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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nChiral Perturbation Theory (ChPT) is the effective field theory of QCD at low energies \\cite{weinberg75,sielder}. \nIts  paradigmatic application is the purely mesonic sector in $SU(2)$.\\footnote{Which even presents one corner of \nconcern due to the enhanced role of the right-hand-cut in the isoscalar scalar pion-pion scattering \n\\cite{npa620,nd,zeros,gamma,alba12,bernasigma,truong1988}, with an important \nimpact as well in the pion-nucleon ($\\pi N$) sector \\cite{sainiosumrule,aco2013,sigma2012}.}\nIts extension to the one-baryon sector presents some complications due to the \nlarge nucleon mass that does not vanish in the chiral limit \\cite{sainio88,manohar91}, which posed interesting problems to \nthe theory.\\footnote{A faster stabilization of the chiral series in this case has been recently accomplished \\cite{sigma2012,aco2013,strangeness14} by combining the covariant formalism of the Extended on Mass Shell Regularization Scheme (EOMS) \\cite{eoms} with the explicit inclusion \nof the $\\Delta(1232)$ in the $\\delta$-counting \\cite{delta-c} .}\n  For reviews on ChPT on these topics see e.g. \\cite{bijnens09,ecker,bernard,pich,ulf}.\n\nThe extension of ChPT to systems with a larger  baryonic number was  \nconsidered in  Ref.~\\cite{weinn}, where the chiral counting is applied to the calculation \nof the multi-nucleon potential. In these cases one also has to face the problem associated with the \ninfrared enhancement associated with the small nucleon kinetic energies, which requires \nto resum the infinite string of diagrams due to the iteration of  intermediate multi-nucleon states. \nThe extension of the chiral power counting to finite density system, including the contributions of multi-nucleon \nreducible diagrams, is given in Ref.~\\cite{finiteden}. For related reviews see e.g. \n\\cite{Epelbaum:2008ga,Machleidt:2011zz,Epelbaum:2005pn,Bedaque:2002mn,vanKolck:1999mw}.\n\n The application of the set up of Ref.~\\cite{weinn} to nucleon-nucleon ($NN$) scattering \nhas been phenomenologically successful \\cite{ordo94,entem,thesis,epe042}. However, the sensitivity \nof the results on the values of the cutoff taken to solve the associated  Lippmann-Schwinger equation for the \niteration of two-nucleon intermediate states has given rise to a flurry of publications, whose fair \n and comprehensive consideration is beyond this introduction. For more detailed accounts on this \nrespect the reader is referred to \\cite{Machleidt:2011zz,pavon06,nogga,kswa,phillipssw,pavon11,longyang,zeoli,Epelbaum:2008ga}.\n\n\nWe continue here  the application of the $N\/D$ method \\cite{chew} to $NN$ scattering \n extending the previous work of Refs.~\\cite{paper1,paper2,gor2013}. For this method the dynamical \ninput is not the $NN$ potential but the discontinuity of a $NN$ partial-wave amplitude along the \nleft-hand-cut (LHC), which is denoted in the following by $2i\\Delta(A)$. Here $A$ is   \nthe center of mass (c.m.) three-momentum squared of a $NN$ state. In other words, \n$\\Delta(A)$ is the imaginary part of a $NN$ partial-wave amplitude along the LHC, that \n extends for real $A$ with $A<-M_\\pi^2\/4$, being $M_\\pi$ the pion ($\\pi$) mass.  \nThe function $\\Delta(A)$  is due to the multi-exchange of pions driving the finite-range nuclear \nforces, while in a low-energy effective field theory the short-range nuclear forces are accounted for by \nlocal interactions of zero range that do not contribute to $\\Delta(A)$ for finite $A$.\n The two-nucleon irreducible contributions to $\\Delta(A)$ are amenable to a straightforward ChPT expansion, \nin much the same way as discussed  in Ref.~\\cite{weinn} for the calculation of the chiral $NN$ potential.\nHowever, $\\Delta(A)$ has also contributions from two-nucleon reducible diagrams but, \nas explained in Ref.~\\cite{gor2013},  these contributions require to cut all \nthe pion lines simultaneously when iterating one-pion exchange (OPE). \nIn this way, when including an extra $NN$ intermediate state in the iteration of the unitarity two-nucleon \ndiagrams their contribution to $\\Delta(A)$ starts further away in the LHC. It then results that the $n$th iteration of \n two-nucleon intermediate states, which at least requires $n+1$ OPE ladders, \n gives contribution to $\\Delta(A)$ only for $A<-(n+1)^2 M_\\pi^2\/4$. \nThis makes that its relevance \nfor physical values of $A$ ($A\\geq 0$) in the low-energy region clearly dismisses \nwith increasing $n$. As a result, because of the chiral expansion together with this other effect that\n numerically suppresses the proliferation of two-nucleon reducible diagrams in the calculation of $\\Delta(A)$,  \none can determine this function reliably in ChPT.\\footnote{Notice that the suppression \nof the iteration of two-nucleon reducible diagrams only occurs for $\\Delta(A)$, and it does not occur to any \nother ``component'' of a $NN$ partial wave amplitude.}\n\n In Refs.~\\cite{paper1,paper2} the $N\/D$ method was \nsolved with $\\Delta(A)$  calculated at leading order (LO) from OPE, while in Ref.~\\cite{gor2013}\nthe NLO contributions to  $\\Delta(A)$ were also included. \nThese contributions comprise two-nucleon irreducible two-pion exchange and once-iterated OPE, whose sum gives \nthe  leading two-pion exchange (TPE). \nReference~\\cite{gor2013} obtained a clear improvement in the reproduction of the phase shifts and mixing angles  \n given by the Nijmegen partial-wave analysis (PWA) \\cite{Stoks:1994wp} as compared with the LO \nstudy, so that a  global and rather good agreement  is achieved at NLO. \n We want to give one step forward and consider here the next-to-next-to-leading (NNLO) contributions to \n$\\Delta(A)$, which are given by the imaginary part along the LHC of the \ntwo-nucleon irreducible TPE diagrams with a NLO $\\pi N$ vertex in Heavy-Baryon ChPT (HBChPT) \\cite{peripheral}. \nWe see that the chiral expansion within our approach is well behaved, so that there is a steady improvement \nin the reproduction of the Nijmegen PWA results when passing from LO to NLO and then to NNLO, where a quite good\n reproduction of the Nijmegen PWA is  finally obtained. \nThis is accomplished in a progressive and smooth way, without violent variations in the results obtained at \nevery order.\\footnote{This was not the case in previous studies, e.g. in the model calculation of $NN$ scattering by Ref.~\\cite{lutz} \nthat uses a modified version of the $N\/D$ method by truncating the integrals along the LHC with a sharp cutoff.}\n In addition, we deal with convergent integrals by taking  enough number of subtractions so that the above \nreferred regulator dependence that arises when solving the Lippmann-Schwinger equation with a chiral $NN$ potential \nis avoided by construction in our approach. An interesting outcome from our study is that we\n corroborate the long-range correlations between the effective range and scattering length for each of the $NN$ $S$ waves,  \n$^1S_0$ and $^3S_1$, when only the corresponding scattering length is taken as experimental input. \nThese correlations, first noticed in Ref.~\\cite{pavon06}, were also obtained in the NLO $N\/D$ study of Ref.~\\cite{gor2013}, and \nwithin our approach they are deduced solely from basic principles of $NN$ partial-wave amplitudes, namely, \n chiral symmetry, unitarity and analyticity.\nThey  are typically fulfilled at the level of around a $10\\%$ when comparing with \nthe experimental values for the effective ranges.  \nWe should say that we can proceed further and include more subtractions, so that we can implement within our formalism \nthe exact values of the effective ranges, something not possible in the tight scheme of Ref.~\\cite{pavon06}.\n \nRegarding the subtraction constants we elaborate below a chiral power counting for them, by \ntaking into account the change in their values due to variations in the subtraction point.\n We show that at NLO and NNLO in the calculation of $\\Delta(A)$ one properly takes twice-subtracted dispersion relations (DRs). \nNevertheless, on top of this criterion we impose that one should obtain the proper threshold behavior \nfor higher partial waves, as well as having meaningful solutions of the integral equations (IEs) \nthat result from the corresponding DRs.\\footnote{By a meaningful solution we mean here a mathematical solution to the IE \nthat does not depend on the the number of points employed and \nin the arbitrary large extension of the LHC on which they lie  \n when performing the numerical discretization to solve the IE.} \nThese two requirements often imply the necessity of taking  more than two subtractions \nin the corresponding DRs relations. \nRegarding the number of subtractions used to guarantee the threshold behavior \nfor higher partial waves we use here the formalism developed in Ref.~\\cite{gor2013}, so that partial waves with \norbital angular momentum $\\ell\\geq 1$ and mixing partial waves with total angular momentum $J\\geq 1$ vanish  \nat threshold as $ A^\\ell$ and $A^J$, respectively. \nThis requires to take at least $\\ell$ or $J$ subtractions, in order, \nwith $\\ell-1$ or $J-1$ free parameters, respectively. \nBut at the end, as emphasized in Ref.~\\cite{gor2013}, \nnone or only one  of the resulting subtraction constants for a given partial wave with $\\ell>1$ (or $J>1$ for a mixing wave) is necessary to reproduce data. \nThis interesting point, which allows to treat easily higher partial waves,  is called in Ref.~\\cite{gor2013} the \nprinciple of maximal smoothness.\n\n  In our study we have also paid special attention to the issue concerning the impact on the results of the rather large size \nof the NLO $\\pi N$ counterterms, typically denoted by $c_i$ \\cite{ulf},  which first appear \n in the calculation of $\\Delta(A)$ at NNLO. \nIt is discussed in Ref.~\\cite{epe04} that the $\\pi N$ monomials, proportional to the $c_i$ counterterms, \n produce a too large contribution to the  $NN$ potential at medium and short distances   when it is \ncalculated at NNLO in dimensional regularization, which worsens the properties of the chiral expansion. \nBecause of this Ref.~\\cite{epe04} argued to better use  a cutoff regularization to calculate the NNLO potential, or equivalently, \nto cut  the energy spectral representation of the NNLO $NN$ potential at around the chiral symmetry breaking scale. \nThis last point would be equivalent to truncate the full extent of the LHC in our dispersive integrals. \n However, it is interesting to remark that we do not need to do that in order to obtain a good reproduction of the Nijmegen \nPWA when employing $\\Delta(A)$ determined up to NNLO. \nIn fact,  we observe that the definitive improvement of our results compared \nwith the Born approximation does not arise by \nmodifying  the two-nucleon \nirreducible diagrams at NNLO,  but by performing the iteration of two-nucleon unitarity diagrams as required\n by analyticity and  unitarity  in a well-defined way.\n\n\nAfter this introduction we review the $N\/D$ method for coupled and uncoupled partial waves in \nSec.~\\ref{unformalism}.\n The function $\\Delta(A)$, calculated in ChPT up to NNLO, is discussed in Sec.~\\ref{delta}, \nwhere we also elaborate the chiral power counting for the subtraction constants.\n Sections \\ref{1s0} to \n\\ref{gi5w} are devoted to discuss the application of the $N\/D$ method to \n the different $NN$ partial waves up to $J=5$.\n There it is shown that a quite good reproduction of the Nijmegen PWA phase shifts and mixing angles results.\n In these sections we also compare \nwith the Born approximation for higher partial waves and discuss on the relative importance of the different contributions \nto $\\Delta(A)$.\n Our concluding remarks are given in Sec.~\\ref{conc}. \nFinally, we discuss in Appendix \\ref{appen:vs} a method to calculate higher order shape parameters \n of the $NN$ $S$ waves. \n\n\n\n\\section{The $N\/D$ method}\n\\label{unformalism}\n\nA detailed presentation of the formalism for the $N\/D$ method \\cite{chew}\n can be found in Ref.~\\cite{gor2013}.\n Here we only reproduce the main facets of the approach. \n\n\\subsection{Uncoupled partial waves}\n\\label{upw}\nAn uncoupled $NN$ partial wave is written as the quotient of two functions, where the\n numerator is the function $N(A)$ and the denominator is $D(A)$. \nThen, one writes\n\\begin{align}\nT(A)&=\\frac{N(A)}{D(A)}~,\n\\label{eq.ta}\n\\end{align}\nwith $T(A)$ the corresponding $NN$ partial wave in the c.m. frame. In the following we use the spectroscopic notation and denote by \n$^{2S+1}L_J$ the different $NN$ partial waves with $S$ the total spin, $L$ the orbital angular momentum and $J$ the \ntotal angular momentum. The point for the splitting of $T(A)$ in two functions is because $N(A)$ has only LHC while $D(A)$ \nhas only right-hand cut (RHC), also called unitarity cut.\n The following expressions for the discontinuities of the functions $N(A)$ and $D(A)$ \nalong their respective cuts then arise,\n\\begin{align}\n\\mathrm{Im} D(A)=-\\rho(A) N(A)~,~A>0~,\\nonumber \\\\\n\\mathrm{Im} N(A)=\\Delta(A) D(A)~,~A<L~.\n\\label{disconts}\n\\end{align}\nHere $L=-M_\\pi^2\/4$  and it represents the onset of the LHC for $A<L$ due to OPE, \nand $\\rho(A)$ is the phase space factor\n\\begin{align}\n\\rho(A)=\\frac{m \\sqrt{A}}{4\\pi}~~,\n\\label{rhodef}\n\\end{align}\nwhere $m$ is the nucleon mass. \n  The first of the relations in Eq.~\\eqref{disconts} is a consequence of elastic unitarity for a $NN$ partial wave, which reads \n\\begin{align}\n\\mathrm{Im}T(A)=\\rho |T(A)|^2~,~A>0~.\n\\end{align}\n In terms of $1\/T(A)$ this can be recast simply as \n\\begin{align}\n\\mathrm{Im}\\,\\frac{1}{T(A)}=-\\rho(A)~,~ A>0~.\n\\label{invTun}\n\\end{align} \nWith this normalization the relation between the $T$ and $S$ matrices is $S(A)=1+2i\\rho(A) T(A)$.  \nThe  discontinuity of a $NN$ partial wave \n$T(A)$ along the LHC is given by $2i \\Delta(A)$, which directly implies the second expression in Eq.~\\eqref{disconts}. \n\n\n Standard DRs for the functions $D(A)$ and $N(A)$ are derived in Ref.~\\cite{gor2013} under the assumption that \nthe function $D(A)$  does not diverge faster than a polynomial of degree $n_0$  for $A\\to \\infty$. Then for $n>n_0$ one \ncan write \\cite{gor2013}\n\\begin{align}\nD(A)&=\\sum_{i=1}^n \\delta_i (A-C)^{i-1}-\\frac{(A-C)^n}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)N(q^2)}{(q^2-A)(q^2-C)^n}~,\\nonumber\\\\\nN(A)&=\\sum_{i=1}^n \\nu_i (A-C)^{i-1}+\\frac{(A-C)^n}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2-C)^n}~,\n\\label{standardr}\n\\end{align}\nwhere $C$ is the subtraction point. Notice that the same number of subtractions is taken both in $D(A)$ and $N(A)$. \nThe argument given in Ref.~\\cite{gor2013} makes use of the fact that $N(A)=T(A)D(A)$ and $T(A)$,\n because of unitarity, vanish at least as $A^{-1\/2}$ for $A\\to +\\infty$. \n As a result if $D(A)$ diverges at most as $A^{n_0}$ then $N(A)$ does not diverge faster than $A^{n_0-1\/2}$. \nHere we take into account the\nSugawara and Kanazawa theorem \\cite{barton,suga}, as a consequence of which any function \nlike $D(A)$ or $N(A)$ with only one cut  of infinite extent along the\n real axis has the same limit for $A\\to \\infty$ in any direction of the $A$-complex plane. \n In addition, it is clear from Eq.~\\eqref{standardr} and the standard theory of DRs \\cite{spearman}, \nthat we can take different values for the corresponding subtraction points for each function \nseparately. Indeed, for many partial waves  we will take the subtractions for the function $D(A)$   in two \ndifferent subtraction points, one at $C=0$ and the other at $C=-M_\\pi^2$. This is motivated by the fact that \nwe impose the normalization \n\\begin{align}\nD(0)=1~,\n\\label{nor.da}\n\\end{align}\nwhich can always be done by dividing simultaneously $D(A)$ and $N(A)$ by a constant without altering their ratio \ncorresponding to $T(A)$, Eq.~\\eqref{eq.ta}. \n In this way,  one subtraction for $D(A)$ is always taken at $C=0$ in order \nto guarantee straightforwardly the normalization Eq.~\\eqref{nor.da}.\n\nTo solve $D(A)$ in terms of the input $\\Delta(A)$ and the subtraction constants  we substitute   in Eq.~\\eqref{standardr} the expression \nfor $N(A)$ into the DR of $D(A)$, so that we end with the following IE for $D(A)$ with $A<L$,\n\\begin{align}\nD(A)&=\\sum_{i=1}^n \\delta_i (A-C)^{n-i}-\\sum_{i=1}^n \\nu_i\\frac{(A-C)^n}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2-C)^{n-i+1}}\\nonumber\\\\\n&+\\frac{(A-C)^n}{\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-C)^n}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2-k^2)}~.\n\\label{inteq1}\n\\end{align}\nThe key point of the method is to solve this IE numerically which provides the knowledge of $D(A)$ for $A<L$. \nOnce $D(A)$ is known  along the LHC  we can calculate all the \nfunctions $D(A)$, $N(A)$ and $T(A)$ in the whole $A$-complex plane. To obtain $D(A)$ one can use \nEq.~\\eqref{inteq1} and for $N(A)$ one has the second of the DRs in Eq.~\\eqref{standardr}. \nOnce $N(A)$ and $D(A)$ are known one can calculate $T(A)$ by applying Eq.~\\eqref{eq.ta}.\n\n Notice also that \nthe integrations along the RHC in Eq.~\\eqref{inteq1} can be done algebraically \nin terms of the function\n\\begin{align}\ng(A,k^2)&\\equiv \\frac{1}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2-k^2)}=\\frac{i m \/ 4\\pi}{\\sqrt{A+i0^+}+\\sqrt{k^2+i0^+}}~.\n\\label{gdef}\n\\end{align}\n The term $+i0$  is necessary  for negative $A$ or $k^2$, with the prescription \n $\\sqrt{-1\\pm i0}=\\pm i\\pi$. \nWe can calculate the other RHC integrals \n of Eq.~\\eqref{inteq1} with higher powers of the factor $(q^2-C)$ in the denominator by simple differentiation \nwith respect to $C$ of the function $g(A,k^2)$, \n\\begin{align}\n\\frac{\\partial^{p-1}\\,g(A,C)}{\\partial C^{p-1}}=\\frac{(p-1)!}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2-C)^p}~.\n\\label{derg}\n\\end{align}\n\n\\subsection{Coupled partial waves}\n\\label{cpw}\n\nFor the case of the triplet partial waves with total angular momentum $J$ we have the mixing between the partial waves \nwith $\\ell=J-1$ and $\\ell'=J+1$, except for the $^3P_0$. In this case we denote the different coupled partial waves \nby $t_{ij}(A)$ with $i,~j=1,~2$, where 1 labels the lower angular momentum $\\ell\\equiv \\ell_1$ and 2 the higher one \n$\\ell' \\equiv \\ell_2$. All of \nthem are gathered together in the $2\\times 2$ matrix $T(A)$, in terms of which the $S$-matrix reads\n\\begin{align}\nS(A)&=I+2i\\rho(A)T(A)\\nonumber\\\\\n& = \\left(\n \\begin{array}{cc}\n \\cos 2\\epsilon_J\\ e^{2i\\delta_1}            & i\\sin 2\\epsilon_J\\ e^{i(\\delta_1+\\delta_2)} \\\\ \ni\\sin 2\\epsilon_J\\ e^{i(\\delta_1+\\delta_2)} &   \\cos 2\\epsilon_J\\ e^{2i\\delta_2}\n \\end{array} \\right)~,\n\\label{relst}\n\\end{align}\n where $I$ is the $2\\times 2 $ unit matrix, $\\epsilon_J$ is the mixing angle, and $\\delta_{1}$ and $\\delta_2$ are the phase\n shifts for the channels with orbital angular momentum $\\ell$ and $\\ell'$, in this order. Equation \n\\eqref{relst} corresponds to the Stapp parameterization \\cite{stapp}.\n\nNow, the $N\/D$ method explained for the uncoupled waves in Sec.~\\eqref{unformalism} is extended to the coupled channel case \n\\cite{paper2,gor2013} by writing down three $N\/D$ equations, one for every $t_{ij}(A)$ [notice that because of time reversal \n$t_{12}(A)=t_{21}(A)$]. \n The main difference with respect to the uncoupled case is that now the discontinuity along the RHC \nof the inverse of $t_{ij}(A)$ does not \nsimply correspond to $-\\rho(A)$, but it also contains information on the other coupled partial waves. \n In the following let us employ the notation \n\\begin{align}\n\\mathrm{Im} \\frac{1}{t_{ij}(A)} \\equiv -\\nu_{ij}(A)~,A>0~.\n\\label{nuij.def}\n\\end{align}\n From Eq.~\\eqref{relst} it is straightforward to obtain  the following expressions for the $\\nu_{ij}(A)$ \\cite{paper2,gor2013},  \n\\begin{align}\n\\nu_{11}(A) & =   \\rho(A) \\left[ 1- \\frac{\\frac{1}{2}\\sin^2 2\\epsilon_J}{1-\\cos 2\\epsilon_J \\cos 2\\delta_1} \\right]^{-1} ~,\\nonumber\\\\\n\\nu_{22}(A) & =   \\rho(A) \\left[ 1- \\frac{\\frac{1}{2}\\sin^2 2\\epsilon_J}{1-\\cos 2\\epsilon_J \\cos 2\\delta_2} \\right]^{-1}~,\\nonumber \\\\\n\\nu_{12}(A) & = 2 \\rho(A) \\frac{\\sin(\\delta_1 + \\delta_2)}{\\sin 2\\epsilon_J} \\label{nuij}~.\n\\end{align}\nIn terms of them we have the analogous  DRs for $D(A)$ and $N(A)$ of Eq.~\\eqref{standardr}, but now distinguishing \nbetween the different $D_{ij}(A)$ and $N_{ij}(A)$ such that $t_{ij}(A)=N_{ij}(A)\/D_{ij}(A)$, and employing $ \\nu_{ij}(A)$ \ninstead of simply $\\rho(A)$. The following expressions are obtained \\cite{gor2013}:\n\\begin{align}\nD_{ij}(A)&=\\sum_{p=1}^n \\delta^{(ij)}_p (A-C)^{p-1}-\\sum_{p=1}^n \\nu^{(ij)}_p\\frac{(A-C)^n}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{ij}(q^2)}{(q^2-A)(q^2-C)^{n-p+1}}\\nonumber\\\\\n&+\\frac{(A-C)^n}{\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2-C)^n}\\int_0^\\infty dq^2\\frac{\\nu_{ij}(q^2)}{(q^2-A)(q^2-k^2)}~,\\\\\nN_{ij}(A)&=\\sum_{p=1}^n \\nu^{(ij)}_p (A-C)^{p-1}+\\frac{(A-C)^n}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2-A)(k^2-C)^n}~.\n\\label{standardrcc}\n\\end{align}\n Here, we also impose the normalization condition at $A=0$,\n\\begin{align}\nD_{ij}(0)=1~.\n\\end{align}\nOf course, the same remark concerning the subtraction point as done in Sec.~\\ref{upw} is also in order here. Namely, \nwe can use different subtraction points for the functions $D_{ij}(A)$ and $N_{ij}(A)$, as well as to use even different subtraction \npoints in the same function, as we will do below for $D_{ij}(A)$. \n\n\\subsection{Higher partial waves}\n\\label{hpw}\n\nAn uncoupled $NN$ partial wave with $\\ell\\geq 1$ should vanish at threshold as $A^\\ell$.\n Similarly for a coupled partial wave we have the analogous results but in terms of $\\ell_{ij}\\equiv (\\ell_i+\\ell_j)\/2$, with   $i,~j=1,~2$. \n As discussed in Ref.~\\cite{gor2013} this threshold behavior is enforced by taken at least $\\ell$ \nor $\\ell_{ij}$ subtractions\n at $C=0$ in the DR for $N(A)$ in  Eq.~\\eqref{standardr} or Eq.~\\eqref{standardrcc}, respectively, \nand setting $\\nu_p=0$ ($\\nu_p^{(ij)}=0$) for $p=1,\\ldots,\\ell$ ($\\ell_{ij}$). In this way we end with the DRs:\n\\begin{align}\n&\\underline{\\mathrm{Uncoupled~ case}:}\\nonumber \\\\\n\\label{highd}\nD(A)&=1+\\sum_{p=2}^{\\ell}\\delta_p A^{p-1}+\\frac{A^\\ell}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^\\ell}g(A,k^2)~,\\\\\n\\label{highn}\nN(A)&=\\frac{A^\\ell}{\\pi}\\int_{-\\infty}^\\ell dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^\\ell (k^2-A)}~,\\\\\n\\label{tayloruc}\n\\delta_p&=\\frac{1}{(p-1)!}D^{(p-1)}(0)~,~p=2,3,\\ldots\\\\\n&\\underline{\\mathrm{Coupled~ case}:}\\nonumber \\\\\n\\label{highdcc}\nD_{ij}(A)&=1+\\sum_{p=2}^{\\ell_{ij}}\\delta^{(ij)}_p A(A-C)^{p-2}  +\\frac{A(A-C)^{\\ell_{ij}-1}}{\\pi} \n\\int_{-\\infty}^L\\!\\! dk^2 \\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^{\\ell_{ij}}} g_{ij}(A,k^2,C;\\ell_{{ij}-1})~,\\\\\n\\label{highncc}\nN_{ij}(A)&=\\frac{A^{\\ell_{ij}}}{\\pi}\\int_{-\\infty}^L \\!\\!dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^{2})^{\\ell_{ij}}(k^2-A)}~,\\\\\n\\delta_p^{(ij)}&=\\frac{(-1)^p}{C^{p-1}}\\left[\n\\sum_{n=0}^{p-2}\\frac{(-1)^n}{n!}C^n D^{(n)}_{ij}(C)-1\n\\right]~,~p=2,3,\\ldots\n\\label{taylor}\n\\end{align}\nwhere we have denoted the derivative of $D(A)$ of order $n$ by $D^{(n)}(A)$. In addition, we have introduced the function $g_{ij}(A,k^2,C;m)$ defined \nas\n\\begin{align}\ng_{ij}(A,k^2,C;m)&=\\frac{1}{\\pi}\n\\int_0^\\infty\\!\\! dq^2\n\\frac{ \\nu_{ij}(q^2) (q^2)^m }{ (q^2-A) (q^2-k^2) (q^2-C)^{m}}~,\n\\label{gij.a.k2.c} \n\\end{align}\nwhich can be expressed algebraically as a combination of $g(A,B)$'s, Eq.~\\eqref{gdef}, with \ndifferent arguments.\n\nAlthough in this way there is  a proliferation of subtraction constants \n(which are not constrained) in the function $D(A)$ as $\\ell$ ($\\ell_{ij}$)\n grows, most of them play a negligible role. \nThis is so because $NN$ partial waves with $\\ell$ or $\\ell_{ij}$ greater than 2 are \n quite perturbative \\cite{peripheral,gor2013}.  \nIn practical terms we have found in our NNLO study, as well as in the previous \n one at NLO \\cite{gor2013}, that for higher partial waves\n only $\\delta_{\\ell}$ (or $\\delta^{(ij)}_{\\ell_{ij}}$), if  any,  \n is needed to fit data, with the rest of them fixed to zero. Furthermore,\n  no significant improvement in the reproduction of data or in the fitted values is observed \nby releasing  $\\delta_i$ or $\\delta_i^{(ij)}$ with $i<\\ell$ or $\\ell_{ij}$, respectively, so that the fit is stable. This is called in Ref.~\\cite{gor2013} \nthe {\\it principle of  maximal smoothness} because it implies for the uncoupled case \nthat  the derivatives of $D(A)$ at $A=0$ with order $< \\ell-1$ are zero, as it follows from Eqs.~\\eqref{highd} and \n\\eqref{tayloruc}. Similarly, for the coupled case\n it implies that $D_{ij}(C)=1$ and $D_{ij}^{(n)}(C)=0$ for $1\\leq n \\leq \\ell_{ij}-3$, cf. Eqs.~\\eqref{highdcc} and \\eqref{taylor}. \n In some cases, it happens that  $\\delta_\\ell$ or $\\delta_{\\ell_{ij}}^{(ij)}$ is also zero and then we say that for this partial wave the subtraction constants have the {\\it pure perturbative values}. \n\nWe further illustrate in this work the perturbative character of $NN$ partial waves with $\\ell\\,(\\ell_{ij})\\geq 3$ by \ncomparing the full outcome from the $N\/D$ method with the perturbative result corresponding to the leading Born \napproximation, cf. Sec.~\\ref{born}. In this case there is no dependence on any of the \nsubtraction constants $\\delta_p$ or $\\delta_p^{(ij)}$ and, indeed, we show below  that the results are typically \nrather similar to the full ones, although the latter reproduce closer the Nijmegen PWA, as one should expect.\n\n\n\\section{The input function $\\Delta(A)$}\n\\label{delta}\nThe discontinuity along the LHC of a NN partial wave, $2i \\Delta(A)$, is taken from the  \ncalculation of Ref.~\\cite{peripheral} in Baryon ChPT (BChPT)  up to ${\\cal O}(p^3)$ or NNLO, which includes OPE plus leading and subleading TPE.\n At this order $\\Delta(A)$ \nfor a given partial wave diverges at most as $\\lambda (-A)^{3\/2}$ for $A\\to-\\infty$, with $\\lambda$ a constant. \nAs discussed in Ref.~\\cite{gor2013}, when $\\lambda<0$ one can  have solutions for the integral \nequation providing $D(A)$ for $A<L$ in the once-subtracted case even  with a divergent $\\Delta(A)$ for  \n$A\\to-\\infty$. \nHowever, as we will see below $\\lambda$ is not always negative and more subtractions are then required.\n\n\\subsection{NLO $\\pi N$ counterterms}\nAt NNLO the function $\\Delta(A)$ is sensitive to the NLO $\\pi N$ ChPT low-energy constants (LECs) \n $c_1$, $c_3$ and $c_4$. \n We take their values from different works in the literature, that are summarized in Table~\\ref{tab:cis}. \nWithin the same reference we distinguish, when appropriate, between those values obtained by \nfitting phase shifts from the  Karlsruhe-Helsinki group (KH) \\cite{kh} or the George Washington University group (GW) \\cite{gw}. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|r|l|l|l|}\n\\hline\n  Analysis      & $c_1$ [GeV$^{-1}$]   &   $c_3$ [GeV$^{-1}$]   &   $c_4$ [GeV$^{-1}$]  \\\\  \n\\hline   \nGW-HBChPT \\cite{epe12} & $-1.13$  & $-5.51$ & $3.71$\\\\\nKH-HBChPT \\cite{epe12} & $-0.75$ & $-4.77$ & $3.34$ \\\\\n\\hline\nKH \\cite{buttiker} & $-0.81\\pm 0.12$ & $8\\pm 57$ & $3.40\\pm 0.04$ \\\\\n\\hline\nGW-EOMS \\cite{aco2013} & $-1.50\\pm 0.007$ & $-6.63\\pm 0.31$ & $3.68\\pm 0.14$ \\\\\nKH-EOMS \\cite{aco2013} & $-1.26\\pm 0.14$ & $-6.74\\pm 0.38$ & $3.74\\pm 0.16$ \\\\\n\\hline\nGW-IR\\cite{ir2011}  & $-1.32\\pm 14$ & $-6.9\\pm 6 $ & $3.66\\pm 0.33$ \\\\\nKH-IR \\cite{ir2011}  & $-1.08\\pm 0.15$ & $-7.0\\pm 0.7$ & $3.72\\pm 0.32$ \\\\  \n\\hline\nNN data \\cite{rentmeester2003} & $-0.76\\pm 0.7$ & $-4.78\\pm 0.10$ & $3.96\\pm 0.22$ \\\\\n\\hline\nGW-UChPT \\cite{aco2013}& $-1.11\\pm 0.02$ & $-4.78\\pm 0.04$ & $3.04\\pm 0.02$ \\\\\nKH-UChPT \\cite{aco2013} & $-1.04 \\pm 0.02$ & $-4.48\\pm 0.05$ & $3.00\\pm 0.02$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Different sets of values for the ${\\cal O}(p^2)$ $ \\pi N$ LECs $c_1$, $c_3$ and $c_4$.\n\\label{tab:cis}}\n\\end{center}\n\\end{table}\n\nReference~\\cite{epe12} performs an ${\\cal O}(p^4)$ HBChPT study of $\\pi N$ scattering data. \nWe take its values instead of the ones from the older HBChPT studies at ${\\cal O}(p^3)$ and ${\\cal O}(p^4)$ \\cite{fettes98}. \n We include too the values from Lorentz covariant BChPT obtained in Ref.~\\cite{aco2013} \nby fitting $\\pi N$ phase shifts making use of EOMS at ${\\cal O}(p^3)$. \n Furthermore, we show in the table the $c_i$'s obtained in the covariant ${\\cal O}(p^3)$ BChPT study of Ref.