda_G$ (but \\il{{\\overline{m}}_X \\gg M_c}), we see a scaling behavior\nof the form\n\\begin{equation}\n     {{\\overline{\\rho}}\\over {\\overline{\\rho}}_{\\rm 4D}} ~\\sim~ {\\hat f_X\\over \\Lambda_G^2}~. \n\\end{equation}\nHowever, as $\\Lambda_G$ increases (and thus $t_G$ decreases),\nthis scaling behavior eventually breaks down \nand the behavior of ${\\overline{\\rho}}\/{\\overline{\\rho}}_{\\rm 4D}$  \nis increasingly determined solely by the behavior of the denominator ${\\overline{\\rho}}_{\\rm 4D}$, with\ncontours of constant ${\\overline{m}}_X t_G$.\nOtherwise,\noutside the ${\\overline{m}}_X \\gtrsim M_c$ and \\il{{\\overline{m}}_X t_G \\gtrsim 1} region,\nthe energy density at late times is very similar to that in the 4D limit.\n\nIt is also instructive to compare our KK results for \\hbox{${\\overline{\\rho}}\/{\\overline{\\rho}}(\\delta_G=0)$} along the left column of\nFig.~\\ref{fig:KKaxion1} with the analogous 4D results along the left column of Fig.~\\ref{fig:4Daxion}.\nFor the full KK tower, we no longer find contours of constant $({\\overline{m}}_X t_G)^2$;  instead we find a region \nexhibiting contours with approximately constant $\\hat f_X$.\nLikewise, the enhancement region that we found in the 4D limit spanned the entire\n\\il{\\lbrace \\Lambda_G, \\hat f_X\\rbrace} parameter space for \\il{{\\overline{m}}_X t_G \\sim 1}, whereas this region \nfor the KK tower exists only for \\il{{\\overline{m}}_X\\lsim M_c}. \n\nNext, we turn to Fig.~\\ref{fig:KKaxion2} where we plot the absolute magnitude of the \ntotal late-time energy density of the KK axion tower in units\nof ${\\textstyle{1\\over 2}} \\langle \\phi\\rangle^2 t^{-2}$.\nTo help understand the features of these plots and how they evolve as a function of $\\delta_G$, we \nhave included a panel with \\il{\\delta_G=0}\nwhose features can be understood directly via the abrupt approximation.\nIndeed, as expected, we see that all ${\\overline{\\rho}}$ contours in the \\il{\\delta_G=0} case\nare contours of either constant ${\\overline{m}}_X t_G$, constant ${\\overline{m}}_X\/M_c$,\nor constant $M_c t_G$.\nThe subsequent panels then illustrate how these contours deform as the width $\\delta_G$\nof our phase transition increases.\n\nLikewise, in Fig.~\\ref{fig:KKaxion3}, we plot the late-time tower fraction ${\\overline{\\eta}}$\nof our KK axion tower.\nOnce again, we see a consistent picture emerging as a function of $\\delta_G$. \nIndeed, just as for ${\\overline{\\rho}}$ in Fig.~\\ref{fig:KKaxion2}, we see that all ${\\overline{\\eta}}$ contours for \\il{\\delta_G=0} \nare contours of either constant ${\\overline{m}}_X t_G$, constant ${\\overline{m}}_X\/M_c$,\nor constant $M_c t_G$, \nwith the subsequent panels once again illustrating how these contours deform \nas $\\delta_G$ increases.\nFrom the left panel of Fig.~\\ref{fig:KKaxion3} we see that \nwhen the phase transition is abrupt, \nthe total energy density of the tower\nis distributed amongst a significant number of KK modes in\nthe small-$\\Lambda_G$, small-$\\hat f_X$ corner of parameter space. \nHowever, as our mass-generating phase transition unfolds over a longer period of time,\nthe energy contributions from the higher modes are suppressed and ${\\overline{\\eta}}$ begins to fall. \n\nOne interesting feature apparent in Fig.~\\ref{fig:KKaxion3}\nis that the value of ${\\overline{\\eta}}$ is non-monotonic\nas a function of $\\Lambda_G$, first increasing and reaching a maximum near \\il{\\Lambda_G\\approx 3}~GeV\nbefore decreasing again. \nThis can be understood as follows.\nThe vanishing of ${\\overline{\\eta}}$ for \\il{{\\overline{m}}_X \\ll M_c} is understood from the distribution of initial abundances as described \nin Fig.~\\ref{fig:initialabruptabundance}.\nIndeed, in this region, nearly all of the abundance of the KK tower is contributed by the lowest KK mode because\nvery little mixing is generated in the phase transition.\nBy contrast, the decline in ${\\overline{\\eta}}$ which occurs as $\\Lambda_G$ increases (or equivalently as $t_G$ decreases) \ncan be understood as corresponding to our entrance \ninto what in Refs.~\\cite{DDM1,DDM2,DDMAxion}\nwas called the ``staggered'' regime wherein the heavier KK modes\n          are oscillating (and thus already dissipating their energy density) \nas soon as our phase transition occurs, whereas the lighter KK modes remain overdamped for a long time after $t_G$.\nThis then results in  a total abundance composed primarily of contributions from those lighter modes.\n\nComparing the results in Figs.~\\ref{fig:KKaxion2} and~\\ref{fig:KKaxion3}, we see that the \nregions of parameter space which produce the largest total energy densities $\\bar\\rho$ almost \ncoincide exactly with the regions that produce the largest tower fractions $\\bar\\eta$.   \nIndeed, both regions have very small $\\hat f_{X}$, and the only difference is that the \nformer region has small $\\Lambda_G$ while the latter region has \\il{\\Lambda_G\\approx 3}~GeV.   \nAt first glance, this difference may appear to violate the claims made in connection with \nFig.~\\ref{fig:rhofracpiesgrand}, namely that these two regions should coincide completely.  However, \nthe correlation observed in connection with Fig.~\\ref{fig:rhofracpiesgrand} holds when $t_G$ \nis held fixed.   By contrast, the results in Figs.~\\ref{fig:KKaxion2} and~\\ref{fig:KKaxion3} \nare generated with $t_G$ implicitly varying throughout the parameter space shown.\n\nHaving focused in this section on the specific situation in which \nour $\\Phi$ field is an axion, one possible next step would be to place\nphenomenological bounds on the parameter spaces we have considered.\nSuch bounds would in principle mirror those which are normally applied\nto the case of a traditional QCD axion, and come from a variety of considerations\nincluding\nsupernova cooling,\nblack hole superradiance,\noverclosure constraints,\ndark-matter and dark-energy constraints,\ntraditional axion searches (such as light shining through walls), {\\it etc}\\\/.\nWithin the context of the abrupt (\\il{\\delta_G=0}) limit, such bounds on KK axion towers\nare discussed in detail in Ref.~\\cite{DDMAxion}. \nAlthough the determination of such bounds for general $\\delta_G$ is beyond the scope of this paper,  \nthe results we have obtained here concerning the late-time energy densities of these\naxion systems should play a critical role in helping to determine exactly\nwhere these bounds lie for general $\\delta_G$, and the extent to which the suppressions and enhancements\nwe have observed translate into a loosening or strengthening of those bounds \nbeyond traditional expectations.\n\n\n\\FloatBarrier\n\\section{Discussion and Conclusions\\label{sec:Conclusions}}\n\n\n\nIn this paper we have investigated the effects of dynamical mass generation on \nthe cosmological abundances of the Kaluza-Klein modes associated with a \nbulk scalar field.  In particular, we have examined the non-trivial case in which \na phase transition localized on a brane leads to time-dependent masses and mixings among\nthese KK modes.  We have found that both the total energy density of the full KK tower\nand the distribution of that energy density across the individual KK modes \nare extremely sensitive to the details of the phase transition.  As a result, \nwithin different regions of model parameter space, these abundances can be \nsignificantly enhanced or suppressed relative to standard expectations ---\nsometimes by many orders of magnitude.  We have also derived a variety of approximate \nscaling behaviors and analytic expressions for the energy densities of the KK modes \nas functions of the relevant model parameters.  \nIn order to illustrate the potentially significant impact \nthat these effects can have on the late-time abundances of our scalars within the context \nof a concrete model, we have also applied our general results to the case of a bulk axion\/\naxion-like field.\nFinally, as a by-product of our analysis, we have also developed an alternate ``UV-based''\neffective truncation of KK theories which is physically different from the more traditional\n``IR-based'' truncation commonly employed in the literature, yet yields \nthe same higher-dimensional theory as the truncation\nis lifted.\n \nDepending on the identity of the scalar field in question,\nour results can have a variety of phenomenological implications.\nFor example, in the case in which \nour bulk scalar is an axion or axion-like field,\none can easily imagine a number of \nphenomenological consequences.  