~\\cite{ir2011} \nwithin Infrared Regularization (IR).\n However, due to the better convergence of the $\\pi N$ \nscattering amplitude in EOMS than in IR \\cite{sigma2012,aco2013} we give results only for\n the values obtained within EOMS \\cite{aco2013}. \nThe resulting uncertainty band is already wide enough to \ntake into account further uncertainties by considering explicitly the $c_i$'s  from the IR study of \nRef.~\\cite{ir2011}, which indeed are rather close to those obtained in EOMS \\cite{aco2013}. \n We also notice that the values from Ref.~\\cite{rentmeester2003}, obtained in a $NN$ scattering study, \nare very similar to those of KH \\cite{epe12}, so that in the following we  consider only the latter ones. \nAgain the uncertainty estimated  takes into account the variation in the results by \nemploying the $c_i$'s from Ref.~\\cite{rentmeester2003}.\n The work \\cite{buttiker} fixes $c_4$ accurately but its analysis is insensitive to $c_3$, \nprecisely the ${\\cal O}(p^2)$ $\\pi N$ LEC on which our results mostly depend. \nThis is  why we do not show results for this set of $c_i$'s either. \nFinally, we also give the resulting values from the fits to $\\pi N$ data within Unitarized  EOMS BChPT\n obtained in Ref.~\\cite{aco2013}.  \nThese are the fits  that provide more stable values under the change of the data between KH and GW. \nThese values are quite similar to those from the set KH \\cite{epe12}. \nIn summary, when discussing our results we will take into account the \n values for the LECs $c_i$  obtained in Refs.~\\cite{epe12} and \\cite{aco2013}, namely, the rows 2, 3, 5, 6, 10 and 11 \nin Table~\\ref{tab:cis}.  \n\n\\subsection{Number of subtractions in the chiral expansion of $\\Delta(A)$}\n\\label{nschpt}\n\nAn interesting point to discuss is   the appropriate number of subtractions for a given chiral \norder in the calculation of $\\Delta(A)$. In other terms, we want to establish a chiral power counting \nfor the subtraction constants involved in the calculation of the functions $D(A)$ and $N(A)$. \n\nIn the previous works in which we applied the $N\/D$ method to study $NN$ interactions \\cite{paper1,paper2,gor2013} our main \ncriterion for fixing the number of subtractions was to end with a well-defined IE for $D(A)$ with $A<L$. \n We could also add more subtractions and fit low-energy data with more precision by having more free parameters\n at our disposal, a point actually  used in these works too. \n However, by having a chiral power counting for the subtraction constants \none has a connection between the number of subtraction  and the chiral order for the \ncalculation of $\\Delta(A)$. \n\n \nA chiral power counting for the subtraction constants can be established by studying their variation when changing the \nsubtraction point in the low-energy region.\n Let us consider first the chiral order for the subtraction constants appearing in $N(A)$,  \ndenoted by $\\nu_i$ in Eq.~\\eqref{standardr}. \nFor definiteness  let us employ a twice-subtracted DR, which reads \n\\begin{align}\nN(A)&=\\nu_1+\\nu_2 A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2(k^2-A)}~.\n\\label{sc.co1}\n\\end{align}\nNow, let us move the subtraction point from zero to $C={\\cal O}(M_\\pi^2)$. It is then straightforward to show that the \nprevious DR can be rewritten as\\footnote{To show this, one can rewrite the factor $A^2\/k^2$ in the integral \nof Eq.~\\eqref{sc.co1} as  $([A-C]+C)^2\/(k^2-C)^2\\times (k^2-C)^2\/(k^2)^2$ and then isolate the term $(A-C)^2\/(k^2-C)^2$. \nThe rest of terms can be reabsorbed in the polynomial on the right-hand side (r.h.s.) of Eq.~\\eqref{sc.co1}.}\n\\begin{align}\nN(A)&=\\nu'_1+\\nu'_2 A+\\frac{(A-C)^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-C)^2(k^2-A)}~,\\nonumber\\\\\n\\nu'_1&=\\nu_1-\\frac{C^2}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta(k^2)D(k^2)}{(k^2-C)^2 k^2}~,\\nonumber\\\\\n\\nu'_2&=\\nu_2+\\frac{C}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta(k^2)D(k^2)}{(k^2-C)^2 k^2}\\frac{2k^2-C}{k^2}~.\n\\label{sc.co2}\n\\end{align}\nFor  $C={\\cal O}(p^2)$, $k^2={\\cal O}(p^2)$ because the result of the convergent integral  at low-energies \nis dominated by the low-energy region of the integrand and $D(k^2)=1+\\ldots={\\cal O}(p^0)$. Furthermore, since at \nleading order $\\Delta(k^2)={\\cal O}(p^0)$,  it  follows then from Eq.~\\eqref{sc.co2} that \n$\\nu_1={\\cal O}(p^0)$ and $\\nu_2= {\\cal O}(p^{-2})$. This\n procedure can be easily generalized so that $ \\nu_n={\\cal O}(p^{-2(n-1)})$. By increasing the chiral order \nin the calculation of $\\Delta(A)$ up to ${\\cal O}(p^m)$ the $\\nu_n$ will receive an extra contribution \nstarting at ${\\cal O}(p^{-2(n-1)+m})$, as it is also clear from Eq.~\\eqref{sc.co2}.\n Now, the point is to demand that for a given $m$ the maximum value of $n$, denoted by $n_0$,\n should not be so large that $-2(n_0-1)+m<0$. \nBy this condition we are  requiring that the chiral dimension for a given subtraction  constant\n with $n\\leq n_0$  be positive or zero, since \nshort-distance physics gives rise to contributions that do not vanish in the chiral limit.\\footnote{As a \nresult they are counted as ${\\cal O}(p^0)$} Then  the raising in the chiral dimension of $\\nu_n$ until the nominal one, \n$-2(n-1)+m\\geq 0$,  must come from powers of $M_\\pi$, $|C|^{\\frac{1}{2}}\\sim  M_\\pi$.\\footnote{One could ask about the fact that the chiral dimension for \nthe other contributions to $\\Delta(A)$ of order $m'<m$ could imply a negative $-2(n-1)+m'$ with $n\\leq n_0$.\n This already occurs e.g. in \nthe paradigmatic example of ChPT, namely, meson-meson scattering. The point is to realize that these extra long-range physics \ncontributions cancel explicitly with other contributions stemming from the rearrangement of the dispersive integral, \nwhich was done already with less subtractions when including only lower orders in $\\Delta(A)$.} This power \ncounting coincides with the standard Weinberg chiral power counting \\cite{weinn}, that is applied to the calculation\n of the $NN$ potential. It is also  worth noticing \nthat $\\nu_n$ is multiplied by $(A-C)^{n-1}$, so that the chiral order of the product is always $m$ for any $n$, which \ncorresponds to the chiral order of the dispersive integral with the ${\\cal O}(p^m)$ contribution of $\\Delta(A)$. \n\nOne can proceed analogously also for the function $D(A)$. \nWe also exemplify it by writing down a \ntwice-subtracted DR for $D(A)$, \n\\begin{align}\nD(A)&=1+\\delta_2 A-\\frac{A(A-C)}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)N(q^2)}{q^2(q^2-C)(q^2-A)}~.\n\\label{sc.co3}\n\\end{align}\nWe now move the subtraction point from $C$ to $E$. \nLet us recall that the normalization $D(0)=1$ is fixed \nand this is why we do not change the position of the first subtraction taken at $A=0$.\n As a result of this rewriting we obtain the evolution \n\\begin{align}\n\\delta_2&\\rightarrow \\delta_2+\\frac{C-E}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)N(q^2)}{q^2(q^2-E)(q^2-C)}\n\\label{sc.co4}\n\\end{align}\nFor ascribing the chiral order to $\\delta_n$, $n\\geq 2$, we have, as before, that\n $C\\sim E\\sim M_\\pi^2$, $q^2={\\cal O}(p^2)$. \nAdditionally, we count $\\rho(q^2)={\\cal O}(p^0)$ because it involves the product $m \\sqrt{q^2}$ with $m\\gg M_\\pi$, a large number.  Let us also recall at this point that along the RHC, the extent of the integral in Eq.~\\eqref{sc.co3},\n the strong effects due to the infrared enhancement of the $NN$ \nintermediate states \\cite{weinn}, which is directly related with \nthe large nucleon mass,   should be resummed.\n We then conclude from Eq.~\\eqref{sc.co4} that\n $\\delta_2={\\cal O}(p^{-2})$ for the LO contribution of $N(A)={\\cal O}(p^0)$. \nThis result can be generalized \neasily to more subtractions so that $\\delta_n={\\cal O}(p^{-2(n-1)})$. \nHowever, this chiral order increases when considering higher orders contributions to $N(A)$ stemming in turn \nfrom  higher orders in the calculation of $\\Delta(A)$.\n As just discussed above in connection with Eq.~\\eqref{sc.co2}, these ${\\cal O}(p^m)$ contributions to\n $\\Delta(A)$   give rise to contributions of the same order in $N(A)$. \nThus, once they are taken into account, one has the corresponding rise in the chiral order of $\\delta_n$ so that now \nit counts as $\\delta_n={\\cal O}(p^{-2(n-1)+m})$.\n In this way,  the chiral orders of $\\nu_n$ and $\\delta_n$ are the same for the same $n$ and $m$. \nIndeed, this is a necessary result because according with the general \nformalism of Sec.~\\ref{unformalism}  the same number of subtractions are taken \nboth in $D(A)$ and $N(A)$.  \nTo satisfy this requirement is also another reason for taking  $\\rho={\\cal O}(p^0)$ in the \nchiral counting.  \n We also stress that the chiral power counting that we have established for the subtraction \nconstants $\\delta_n$ corresponds to two-nucleon reducible diagrams,\\footnote{As it is apparent \nfrom the factor $q^2-A$ in the denominator of the RHC integral in Eq.~\\eqref{sc.co3}.} \nwhile the standard Weinberg chiral power counting for  nuclear interactions \\cite{weinn} only involves \ntwo-nucleon irreducible diagrams.\n\nAlthough we have offered here the arguments for the uncoupled case the same results follow \nfor the coupled-channel partial waves because the function $\\nu_{ij}(A)$, Eq.~\\eqref{nuij.def},\n share the same chiral counting \nas $\\rho(A)$, since the $T$-matrix is ${\\cal O}(p^0)$.\\footnote{With $\\rho={\\cal O}(p^0)$ it is also true that $\\text{Im}t_{ij}={\\cal O}(p^0)$ because \nof unitarity  for $A\\geq 0$, $\\text{Im}t_{ij}=\\rho\\sum_k t_{ik}t_{jk}^*$~.}  \nIn summary, for $\\Delta(A)$ calculated up to ${\\cal O}(p^m)$ we have \nthe following power counting for \nthe subtractions constants, \n\\begin{align}\n\\nu_n~,~\\delta_n&\\sim {\\cal O}(p^{-2(n-1)+m})~.\n\\label{summarypwc}\n\\end{align}\n\nNow, by applying the requirement that $-2(n-1)+m\\geq 0$  it results \nthat in our present study at NNLO  one should properly take two subtractions ($n=2$) since $m=3$. \nHowever, on top of this criterion we first require that the resulting IE  has  \nwell-defined solutions and for this to happen it is necessary to introduce more than two subtractions in \nsome $NN$ partial waves, as discussed below. \nIn addition, we have to satisfy the right threshold behavior for higher partial waves, which for $\\ell\\geq 3$ ($J\\geq 3$ for \nthe mixing partial waves)\nrequires to take $\\ell>2$ ($J>2$ subtractions), cf. Sec.~\\ref{hpw}.\n\n\n\\section{Uncoupled $^1 S_0$ wave}\n\\label{1s0} \n\nIn this section we study the $^1 S_0$ partial wave. We first take the once-subtracted DRs:\n\\begin{align}\n\\label{onceD}\nD(A)&=1-\\nu_1 A  g(A,0)+\\frac{A}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{k^2}g(A,k^2)~,\\\\\n\\label{onceN}\nN(A)&=\\nu_1+\\frac{A}{\\pi}\\int_{-\\infty}^Ldk^2\\frac{\\Delta(k^2)D(k^2)}{k^2(k^2-A)}~.\n\\end{align}\nWe have one free parameter $ \\nu_1$ that can be fixed in terms of the $^1S_0$ scattering length $a_s$ \n\\begin{align}\n\\nu_1=-\\frac{4\\pi a_s}{m}~,\n\\label{1s0nu1}\n\\end{align}\nwith the experimental value $a_s=-23.76\\pm 0.01$~fm \\cite{thesis}.  \n\nThe phase shifts obtained by solving the IE of Eq.~\\eqref{onceD} are shown in \n Fig.~\\ref{fig:1fp1s0} as a function of the c.m. three-momentum, denoted by $p$ ($p= \\sqrt{A}$)  \nin the axis of abscissas. \nThe (red) hatched area   corresponds to our results from Eqs.~\\eqref{onceD}-\\eqref{1s0nu1}\n with $\\Delta(A)$ calculated up-to-and-including ${\\cal O}(p^3)$ contributions and by taking into account the variation in the results from the different values employed for the NLO $\\pi N$ ChPT counterterms in Table~\\ref{tab:cis}.\nOur present results are compared with \n the neutron-proton ($np$) $^1S_0$ phase shifts of the Nijmegen PWA \\cite{Stoks:1994wp}  (black dashed line), \n the OPE results  of Ref.~\\cite{paper1}  (blue dotted line) and the NLO results of Ref.~\\cite{gor2013} (magenta solid line).\n As we see,  the Nijmegen PWA phase shifts are better reproduced at lower energies at NNLO than  at smaller orders, \nthough one also observes an excess of repulsion at this order.  \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.7\\textwidth]{.\/1s0.ps} \n\\end{center}\n\\caption[pilf]{\\protect {  (Color online.) Phase shifts of the $^1S_0$ $NN$ partial wave where the number \nof subtractions taken is indicated by the value of $n$ given in the legend of each type of line. \n The once-subtracted DR results are shown by the (red) hatched areas at NNLO, the \n(magenta) solid lines at NLO \\cite{gor2013} and the (blue) dotted lines at LO (OPE) \\cite{paper1}. \nThe twice-subtracted DR results correspond to the (cyan) band at NNLO and \nthe (green) dash-dotted  line at NLO  \\cite{gor2013}. \n The Nijmegen PWA phase shifts are shown by the (black) dashed lines.\n}\n\\label{fig:1fp1s0}\n}\n\\end{figure}\n\nNext, we work out the effective range expansion (ERE) parameters for the $^1S_0$ . \nBy taking into account the relation in our normalization\n\\begin{align}\n\\frac{4\\pi}{m}\\frac{D}{N}=-\\frac{1}{a_s}+\\frac{1}{2}r_s A+\n\\sum_{i=1}^{10}v_i A^i - i\\sqrt{A}+{\\cal O}(A^{11})~,\n\\label{efr1}\n\\end{align}\nwith $r_s$ the $^1S_0$ effective range and the shape parameters $v_i$, $i=2,\\ldots,10$. \n We designate by $I_m$, $m=1,2,\\ldots$, the  integral along the LHC,\n\\begin{align}\n I_{2n}&=\\int_{-\\infty}^Ldk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^n}~,\\nonumber\\\\\n I_{2n+1}&=\\int_{-\\infty}^Ldk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^n\\sqrt{-k^2}}~.\\nonumber\\\\\n\\end{align}\nFrom Eqs.~\\eqref{onceD}, \\eqref{onceN} and \\eqref{efr1} we derive the following expressions for \n$r_s$ and the shape parameters in the ERE up to $i=4$\n\\begin{align}\nr_s&=-\\frac{m (a_s I_3+I_4)}{2 \\pi ^2 a_s^2} ~,\\nonumber\\\\\nv_2&=-\\frac{m \\left(I_4 m (a_s I_3+I_4)+4 \\pi ^2 a_s (a_s\n   I_5+I_6)\\right)}{16 \\pi ^4 a_s^3}~,\\nonumber\\\\\nv_3&=-\\frac{m \\left[16 \\pi ^4 a_s^2 (a_s I_7+I_8)+I_4^2 m^2 (a_s I_3+I_4)+4\n   \\pi ^2 a_s m (a_s I_3 I_6+a_s I_4 I_5+2 I_4 I_6)\\right]}{64 \\pi\n   ^6 a_s^4}~\\,\\\\\nv_4&=-\\frac{m}{256 \\pi ^8 a_s^5}\\left[64 \\pi ^6 a_s^3 (a_s I_9+I_{10})+16 \\pi ^4 a_s^2 m \\left(a_s (I_3\n   I_8+I_4 I_7+I_5 I_6)+2 I_4 I_8+I_6^2\\right)+I_4^3 m^3\n   (a_s I_3+I_4)\\right.\\nonumber\\\\\n&\\left.+4 \\pi ^2 a_s I_4 m^2 (2 a_s I_3 I_6+a_s I_4\n   I_5+3 I_4 I_6)\\right]~.\n\\label{ere.1s0}\n\\end{align}\nFor higher order shape parameters is more efficient to use the numerical method developed in Appendix \\ref{appen:vs}, \nto which we refer.\n\nThe resulting values for $r_s$ and \n the shape parameters $v_i$, $i=1,\\ldots,6$, are given in \nTable~\\ref{table:vs1s0a} and for $v_i$, $i=7,\\ldots, 10$ are shown \nin Table~\\ref{table:vs1s0b} in the second and third rows for NLO and \nNNLO, respectively. The latter are indicated by NNLO-I.\n These results are compared \nwith the results from the calculation based on the NNLO $NN$ potential of Refs.~\\cite{epe04} and \\cite{thesis},\n and with the Nijmegen PWA values.\n Our results for $v_3$ and $v_4$ are very similar to those obtained in Ref.~\\cite{thesis}. The difference between \n\\cite{thesis} and \\cite{epe04} stems from the fact that in the latter reference a different method to regularize pion exchanges was introduced, the so-called spectral function regularization, instead of the dimensional regularization used in Ref.~\\cite{thesis}. We also observe a clear improvement in the reproduction of the ERE parameters from NLO to NNLO. \nAt NLO the errors in Tables~\\ref{table:vs1s0a} and \\ref{table:vs1s0b} reflect the numerical uncertainty in the \ncalculation of higher order derivatives. At NNLO in addition they take into account the spread in the results \nfrom the different sets of $c_i$'s used. \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|}\n\\hline\n    &$r_s$   & $v_2$  & $v_3$   & $v_4$ & $v_5$ & $v_6$    \\\\\n\\hline\nNLO & $2.32$ & $-1.08$ & 6.3  & $-36.2$  & 225 & $-1463$  \\\\\n\\hline\nNNLO-I & $2.92(6)$ & $-0.32(8)$ & $4.9(1)$ & $-27.7(8)$ & $177(4)$  & $-1167(30)$  \\\\\n\\hline\nNNLO-II & $2.699(4)$ & $-0.657(3)$ & $5.20(2)$  & $-30.39(9)$  & $191.9(6)$ & $-1263(3)$  \\\\\n\\hline\nRef.~\\cite{thesis} & $2.68$ & $-0.61$ & $5.1$ & $-30.0$ & & \\\\\n\\hline\nRef.~\\cite{epe04} & $2.62\\sim 2.67$ & $-0.52\\sim -0.48$ & $4.0\\sim 4.2$ & $-20.5\\sim -19.9$ & & \\\\\n\\hline\nRef.~\\cite{Stoks:1994wp} & $2.68$ & $-0.48$ & $4.0$ & $-20.0$ & & \\\\\n\\hline\n\\end{tabular}\n\\caption{Values for effective range $r_s$ [fm] and the shape parameters \n$v_i$, $i=2,\\ldots,6$ in units of fm$^{2i-1}$\n for our present results at NNLO with once-subtracted DRs [Eq.~\\eqref{onceD}] (NNLO-I in the third row) \nand with twice-subtracted DRs [Eq.~\\eqref{twiceD}] (NNLO-II in the fourth row). \nThe second row shows the results at  NLO with once-subtracted DRs [Eq.~\\eqref{onceD}].\n We also give the values obtained by using the NNLO $NN$ potential in \nRefs.\\cite{thesis} and \\cite{epe04} (fifth and\n sixth  rows, respectively). The values corresponding to the Nijmegen PWA \\cite{Stoks:1994wp}, \n as obtained in Refs.~\\cite{epe04,thesis}, are given in the last row.  \n\\label{table:vs1s0a}}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n    &  $v_7\\times 10^{-1}$ & $v_8\\times 10^{-2}$  & $v_{9}\\times 10^{-3}$  & $v_{10}\\times 10^{-4}$   \\\\\n\\hline\nNLO & $985$ & $-681$ & 480  & $-344(3)$  \\\\\n\\hline\nNNLO-I & $795(18)$ & $-554(12)$ & $393(8)$ & $-284(6)$  \\\\\n\\hline\nNNLO-II & $857.1(1.9)$ & $-595.7(1.3)$ & $421.7(9)$ & $-304(3)$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Values for the shape parameter $v_i$, $i=7,\\ldots,10$ in units of fm$^{2i-1}$. \nFor the meanings of the rows see Table~\\ref{table:vs1s0a}.\n\\label{table:vs1s0b}}\n\\end{center}\n\\end{table}\n\n\n\nFrom Eq.~\\eqref{ere.1s0}  we can also derive a power series expansion of the ERE parameters as a function of $a_s$, as it was done previously for $r_s$ in Ref.~\\cite{gor2013} at NLO. We refer to that reference for further details. The important point is that $D(A)$ satisfies the linear IE of  Eq.~\\eqref{onceD} with an inhomogeneous term that is a polynomial of first degree in $a_s$. As a result,  $D(A)=D_0(A)+a_s D_1(A)$, with $D_0(A)$ and $D_1(A)$ independent of $a_s$. This also implies that the different $I_n$ can be expressed as $I_n^{(0)}+a_s I_n^{(1)}$ with $I_n^{(0)}$ and $I_n^{(1)}$ independent of $a_s$. \nIn this way, the ERE parameters satisfies the following expansions \n\\begin{align}\nr_s&=\\alpha_0+\\frac{\\alpha_{-1}}{a_s}+\\frac{\\alpha_{-2}}{a_s^2}~,\\nonumber\\\\\nv_n&=\\sum_{m=-n-1}^0 \\frac{v_n^{(m)}}{a_s^m}~,\n\\label{expvs.1s0}\n\\end{align}\nwith the coefficients $\\alpha_i$ and $v_n^{(i)}$ independent of $a_s$. \nThe relation between $r_s$ and $a_s$ was first realized in Ref.~\\cite{pavon06} in the context of $NN$ scattering.\\footnote{The correlation between the effective range and the scattering length\n in Eq.~\\eqref{expvs.1s0} was derived earlier in atomic physics for Van der Waals potentials\n \\cite{flambaum}, and throughly confronted    with data \\cite{calle2010}.} The explicit expressions \nof $\\alpha_i$ ($i=-2,-1,0)$ in terms of $D_0(A)$ and $D_1(A)$ were given in Ref.~\\cite{gor2013}.\n Its values at NNLO are\n\\begin{align}\n\\alpha_0&= 2.61\\sim 2.73~\\text{fm}~,\\nonumber\\\\\n\\alpha_{-1}&=-5.93\\sim -5.65~\\text{fm}^2~,\\nonumber\\\\\n\\alpha_{-2}&=5.92\\sim 6.12~ \\text{fm}^3~.\n\\label{exp.1s0}\n\\end{align}\n The expressions for the coefficients $v_n^{(m)}$ in Eq.~\\eqref{expvs.1s0} can also be worked straightforwardly \nin terms of $D_0(A)$ and $D_1(A)$ by the interested reader. For conciseness we do not reproduce them here. \nThe  results in Eq.~\\eqref{exp.1s0}  are perfectly compatible with those obtained in the first entry of \nRef.~\\cite{pavon06}, $\\alpha_0=2.59\\sim 2.67$~fm, $\\alpha_{-1}=-5.85\\sim -5.64$~fm$^2$ and $\\alpha_{-2}=5.95\\sim \n6.09$~fm$^3$. This reference employs the chiral $NN$ potential in a Lippmann-Schwinger equation that is \n renormalized  with boundary conditions and imposing \nthe hypothesis of orthogonality of the wave functions determined with\n different energy.\\footnote{Since the potentials involved are singular this orthogonality condition is imposed in \n the formalism of Ref.~\\cite{pavon06}.} \nIn our case, however, the expansions in Eq.~\\eqref{expvs.1s0} are consequences of basic principles of a $NN$ \npartial wave like unitarity, analyticity and chiral symmetry.  \n The resulting phase shifts in Fig.~\\ref{fig:1fp1s0} from Eq.~\\eqref{onceD}, and shown by the (red) hatched area,\n are also coincident with those obtained by Ref.~\\cite{pavon06}.  \nThey are also rather similar to those obtained when employing only one contact term in the third entry of Ref.~\\cite{phillipssw},\n which studies the independence of its results as a function of the cutoff used to solve the Lippmann-Schwinger equation.\n Nevertheless, in this case  the NNLO chiral potential is calculated by truncating its spectral representation \\cite{epe04},  \n while Ref.~\\cite{pavon06} uses the dimensional regularized result \n(which requires to take to infinity the cutoff(s) used in Ref.~\\cite{phillipssw}.)\n \n\n\nNext,  we consider the twice-subtracted DRs: \n\\begin{align}\n\\label{twiceD}\nD(A)&=1+\\delta_2 A-\\nu_1\\frac{A(A+M_\\pi^2)}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2+M_\\pi^2)q^2}\n-\\nu_2 A(A+M_\\pi^2) g(A,-M_\\pi^2)\\nonumber\\\\\n&+\\frac{A(A+M_\\pi^2)}{\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2} \n\\int_0^\\infty dq^2\\frac{\\rho(q^2)q^2}{(q^2-A)(q^2+M_\\pi^2)(q^2-k^2)}\n~, \\\\\n\\label{twiceN}\nN(A)&=\\nu_1+\\nu_2 A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2)^2}~,\n\\end{align}\nwhere the two subtractions in the function $N(A)$ and one  for $D(A)$ are taken at $C=0$, while\n the other subtraction in $D(A)$ is placed at $C=-M_\\pi^2$. \nTaking into account Eq.~\\eqref{gdef} it is straightforward to rewrite\n\\begin{align}\n\\frac{1}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)q^2}{(q^2-A)(q^2-k^2)(q^2-C)}=\\frac{C g(A,C) - k^2 g(A,k^2)}{C-k^2}~.\n\\end{align} \n The subtraction constant $\\nu_1$ is given by Eq.~\\eqref{1s0nu1}, while $\\nu_2$ and $\\delta_2$ \nare directly fitted to \n the $np$ Nijmegen PWA phase shifts.\\footnote{Since Ref.~\\cite{Stoks:1994wp} does not provide \nerrors we always perform a least square fit,  without weighting.}  \n The best fit occurs for\n\\begin{align}\n\\nu_2&=-23(1)~M_\\pi^{-4}\\nonumber\\\\\n\\delta_2&=-8.0(3)~M_\\pi^{-2}~,\n\\label{nu2delta2}\n\\end{align}        \nwhere the intervals of values stem from the uncertainty due to the different values of $c_i$'s taken. \nThe reproduction of data is very good, as shown by the (cyan) filled area in Fig.~\\ref{fig:1fp1s0} which \nlies on top of the Nijmegen PWA $np$ phase shifts.  In the same figure \n  we show by the (green) dash-dotted line \nthe twice-subtracted DR result at NLO,  which reproduces  \nthe  Nijmegen data equally well as obtained at NNLO, with  \n the fitted values $\\nu_2=-11.9$~$M_\\pi^{-4}$ and $\\delta_2=-4.6~M_\\pi^{-2}$. \n   The resulting ERE shape parameters for the fit in Eq.~\\eqref{nu2delta2} \nare shown in the fourth rows of Tables~\\ref{table:vs1s0a} and \\ref{table:vs1s0b}, where \n we observe a remarkable good agreement with Ref.~\\cite{thesis}. \nWe predict $r_s=2.70$~fm which is compatible with its experimental value \n$r_s=2.75\\pm 0.05$~fm \\cite{thesis}. A similar good reproduction of the $^1S_0$ phase shifts is also achieved by Ref.~\\cite{phillipssw} in \nterms of two contact terms, although in this case there is a strong sensitivity   on the cutoff employed to \nregularize the Lippmann-Schwinger equation near those values that give rise to poles in the domain of validity \nof the effective field theory. \n\n The value of $\\nu_2$ in \nEq.~\\eqref{nu2delta2} is rather large, of similar size in absolute value to $\\nu_1\\simeq 31~M_\\pi^{-2}$, Eq.~\\eqref{1s0nu1}. A linear correlation between $\\nu_2$ and $\\delta_2$ can be observed in  a $\\chi^2$ contour plot,  along which \n there is an absolute minimum corresponding to the parameters given \nin Eq.~\\eqref{nu2delta2}.  \nThe subtraction constant $\\nu_2$ that results from the once-subtracted DR \n Eq.~\\eqref{onceN}, and that we denote by $\\nu_2^{\\mathrm{pred}}$, is given by the \nexpression\n\\begin{align}\n\\nu^{\\mathrm{pred}}_2&=\\frac{1}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2}~,\n\\label{nu2predicted}\n\\end{align}\nwith the numerical value $\\nu_2^{\\mathrm{pred}}\\simeq -6.0$, $-6.4$ and $-7.5\\pm 0.2~M_\\pi^{-4}$ when \n $\\Delta(A)$ is calculated up to ${\\cal O}(p^0)$, ${\\cal O}(p^2)$ and ${\\cal O}(p^3)$, respectively.\n The difference between the predicted and fitted values for $\\nu_2$   at NLO  \n is denoted by $\\delta\\nu_2^{(0)}$.\n The superscript takes into account the chiral order for $\\nu_2$, ${\\cal O}(p^{-2+m})$ according to the new \ncontribution to $\\Delta(A)$ of ${\\cal O}(p^m)$, Eq.~\\eqref{summarypwc}. \n  The value obtained is $\\delta \\nu_2^{(0)}\\simeq -5.5~M_\\pi^{-4}$. At NNLO in order \nto calculate  $\\delta \\nu_2^{(1)}$ one has to subtract  $\\delta\\nu_2^{(0)}$ to the difference \nbetween the fitted value in Eq.~\\eqref{nu2delta2} and the predicted one from Eq.~\\eqref{nu2predicted}. \n Then, one has $\\delta\\nu_2^{(1)}\\simeq -15+5.5=-9.5~M_\\pi^{-4}$.\n This implies that in order to overcome the excess of repulsion at NNLO   one needs\n to incorporate a significant contribution from short-distance physics to give account \nof ``missing physics'', beyond the pure long-range physics\\footnote{We mean here the physics driven by the \n multi-pion exchanges giving rise to the LHC and  to $\\Delta(A)$.}\n that stems from the once-subtracted DR case and that is not able to provide an accurate \nreproduction of data as shown in Fig.~\\ref{fig:1fp1s0} by the (red) hatched areas. \n  The large value for $\\delta\\nu_2^{(1)}$ is mainly due to the ${\\cal O}(p^2)$ $\\pi N$ counterterms \n$c_i$'s, which in turn are dominated by the  $\\Delta(1232)$ resonance contribution \\cite{bernard93,aco2012}. \n  This can be easily seen by performing a fit to data in which we set  $c_i=0$ for all of them.\n A good reproduction \nof the Nijmegen PWA phase shifts results but now $\\delta\\nu_2^{(1)}\\simeq -1.5~M_\\pi^{-4}$, which is \nmuch smaller than $\\delta\\nu_2^{(0)}$, with a ratio $\\delta\\nu_2^{(1)}\/\\delta\\nu_2^{(0)}\\sim 30\\% \\sim \n{\\cal O}(p)$. This indicates that once the large  contributions \n that stem from the  $c_i$ coefficients are discounted a quite natural (baryon) chiral expansion emerges. \n \nRegarding the absolute value of $\\delta\\nu_2^{(0)}$ one should expect on dimensional grounds that\n\\begin{align}\n|\\delta\\nu_2^{(0)}|\\sim  \\frac{4\\pi\\,|a_s|}{m \\Lambda^2}~,\n\\label{abvalue_v2}\n\\end{align}\nwith $\\Lambda$ the expansion scale. The factor $4\\pi\/m$ is due to our normalization, cf. Eq.~\\eqref{efr1}. \nThere should be also another contribution to $\\delta\\nu_2^{(0)}$ not proportional to $a_s$, but since \nthe scattering length is so large the contribution shown in Eq.~\\eqref{abvalue_v2}\n is expected to be the most important. For $\\Lambda\\simeq 350$~MeV, one would have \n$|\\delta\\nu_2^{(0)}|\\sim 5~M_\\pi^{-4}$, which is very similar indeed to the reported value above. This \nvalue of $\\Lambda$ is also consistent with the ratio $\\delta\\nu_2^{(1)}\/\\delta\\nu_2^{(0)}\\sim 1\/3$ given above \nas $M_\\pi\/\\Lambda \\sim 1\/3$.\n\nLet us consider now the relevance  of the different contributions to $\\Delta(A)$ by evaluating the double\n integral in Eq.~\\eqref{twiceD}, namely,\n\\begin{align}\n\\frac{A(A+M_\\pi^2)}{\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2}\\int_0^\\infty dq^2\\frac{\\rho(q^2)q^2}{(q^2-A)(q^2+M_\\pi^2)(q^2-k^2)}~,\n\\label{1s0.quanty}\n\\end{align}\n with the full result for $D(A)$ but with $\\Delta(A)$ in the integrand of Eq.~\\eqref{1s0.quanty} evaluated \npartially with some contributions or all of them. The result of this exercise is given\n in the left panel of Fig.~\\ref{fig:1s0quanty} for the $c_i$ coefficients of Ref.~\\cite{epe12}, \n collected in the first row of Table~\\ref{tab:cis}. \n In turn,  we show directly $\\Delta(A)$ along the LHC in the right panel of Fig.~\\ref{fig:1s0quanty}. \n The (black) dash-dotted lines correspond to OPE, \nthe  (blue) dotted lines take into account the full ${\\cal O}(p^2)$ TPE,\n including both two-nucleon reducible and irreducible TPE, and  \n the (cyan) double-dotted lines contain  the ${\\cal O}(p^3)$  two-nucleon irreducible TPE. \nIn the right panel we show by the (cyan) filled area the variation in the ${\\cal O}(p^3)$ irreducible TPE \ncontribution by varying between the different sets of $c_i$'s from Refs.