Note that\nin general,\ncosmological abundances in this paper\nhave been generated through a two-step process:\nthe assumption of a non-zero VEV for one or more modes \nin the KK tower\nfollowed by a mass-generating phase transition.\nAs such, given the specific form of the initial conditions \nwe have adopted in Eq.~(\\ref{eq:initialconditions}),\nthe method of abundance generation we have studied in this paper\nis tantamount to misalignment production.\nHowever, misalignment production is not the only mechanism through which a population of \naxion-like particles can be produced in the early universe.  For example,\nspontaneous breaking of the global $U(1)_X$ symmetry of Sect.~\\ref{sec:AxionintheBulk} \ncan lead to the formation of a network of cosmic strings and other topological \ndefects~\\cite{DavisAxionsProdCosmicString1}.  If this breaking occurs before cosmic \ninflation, these defects are simply inflated away.  By contrast, if the breaking of \n$U(1)_X$ occurs after inflation, these defects retain a non-negligible energy density \nuntil late times, and their decays can therefore generate a potentially significant \ncontribution to the relic abundance of the corresponding axion-like particles.  Indeed, \nthis topic has been studied in detail for the specific case of a QCD \naxion~\\cite{DavisAxionsProdCosmicString2,\nDabholkarCosmicStringAxionBounds,AxionProdStringsWalls,AxionProdDomainWalls}, but\nin principle applies to other axion-like particles as well.\n \nThese considerations are important because \nthe phase-space distribution of a \npopulation of axions or axion-like particles produced via the decays of \ntopological defects is significantly different \nfrom that generated via misalignment \nproduction~\\cite{ChangHagmannSikivie,AxionProdStringsWalls,AxionProdDomainWalls}.\nThus, in cases in which the misalignment contribution to the \noverall late-time axion abundance is suppressed by the time-dependent masses and mixings \nwe have discussed here, \nthe phase-space distribution\nof relic axions can be significantly altered --- especially\nif the contribution from topological defects ends up dominating the total axion abundance.\nMoreover, the suppression of the\nmisalignment contribution to this overall axion abundance can also serve to weaken\nthe overclosure bound on the axion-decay constant, and thus could potentially enlarge the allowed \nregion of parameter space for the QCD axion.   \n\nOur results concerning the effects of a mass-generating phase transition can also have significant implications \nfor other new-physics scenarios.  For example, our results may provide a way of mitigating \nthe cosmological moduli problem which arises in supergravity~\\cite{ModuliProblemSUGRA} and in \nstring theory~\\cite{BanksModuliProblem1,deCarlosModuliProblem,BanksModuliProblem2}.\nIndeed, such theories generically predict large numbers of neutral scalar \nfields --- so-called moduli --- with flat potentials, long lifetimes, and \nlarge (and even Planck-scale) VEVs.  On the one hand, a number of \nphenomenological considerations imply that some mechanism must exist through which \na potential is generated for these moduli, rendering them massive.  On the other \nhand, once these fields acquire masses, they can potentially overclose\nthe universe or precipitate an unacceptably late period of reheating.  A suppression\nof the collective energy density of such moduli due to time-dependent mixing effects\nof the sort we have discussed here\ncould potentially provide a way of addressing these issues.   \n\nOur results also have implications within the context of the Dynamical\nDark Matter (DDM) framework~\\cite{DDM1,DDM2}, an alternative framework for dark-matter\nphysics in which a potentially vast ensemble of unstable particles contribute to the \npresent-day dark-matter abundance and in which phenomenological constraints on the dark sector\nare satisfied through a balancing between constituent decay widths and cosmological abundances \nacross the ensemble.  \nDark-matter ensembles with these properties emerge naturally in a variety of contexts~\\cite{DDM1,DDM2,Chialva:2012rq,Dienes:2016kgc,Dienes:2016vei,Jaketoappear},\nand detection strategies for such ensembles are discussed in \nRefs.~\\cite{Dienes:2012yz,Dienes:2012cf,Dienes:2013xff,Dienes:2014via,Dienes:2014bka,Boddy:2016fds,Boddy:2016hbp}.\nIndeed, it has been shown that the KK modes associated with a bulk axion-like\nfield which receives its abundance via misalignment production constitute a viable DDM\nensemble~\\cite{DDM2,DDMAxion}, \nThe results we have obtained here are thus directly applicable\nto DDM ensembles of this sort and lead to a suppression of the overall relic abundance\nof the ensemble for large $\\delta_G$ relative to the relic abundance which arises for the abrupt case considered in \nRefs.~\\cite{DDM2,DDMAxion}.  This suppression can potentially widen the phenomenologically allowed\nparameter space of such DDM models.  Moreover, we also note from the results \nin Fig.~\\ref{fig:rhofracpiesgrand}\nthat there exists a correlation \nbetween the overall magnitude of the late-time energy density of the KK tower\nand the degree to which its distribution across the different KK modes is\nparticularly ``DDM-like'' ({\\it i.e.}\\\/, shared non-trivially across many different KK modes), with larger\ntotal abundances tending to correlate with increased DDM-like behavior at late times.\nThus, we see that we can control the degree to which our ensemble of KK states\nis truly DDM-like at late times simply by adjusting phenomenological parameters \nsuch as ${\\overline{m}}$ and $\\delta_G$ \nassociated with our mass-generating phase transition. \n\nThe suppression of late-time scalar abundances due to time-dependent masses and mixings \nalso has potentially\nimportant implications for a broad range of additional scenarios involving weakly-coupled scalar \nparticles which receive a non-negligible contribution to their relic abundances from \nnon-thermal mass-generating phase transitions of the sort we have discussed here.\nThe reason is that in such scenarios,   \na non-negligible population of these particles can also be produced\nthermally from scattering processes involving SM particles in the radiation bath.  \nThis production mechanism is often referred to as ``freeze-in''~\\cite{FreezeIn}, \nand it also gives rise to a population of particles whose phase-space distribution \ndiffers significantly from that of the population generated \nby non-thermal mass-generating phase transitions.\nFor the QCD axion, the freeze-in contribution to the total relic abundance   \nis typically quite small~\\cite{TurnerThermalAxion,MassoThermalAxion,GrafThermalAxion} \ncompared to the contributions from other sources, such as misalignment production \nand cosmic-string and domain-wall decay.  By contrast, for other exotic scalars, the\nfreeze-in contribution can be significant.  Thus, in scenarios in which both misalignment\nproduction and freeze-in production are na\\\" ively expected to generate comparable \ncontributions to the overall abundance of a particular weakly-coupled scalar particle, a\nsuppression of the former contribution due to time-dependent mixing could \nboth modify overclosure bounds and alter the expected \nphase-space distribution of the relic scalar population.\n\nIn this paper we have focused on the contribution to the total abundance\nof a collection of KK scalar modes that arises due to a time-dependent\nmass-generating phase transition.  In so doing, however, we have disregarded the effects \nof the quantum fluctuations that arise for these fields during the inflationary epoch.\nIn general, fluctuations in the long-wavelength modes of light fields --- and, in particular, those \nmodes whose wavelengths exceed the Hubble length during that epoch --- behave like \nvacuum energy until after inflation ends.  Thus, the energy density in these \nfluctuations survives inflation and yields an additional contribution to the \nabundance of any field with mass \\il{\\lambda_k \\lesssim H_I}, where $H_I$ is the value \nof the Hubble parameter during inflation.  This additional contribution depends \nsensitively on the parameters of the inflationary model (for a review, see, {\\it e.g.}\\\/, \nRef.~\\cite{InflationFluctuations}) and in particular on $H_I$ itself.  In \nRef.~\\cite{DDMAxion}, it was shown that for a KK tower of axion-like fields, the \ncontribution to the relic abundance from vacuum misalignment dominates over this  \nadditional contribution from fluctuations during inflation for sufficiently small \n$H_I$.  