~\\cite{epe04} and \\cite{aco2013}, as discussed above. \nThis band indicates a large source of uncertainty  in $\\Delta(A)$.\n  In the left panel the (red) solid line  results by keeping  all the contributions to $\\Delta(A)$, and \none can quantify from this panel the fact that   the ${\\cal O}(p^3)$ irreducible TPE is the largest subleading contribution.\n At $\\sqrt{A}=100$~MeV it is around 28\\% of the OPE contribution, and it raises with energy\n so that at $\\sqrt{A}=200$~MeV it is 44\\% and at 300~MeV it becomes 66\\%. The increase in energy \nof the relative size of the subleading TPE contribution should be expected because at low energies\n the suppression mechanism due to the earlier onset of the OPE source of $\\Delta(A)$ along the LHC at $L$ is more efficient.\n In addition, it is well-known that the $\\Delta(1232)$ plays a prominent role in $\\pi N$ scattering because\n its proximity to the $\\pi N$ threshold and its strong coupling to this channel.\n This manifests in the large size of the LECs $c_3$ and $c_4$ in Table~\\ref{tab:cis} due to the $\\Delta(1232)$ contribution \nto them, evaluated in Refs.~\\cite{bernard93,aco2012}. \nThe large impact of the $\\Delta(1232)$ is  the well-known reason for the large size of subleading TPE,\n but once its leading effects are taken into account at $ {\\cal O}(p^3)$ the chiral expansion\n stabilizes \\cite{entem,epe04}, as we have also concluded  in the discussion following Eq.~\\eqref{nu2predicted}. \n  In the following, we skip the discussion on the relative importance of the different contributions \nto $\\Delta(A)$ for those $NN$ partial waves with a similar situation to the one discussed \nconcerning the $^1S_0$. \n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/lhcd1s0.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/dis.lhc1s0.ps}\n\\end{tabular}\n\\caption{ { (Color online.) Left panel: different contributions to the integral in Eq.~\\eqref{1s0.quanty} for the $^1S_0$.\n Right panel: contributions to $\\Delta(A)$. These contributions comprise OPE (black dash-dotted line), \nleading TPE (blue dotted line) and \nthe subleading TPE contribution, shown by the (cyan) double-dotted line in the left panel \nand by the (cyan) filled area in the right one. The total result, only shown for the left panel, \nis the (red) solid line.}\n\\label{fig:1s0quanty}}\n\\end{center}\n\\end{figure}\n\n\\section{Uncoupled $P$ waves}\n\\label{pw} \nIn this section we discuss the application of the method to the  uncoupled $P$ waves. \n At NNLO one has for these waves that \n\\begin{align}\n\\lambda_P=\\lim_{A\\to-\\infty}\\frac{\\Delta(A)}{(-A)^{(3\/2)}}>0~,\n\\label{unlambdap}\n\\end{align}\nso that, according to the results of Ref.~\\cite{gor2013}, its Proposition 4, a once-subtracted DR for $D(A)$, \nEq.~\\eqref{standardr}, does not converge and more subtractions should be taken. Then, we directly discuss \nthe twice- and three-time subtracted DRs. \n\nThe twice-subtracted DRs are given by:\n\\begin{align}\n\\label{twiceDNl}\nD(A)&=1+\\delta_2 A-\\nu_2 A^2 g(A,0)\n+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2}g(A,k^2)~,\\nonumber \\\\\nN(A)&=\\nu_2 A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2)^2}~,\n\\end{align}\nwith all the subtractions in Eq.~\\eqref{standardr} taken at $C=0$. \n The three-time subtracted DRs are:\n\\begin{align}\nD(A)&=1+\\delta_2 A + \\delta_3 A(A+M_\\pi^2)+(\\nu_2-\\nu_3 M_\\pi^2)  A(A+M_\\pi^2)^2  \\frac{\\partial g(A,-M_\\pi^2)}{\\partial M_\\pi^2}\n-\\nu_3 \\, A(A+M_\\pi^2)^2 g(A,-M_\\pi^2)\\nonumber\\\\\n&+\\frac{A(A+M_\\pi^2)^2}{\\pi}\\int_{-\\infty}^Ldk^2 \\frac{\\Delta(k^2)D(k^2)}{(k^2)^3}g(A,k^2,-M_\\pi^2;2)~,\\nonumber\\\\\nN(A)&=\\nu_2 A+ \\nu_3 A^2+\\frac{A^3}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2)^3}~.\n\\label{pw.tdr}\n\\end{align}\nHere all the subtractions in $N(A)$ and one in $D(A)$ are taken at $C=0$, while the other two subtractions \nin $D(A)$ are taken at $C=-M_\\pi^2$. This is done in order to avoid handling an infrared diverging \nintegral along the RHC multiplying $\\nu_2$ that would result if all the subtractions were taken at $C=0$. \nThe function $g(A,k^2,C;m)$ appearing in Eq.~\\eqref{pw.tdr} is defined as\n\\begin{align}\ng(A,k^2,C;m)=\\int_0^\\infty\\!\\! dq^2\\frac{\\rho(q^2) (q^2)^m}{(q^2-A)(q^2-k^2)(q^2-C)^m}~.\n\\label{def.el.gm}\n\\end{align}\n\nIn all the cases the subtraction constant $\\nu_2$ is fixed in terms of the \nscattering volume, $a_V$, \n\\begin{align}\n\\nu_2=4\\pi a_V\/m~.\n\\label{pnu2fix}\n\\end{align}\nFor $a_V$ we take the values $0.890$, $-0.543$ and $-0.939~M_\\pi^{-3}$ for the partial waves $^3P_0$, $^3P_1$ and $^1P_1$, \nin order, as deduced from Ref.~\\cite{Stoks:1994wp}.\n\n\\subsection{$^3P_0$ wave}\n\\label{3p0}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{.\/3p0.ps} \n\\caption{ {\\small }\n\\label{fig:3p0}\n Phase shifts of the $^3P_0$ $NN$ partial wave. \n The three-time subtracted DR results at NNLO are shown by the (red) hatched area and the twice-subtracted DR results  at NLO \\cite{gor2013} are given by the (magenta) solid line. \nThe (blue) dotted line corresponds to the OPE results \\cite{paper1} and \nthe Nijmegen PWA phase shifts are shown by the (black) dashed lines.}\n\\end{center}\n\\end{figure}\n\n \n For the $^3P_0$ wave the twice-subtracted DRs at NNLO, Eq.~\\eqref{twiceDNl}, do not provide \nstable results under the increase in absolute value \nof the lower limit of integration along the LHC. \nHowever, the three-time subtracted DRs, Eq.~\\eqref{pw.tdr}, are convergent and provide meaningful results. \n Notice that, as stated in Sec.~\\ref{nschpt}, \non top of the number of subtractions required by the chiral counting, two at NNLO,\n we impose the requirement of having well-defined IEs providing stable \n solutions. \nRegarding the subtractions constants $\\nu_3$, $\\delta_2$ and $\\delta_3$ in Eq.~\\eqref{pw.tdr}, \nwe can fix $\\nu_3=0$ because it plays a negligible role in the fits and, if released,  the fit remains stable. \nThe fitted values for $\\delta_2$ and $\\delta_3$  are\n\\begin{align}\n\\delta_2&= 2.82(5)~M_\\pi^{-2}\\nonumber\\\\\n\\delta_3&=0.18(6) ~M_\\pi^{-4}~,\n\\end{align}\nwhere the intervals of values take into account the dispersion in the results \nthat stems from the different sets of $c_i$'s in Table~\\ref{tab:cis}.\n The phase shifts calculated, shown by the (red) hatched area \nin Fig.~\\ref{fig:3p0}, reproduce exactly the Nijmegen PWA phase shifts \\cite{Stoks:1994wp}, \ngiven by the (black) dashed line. Indeed, the two lines overlap each other. \nThe results with different sets of values for the $c_i$ counterterms cannot be distinguished  either between each other. \n The (magenta) solid line shows the results with twice-subtracted DRs at NLO \\cite{gor2013}, \nwhich are already almost on top of the data, and the OPE results \\cite{paper1} are shown by the (blue) dotted line. \n We have also checked that a tree-time-subtracted DR at LO and NLO provide already a prefect reproduction of data as well. \nThen, the wave $^3P_0$ studied at ${\\cal O}(p^3)$ is not a good partial wave to learn above chiral dynamics, because \n independently of order up to which $\\Delta(A)$ is calculated the reproduction of data is excellent when \nthree-subtractions are taken. \n\n\n\n\\subsection{$^3P_1$ wave}\n\\label{3p1}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{.\/3p1.ps} \n\\caption{ {\\small }\n\\label{fig:3p1}\n Phase shifts of the $^3P_1$ $NN$ partial wave. \n The three-time subtracted DR results at NNLO are shown by the (red) hatched area. \nThe (blue) dotted line corresponds to the OPE results \\cite{paper1} and\n the Nijmegen PWA phase shifts are shown by the (black) dashed lines.}\n\\end{center}\n\\end{figure}\n\nFor this partial wave the situation is similar to that discussed for  the $^3P_0$. \nThe twice-subtracted DRs, Eq.~\\eqref{twiceDNl}, do not provide stable results and we have \nto consider then the three-time subtracted DRs, Eq.~\\eqref{pw.tdr}. \n The free parameters are $\\delta_2$ and $\\delta_3$, with $\\nu_3$ fixed to 0 (the fit is stable if this \nsubtraction constant is released). The fitted values  are\n\\begin{align}\n\\delta_2& = 2.7(1)~M_\\pi^{-2},\\nonumber\\\\\n\\delta_3& = 0.47(3)~M_\\pi^{-4}~.\n\\end{align}\n\nThe resulting phase shifts are  shown in  Fig.~\\ref{fig:3p1} by the (red) hatched area and \n reproduce perfectly the Nijmegen PWA phase shifts (shown by \nthe black dashed line),  independently of the set of values\n for the $c_i$'s chosen from Refs.~\\cite{epe12,aco2013} in Table~\\ref{tab:cis}. \n At NLO \\cite{gor2013} it is also necessary to take three-subtracted DRs in order to obtain stable results and the reproduction \nof data is equally perfect. \nThis is why we have not included the NLO results in Fig.~\\ref{fig:3p1}. \n Similarly to the $^3P_0$ case, we cannot discern the impact of chiral dynamics at ${\\cal O}(p^3)$ once three-time subtracted DRs \nare considered.\n\n\\subsection{$^1P_1$ wave}\n\\label{1p1}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{.\/1p1.ps} \n\\caption{ {\\small }\n\\label{fig:1p1}\n Phase shifts of the $^1P_1$ $NN$ partial wave. \n The twice subtracted DR results at NNLO are shown by the (red) hatched area, while at NLO \\cite{gor2013}\ncorrespond to  the (magenta) solid line. \nThe (blue) dotted line corresponds to the OPE results \\cite{paper1} and \nthe Nijmegen PWA phase shifts are shown by the (black) dashed lines.}\n\\end{center}\n\\end{figure}\n \nFor this partial wave the twice-subtracted DR results from Eq.~\\eqref{twiceDNl} are quite stable  at low energies. \nThe free parameters are now $\\nu_2$ and $\\delta_2$. \nThe resulting fitted value for $\\delta_2$ to  the Nijmegen PWA phase shifts  is  \n\\begin{align}\n\\delta_2 =0.4(1)~M_\\pi^{-2}~,\n\\end{align}\nwith the variation in the value due to the set of $c_i$'s taken [$\\nu_2$ is given by Eq.~\\eqref{pnu2fix}]. \nWe show by the (red) hatched area in Fig.~\\ref{fig:1p1} our results by employing the different $c_i$ sets of values. \nFor this case the curves obtained with the $c_i$ from \\cite{aco2013}, by reproducing the $\\pi N$ phase shifts with \nLorentz covariant EOMS BChPT, are the closest to data and determine the upper\n limit of the hatched area in Fig.~\\ref{fig:1p1}. \n The improvement in the reproduction of data for the $^1P_1$ partial wave by the twice-subtracted DRs at NNLO compared with the \nresults obtained at NLO with the same number of subtractions \n  (hatched area versus (magenta) solid line)  is a notorious effect from $\\pi N$ physics. One should \nnotice that for the $^1P_1$ wave the dispersive integral on the r.h.s. of Eq.~\\eqref{twiceDNl} for the function $D(A)$ is clearly \ndominated by the OPE contribution\n This is the reason why for the $^1P_1$ one does not need to take three subtractions but two are enough. \n Although, as much as for the other partial waves discussed until now, the ${\\cal O}(p^3)$ two-nucleon irreducible \nTPE is the dominant contribution between the subleading effects to $\\Delta(A)$. \n\n\n\n\n\\section{Uncoupled $D$ waves}\n\\label{dw}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/1d2.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3d2.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) Phase shifts for $^1D_2$ (left panel) and $^3D_2$ (right panel). \n The (red) hatched areas  correspond to the NNLO results while the (magenta) solid lines are \nthe NLO outcome \\cite{gor2013}. In both cases twice-subtracted DRs are used. The \nphase shifts in the Born approximation   are shown by the (cyan) filled bands, \nthe OPE result from Ref.~\\cite{paper1} is the (blue) dotted lines and \n the Nijmegen PWA phase shifts are given by the (black) dashed lines.}\n\\label{fig:dw} }\n\\end{center}\n\\end{figure}\n\n\nHere, we discuss the $D$ waves. In order to preserve the right threshold behavior we \nemploy the twice-subtracted DRs of Eqs.~\\eqref{highd} and \\eqref{highn} with $\\ell=2$. \nFor the uncoupled $D$ waves one has that\n\\begin{align}\n\\lambda_D&=\\lim_{A\\to -\\infty}\\frac{\\Delta(A)}{(-A)^{3\/2}}<0\n\\end{align}\nand for this sign we do not have numerical problems \nin the solution of the resulting IE even for diverging $\\Delta(A)$  \\cite{gor2013}.\n\n\\begin{align}\n\\label{twiceDNl_Dw}\nD(A)&=1+\\delta_2 A\n+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2}g(A,k^2)~,\\nonumber \\\\\nN(A)&=\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2)^2}~.\n\\end{align}\n\nThe only free parameter per partial wave is $\\delta_2=D^{(1)}(0)$ which is  fitted to the \nNijmegen PWA phase shifts. Taking into account the \ndifferent sets of values for the $c_i$ counterterms we have the following \nresults,\n\\begin{align}\n^1D_2:& ~D^{(1)}(0)= 0.07(1) ~M_\\pi^{-2},\\nonumber\\\\\n^3D_2:& ~D^{(1)}(0)= -0.017(3)~M_\\pi^{-2}~.\n\\end{align} \nThe reproduction of data is excellent as shown by the (red) \nhatched areas in Fig.~\\ref{fig:dw}, where the phase shifts for the $^1D_2$ are given in the left panel \nand those of the $^3D_2$ in the right one. Our results indeed overlap the Nijmegen PWA phase shifts given by \nthe (black) dashed lines.  \nReference~\\cite{gor2013} obtained the (magenta) solid line making use also of twice-subtracted DRs at NLO. \n We see a remarkable   improvement from NLO to NNLO due to the inclusion of NLO  $\\pi N$ dynamics, \nparticularly for the $^1D_2$ partial wave.  \n\n\\subsection{Perturbative and Born approximation phase shifts}\n\\label{born}\n\nThe higher is the orbital angular momentum $\\ell$ the more perturbative is expected to be the \ncorresponding $NN$ partial wave. \nThis statement was studied in detail in the perturbative study of  \n Ref.~\\cite{peripheral} by making use \nof the one-loop approximation in BChPT.\n Indeed, we can easily obtain  from \nour formalism both the leading perturbative solution to the \nIEs of the $N\/D$ method in powers of $\\Delta(A)$,  as well as the leading term in the Born series approximation \nfor the chiral $NN$ amplitude calculated up to ${\\cal O}(p^3)$ in Ref.~\\cite{peripheral}. \nThe point is that for a \nweak  interaction  (small $\\Delta(A)$ at low three-momentum) one can expect that $D(A)\\simeq 1$ at low energies.\n It is then reasonable to consider that substituting $D(A)\\to 1$ in the integral on the r.h.s.  of \nEq.~\\eqref{highn} would be meaningful to calculate $N(A)$, \n because we have a rapid converging integral due to the factor $(k^2)^\\ell$ in the denominator for a sufficiently \nlarge value of $\\ell$.\\footnote{Of course, the precise meaning of this statement could vary from one case to \nthe other due to characteristic facets of the considered  partial wave.}  The perturbative \nresult for $N(A)$, denoted by $N_p(A)$, is then\n\\begin{align}\nN^{(p)}(A)=\\frac{A^\\ell}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)}{(k^2)^\\ell (k^2-A)}~.\n\\label{eq.hw.per}\n\\end{align}\nHad we included only the two-nucleon irreducible contributions to  $\\Delta(A)$, which is then \ndenoted as $\\Delta_B(A)$,   \nthe previous integral becomes  the DR representation of the $NN$ potential that \nwe denominate $N_B(A)$,\n\\begin{align}\nN_B(A)=\\frac{A^\\ell}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_B(k^2)}{(k^2)^\\ell (k^2-A)}~.\n\\label{eq.nborn}\n\\end{align}\nThis is due to the fact that the $NN$ potential projected in a given \npartial wave  is an analytical function that only has LHC and it can be \nwritten in terms of a  DR  along the latter cut. \n We have checked numerically that  the DR representation \nEq.~\\eqref{eq.nborn} for the $NN$ potential coincides\n with its explicit partial wave decomposition taking into account \nthe expressions given in Ref.~\\cite{peripheral}.  \n In our notation the relation between $N_B(A)$ and the phase shifts in the Born approximation, $\\delta_B(A)$, reads\n\\begin{align}\n\\delta_B(A)=\\rho(A) N_B(A)~.\n\\label{deltab}\n\\end{align}\nAn analogous  expression holds for the perturbative phase shifts $\\delta^{(p)}(A)$ calculated in terms of $N^{(p)}(A)$. \n The difference between the perturbative phase shifts and the Born approximation ones \nfor $\\ell\\geq 2$ is typically not very significant and quite small. \nIn the following we compare our full results  with $\\delta_B(A)$, \nsince these phase shifts can be also calculated straightforwardly in potential models.\nWe proceed in the same way for the coupled channel case as well by evaluating $N_{ij}(A)$ in the Born approximation \nby substituting  $D_{ij}(A)\\to 1$ in Eq.~\\eqref{highncc}, and keeping only the two-nucleon irreducible contributions \nto $ \\Delta_{ij}(A)$.\n\nTurning back to the uncoupled $D$ waves we also show in Fig.~\\ref{fig:dw} the leading Born approximation phase shifts \nobtained from the NNLO two-nucleon  irreducible contributions to $\\Delta(A)$ by the (cyan) filled areas.\n One observes that these curves are quite different from our full results \ngiven by the hatched areas. This clearly indicates that the perturbative \ntreatment of the $NN$ $D$ waves is not accurate.\n\n\\section{Uncoupled $F$ waves}\n\\label{fw}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/1f3.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3f3.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) Phase shifts for $^1F_3$ (left panel) and $^3F_3$ (right panel). \n The (red) hatched areas  correspond to the NNLO results while the (magenta) solid lines are \nthe NLO outcome. In both cases three-time-subtracted DRs are used.\nThe (cyan) filled bands give $\\delta_B(A)$, \nthe OPE result from Ref.~\\cite{paper1} is the (blue) dotted lines and \nthe Nijmegen PWA phase shifts correspond to the (black) dashed lines.}\n\\label{fig:fw} }\n\\end{center}\n\\end{figure}\n\nFor the $F$ waves we have three subtractions with two free parameters $\\delta_2$ and $\\delta_3$. We fix $\\delta_2=0$ in the following (according to the principle of maximal smoothness) and fit $\\delta_3$ to data. \n At NNLO the fitted values for $D^{(2)}(0)=2 \\delta_3$, Eq.~\\eqref{tayloruc}, are: \n\\begin{align}\n^1F_3:~& D^{(2)}(0)= 0.057(3)~M_\\pi^{-4}~,\\nonumber\\\\\n^3F_3:~& D^{(2)}(0)= 0.035(5)~M_\\pi^{-4}~,\n\\end{align}\nwhere the variation in the values is due to the different sets of $c_i$ counterterms employed. \nThe NNLO results are shown by the (red) hatched areas in Fig.~\\ref{fig:fw} which reproduce the \nNijmegen PWA phase shifts (black dashed line) better than the NLO results (magenta lines) and \nthe perturbative phase shifts (cyan filled areas).  This improvement \nis particularly noticeable for the $^3F_3$ partial wave.\n\nWe also observe that for the $F$ waves the phase shifts in the leading Born approximation, Eq.~\\eqref{deltab}, \nrun much closer to our full results than for the $D$ waves, which clearly indicates that \n$F$ waves are more perturbative. \n Nevertheless, the relative deviation of the perturbation results compared \nwith the full solution is still around a 50\\% at the end of the interval shown in Fig.~\\ref{fig:fw}. \n A similar conclusion on the more perturbative nature of the $F$ waves \nwas also reached in the pure perturbative study of Ref.~\\cite{peripheral} by comparing with experimental data.  \n However, here we can also compare with the full unambiguous solution of the corresponding IE. \n For example, we can learn from Fig.~\\ref{fig:fw}\n that the widths of the (cyan) filled bands for the Born approximation results \nreflect a much larger dependence on the $c_i$ \ncoefficients than the one corresponding to the full nonperturbative results given by the (red) hatched\n areas.\n Thus, within our approach the failure reported in Refs.~\\cite{epe04,thesis} to reproduce simultaneously \nthe $D$ and $F$ waves by using the NNLO chiral potential \ncalculated in dimensional regularization in Ref.~\\cite{peripheral}  \n because the large values of the $c_i$  counterterms does not happen. \nNamely, we are able to \ndescribe properly both the  uncoupled $D$ and $F$ waves, Figs.~\\ref{fig:dw} and \\ref{fig:fw}, respectively, \nand the dependence on the precise set of $c_i$'s taken is quite mild for the full results. \nIndeed our calculation at NNLO describe the Nijmegen PWA phase shifts better than  the NLO ones \\cite{gor2013},\n which is not the case for all of these waves in  Ref.~\\cite{thesis} \nbased on the (modified) Weinberg approach when comparing their NLO and NNLO results.\n\n \n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/lhcd1f3.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/dis.lhc.1f3.ps} \\\\\n\\includegraphics[width=.4\\textwidth]{.\/lhcd3f3.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/dis.lhc.3f3.ps} \\\\\n\\end{tabular}\n\\caption{ { (Color online.) Left panel: different contributions to the integral on the \nr.h.s. of Eq.~\\eqref{highd} for $\\ell=3$.\n The meanings of the lines are the same as in Fig.~\\ref{fig:1s0quanty}. \nFor definiteness we consider the $c_i$'s given in the last row of Table~\\ref{tab:cis}.  }\n\\label{fig:unFquanty}}\n\\end{center}\n\\end{figure}\n\nThe increase in the perturbative character of the $F$ waves can also be seen by considering the \nrelevance of the different contributions of $\\Delta(A)$ to the integral on the r.h.s. of\n Eq.~\\eqref{highd}, proceeding in a similar way to that already performed for the $^1S_0$ partial wave \nin Sec.~\\ref{1s0}.\n The result is shown in the left panels of Fig.~\\ref{fig:unFquanty}, where\n the first row corresponds to $^1F_3$ and the second to $^3F_3$.\n In the right panels we show directly the different contributions to $\\Delta(A)$. \nThe meanings of the lines \nin Fig.~\\ref{fig:unFquanty} are the same as in Fig.~\\ref{fig:1s0quanty}, though here \n the $c_i$'s are taken from Ref.~\\cite{aco2013}, given in the last row of Table~\\ref{tab:cis}, \nwhich is enough for the present purposes. \nNotice, that now a qualitative different situation is found with respect to what is shown in \nFig.~\\ref{fig:1s0quanty}, that also holds for the $P$ and $D$ waves discussed in Secs.~\\ref{pw} and \\ref{dw}. \n For the $F$ and higher waves the subleading two-nucleon irreducible TPE contribution is much less important and \nOPE is by far the dominant contribution, as it should correspond to a perturbative high-$\\ell$ wave.\n \n\n \n\n\n\\section{Uncoupled $G$ waves}\n\\label{gw}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/1g4.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3g4.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) Phase shifts for $^1G_4$ (left panel) and $^3G_4$ (right panel). \n The (red) hatched areas  correspond to the NNLO results while the (magenta) solid lines are \nthe NLO outcome. In both cases four-time--subtracted DRs are used. \n The  (cyan) filled areas represent\n the outcome from the leading Born approximation,  \n the OPE result from Ref.~\\cite{paper1} is the (blue) dotted lines \nand the Nijmegen PWA analysis is the (black) dashed lines.}\n\\label{fig:gw} }\n\\end{center}\n\\end{figure}\n\nFor the $G$ waves we have four subtractions of which $\\delta_i$ $(i=2,3,4)$ are free but, according to the \nprinciple of maximal smoothness, all of them are fixed to 0 except \n$\\delta_4=D^{(3)}(0)\/3!$ that is fitted to data. \nAt  NNLO the fitted values for $D^{(3)}(0)$ are: \n\\begin{align}\n^1G_4:~& D^{(3)}(0)=-0.014(2)~M_\\pi^{-6}~,\\nonumber\\\\\n^3G_4:~& D^{(3)}(0)=-0.055(5)~M_\\pi^{-6}~,\n\\end{align}\nwhere the variation in the values is due to the different sets of $c_i$ counterterms employed. \nThe corresponding results are shown by the (red) hatched areas in Fig.~\\ref{fig:gw}. \nFor both partial waves the actual dependence on the $c_i$ coefficients for the resulting phase shifts \nis almost negligible and the hatched  areas degenerate to lines. \n  The low-energy results are very similar at NLO and NNLO and reproduce the Nijmegen \nPWA phase shifts quite well.\n These results are better than the perturbative ones in the Born approximation,  Eq.~\\eqref{deltab}, which  \nare shown by the (cyan) filled areas. \nAs indicated for the uncoupled $F$ waves here OPE overwhelmingly dominates \n the different contribution to the dispersive integral on the r.h.s. of Eq.~\\eqref{highd}. \nThis indicates that these waves are rather perturbative, though still we observe differences \naround 30\\% for $p\\lesssim 300$~MeV in Fig.~\\ref{fig:gw} between the full and perturbative results.\n\n\n\n\n\\section{Uncoupled $H$ waves}\n\\label{hw}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/1h5.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3h5.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) Phase shifts for $^1H_5$ (left panel) and $^3H_5$ (right panel). \n The (red) hatched areas  correspond to the NNLO results while the (magenta) solid lines are \nthe NLO outcome. \n The (cyan) filled bands correspond to $\\delta_B(A)$, \nthe OPE result from Ref.~\\cite{paper1} is the (blue) dotted lines and  \nthe Nijmegen PWA is the (black) dashed lines.}\n\\label{fig:hw} }\n\\end{center}\n\\end{figure}\n\nFor the case of the uncoupled $H$ waves, $^1H_5$ and $^3H_5$, we apply the five-time subtracted DRs of  \nEqs.~\\eqref{highd} and \\eqref{highn} with $\\ell=5$. \nWe fit $\\delta_5=D^{(4)}(0)\/4 !$ to the Nijmegen PWA phase shifts, which for $\\ell\\geq 5$ \ncorrespond to  those obtained from the $NN$ potential model of Ref.~\\cite{obe}, while $\\delta_{2,3,4}$ are fixed to 0 (principle of maximal smoothness). \nWe obtain the fitted values: \n\\begin{align}\n^1H_5:~D^{(4)}(0)&=0.156~ M_\\pi^{-8}~,\\nonumber\\\\\n^3H_5:~D^{(4)}(0)&=0.066~M_\\pi^{-8}.\n\\end{align} \nThe resulting fit is stable if  we release $\\delta_i$ $(i=2,3,4)$. \nThe  phase shifts obtained are shown by the (red) hatched areas in  Fig.~\\ref{fig:hw}  by taking into account the spread of the results \ndepending of the set of $c_i$'s chosen. \nIn this figure the left panel corresponds to  $^1H_5$ and  the right one to $^3H_5$.  \n For the former the resulting curve indeed  overlaps the Nijmegen PWA phase shifts \\cite{Stoks:1994wp}. \nWe also show by the (cyan) filled bands the phase shifts in the leading Born approximation\n which run rather close to \nthe full results, indeed for the $^3H_5$ case the (cyan) filled band is overlapped by the (red) hatched one.   \nThis clearly indicates the perturbative nature for the $H$ waves. \nFor them it is also true that OPE overwhelmingly dominates \nthe dispersive integral on the r.h.s. of Eq.~\\eqref{highd}, which is also the expected behavior \nfor a perturbative partial wave.\nNotice that for the $^1H_5$ wave the dependence on the actual values of the \n$c_i$ coefficients is so small that at the end the hatched and filled areas collapse to lines.\n For the $^3H_5$ case there is a visible, albeit small, dependence on the set of $c_i$'s employed. \nIn both cases the NNLO results reproduce the Nijmegen PWA phase shifts closer than the NLO and OPE results. \n\n \n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/N3h5.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/D3h5.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) The functions $N(A)$ and $N_p(A)$ are shown by the (red) solid and (blue) \ndash-dotted lines in the left panel, respectively. The real part of the function $D(A)$ is plotted in the right panel. }\n\\label{fig:hw.nd} }\n\\end{center}\n\\end{figure}\n\nIt is interesting to discuss in this case the behavior of the function $N(A)$ compared with $N_p(A)$, given in Eq.~\\eqref{eq.hw.per}. \nThe main point is that here both $N(A)$ and $D(A)$ have a zero at around 450~MeV.\n We consider only the $^3H_5$ wave because a similar discussion would follow for $^1H_5$ as well, \nthat we skip for brevity.\n In the left panel of Fig.~\\ref{fig:hw.nd} we show by the (red) solid line the full $N(A)$ \n and by the (blue) dashed line the perturbative result $N_p(A)$. \nWe see that they  are very similar, as expected for a partial wave with an $\\ell$ as high as 5.\n In addition, we display  in the right panel of the same figure the real part of $D(A)$ from Eq.~\\eqref{highd}, \nwhich is very close to 1, as expected for a situation with a weak interaction as well. \nAll these curves are obtained by employing the  $c_i$'s from Ref.\\cite{aco2013}.\n A bit higher in energy both  $N_p(A)$ and  $N(A)$ have a zero \n at around $\\sqrt{A}=450$~MeV.\n Since $T(A)=N(A)\/D(A)$ this would imply that $T(A)=0$ at that energy, which is at odds \nwith the values of the phase shifts given by the Nijmegen PWA \\cite{Stoks:1994wp} that do not vanish at this point. \nThe only remedy is that  $D(A)$ is also zero at the same point so that \n one had a limit 0\/0 that is finally finite. \nThis is indeed the case and it is is the reason why \n $D(A)$ starts to decrease for $\\sqrt{A}>200$~MeV in Fig.~\\ref{fig:hw.nd}.