However, we note that in general, this contribution exists and for \nhigher-scale inflationary models may have a significant impact on the overall\nrelic abundance of the KK tower.  \n\nIn the course of our analysis in this paper, we also derived results for truncated\nKK towers consisting of only $N$ modes, where \\il{N<\\infty}.\nIn so doing,\nwe presented the results of this finite-$N$ case merely as a stepping\nstone on the way towards understanding the properties of the full KK tower that emerges in the \n\\il{N\\rightarrow\\infty} limit.  However, the results of our finite-$N$\nanalysis are also interesting in their own right.  Indeed, the mass-squared \nmatrix for a tower of $N$ modes is similar to the mass matrix which emerges \nin certain ``moose''~\\cite{Moose} or ``quiver''~\\cite{Quiver} gauge theories --- \ntheories which also contain only a finite number of modes yet which yield \na deconstructed extra spacetime dimension in the \\il{N\\rightarrow\\infty} \nlimit~\\cite{Deconstruction}.  Thus, the results we have obtained here for such finite-$N$ \ntheories should provide guidance as to how time-dependent mass-generating phase transitions\nand time-dependent mixings will affect the late-time \nenergy densities for the collections of scalars which emerge in such moose or quiver\ntheories.\n\nIt is also worth emphasizing that the results we have derived in\nthis paper reflect the particular KK-derived mixing \nstructure of our model.  As a result, we did not observe certain phenomena which may in principle \narise for more general systems of scalar fields in the presence of time-dependent \nmass-generating phase transitions and mixings.\nFor example, it has been shown~\\cite{TwoTimescales} that in similar scalar systems with\nmore general mixing patterns, parametric resonances can arise which \nare extremely sensitive to the widths and mixings associated with mass-generating \nphase transitions and which \n``pump'' energy density into a particular field or fields during such phase transitions.  \nSuch systems may also exhibit so-called ``re-overdamping'' phenomena~\\cite{TwoTimescales} in which the\ntotal energy density of our system exhibits a time dependence which transcends those\nnormally associated with vacuum-dominated or matter-dominated cosmologies. \nHowever, in order for such novel effects to arise, the \ndynamically generated contribution $\\overline{\\mathcal{M}}^2_{\\rm gen}$ to the \nmass-squared matrix at late times must satisfy the criterion   \n\\begin{equation}\n  \\det\\overline{\\mathcal{M}}_{\\rm gen}^2 ~<~ 0~,\n\\end{equation}\nand this criterion is not satisfied within the parameter space of our KK model.  \nHowever, it is easy to contemplate early-universe\nscenarios involving multiple scalars in which \nthis condition is satisfied.\nIn such scenarios, parametric resonances and re-overdamping phenomena \ncan then have a significant further impact on both the \ntotal abundance of the scalar fields involved and the distribution of that\ntotal abundance among these scalars. \n\nThere are a number of potentially interesting generalizations\nand extensions of this work.  For example, in this paper we have focused on \nthe case in which the higher-dimensional scalar propagates in a single, flat \nextra dimension.  However, such a flat extra dimension can be viewed as a\nlimit of the more general case of a warped geometry in which the \nextra dimension represents a slice of anti-de Sitter space~\\cite{RS1,RS2}.  Such \nwarped compactifications have a variety of phenomenological applications, such as\nproviding ways of addressing the hierarchy between \nthe weak and Planck scales~\\cite{RS1} as well as the hierarchies amongst the masses of \nthe SM fermions~\\cite{TonyRSYukawas1,HuberShafiRSYukawas}.\nIt would therefore be interesting to examine how our findings generalize to the \ncase of a warped extra dimension.  It would also be interesting to \nconsider the further generalization of our results to the case of multiple extra \ndimensions, both flat and warped.  \nOne could also imagine adopting more general initial conditions for our KK modes\nbeyond those in Eq.~(\\ref{eq:initialconditions}), such as might arise if\nthere is an earlier phase of non-trivial dynamics (either in the bulk\nor on the brane) prior to our mass-generating phase transition.\n\nAnother possible avenue for generalization concerns the nature of the cosmological\nepoch during which we are presuming our dynamics to take place.\nFor example, throughout this paper we have focused primarily on the case in which the \nuniverse is effectively matter-dominated ({\\it i.e.}\\\/, with \\il{\\kappa = 2}) throughout the\nperiod of mass generation.   Indeed, as discussed in Sect.~\\ref{sec:TheModel},\nphenomenological constraints on theories with large extra spacetime dimensions \nare more easily satisfied in LTR scenarios, in which the universe remains \nmatter-dominated until very late times.  However, these constraints are \nconsiderably weaker in cases in which the compactification scale $M_c$ is \n$\\mathcal{O}(\\mathrm{TeV})$ or above.  Thus, it would also be interesting to \nconsider the case in which the universe is effectively radiation-dominated \n({\\it i.e.}\\\/, with \\il{\\kappa = 3\/2}) during the mass-generation epoch.\nCosmologies with other values of $\\kappa$ can be considered as well.\n\nFinally, other interesting extensions of this work involve considering a broader\nvariety of cosmological contexts in which mass generation for\nour KK modes might take place.\nThroughout this paper, we have implicitly operated under the assumption that\nthe total energy \ndensity $\\rho$ of the KK tower is negligible compared to the critical density \n$\\rho_{\\mathrm{crit}}$ during the mass-generation epoch.  Under such an assumption, the \nback-reaction of the KK tower on the evolution of the spacetime metric is therefore \nnegligible during this epoch.  Thus, to a very good approximation, we may \nview the dynamics of the KK modes studied in this paper\nas taking place within the context of a particular \n``background'' cosmology which is effectively decoupled from this dynamics and which\nis therefore specified by a fixed functional form for the Hubble parameter $H(t)$ as\na function of $t$.  Indeed, such assumptions are valid for a variety of light scalars which\nreceive their abundances from misalignment production --- including, by necessity,\nany such fields which serve as dark-matter candidates.  \nHowever, one could alternatively consider the opposite regime in which \n\\il{\\rho\\sim\\rho_{\\mathrm{crit}}} during the mass-generation epoch.  Under such conditions,\nthe back-reaction of the KK tower on $H(t)$ can be significant and must be \nincorporated into the equations of motion for the KK modes.  This situation \ncan arise, for example, in scenarios in which these scalar particles play a role\nin inflationary dynamics or in which they are responsible for additional, later \nperiods of reheating after inflation in cosmologies with non-thermal\nhistories~\\cite{WatsonNonThermalHistories}.  It would therefore be interesting\nto study the evolution of the energy densities of the KK modes of our theory\nin such scenarios.\n\n\\bigskip\n\n\n\n\\section*{Acknowledgments}\n\n\n\nWe would like to thank S.~Watson for useful discussions. \nThe research activities of KRD and JK were supported in part by the Department\nof Energy under Grant DE-FG02-13ER41976 (DE-SC0009913), while\nthe research activities of KRD were also supported in part by\nthe National Science Foundation through its employee IR\/D program.\nThe opinions and conclusions\nexpressed herein are those of the authors, and do not represent any funding agencies.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section*{Acknowledgements}\nThis work was carried out in part on the High-Performance Computing resources at Khalifa University.\nBesides, this work is supported in part by the Center for Cyber Security (CCS) at New York University Abu Dhabi (NYUAD).\n\n\n\n\\section{Introduction}\n\\label{sec:Introduction}\n\nLogic locking (LL) is a design-for-trust technique that aims to protect the design intellectual property throughout the supply chain. \nThe design's functionality is locked with a secret key (driven from an on-chip tamper-proof memory) and a set of added key-gates. \nThe locked design functions correctly only when the correct secret key is in place. \nEarlier works in LL focused on the placement of key-gates and high output corruptibility~\\cite{epic_journal,JV-Tcomp-2013,dupuis2014novel}.\nHowever, these schemes were deemed vulnerable by the Boolean satisfiability (SAT)-based attack~\\cite{Subramanyan_host_2015}. \n\n\nProvably secure logic locking (PSLL) ensures an exponential number of SAT calls for a successful key recovery~\\cite{xie2016mitigating,yasin_glsvlsi_2017,yasin_CCS_2017,mengli_provably_secure_camo,shakya2020cas}. \nIn principle, a \\textit{perturb} unit injects errors into the design for some protected input patterns. \nThese errors are canceled out by a \\textit{restore} unit when the correct key is applied. \nRecently, structural\/functional \\textit{oracle-less} attacks have emerged to circumvent PSLL~\\cite{yasin_2017_sps,alrahis2019functional,yang2019stripped,sirone2020functional}.\\footnote{\\textit{Oracle-guided} attacks are successful in approximately circumventing specific instances of PSLL~\\cite{shen2017double,shamsi2017appsat,xiaolin_bypass_attack_ches17}.\nHowever, these attacks require a working chip for functional verification. \nWe focus on the \\textit{oracle-less} attacks as they pose a more significant and realistic threat, as highlighted by the LL community.} \\textbf{Each attack aims to break a specific locking scheme, under restricted parameter settings and circuit formats}, resulting in a large set of attack strategies for different PSLL implementations. \nDeveloping a holistic attack on PSLL is an open research problem.\nTable~\\ref{tab:comparison} outlines the drawbacks of oracle-less attacks on PSLL.\n\n\n\\begin{table}[tb]\n\\centering\n\\scriptsize\n\\vspace{-2mm}\n\\caption{\\textsc{Capabilities offered by oracle-less attacks}}\n\\label{tab:comparison}\n\\resizebox{\\textwidth}{!}{%\n\\setlength\\tabcolsep{1.9pt}\n\\renewcommand\\arraystretch{0.9}\n\\begin{tabular}{cccc}\n\\hline\n\\textbf{Attacks} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Different\\\\ Circuit Formats\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Different\\\\ Locking Schemes\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Different\\\\ Parameter Settings\\end{tabular}} \\\\\\hline\nSPS~\\cite{yasin_2017_sps} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} & \\color{cadmiumgreen}{\\ding{51}}\n\\\\ \\hline\nRE-based~\\cite{alrahis2019functional} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} & \\color{cadmiumgreen}{\\ding{51}} \n\\\\ \\hline\nFALL~\\cite{sirone2020functional} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} \n\\\\ \\hline\nSFLL-HD-Unlocked~\\cite{yang2019stripped} & \n\\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} & \\color{red}{\\ding{55}} \n\\\\ \\hline\n\\textbf{GNNUnlock} & \n\\color{cadmiumgreen}{\\ding{51}} & \n\\color{cadmiumgreen}{\\ding{51}} & \n\\color{cadmiumgreen}{\\ding{51}} \n\\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\n\\subsection{Motivation and Research Challenges}\n\n\\textbf{Restricted Parameter Settings:} Stripped functionality logic locking (\\textit{SFLL-HD$^h$})~\\cite{yasin_CCS_2017} showed resilience against all known attacks on LL.\nSFLL-HD$^h$ protects all input patterns from a Hamming distance $h$ from the secret key $K$. \nSirone \\textit{et al.}~\\cite{sirone2020functional} developed functional analysis attacks (FALL) on SFLL-HD$^h$ which retrieved the secret key \\textit{without} requiring an oracle. \nThe attack algorithms are based on a set of derived Lemmas defining functional properties of the protection logic used in SFLL-HD$^h$. \nThe derived Lemmas hold for specific ranges of $h$ values. \nHence, by definition, the attack algorithms cannot break all cases of SFLL-HD$^h$. \nFor example, their \\textit{AnalyzeUnateness} algorithm is only applicable when $h=0$, and their \\textit{Hamming2D} algorithm is only applicable when $h \\leq K\/4$. \nTheir third algorithm \\textit{SlidingWindow} should apply to larger $h$ values in principle, but it does not fare well since it requires computationally hard SAT calls. \nYang \\textit{et al.}~\\cite{yang2019stripped} proposed the \\textit{SFLL-HD-Unlocked} attack that performs connectivity analysis on the locked circuit followed by a Gaussian-elimination-based analysis to recover the secret key \\textit{without} requiring an oracle. \nBecause of the use of Gaussian-elimination, the attack does not work when $h\\leq4$ due to the composition of singular matrices.\n\nTo demonstrate the shortcomings of existing attacks, we lock selected ISCAS-85 and ITC-99 benchmarks using SFLL-HD$^h$ with $K\/h=2$, multiple times, generating \\texttt{192} locked designs. \n\\textit{Such a locking ratio is critical as it achieves the highest resilience to removal attacks, as indicated in~\\cite{yasin_CCS_2017}}. \nWhen the FALL attacks~\\cite{sirone2020functional} were launched on these locked benchmarks; they reported \\texttt{0} keys, failing to break any of the designs. \nSimilarly, SFLL-HD-Unlocked attack~\\cite{yang2019stripped} also was unable to recover the key for any design. \n\\textit{This analysis highlights the need for a holistic approach that can break SFLL-HD$^h$ and other PSLL schemes under all the possible scenarios}.\n\n\n\\textbf{Restricted Circuit Formats:} The prior attacks on PSLL accept restricted and non-standard circuit formats. \nFor example, the work in~\\cite{alrahis2019functional} showcased that functional reverse engineering (RE) aids in the detection and subsequent removal of the protection logic added by SFLL-HD$^h$. \nHowever, the approach is not generic and will only work on designs synthesized using only 2-input gates. \nThis attack achieves low accuracy results when launched on benchmarks synthesized with a standard cell library. \nFor example, the \\textit{restore} unit detection percentage in the ISCAS-85 benchmark \\texttt{c7552} locked with $K=32$ and $h=0$ drops from $100\\%$ to $0\\%$. Moreover, both attacks~\\cite{yang2019stripped,sirone2020functional} accept circuits in bench format (the latter~\\cite{sirone2020functional} requires topologically sorted AND-OR-INVERT gates), which is a non-industry format, as opposed to \\textit{Verilog\/VHDL}. \nThus, these techniques cannot be employed in the real-world design flow.\n\n\\textbf{Locking Scheme-specific Attacks:} All the aforementioned oracle-less attacks are locking scheme-specific. \nThey target a distinct protection implementation (SFLL-HD$^h$ in this case) and are not scalable to other variants (such as \\textit{Anti-SAT}~\\cite{xie2016mitigating}). \nThe signal probability skew (SPS) attack~\\cite{yasin_2017_sps} is another oracle-less attack targeting Anti-SAT.\nThe SPS attack looks for two oppositely skewed nets feeding an AND gate, a property specific to the Anti-SAT protection and not to other PSLL schemes, and hence it is another scheme-specific attack.\n\n\n\\textbf{Associated Research Challenges:} Automating the detection of the protection logic added by PSLL while accounting for all possible variations among the implementations is an open research problem posing the following important challenges.\n\n\\begin{enumerate}[leftmargin=*]\n\n\\item \\textit{Recognizing all implementation variants:} The protection logic structure depends on (i)~the key-value, (ii)~the Hamming distance value (for the case of SFLL-HD), (iii)~the choice of the logic blocks (for the case of Anti-SAT), and (iv)~the key-size. Hence, different settings will result in distinct topologies. \nThus, utilizing an exact matching algorithm would be a naive approach. \n\n\\item \\textit{Handling different synthesis settings and circuit formats:} The structure of the protection logic and its integration with the original design depends on the synthesis procedures and the target technology library. \nThe attack model should be able to understand and process different circuit formats without having to modify the attack tactic.\n\n\\end{enumerate}\n\n\\subsection{Our Novel Concept and Contributions}\n\nTo address the above challenges, we propose the \\textbf{\\textit{GNNUnlock} framework} that identifies all variants of the targeted protection logic. \nIt is the first-ever concept to leverage graph neural networks (GNNs) along with node connectivity analysis to identify all the gates in a given locked netlist that form the protection logic, as depicted in Fig.~\\ref{fig:GNN_attack}. \nThis work is inspired by the ability of advanced machine learning (ML) models to adapt to new unseen data and the fact that a circuit is a graph with an inherent structure to it. \nUnlike other ML models, GNNs leverage the graph structure and node (gate) features to learn a representation for each node capturing its neighborhood characteristics.\\footnote{We use the terms interchangeably gate or node, and circuit or graph.}\nGNNs are a fitting choice for identifying the protection logic, which can be conceived as a sub-graph with specific properties.