\n\n\nAnother question of interest to think about is  what have we  gained by solving exactly Eq.~\\eqref{highd} instead of \n using only the perturbative solution, Eq.~\\eqref{eq.hw.per}, or the Born approximation, Eq.~\\eqref{eq.nborn}, with \nthe related $ \\delta_B(A)$, Eq.~\\eqref{deltab}?.  \nThe main point that one should consider \n in connection with this question is that by solving the full and nonperturbative Eq.~\\eqref{highd} \n(furthermore, in good agreement with data) one can then state that Eq.~\\eqref{eq.hw.per} is a  perturbation of a \n well-defined and existing nonperturbative solution. \nBy solving exactly Eq.~\\eqref{highd} we have needed to consider explicitly $\\delta_5$ as a free parameter  for the uncoupled \n$H$ waves  and fit it to the Nijmegen PWA. \nIndeed, $\\delta_5$ is not only necessary for a good fit, \n but it is also required in order to keep $D(A)\\simeq 1$ at low \nthree-momentum. \nOtherwise, the contribution from the dispersive integral to $D(A)$ on the r.h.s. of Eq.~\\eqref{highn} would \nbe too large and negative and would render a too strong function $N(A)$  in plain disagreement with $N_p(A)$. \nNotice as well that in the case of the partial wave $^1H_5$ a better reproduction of data is achieved than with $\\delta_B(A)$. \n It is also worth recalling the previous finding in Sec.~\\ref{fw} for the $F$ waves, \nwhere the full results show a much smaller dependence on \nthe set of $c_i$ coefficients used than the perturbative or Born approximation phase shifts, cf. Fig.~\\ref{fig:fw}.\n\n\n\\section{Coupled $^3S_1-{^3D_1}$ waves}\n\\label{sd12w}\n\n\nWe start our study of the $^3S_1-{^3D_1}$ coupled-partial-wave system in terms of just one free parameter, \nthat we choose as the pole position of the deuteron in the $A$-complex plane, $k^2_d=-m E_d$, with $E_d= 2.225$~MeV \nthe deuteron binding energy. \n Thus we implement once-subtracted DRs for the $^3S_1$ and twice-subtracted ones for the $^3D_1$. \nIn the case of the mixing partial wave we have a mixed situation with a once-subtracted DR \nfor $N_{12}(A)$ and a twice-subtracted one for $D_{12}(A)$.\n In this way we guarantee both the right threshold behavior as well as the experimental \n deuteron-pole position in all the partial waves. \nWe write now explicitly  the DRs considered. For the $^3S_1$ one has,\n\\begin{align}\n\\label{3s1_a}\nD_{11}(A)&=1-\\frac{A}{k_d^2}\\frac{g_{11}(A,0)}{g_{11}(k_d^2,0)}\n+\\frac{A}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{k^2}\n\\Bigg[g_{11}(A,k^2)-g_{11}(A,0)\\frac{g_{11}(k_d^2,k^2)}{g_{11}(k_d^2,0)}\\Bigg]~,\\nonumber\\\\\nN_{11}(A)&=\\nu_1^{(11)}+\\frac{A}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{k^2(k^2-A)}~,\n\\end{align}\nwith all the subtractions taken at $A=0$ and the new function $g_{ij}(A)$ is defined as\n \\begin{align}\ng_{ij}(A,k^2)&=\\frac{1}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{ij}(q^2)}{(q^2-A)(q^2-k^2)}~,\n\\label{gij}\n\\end{align}\n The subtraction constant $\\nu_1$ in $N_{11}(A)$ is fixed by imposing that $D_{11}(k_d^2)=0$,\n\\begin{align}\n\\nu_1^{(11)}&=\\frac{1}{k_d^2 \\,g_{11}(k_d^2,0)}\\Bigg[\n1+\\frac{k_d^2}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{11}(k^2) D_{11}(k^2)}{k^2}g_{11}(k^2,k_d^2)\n\\Bigg]~,\n\\label{once_nu0}\n\\end{align}\na result that is  already implemented in Eq.~\\eqref{3s1_a} for $D_{11}(A)$.\n \nThe corresponding DRs for the $^3D_1$ and the mixing wave can be grouped together in the same form,\n\\begin{align}\n\\label{3sd1_a}\nD_{ij}(A)&=1-\\frac{A}{k_d^2}+\\frac{A(A-k_d^2)}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^{\\ell_{ij}}}\ng_{ij}^{(d)}(A,k^2;\\ell_{ij})~,\\nonumber\\\\\nN_{ij}(A)&=\\frac{A^{\\ell_{ij}}}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^{\\ell_{ij}}(k^2-A)}~.\n\\end{align}\nwhere $\\ell_{12}=1$ and $\\ell_{22}=2$ and all the subtractions for the $N_{ij}(A)$ are taken at $A=0$,\n while in the function $D(A)$ one is taken at $A=0$ and \nthe other at $A=k_d^2$. The  function $g^{(d)}_{ij}(A,k^2;m)$ is defined as\n\\begin{align}\ng^{(d)}_{ij}(A,k^2;m)&=\\frac{1}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{ij}(q^2)(q^2)^{m-1}}{(q^2-A)(q^2-k^2)(q^2-k_d^2)}~.\n\\label{gijd}\n\\end{align}\n\nThe results obtained by solving the IEs for the functions $D_{ij}(A)$ along the LHC\n from Eqs.~\\eqref{3s1_a} and \\eqref{3sd1_a} are \nshown in Fig.~\\ref{fig:3sd1_a} by the (cyan) filled areas.\n These results are indicated as NNLO-I and \n all the subtraction constants are fixed in terms of $k_d^2$, without any other freedom. \n The spread in the results originates by taking  different  sets of $c_i$'s from Refs.~\\cite{epe12,aco2013} \nand varying the input in the iterative procedure.\n The present NNLO calculation from Eqs.~\\eqref{3s1_a} and \\eqref{3sd1_a}\n reproduces the  Nijmegen PWA  mixing angle $\\epsilon_1$ much better than the NLO \nresult from the same set of equations,  which is shown by the (magenta) dot-dashed lines.\n This improvement in the description of $\\epsilon_1$ \nwhen passing from NLO to NNLO is also seen in Ref.~\\cite{thesis} by \nemploying the Weinberg scheme. \nThe $^3S_1$ phase shifts are also reproduced better at NNLO than at NLO, while \nthe $^3D_1$ phase shifts are somewhat worse described by the former. \nOur results for the  $^3S_1$ and $^3D_1$ phase shifts are quite similar to those obtained \nin Ref.~\\cite{pavon06}, but not for $\\epsilon_1$ where our outcome is closer to the Nijmegen PWA. \nThe comparison is not so straightforward with the results of Ref.~\\cite{phillipssw},  which depend\n very much on the type of chiral $NN$ potential used.\nFor the $^3S_1-{^3D_1}$ coupled partial waves we do not show the Born approximation results  in Fig.~\\ref{fig:3sd1_a} because \nthey are specially poor, see e.g. Refs.\\cite{epe04,peripheral} for the $^3D_1$ phase shifts.\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3s1_all.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3d1_all.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e1_all.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From left to right and top to bottom:\n Phase shifts for $^3S_1$, $^3D_1$ and the mixing angle  $\\epsilon_1$, respectively.\nThe (cyan) filled areas  correspond to the NNLO-I outcome obtained by solving \n Eqs.~\\eqref{3s1_a} and \\eqref{3sd1_a}. The  hatched areas with (red) crossed lines are the NNLO-II results \n that stem from Eqs.~\\eqref{2subs.3s1}, \\eqref{2subs.mix} and \\eqref{3sd1_a}.  \nIn addition, for the $^3D_1$ we show by the hatched areas with (gray) parallel lines the results obtained \nby employing three-time subtracted DRs for $^3D_1$, Eq.~\\eqref{3d1.extra}. \nAs usual, the (magenta) dot-dashed lines are the NLO phase shifts and mixing angle, \n the  LO ones are given by the  (blue) dotted lines and the \nNijmegen PWA results correspond to the (black) dashed lines.}\n\\label{fig:3sd1_a} }\n\\end{center}\n\\end{figure}\n\nWe can also predict from Eqs.\\eqref{3s1_a} and \\eqref{3sd1_a} the $^3S_1$ scattering length ($a_t$) and effective range ($r_t$). The former is given in terms of $\\nu_1^{(11)}$, Eq.~\\eqref{once_nu0}, as \n\\begin{align}\na_t&=-\\frac{m \\nu_1^{(11)}}{4\\pi}~.\n\\end{align} \nRegarding $r_t$ we can proceed similarly as discussed in detail in Ref.~\\cite{gor2013} \nwhere the following expression  is derived, \n\\begin{align}\nr_t=-\\frac{m}{2\\pi^2 a_t}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^2}\\left\\{\\frac{1}{a_t}+\\frac{4\\pi k^2}{m}g_{11}(0,k^2)  \\right\\}-\\frac{8}{m}\\int_0^\\infty dq^2\\frac{\\nu_{11}(q^2)-\\rho(q^2)}{(q^2)^2}~,\n\\label{3s1.rt}\n\\end{align}\nThis equation also exhibits a correlation between $r_t$ and $a_t$, although in a more complicated manner than \nfor the $^1S_0$ partial wave, as shown in Eq.~\\eqref{expvs.1s0}, because $\\nu_{11}(A)$ depends nonlinearly on $D_{11}(A)$. \n \nAnother observable that we also consider is the slope at threshold of $\\epsilon_1$, \nindicated as $a_\\varepsilon$, and defined by  \n\\begin{align}\na_\\varepsilon=\\lim_{A\\to 0^+}\\frac{\\sin 2\\epsilon_1}{A^\\frac{3}{2}}=1.128~M_\\pi^{-3}~,\n\\label{avarepsilon}\n\\end{align}\nwhere the numerical value is deduced from the Nijmegen PWA phase shifts. From the DRs in Eq.~\\eqref{3sd1_a} we obtain \nthe following expression for $a_\\varepsilon$,\n\\begin{align}\na_\\varepsilon&=\\frac{m}{4\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^2}~.\n\\label{aep.predicted}\n\\end{align}\n\nIt is also interesting to diagonalize the $^3S_1-{^3D_1}$ $S$-matrix around the deuteron pole position. This can \nbe done by means of a real orthogonal matrix \\cite{swart},\n\\begin{align}\n{\\cal O}&=\\left(\n\\begin{array}{ll}\n \\cos \\varepsilon_1 & -\\sin\\varepsilon_1 \\\\\n\\sin\\varepsilon_1 & \\cos \\varepsilon_1\n\\end{array}\n\\right)~.\n\\label{mat.ort}\n\\end{align}\nSuch that\n\\begin{align}\nS&={\\cal O}\\left(\n\\begin{array}{ll}\nS_0 & 0 \\\\\n0 & S_2\n\\end{array}\n\\right) {\\cal O}^T~,\n\\label{eigen3s1}\n\\end{align}\nwith $S_0$ and $S_2$ the $S$-matrix eigenvalues.  The asymptotic $D\/S$ ratio of the deuteron, $\\eta$, \ncan be expressed in terms of $\\varepsilon_1$ as\n\\begin{align}\n\\eta=-\\tan \\varepsilon_1~.\n\\label{eta.ep}\n\\end{align}\nThe residue of $S_0$ at the deuteron pole position is denoted by $N_p^2$,\n\\begin{align}\nN_p^2&=\\lim_{A\\rightarrow k_d^2} \\left(\\sqrt{-k_d^2}+i\\sqrt{A}\\right) S_0~.\n\\end{align} \n\nAs discussed in  Ref.~\\cite{cohen}  the shape parameters are a good testing ground for the range of \napplicability of the underlying EFT. \nWe then study our results for the shape parameters of the lowest \neigenphase $\\delta_0$ (also called $^3S_1$ eigenphase), Eq.~\\eqref{eigen3s1}, with the diagonalization of the \n$S$-matrix performed in the physical region $A\\geq 0$,\\footnote{This can also be done in terms of an \northogonal matrix Eq.~\\eqref{mat.ort} because of two-body unitarity.}\n \\begin{align}\n\\sqrt{A}\\,\\text{cot}\\delta_0=-\\frac{1}{a_t}+\\frac{1}{2}r_t A +\\sum_{i=2}^{10}v_i A^i+{\\cal O}(A^{11})~.\n\\label{3s1.ere}\n\\end{align}\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n    & $a_t$ [fm]  & $r_t$ [fm]  & $\\eta$ & $ N_p^2$ [fm$^{-1}$] & $a_\\varepsilon$ [$M_\\pi^{-3}$] \\\\\n\\hline\nNLO     & 5.22 & 1.47 & 0.0295 & 0.714 & 1.372 \\\\\n\\hline\nNNLO-I & $5.52(3)$ & $1.89(3)$ & $0.0242(3)$ & $0.818(10)$& $1.270(9)$\\\\\n\\hline\nNNLO-II & $5.5424^\\star$ & $ 1.759^\\star$ & $0.02535(13)$ & $0.78173(2)$ & $1.293(8)$\\\\\n\\hline\nRef.~\\cite{swart}& $5.4194(20)$ & $1.7536(25)$ & $0.0253(2)$ &  $0.7830(15)$ & \\\\\n\\hline\nRef.~\\cite{thesis} & $5.424$ & $1.753$ & 0.0245 & & \\\\ \n\\hline\n\\end{tabular}\n\\caption{Values for $a_t$, $r_t$, $\\eta$, $N_p^2$ and $a_\\varepsilon$. The results \n predicted from Eqs.~\\eqref{3s1_a} \n and \\eqref{3sd1_a} are given in the second (NLO) and third row (NNLO-I). \nThe values given in the fourth row (NNLO-II) are obtained once $a_t$ and $r_t$ are fixed to the experimental \n figures, which is indicated by a star on top of the values. \nWe also show the results from Refs.~\\cite{swart} and \\cite{thesis} in the fifth and sixth rows, respectively. \n\\label{table:eta}}\n\\end{center}\n\\end{table}\n\nThe scattering length and effective range in the previous equation are the same as given above because \ncoupled-wave effects with the $^3D_1$ only affects the shape parameters $v_i$, $i\\geq 2$. The values obtained at NLO and NNLO from Eqs.~\\eqref{3s1_a} and \\eqref{3sd1_a} for $a_t$, $r_t$, $\\eta$, \n$N_p^2$ and $a_\\varepsilon$ are shown in Table~\\ref{table:eta} in the \nsecond and third rows, respectively. \nWe observe that the numbers at NNLO (indicated by NNLO-I) are already \n rather close to those of Ref.~\\cite{swart}, obtained from the Nijmegen PWA of $n p$ data,\n and Ref.~\\cite{thesis}. \nIt is interesting to remark that our value for $r_t$ is a prediction  in terms of only one  \nsubtraction constant (fixed by the deuteron pole position) and \n$NN$ forces stemming from $\\pi N$ physics. This value deviates from experiment \n $r_t=1.759\\pm 0.005$~fm \\cite{thesis} around a $10\\%$ at NNLO ($\\sim 20\\%$  at NLO), while the relative \n experimental error is  around $3\\%$. \n Other determinations for the parameter $\\eta$,  not shown in Table~\\ref{table:eta}, are $\\eta=0.0256(4)$ \\cite{rodning}, \n$\\eta=0.0271(4)$ \\cite{ericson:82}, \n$\\eta=0.0263(13)$ \\cite{conzett:79} and \n$\\eta=0.0268(7)$ \\cite{martorell}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n    & $v_2$  & $v_3$   & $v_4$ & $v_5$ & $v_6$    \\\\\n\\hline\nNLO & -0.10572(12)& 0.8818(11) & $-5.427(11)$  & 36.73(11) & $-259.9(1.1)$ \\\\\n\\hline\nNNLO-I & 0.157(22)& 0.645(9) & $-3.41(13)$  & 23.2(8) & $-161(6)$ \\\\\n\\hline\nNNLO-II & $0.0848(4)$ & $0.762(7)$ & $-4.33(2)$ & $29.0(2)$ & $-198(2)$\\\\\n\\hline\nRef.~\\cite{swart} & $0.040(7)$ & $0.673(2)$ & $-3.95(5)$ & $27.0(3)$ & \\\\\n\\hline\nRef.~\\cite{thesis} & $0.046$ & $0.67$ & $-3.9$ & & \\\\\n\\hline\n\\end{tabular}\n\\caption{Values for the shape parameters $v_i$, $i=2,\\ldots,6$ in units of fm$^{2i-1}$.\n The results \n predicted from Eqs.~\\eqref{3s1_a}\n and \\eqref{3sd1_a} are given in the second  (NLO) and third row (NNLO-I). \nThe errors for the NLO results correspond  entirely to the numerical accuracy in the calculation. \nThose values corresponding to NNLO-II are given in the fourth row. \nThe values from Refs.~\\cite{swart} and \\cite{thesis} appear in the fifth and sixth rows, in order.\n\\label{table:vs3s1a}}\n\\end{center}\n\\end{table}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n    &  $v_7$ & $v_8\\times 10^{-1}$  & $v_{9}\\times 10^{-2}$  & $v_{10}\\times 10^{-3}$   \\\\\n\\hline\nNLO &  1867(11) & $-1375(11)$ & $1008(11)$ & $-760(12)$ \\\\\n\\hline\nNNLO-I & 1161(41) & $-840(30)$ & $625(22)$ & $-463(17)$ \\\\ \n\\hline\nNNLO-II& $1426(13)$ & $-1015(15)$ & $764(17)$ & $-545(20)$\\\\\n\\hline \n\\end{tabular}\n\\caption{Values for the shape parameters $v_i$, $i=7,\\ldots,10$ in units of fm$^{2i-1}$. \nFor the meanings of the rows see Table~\\ref{table:vs3s1a}.\n\\label{table:vs3s1b}}\n\\end{center}\n\\end{table}\n\nThe values for the shape parameters $v_i$, $i=2,\\ldots,6$, are given in Table~\\ref{table:vs3s1a} and \nfor $i=7,\\ldots,10$ in Table~\\ref{table:vs3s1b}. Up to \nour knowledge the values of the shape parameters with $i>5$ were not given before. We detailed in \nAppendix \\ref{appen:vs} the numerical method that allows us to perform the appropriate \nderivatives up to so high order.\\footnote{For example in Ref.~\\cite{swart} it is stated that their numerical \nset up is not precise enough to calculate  $v_6$ and that it already casts doubts about the numerical \n accuracy for $v_5$.}  \n We could  have also given shape parameters of even higher orders\n within a numerical precision of a few per cent, \nbut this is skipped because its apparent little relevance in practice.\n One can appreciate the numerical precision in the calculation of the shape parameters by considering the errors in \n Tables~\\ref{table:vs3s1a} and \\ref{table:vs3s1b} for the NLO results, which entirely  correspond to  the numerical accuracy.\n Notice that for the highest shape parameter shown,  $v_{10}$, its relative error is 1.5$\\%$, \njust slightly worse than for $v_9$ with a relative error of 1.1\\%. \nWe then see that by increasing the order of the shape parameter \nthe numerical accuracy only worsens little by little. \nMorever,  the errors at NNLO take into account additionally \nthe variation in the results from the different sets of $c_i$'s employed and the dependence in the \ninput for starting the iterative process.\n For the shape parameters with large order, $i\\geq 5$,  their absolute values increase \ntypically as ${\\cal O}(1\/M_\\pi)^{2i-1}$, which is the expected behavior for long-range interactions\n mediated by OPE. It is clear from Table~\\ref{table:vs3s1a} that the shape parameters $v_i$, $i=2,\\ldots,5$ \npredicted by the NNLO-I calculation (third row) are typically\n closer to the values of Refs.~\\cite{swart,thesis} than those at NLO (second row). \nThis is a positive feature indicating a well-behaved expansion of the results obtained by \napplying the $N\/D$ method with the discontinuity $\\Delta(A)$ expanded in BChPT.\n\n\nAccording to  the power counting for the subtraction constants, Eq.~\\eqref{summarypwc}, at NNLO it is appropriate\n to consider twice-subtracted DRs. \nFor the $^3S_1-{^3D_1}$ system this implies to take into account two more free parameters for the $^3S_1$ \nwave and one more for the mixing partial wave.\n The three parameters for the $^3S_1$ wave are fixed in terms \nof the experimental values of $k_d^2$, $r_t$ and $a_t$.\nThe DR for the $^3D_1$ wave is the same as in Eq.~\\eqref{3sd1_a}. \n The twice-subtracted DRs taken now regarding the $^3S_1$ partial wave are\n\\begin{align}\nD_{11}(A)&=1-\\frac{A}{k_d^2}-\\nu_1^{(11)} \\,A(A-k_d^2) g_{11}^{(d)}(A,0;1)\n-\\nu_2^{(11)} \\,A(A-k_d^2) g_{11}(A,k_d^2)\\nonumber\\\\\n&+\\frac{A(A-k_d^2)}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^2} \ng_{11}^{(d)}(A,k^2;2)~,\\nonumber\\\\\nN_{11}(A)&=\\nu_1^{(11)}+\\nu_2^{(11)}\\,A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^2(k^2-A)}~,\\nonumber\\\\\n\\nu_1^{(11)}&=-\\frac{4\\pi a_t}{m}~,\\nonumber\\\\\n\\nu_2^{(11)}&=\\frac{\\nu_1^{(11)}}{\\nu_1^{(11)} \\,k_d^2\\, g_{11}(0,k_d^2)-1}\\left\\{\n\\frac{1}{k_d^2}+a_t\\Bigg(\n\\frac{4 k_d^2}{m}\\int_0^\\infty dq^2\\frac{\\nu_{11}(q^2)-\\rho(q^2)}{(q^2)^2(q^2-k_d^2)}+\\frac{1}{\\sqrt{-k_d^2}}\n-\\frac{r_t}{2}\\Bigg)\\right. \\nonumber\\\\\n&\\left. +\\frac{k_d^2}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^2}g_{11}(k_d^2,k^2)\n\\right\\}~.\n\\label{2subs.3s1}\n\\end{align}\n\nFor the mixing partial wave the DRs are \n\\begin{align}\nD_{12}(A)&=1-\\frac{A}{k_d^2}-\\nu_2^{(12)} A(A-k_d^2)g_{12}(A,k_d^2)\n+\\frac{A(A-k_d^2)}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{12}(k^2)D_{12}(k^2)}{(k^2)^2}g_{12}^{(d)}(A,k^2;2)~,\\nonumber\\\\\nN_{12}(A)&=\\nu_2^{(12)} A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{12}(k^2)D_{12}(k^2)}{(k^2)^2(k^2-A)}~,\n\\label{2subs.mix}\n\\end{align}\n The results obtained by solving the IEs of Eqs.~\\eqref{2subs.3s1}, \\eqref{2subs.mix} and Eq.~\\eqref{3sd1_a} \nwith $\\ell_{22}=2$ are denoted in the following by NNLO-II and correspond to the (red) hatched areas with crossed lines \nin Fig.~\\ref{fig:3sd1_a}. \nIt turns out that we cannot obtain a solution of the resulting IE for $D_{12}(A)$ by \nimplementing any arbitrary value for $\\nu_2^{(12)}$.\n We have further checked this statement by employing the following expression for $\\nu_2^{(12)}$,\n\\begin{align}\n\\nu_{2}^{(12)}&=\\frac{\\Theta}{2\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^2}~.\n\\end{align}\nHere, the integral is the same as in Eq.~\\eqref{aep.predicted}, so that if we take $ \\Theta=1$ we would simply \nrewrite the IE of Eq.~\\eqref{3sd1_a} in terms of twice-subtracted DRs. Then, we  vary $\\Theta$ and whenever we find \na meaningful solution the obtained value for $a_\\varepsilon=m\\nu_2^{(12)}\/2\\pi$ is always basically the same,  \n$a_\\varepsilon\\simeq 1.30~M_\\pi^{-3}$.\n In our opinion this difficulty in our approach to reproduce the value for $a_\\varepsilon$ \n that follows from the Nijmegen PWA, Eq.~\\eqref{avarepsilon}, casts doubts on this number. \n  Notice that  the calculated values for $\\epsilon_1$ at low momentum, \ne.g. for $\\sqrt{A}\\lesssim 100$~MeV, lie on top of the curve for the Nijmegen PWA results\n as shown in the third panel of Fig.~\\ref{fig:3sd1_a} by the coincident hatched and filled areas that overlap the Nijmegen PWA line. \n  The phase shifts and $\\epsilon_1$  are quite similar to the NNLO-I results in terms of just one free parameter.\nNevertheless,  the $^3S_1$ phase shifts for  NNLO-II are closer to the Nijmegen PWA ones  \nat lower energies, but the change for this S-wave by going from once- to twice-subtracted DRs\n is much less notorious than in the case of the partial wave $^1S_0$, discussed in Sec.~\\ref{1s0}. \nWe can also see in the fourth row of\n Table~\\ref{table:eta} that the NNLO-II values for $\\eta$ and $N_p^2$ are compatible \nwith those of  Ref.~\\cite{swart}, which is quite remarkable.\n The value for $a_\\varepsilon$ mentioned above is shown in \nthe last column of the same table.\n The shape parameters are shown in the forth rows of Tables~\\ref{table:vs3s1a} \nand \\ref{table:vs3s1b}, where we observe \na better agreement with the numbers given in Ref.~\\cite{swart} for \n $v_4$ and $v_5$ than for $v_2$ and $v_3$. The variation of the values\n between NNLO-I and NNLO-II for the higher order shape parameters allows us to guess \nin a  conservative way the systematic uncertainty affecting their calculation. \n\nOn the other hand, we would like  to elaborate further on the fact that at NNLO the results for the $^3D_1$ phase \nshifts do not still offer a good reproduction of the Nijmegen PWA ones, being even worse than those obtained \nat NLO.\n In Ref.~\\cite{epe04} one can find a discussion on the difficulties arisen in their calculation \n because of the large values of the  NLO $\\pi N$ \ncounterterms, namely $c_3$ and $c_4$, in order to reproduce simultaneously the $D$ and $F$ waves \nwithin the Weinberg scheme using the NNLO  chiral potential  \ncalculated in dimensional regularization. \n Considering this observation \n we obtain that when all the $c_i=0$ our NNLO result \nfor $\\delta_2$ is then essentially the same as the NLO one in Fig.~\\ref{fig:3sd1_a}, \ncorresponding to the (magenta) dot-dashed line. \n In view of this, we study now the influence in the results by including one more subtraction in the DRs for $^3D_1$ \nwith the aim of determining whether this worsening \nis an effect that can be counterbalanced in a natural way at ${\\cal O}(p^4)$. \n In this way we use the same twice-subtracted DRs for $^3S_1$ and the mixing partial wave \ngiven in Eqs.~\\eqref{2subs.3s1} and \\eqref{2subs.mix}, respectively, while  \n the following three-time subtracted DRs are used for the $^3D_1$\n\\begin{align}\nD_{22}(A)&=1-\\frac{A}{k_d^2}+\\delta_3^{(22)} A(A-k_d^2)\n-\\nu_3^{(22)} A(A-k_d^2)^2\\frac{\\partial g_{22}^{(d)}(A,0;2)}{\\partial k_d^2}\\nonumber\\\\\n&+\\frac{A(A-k_d^2)^2}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{22}(k^2)D_{22}(k^2)}{(k^2)^3} \n\\frac{\\partial g_{22}^{(d)}(A,k^2;3)}{\\partial k_d^2}~,\\nonumber\\\\\nN_{22}(A)&=\\nu_3^{(22)} A^2+\\frac{A^3}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{22}(k^2)D_{22}(k^2)}{(k^2)^3(k^2-A)}~,\n\\label{3d1.extra}\n\\end{align}\nwith two additional subtraction constants  $\\delta_3^{(22)}$ and $\\nu_3^{(22)}$. \nConsidering the results  obtained from the twice-subtracted DRs for all the waves in the system\n $^3S_1-{^3D_1}$, and denoting by  $\\hat{D}_{22}(A)$ the function $D_{22}(A)$ obtained then, we have \nthe following predictions for the subtraction constants $\\delta_3^{(22)}$ and $\\nu_3^{(22)}$, \n\\begin{align}\n\\nu_3^{\\mathrm{pred}}&=\\frac{1}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{22}(k^2)\\hat{D}_{22}(k^2)}{(k^2)^3}~,\\nonumber\\\\\n\\delta_3^{\\mathrm{pred}}&=\\frac{1}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{22}(k^2) \\hat{D}_{22}(k^2)}{(k^2)^2}g_{22}^{(d)}(k^2,k_d^2;2)~.\n\\label{nudelta2pre}\n\\end{align}\nThe numerical values that stem from the previous expressions are \n$\\delta_3^{\\mathrm{pred}}\\simeq 1~m_\\pi^{-4}$ and $\\nu_3^{\\mathrm{pred}}\\simeq -2.5~m_\\pi^{-6}$~. A fit \nto the $^3D_1$ phase shifts only requires to vary $\\nu_3^{(22)}$ around that value with the final \nresult $\\nu^{(22)}_3= -2.05(5)$~$m_\\pi^{-6}$, while \n$\\delta^{(22)}_3$ stays put.\n Then, it is only necessary a relatively small change of around 20\\% in \n $\\nu_3^{(22)}$ \n from the one predicted by the twice-subtracted DRs in Eq.~\\eqref{nudelta2pre} in order to end with a much better\n reproduction of the \n$^3D_1$ phase shifts that is compatible with the Nijmegen PWA, as shown \n by the hatched areas with (gray) parallel lines in Fig.~\\ref{fig:3sd1_a} (denoted as \nNNLO-III results).\n Since the reproduction of \nthe $^3S_1$ phase shifts and mixing angle $\\epsilon_1$ is the same as the one obtained already in terms of the \ntwice-subtracted DRs, the so-called NNLO-II results, we do not show them \nnor the values for the other parameters given  in Tables~\\ref{table:eta}, \\ref{table:vs3s1a} and \\ref{table:vs3s1b}, \nthat would be also basically coincident with the NNLO-II ones in these tables. \n\n We now elaborate on the difference between the value of $\\nu_3^{(22)}$ fitted and the one predicted, $\\nu_3^{\\mathrm{pred}}$. \nAccording to the power counting of Sec.~\\ref{nschpt}, cf. Eq.~\\eqref{summarypwc}, $\\nu_3^{(22)}={\\cal O}(p^{-1})$\n in our present NNLO calculation. If we consider that this difference is an effect that stems from the ${\\cal O}(p^4)$ \ncontributions to $\\Delta(A)$, which are not considered here yet, one would have that  \n$\\delta\\nu_3 \\equiv \\nu_3^{(22)}-\\nu_3^{\\mathrm{pred}}\\simeq 0.6~M_\\pi^{-6} \n={\\cal O}(p^0)$. It also follows then that nominally\n $\\delta \\nu_3\/\\nu_3^{\\mathrm{pred}}={\\cal O}(p)$ and taking into account the \n numerical values\n\\begin{align}\n\\frac{\\delta \\nu_3}{\\nu_3^{\\mathrm{pred}}}=0.23\\sim \\frac{M_\\pi}{\\Lambda}~,\n\\end{align}\nwe can estimate that $\\Lambda\\sim 4 M_\\pi$, which is \n similar to the estimate of $\\Lambda$ obtained in Sec.~\\ref{1s0} \nfor  the  $^1S_0$ partial wave.\n As a result, $\\delta \\nu_3$ is consistent with a naturally sized ${\\cal O}(p^4)$ effect.\n\nThe fact that the matrix of limiting values\n\\begin{align}\nM_{ij}=\\lim_{A\\to -\\infty}\\frac{\\Delta_{ij}(A)}{(-A)^{3\/2}}\n\\label{mij.3s1}\n\\end{align}\nhas two negative eigenvalues  is certainly related with the possibility of obtaining \nmeaningful DRs with only one free parameter as first obtained in this section. \nWe base this statement on the necessity condition of Ref.~\\cite{gor2013} \nin order to obtain meaningful once-subtracted DRs for $\\lambda<0$,\n a condition also introduced in Sec.~\\ref{nschpt}. \nIndeed, since the mixing between different partial waves is very small \nthese eigenvalues are given in good approximation by $M_{11}$ and $M_{22}$; this rule applies \nindeed  not only to the $^3S_1-{^3D_1}$ coupled waves but to any other one.\n\n\\section{Coupled $^3P_2-{^3F_2}$ waves}\n\\label{pf2w}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3p2.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3f2.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e2.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From top to bottom and left to right: Phase shifts for $^3P_2$, $^3F_2$ and the mixing angle  $\\epsilon_2$, respectively.\nThe (red) hatched areas  correspond to the NNLO results and the (cyan) filled bands are the leading Born approximation results.\n The NLO phase shifts and mixing angle are shown by the (magenta) dot-dashed lines and the LO ones are given by the \n (blue) dotted lines.\n The Nijmegen PWA phase shifts correspond to the (black) dashed lines.\n} \n\\label{fig:3pf2} }\n\\end{center}\n\\end{figure}\n\nWe dedicate this section to the study of the coupled wave system $^3P_2-{^3F_2}$. By direct computation \none has in this case that \n\\begin{align}\n\\lambda_{11}=\\lim_{A\\to -\\infty}\\frac{\\Delta_{11}(A)}{(-A)^{3\/2}}>0~,\n\\label{couplambdap}\n\\end{align}\nwhich requires one to consider DRs with more than one subtraction for the $^3P_2$ wave \\cite{gor2013}.\n Indeed, similarly to the $^3P_0$ and $^3P_1$ partial waves, studied in Secs.~\\ref{3p0} and \\ref{3p1}, respectively,  \nwe need to take at least three subtractions in the DRs for the $^3P_2$ wave in order to obtain \nstable and meaningful results.\n Thus,   we have the following  three-time subtracted DRs for the $^3P_2$ wave, \n\\begin{align}\n\\label{d.3p2}\nD_{11}(A)&=1+\\delta^{(11)}_2 A+\\delta^{(11)}_3 A(A-C)-\\nu^{(11)}_{2}\\frac{A(A-C)^2}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{11}(q^2)}{(q^2-A)(q^2-C)^2}\\nonumber\\\\\n&-\\nu^{(11)}_3\\frac{A(A-C)^2}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{11}(q^2)q^2}{(q^2-A)(q^2-C)^2}\\nonumber\\\\\n&+\\frac{A(A-C)^2}{\\pi}\\int_{-\\infty}^L dk^2\n\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^3} g_{11}(A,k^2,C;2)~,\\\\\n\\label{n.3p2}\nN_{11}(A)&=\\nu^{(11)}_2 A+\\nu^{(11)}_3 A^2+\\frac{A^3}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^3(k^2-A)}~.\n\\end{align}\nWith respect to the mixing and $^3F_2$  partial waves we use the standard formalism \n for the coupled waves  given in Eqs.~\\eqref{highdcc} and \\eqref{highncc} with $\\ell_{12}=2$ and $\\ell_{22}=3$, respectively. \nAs a result  2 and 3 subtractions are taken in order.\n\n As usual for the $P$ waves, we fix $\\nu_2^{(11)}=4\\pi a_V\/m$  by requiring the exact reproduction  of the $^3P_2$ scattering volume \nextracted from the Nijmegen PWA  \\cite{Stoks:1994wp},\n\\begin{align}\na_V=0.0964~M_\\pi^{-3}~,\n\\end{align}\nwhile $\\nu_3^{(11)}$ is fitted to the results of this PWA. \nRegarding the subtraction constants $\\delta_i^{(11)}$, $i=1,$~2, we  follow the principle of maximal smoothness in virtue of which we  fix $\\delta^{(11)}_{2}=0$\n and fit $D_{11}^{(1)}(-M_\\pi^2)$.\\footnote{In the following we use \n$D_{ij}^{p-2}(-M_\\pi^2)$ as free parameter in terms of which one can calculate $\\delta_{p}^{(ij)}$ from Eq.~\\eqref{taylor}.}  \n The resulting fitted values are: \n\\begin{align}\nD_{11}^{(11)}(-M_\\pi^2)&=0.025(5)~M_\\pi^{-2}~,\\nonumber\\\\\n\\nu_3^{(11)}&=0.155(5)~M_\\pi^{-6}~,\\\\\nD_{22}^{(11)}(-M_\\pi^2)&=0.011(4)~M_\\pi^{-2}~,\n\\end{align}\n with the interval of values reflecting the dependence on the $c_i$'s chosen.\n The free parameter associated with  the mixing wave is fixed to its pure\n perturbative value, cf. Sec.~\\ref{hpw}, $D_{12}(-M_\\pi^2)=1$.\n\nAll in all the resulting phase shifts are shown by the (red) hatched areas in Fig.~\\ref{fig:3pf2}. \nThere we see a clear improvement at NNLO in the reproduction \nof the $^3P_2$ phase shifts  compared with the results at NLO, given by the (magenta) dot-dashed lines, so that now the (red) hatched area overlaps the Nijmegen PWA phase shifts.\n The $^3F_2$ phase shifts and mixing angle $\\epsilon_2$ are reproduced with a similar quality to that already achieved at NLO.\n We also give by the (cyan) filled bands the results obtained by the leading Born approximation, Eq.~\\eqref{deltab}, \nwith $\\Delta(A)$ calculated at NNLO. Due to the fact that the latter diverges as $(-A)^{3\/2}$ for $A\\to-\\infty$ at least two \nsubtractions have to be taken in the DR for $N_B(A)$, Eq.~\\eqref{eq.nborn}. \nThis is immediately accomplished for the $D$ and \nhigher partial waves but for a $P$-wave with $\\ell=1$ one needs to include one \nextra subtraction.\n In particular, for our present case we use Eq.~\\eqref{n.3p2} with $D_{11}(A)\\to 1$ and with $\\Delta_{11}(k^2)$\n restricted to its two-nucleon irreducible contributions, with the subtraction constants $\\nu^{(11)}_{2}$ and $\\nu^{(11)}_{3}$ \ntaking the same values as discussed before. \nWe see that our full results provide a clear improvement in the reproduction of the Nijmegen PWA phase shifts and \nmixing angle  with respect to the Born approximation. \n One should mention that the Born approximation  phase shifts for \n$^3F_2$ and $^3F_3$ have a striking resemblance to the full NNLO results of Ref.~\\cite{thesis} obtained within the Weinberg \nscheme. \n We have obtained this improvement without dismissing the strength of the TPE at NNLO, as advocated in Ref.~\\cite{epe04}. \n This makes that our full results are not so much sensitive to the particular set of $c_i$'s taken as \npreviously thought in the literature from the results of Refs.~\\cite{thesis,epe04}.\n\n\\section{Coupled $^3D_3-{^3G_3}$ waves}\n\\label{dg3w}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3d3.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3g3.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e3.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From top to bottom and left to right: Phase shifts for $^3D_3$, $^3G_3$ and the mixing angle  $\\epsilon_3$, in order.\nThe (red) hatched areas correspond to the NNLO results and the (cyan) filled bands are the leading order Born approximation.\n The NLO results are shown by the (magenta) dot-dashed lines and the LO ones are given by the \n (blue) dotted lines. The Nijmegen PWA phase shifts correspond to the (black) dashed lines.}\n\\label{fig:3dg3} }\n\\end{center}\n\\end{figure}\n\nFor the study of the $^3D_3-{^3G_3}$ coupled waves we follow the formalism for coupled waves, Eqs.~\\eqref{highdcc} and \n\\eqref{highncc}, with $\\ell_{11}=2$, $\\ell_{12}=3$ and $\\ell_{22}=4$, so that $\\ell_{ij}$ subtractions are taken in the DRs for the  \ncoupled wave $ij$. \n Regarding  the free parameters we follow the principle of maximal smoothness, although for the mixing wave \nthe subtraction constants take their pure perturbative values. So that  we fit to data \n$D_{11}(-M_\\pi^2)$ and $D_{22}^{(2)}(-M_\\pi^2)$, with the resulting values:\n\\begin{align}\nD_{11}(-M_\\pi^2)&=0.90(5)~,\\nonumber\\\\\nD_{22}^{(2)}(-M_\\pi^2)&=-0.09(1)~M_\\pi^{-4}~,\n\\label{fit.3dg3}\n\\end{align}\n The interval of values in Eq.~\\eqref{fit.3dg3} reflect the dependence on the set of  values considered for the $c_i$'s. \nThe resulting phase shifts are shown by the (red) hatched areas in Fig.~\\ref{fig:3dg3}.\n Importantly at NNLO the phase shifts for the $^3D_3$ wave   follow  closely the Nijmegen PWA phase shifts \nso that a remarkable improvement is obtained in comparison with both \nthe NLO and Born results.\n  Notice that this is accomplished without any need of dismissing the strength of\n TPE as directly obtained from the NLO $\\pi N$ amplitudes. \nWe have been able to improve the situation by taking into account the subtraction constant $ \\delta_2$ or $D_{11}(-M_\\pi^2)$, \nwhose presence is required by the nonperturbative unitarity implementation\\footnote{In more general terms, by generating the analytical properties \nassociated with the RHC while respecting unitarity in the full amplitudes.} at NNLO, cf. Eq.~\\eqref{summarypwc}.\n  We also observe a good reproduction of the Nijmegen PWA results for the waves $^3G_3$ and $\\epsilon_3$, \nwhich are already well reproduced at NLO \\cite{gor2013} as shown  by the (magenta) dot-dashed lines.\n \n\n\n\n\\section{Coupled $^3F_4-{^3H_4}$ waves}\n\\label{gh4w}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3f4.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3h4.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e4.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From top to bottom and left to right: Phase shifts for $^3F_4$, $^3H_4$ and the mixing angle  $\\epsilon_4$, in order.\nThe (red) hatched areas correspond to the NNLO results and the (cyan) filled ones to the leading\nBorn approximation.\n The NLO results are shown by the (magenta) solid line and the LO ones are given by the \n (blue) dotted lines. \nThe Nijmegen PWA phase shifts are given by the (black) dashed lines.}\n\\label{fig:3fh4} }\n\\end{center}\n\\end{figure}\n\nThe discussion of the $^3F_4-{^3H_4}$ coupled-wave system follows the  standard formalism for coupled waves, Eq.~\\eqref{highdcc} and \\eqref{highncc},\n with $\\ell_{11}=3$, $\\ell_{12}=4$ and $\\ell_{22}=5$. The free parameters are then fitted to data according to the \n principle of maximal smoothness. However, for $^3H_4$ and the mixing partial wave there is no improvement in the reproduction of data\n with respect to the situation in which the pure perturbative values are taken, so that \nat the end we only have to fit $D_{11}^{(1)}(-M_\\pi^2)$  to the Nijmegen PWA results. \nThe fitted value is\n\\begin{align}\nD_{11}^{(1)}(-M_\\pi^2)&= -0.009(3)~M_\\pi^{-2}~.\n\\label{free.fh4}\n\\end{align}\n The resulting phase shifts and mixing angle are shown by the (red) hatched areas  in  Fig.\\ref{fig:3fh4},\nwith the width of the band reflecting the dependence on values for the $\\pi N$ NLO counterterms. \nOne can observe  a clear improvement in the description of the $^3F_4$ phase shifts compared with the results from OPE (blue dotted lines), NLO \n(magenta dot-dashed lines) and leading Born approximation (cyan filled areas). \n Similarly to the $^3D_3$ wave in the previous section, \n this improvement is related with the effect of the subtraction constant $\\delta_3^{(11)}$ which  \nis not directly related with an improvement in  the \ncalculation of $\\Delta_{11}(A)$, and hence of the $NN$ potential. \nLet us recall that the subtraction constants $\\delta_p^{(ij)}$ arise because of the rescattering process that the \n$N\/D$  method allows to treat in a clear and well-defined way, \novercoming the obscurities that still remain in the literature associated with the use of the cutoff regularized Lippmann-Schwinger  with \na higher-order $NN$ potential. \nFor the  mixing angle $\\epsilon_4$ the quality in the reproduction of data is  similar to that obtained \nby the other approximations just quoted. \nHowever, for the  $^3 H_4$ phase shifts the outcome at NNLO is a bit worse than at \nNLO and OPE, though one should also notice the tiny values for the $^3H_4$ phase shifts so that this discrepancy \nis certainly small  in absolute value. \nWe have also checked that it cannot be removed by releasing the other subtraction constants $\\delta_p^{(22)}$, with \n$p=2$, 3 and 4. \nLikely, the origin of this difference in the $^3H_4$ phase shifts  between our full results and the Nijmegen PWA \ncan be tracked back to the change in the leading Born approximation once the ${\\cal O}(p^3)$ two-nucleon irreducible \ncontributions are included in $\\Delta_{22}(A)$.\n\n\n\\section{Coupled $^3G_5-{^3I_5}$ waves}\n\\label{gi5w}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3g5.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3i5.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e5.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From top to bottom and left to right: Phase shifts for $^3G_5$, $^3I_5$ and the mixing angle  $\\epsilon_5$, in order.\nThe (red) hatched areas correspond to the NNLO results and the filled ones to the leading\nBorn approximation.\n The NLO results are shown by the (magenta) dot-dashed line and the LO ones are given by the \n (blue) dotted lines.\n The Nijmegen PWA phase shifts are shown by the (black) dashed lines.}\n\\label{fig:3gi5} }\n\\end{center}\n\\end{figure}\n\n\nThe standard formalism for coupled waves with high angular momentum, Eqs.~\\eqref{highdcc} and \\eqref{highncc}, is followed here \nwith  $\\ell_{11}=4$, $\\ell_{12}=5$ and $\\ell_{22}=6$.\n The application of the  principle of maximal smoothness \nto fit the free parameters  provides a good reproduction of the Nijmegen PWA phase shifts \\cite{Stoks:1994wp}.\\footnote{For $5\\leq J\\leq 8$ the Nijmegen PWA phase shifts \\cite{Stoks:1994wp} are those obtained from the $NN$ potential model of Ref.~\\cite{obe}.}\nThe range of values obtained for the free parameters $D_{11}^{(2)}(-M_\\pi^2)$ and $D_{22}^{4}(-M_\\pi^2)$ is\n\\begin{align}\nD_{11}^{(2)}(-M_\\pi^2)&=-0.0025(5)~M_\\pi^{-4}~,\\nonumber\\\\\nD_{22}^{(4)}(-M_\\pi^2)&=-0.0125(5)~M_\\pi^{-8}~,\n\\label{free.gi5}\n\\end{align}\nwhile basically the same results are obtained for any  $D_{12}^{(3)}(-M_\\pi^2)\\leq 0~M_\\pi^{-6}$. \n The results are shown in Fig.\\ref{fig:3gi5} by the (red) hatched areas whose widths take into account the \nuncertainty from the set of $c_i$'s   taken and some numerical noise from the iterative process. \n A clear improvement results in the description of the $^3G_5$ phase shifts compared with the OPE (blue dotted lines), NLO (magenta dot-dashed lines)\n and leading Born approximation results (cyan filled areas). \nIt is worth stressing that this partial wave cannot be well reproduced\neven at NNNLO in the Weinberg potential scheme  neither by \nkeeping a finite value for the three-momentum cutoff entering \nin the solution of the Lippmann-Schwinger equation \\cite{epe042}, \nnor by  sending it to $\\infty$ as in Ref.~\\cite{zeoli}.\n A similar situation occurs too  \nfor the leading Born approximation results at NNLO, as shown by the (cyan) filled area in the first panel, \n a result also obtained in Ref.~\\cite{epe04}. \nEven more, the modification of the TPE mechanism proposed in this reference by making use of the \nso-called spectral-function regularization  is inoperative here to provide \nan improvement in the Born approximation results.\n A similar problem was also observed in the perturbative calculation at NNNLO in \nRef.~\\cite{entem}.\n From ours results this is not surprising because the improvement in the reproduction of the  \nNijmegen PWA phase shifts for the $^3G_5$ wave is accomplished through the subtraction constant $\\delta_4^{(11)}$. \nThis constant is directly related to the $NN$ rescattering (from which the final function $D_{11}(A)$ stems nonperturbatively) \nand not to the $NN$ potential or $\\Delta_{22}(A)$.\n In the case of the mixing angle $\\epsilon_5$ and the \n$^3I_5$ phase shifts  there is a slight worsening in the reproduction of Nijmegen \nPWA compared with the NLO ones, but still our results run very close to the Nijmegen PWA ones.\n\n\n\\begin{table}\n\\begin{center}\n{\\small\n\\begin{tabular}{|l|l|l|}\n\\hline\nWave & Type of DRs & Parameters \\\\\n\\hline\n$^1S_0$ & 1DR & $\\nu_1=30.69$ \\\\\n       &  2DR & $\\nu_1=30.69$~,~$\\nu_2=-23(1)$, $\\delta_2=-8.0(3)$ \\\\\n\\hline\n$^3P_0$ & 3DR & $\\nu_2=1.644$~,~$\\delta_2=2.82(5)$~,~$\\delta_3=0.18(6)$ \\\\\n\\hline\n$^3P_1$ & 3DR & $\\nu_2=-1.003$~,~$\\delta_2=2.7(1)$~,~$\\delta_3=0.47(3)$ \\\\\n\\hline\n$^1 P_1$ & 2DR & $\\nu_2=-1.723$~,~$\\delta_2=0.4(1)$ \\\\\n\\hline\n$^1D_2$ &  LTS & $D^{(1)}(0)=0.07(1)$  \\\\\n\\hline\n$^3D_2$ & LTS &   $D^{(1)}(0)=-0.017(3)$  \\\\\n\\hline\n$^1F_3$ & LTS & $D^{(2)}(0)=0.057(3)$  \\\\\n\\hline\n$^3F_3$ & LTS & $D^{(2)}(0)= 0.035(5)$  \\\\\n\\hline\n$^1G_4$ & LTS & $D^{(3)}(0)=-0.014(2)$  \\\\\n\\hline\n$^3G_4$ & LTS & $D^{(3)}(0)=-0.055(5)$  \\\\\n\\hline\n$^1H_5$ & LTS & $D^{(4)}(0)=0.156$  \\\\\n\\hline\n$^3H_5$ & LTS & $D^{(4)}(0)=0.066$  \\\\\n\\hline\n$^3S_1-{^3D_1}$ & $1$DR $^3S_1$, 2DR $^3D_1$, mixing & $E_d$ \\\\\n               & 2DR all & $a_t$, $r_t$, $E_d$ \\\\\n               & 2DR $^3S_1$, mixing, 3DR $^3D_1$ &  $a_t$, $r_t$, $E_d$, $\\nu_3^{(22)}=-2.05(5)$ \\\\  \n\\hline\n$^3P_2-{^3F_2}$ & 3DR for $^3P_2$ and LTS for the others & $\\nu^{(11)}_2=0.178$~,~$D^{(1)}_{11}(-M_\\pi^2)=0.025(5)$~,~\n$\\nu_3^{(11)}=0.155(5)$\\\\\n               &  &  $D_{22}(-M_\\pi^2)=0.011(4)$\\\\\n\\hline\n$^3D_3-{^3G_3}$ & LTS & $D_{11}(-M_\\pi^2)=0.90(5)$~,~$D^{(2)}_{22}(-M_\\pi^2)=-0.09(1)$  \\\\\n\\hline\n$^3F_4-{^3H_4}$ & LTS & $D_{11}^{(1)}(-M_\\pi^2)=-0.009(3)$  \\\\\n\\hline\n$^3G_5-{^3I_5}$ & LTS & $D_{11}^{(2)}(-M_\\pi^2)=-0.0025(5)$~,~$D_{22}^{(4)}(-M_\\pi^2)=-0.0125(5)$ \\\\\n\\hline\n\\end{tabular} }\n\\caption[pilf]{\\protect {  We give in the columns from left to right, in order,  \nthe partial wave, the type of DRs employed to study it\n and the values for the free parameters involved.  }\n\\label{tab:allparam} }\n\\end{center}\n\\end{table}\n\nFinally, we give in Table~\\ref{tab:allparam} the values of the free parameters employed in the different partial waves according \nto the type of DRs employed, which is  indicated in the second column.\n This is done by following the notation, already introduced in Ref.~\\cite{gor2013}, \n $m$DR with $m=1,2,\\ldots$, and it should be read as $m$-time subtracted DR.\n For the higher $NN$ partial waves we use the abbreviation  LTS to indicate that \n$\\ell$ (or $J$ for the mixing partial waves) subtractions are taken  to satisfy the threshold behavior, following \nthe standard formalism explained in Sec.~\\ref{hpw}.\n According to the \nprinciple of maximal smoothness only the highest derivative $D^{(n)}(C)$ \nrelated to the subtraction constants in $D(A)$ is not fixed to its perturbative value \n(1 for $n=0$ and 0 for $n\\neq 0$) and released, \nif appropriate.  The units correspond to appropriate powers of $M_\\pi^2$, although they are not explicitly shown. \nThere is a proliferation of free parameters for the $P$ waves because for them $\\lambda>0$, Eqs.~\\eqref{unlambdap} \nand \\eqref{couplambdap}, so that, except for the $^1P_1$ wave,  three-time-subtracted DRs are needed.\n This could be a specific feature for the NNLO calculation of $\\Delta(A)$ that has to be investigated for\n higher-orders.\\footnote{If then $\\lambda<0$ \none would need to invoke  less free parameters for the $P$ waves than in Table~\\ref{tab:allparam}.}\n \n\n\\section{Conclusions}\n\\label{conc}\n\nWe have discussed in this paper the application of the $N\/D$ method when its dynamical input, namely, \nthe imaginary part of the $NN$ partial waves along the LHC, is calculated in ChPT up to NNLO.  \nIt then comprises OPE, leading and subleading two-nucleon irreducible TPE and once-iterated OPE ~\\cite{peripheral}. \nWe have obtained a quite good reproduction of the Nijmegen PWA phase shifts and mixing angles, in better agreement \nthan the one achieved in the previous lower order studies at LO \\cite{paper1,paper2} and NLO \\cite{gor2013}.   In particular, our NNLO results are able to reproduce \n the phase shifts for the triplet waves with $\\ell_{11}=J-1$, $^3P_2$, $^3D_3$, $^3F_4$ \nand $^3G_5$,  while at NLO they were not properly accounted for.\nWe do not need to modify the NNLO two-nucleon irreducible diagrams (or chiral $NN$ potential) in order to obtain such a good agreement \nwith the Nijmegen PWA, contrary to common wisdom.  \nThe point that stems from our study is that one should perform in a well-defined way\n the iteration of diagrams along the RHC, which are responsible \nfor unitarity and analyticity attached to this cut, rather than reshuffling  the $NN$ potential with contributions  \nfrom higher orders. \nIn this respect, the use of DRs allows one to perform the iteration of two-nucleon intermediate states independently of regulator. We have also compared our full results for the higher partial waves with the Born approximation. From this comparison, as well as from \nthe direct study \nof the importance of the different contributions of $\\Delta(A)$ to the dispersive integrals,  \nit follows that  the $NN$  $D$ waves cannot be treated perturbatively.\n\nIt is also worth remarking that up to the order studied here we reproduce the long-range correlation between the \neffective ranges and the scattering lengths for the $NN$ $S$ waves when only once-subtracted DRs are applied. \nIn this way one can predict values for the $S$-wave effective ranges in agreement with experiment up to around a $10 \\%$. \nWe have also elaborated a chiral power counting for the subtraction constants, so that twice-subtracted DRs \nare appropriate when $\\Delta(A)$ is calculated at NLO and NNLO.  \nFrom these considerations it turns out also that the chiral power expansion is made over a scale $\\Lambda\\sim 400$~MeV.  \nOne should consider further  the impact of higher orders in $\\Delta(A)$, which are partially calculated already in \nthe literature, as an interesting extension of the present work in order to settle the applicability of the $N\/D$\nmethod to $NN$ scattering in ChPT with a high degree of accurateness.\n\n\n\n\\section*{Acknowledgments}\n This work is partially funded by the grants MINECO (Spain) and ERDF (EU), grant FPA2010-17806 and the Fundaci\\'on S\\'eneca 11871\/PI\/09.\n We also thank the financial support from the EU-Research Infrastructure\nIntegrating Activity\n ``Study of Strongly Interacting Matter\" (HadronPhysics2, grant n. 227431)\nunder the Seventh Framework Program of EU and   \nthe Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, we study  the following problem raised by Coman-Guedj-Zeriahi.\n\\begin{prob}[\\cite{CGZ13}]\\label{prob:1}\nLet $(X,\\omega)$ be a compact K\\\"ahler manifold of complex dimension $n$, equipped with a K\\\"ahler metric $\\omega$. Let $V\\subset X$ be a complex submanifold of complex dimension $k>0$. Is the following holds\n\\begin{align*}\n\\mbox{Psh}(V,\\omega|_V)=\\mbox{Psh}(X,\\omega)|_V?\n\\end{align*}\n\\end{prob}\n\nRecently, there are many progress towards this problem. \n\\begin{itemize}\n\\item When $\\omega$ is a Hodge metric and $\\varphi$ is a smooth quasi-psh function on $V$, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>0$, then Problem \\ref{prob:1} has a positive answer by Schumacher \\cite{Sch98}.\n\\item When $\\omega$ is a Hodge metric, then Problem \\ref{prob:1} has a positive answer by Coman-Guedj-Zeriahi \\cite{CGZ13}, and when $\\omega$ is a K\\\"ahler metric and $\\varphi$ is a smooth quasi-psh function on $V$, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>0$, then Problem \\ref{prob:1} has a positive answer in the same paper \\cite{CGZ13}.\n\\item When  $\\omega$ is a K\\\"ahler metric and $\\varphi$ is a  quasi-psh function  on $V$, which has analytic singularities, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>\\epsilon\\omega|_V$ for some $\\epsilon>0$, there is a quasi-psh function $\\Phi$ on $X$, such that $\\Phi|_V=\\varphi$ and $\\omega+\\sqrt{-1}\\partial\\bar\\partial \\Phi>\\epsilon'\\omega$ on $X$ by Collins-Tosatti \\cite{CT14}.\n\\item When  $\\omega$ is a K\\\"ahler metric and $\\varphi$ is a  quasi-psh function with arbitrary singularity  on $V$, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>\\epsilon\\omega|_V$ for some $\\epsilon>0$. Suppose that $V$ has a holomorphic tubular neighborhood in $X$, then there is a quasi-psh function $\\Phi$ on $X$, such that $\\Phi|_V=\\varphi$ and $\\omega+\\sqrt{-1}\\partial\\bar\\partial \\Phi>\\epsilon'\\omega$ on $X$ for some $\\epsilon'>0$ by Wang-Zhou \\cite{WZ20}.\n\\end{itemize}\n\nThe main theorem of this paper is as follows.\n\\begin{thm}\\label{thm:main}\nLet $(X,\\omega)$ be a compact K\\\"ahler manifold of complex dimension $n$, equipped with a K\\\"ahler metric $\\omega$. Let $V\\subset X$ be a complex submanifold of complex dimension $k>0$. Let $\\varphi$ be  a  quasi-psh function with arbitrary singularity  on $V$, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>\\epsilon\\omega|_V$ for some $\\epsilon>0$.  Suppose that there is a open neighborhood $U$ of $V$ in $X$, and  a holomorphic retraction $\\pi:U\\rightarrow V$. Then  there is a quasi-psh function $\\Phi$ on $X$, such that $\\Phi|_V=\\varphi$ and $\\omega+\\sqrt{-1}\\partial\\bar\\partial \\Phi>\\epsilon'\\omega$ on $X$  for some $\\epsilon'>0$.\n\\end{thm}\n\n\\begin{rem}\nThe main theorem is a slightly stronger than the result in \\cite{WZ20}, by weakening the assumption that $V$ has a holomorphic tubular neighborhood structure in $X$ to the assumption that $V$ has a holomorphic retraction structure in $X$. By a holomorphic retraction, we mean that there is an  open neighborhood $U$ of $V$ in $X$, and  a holomorphic map $\\pi:U\\rightarrow V$, such that $\\pi|_V:V\\rightarrow V$ is the identity map.  Without the holomorphic tubular neighborhood structure,  we need to compute the complex Hessian of the square of the distance function to $V$ on $X$. The main advantage of this generalization is that there many examples of compact  K\\\"ahler  manifolds  which are  not necessarily projective.\n\\end{rem}\nWe also consider the extension of K\\\"ahler currents in a big class.\n\\begin{thm}\nLet $(X,\\omega)$ be a compact K\\\"ahler manifold of complex dimension $n$, and $V\\subset X$ be a complex submanifold of positive dimension. Suppose that $V$ has a holomorphic retraction structure  in $X$.\nLet $\\alpha\\in H^{1,1}(X,\\mathbb R)$ be a big class and $E_{nK}(\\alpha)\\subset V$. \nThen any  K\\\"ahler current in  $\\alpha|_V$ is the restriction of a K\\\"ahler current in $\\alpha$.\n\\end{thm}\n\n\\subsection*{Acknowledgement}\nThe second author would like to thank G. Hosono and T. Koike for helpful discussions, and especially T. Koike for  sharing the note \\cite{Ko21}.\n\n\n\n\n\n\n\n\\section{Complex Hessian of square of  distance to  a complex  submanifold}\\label{sect:dist}\nWe follow  Matsumoto's notations in \\cite{Ma}.\nLet $(M,g)$ be a $C^\\infty$ Rimannian manifold of dimension $n$.\nFor $x,y\\in M$, we denote by $\\delta(x,,y)$ the distance between $x$ and $y$ induced by the metric $g$.\n\nIt is known that for any $p\\in M$, there is an open coordinate neighborhood $U$ of $p$,\nchoose a coordinate $x_1,x_2,\\cdots,x_n$ on $U$, with $x(p)=0$ and\n$g_{ij}=g(\\frac{\\partial}{\\partial x_i},\\frac{\\partial}{\\partial x_j}),\\, i,j=1,2,\\cdots,n$.\nFor $v=(v_1,v_2,\\cdots,v_n)\\in\\mathbb{R}^n$,\nwe may view $v\\in T_xM$ as $\\sum_{j=1}^{n}v_j\\frac{\\partial}{\\partial x_j}|_x$.\nWe may shrink $U$ if necessary, there is an open neighborhood $B\\subset \\mathbb{R}^n$ of $0$,\nsuch that $\\Phi(x,v)=(x,\\exp_x(v))$\nis bijection from $U\\times B$ to $\\Phi(U\\times B)$, both $\\Phi$ and $\\Phi^{-1}$ are $C^\\infty$.\nAs $\\Phi(x,0)=(x,x)$, and from the property of exponential mapping $y=\\exp_x v$,\nwe can get\n\\begin{equation}\\label{eq1}\nJ\\Phi(0,0)\n=\\left[\n\\begin{array}{cc}\nI & 0\\\\\nI & I\n\\end{array}\n\\right].\n\\end{equation}\n\nAs $\\Phi(U\\times B)$ is an open neighborhood of $(p,p)$,\nwe may take an open set $V\\subset U$, such that $p\\in V$, and  $\\Phi(U\\times B)\\supset V\\times V$.\nWrite $(x,v(x,y))=\\Phi^{-1}(x,y)$, then\n$$y=\\exp_x(v(x,y)),\\, \\delta(x,y)^2=\\sum_{i,j=1}^n g_{ij}(x)v_i(x,y)v_j(x,y)$$\nand from (\\ref{eq1}), we have\n\\begin{equation}\\label{eq2}\nv(0,0)=0, \\,\\, \\frac{\\partial v_i}{\\partial y_j}(0,0)=-\\frac{\\partial v_i}{\\partial x_j}(0,0)=\\delta_{ij},\\, 1\\leq i,j\\leq n.\n\\end{equation}\nLet $S\\in M$ be a $C^\\infty$ submanifold of $M$ with $dim S=k$, $0\\leq r<n$, define\n$$d(x)=\\delta(x,S)=inf\\{\\delta(x,y):y\\in S\\}, \\; x\\in M.$$\nLet $h(x)=d(x)^2$, it is  proved in \\cite{Ma} that $h(x)$ is $C^\\infty$ in a tube neighborhood of $S$.\nAs we need do calculation on $h$, and for completeness, we introduce Matsumoto's proof of smoothness.\n\nGiven $p\\in S$, we can choose a coordinate $(U,x=(x_1,x_2,\\cdots,x_n))$ around $p$, such that\n$x(p)=0$, $S\\cap U=\\{x_{k+1}=\\cdots=x_n=0\\}$, and\n\\begin{equation}\\label{eq3}\ng_{ij}(0)=g(\\frac{\\partial}{\\partial x_i},\\frac{\\partial}{\\partial x_j})(0)=\\delta_{ij},\\, i,j=1,2,\\cdots,n.\n\\end{equation}\nTake a small neighborhood $V\\subset U$ of $p$, such that $d(x)=\\delta(x,S\\cap U)$ for any $x\\in V$.\nLet $$f(x,y)=\\delta(x,y)^2=\\sum_{i,j=1}^{n}g_{ij}(x)v_i(x,y)v_j(x,y)$$\nand $$F_\\mu(x,y)=\\frac{\\partial f}{\\partial y_\\mu}(x,y),\\; \\mu=1,2,\\cdots,k$$\nfor $x\\in V$ and $y=(y_1,\\cdots,y_k,0,\\cdots,0)\\in S\\cap U$. Notice that $F_\\mu(0,0)=0, \\mu=1,2,\\cdots,k$.\n\n\nFrom (\\ref{eq2}), (\\ref{eq3}), we can get\n\\begin{equation}\\label{eq4}\n\\begin{split}\n&\\frac{\\partial F_\\mu}{\\partial y_\\nu}(0,0)\\\\\n=&2\\sum_{i,j=1}^{n}g_{ij}(0)\\frac{\\partial v_i}{\\partial y_\\mu}(0,0)\\frac{\\partial v_j}{\\partial y_\\nu}(0,0)\\\\\n=&2\\delta_{\\mu\\nu}.\n\\end{split}\n\\end{equation}\n\nTherefore, from the implicit function theorem, we can find a neighborhood $V_0\\subset V$ of $p$,\nso that each $x\\in V_0$ has a unique solution $y=y(x)\\in S\\cap U$ of equations $F_\\mu(x,y)=0, \\mu=1,2,\\cdots,n$,\n$y(0)=0$ and $y=y(x)$ is $C^\\infty$ on $V_0$. As for each $x\\in V_0$, there is at least one point $y\\in S\\cap U$, \nsuch that $d(x)=\\delta(x,S\\cap U)=\\delta(x,y)$. Therefore, the point $y$ is uniquely determined by $x$ and \nit must coincide to $y(x)$ because $f(x,y)=\\delta(x,y)^2$ is minimal at $y=y(x)$ for each $x$.\n\n\nHence $$h(x)=\\delta(x,S\\cap U)^2=\\delta(x,y(x))^2=f(x,y(x))$$\nfor $x\\in V_0$.\n\nFrom $F_\\mu(x,y(x))=0$, taking the partial derivatives of this equation, we can get \n$$\\frac{\\partial F_\\mu}{\\partial x_j}(0,0)+\\sum_{\\nu=1}^{k}\\frac{\\partial F_\\mu}{\\partial y_\\nu}(0,0)\n\\frac{\\partial y_\\nu}{\\partial x_j}(0)=0.$$ \nCombining with  equation (\\ref{eq4}),\nwe can get $\\frac{\\partial y_\\mu}{\\partial x_j}(0)=-\\frac{1}{2}\\frac{\\partial F_\\mu}{\\partial x_j}(0,0)$\nfor $\\mu=1,2,\\cdots,k$ and $j=1,2,\\cdots,n$.\n\nFrom (\\ref{eq2}), (\\ref{eq3}), we have\n\n\\begin{equation*\n\\begin{split}\n&\\frac{\\partial F_\\mu}{\\partial x_l}(0,0)\\\\\n=&\\sum_{i,j=1}^{n}g_{ij}(0)\\frac{\\partial v_i}{\\partial y_\\mu}(0,0)\\frac{\\partial v_j}{\\partial x_l}(0,0)+\n\\sum_{i,j=1}^{n}g_{ij}(0)\\frac{\\partial v_i}{\\partial x_l}(0,0)\\frac{\\partial v_j}{\\partial y_\\mu}(0,0)\\\\\n=&-2\\delta_{\\mu l}.\n\\end{split}\n\\end{equation*}\n\nHence, \n\\begin{equation}\\label{eq d}\n\\frac{\\partial y_\\mu}{\\partial x_j}(0)=\\delta_{\\mu j}\n\\; \\text{for}\\; \\mu=1,2,\\cdots,k  ; j=1,2,\\cdots,n.\n\\end{equation}\n\nLet $a_j(x)=v(x,y(x)), j=1,2,\\cdots,n$, then $a_j(0)=0$, and \n\\begin{equation}\\label{eq6}\n\\begin{split}\n\\frac{\\partial a_j}{\\partial x_l}(0)&=\\frac{\\partial v_j}{\\partial x_l}(0,0)+\n\\sum_{\\mu=1}^{k}\\frac{\\partial v_j}{\\partial y_\\mu}(0,0)\\frac{\\partial y_\\mu}{\\partial x_k}(0)\\\\\n&=\\sum_{\\mu=1}^{k}\\delta_{j\\mu}\\delta_{\\mu k}-\\delta_{jk}.\n\\end{split}\n\\end{equation}\nAs $h(x)=\\sum_{i,j=1}^{n}g_{ij}(x)v_i(x,y(x))v_j(x,y(x))=\\sum_{i,j=1}^{n}g_{ij}(x)a_i(x)a_j(x)$, from (\\ref{eq6}), then\n\\begin{equation}\\label{eq 7}\n\\begin{split}\n\\frac{\\partial^2h}{\\partial x_s\\partial x_t}(0)\n&=2\\sum_{i,j=1}^{n}g_{ij}(0)\\frac{\\partial a_i}{\\partial x_s}(0)\\frac{\\partial a_j}{\\partial x_t}(0)\\\\\n&=\n\\begin{cases}\n0 \\quad\\quad s \\; \\text{or} \\; t\\leq k \\\\\n2\\delta_{st} \\quad s,t>k\n\\end{cases}\n\\end{split}\n\\end{equation}\n\nNow we let $(X,\\omega)$ be a compact Hermitian manifold with  a Hermitian metric $\\omega$. Let $g$ be the Riemannian metric on $X$ induced by $\\omega$. Let $V\\subset X$ be a complex submanifold of complex dimension $r>0$.  