\nThe GNN learns the trend of the protection logic and not a syntactic implementation of it, and hence GNNUnlock deals with variations naturally. \nOur GNNUnlock framework employs the following techniques:\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figures\/fig1.pdf}\n\\vspace{-2mm}\n\\caption{High-level view of our work: Breaking PSLL using GNN.\\vspace{-5pt}}\n\\label{fig:GNN_attack}\n\\end{figure}\n\n\\begin{enumerate}[leftmargin=*]\n\n\\item A framework for the \\textbf{netlist-to-graph transformation (Section~\\ref{Sec:transfromation})} is developed which implements automatic \\& efficient feature extraction to capture each gate's functionality and connectivity in the gate-level netlist (in any format). \n\n\\item \\textbf{GNN learning on locked circuits is achieved (Section~\\ref{Sec:attack_gnn})}, allowing the GNN to identify the structural features of all the nodes in the targeted protection logic.\n\n\\item \\textbf{Post-processing rectification procedure (Section~\\ref{Sec:post_processing})} guided by the connectivity of the circuit and by the predictions of the GNN (on the under-attack circuit) is developed to remove the identified protection logic effectively.\n\n\\end{enumerate}\n\nThe effectiveness of GNNUnlock is showcased by breaking \\texttt{564} benchmarks locked using three different PSLL techniques -- SFLL-HD~\\cite{yasin_CCS_2017}, \\textit{TTLock}~\\cite{yasin_glsvlsi_2017}, and Anti-SAT~\\cite{xie2016mitigating}.\nThe effect of different $K$, different $h$ values (for the case of SFLL-HD, and other technology libraries are considered when evaluating GNNUnlock comprehensively. \nAll the files locked at the register-transfer level (RTL) are synthesized following the commercial ASIC-design flow using Synopsys tools.\n\\textbf{We also make our attack framework available online at~\\cite{webinterface}.}\n\\section{Background and Related works}\n\\label{sec:backgroud_RW}\n\n\\subsection{Provably Secure Logic Locking (PSLL)} \n\\subsubsection{Anti-SAT~\\cite{xie2016mitigating}} \n\nIn the Anti-SAT block, the outputs of two complementary logic functions ($gl1$ and $\\overline{gl2}$), locked with two sets of keys are fed into an AND gate, as shown in Fig.~\\ref{fig:SFLL}a. \nWhen the correct key is in place the output of the AND gate $Y$ is \\texttt{0}. For a wrong key-input, the output signal $Y$ could be \\texttt{0} or \\texttt{1} depending on the input $X$. \n$Y$ is XORed with an internal net in the original netlist, locking it. \nHaving the $Y$ signal highly skewed to \\texttt{0} ensures resilience against the SAT-based attack. \nNevertheless, Anti-SAT is vulnerable to structural analysis-based removal attacks~\\cite{yasin_2017_sps}.\n\n\\subsubsection{TTLock~\\cite{yasin_glsvlsi_2017} and SFLL-HD~\\cite{yasin_CCS_2017}} modify the original design, as shown in Fig.~\\ref{fig:SFLL}b, to achieve resilience against removal attacks.\nTTLock protects an input pattern that is the same as the locking key. Tracing the key-inputs (KIs) helps identify the \\textit{restore} logic of TTLock, which is a basic comparator. \nHowever, the \\textit{perturb} unit structure depends on the selected secret key, and not controlled by the external KIs, as demonstrated in Fig.~\\ref{fig:SFLL}c. \nSFLL-HD$^h$ protects all input patterns that are an $h$ Hamming distance away from the secret key. \nTTLock is equivalent to SFLL-HD$^0$. For SFLL-HD$^{h>0}$, the \\textit{restore} and \\textit{perturb} units are Hamming distance checkers ($G$), as illustrated in Fig.~\\ref{fig:SFLL}d. \nThe structure of the \\textit{perturb} unit depends on the hard-coded key, making it difficult to be traced.\n\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figures\/fig2.pdf}\n\\vspace{-2mm}\n\\caption{Logic locking using Anti-SAT~\\cite{xie2016mitigating}, TTLock~\\cite{yasin_glsvlsi_2017}, and SFLL-HD~\\cite{yasin_CCS_2017}. $G$ block in (d) performs Hamming distance checking using adders and comparators. \\vspace{-5pt}}\n\\label{fig:SFLL}\n\\end{figure*}\n\n\\vspace{-0.5ex}\n\\subsection{Graph Neural Network (GNN)} \n\nIn GNNs, nodes aggregate information from their neighbors using neural networks. \nEach node gathers information from its neighbors in a previous layer, where each one of these neighbors collects information from the previous layer. \nThe aggregation functions in \\textit{GraphSAGE}~\\cite{hamilton2017inductive} approach learn to aggregate feature information to obtain embeddings for the nodes, which are then used to perform the desired task such as node classification. \nThe same aggregation parameters are shared for all nodes. \nHence, once the model is trained, those parameters can generate embeddings on entirely unseen graphs. \nWe follow such an inductive approach since we train on a set of locked benchmarks and \\textit{test the model on unseen benchmarks locked with unknown key-values}. \nWe adopt the GraphSAGE architecture in which each node's previous state is concatenated with its neighbors' current state. The mean aggregation function is used to update the state of each node. \n\nDue to neighborhood aggregation, the deeper the GNN model is, the more multi-hop neighbors get incorporated for the root node's computation. \nHence, training time could grow exponentially with GNN depth~\\cite{zeng2019graphsaint}. \nSeveral methods, including GraphSAGE, perform layer sampling to ensure a limited set of neighbors is considered for a node in the next layer. \nSuch sampling speeds up training but suffers from scalability issues.\nWe leverage the graph sampling method \\textit{GraphSAINT}~\\cite{zeng2019graphsaint} to construct mini-batches for training. \nRather than building a GNN on full training graph and then perform layer sampling, GraphSAINT samples the training graph and then builds a GNN on the sampled graph for each mini-batch,\\footnote{In each mini-batch, forward and backward propagation is performed iteratively on the sampled GNN to update weights via stochastic gradient descent.} ensuring scalability concerning graph size and GNN depth. \nTheir random walk-based sampler is used in our experiments.\n\\section{Attacker Model}\n\\label{sec:threat_model}\n\nWe assume that the attacker will be present in the untrusted foundry with access to the GDSII mask information and reverse engineering tools that facilitate the gate-level netlist extraction.\nBesides, the attacker is aware of the locking algorithm and can distinguish between the regular primary inputs (PIs) and KIs. \nThe attacker also knows the usage of the technology library and the usage of different synthesis settings. \nWe do not assume the attacker to have access to an unlocked chip (i.e., an \\textit{oracle-less} setting). \nFor SFLL-HD$^h$~\\cite{yasin_CCS_2017}, the attacker knows the Hamming distance value, while the type of logic function ($gl$) is known for the case of Anti-SAT~\\cite{xie2016mitigating}. \nNote, all these assumptions are in line with the \\textit{Kerckhoffs's principle}, which states that everything about the system should be known to an attacker \\textit{except} for the value of the secret key.\n\\section{Proposed GNNUnlock Framework}\n\\label{sec:Proposed_attack}\n\nFig.~\\ref{fig:GNNUnlock_flow} shows an overview of our methodology, with key steps discussed in the following subsections.\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figures\/fig3.pdf}\n\\vspace{-2mm}\n\\caption{Proposed GNNUnlock methodology. \nThe \\textit{Anti-SAT}, \\textit{restore}, \\textit{perturb}, and \\textit{design} nodes are indicated by AN, RN, PN, and DN, respectively.\n\\vspace{-1pt}\n}\n\\label{fig:GNNUnlock_flow}\n\\end{figure*}\n\n\\subsection{Dataset Generation} \n\nUsing the underlying locking scheme, we generate a set of locked benchmarks for training and validation. \nEach benchmark is locked several times with randomly generated key-values and different key-sizes $K$, which results in an extensive training set. \nThe training process is not aware of the correct key-values. \nHowever, different key-values cause the generation of varying \\textit{perturb} and \\textit{Anti-SAT} structures, and hence variants are accounted for during training. GNNUnlock transforms locked benchmarks into graphs (Section~\\ref{Sec:transfromation}) and labels the nodes. \nFor the case of SFLL-HD$^{h>0}$\/TTLock, a node belongs to the \\textit{perturb}, the \\textit{restore}, or the \\textit{design} logic. For the case of Anti-SAT, a node belongs to the \\textit{Anti-SAT} block or the \\textit{design} logic.