Fix any  $p\\in V$. There is a holomorphic coordinate  $(U, z=(z_1,\\cdots, z_k, z_{k+1}, \\cdots,z_n))$ centered at $p$ in $X$, such that $U\\cap V=\\{z_{k+1}=\\cdots=z_n=0\\}$, and $g_{ij}(0)=\\delta_{ij}$ for $i,j=1,\\cdots, 2n$, here we write $z_i=x_{2i-1}+\\sqrt{-1}x_{2i}$. Since  $$\\frac{\\partial}{\\partial z_i}=\\frac{1}{2}(\\frac{\\partial}{\\partial x_{2i-1}}-\\sqrt{-1}\\frac{\\partial}{\\partial x_{2i}}), \\frac{\\partial}{\\partial\\bar z_i}=\\frac{1}{2}(\\frac{\\partial}{\\partial x_{2i-1}}+\\sqrt{-1}\\frac{\\partial}{\\partial x_{2i}}),$$\nwe get that \n\\begin{align}\\label{equ: chessian}\n\\frac{\\partial^2h}{\\partial z_i\\partial \\bar z_j}=\\frac{1}{4}\\left(\\frac {\\partial^2h}{\\partial x_{2i-1}\\partial x_{2j-1}}       +\\frac{\\partial^2h}{\\partial x_{2i}\\partial x_{2j}}-\\sqrt{-1}\\frac{\\partial^2h}{\\partial x_{2j}\\partial x_{2j-1}} +\\sqrt{-1}\\frac{\\partial^2h}{\\partial x_{2j-1}\\partial x_{2j}}    \\right).\n\\end{align}\n\nCombining (\\ref{eq 7}) and (\\ref{equ: chessian}), we obtain the following \n\n\\begin{prop}\\label{prop:hessian}Let $(X,\\omega)$ be a complex $n$-dimensional  Hermitian manifold with a Hermitian metric $\\omega$. Let $V\\subset X$ be a complex submanifold of complex dimension $k$. Let $p\\in V$ be an arbitrarily fixed point in $V$, then there is a holomorphic coordinate chart  $(U,z=(z_1,\\cdots,z_k,z_{k+1},\\cdots, z_n))$ centered at $p$ such that $U\\cap V=\\{z_{k+1}=\\cdots=z_n=0\\}$ and \n\t\\begin{align*}\n\t\\frac{\\partial^2h}{\\partial z_i\\partial \\bar z_j}(0)=\n\t\\begin{cases}\n\t0 \\quad\\quad i \\; \\text{or} \\; j\\leq k \\\\\n\t2\\delta_{ij} \\quad i,j>k.\n\t\\end{cases}\n\t\\end{align*}\n\t\\end{prop}\n\n\\section{Proof of the main theorem}\n\nIn this section, we give the proof of Theorem \\ref{thm:main}. The idea of the proof is similar with that in \\cite{WZ20}. The main difference lies in the construction of the local uniform extension. For the sake of completeness, we give the detailed proof.\n\n\\begin{lemma}[\\cite{BK07, WZ20}]\\label{key lemma}\n\tLet $\\varphi$ be a quasi-psh function on a compact Hermitian  manifold $(X,\\omega)$,  such that $\\omega+\\sqrt{-1}\\partial\\bar{\\partial}\\varphi\\geq \\varepsilon\\omega$ and $\\varphi<C<0$.\n\tThen there is a  sequence of smooth functions $\\varphi_m$ and a decreasing sequence $\\varepsilon_m>0$ converging to $0$, satisfying the following\n\t\\begin{itemize}\n\t\t\\item [(a)] $\\varphi_m\\searrow \\varphi$;\n\t\t\\item [(b)]$\\omega+\\sqrt{-1}\\partial\\bar{\\partial}\\varphi_m\\geq (\\varepsilon-\\varepsilon_m)\\omega$;\n\t\t\\item[(c)] $\\varphi_m\\leq -\\frac{C}{2}$.\n\t\\end{itemize}\n\\end{lemma}\n\n\\begin{lemma}[c.f. \\cite{DP04}]\\label{reference function}\n\tThere exists a function $F:X\\rightarrow [-\\infty, +\\infty)$ which is smooth on $X\\setminus V$, with logarithmic singularities along $V$, and such that $\\omega+\\sqrt{-1}\\partial\\bar{\\partial}F\\geq \\varepsilon \\omega$ is  a K\\\"{a}hler current on $X$.\n\tBy subtracting a large constant, we can make that $F<0$ on $X$.\n\\end{lemma}\n\n\nLet $T=\\omega|_V+\\sqrt{-1}\\partial\\bar{\\partial}\\varphi\\geq \\varepsilon\\omega|_V$ be the given K\\\"{a}hler current  in the K\\\"{a}hler class $[\\omega|_V]$, where $\\varphi$ is a strictly $\\omega|_V$-psh function.\nBy subtracting a large constant, we may assume that $\\sup_V \\varphi<-C$ for some positive constant $C$.\n\nBy  Lemma \\ref{key lemma}, we have that there is a sequence of non-increasing  smooth strictly $\\omega|_V$-psh functions $\\varphi_{m}$ on $V$,\nand a decreasing sequence of positive numbers $\\varepsilon_m$ such that as $m\\rightarrow \\infty$\n\\begin{itemize}\n\n\t\\item $\\varphi_{m} \\searrow \\varphi$;\n\t\\item$\\omega|_V+\\sqrt{-1}\\partial\\bar{\\partial}\\varphi_m> \\frac{\\varepsilon}{2}\\omega|_V$;\n\t\\item $\\varphi_m\\leq -\\frac{C}{2}$.\n\\end{itemize}\n\nWe say a smooth strictly $\\omega|_V$-psh function $\\phi$ on $V$ satisfies \\textbf{assumption $\\bigstar_{\\varepsilon, C}$}, if $\\omega|_V+\\sqrt{-1}\\partial\\bar{\\partial}\\phi>\\frac{\\varepsilon}{2}\\omega|_V$ and $\\phi<-\\frac{C}{2}$.\n\nNote that for all $m\\in \\mathbb{N}^+$, $\\varphi_m$ satisfy  \\textbf{assumption $\\bigstar_{\\varepsilon, C}$}.  In the following, we will extend all the $\\varphi_m$ simultaneously to non-increasing  strictly $\\omega$-psh functions on   the ambient manifold $X$.\n\n\\noindent\\textbf{Step1: Local uniform extensions of $\\varphi_m$ for all $m$.} Let $U\\subset X$ be an open neighborhood of $V$ and let $r:U\\rightarrow V$ be a holomorphic retraction. Let $\\phi$ be a function satisfying  \\textbf{assumption $\\bigstar_{\\varepsilon, C}$}.  Let $h$ be the square of the distance function , which is a smooth function defined in \\S\\ref{sect:dist}.\n We define \t\n\\begin{align*}\n\\bar{\\phi}:=\\phi\\circ r+Ah\n\\end{align*}\nwhere $A$ is a positive constant to be determined later. \n\nFix arbitrary $p\\in V$, choose a holomorphic coordinate chart $(W_p,z=(z_1,\\cdots,z_n))$ centered at $p$ and $W_p\\cap V=\\{z_{k+1}=\\cdots=z_n=0\\}$, $g_{i\\bar j}(0)=\\delta_{ij}$  and $W\\subset U$, where $\\omega=\\sum_{i,j=1}^ng_{i\\bar j}dz_i\\wedge d\\bar z_j$. Then on $W_p$, we have that \n\\begin{align*}\n\\bar{\\phi}(z):=(\\phi\\circ r)(z)+Ah(z).\n\\end{align*}\nNote that on $W_p$, \n\\begin{align*}\n\\omega+\\sqrt{-1}\\partial\\bar\\partial \\bar\\phi(z)&=(\\omega-r^*(\\omega|_V))+r^*(\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial\\phi)+A\\sqrt{-1}\\partial\\bar\\partial h\\\\\n&\\geq (\\omega-r^*(\\omega|_V))+\\varepsilon r^*(\\omega|_V)+A\\sqrt{-1}\\partial\\bar\\partial h\n\\end{align*}\nThe second inequality follows from the fact that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial\\phi\\geq \\varepsilon\\omega|_V$  on $V$ and $r$ is a holomorphic retraction map.  The key point is that the last term in above inequality  is independent of $\\phi$.\n\n\n\n\\begin{claim}\\label{claim: 1}There is  an open neighborhood  $W_p$ (independent of $\\phi$), of $p$ in $U$, and  positive constants $A>0$ and  $\\varepsilon'>0$ (independent of $\\phi$), such that on $W_p$,\n\t\n\\begin{align*}\n\\omega+\\sqrt{-1}\\partial\\bar\\partial \\bar\\phi\\geq \\frac{\\varepsilon'}{2}\\omega \\mbox{~~and~~} \\bar\\phi\\leq -\\frac{C}{4}.\n\\end{align*}\n\\end{claim}\n\\begin{proof}Under the local coordinate chosen as above, one can see that \n\t\\begin{align*}\n\t&r(z)=(r_1(z_1,\\cdots,z_n),\\cdots, r_k(z_1,\\cdots,z_n),0,\\cdots,0);\\\\\n&\tr(z_1,\\cdots, z_k,0,\\cdots, 0)=(z_1,\\cdots,z_k,0,\\cdots,0);\\\\\n&dr_i(z_1,\\cdots,z_k,0,\\cdots,0)=dz_i+\\sum_{k+1\\leq j\\leq n}\\frac{\\partial r_i}{\\partial z_j}(z_1,\\cdots, z_k,0\\cdots,0)dz_j.\n\t\\end{align*}\nSince $\\omega|_V=\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}(z_1,\\cdots,z_k,0,\\cdots,0)dz_i\\wedge d\\bar z_j$, it follows that at $(z_1,\\cdots, z_k,0,\\cdots,0)$, \n\\begin{align*}\nr^*(\\omega|_V)=&\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}(dz_i+\\sum_{k+1\\leq l\\leq n}\\frac{\\partial r_i}{\\partial z_l}dz_l)\\wedge(d\\bar z_j+\\sum_{k+1\\leq m\\leq n}\\frac{\\partial \\bar r_j}{\\partial \\bar z_m} d\\bar z_m)\\\\\n=&\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}dz_i\\wedge d\\bar z_j+\\sum_{1\\leq i\\leq k,k+1\\leq m\\leq n}\\sum_{1\\leq j\\leq k}g_{i\\bar j}\\frac{\\partial \\bar r_j}{\\partial \\bar z_m}dz_i\\wedge d\\bar z_m\\\\\n&+\\sum_{1\\leq j\\leq k,k+1\\leq l\\leq n}\\sum_{1\\leq i\\leq k}g_{i\\bar j}\\frac{\\partial  r_i}{\\partial \\bar z_l}dz_i\\wedge d\\bar z_l\n+\\sum_{k+1\\leq l,m\\leq n}\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}\\frac{\\partial  r_i}{\\partial  z_l}\\frac{\\partial  \\bar r_j}{\\partial \\bar z_m}dz_l\\wedge d\\bar z_m.\n\\end{align*}\n\nThus,  at $(z_1,\\cdots,z_k,0,\\cdots,0)$, we get the following\n\\begin{align*}\n\\omega+\\sqrt{-1}\\partial\\bar\\partial \\bar\\phi(z)\\geq &\\sum_{1\\leq i,j\\leq k}(\\varepsilon g_{i\\bar j}+Ah_{i\\bar j})dz_i\\wedge d\\bar z_j+\\sum_{1\\leq i\\leq k,k+1\\leq m\\leq n}(g_{i\\bar m}+Ah_{i\\bar m}+(\\varepsilon-1)\\sum_{1\\leq j\\leq k}g_{i\\bar j}\\frac{\\partial \\bar r_j}{\\partial \\bar z_m})dz_i\\wedge d\\bar z_m\\\\\n&+\\sum_{1\\leq j\\leq k,k+1\\leq l\\leq n}(g_{j\\bar l}+Ah_{j\\bar l}+(\\varepsilon-1)\\sum_{1\\leq i\\leq k}g_{i\\bar j}\\frac{\\partial  r_i}{\\partial \\bar z_l})dz_i\\wedge d\\bar z_l\\\\\n&+\\sum_{k+1\\leq i,j\\leq n}(g_{i\\bar j}+Ah_{i\\bar j}+(\\varepsilon-1)\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}\\frac{\\partial  r_i}{\\partial  z_l}\\frac{\\partial  \\bar r_j}{\\partial \\bar z_m})dz_i\\wedge d\\bar z_j.\n\\end{align*}\nSince $(g_{i\\bar j})_{1\\leq i,j\\leq k}$ is positive definite, from Proposition \\ref{prop:hessian}, we can see that when $A>0$ is sufficiently large (independent of $\\phi$), there is an open neighborhood $W_p$ (independent of $\\phi$), such that the conclusion of Claim \\ref{claim: 1} holds.\n\n\n\n\t\\end{proof}\n\nTo emphasis the uniformity, it is worth to point out again that the chosen of the open set $W_p$, and  the constant $\\varepsilon'$ is independent of $\\phi$, as long as $\\phi$ satisfies \\textbf{assumption $\\bigstar_{\\varepsilon,C}$}.\nWe call the above data $ (W_p,\\varepsilon',-\\frac{C}{4},\\bar\\phi)$ an \\textbf{admissible local extension} of $\\phi$.\n\n\nSince all the $\\varphi_m$ satisfy the same \\textbf{assumption $\\bigstar_{\\varepsilon,C}$}, thus near $p$, we can choose a \\textbf{uniform admissible local extension $ (W_p,A,\\varepsilon',-\\frac{C}{4},\\bar\\varphi_m)$} of $\\varphi_m$, for all $m\\in\\mathbb{N}^+$.\nSince $V$ is compact, one may choose an  open neighborhood $W$ of $V$ in $X$, and universal constants $A>0$ and $\\varepsilon'>0$, such that the functions $\\widetilde \\varphi_m:=\\varphi_m\\circ r+Ah$ are defined on $W$, such that $\\omega+i\\partial\\bar\\partial \\widetilde \\varphi_m\\geq \\varepsilon'\\omega$ on $W$ for all $m$. Since $\\{\\varphi_m\\}$ is a non-increasing sequence, one obtains that $\\{\\widetilde{\\varphi}_m\\}$ is a non-increasing sequence.\n\n\\noindent\\textbf{Step 2: Global extensions of $\\varphi_m$ for all $m$.} Up to shrinking, we may assume that $\\widetilde{\\varphi}_m$ are defined on the closure of $W$ for all $m\\in \\mathbb{N}^+$.\nLet $F$  be the quasi-psh function in Lemma \\ref{reference function}.\nNear $\\partial W$ (the boundary of $W$), the function $F$ is smooth, and $\\sup_{\\partial W}\\widetilde{\\varphi}_{1}=-C''$ for some positive constant $C''>0$.\nNow we choose a small positive $\\nu$, such that $\\inf_{\\partial W}(\\nu F)>-\\frac{C''}{2}$ and   $\\omega+i\\partial\\bar{\\partial}\\nu F\\geq\\varepsilon'\\omega$.\nThus $\\nu F >\\widetilde{\\varphi}_{1}\\geq \\widetilde{\\varphi}_m$ in a neighborhood of $\\partial W$ for all $m\\in \\mathbb{N}^+$, since $\\widetilde{\\varphi}_m$ is non-increasing.\nTherefore, we can finally define\n\\begin{align*}\n\\Phi_m=\\left\\{\n\\begin{array}{ll}\n\\max\\{\\widetilde{\\varphi}_m, \\nu F\\}, & \\hbox{on $W$;} \\\\\n\\nu F, & \\hbox{on $X\\setminus W$,}\n\\end{array}\n\\right.\n\\end{align*}\nwhich is defined on the whole of $X$. It  is easy to check that $\\Phi_m$ satisfies the following properties:\n\\begin{itemize}\n\t\\item $\\Phi_m$ is non-increasing in  $m$,\n\t\\item $\\Phi_m\\leq 0$ for all $m\\in \\mathbb{N}^+$,\n\t\\item $\\omega+i\\partial\\bar{\\partial}\\Phi_m\\geq \\varepsilon'\\omega$ for all $m\\in \\mathbb{N}^+$,\n\t\\item $\\Phi_m|_V=\\varphi_m$ for all $m\\in \\mathbb{N}^+$.\n\\end{itemize}\n\n\\noindent\\textbf{Step 3: Taking limit to complete the proof of Theorem \\ref{thm:main}.}\nFrom above steps, we get a sequence of non-increasing, non-positive strictly $\\omega$-psh functions $\\Phi_m$ on $X$. Then  either $\\Phi_m\\rightarrow -\\infty $ uniformly on $X$, or $\\Phi:=\\lim\\limits_m\\Phi_m\\in$ Psh$(X,\\omega)$.\nBut $\\Phi_m|_V=\\varphi_m\\searrow \\varphi\\not\\equiv -\\infty$, the first case will not appear.\nMoreover,   we can see that $\\Phi:=\\lim\\limits_m\\Phi_m$ is a strictly $\\omega$-psh function on $X$ from the property $\\omega+i\\partial\\bar{\\partial}\\Phi_m\\geq \\varepsilon'\\omega$ for all $m\\in \\mathbb{N}^+$, and $\\Phi|_V=\\lim\\limits_m\\Phi_m|_V=\\lim\\limits_m\\varphi_m=\\varphi$.\nIt follows that   $(\\omega+i\\partial\\bar{\\partial}\\Phi)|_V=\\omega|_V+i\\partial\\bar{\\partial}\\varphi$.\nThus we complete the proof of Theorem \\ref{thm:main}.\n\n\n\\begin{rem} By similar arguments as in \\cite{WZ20}, we can get the following extension results for K\\\"ahler currents in a big class.\n\t\\begin{thm}\n\t\tLet $(X,\\omega)$ be a compact K\\\"ahler manifold of complex dimension $n$, and $V\\subset X$ be a complex submanifold of positive dimension. Suppose that $V$ has a holomorphic retraction structure  in $X$.\n\t\tLet $\\alpha\\in H^{1,1}(X,\\mathbb R)$ be a big class and any of the irreducible components of $E_{nK}(\\alpha)$ either does not intersect with $V$, or is contained in $V$. \n\t\tThen any  K\\\"ahler current in  $\\alpha|_V$ is the restriction of a K\\\"ahler current in $\\alpha$.\n\t\\end{thm}\n\t\\end{rem}\n\n\\section{Examples}\n\nIn \\cite{HK20}, Hosono-Koike point out that in Nakayama's example and Zariski's example, the submanifold  have holomorphic tubular neighborhood structure in  the ambient manifold, thus have  holomorphic retraction structure. \n\n\\noindent{\\textbf{Product manifold.}} Let $Y_1$ and $Y_2$ be two compact K\\\"ahler manifold and set $X:=Y_1\\times Y_2$. Fix an arbitrary point $p\\in Y_2$, let $V=Y_1\\times p$, then the natural map $\\pi:Y_1\\times Y_2\\rightarrow Y_1\\times p$  serves as a holomorphic retraction map. \n\n\nAn interesting  example of non-product manifold, communicated to us by Koike \\cite{Ko21}, is  the following famous example of Serre.\n\n\\noindent{\\textbf{Serre's example.}} Let $X:=\\mathbb P_{[x;y]}\\times \\mathbb C_z\/\\sim$, where $\\tau\\in \\mathbb H$ with $\\mathbb H$ be the upper half plane, and $$([x;y],z)\\sim ([x;y+x],z+1)\\sim ([x;y+\\bar\\tau\\cdot x],z+\\tau).$$\nLet $V:=\\{x=0\\}\\subset X$ as a submanifold of $X$ which is obviously isomorphic to the elliptic curve $\\mathbb C\/\\langle 1,\\tau\\rangle$.  It is easy to check  that  the projection map $\\pi: X\\rightarrow \\mathbb C\/\\langle 1,\\tau\\rangle=:V$ is a holomorphic retraction. It can also be verified that $V$ does not have a holomorphic tubular neighborhood structure in $X$.\n\\begin{rem}\n\tIn \\cite{Ko21}, Koike gives a very interesting  proof of Theorem \\ref{thm:main} for Serre's example,  which however seems not applicable to general case treated in this paper.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{{\\bf Introduction}}\n\n\nThe loop space $LM$ of a manifold $M$ appears frequently in mathematics and\n mathematical physics. In this paper, using an infinite dimensional version of\n Chern-Simons theory associated to the Wodzicki residue for\n pseudodifferential operators ($\\Psi{\\rm DO}$s), we develop a  computable theory of secondary\n characteristic classes  on the tangent\n bundle to loop spaces.   We apply these secondary classes to distinguish circle actions\n on $S^2\\times S^3$, and we prove that $\\pi_1({\\rm Diff} (S^2\\times S^3))$ is infinite.  To our knowledge, these applications are the first examples of nonzero Wodzicki-type characteristic classes. \n\n\nSince Chern-Weil and Chern-Simons theory are geometric, it is necessary\nto understand connections and curvature on loop spaces.  A  Riemannian metric\n$g$ \non $M$ induces a family of metrics $g^s$ on $LM$ parametrized by a Sobolev\nspace parameter $s \\geq 0$, where $s=0$ gives the usual $L^2$ metric, and the\nsmooth case is a kind of limit as $s\\to\\infty.$  Thus we think of\n$s$ as a regularizing parameter, and pay attention to  the parts of the theory which\nare independent of $s$. \n\nIn Part I, we compute the connection and curvature  for the Levi-Civita\nconnection for $g^s$ for $s>\\frac{1}{2}$.  The \nclosed form expressions obtained for the Levi-Civita connection for general $LM$ \nextend Freed's results for loop groups \\cite{Freed}.  The connection and \ncurvature forms take values in\nzeroth order $\\Psi{\\rm DO}$s acting on a trivial bundle over $S^1$. \nFor Wodzicki-Chern-Simons classes, we only need\nthe principal and subprincipal symbols for these\nforms, which we calculate.\n\n\n\n\nIn Part II, we develop a theory of Chern-Simons classes on loop spaces.\nThe structure group for the Levi-Civita connection for\n$(LM, g^s)$ is the set of invertible zeroth order $\\Psi{\\rm DO}$s, so we need\n invariant polynomials on the corresponding Lie algebra.  The naive choice is\nthe standard polynomials $\\operatorname{Tr}(\\Omega^k)$ of the curvature $\\Omega = \\Omega^s$,\nwhere Tr is the operator trace.  However, $\\Omega^k$ is zeroth order\nand hence not trace class, and in any case the operator trace\nis impossible to compute in general.  Instead, as in \\cite{P-R2} we use the \nWodzicki residue, the only trace on the full\nalgebra of $\\Psi{\\rm DO}$s.  \nFollowing Chern-Simons\n \\cite{C-S} as much as possible, we build a theory of\nWodzicki-Chern-Simons (WCS) classes, which gives classes in $H^{2k-1}(LM^{2k-1})$ associated to partitions of $k$.\n\nThere are two main differences from the finite\ndimensional theory. The absence of a Narasimhan-Ramanan universal connection\ntheorem means that we do not have a theory of differential characters\n\\cite{Ch-Si}.  However, since we have a family of connections on $LM$, we can define real valued, not just ${\\mathbb R}\/{\\mathbb Z}$-valued, WCS classes. \n\n In contrast to the operator trace, the Wodzicki residue is locally\n computable, so we can write explicit expressions for the WCS classes.\nIn particular, we can see how the WCS classes depend on the Sobolev parameter $s$, and \nhence define a  ``regularized\" or $s$-independent WCS classes.  \nThe local expression also yields some vanishing results for WCS classes.  More importantly,\nwe produce a nonvanishing \nWCS class on $L(S^2\\times S^3).$  This leads to the topological results described in the first paragraph.\n\nFor related results on characteristic classes on infinite rank bundles with a group of $\\Psi{\\rm DO}$s as structure group, see \n \\cite{lrst, P-R2}.\n \n\n\\medskip\nThe paper is organized as follows.  Part I treates the family of metrics $g^s$ on $LM$\nassociated to $(M,g)$.  \\S2 discusses connections associated to $g^s.$  After some preliminary material,\nwe compute the Levi-Civita connection for $s=0$ (Lemma \\ref{lem:l2lc}), $s=1$\n(Theorem \\ref{old1.6}),  $s\\in {\\mathbb Z}^+$ (Theorem \\ref{thm:sinz}), and general \n$s>\\frac{1}{2}$ (Theorem \\ref{thm25}).   These connections allow us to track how the geometry\nof $LM$ depends on $s$.  \n\nBoth the Levi-Civita and $H^s$ connections have connection and curvature forms taking \nvalues in $\\Psi{\\rm DO}$s of order zero.  In \\S3, we compute the symbols of these forms needed in Part II.  In \\S4, we show that our results extend Freed's on loop groups \\cite{Freed}.\n\nPart II covers Wodzicki-Chern-Simons classes.  In \\S5, we review the finite dimensional\nconstruction of Chern and Chern-Simons classes, and use the Wodzicki residue to define Wodzicki-Chern (WC) and WCS classes (Definition \\ref{def:WCS}).  We prove the necessary vanishing\nof the WC classes for mapping spaces (and in particular for $LM$) in Proposition\n\\ref{prop:maps}.  In Theorem \\ref{thm:5.5}, we give the explicit local expression for the\nrelative WCS class $CS_{2k-1}^W(g)\\in H^{2k-1}(LM^{2k-1})$ associated to the trivial\npartition of $k$.  We then define the regularized or $s$-independent WCS class.  \nIn Theorem \\ref{WCSvan}, we give a vanishing result\nfor WCS classes.  \n\nIn particular, the WCS class which is the analogue of the classical dimension three Chern-Simons class vanishes on loop spaces of \n $3$-manifolds, so we look for nontrivial examples on $5$-manifolds.\n In \\S\\ref{dimfive}, we use a Sasaki-Einstein\nmetric constructed in \\cite{gdsw} to produce a nonzero WCS class  $CS_5^W\\in\nH^5(L(S^2\\times\nS^3)).$  We prove $CS_5^W\\neq 0$ by an exact computer calculation showing\n $\\int_{[a^L]} CS_5^W \\neq 0$, where\n$[a^L]\\in H_5(LM)$ is a cycle associated to a simple\ncircle action on $S^2\\times S^3.$  From this\nnonvanishing, we conclude both that  the circle action is not \nsmoothly homotopic to the trivial action and that $\\pi_1({\\rm Diff} (S^2\\times S^3))$ is infinite.\nWe expect other similar results in the future.  \n\n\n\nOur many discussions with Sylvie Paycha are gratefully\nacknowledged. We also thank Kaoru Ono and Dan Freed for pointing out errors in  previous versions of \nthe paper.\n\n\n\n\n\\bigskip\n\n\n\\large\n\\noindent {{\\bf Part I. The Levi-Civita Connection on the Loop Space $LM$}}\n\\normalsize\n\n\\bigskip\n\nIn this part, we compute the Levi-Civita connection on $LM$\nassociated to a Riemannian metric on $M$ and a Sobolev parameter $s=0$ \nor $s>\\frac{1}{2}.$  The standard $L^2$ metric on $LM$ is the case $s=0$, and otherwise we avoid technical issues by assuming that $s$ is greater than the critical exponent $\\frac{1}{2}$ for analysis on bundles over $S^1.$  \n In \\S\\ref{LCconnection}, the\nmain results are  Lemma \\ref{lem:l2lc}, Theorem \\ref{old1.6}, \nTheorem \\ref{thm:sinz}, and Theorem \\ref{thm25},\nwhich compute the Levi-Civita connection for $s =0$, $s=1$, \n$s\\in {\\mathbb Z}^+$,  and general $s >\\frac{1}{2},$ respectively.\n\n\nIn\n\\S3, we compute the relevant symbols of the connection one-forms and the\ncurvature two-forms.  In \\S4, we compare our results with work of Freed\n\\cite{Freed} on loop groups.    \n\n\n\n\\section{{\\bf The Levi-Civita Connection for Sobolev Parameter $s\\geq 0$}}\n\\label{LCconnection}\n\nThis section covers background material and computes the Levi-Civita\nconnection on $LM$ for Sobolev parameter $s=0$ and $s>\\frac{1}{2}$.  \nIn \\S2.1, we review\nmaterial on $LM$, and in \\S2.2 we review pseudodifferential operators and the\nWodzicki residue.  In \\S2.3, we give the crucial computations of the Levi-Civita connections\nfor $s=0,1$.\nThis computation is extended to $s\\in {\\mathbb Z}^+$ in \\S2.4, and to general $s>\\frac{1}{2}$ in\n\\S2.5.    In \\S2.6, we discuss how the geometry of $LM$ forces an extension of\nthe structure group of $LM$ from a gauge group to a group of bounded\ninvertible $\\Psi{\\rm DO}$s.\n     \n\\subsection{{\\bf Preliminaries on $LM$}}\n\n${}$\n\\medskip\n\nLet $(M, \\langle\\ ,\\ \\rangle)$ \nbe a closed, connected,  oriented Riemannian $n$-manifold with  loop space $LM\n= C^\\infty(S^1,M)$ of smooth loops. \n$LM$ is a smooth infinite dimensional Fr\\'echet manifold, but it is\n technically simpler \nto work \nwith the smooth Hilbert manifold $H^{s'}(S^1,M)$ of loops in some Sobolev class $s' \\gg 0,$\nas we now recall. For $\\gamma\\in LM$, the formal\ntangent space $T_\\gamma LM$ is \n$\\Gamma(\\gamma^*TM)$, the space\n of smooth sections of the pullback bundle $\\gamma^*TM\\to\nS^1$.   The actual tangent space of $H^{s'}(S^1, M)$ at $\\gamma$ is \n$H^{s'-1}(\\gamma^*TM),$     the sections of $\\gamma^*TM$ of Sobolev class $s'-1.$\nWe will fix $s'$ and use $LM, T_\\gamma LM$ for $H^{s'}(S^1, M), H^{s'-1}(\\gamma^*TM)$, respectively.\n\nFor each $s>1\/2,$ we can complete $\\Gamma(\\gamma^*TM\\otimes {\\mathbb C})$\n with respect to the Sobolev inner product\n \\begin{equation}\\label{eq:Sob1}\n\\langle X,Y\\rangle_{s}=\\frac{1}{2\\pi}\\int_0^{2\\pi} \\langle(1+\\Delta)^{s}\nX(\\alpha),Y(\\alpha)\n\\rangle_{\\gamma (\\alpha)}d\\alpha,\\  X,Y\\in \\Gamma(\\gamma^*TM).\n\\end{equation}\nHere $\\Delta=D^*D$, with $D=D\/d\\gamma$ the covariant derivative along\n$\\gamma$. (We use this notation instead of the classical $D\/dt$ to keep track\nof $\\gamma$.)\nWe need the complexified pullback bundle $\\gamma^*TM\\otimes {\\mathbb C}$, denoted from now on\njust as $\\gamma^*TM$, in order to apply the\npseudodifferential operator  $(1+\\Delta)^{s}.$\nThe construction of $(1+\\Delta)^{s}$ is reviewed in\n\\S\\ref{pdoreview}.  \nWe denote this completion by  $H^{s'}(\\gamma^*TM)$.   We can consider the\n$s$ metric on $TLM$ for any $s\\in {\\mathbb R}$, but we will only consider \n$s=0$ or $1\/2 < s\\leq s'-1.$\n\n\nA small real neighborhood $U_\\gamma$ \nof the zero section in $H^{s'}(\\gamma^*TM)$ is a\ncoordinate chart near $\\gamma\\in LM$ \nvia the pointwise exponential map\n\\begin{equation}\\label{pointwiseexp}\n\\exp_\\gamma:U_\\gamma\n\\to L M, \\ X \\mapsto \n\\left(\\alpha\\mapsto \\exp_{\\gamma(\\alpha)} X(\\alpha)\\right).  \n\\end{equation}\nNote that the domain of the exponential map is not contained in $T_\\gamma LM.$\nThe differentiability of the transition functions $\\exp_{\\gamma_1}^{-1}\\cdot\n\\exp_{\\gamma_2}$ is proved in\n\\cite{E} and \\cite[Appendix A]{Freed1}.\nHere $\\gamma_1, \\gamma_2$ are close loops in the sense that\na geodesically convex neighborhood of $\\gamma_1(\\theta)$ contains\n$\\gamma_2(\\theta)$ and vice versa for all $\\theta.$\nSince \n$\\gamma^*TM$\nis (noncanonically) isomorphic to the trivial bundle ${\\mathcal R} =\nS^1\\times {\\mathbb C}^n\\to S^1$, \nthe model space for $LM$ is the set of \n$H^{s'}$ sections of this trivial bundle.  \nThe $s$ metric is a weak Riemannian metric for $s<s'$ in the sense that the topology induced on $H^{s'}(S^1, M)$ by the exponential map applied to $H^{s}(\\gamma^*TM)$ is weaker than the $H^{s'}$ topology.\n\n\nThe complexified tangent bundle\n$TLM$ has\ntransition functions $d(\\exp_{\\gamma_1}^{-1} \n\\circ \\exp_{\\gamma_2})$.  Under\nthe isomorphisms $\\gamma_1^*TM \\simeq {\\mathcal R} \\simeq\n\\gamma_2^*TM$, the transition functions lie in the gauge group\n${{\\mathcal G}}({\\mathcal R})$, so this is the structure group of $TLM.$\n\n\n\n\n\n\n\\subsection{{\\bf Review of $\\Psi{\\rm DO}$ Calculus}}\\label{pdoreview}\n${}$\n\\medskip\n\nWe recall the construction of classical\npseudodifferential operators ($\\Psi{\\rm DO}$s)\n on a closed \nmanifold $M$ from \\cite{gilkey, See}, assuming knowledge of $\\Psi{\\rm DO}$s on\n ${\\mathbb R}^n$ (see e.g. \\cite{hor, shu}).   \n \n\n\n\n\n A linear operator $P:C^\\infty(M)\\to C^\\infty(M)$\nis a $\\Psi{\\rm DO}$ of  order $d$ if for\nevery open chart $U\\subset M$ and functions $\\phi,\\psi\\in C_c^\\infty(U)$,\n$\\phi P\\psi$ is a $\\Psi{\\rm DO}$ of order $d$ on ${\\mathbb R}^n$, where we do not\n distinguish between $U$ and its diffeomorphic image in ${\\mathbb R}^n$.\n Let\n  $\\{U_i\\}$ be a finite cover of $M$\nwith subordinate partition of unity\n  $\\{\\phi_i\\}.$  Let $\\psi_i\\in C^\\infty_c(U_i)$ have $\\psi_i \\equiv 1$ on\n  supp$(\\phi_i)$  and set $P_i = \\psi_iP\\phi_i.$\nThen\n  $ \\sum_i \\phi_iP_i\\psi_i$ is a $\\Psi{\\rm DO}$ on $M$, and\n$P$ differs from $ \\sum_i \\phi_iP_i\\psi_i$ by a smoothing operator, denoted\n$P\\sim \\sum_i \\phi_iP_i\\psi_i$. \n In particular, this sum is independent of the\n  choices up to smoothing operators.\nAll this carries over to $\\Psi{\\rm DO}$s acting on sections of a bundle over $M$.\n\nAn example is the $\\Psi{\\rm DO}$ $(1+\\Delta-\\lambda)^{-1}$ for $\\Delta$ a positive\norder nonnegative elliptic $\\Psi{\\rm DO}$ and $\\lambda$\noutside the spectrum of $1+\\Delta.$  In each $U_i$, we construct a parametrix\n$P_i$ for $A_i = \\psi_i\n(1+\\Delta-\\lambda)\\phi_i$ by formally inverting $\\sigma(A_i)$ and then\nconstructing a $\\Psi{\\rm DO}$ with the inverted symbol.  By \\cite[App.~A]{A-B2},\n$ B = \\sum_i \\phi_iP_i\\psi_i$ is a parametrix for $(1+\\Delta-\\lambda)^{-1}$.\n  Since\n  $B \\sim (1+\\Delta-\\lambda)^{-1}$,\n  $(1+\\Delta-\\lambda)^{-1}$ is itself a $\\Psi{\\rm DO}$.\nFor $x\\in U_i$, by definition\n$$\\sigma((1+\\Delta-\\lambda)^{-1})(x,\\xi) = \\sigma(P)(x,\\xi) = \\sigma(\\phi\nP\\phi)(x,\\xi),$$\nwhere $\\phi$ is a bump function with $\\phi(x) = 1$ \\cite[p.~29]{gilkey};  \nthe symbol depends on the choice of $(U_i, \\phi_i).$  \n\nThe operator $(1+\\Delta)^s$ for Re$(s) <0$,\nwhich exists as a bounded\noperator on $L^2(M)$ by the functional calculus, is also a $\\Psi{\\rm DO}$.\nTo see this,\n we construct the putative symbol $\\sigma_i$\nof $ \\psi_i(1+\\Delta)^s\\phi_i$ in each $U_i$\nby a contour integral $\\int_\\Gamma \\lambda^{s}\\sigma[(1+\\Delta-\\lambda)^{-1}]\nd\\lambda$\naround the spectrum of $1+\\Delta$.\nWe then\nconstruct a $\\Psi{\\rm DO}$ $Q_i$ on $U_i$ with $\\sigma(Q_i) = \\sigma_i$,\nand set $Q  = \\sum_i\\phi_i Q_i\\psi_i.$\nBy arguments in \\cite{See},\n$(1+\\Delta)^s \\sim Q$,\nso $(1+\\Delta)^s$ is a $\\Psi{\\rm DO}$.\n\n\n\n\n Recall that the {\\it Wodzicki residue} of a $\\Psi{\\rm DO}$ $P$ on sections\n  of a bundle $E\\to\nM^n$ is\n\\begin{equation}\\label{wresdef} {\\rm res}^{\\rm w}(P) = \n\\int_{S^*M} \\operatorname{tr}\\ \\sigma_{-n}(P)(x,\\xi) d\\xi dx,\\end{equation}\nwhere $S^*M$ is the unit cosphere bundle for some metric.  The Wodzicki\nresidue is independent of choice of local coordinates, and up to scaling is \nthe unique trace on the algebra of $\\Psi{\\rm DO}$s if dim$(M)>1$ (see\ne.g.~\\cite{fgl} in general and \\cite{ponge} for the case $M=S^1.$).  \n\n\nThe Wodzicki residue will be used in\nPart II to define characteristic classes on $LM$.  In our particular case, the operator $P$ \nwill be an $\\Psi{\\rm DO}$ of order $-1$ acting on sections of a bundle over $S^1$\n (see (\\ref{cswint})), so\n$\\sigma_{-1}(P)$ is globally defined. Of course, $\\int_{S^*S^1}\n\\operatorname{tr}\\sigma_{-1}(P) d\\xi d\\theta = 2\\int_{S^1}\\operatorname{tr}\\sigma_{-1}(P)d\\theta$.  It is\neasy to check that this integral,\nwhich strictly speaking involves a choice of cover of $S^1$ and a partition of unity,\nequals the usual $2\\int_0^{2\\pi} \\operatorname{tr}\\sigma_{-1}(P) d\\theta.$\n\n\n\n\n\n\n\n\n\n\n\\subsection{The Levi-Civita Connection for $s=0, 1$}\n${}$\n\\medskip\n\n\nThe smooth Riemannian manifold $LM = H^{s'}(S^1,M)$ has tangent bundle $TLM$ with \n$T_\\gamma LM = H^{s'-1}(\\gamma^*TM).$  For the $s'-1$ metric on $TLM$ (i.e., \n$s = s'-1$ in (\\ref{eq:Sob1})), \nthe \nLevi-Civita connection exists and is determined by the six term formula\n\\begin{eqnarray}\\label{5one}\n2\\ip{\\con{s}{X}Y,Z}_{s} &=& X\\ip{Y,Z}_{s}+Y\\ip{X,Z}_{s}-Z\\ip{X,Y}_{s}\\\\\n&&\\qquad +\\ip{[X,Y],Z}_{s}+\\ip{[Z,X],Y}_{s}-\\ip{[Y,Z],X}_s\\nonumber\n\\end{eqnarray}\n\\cite[Ch. VIII]{lang}.  The point is that each term on the RHS of (\\ref{5one}) \nis\na {\\it continuous} linear functional $T_i:H^{s=s'-1}(\\gamma^*TM) \\to {\\mathbb C}$ in $Z$.  Thus \n$T_i(Z) = \\ip{T_i'(X,Y),Z}_s$ for a unique $T'(X,Y)\\in H^{s'-1}(\\gamma^*TM)$, and $\\con{s}{Y}X\n= \\frac{1}{2}\\sum_i T'_i.$  \n\nIn general, the Sobolev parameter $s$ in (\\ref{eq:Sob1}) differs from the parameter $s'$ defining the loop space.  We discuss how this affects the existence of a Levi-Civita connection. \n\n\\begin{rem}\\label{lcrem}  For general $s >\\frac{1}{2}$, the Levi-Civita connection for the $H^s$ \nmetric is guaranteed to exist on the bundle $H^s(\\gamma^*TM)$, as above.  However, it is inconvenient to have the bundle depend on the Sobolev parameter, for several reasons:  \n(i) $H^s(\\gamma^*TM)$ is strictly speaking not the tangent bundle of $LM$, (ii) for the\n$L^2$ ($s=0$) metric, the Levi-Civita connection should be given by the Levi-Civita connection on $M$ applied pointwise along the loop (see Lemma \\ref{lem:l2lc}), and on $L^2(\\gamma^*TM)$  this would have to be interpreted in the distributional sense; (iii) to compute Chern-Simons classes on\n$LM$ in Part II, we need to compute with a pair of connections corresponding to $s=0, s=1$ on the\nsame bundle.  These problems are not fatal: (i) and (ii) are essentially  aesthetic issues,\nand for (iii), the connection one-forms will take values in zeroth order $\\Psi{\\rm DO}$s, which are bounded operators on any \n$H^{s'-1}(\\gamma^*TM)$, so $s' \\gg 0$ can be fixed.  \n\nThus it is more convenient\nto fix $s'$ and consider the family of $H^s$ metrics on $TLM$ for \n$\\frac{1}{2} < s < s'-1$. \n  However, the existence of the Levi-Civita connection for the $H^s$ metric is trickier.\n  For a sequence $Z\\in H^{s'-1} = H^{s'-1}(\\gamma^*TM)$ with $Z\\to 0$\n  in $H^{s'-1}$ or in \n  $H^s$, the RHS of (\\ref{5one}) goes to $0$ for fixed $X, Y\\in H^s.$  Since\n  $H^{s'-1}$ is dense in $H^{s}$, the RHS of (\\ref{5one}) extends to a continuous linear functional on $H^s$.  Thus the RHS of (\\ref{5one}) is given by\n  $\\langle L(X,Y), Z\\rangle_s$ for some $L(X,Y)\\in H^s.$  We set $\\nabla^{s}_YX = \n  \\frac{1}{2}L(X,Y)$.  Note that even if we naturally demand that\n  $X, Y\\in H^{s'-1}$, we only get $\\nabla^s_YX\\in H^s\\supset H^{s'-1}$ without additional work.  Part of the content of Theorem \\ref{thm25} is that the Levi-Civita connection exists in the {\\it strong sense}:  given a tangent vector $X\\in H^{s'-1}(\\gamma^*TM)$ and a smooth vector field\n$Y(\\eta)\\in H^{s'-1}(\\eta^*TM)$ for all $\\eta$,\n   $\\nabla^s_XY(\\gamma)\\in H^{s'-1}(\\gamma^*TM).$  See Remark 2.6.\n  \n \n\n\\end{rem}\n\n\n\n\nWe need to discuss local coordinates on $LM$.\nFor motivation, recall that\n\\begin{equation}\\label{lie}[X,Y]^a = X(Y^a)\\partial_a - Y(X^a)\\partial_a\n\\equiv \\delta_X(Y) -\\delta_Y(X)\n\\end{equation}\nin local coordinates on a finite dimensional manifold.  Note that\n$X^i\\partial_iY^a = X(Y^a) =\n(\\delta_XY)^a$ in this notation.\n\nLet $Y$ be a vector field on $LM$, and let $X$ be a tangent vector at\n$\\gamma\\in LM.$  The local variation $\\delta_XY$ of $Y$ in the direction of $X$ at $\\gamma$ is \ndefined as usual: let $\\gamma(\\varepsilon,\\theta)$ be a family of loops in $M$\nwith $\\gamma(0,\\theta) = \\gamma(\\theta), \\frac{d}{d\\varepsilon}|_{_{\\varepsilon=0}}\n\\gamma(\\varepsilon,\\theta) = X(\\theta).$  Fix $\\theta$, and let $(x^a)$ be\ncoordinates near $\\gamma(\\theta)$.  We call these coordinates \n{\\it manifold coordinates.} Then\n$$\\delta_XY^a(\\gamma)(\\theta) \\stackrel{{\\rm def}}{=}\n\\frac{d}{d\\varepsilon}\\biggl|_{_{_{\\varepsilon =0}}} Y^a(\\gamma(\\varepsilon,\\theta)).$$\nNote that $\\delta_XY^a = (\\delta_XY)^a$ by definition.\n\n\\begin{rem} Having $(x^a)$ defined only near a fixed $\\theta$ is inconvenient.\nWe can find coordinates that work for all $\\theta$ as follows. For\n  fixed $\\gamma$, there is an $\\varepsilon$ such that for all $\\theta$,\n  $\\exp_{\\gamma(\\theta)} X$ is inside the cut locus of $\\gamma(\\theta)$ if\n  $X\\in T_{\\gamma(\\theta)}M$ has $|X|<\\varepsilon.$  Fix such an $\\varepsilon.$ Call \n  $X\\in H^{s'-1}(\\gamma^*TM)$ {\\it\n  short} if $|X(\\theta)|<\\varepsilon$ for all $\\theta.$  Then\n$$U_\\gamma = \\{\\theta \\mapsto \\exp_{\\gamma(\\theta)}X(\\theta) | X\\ {\\rm is\\\n    short}\\}\\subset LM$$\nis a coordinate neighborhood of $\\gamma$ parametrized by $\\{ X: X\\  {\\rm is\\ \n    short}\\}.$  \n\n We know\n$H^{s'-1}(\\gamma^*TM) \\simeq H^{s'-1}(S^1\\times {\\mathbb R}^n)$ noncanonically, so\n$U_\\gamma$ is parametized by short sections of $H^{s'-1}(S^1\\times {\\mathbb R}^n)$ for\na different $\\varepsilon.$  In particular, we have a smooth diffeomorphism $\\beta$ from\n$U_\\gamma$ to short sections of $H^{s'-1}(S^1\\times {\\mathbb R}^n)$.\n\nPut coordinates $(x^a)$ on ${\\mathbb R}^n$, which we identify canonically with the fiber ${\\mathbb R}^n_\\theta$\nover $\\theta$ in $S^1\\times {\\mathbb R}^n$. For $\\eta\\in\nU_\\gamma$, we have $\\beta(\\eta) = (\\beta(\\eta)^1(\\theta),...,\\beta(\\eta)^n(\\theta)).$\nAs with finite dimensional coordinate systems, we will drop $\\beta$ and just\nwrite\n$\\eta = (\\eta(\\theta)^a).$ These coordinates work for all\n$\\eta$ near $\\gamma$ and for all $\\theta.$  The definition of $\\delta_XY$ above carries over to exponential coordinates.\n\nWe will call these coordinates {\\it exponential coordinates}.\n\\end{rem}\n\n(\\ref{lie}) continues to hold\n for vector fields on $LM$, in either \n manifold or exponential coordinates.\n   To see this, one checks that the coordinate-free proof that $L_XY(f) =\n [X,Y](f)$ for $f\\in C^\\infty(M)$ (e.g.~\\cite[p.~70]{warner}) carries over to\n functions on $LM$.  In brief, the usual proof involves a map $H(s,t)$ of a\n neighborhood of the origin in ${\\mathbb R}^2$ into $M$, where $s,t$ are parameters for\n the flows of $X, Y,$ resp.  For $LM$, we have a map $H(s,t,\\theta)$, where\n $\\theta$ is the loop parameter.  \nThe  usual proof uses\n only $s, t$ differentiations,\n so $\\theta$ is unaffected.    The point is that the $Y^i$ are local functions\n on the $(s,t,\\theta)$  parameter space, whereas\nthe $Y^i$ are not\n  local functions on $M$ at points where loops cross or self-intersect.\n\nWe first compute the $L^2$ ($s=0$) Levi-Civita connection invariantly and in \nmanifold coordinates.\n\n\n\n\\begin{lem} \\label{lem:l2lc} Let $\\nabla^{LC}$ be the Levi-Civita connection on $M$.  \n Let $\\operatorname{ev}_\\theta:LM\\to M$ be $\\operatorname{ev}_\\theta(\\gamma) = \\gamma(\\theta).$ \n Then $D_XY(\\gamma)(\\theta) \\stackrel{\\rm def}{=} \n (\\operatorname{ev}_\\theta^*\\nabla^{LC})_XY(\\gamma)(\\theta)$ is the $L^2$ Levi-Civita connection on $LM$.  In manifold coordinates,\n \\begin{equation}\\label{l2lc} (D_XY)^a(\\gamma)(\\theta) = \\delta_XY^a(\\gamma)(\\theta) +\n  \\cch{bc}{a}(\\gamma(\\theta))X^b(\\gamma)(\\theta) Y^c(\\gamma)(\\theta).\n  \\end{equation}\n\\end{lem}\n\\medskip\n\nAs in Remark \\ref{lcrem}, we may assume that\n$X, Y\\in H^{s'-1}(\\gamma^*TM)$ with $s' \\gg 0$, so (\\ref{l2lc}) makes sense.\n\n\\begin{proof} $\\operatorname{ev}_\\theta^*\\nabla^{LC}$ is a connection on\n$\\operatorname{ev}_\\theta^*TM\\to LM$. We have\n$\\operatorname{ev}_{\\theta,*}(X) = X(\\theta)$.  If $U$ is a coordinate\nneighborhood on $M$ near some $\\gamma(\\theta)$, then on $\\operatorname{ev}_\\theta^{-1}(U)$, \n\\begin{eqnarray*}(\\operatorname{ev}_\\theta^*\\nabla^{LC})_XY^a(\\gamma)(\\theta) &=& (\\delta_{X}Y)^a(\\gamma)(\\theta) +\n((\\operatorname{ev}_\\theta^*\\omega^{LC}_{X})Y)^a (\\theta)\\\\\n&=& (\\delta_{X}Y)^a(\\gamma)(\\theta) + \n  \\chw{b}{c}{a}(\\gamma(\\theta))X^b(\\gamma)(\\theta) Y^c(\\gamma)(\\theta)\n\\end{eqnarray*}\nSince $\\operatorname{ev}_\\theta^*\\nabla^{LC}$ is a connection, for each fixed $\\theta$, $\\gamma$ and $X\\in\n T_\\gamma LM$, \n $Y\\mapsto$\\\\\n $ (\\operatorname{ev}^*_\\theta\\nabla^{LC})_XY(\\gamma)$\n has Leibniz rule with respect to\nfunctions on $LM$.   Thus $D$ is a connection on $LM.$\n\n\n$D$ is torsion free, as from the local expression\n  $D_XY - D_YX = \\delta_XY - \\delta_YX = [X,Y].$\n\nTo show that  $D_XY$ is compatible with the $L^2$\n  metric, first recall that  for a function $f$ on $LM$, $D_Xf = \\delta_Xf =\n  \\frac{d}{d\\varepsilon}|_{_{\\varepsilon=0}}f(\\gamma(\\varepsilon,\\theta))$ for $X(\\theta)\n   = \\frac{d}{d\\varepsilon}|_{_{\\varepsilon=0}}\\gamma(\\varepsilon, \\theta).$\n  (Here $f$ depends only on\n  $\\gamma$.)  Thus (suppressing the partition of unity, which is independent of $\\varepsilon$)\n\\begin{eqnarray*} D_X\\langle Y,Z\\rangle_0 &=& \n  \\frac{d}{d\\varepsilon}\\biggl|_{_{_{\\varepsilon=0}}}\\int_{S^1} g_{ab}(\\gamma(\\varepsilon,\\theta))\nY^a(\\gamma(\\varepsilon,\\theta))Z^b(\\gamma(\\varepsilon,\\theta))d\\theta\\\\\n&=& \\int_{S^1}\\partial_c\ng_{ab}(\\gamma(\\varepsilon,\\theta))\nX^cY^a(\\gamma(\\varepsilon,\\theta))Z^b(\\gamma(\\varepsilon,\\theta))d\\theta\\\\\n&&\\qquad + \\int_{S^1} \ng_{ab}(\\gamma(\\varepsilon,\\theta))\n(\\delta_XY)^a(\\gamma(\\varepsilon,\\theta))Z^b(\\gamma(\\varepsilon,\\theta))d\\theta\\\\\n&&\\qquad \n+ \\int_{S^1} \ng_{ab}(\\gamma(\\varepsilon,\\theta))\nY^a(\\gamma(\\varepsilon,\\theta))(\\delta_XZ)^b(\\gamma(\\varepsilon,\\theta))d\\theta\\\\\n&=& \\int_{S^1}\\Gamma{}_{c a}^{e}g_{eb} X^cY^aZ^b +\n\\Gamma{}_{c b}^{e}g_{ae}X^cY^aZ^b\\\\\n&&\\qquad +g_{ab}(\\delta_XY)^aZ^b + g_{ab}Y^a(\\delta_X Z)^bd\\theta\\\\\n&=& \\langle D_XY,Z\\rangle_0 + \\langle Y, D_XZ\\rangle_0.\n\\end{eqnarray*}\n\\end{proof}\n\n\n\\begin{rem} The local expression for $D_XY$ also holds in exponential coordinates. More precisely, let $(e_1(\\theta),...,e_n(\\theta))$\nbe a global frame of $\\gamma^*TM$ given by the trivialization of\n$\\gamma^*TM.$  Then $(e_i(\\theta))$ is also naturally a frame of\n$T_XT_{\\gamma(\\theta)}M$ for all $X\\in T_{\\gamma(\\theta)}M.$  We use\n$\\exp_{\\gamma(\\theta)}$ to pull back the metric on $M$ to a metric on\n$T_{\\gamma(\\theta)}M$: \n$$g_{ij}(X) = (\\exp^*_{\\gamma(\\theta)}g)(e_i, e_j) =\n  g(d(\\exp_{\\gamma(\\theta)})_X (e_i),   d(\\exp_{\\gamma(\\theta)})_X\n  (e_j))_{\\exp_{\\gamma(\\theta)}X}.$$\nThen the Christoffel symbols  \n$\\Gamma_{b c}^{a}(\\gamma(\\theta))$\nare computed with respect to\n  this metric.  For example, the term $\\partial_\\ell g_{bc}$ means $e_\\ell\n  g(e_a, e_b)$, etc.  The proof that $D_XY$ has the local expression (\\ref{l2lc}) \n  then carries over to exponential coordinates.\n\n\\end{rem}\n\nThe  $s=1$ Levi-Civita connection on $LM$ is given as follows.\n\n\\begin{thm} \\label{old1.6}\nThe $s=1$ Levi-Civita connection $\\nabla = \\nabla^1$ on $LM$ is given at the loop\n$\\gamma$ by\n\\begin{eqnarray*}  \\nabla_XY &=& D_XY + \\frac{1}{2}(1+\\Delta)^{-1}\\left[\n-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) - R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y\\right.\\\\\n&&\\qquad \\left. -\\nabla_{\\dot\\gamma}(R(Y,\\dot\\gamma)X) - R(Y,\\dot\\gamma)\\nabla_{\\dot\\gamma} X\\right.\\\\\n&&\\qquad \\left. +R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma - R(\\nabla_{\\dot\\gamma} X, Y)\\dot\\gamma\\right].\n\\end{eqnarray*}\n\\end{thm}\n\n\nWe prove this in a series of steps.  The assumption in the next Proposition will be dropped later.\n\n\n\\begin{prop} \\label{old1.3}\nThe Levi-Civita connection for the $s=1$ metric is given by\n$$\\nabla_X^1Y = D_XY + \\frac{1}{2}(1+\\Delta)^{-1}[D_X, 1+\\Delta]Y +\n  \\frac{1}{2}(1+\\Delta)^{-1}[D_Y, 1+\\Delta]X \n+ A_XY,$$\nwhere  we assume that for $X, Y\\in H^{s'-1}$, $A_XY$ is well-defined by\n\\begin{equation}\\label{insert2}-\\frac{1}{2}\\langle [D_Z,1+\\Delta]X,Y\\rangle_0 = \\langle A_XY,Z\\rangle_1.\n\\end{equation}\n\\end{prop}\n\n\n\n\n\\begin{proof} By Lemma \\ref{lem:l2lc},\n\\begin{eqnarray*} X\\langle Y,Z\\rangle_1 &=& X\\langle (1+\\Delta)Y,Z\\rangle_0 =\n  \\langle D_X((1+\\Delta)Y),Z\\rangle_0 + \\langle (1+\\Delta)Y, D_XZ\\rangle_0\\\\\nY\\langle X,Z\\rangle_1 &=& \\langle D_Y((1+\\Delta)X),Z\\rangle_0 + \\langle (1+\\Delta)X,\nD_YZ\\rangle_0\\\\\n-Z\\langle X,Y\\rangle_1 &=& -\\langle D_Z((1+\\Delta)X),Y\\rangle_0 - \\langle (1+\\Delta)X,\nD_ZY\\rangle_0\\\\\n\\langle [X,Y],Z\\rangle_1 &=& \\langle(1+\\Delta)(\\delta_XY - \\delta_YX), Z\\rangle_0\n= \\langle (1+\\Delta)(D_XY - D_YX),Z\\rangle_0\\\\\n\\langle[Z,X],Y\\rangle_1 &=& \\langle(1+\\Delta)(D_ZX-D_XZ),Y\\rangle_0\\\\\n-\\langle[Y,Z],X\\rangle_1 &=& -\\langle(1+\\Delta)(D_YZ-D_ZY),X\\rangle_0.\n\\end{eqnarray*}\nThe six terms on the left hand side must sum up to $2\\langle \\nabla^1_XY,Z\\rangle_1$ \nin the sense of Remark \\ref{lcrem}.\nAfter some cancellations, for $\\nabla =\\nabla^1$ we get\n\\begin{eqnarray*} 2\\langle\\nabla_XY,Z\\rangle_1 &=& \\langle D_X((1+\\Delta)Y),Z\\rangle_0 +\n  \\langle D_Y((1+\\Delta)X),Z\\rangle_0\\nonumber\\\\\n&&\\qquad + \\langle (1+\\Delta)(D_XY - D_YX),Z\\rangle_0 - \\langle\n  D_Z((1+\\Delta)X),Y\\rangle_0\\nonumber\\\\\n&&\\qquad +\\langle(1+\\Delta)D_ZX),Y\\rangle_0\\nonumber\\\\\n&=& \\langle (1+\\Delta)D_XY,Z\\rangle_0 + \\langle [D_X,1+\\Delta] Y, Z\\rangle_0\\\\\n&&\\qquad + \\langle (1+\\Delta)D_YX,Z\\rangle_0 + \\langle [D_Y,1+\\Delta] X, Z\\rangle_0\\nonumber\\\\\n&&\\qquad + \\langle (1+\\Delta)(D_XY - D_YX),Z\\rangle_0 -\\langle\n  [D_Z,1+\\Delta]X,Y\\rangle_0\\\\\n&=& 2\\langle D_XY,Z\\rangle_1 + \\langle (1+\\Delta)^{-1}[D_X,1+\\Delta]Y,Z\\rangle_1\\\\\n&&\\qquad + \\langle (1+\\Delta)^{-1}[D_Y,1+\\Delta]X,Z\\rangle_1 +2\n\\langle A_XY,Z\\rangle_1.\n\\end{eqnarray*}\n\n\\end{proof}\n\nNow we compute the bracket terms in the Proposition.  We have $[D_X,1+\\Delta] =\n[D_X,\\Delta]$. Also,\n$$0 = \\dot\\gamma\\langle X, Y\\rangle_0 = \\langle\\nabla_{\\dot\\gamma}X,Y\\rangle_0\n+ \\langle X,\\nabla_{\\dot\\gamma}Y\\rangle_0,$$\nso \n\\begin{equation}\\label{one}\\Delta = \\nabla_{\\dot\\gamma}^* \\nabla_{\\dot\\gamma}\n  = -\\nabla_{\\dot\\gamma}^2.\n\\end{equation}\n\n\\begin{lem} $[D_X,\\nabla_{\\dot\\gamma}]Y = R(X,\\dot\\gamma)Y.$\n\\end{lem}\n\n\\begin{proof} \nNote that $\\gamma^\\nu, \\dot\\gamma^\\nu$ are locally defined functions on \n$S^1\\times LM.$\nLet $\\tilde\\gamma:\n[0,2\\pi]\\times (-\\varepsilon,\\varepsilon)\\to M$ be a smooth map with $\\tilde\\gamma(\\theta,0) =\n\\gamma(\\theta)$, and\n$\\frac{d}{d\\tau}|_{\\tau = 0}\\tilde\\gamma(\\theta,\\tau) = Z(\\theta).$\nSince $(\\theta,\\tau)$ are coordinate functions on\n$S^1\\times (-\\varepsilon,\\varepsilon)$, we have\n\\begin{eqnarray}\\label{badterms} Z(\\dot\\gamma^\\nu) &=& \\delta_Z(\\dot\\gamma^\\nu) = \\ptau{Z}(\\dot\\gamma^\\nu) =\n\\dtau\\right.\\left(\\frac{\\partial}{\\partial\\theta}\n(\\tilde\\gamma(\\theta,\\tau)^\\nu\\right)\\\\\n&=& \\frac{\\partial}{\\partial\\theta}\n\\dtau\\right. \\tilde\\gamma(\\theta,\\tau)^\\nu = \\partial_\\theta Z^\\nu \\equiv\n\\dot Z^\\nu.\\nonumber\n\\end{eqnarray}\n\n We compute\n\\begin{eqnarray*} \n(D_X\\nabla_{\\dot\\gamma} Y)^a\n&=& \\delta_X(\\nabla_{\\dot\\gamma} Y)^a +\n  \\chw{b}{c}{a}X^b\\nabla_{\\dot\\gamma} Y^c\\\\\n&=& \\delta_X(\\dot\\gamma^j\\partial_jY^a + \\chw{b}{c}{a}\\dot\\gamma^bY^c\n+ \\chw{b}{c}{a}X^b(\\dot\\gamma^j\\partial_jY^c + \\chw{e}{f}{c}\\dot\\gamma^e Y^f)\\\\\n&=& \\dot X^j\\partial_jY^a + \\dot\\gamma^j\\partial_j\\delta_XY^a +\n  \\partial_m\\chw{b}{c}{a}X^m\\dot\\gamma^bY^\n+ \\chw{b}{c}{a}\\dot X^bY^c + \\chw{b}{c}{a}\\dot\\gamma^b\\delta_XY^c\\\\\n&&\\qquad \n+ \\chw{b}{c}{a}X^b\\dot\\gamma^j\\partial_jY^c +\n  \\chw{b}{c}{a}\\chw{e}{f}{c}X^b\\dot\\gamma^eY^f.\\\\\n(\\nabla_{\\dot\\gamma} D_XY)^a &=& \\dot\\gamma^j(\\partial_j(D_XY)^a +\n  \\chw{b}{c}{a}\\dot\\gamma^b (D_XY)^c)\\\\\n&=& \\dot\\gamma^j\\partial_j(\\delta_XY^a + \\chw{b}{c}{a}X^b Y^c) \n+ \\chw{b}{c}{a}\\dot\\gamma^b(\\delta_XY^c + \\chw{s}{f}{c}X^eY^f)\\\\\n&=& \\dot\\gamma^j\\partial_j\\delta_XY^a + \\dot\\gamma^j\\partial_j\\chw{b}{c}{a}X^bY^c +\n  \\chw{b}{c}{a}\\dot X^bY^c\n+ \\chw{b}{c}{a}X^b\\dot Y^c + \\chw{b}{c}{a}\\dot\\gamma^b\\delta_XY^c \\\\\n&&\\qquad + \\chw{b}{c}{a}\\chw{e}{f}{c}\\dot\\gamma^b X^eY^f.\n\\end{eqnarray*}\nTherefore\n\\begin{eqnarray*} (D_X\\nabla_{\\dot\\gamma}Y - \\nabla_{\\dot\\gamma}D_XY)^a &=& \\partial_m\n  \\chw{b}{c}{a}X^m\\dot\\gamma^bY^c - \\partial_j \\chw{b}{c}{a}\\dot\\gamma^j X^bY^c\n  + \\chw{b}{c}{a}\\chw{e}{f}{c}X^b\\dot\\gamma^e Y^f   \\\\\n&&\\qquad \n-\\chw{b}{c}{a}\\chw{e}{f}{c}\\dot\\gamma^b X^e Y^f \\\\\n&=& (\\partial_j \\Gamma_{bc}^{a} - \\partial_b \\chw{j}{c}{a} +\\chw{j}{e}{a}\\chw{b}{c}{e}-\n\\chw{b}{e}{a}\\chw{j}{c}{e})\\dot\\gamma^b X^j Y^c \\\\\n&=& R_{jbc}^{\\ \\ \\ a}X^j\\dot\\gamma^b Y^c,\n\\end{eqnarray*}\nso \n$$D_X\\nabla_{\\dot\\gamma}Y - \\nabla_{\\dot\\gamma}D_XY =  R(X,\\dot\\gamma)Y.$$\n\\end{proof}\n\n\\begin{cor}\\label{cor:zero}\n At the loop $\\gamma$, $[D_X,\\Delta]Y = -\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) -\n  R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}Y.$  In particular, $[D_X,\\Delta]$ is a zeroth order\n  operator.\n\\end{cor}\n\n\\begin{proof}  \n\\begin{eqnarray*} [D_X,\\Delta]Y &=& (-D_X\\nabla_{\\dot\\gamma}\\ndg + \\nabla_{\\dot\\gamma}\\ndg D_X)Y \\\\\n&=& -(\\nabla_{\\dot\\gamma} D_X\\nabla_{\\dot\\gamma} Y+ R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y) +\\nabla_{\\dot\\gamma}\\ndg D_XY\\\\\n&=& -(\\nabla_{\\dot\\gamma}\\ndg D_XY + \\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) + R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y) \n+\\nabla_{\\dot\\gamma}\\ndg D_XY\\\\\n&=& -\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) - R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y.\n\\end{eqnarray*}\n\\end{proof}\n\nNow we complete the proof of Theorem \\ref{old1.6}, showing in the process that $A_XY$ exists. \n\\medskip\n\n\\noindent {\\it Proof of Theorem \\ref{old1.6}.}\n By Proposition \\ref{old1.3} and Corollary \\ref{cor:zero}, we have\n\\begin{eqnarray*} \\nabla_XY &=& D_XY + \\frac{1}{2}(1+\\Delta)^{-1}[D_X,1+\\Delta]Y +\n  (X\\leftrightarrow Y) + A_XY\\\\\n&=& D_XY + \\frac{1}{2}(1+\\Delta)^{-1}(-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) - R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y) + \n  (X\\leftrightarrow Y) + A_XY,\n\\end{eqnarray*}\nwhere $(X\\leftrightarrow Y)$ denotes the previous term with $X$ and $Y$ switched.\n\nThe curvature tensor satisfies \n$$-\\langle Z, R(X,Y)W\\rangle = \\langle R(X,Y)Z,W\\rangle = \\langle R(Z,W)X,Y\n\\rangle$$\n pointwise, so\n\\begin{eqnarray*} \\langle A_XY,Z\\rangle_1 &=&\n  -\\frac{1}{2}\\langle[D_Z,1+\\Delta]X,Y\\rangle_0\\\\\n&=& -\\frac{1}{2}\\langle (-\\nabla_{\\dot\\gamma}(R(Z,\\dot\\gamma)X) - R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma} X,Y\\rangle_0\\\\\n&=& -\\frac{1}{2} \\langle R(Z,\\dot\\gamma)X,\\nabla_{\\dot\\gamma} Y\\rangle_0 + \\frac{1}{2}\\langle\n  R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma} X,Y\\rangle_0\\\\\n&=& -\\frac{1}{2} \\langle R(X,\\nabla_{\\dot\\gamma} Y)Z,\\dot\\gamma\\rangle_0 + \\frac{1}{2}\\langle R(\\nabla_{\\dot\\gamma} X,\n  Y)Z,\\dot\\gamma\\rangle_0\\\\\n&=& \\frac{1}{2}\\langle Z, R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma\\rangle_0 - \\frac {1}{2} \\langle Z,\n  R(\\nabla_{\\dot\\gamma} X,Y)\\dot\\gamma\\rangle_0\\\\\n&=&\\frac{1}{2}\\langle Z, (1+\\Delta)^{-1}(R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma - R(\\nabla_{\\dot\\gamma} X, Y)\\dot\\gamma)\\rangle_1.\n\\end{eqnarray*}\nThus $A_XY$ must equal $\\frac{1}{2} (1+\\Delta)^{-1}(R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma - R(\\nabla_{\\dot\\gamma} X, Y)\\dot\\gamma).$  \nThis makes sense: for $X, Y\\in H^{s'-1}$, \n $A_XY\\in H^{s'}\\subset H^1,$ since $R$ is zeroth order.  \n\n\\hfill$\\Box$\n\\medskip \n\n\n\n\\begin{rem} Locally on $LM$, we should have \n$D_XY = \\delta_X^{LM}Y +  \\omega_X^{LM}(Y)$.  \nNow\n$\\delta_X^{LM}Y$  can only mean $\\frac{d}{d\\tau}|_{\\tau =\n  0}\\frac{d}{d\\epsilon}|_{\\epsilon = 0}\\gamma(\\epsilon,\\tau,\\theta)$, where\n$\\gamma(0,0,\\theta) = \\gamma(\\theta)$, ${d\\epsilon}|_{\\epsilon =\n  0}\\gamma(\\epsilon,0,\\theta) = X(\\theta)$, \n${d\\tau}|_{\\tau = 0}\\gamma(\\epsilon,\\tau,\\theta) = Y_{\\gamma(\\epsilon, 0,\\cdot)}(\\theta).$\n In other words, $\\delta_X^{LM}Y$ equals $ \\delta_XY$.\nSince $D_XY^a = \\delta_XY^a + \\chw{b}{c}{a}(\\gamma(\\theta))$, the connection one-form for the $L^2$ Levi-Civita connection on $LM$ is given by\n$$\\omega^{LM}_X(Y)^a(\\gamma)(\\theta) = \\chw{b}{c}{a}(\\gamma(\\theta))X^bY^c\n= \\omega^M_X(Y)^a(\\gamma(\\theta)).$$\n\\end{rem}\n\nBy this remark, we get\n\\begin{cor}\\label{cor2} The connection one-form $\\omega = \\omega^1$ for $\\nabla^1$ in \nexponential coordinates is\n\\begin{eqnarray}\\label{two}\\omega_X(Y)(\\gamma)(\\theta) &=& \\omega^M_X(Y)(\\gamma(\\theta))\n  +  \n\\frac{1}{2}\\bigl\\{(1+\\Delta)^{-1}\\left[\n-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) - R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y\\right.\\nonumber\\\\\n&&\\qquad \\left. -\\nabla_{\\dot\\gamma}(R(Y,\\dot\\gamma)X) - R(Y,\\dot\\gamma)\\nabla_{\\dot\\gamma} X\\right.\\\\\n&&\\qquad  \\left.+R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma - R(\\nabla_{\\dot\\gamma} X, Y)\\dot\\gamma\\right]\\bigr\\}(\\theta).\\nonumber\n\\end{eqnarray}\n\\end{cor}\n\n\n\\subsection{The Levi-Civita Connection for $s\\in{\\mathbb Z}^+$}\n${}$\n\\medskip\n\nFor $s>\\frac{1}{2}$, the proof of Prop.~\\ref{old1.3} extends directly to give\n\n\\begin{lem} \\label{lem: LCs}\nThe Levi-Civita connection for the $H^s$ metric is given by\n$$\\nabla_X^sY = D_XY + \\frac{1}{2}(1+\\Delta)^{-s}[D_X, (1+\\Delta)^s]Y +\n  \\frac{1}{2}(1+\\Delta)^{-s}[D_Y, (1+\\Delta)^s]X \n+ A_XY,$$\nwhere we assume that for $X, Y\\in H^{s'-1}$, $A_XY\\in H^s$ is characterized by\n\\begin{equation}\\label{axy}\n-\\frac{1}{2}\\langle [D_Z,(1+\\Delta)^s]X,Y\\rangle_0 = \\langle A_XY,Z\\rangle_s.\n\\end{equation}\n\\end{lem}\n\\bigskip\n\nWe now compute the bracket terms.\n\n\\begin{lem}\\label{bracketterms}\nFor $s\\in {\\mathbb Z}^+$, at the loop $\\gamma$,\n\\begin{equation}\\label{bracket}\n[D_X,(1+\\Delta)^s]Y = \\sum_{k=1}^s(-1)^k\\left(\\begin{array}{c}s\\\\k\\end{array}\\right)\n\\sum_{j=0}^{2k-1} \\nabla_{\\dot\\gamma}^j(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}Y).\n\\end{equation}\nIn particular, $[D_X,(1+\\Delta)^s]Y$ is a $\\Psi{\\rm DO}$ of order $2s-1$ in either $X$ or $Y$.\n\\end{lem}\n\n\\begin{proof}  The sum over $k$ comes from the binomial expansion of $(1+\\Delta)^s$, so\nwe just need an inductive formula for \n$[D_X,\\Delta^s].$   \nThe case $s=1$ is Proposition \\ref{old1.3}. For the induction step, we have\n\\begin{eqnarray*} [D_X,\\Delta^s] &=& D_X\\Delta^{s-1}\\Delta - \\Delta^sD_X\\\\\n&=& \\Delta^{s-1}D_X\\Delta + [D_X,\\Delta^{s-1}]\\Delta - \\Delta^sD_X\\\\\n&=& \\Delta^sD_X +\\Delta^{s-1}[D_X,\\Delta] + [D_X,\\Delta^{s-1}]\\Delta\n-\\Delta^sD_X\\\\\n&=& \\Delta^{s-1}(-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) -R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}Y)\\\\\n&&\\qquad  -  \n\\sum_{j=0}^{2s-3}(-1)^{s-1}\n\\nabla^j_{\\dot\\gamma}(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-j-1}(-\\nabla^2_{\\dot\\gamma}Y)\\\\\n&=& (-1)^{s-1}(-\\nabla_{\\dot\\gamma}^{2s-1}(R(X,\\dot\\gamma)Y) - (-1)^{s-1}\\nabla_{\\dot\\gamma}^{2s-2}(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}Y)\\\\\n&&\\qquad + \\sum_{j=0}^{2s-3}(-1)^{s}\n\\nabla^j_{\\dot\\gamma}(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-j-1}(-\\nabla^2_{\\dot\\gamma}Y)\\\\\n&=& \\sum_{j=0}^{2s-1}(-1)^s \\nabla_{\\dot\\gamma}^j(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}Y).\n\\end{eqnarray*} \n\\end{proof}\n\nWe check that $A_XY$ is a $\\Psi{\\rm DO}$ in $X$ and $Y$ for $s\\in {\\mathbb Z}^+.$\n\n\\begin{lem} \\label{insert3} For $s\\in{\\mathbb Z}^+$ and fixed $X, Y\\in H^{s'-1}$, $A_XY$ in (\\ref{axy})\n is an explicit $\\Psi{\\rm DO}$ in $X$ and $Y$ of order at most $-1.$\n\n\\end{lem}\n\n\\begin{proof}  By (\\ref{bracket}), for  $j, 2k-1-j \\in \\{0,1,...,2s-1\\}$, a typical term on\nthe left hand side of (\\ref{axy}) is\n\\begin{eqnarray*}   \\ipo{\\nabla^j_{\\dot\\gamma}(R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}X)}{Y} &=& \n(-1)^j\n \\ipo{R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}X}{\\nabla^j_{\\dot\\gamma} Y}\\\\\n&=& (-1)^j\\int_{S^1} g_{i\\ell} (R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}X)^i(\\nabla^j_{\\dot\\gamma} Y)^\\ell d\\theta\\\\\n&=& (-1)^j\\int_{S^1} g_{i\\ell} Z^k R_{krn}^{\\ \\ \\ i}\\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell d\\theta\\\\\n&=& (-1)^j\n \\int_{S^1} g_{tm}g^{kt}  g_{i\\ell} Z^m R_{krn}^{\\ \\ \\ i}\\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell d\\theta\\\\\n&=& (-1)^j \\ipo{Z}\n{g^{kt} g_{i\\ell} R_{krn}^{\\ \\ \\ i}\\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell\\partial_t}\\\\ \n&=& (-1)^j\\ipo{Z}{R^t_{\\ rn\\ell} \\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell\\partial_t}\\\\ \n&=&(-1)^{j+1} \\ipo{Z}{R^{\\ \\ \\ t}_{n\\ell r} \\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell\\partial_t}\\\\\n&=& (-1)^{j+1}\\ipo{Z}{R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma}\\\\\n&=& (-1)^{j+1} \\ips{Z}{(1+\\Delta)^{-s} R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma}.\n  \\end{eqnarray*}\n  (In the integrals and inner products, the local expressions are in fact globally defined one-forms on $S^1$, resp.~vector fields along $\\gamma$, so we do not need a partition of unity.)\n$(1+\\Delta)^{-s} R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma$ is of order at most $-1$ in either $X$ or $Y$, so this term is in \n$H^{s'}\\subset H^s.$  Thus the last inner product is well defined.  \n\\end{proof}\n\nBy  (\\ref{axy}), (\\ref{bracket}) and the proof of Lemma \\ref{insert3}, we get\n$$A_XY = \\sum_{k=1}^s(-1)^k\\left(\\begin{array}{c}s\\\\k\\end{array}\\right)\n\\sum_{j=0}^{2k-1} (-1)^{j+1}   (1+\\Delta)^{-s} R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma.$$\nThis gives:\n\n\\begin{thm} \\label{thm:sinz}\nFor $s\\in{\\mathbb Z}^+$, the Levi-Civita connection for the $H^s$ metric at the\nloop $\\gamma$ is given by\n\\begin{eqnarray*} \\nabla_X^sY(\\gamma) &=& D_XY(\\gamma) + \\frac{1}{2}(1+\\Delta)^{-s}\n \\sum_{k=1}^s(-1)^k\\left(\\begin{array}{c}s\\\\k\\end{array}\\right)\n\\sum_{j=0}^{2k-1} \\nabla_{\\dot\\gamma}^j(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}Y)\\\\\n&&\\qquad + (X\\leftrightarrow Y)\\\\\n  &&\\qquad \n +  \\sum_{k=1}^s(-1)^k\\left(\\begin{array}{c}s\\\\k\\end{array}\\right)\n\\sum_{j=0}^{2k-1} (-1)^{j+1}   (1+\\Delta)^{-s} R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma.\n\\end{eqnarray*}\n\\end{thm}\n\n\\subsection{The Levi-Civita Connection for General $s>\\frac{1}{2}$}\n\n${}$\n\\medskip\n\nIn this subsection, we show that the $H^s$ Levi-Civita connection for general $s>\\frac{1}{2}$ exists in the strong sense of Remark \\ref{lcrem}.\nThe formula is less explicit than in the \n$s\\in {\\mathbb Z}^+$ case, but is good enough for symbol calculations.\n\n\n\n\nBy Lemma \\ref{lem: LCs}, we have to examine the term $A_XY$, which, if it exists, is\n characterized by (\\ref{axy}):\n$$-\\frac{1}{2}\\ipo{[D_Z,(1+\\Delta)^s]X}{Y} = \\ips{A_XY}{Z}$$\nfor $Z\\in H^s$.  As explained in Remark \\ref{lcrem}, we may take\n$X, Y\\in H^{s'-1}.$\nThroughout this section we assume that $s'\\gg s$.\n\nThe following lemma extends Lemma \\ref{bracketterms}.