\n\n\nGNNUnlock attacks each design independently by excluding its corresponding graphs from training\/validation. For example, when attacking \\texttt{b17\\_C} from ITC-99, only the graphs of \\texttt{b14\\_C}, \\texttt{b15\\_C}, \\texttt{b20\\_C}, and \\texttt{b21\\_C} are used in training, while the graphs of \\texttt{b22\\_C} are used for validation to evaluate the model on unseen data, thereby avoiding biasing.\n\n\n\\subsection{Netlist-to-Graph Transformation}\n\\label{Sec:transfromation}\n\nWe model connectivity between the gates in a design using an undirected graph $G(I, J)$. \nThe set $I$ of nodes represents all the gates, while the set $J$ of edges represents the wires. \nAn example of a logic locked netlist and the corresponding graph is shown in Fig.~\\ref{fig:GNNUnlock_flow}b. \nThe set $I$ does not contain PIs, KIs, or primary outputs (POs) of the design. \nEach node in the graph is associated with a feature vector $\\hat{f}$ that contains information describing the node's important characteristics, such as its in-degree ($IN$) and out-degree ($OUT$). \nIt also contains information on whether the node is connected to a PI, a PO, or a KI. \nAdditionally, $\\hat{f}$ captures the type and number of gates appearing in the neighborhood of the node (all nodes within two-hops away). \nFor example, $\\hat{f_{i}}$ shows that gate $i$ is connected to a PO with $IN=3$ and $OUT=1$ (Fig.~\\ref{fig:GNNUnlock_flow}b). \nIt also indicates that three XOR gates and one XNOR gate exist in $i$'s neighborhood. \nThe length of the feature vector $|\\hat{f}|$ depends on the number of logic gates in the target library. \nEach locked design is represented by a graph instance with a corresponding adjacency matrix.\nTo feed multiple graphs of different sizes to the GNN, a block-diagonal matrix is created for each dataset. \nEach block represents the adjacency of one locked design.\n\n\\vspace{-1ex}\n\\subsection{GNN Topology}\n\\label{Sec:attack_gnn}\n\nIn all experiments reported in Section~\\ref{sec:results}, graph sampling using the GraphSAINT method is performed.\nA two-layer GNN is constructed using GraphSAGE architecture, with mean and concatenation aggregation, and the \\textit{ReLU} activation function. Training stops after a fixed number of epochs based on convergence. \nThe model with the best performance on the validation set is used to evaluate the test set accuracy. \nDetails regarding the GNN model and sampling are shown in Table~\\ref{tab:GNN_info}.\n\n\\subsection{Post-processing}\n\\label{Sec:post_processing}\n\nAlthough our trained GNN achieves high node classification accuracy on its own (with an average of $99.99\\%$, $99.95\\%$, and $99.92\\%$, for the case of Anti-SAT, TTLock, and SFLL-HD$^2$, respectively), we propose a post-processing algorithm to rectify any potential misclassifications to enhance the accuracy further.\nThe algorithm considers predictions made by the GNN, connectivity of the circuit, and known properties of the protection logic, which is explained further in the next subsections.\n\n\\subsubsection{Anti-SAT} Every node in the \\textit{Anti-SAT} block must have at least one KI in its fan-in cone. \nIf a node without KIs in its fan-in cone is predicted as an \\textit{Anti-SAT} node, then the prediction will be dropped. \nFor each \\textit{predicted design} node, we check its fan-in cone. If there are only other \\textit{predicted Anti-SAT} nodes in the fan-in cone, then the node initially classified as a \\textit{design} node is considered part of the Anti-SAT block. The flow of our post-processing algorithm is presented in Fig.~\\ref{fig:GNNUnlock_flow}c. \n\n\\subsubsection{TTLock and SFLL-HD$^{h>0}$} Our post-processing algorithm for the case of SFLL-HD$^{h>0}$ and TTLock is presented in Fig.~\\ref{fig:GNNUnlock_flow}d. \nThe \\textit{predicted restore} nodes are visited to identify possible protected inputs (set $X$). The following properties then guide the post-processing procedure. (i)~All \\textit{restore} nodes have KIs in their fan-in cone. \n(ii)~All \\textit{perturb} nodes have connections with the \\textit{restore} nodes in the netlist and are controlled solely by protected inputs $X$. \nIf a \\textit{predicted restore} node has KIs in its fan-in cone, then the prediction is confirmed. \nOtherwise, the algorithm checks if the node is a \\textit{perturb} node. \nA \\textit{predicted perturb} node is verified if it has a connection to other \\textit{predicted restore} nodes in the netlist and if the node has $X$ (and no other PIs) in its fan-in cone. \nIn order not to misclassify \\textit{perturb} nodes as \\textit{design} nodes, the algorithm checks the fan-in cone of the predicted \\textit{design} nodes. \nIf a predicted \\textit{design} node has $X$ and other predicted \\textit{perturb} nodes in its fan-in, it is a \\textit{perturb} node.\n\n\\begin{table}[tb]\n\\centering\n\\vspace{-2mm}\n\\caption{\\textsc{GNN configuration and sampling details. The \\#classes for SFLL-HD$^{h>0}$ and TTLock is \\texttt{3}, while for Anti-SAT it is \\texttt{2}}}\n\\label{tab:GNN_info}\n\\resizebox{\\textwidth}{!}{%\n\\renewcommand\\arraystretch{0.9}\n\\begin{tabular}{cccc}\n\\hline\n\\multicolumn{2}{c}{\\textbf{Architecture}} & \\multicolumn{2}{c}{\\textbf{Training and Sampling}} \\\\ \\hline\n\\textbf{Input Layer} & {[}$|\\hat{f}|$,$512${]} & \\textbf{Optimizer} & Adam \\\\ \\hline\n\\textbf{Hidden Layer 1} & {[}$1024$,$512${]} & \\textbf{Learning Rate} & $0.01$ \\\\ \\hline\n\\textbf{Hidden Layer 2} & {[}$1024$,$512${]} & \\textbf{Dropout} & $0.1$ \\\\ \\hline\n\\textbf{Output Layer} & {[}$512$,\\#classes{]} & \\textbf{Sampler} & Random Walk \\\\ \\hline\n\\textbf{Aggregation} & Mean with concatenation & \\textbf{Walk Length} & $2$ \\\\ \\hline\n\\textbf{Activation} & ReLU & \\textbf{Root Nodes} & $3000$ \\\\ \\hline\n\\textbf{Classification} & Softmax & \\textbf{Max \\# Epochs} & $2000$ \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\\section{Experimental Investigations}\n\\label{sec:results}\n\n\\subsection{Evaluation Setup, Tool Flow, and Evaluation Metrics} \n\nGNNUnlock is evaluated on selected ISCAS-85 and ITC-99 benchmarks. \nNetlist-to-graph transformation is implemented in Perl\/Python3. \nTensorflow with Python3 implementation of GraphSAINT is used for GNN training~\\cite{zeng2019graphsaint}. \nTraining is performed on a single node with 24 cores (2x Intel Xeon CPUs E5-2695 v2@2.4 GHz), 256GB RAM, and one NVIDIA Tesla K20m GPU (2,496 CUDA cores and 5GB of GDDR5 memory). Fig.~\\ref{fig:GNNUnlock_exp} summarizes the experimental setup.\n\n\\subsubsection{Dataset Generation for Anti-SAT}\n\nThe benchmarks (in bench format) were locked using the Anti-SAT locking binary provided by the authors.\nEach ISCAS-85 benchmark is locked twice with $K:\\{8, 16, 32, 64\\}$, except for \\texttt{c3540}, where $K=64$ is not considered due to the limited number of PIs in the design. \nIn total, \\texttt{30} Anti-SAT locked ISCAS-85 benchmarks are obtained. \n$K=8$ is used to evaluate the effectiveness of GNNUnlock in isolating the Anti-SAT block when its size is very small compared to that of the original design.\nEach ITC-99 benchmark is locked twice with $K:\\{32, 64, 128\\}$ resulting in a total of \\texttt{36} Anti-SAT locked ITC-99 benchmarks. Two labeled datasets for Anti-SAT block identification have been created, one for locked ISCAS-85 benchmarks and one for locked ITC-99 benchmarks. A feature vector $|\\hat{f}|=13$ is associated with each node. \n\\textit{The Anti-SAT locking binary only accepts circuits in bench format, which includes a restricted set of logic gates (\\texttt{8} gates)}. \nHence, only \\texttt{8} out of the \\texttt{13} features are required to represent the neighborhood of each node.\nRest of the features represent $IN$, $OUT$, and the connectivity to PIs, KIs, or POs.\n\\subsubsection{Dataset Generation for TTLock and SFLL-HD$^{h>0}$}\nSFLL-HD$^h$ is implemented in Perl as described in~\\cite{yasin_CCS_2017} to lock the benchmarks at RTL -- TTLock is the case when $h=0$. Each ISCAS-85 benchmark is locked thrice with $K:\\{8, 16, 32, 64\\}$ for $h:\\{0,2\\}$, except for the \\texttt{c3540} benchmark where $K=64$ is not considered due to the limited number of PIs in the design. \nIn total, \\texttt{45} locked ISCAS-85 benchmarks for each $h$ are obtained. \nEach ITC-99 benchmark is locked thrice with $K:\\{32, 64, 128\\}$ for $h:\\{0,2,4\\}$, resulting in a total of \\texttt{54} locked ITC-99 benchmarks for each setting of $h$. \nLocked RTL files are synthesized using the standard ASIC design flow for the $65nm$ LPe technology. \nSynthesis is performed using Synopsys Design Compiler. \n$|\\hat{f}|=34$ is associated with each node. Since a full standard cell library is used for synthesis, a larger $|\\hat{f}|$ (than the case of Anti-SAT) is required to capture all the possible combinational logic gates.\n\nTo verify that GNNUnlock handles different circuit formats and technologies, the Nangate $45nm$ open cell library is also used for synthesis when $h=2$. Varying the circuits' format affects $|\\hat{f}|$ only. For this case, $|\\hat{f}|=18$ is used. \nWe also consider other corner cases for when $h:\\{16, 32, 64\\}$ with corresponding $K:\\{32,64,128\\}$, more details are in Section~\\ref{sec:corner_cases}. The characteristics of all the datasets are listed in Table~\\ref{tab:Datasets}.\n\n\\subsubsection{Evaluation Methods and Metrics}\n\nGNNUnlock attacks each design independently, by excluding its corresponding graphs from training and validation. \nFor example, when attacking the TTLock-ed \\texttt{b17\\_C} design from ITC, only the graphs of \\texttt{b14\\_C}, \\texttt{b15\\_C}, \\texttt{b20\\_C}, and \\texttt{b21\\_C} are included in the training set, resulting in \\texttt{161,942} training nodes (\\texttt{36} graphs), while the graphs of \\texttt{b22\\_C} are included only in the validation set (to evaluate the model on unseen data), resulting in \\texttt{74,753} validation nodes (\\texttt{9} graphs). And only the graphs of \\texttt{b17\\_C} are tested on, resulting in \\texttt{110,685} testing nodes (\\texttt{9} graphs).\n\nThe predictions of the GNN are compared with the true labels of the nodes to evaluate the performance of the attack. \nWe report the accuracy and the non-averaged precision, recall, and F1-score for each classifier. \nWe further evaluate GNNUnlock by removing the predicted protection logic to retrieve the unlocked design without accessing the true labels (Section~\\ref{Sec:post_processing}). \nThe recovered benchmark is then compared with the original benchmark via circuit-equivalence using Synopsys Formality.\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.97\\textwidth]{figures\/fig4.pdf}\n\\vspace{-2mm}\n\\caption{Evaluation setup showing different components and platforms. TE, TR, and VA stand for testing, training, and validation, respectively.\\vspace{-5pt}}\n\\label{fig:GNNUnlock_exp}\n\\end{figure}\n\n\n\\subsection{Breaking Anti-SAT with GNNUnlock}\n\\label{sec:breaking_antisat}\n\nFor the ISCAS-85 dataset, the GNN gave $100\\%$ node classification accuracy for \\texttt{25} of the tested graphs, and an average of $99.98\\%$ accuracy across all \\texttt{30} graphs. \nFor the remaining \\texttt{5} graphs, only \\texttt{6} nodes, were misclassified and then rectified during post-processing. \nFor example, two \\textit{design} nodes feeding the XOR gate connecting the Anti-SAT block with the rest of the circuitry got misclassified as part of the Anti-SAT block. \nHowever, having no KIs in their fan-in, the prediction is ignored. \nFor the ITC-99 dataset, the GNN gave $100\\%$ node classification accuracy for \\texttt{29} out of \\texttt{30} tested graphs. \nOnly one node was misclassified in one of the \\texttt{b21\\_C} graphs locked with $K=128$, which was rectified during post-processing. \nDetailed results are reported in Table~\\ref{tab:Anti_SAT_results}.\n\n\\begin{table}[tb]\n\\centering\n\\caption{\\textsc{Summary of Generated Datasets}}\n\\label{tab:Datasets}\n\\resizebox{\\textwidth}{!}{\n\\setlength\\tabcolsep{1.9pt} \n\\renewcommand\\arraystretch{0.9}\n\\begin{tabular}{ccccccc}\n\\hline\n\\textbf{Dataset} & \\textbf{Benchmarks} & \\textbf{Circuit Format} & \\textbf{\\#Classes} & \\textbf{$|\\hat{f}|$} & \\textbf{\\#Nodes} & \\textbf{\\#Circuits} \\\\ \\hline\n\\multirow{2}{*}{\\textbf{Anti-SAT}} & ISCAS-85 & Bench & $2$ & $13$ & $69,468$ & $30$ \\\\ \\cline{2-7} \n & ITC-99 & Bench & $2$ & $13$ & $450,359$ & $36$ \\\\ \\hline\n\\multirow{2}{*}{\\textbf{TTLock}} & ISCAS-85 & Verilog netlist $65nm$ & $3$ & $34$ & $29,745$ & $45$ \\\\ \\cline{2-7} \n & ITC-99 & Verilog netlist $65nm$ & $3$ & $34$ & $347,380$ & $54$ \\\\ \\hline\n\\multirow{3}{*}{\\textbf{SFLL-HD$^2$}} & ISCAS-85 & Verilog netlist $65nm$ & $3$ & $34$ & $30,178$ & $45$ \\\\ \\cline{2-7} \n & \\multirow{2}{*}{ITC-99} & Verilog netlist $65nm$ & $3$ & $34$ & $357,374$ & $54$ \\\\ \\cline{3-7} \n & & Verilog netlist $45nm$ & $3$ & $18$ & $391,411$ & $54$ \\\\ \\hline\n\\textbf{SFLL-HD$^4$} & ITC-99 & Verilog netlist $65nm$ & $3$ & $34$ & $356,420$ & $54$ \\\\ \\hline\n\\textbf{SFLL-HD$^{16}$} & ISCAS-85 & Verilog netlist $65nm$ & $3$ & $34$ & $33,354$ & $48$ \\\\ \\hline\n\\textbf{SFLL-HD$^{32}$} & \\multirow{2}{*}{ITC-99} & Verilog netlist $65nm$ & $3$ & $34$ & $465,032$ & $72$ \\\\ \\cline{1-1} \\cline{3-7} \n\\textbf{SFLL-HD$^{64}$} & & Verilog netlist $65nm$ & $3$ & $34$ & $483,777$ & $72$ \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table}[tb]\n\\centering\n\\vspace{-2mm}\n\\caption{\\textsc{Results of GNNUnlock on Anti-SAT. \nA \\textit{design} node and an \\textit{Anti-SAT} node are indicated by DN and AN, respectively. \nMN stands for misclassified nodes}}\n\\label{tab:Anti_SAT_results}\n\\resizebox{\\textwidth}{!}{%\n\\setlength\\tabcolsep{1.9pt} \n\\renewcommand\\arraystretch{0.8}\n\\begin{tabular}{ccccccccccc}\n\\hline\n\\multirow{2}{*}{\\textbf{Test}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}\\#Test\\\\ Graphs\\end{tabular}}}& \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}GNN\\\\ Acc. (\\%)\\end{tabular}}}& \\multicolumn{2}{c}{\\textbf{Prec. (\\%)}} & \\multicolumn{2}{c}{\\textbf{Rec. (\\%)}} & \\multicolumn{2}{c}{\\textbf{F1-Score (\\%)}} & \\multirow{2}{*}{\\textbf{\\#MN}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Removal\\\\ Success (\\%)\\end{tabular}}} \\\\ \\cline{4-9}\n & & & \\textbf{AN} & \\textbf{DN} & \\textbf{AN} & \\textbf{DN} & \\textbf{AN} & \\textbf{DN} & & \\\\ \\hline\n\\textbf{c2670}&$8$ &$99.98$ &$99.79$ & $100$ & $100$ &$99.98$ &$99.90$ &$99.99$ & \n\\begin{tabular}[c]{@{}c@{}}$2$ DN \\end{tabular}& $100$ \\\\ \\hline\n\n\\textbf{c3540}&$6$ &$99.98$ &$99.55$ &$99.99$ &$99.78$ &$99.98$ &$99.67$ &$99.99$ & \\begin{tabular}[c]{@{}c@{}}$2$ DN\\\\$1$ AN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{c5315}&$8$ &$99.99$ &$99.90$ & $100$ & $100$ &$99.99$ &$99.95$ &$99.99$ & $1$ DN & $100$ \\\\ \\hline\n\\textbf{c7552}&$8$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\textbf{b14\\_C}&$6$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\textbf{b15\\_C}&$6$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\textbf{b20\\_C}&$6$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\textbf{b21\\_C}&$6$ &$99.99$ & $100$ &$99.99$ &$99.92$ & $100$ &$99.96$ &$99.99$ & $1$ AN & $100$ \\\\ \\hline\n\n\\textbf{b17\\_C}&$6$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $-$ & $100$ \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\n\\begin{table*}[tb]\n\\centering\n\\vspace{-2mm}\n\\caption{\\textsc{Results of GNNUnlock on SFLL-HD$^2$. \nThe \\textit{design}, \\textit{perturb}, and \\textit{restore} nodes are indicated by DN, PN, and RN, respectively}}\n\\label{tab:TTLOCK_SFLL_HD_results}\n\\scriptsize\n\\resizebox{0.97\\textwidth}{!}{%\n\\renewcommand\\arraystretch{0.85}\n\\begin{tabular}{ccccccccccccccc}\n\\hline\n\\multirow{2}{*}{\\textbf{Test Set}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}\\#Test\\\\ Graphs\\end{tabular}}}& \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}GNN\\\\ Acc. (\\%)\\end{tabular}}} & \\multicolumn{3}{c}{\\textbf{Prec. (\\%)}} & \\multicolumn{3}{c}{\\textbf{Rec. (\\%)}} & \\multicolumn{3}{c}{\\textbf{F1-Score (\\%)}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}\\#Misclassified\\\\ Nodes\\end{tabular}}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Removal\\\\ Success (\\%)\\end{tabular}}} \\\\ \\cline{4-12}\n & & &\\textbf{RN} & \\textbf{PN} & \\textbf{DN} & \\textbf{RN} & \\textbf{PN} & \\textbf{DN} & \\textbf{RN} & \\textbf{PN} & \\textbf{DN} & & \\\\ \\hline\n\\textbf{c2670} &$12$&$99.53$ &$99.55$ & $96.36$ & $100$ & $100$ &$99.37$ &$99.46$ &$99.77$ & $97.84$ &$99.73$ & \\begin{tabular}[c]{@{}c@{}}$24$ DN as PN\\\\ $4$ PN as RN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{c3540}&$9$ &$99.79$ & $98.31$ & $98.97$ & $100$ & $100$ & $98.30$ &$99.88$ &$99.15$ & $98.63$ &$99.94$ & \\begin{tabular}[c]{@{}c@{}}$5$ PN as RN\\\\ $3$ DN as PN\\\\ $2$ DN as RN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{c5315 } &$12$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ &$-$& $100$ \\\\ \\hline\n\\textbf{c7552 } &$12$ &$99.99$ & $100$ & $100$ &$99.99$ & $100$ &$99.85$ & $100$ & $100$ &$99.93$ &$99.99$ & $1$ PN as DN & $100$ \\\\ \\hline\n\\textbf{b14\\_C } &$9$ &$99.97$ &$99.88$ & $100$ &$99.98$ & $100$ &$99.43$ & $100$ &$99.94$ &$99.72$ &$99.99$ & \\begin{tabular}[c]{@{}c@{}}$2$ PN as RN\\\\ $5$ PN as DN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{b15\\_C } &$9$ &$99.99$ &$99.94$ & $100$ &$99.99$ & $100$ &$99.68$ & $100$ &$99.97$ &$99.84$ &$99.99$ & \\begin{tabular}[c]{@{}c@{}}$3$ PN as DN\\\\ $1$ PN as RN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{b20\\_C } &$9$ &$99.98$ &$99.46$ & $100$ &$99.99$ & $100$ &$99.04$ & $100$ & \\multicolumn{1}{l}{$99.73\\%$} & \\multicolumn{1}{l}{$99.52\\%$} & \\multicolumn{1}{l}{$99.99\\%$} & \\begin{tabular}[c]{@{}c@{}}$9$ PN as RN\\\\ $2$ PN as DN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{b21\\_C } &$9$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ & $100$ &$-$& $100$ \\\\ \\hline\n\\textbf{b22\\_C } &$9$ &$99.96$ &$99.94$ & $97.83$ &$99.99$ & $100$ &$99.67$ &$99.96$ &$99.97$ & $98.74$ &$99.98$ & \\begin{tabular}[c]{@{}c@{}}$1$ PN as RN\\\\ $3$ PN as DN\\\\ $27$ DN as PN\\end{tabular} & $100$ \\\\ \\hline\n\\textbf{b17\\_C } &$9$ &$99.94$ &$99.46$ & $95.60$ & $100$ & $100$ &$99.52$ &$99.95$ &$99.73$ & $97.52$ &$99.98$ & \\begin{tabular}[c]{@{}c@{}}$3$ DN as RN\\\\ $57$ DN as PN\\\\ $6$ PN as RN\\end{tabular} & $100$ \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table*}\n\nHaving $100\\%$ node classification accuracy for a given benchmark locked with different key-sizes (such as the case for \\texttt{c7552}, \\texttt{b14\\_C}, etc.) shows that \\textbf{the ratio of sizes between Anti-SAT block and design does not affect the performance of GNNUnlock and overall recovery of the original design}. \nObtaining $100\\%$ node classification accuracy for Anti-SAT is validated since the Anti-SAT block has a specific structure controlled by external KIs, making it easy to be learned.\n\n\n\\subsection{Breaking SFLL-HD$^{h>0}$ and TTLock with GNNUnlock}\n\\label{sec:breaking_ttlock}\n\n\nFor the SFLL-HD$^2$ ISCAS-85 dataset, the GNN gave $100\\%$ node classification accuracy for \\texttt{20} tested graphs and an average of $99.83\\%$ accuracy across all \\texttt{45} ISCAS-85 graphs. \nIn total, \\texttt{38} nodes were misclassified then rectified using post-processing. \nFor example, \\texttt{27} \\textit{design} nodes, coming from the different \\texttt{25} graphs were misclassified as \\textit{perturb} nodes. \n\\textit{The misclassified nodes belonged to NOR-tree-like structures in the original designs.}\nThis misclassification did not affect the removal of the protection logic because non-protected input patterns controlled these nodes and hence predictions were dropped.\nFor SFLL-HD$^2$ ITC-99 dataset, the GNN gave $100\\%$ node classification accuracy for \\texttt{29} of the tested graphs and an average of $99.97\\%$ accuracy across all \\texttt{54} graphs (Table~\\ref{tab:TTLOCK_SFLL_HD_results}). \n\n\nFor the TTLock ISCAS-85 dataset, the GNN gave $100\\%$ node classification accuracy for $29$ of the tested graphs and an average of $99.94\\%$ accuracy across all $45$ graphs. \nFor the remaining $16$ graphs, only $19$ nodes were misclassified. \nThe results show that \\textbf{GNNUnlock can accurately detect all three types of nodes} with precision and recall values ranging from $93.17\\%$ to $100\\%$. \nFor the TTLock ITC-99 dataset, the GNN gave $100\\%$ node classification accuracy for $16$ of the tested graphs and an average of $99.95\\%$ accuracy across all $54$ graphs. \nComparing accuracy results on ISCAS-85 Vs. ITC-99 (for both SFLL-HD$^{h>0}$ and TTLock) confirms that \\textbf{GNNUnlock has a consistent performance for predicting the protection logic regardless of the benchmark size}.\nTable~\\ref{tab:results_hd4_nan} represents all the averaged results for TTLock datasets.\nIt was observed that the \\textit{restore} predictor is $100\\%$ successful (with $100\\%$ precision and recall) in all ITC-99 test cases. This also confirms that the separation between the \\textit{perturb} and \\textit{design} nodes is challenging.\n\n\\textit{To study the effect of $h$ on the performance of GNNUnlock}, we launch the attack on the SFLL-HD$^4$ ITC-99 dataset. \nNote: due to lack of space, we only report the averaged metrics in Table~\\ref{tab:results_hd4_nan}. \nHere, we also report the results of GNNUnlock on the SFLL-HD$^2$ ITC-99 dataset with a technology node of $45nm$ to evaluate the effect of the target library on the attack's performance. \nThe trend is the same in all cases, while there is a small difference in the values. \nGNNUnlock detects all three types of nodes accurately with average accuracy values of $99.94\\%$, $99.97\\%$, and $99.90\\%$, for SFLL-HD$^2$ in $45nm$, SFLL-HD$^2$ in $65nm$, and SFLL-HD$^4$ in $65nm$ datasets, respectively. \nAfter post-processing, all nodes were classified correctly ($100\\%$), verifying that \\textbf{GNNUnlock can handle different technology nodes and various parameter settings.}\n\n\\subsection{Comparison with State-of-the-art Attacks}\n\\label{sec:corner_cases}\n\nTo demonstrate the superiority of GNNUnlock, a set of ITC-99 benchmarks are locked using SFLL-HD with $K\/h=2$, generating two datasets: one for $K=128$, $h=64$ and one for $K=64$, $h=32$.\nThe ISCAS-85 benchmarks are locked using $K=32$, $h=16$. \nThe characteristics are listed in Table~\\ref{tab:Datasets}. When the FALL attacks~\\cite{sirone2020functional} were launched on these benchmarks, they reported $0$ keys. Next, the attack platform of SFLL-HD-Unlocked~\\cite{yang2019stripped} was launched on the same benchmarks, which failed to identify the perturb signals and did not recover the secret key. \nHowever, our GNNUnlock was $100\\%$ successful in retrieving the unlocked designs in all of the cases.\nThe results are documented in Table~\\ref{tab:results_hd4_nan}.\n\n\\begin{table}[tb]\n\\centering\n\\caption{\\textsc{Examining the effect of $h$ value and technology node on the performance of GNNUnlock. TR stands for training}}\n\\label{tab:results_hd4_nan}\n\\resizebox{\\textwidth}{!}{%\n\\setlength\\tabcolsep{1.9pt}\n\\begin{tabular}{ccccccccc}\n\\hline\n\\textbf{Dataset} &\\textbf{Benchmarks} &\\textbf{\\begin{tabular}[c]{@{}c@{}}Tech.\\\\ Node (nm)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}GNN\\\\ Acc. (\\%)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Avg.\\\\ Prec. (\\%)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Avg.\\\\ Rec. (\\%)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Avg.\\\\ F1-Score (\\%)\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Removal\\\\ Success (\\%)\\end{tabular}}& \\textbf{\\begin{tabular}[c]{@{}c@{}}Avg.\\\\ TR Time ($s$)\\end{tabular}} \n\\\\ \\hline\n\\textbf{TTLock} & ISCAS-85 &$45$ & $99.94$ &\t$99.20$&\t$99.53$\t&$99.33$& $100$&$1,910$\\\\ \\hline\n\\textbf{TTLock} & ITC-99 &$45$ & $99.95$ &\t$97.48$&\t$99.36$&\t$98.33$ & $100$&$22,777$\\\\ \\hline\n\\textbf{SFLL-HD$^2$} &ITC-99& $45$ & $99.94$ & $98.73$ &$99.77$ &$99.21$ & $100$&$24,123$ \\\\ \\hline\n\\textbf{SFLL-HD$^2$} &ITC-99& $65$ & $99.97$ &$99.56$ &$99.85$ &$99.70$ & $100$ &$22,052$\\\\ \\hline\n\\textbf{SFLL-HD$^4$} &ITC-99& $65$ &$99.90$ & $98.40$ &$99.60$ & $98.95$ & $100$&$233,017$ \\\\ \\hline\n\\textbf{SFLL-HD$^{16}$} &ISCAS-85& $65$ &$99.24$ & $97.68$ & $97.30$ & $97.40$ & $100$&$1,907$ \\\\ \\hline\n\\textbf{SFLL-HD$^{32}$} &ITC-99& $65$ &$99.83$ & $97.56$ & $98.45$ & $97.96$ & $100$&$28,049$ \\\\ \\hline\n\\textbf{SFLL-HD$^{64}$} &ITC-99& $65$ &$99.69$&\t$97.60$&\t$97.96$\t&$97.74$&$100$&$32,494$\\\\\\hline\n\\end{tabular}\n}\n\\end{table}\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nIn this paper, we proposed the first-of-its-kind scheme for unlocking provably secure logic locking (PSLL), leveraging a graph neural network (GNN) that learns the common structural features of the protection logic added by such techniques. \nThe GNN does not learn a syntactic implementation but absorbs the protection logic trend, allowing it to handle variants naturally.\nWe also developed a post-processing algorithm to rectify any misclassifications by the GNN, depending solely on the nodes' connectivity, the GNN predictions, and the known properties of the protection logic. \nWe demonstrated the superiority of GNNUnlock by comparing it to two state-of-the-art attacks in unlocking three PSLL techniques, Anti-SAT, TTLock, and SFLL-HD. \nOur GNNUnlock scheme was able to break all locked benchmarks while state-of-the-art attacks struggled with some corner cases.\nWe believe GNNUnlock opens up new frontiers in advancing the state-of-the-art defenses and attacks in logic locking.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}