\n\\begin{lem}\\label{pdo}\n (i)   For fixed $Z\\in H^{s'-1}$,  $[D_Z,(1+\\Delta)^s] X$ is a $\\Psi$DO of\n  order $2s-1$ in $X$. For ${\\rm Re}(s)\\neq 0$, the principal symbol of $[D_Z,(1+\\Delta)^s]$ is \n  linear in $s$.\n  \n  (ii) For fixed $X\\in H^{s'-1}$, $[D_Z,(1+\\Delta)^s]X$ is a $\\Psi{\\rm DO}$ \n  of order $2s-1$   in $Z$.\n\\end{lem}\n\nAs usual, ``of order $2s-1$\" means ``of order at most $2s-1.$\"  \n\n\n\n\\begin{proof}\n(i) For $f:LM\\to {\\mathbb C}$, we get $[D_Z,(1+\\Delta)^s]fX = f[D_Z,(1+\\Delta)^s]X$, since $[f,(1+\\Delta)^s]=0.$  \nTherefore, $[D_Z,(1+\\Delta)^s]X$ depends only on $X|_\\gamma.$\n\nBy Lemma \\ref{lem:l2lc}, $D_Z = \\delta_Z + \\Gamma \\cdot Z$ in shorthand exponential \ncoordinates.  The Christoffel symbol term is zeroth order and $(1+\\Delta)^s$ has scalar leading order symbol, so $[\\Gamma\\cdot Z,(1+\\Delta)^s]$ has order $2s-1.$  \n\nFrom the integral expression for \n$(1+\\Delta)^s$, it is immediate that \n\\begin{eqnarray}\\label{immediate}\n[\\delta_Z,(1+\\Delta)^s]X &=& (\\delta_Z(1+\\Delta)^s) X + (1+\\Delta)^s\\delta_Z X - (1+\\Delta)^s\\delta_ZX\\\\\n&=& (\\delta_Z(1+\\Delta)^s) X.\\nonumber\n\\end{eqnarray}\n$\\delta_Z(1+\\Delta)^s$ is a limit of differences of $\\Psi{\\rm DO}$s on bundles isomorphic to $\\gamma^*TM$.\nSince the algebra of $\\Psi{\\rm DO}$s is closed in the Fr\\'echet topology\nof all $C^k$ seminorms\nof symbols and smoothing terms\non compact sets, $\\delta_Z(1+\\Delta)^s$ is a $\\Psi{\\rm DO}.$\n\nSince $(1+\\Delta)^s$ has order $2s$ and has scalar leading order symbol, \n$[D_Z,(1+\\Delta)^s]$ have order $2s-1$.  For later purposes (\\S3.2), we compute some explicit symbols.  \n\nAssume Re$(s)<0.$  As in the construction of $(1+\\Delta)^s$,\n we will compute what the symbol asymptotics\nof $\\delta_Z(1+\\Delta)^s$ should\nbe, and then construct an operator with these asymptotics.\nFrom the functional calculus for unbounded operators, we have\n\\begin{eqnarray}\\label{funcalc}\n\\delta_Z(1+\\Delta)^s &=& \\delta_Z\\left(\\frac{i}{2\\pi}\\int_\\Gamma\n\\lambda^s(1+\\Delta-\\lambda)^{-1}d\\lambda\\right)\\nonumber\\\\\n&=& \\frac{i}{2\\pi}\\int_\\Gamma\n\\lambda^s\\delta_Z (1+\\Delta-\\lambda)^{-1}d\\lambda\\\\\n&=& -\\frac{i}{2\\pi}\\int_\\Gamma\n\\lambda^s (1+\\Delta-\\lambda)^{-1} (\\delta_Z\\Delta)\n(1+\\Delta-\\lambda)^{-1}d\\lambda,\\nonumber\n\\end{eqnarray}\nwhere $\\Gamma$ is a contour around the spectrum of $1+\\Delta$, and the\nhypothesis on $s$ justifies the exchange of $\\delta_Z$ and the integral.  The\noperator $A =\n(1+\\Delta-\\lambda)^{-1} \\delta_Z\\Delta  (1+\\Delta-\\lambda)^{-1}$ is a $\\Psi{\\rm DO}$\nof order $-3$\n with top order symbol\n\\begin{eqnarray*} \\sigma_{-3}(A)(\\theta,\\xi)^\\ell_j &=&\n(\\xi^2-\\lambda)^{-1}\\delta^\\ell_k (-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{\\mu}{k}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{\\mu}{k}\\dot Z^\\nu) \\xi (\\xi^2-\\lambda)^{-1}\\delta^\\mu_j\\\\\n&=&\n(-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot Z^\\nu)\n\\xi (\\xi^2-\\lambda)^{-2}.\n\\end{eqnarray*}\nThus the top order symbol of $\\delta_Z(1+\\Delta)^s$ should be\n\\begin{eqnarray}\\label{tsmo}\n \\sigma_{2s-1}(\\delta_Z(1+\\Delta)^s)(\\theta,\\xi)^\\ell_j\n&=& -\\frac{i}{2\\pi}\\int_\\Gamma\n\\lambda^s (-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot Z^\\nu)\n\\xi (\\xi^2-\\lambda)^{-2} d\\lambda  \\nonumber\\\\\n&=& \\frac{i}{2\\pi}\\int_\\Gamma s\\lambda^{s-1}\n(-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot Z^\\nu)\n\\xi (\\xi^2-\\lambda)^{-1} d\\lambda \\nonumber\\\\\n&=& s(-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot Z^\\nu)\\xi (\\xi^2-\\lambda)^{s-1}.\n\\end{eqnarray}\nSimilarly, all the terms in the symbol asymptotics for $A$ are of the form\n$B^\\ell_j \\xi^n(\\xi^2-\\lambda)^m$ for some matrices $B^\\ell_j =\nB^\\ell_j(n,m).$   This produces a symbol sequence $\n\\sum_{k\\in {\\mathbb Z}^+}\\sigma_{2s-k}$,  and there exists a  $\\Psi{\\rm DO}$ $P$ with $\\sigma(P) =\n\\sum \\sigma_{2s-k}$.  (As in \\S\\ref{pdoreview}, we \nfirst produce operators $P_i$ on a coordinate cover $U_i$ of $S^1$, \nand then set\n$P = \\sum_i\\phi_iP_i\\psi_i$.) The construction\ndepends on\nthe choice of local coordinates\ncovering $\\gamma$, the partition of unity and cutoff\nfunctions as above, and a cutoff function in $\\xi$; as\nusual, different choices change the operator by a smoothing operator.\nStandard estimates \nshow that $P-\\delta_Z(1+\\Delta)^s$ is a smoothing\noperator, this verifies explicitly that  $\\delta_Z(1+\\Delta)^s$ is a $\\Psi{\\rm DO}$ of order $2s-1.$\n\n\nFor Re$(s) >0$,  motivated by differentiating $(1+\\Delta)^{-s}\\circ(1+\\Delta)^s = {\\rm\n  Id}$, we set\n\\begin{equation}\\label{abc}\n\\delta_Z(1+\\Delta)^s = -(1+\\Delta)^s\\circ\\delta_Z(1+\\Delta)^{-s}\\circ(1+\\Delta)^s.\n\\end{equation}\nThis is again a $\\Psi{\\rm DO}$ of order $2s-1$ with principal symbol linear in $s$. \n\n (ii) As a $\\Psi{\\rm DO}$ of order $2s$, $(1+\\Delta)^s$ has the expression\n $$(1+\\Delta)^s X(\\gamma)(\\theta) = \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi,$$\n where we omit the cover of $S^1$ and its partition of unity on the right hand side.\n Here \n p(\\theta,\\xi)$ is the symbol of $(1+\\Delta)^s$, which has the asymptotic expansion\n $\n p(\\theta,\\xi) \\sim \\sum_{k=0}^\\infty p_{2s-k }(\\theta,\\xi).$$\n The covariant derivative along $\\gamma$ on\n$Y\\in\\Gamma(\\gamma^*TM)$ is given by\n\\begin{eqnarray*}\\frac{DY}{d\\gamma} &=&\n(\\gamma^*\\nabla^{M})_{\\partial_\\theta}(Y) =\n\\partial_\\theta Y + (\\gamma^*\\omega^{M})(\\partial_\\theta)(Y)\\\\\n&=& \\partial_\\theta(Y^i)\\partial_i + \\dot\\gamma^t Y^r\n\\Gamma^j_{tr}\\partial_j,\n\\end{eqnarray*}\nwhere $\\nabla^{M}$ is the Levi-Civita connection on $M$ and $\\omega^{M}$ is the\nconnection one-form in exponential coordinates on $M$. \nFor $\\Delta =\n(\\frac{D}{d\\gamma})^* \\frac{D}{d\\gamma}$, an integration by parts using the\nformula\n$\\partial_tg_{ar} = \\Gamma_{\\ell t}^ng_{rn} + \\Gamma_{rt}^ng_{\\ell n}$  gives\n$$(\\Delta Y)^k = -\\partial^2_\\theta Y^k\n-2\\Gamma_{\\nu\\mu}^k\\dot\\gamma^\\nu\\partial_\\theta Y^\\mu -\\left\n( \\partial_\\theta\\Gamma_{\\nu\\delta}^k\\dot\\gamma^\\nu\n+\\Gamma_{\\nu\\delta}^k\\ddot\\gamma^\\nu +\n\\Gamma_{\\nu\\mu}^k\\Gamma_{\\varepsilon\\delta}^\\mu\\dot\\gamma^\\varepsilon\\dot\\gamma^\\nu\\right)\nY^\\delta.$$\nThus $p_{2s}(\\theta, \\xi) =  |\\xi|^2$ is independent of $\\gamma$, but the lower order symbols  depend on \n derivatives of both $\\gamma$ and the metric on $M$. \n \n We have\n \\begin{eqnarray} [D_Z,(1+\\Delta)^s]X(\\gamma)(\\theta) &=&\n D_Z \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\label{216}\\\\\n &&\\quad - \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) D_ZX(\\gamma)(\\theta')d\\theta' d\\xi. \\label{217}\n \\end{eqnarray}\n In local coordinates, (\\ref{216}) equals\n \\begin{eqnarray} \\label{218}\n \\lefteqn{\n\\left[ D_Z \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a}\\nonumber\\\\\n &=& \\delta_Z\\left[\n \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a(\\theta)\\\\\n &&\\quad + \\Gamma^a_{bc} Z^b(\\gamma)(\\theta) \n \\left[ \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^c(\\theta).\\nonumber\n \\end{eqnarray}\n Here we have suppressed matrix indices in $p$ and $X$.\nWe can bring $\\delta_Z$ past the integral on the right hand side of (\\ref{218}).  If\n$\\gamma_\\epsilon$ is a family of curves with $\\gamma_0 = \\gamma, \\dot\\gamma_\\epsilon = Z$, then\n$$\\delta_Zp(\\theta, \\xi) = \\frac{d}{d\\epsilon}\\biggl|_{_{_{\\epsilon=0}}} p(\\gamma_\\epsilon,\n\\theta,\\xi) = \\frac{d\\gamma_\\epsilon^k}{d\\epsilon}\\biggl|_{_{_{\\epsilon=0}}}\n\\partial_k p(\\gamma,\\theta, \\xi) = Z^k(\\gamma(\\theta))\\cdot \\partial_k p(\\theta,\\xi).$$ Substituting this into (\\ref{218}) gives\n\\begin{eqnarray}\\label{219}\n\\lefteqn{\n\\left[ D_Z \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a}\\nonumber\\\\\n&=& \n\\lefteqn{\n\\left[ \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n Z^k(\\gamma)(\\theta)\\cdot \\partial_k p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a}\\\\\n &&\\quad \n+ \\Gamma^a_{bc} Z^b(\\gamma)(\\theta) \n \\left[ \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^c(\\theta).\\nonumber\\\\\n&&\\qquad + \n \\left[ \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) \\delta_Z X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^c(\\theta).\\nonumber\n\\end{eqnarray}\nSimilarly, (\\ref{217}) equals\n\\begin{eqnarray}\\label{220}\n\\lefteqn{\\left[\\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) D_ZX(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a}\\nonumber\\\\\n &=& \\left[\\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) \\delta_ZX(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a\\\\\n &&\\quad + \n \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi)^a_e\\Gamma^e_{bc}Z^b(\\gamma)(\\theta') X^c(\\gamma)(\\theta')d\\theta' d\\xi.\n \\nonumber\n \\end{eqnarray}\n Substituting (\\ref{219}), (\\ref{220}), into (\\ref{216}), (\\ref{217}), respectively, gives\n \\begin{eqnarray}\\label{221}\n\\lefteqn{( [D_Z,(1+\\Delta)^s]X(\\theta))^a}\\\\\n&=&\nZ^b(\\theta)\\cdot\\left[\\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\xi}\\left(\\partial_bp^a_e(\\theta,\\xi) +\n\\Gamma_{bc}^a(\\gamma(\\theta)p_e^c(\\theta, \\xi)\\right)X^e(\\theta')d\\theta'd\\xi\\right]\\nonumber\\\\\n&&\\quad - \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\xi}p(\\theta,\\xi)^a_e\\Gamma_{bc}^e\n(\\gamma(\\theta'))Z^b(\\theta') X^c(\\theta')d\\theta' d\\xi,\\nonumber\n  \\end{eqnarray}\n where $X(\\theta') = X(\\gamma)(\\theta)$ and similarly for Z.\n \n The first term on the right hand side of (\\ref{221}) is order zero in $Z$; note that\n $0<2s-1$, since $s>\\frac{1}{2}$.  For the last term in (\\ref{221}), we do a change of variables typically used in the proof that the composition of $\\Psi{\\rm DO}$s is a $\\Psi{\\rm DO}.$  Set\n \\begin{equation}\\label{221a}q(\\theta, \\theta', \\xi)^a_b = p(\\theta,\\xi)^a_e \\Gamma_{bc}^e(\\gamma(\\theta'))X^c\n (\\theta'),\n \\end{equation}\n so the last term equals\n \\begin{eqnarray*}\n(PZ)^a(\\theta) &\\stackrel{\\rm def}{=}&  \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\xi}q(\\theta, \\theta', \\xi)^a_b Z^b(\\theta') d\\theta' d\\xi\\\\\n &=& \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\xi}q(\\theta, \\theta', \\xi)^a_b e^{i(\\theta'-\\theta'')\n \\cdot\\eta} Z^b(\\theta'') d\\theta'' d\\eta \\ d\\theta' d\\xi,\n \\end{eqnarray*}\n by applying  Fourier transform and its inverse to $Z$. A little algebra gives\n \\begin{equation}\\label{222}\n (PZ)^a(\\theta) = \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\eta}r(\\theta,\\eta)^a_b Z^b(\\theta')\n d\\theta' d\\eta,\n \\end{equation}\n with \n \\begin{eqnarray*}r(\\theta, \\eta) &=& \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot(\\xi-\\eta)}\n q(\\theta,\\theta', \\xi) d\\theta' d\\xi\\\\\n &=&  \\int_{T^*S^1} e^{it\\cdot\\xi} q(\\theta,\\theta - t, \\eta + \\xi) dt\\  d\\xi.\n \\end{eqnarray*} \n In the last line we continue to abuse notation by treating the integral in local coordinates in \n $t = \\theta-\\theta'$ lying in an interval $I\\subset {\\mathbb R}$ and implicitly\n summing over a cover and partition of unity of $S^1;$ thus we can consider $q$ as a compactly supported function in $t\\in{\\mathbb R}.$\n Substituting in the Taylor expansion of $q(\\theta,\\theta - t, \\eta + \\xi)$ in $\\xi$ gives in local coordinates\n  \\begin{eqnarray}\\label{223a}\n  r(\\theta, \\eta) &=& \\int_{T^*{\\mathbb R}} e^{it\\cdot \\xi} \\left[ \\sum_{\\alpha, |\\alpha|=0}^N\n \\frac{1}{\\alpha!} \\partial_\\xi^\\alpha|_{\\xi=0} q(\\theta,\\theta-t, \\eta+\\xi)\\xi^\\alpha + {\\rm O}\n (|\\xi|^{N+1})\\right] dt \\  d\\xi\\nonumber\\\\\n &=& \\sum_{\\alpha, |\\alpha|=0}^N \\frac{i^{|\\alpha|}}{\\alpha!} \\partial^\\alpha_t\\partial^\\alpha\n _\\xi q(\\theta, \\theta, \\eta) + {\\rm O} (|\\xi|^{N+1}).\n \\end{eqnarray}\n Thus $P$ in (\\ref{222}) is a $\\Psi{\\rm DO}$ with apparent top order symbol \n $q(\\theta, \\theta, \\eta)$, which by (\\ref{221a}) has order $2s.$  The top order symbol can be computed in any local coordinates on $S^1$ and $\\gamma^*TM$.  If we choose\n manifold coordinates (see \\S2.3) which are \nRiemannian normal coordinates centered at $\\gamma(\\theta)$, the Christoffel symbols vanish at this point, and\n so\n $$q(\\theta, \\theta, \\eta)^a_b = p(\\theta,\\xi)^a_e\\Gamma_{bc}^e(\\gamma(\\theta)) X^c(\\theta)\n =0.$$\n Thus $P$ is in fact of order $2s-1$, and so both terms on the right hand side of (\\ref{221}) have order at most $2s-1$.\n\n\\end{proof}\n\n\n\\begin{rem} (i) For $s\\in{\\mathbb Z}^+$,  $\\delta_Z(1+\\Delta)^s$ differs\nfrom the usual definition by a smoothing operator.  \n\n(ii) For all $s$, the proof of Lemma \\ref{pdo}(i) shows that\n$\\sigma(\\delta_Z(1+\\Delta)^s) = \\delta_Z(\\sigma((1+\\Delta)^s)).$\n\n\\end{rem}\n\nWe can now complete the computation of the Levi-Civita connection for general $s.$\n\nLet $[D_\\cdot,(1+\\Delta)^s]X^*$ be the formal $L^2$ adjoint  of $[D_\\cdot,(1+\\Delta)^s]X$.\nWe abbreviate $[D_\\cdot,(1+\\Delta)^s]X^*(Y)$ by $[D_Y,(1+\\Delta)^s]X^*.$\n\n\n\n\\begin{thm}  \\label{thm25} (i) For $s>\\frac{1}{2}$, \nThe Levi-Civita connection for the $H^s$ metric is given by\n\\begin{eqnarray}\\label{quick}\\nabla_X^sY &=& D_XY + \\frac{1}{2}(1+\\Delta)^{-s}[D_X, (1+\\Delta)^s]Y +\n  \\frac{1}{2}(1+\\Delta)^{-s}[D_Y, (1+\\Delta)^s]X \\nonumber\\\\\n&&\\quad -\\frac{1}{2} (1+\\Delta)^{-s}[D_Y,(1+\\Delta)^s]X^*.\n\\end{eqnarray}\n\n(ii) The connection one-form $\\omega^s$ in exponential coordinates is given by\n\\begin{eqnarray}\\label{223}\\lefteqn{\\omega^s_X(Y)(\\gamma) (\\theta)}\\\\\n&=& \\omega^M(Y)(\\gamma(\\theta)) + \n\\left(\\frac{1}{2}(1+\\Delta)^{-s}[D_X, (1+\\Delta)^s]Y +\n  \\frac{1}{2}(1+\\Delta)^{-s}[D_Y, (1+\\Delta)^s]X \\right.\\nonumber\\\\\n  &&\\quad \\left.\n-\\frac{1}{2} (1+\\Delta)^{-s}[D_Y,(1+\\Delta)^s]X^*\\right)(\\gamma)(\\theta).\\nonumber\n\\end{eqnarray}\n\n(iii) The connection one-form takes values in zeroth order $\\Psi{\\rm DO}$s.\n\\end{thm}\n\n\\begin{proof}  Since $[D_Z,(1+\\Delta)^s]X$ is a $\\Psi{\\rm DO}$ in $Z$ of order $2s-1$, its formal adjoint is\na $\\Psi{\\rm DO}$ of the same order.  Thus\n$$\\langle [D_Z,(1+\\Delta)^s]X,Y\\rangle_0 = \\langle Z, [D_\\cdot, (1+\\Delta)^s]X^*(Y)\\rangle\n= \\langle Z, (1+\\Delta)^{-s}[D_Y,(1+\\Delta)^s]X^*\\rangle_s.$$\nThus $A_XY$ in (\\ref{axy}) satisfies\n$A_XY = (1+\\Delta)^{-s}[D_Y,(1+\\Delta)^s]X^*.$  Lemma \\ref{lem: LCs} applies to all $s>\\frac{1}{2}$, \nso (i) follows.  (ii) follows as \nin Corollary \\ref{cor2}.  Since $\\omega^M$ is zeroth order and all \nother terms have order $-1$, (iii) holds as well.\n\\end{proof}\n\n\\begin{rem}  This theorem implies that the Levi-Civita connection exists for the \n$H^s$ metric in the strong sense:  for $X\\in  T_\\gamma LM =H^{s'-1}(\\gamma^*TM)$\nand $Y\\in H^{s'-1}(\\cdot^*TM)$ a smooth vector field on $LM = H^{s'}(S^1,M)$,\n $\\nabla^s_XY(\\gamma)\\in H^{s'-1}(\\gamma^*TM).$  (See Remark\n2.1.)  For each term except $D_XY$ on the right hand side of (\\ref{quick}) is order\n$-1$ in $Y$, and so takes $H^{s'-1}$ to $H^{s'}\\subset H^{s'-1}.$  For $D_XY = \\delta_XY + \\Gamma\\cdot Y$, $\\Gamma$ is zeroth order and so bounded on $H^{s'-1}.$  Finally, \nthe definition of a smooth vector field on $LM$ implies that $\\delta_XY$ stays in $H^{s'-1}$\nfor all $X$.\n\\end{rem}\n\n\n\n\n\n\n\n\\subsection{{\\bf Extensions of the Frame Bundle of $LM$}}\\label{extframe}\n\nIn this subsection we discuss the choice of structure group for the\n$H^s$ and Levi-Civita connections on $LM.$\n\nLet ${\\mathcal H}$ be the Hilbert space\n $H^{s_0}(\\gamma^*TM)$ for \na fixed $s_0$ and $\\gamma.$ \n Let $GL({\\mathcal H})$ be the group of bounded invertible linear\noperators on ${\\mathcal H}$; inverses of elements are bounded by the closed graph\ntheorem.   $GL({\\mathcal H})$ has the subset\ntopology of the norm topology on ${\\mathcal B}({\\mathcal H})$, the bounded linear\noperators on ${\\mathcal H}$.\n$GL({\\mathcal H})$ is an infinite dimensional Banach Lie group, as a group which\nis an open subset of the infinite dimensional Hilbert manifold \n${\\mathcal B}({\\mathcal H})$\n\\cite[p.~59]{Omori}, and has Lie algebra \n${\\mathcal B}({\\mathcal H})$. Let $\\Psi{\\rm DO}_{\\leq 0}, \n\\Psi{\\rm DO}_0^*$ denote the algebra of classical\n$\\Psi{\\rm DO}$s of nonpositive order \nand the group of invertible zeroth order $\\Psi{\\rm DO}$s, respectively,\nwhere all $\\Psi{\\rm DO}$s act on ${\\mathcal H}.$   \nNote that $\\Psi{\\rm DO}_0^*\\subset GL({\\mathcal H}).$\n   \n\\begin{rem} \nThe inclusions of $\\Psi{\\rm DO}_0^*, \\Psi{\\rm DO}_{\\leq 0}$ into $GL({\\mathcal H}), {\\mathcal\n    B}({\\mathcal H})$ are trivially continuous in the subset topology.\nFor the Fr\\'echet topology on $\\Psi{\\rm DO}_{\\leq 0}$, \nthe  inclusion is \ncontinuous as in \\cite{lrst}.\n\\end{rem}\n\nWe recall\nthe relationship between\n the  connection one-form $\\theta$ on the frame bundle $FN$ of a\n manifold $N$\nand\nlocal expressions for the connection on $TN.$ For $U\\subset N$,\n let $\\chi:U\\to FN$ be a local section.\n A metric  connection $\\nabla$ on $TN$ with local\nconnection one-form $\\omega$ determines a connection $\\theta_{FN}\\in\n \\Lambda^1(FN, {\\mathfrak o}(n))$ on $FN$\nby {\\it (i)} $\\theta_{FN}$ is the Maurer-Cartan one-form on each fiber,\nand {\\it (ii) }\n$\\theta_{FN}(Y_u)=\\omega (X_p),$ for $ Y_u=\\chi_*X_p$\n\\cite[Ch.~8, Vol.~II]{Spi}, or equivalently\n$\\chi^*\\theta_{FN} = \\omega.$\n\n\nThis applies to $N=LM.$\nThe frame bundle $FLM\\to LM$ is constructed\nas in the finite dimensional case. The\nfiber over $\\gamma$ is isomorphic to the gauge group ${\\mathcal G}$ of ${\\mathcal R}$\nand fibers are glued by the transition functions for\n$TLM$. Thus the frame bundle is\ntopologically a\n${\\mathcal G}$-bundle.\n\nHowever, by Theorem \\ref{thm25},\nthe Levi-Civita connection one-form $\\omega^s_X$\ntakes\nvalues in $\\Psi{\\rm DO}_{\\leq 0}$. \nThe curvature two-form $\\Omega^{s} = d_{LM}\\omega^{s} + \\omega^{s}\\wedge\n\\omega^s$ also takes values in $\\Psi{\\rm DO}_{\\leq 0}.$  (Here $d_{LM}\\omega^{s}(X,Y)$\nis defined by the Cartan formula for the exterior derivative.)\nThese\nforms should take values in the Lie algebra of the structure\ngroup.  Thus we should extend the structure group to the Fr\\'echet Lie group\n $\\Psi{\\rm DO}_0^*$, since its Lie\nalgebra is $\\Psi{\\rm DO}_{\\leq 0}.$  \nThis leads to an extended\nframe bundles, also denoted $FLM$. The  transition\n functions are unchanged, since \n${\\mathcal G} \\subset \\Psi{\\rm DO}_0^*$.\n Thus $(FLM,\\theta^s)$ as a geometric\nbundle (i.e.~as a  bundle with connection $\\theta^s$ associated to\n$\\nabla^{1,s}$) is a $\\Psi{\\rm DO}_0^*$-bundle.\n\nIn summary, for the Levi-Civita connections we have\n$$ \\begin{array}{ccc}\n{\\mathcal G}&\\longrightarrow &FLM\\\\\n& & \\downarrow\\\\\n& & LM\n\\end{array}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\begin{array}{ccc}\n\\Psi{\\rm DO}_0^*&\\longrightarrow &(FLM,\\theta^s)\\\\\n& & \\downarrow\\\\\n& & LM\n\\end{array}\n$$\n\n\n\\begin{rem}\\label{rem:ext}  If\n we extend the structure group of the frame bundle with\n  connection from $\\Psi{\\rm DO}_0^*$ to\n  $GL({\\mathcal H})$, the frame bundle becomes trivial by Kuiper's theorem.  \n \nThus\nthere is a potential loss of information if \nwe pass to the larger frame\n  bundle. \n\n\nThe situation is similar to the following examples.  Let $E\\to S^1$ be\nthe $GL(1,{\\mathbb R})$ (real line)\n bundle with gluing functions (multiplication by) $1$ at $1\\in\nS^1$ and $2$ at $-1\\in S^1.$  $E$ is trivial as a $GL(1,{\\mathbb R})$-bundle, \nwith global section $f$ with $\\lim_{\\theta\\to -\\pi^+}f(e^{i\\theta}) = 1, \nf(1) = 1,\n\\lim_{\\theta\\to\\pi^-}f(e^{i\\theta}) = 1\/2.$  \nHowever, as a $GL(1,{\\mathbb Q})^+$-bundle, $E$ is nontrivial, as a\nglobal section is locally constant. As a second example,\n let $E\\to M$ be a nontrivial\n$GL(n,{\\mathbb C})$-bundle. Embed ${\\mathbb C}^n$ into a Hilbert space ${\\mathcal H}$, and extend $E$\nto an $GL({\\mathcal H})$-bundle ${\\mathcal E}$ \nwith fiber ${\\mathcal H}$ and with the  transition functions for $E$ (extended by the identity in\ndirections perpendicular to the image of $E$).  Then ${\\mathcal E}$ is\ntrivial.\n\n\\end{rem}\n\n\n\n\\section{{\\bf Local Symbol Calculations}}\\label{localsymbols}\n\nIn this section, we  compute the $0$ and $-1$ order symbols of the\nconnection one-form and the curvature two-form of\n the $s=1$ Levi-Civita connection. \n We also compute the $0$ and $-1$ order symbols of the\nconnection one-form for the general $s>\\frac{1}{2}$ connection, and the $0$ order symbol of the \ncurvature of the general $s$ connection.\n These results are used in the calculations of Wodzicki-Chern-Simons\nclasses in \\S6. The formulas  show that\nthe $s$-dependence of these symbols is\nlinear, which will be used to define regularized Wodzicki-Chern-Simons classes\n(see Definition \\ref{def:regularized}).\n\n\\subsection{{\\bf Connection and Curvature Symbols for $s=1$}}\n${}$\n\\medskip\n\nIn this subsection $\\omega = \\omega^1, \\Omega = \\Omega^1.$\n\nUsing Corollary \\ref{cor2}, we can compute these symbols easily. \n\n\\begin{lem} \\label{old2.1}\n(i) At $\\gamma(\\theta)$,\n$\\sigma_0(\\omega_X)^a_b =  (\\omega^M_X)^a_b = \\chw{c}{b}{a}X^c.$\n\n(ii)  \\begin{eqnarray*}\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\omega_X) &=& \\frac{1}{2}(-2R(X,\\dot\\gamma)\n-R(\\cdot,\\dot\\gamma)X + R(X,\\cdot)\\dot\\gamma).\n\\end{eqnarray*}\nEquivalently,\n\\begin{eqnarray*}\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\omega_X)^a_b &=& \n\\frac{1}{2}\n(-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\  a})X^c\\dot\\gamma^d.\n\\end{eqnarray*}\n\\end{lem}\n\n\\begin{proof} (i) For $\\sigma_0(\\omega_X)$, the only term in (\\ref{two}) of order\n  zero is the Christoffel term.  \n\n(ii) For $\\sigma_{-1}(\\omega_X)$, label the last six terms on the right hand side of\n(\\ref{two}) by (a), ..., (f).  By Leibniz rule for the tensors, the only\nterms of order $-1$ come from:\nin (a), $-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) = -R(X, \\dot\\gamma) \\nabla_{\\dot\\gamma}Y +$ lower order in\n$Y$;\nin (b), the term  $-R(X, \\dot\\gamma) \\nabla_{\\dot\\gamma}Y$;\nin (c), the term $-R(\\nabla_{\\dot\\gamma}Y, \\dot\\gamma)X$;\nin (e), the term $R(X,\\nabla_{\\dot\\gamma}Y)\\dot\\gamma.$\n\n\nFor any vectors $Z, W$, the curvature endomorphism $R(Z, W): TM\\to TM$ has\n$$R(Z,W)^a_b = R_{c d b}^{\\ \\ \\ a}Z^cW^d.$$\n  Also, since $(\\nabla_{\\dot\\gamma}Y)^a =\n\\frac{d}{d\\theta}Y^a $ plus zeroth order terms,\n$\\sigma_{1}(\\nabla_{\\dot\\gamma}) =\ni\\xi\\cdot Id.$\nThus in (a) and (b), \n$\\sigma_1(-R(X, \\dot\\gamma) \\nabla_{\\dot\\gamma})^a_b = -R_{cdb}^{\\ \\ \\ a}X^c\\dot\\gamma^d\\xi.$\n\nFor (c), we have $-R(\\nabla_{\\dot\\gamma}Y, \\dot\\gamma)X = -R_{cdb}^{\\ \\ \\\n  a}(\\nabla_{\\dot\\gamma}Y)^c\\dot\\gamma^d X^b\\partial_a$, so the top order symbol is\n$-R_{cdb}^{\\ \\ \\ a}\\xi\\dot\\gamma^dX^b = -R_{bdc}^{\\ \\ \\ a}\\xi\\dot\\gamma^d X^c.$\n\nFor (e), we have $R(X,\\nabla_{\\dot\\gamma}Y)\\dot\\gamma = R_{cdb}^{\\ \\ \\ a}X^c(\\nabla_{\\dot\\gamma}Y)^d\n\\dot\\gamma^b\\partial_a$, so the top order symbol is \n$R_{cdb}^{\\ \\ \\ a}X^c\\xi \\dot\\gamma^b = R_{cbd}^{\\ \\ \\ a}X^c\\xi \\dot\\gamma^d.$\n\nSince the top order symbol of $(1+\\Delta)^{-1}$ is $|\\xi|^{-2}$, adding these four terms\nfinishes the proof. \n\\end{proof}\n \n We now compute the top symbols of the curvature tensor.  $\\sigma_{-1}(\\Omega)$ involves\n the covariant derivative of the curvature tensor on $M$, but fortunately this symbol\n will not be needed in Part II.\n \n\n\\begin{lem}\\label{old2.2}\n(i) \n$\\sigma_0(\\Omega(X,Y))^a_b =  R^M(X,Y)^a_b = R_{cdb}^{\\ \\ \\ a}X^cY^d.$\n\n(ii) \\begin{eqnarray*}\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\Omega(X,Y)) &=&\n\\frac{1}{2}\\left(\\nabla_X[-2R(Y,\\dot\\gamma) - R(\\cdot,\\dot\\gamma)Y +\n  R(Y,\\cdot)\\dot\\gamma]\\right.\\\\\n&&\\qquad \\left. - (X\\leftrightarrow Y) \\right.\\\\\n&&\\qquad\\left. - [-2R([X,Y],\\dot\\gamma) -R(\\cdot,\\dot\\gamma)[X,Y] + R([X,Y],\\cdot)\\dot\\gamma] \\right).\n\\end{eqnarray*}\nEquivalently, in Riemannian normal coordinates on $M$ centered at $\\gamma(\\theta)$,\n\\begin{eqnarray}\\label{moc}\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\Omega(X,Y))^a_b &=& \\frac{1}{2}\nX[(-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a})\\dot\\gamma^d]Y^c - (X\\leftrightarrow Y)\\nonumber\\\\\n&=&\\frac{1}{2}\nX[-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a}]\\dot\\gamma^dY^c  -(X\\leftrightarrow Y)\\\\\n&&\\qquad +\n\\frac{1}{2}[-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a}]\\dot X^dY^c - (X\\leftrightarrow Y)\\nonumber\n\\end{eqnarray}\n\\end{lem}\n\n\n\n\n\n\\begin{proof} \n(i)\n\\begin{eqnarray*} \\sigma_0(\\Omega(X,Y))^a_b &=& \\sigma_0((d\\omega +\n  \\omega\\wedge\\omega)(X,Y))^a_b\\\\\n&=& [(d\\sigma_0(\\omega) + \\sigma_0(\\omega)\\wedge\\sigma_0(\\omega))(X,Y)]^a_b\\\\\n&=& [(d\\omega^M + \\omega^M\\wedge\\omega^M)(X,Y)]^a_b\\\\\n&=& R^M(X,Y)^a_b = R_{cdb}^{\\ \\ \\ a}X^cY^d.\n\\end{eqnarray*}\n\n(ii) Since $\\sigma_0(\\omega_X)$ is independent of $\\xi$, after dividing by\n$i|\\xi|^{-2}\\xi$ we have\n\\begin{eqnarray*}\\sigma_{-1}(\\Omega(X,Y))^a_b &=& (d\\sigma_{-1}(\\omega)\n  (X,Y))^a_b + \\sigma_0(\\omega_X)^a_c\\sigma_{-1}(\\omega_Y)^c_b\n+ \\sigma_{-1}(\\omega_X)^a_c\\sigma_{0}(\\omega_Y)^c_b\\\\\n&&\\qquad\n-\\sigma_0(\\omega_Y)^a_c\\sigma_{-1}(\\omega_X)^c_b\n+ \\sigma_{-1}(\\omega_Y)^a_c\\sigma_{0}(\\omega_X)^c_b.\n\\end{eqnarray*}\nAs an operator on sections of $\\gamma^*TM$, \n$\\Omega^{LM} - \\Omega^M$ has order $-1$  so $\\sigma_{-1}(\\Omega^{LM})\n= \\sigma_{-1}(\\Omega^{LM} -\\Omega^M)$ is independent of coordinates.\nIn Riemannian normal coordinates at $\\gamma(\\theta)$, $\\sigma_0(\\omega_X) = \\sigma_0(\\omega_Y) = 0$, so\n\\begin{eqnarray*}\\sigma_{-1}(\\Omega(X,Y))^a_b &=&\n  (d\\sigma_{-1}(\\omega)(X,Y))^a_b\\\\\n&=& X(\\sigma_{-1}(\\omega_Y))^a_b - Y(\\sigma_{-1}(\\omega_X))^a_b\n  -\\sigma_{-1}(\\omega_{[X.Y]})^a_b\\\\\n&=& \\frac{1}{2} X[(-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a}]Y^c\\dot\\gamma^d] - (X\\leftrightarrow Y)\\\\\n&&\\qquad -\\frac{1}{2}( -2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a}][X,Y]^c\\dot\\gamma^d.\n\\end{eqnarray*}\nThe terms involving $X(Y^c) - Y(X^c) - [X,Y]^c$ cancel (as they must, since the symbol two-form \ncannot involve derivatives of $X$ or $Y$).  Thus\n$$\\sigma_{-1}(\\Omega(X,Y))^a_b = \\frac{1}{2} X[(-2R_{cdb}^{\\ \\ \\ a} \n-R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\  a})Y^c\\dot\\gamma^d] - (X\\leftrightarrow Y).$$\n\nThis gives the first coordinate expression in (\\ref{moc}). The second expression follows from\n $X(\\dot\\gamma^d) = \\dot X^d $ (see (\\ref{badterms})).\n\nTo convert from the coordinate expression to the covariant expression, we follow the \nusual procedure of changing ordinary derivatives to covariant derivatives and adding bracket terms.  For example,\n\\begin{eqnarray*}\\nabla_X(R(Y,\\dot\\gamma)) &=& (\\nabla_XR)(Y,\\dot\\gamma) + R(\\nabla_XY,\\dot\\gamma)\n  + R(Y,\\nabla_X\\dot\\gamma) \\\\\n&=& X^iR_{cdb\\ ;i}^{\\ \\ \\ a}Y^c\\dot\\gamma^d + R(\\nabla_XY,\\dot\\gamma) + R_{cdb}^{\\ \\ \\\n    a}Y^c(\\nabla_X\\dot\\gamma)^d.\n\\end{eqnarray*}\nIn Riemannian normal coordinates at\n$\\gamma(\\theta)$, we have $X^iR_{cdb\\ ;i}^{\\ \\ \\ a} = X^i\\partial_i R_{cdb}^{\\\n  \\ \\ a} = X(R_{cdb}^{\\ \\ \\ a})$ and $(\\nabla_X\\dot\\gamma)^d =  X(\\dot\\gamma^d).$ \n \n  Thus \n  $$\\nabla_X(R(Y,\\dot\\gamma)) -(X\\leftrightarrow Y) - R([X,Y],\\dot\\gamma) = X(R_{cdb}^{\\ \\ \\ a}\\dot\\gamma^d)Y^c - (X\\leftrightarrow Y).$$\n  The other terms are handled similarly.\n  \\end{proof} \n\n\\subsection{{\\bf Connection and Curvature Symbols for General $s$}}\n${}$\n\\medskip\n\nThe noteworthy feature of these computations is the linear dependence of $\\sigma_{-1}(\\omega^{s})$ on $s$.  \n\nLet $g$ be the Riemannian metric on $M$.\n\n\\begin{lem}\\label{lem:33}\n(i) At $\\gamma(\\theta)$,\n$\\sigma_0(\\omega^s_X)^a_b =  (\\omega^M_X)^a_b = \\chw{c}{b}{a}X^c.$\n\n(ii) $\\sigma_0(\\Omega^s(X,Y))^a_b =  R^M(X,Y)^a_b = R_{cdb}^{\\ \\ \\ a}X^cY^d.$\n\n\n(iii)  $\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\omega^s_X)^a_b = s T(X,\\dot\\gamma, g)$,\nwhere $T(X, \\dot\\gamma, g)$  is tensorial and independent of $s$. \n\\end{lem}\n\n\n\\begin{proof} (i)  By Lemma \\ref{pdo}, the only term of order zero in (\\ref{223}) \n is $\\omega^M_X.$\n\n(ii)  The proof of Lemma \\ref{old2.2}(ii) carries over.  \n\n(iii) By Theorem \\ref{thm25}, we have to compute $\\sigma_{2s-1}$ for $[D_X,(1+\\Delta)^s]$, \n$[D_\\cdot,(1+\\Delta)^s]X$, and $[D_\\cdot,(1+\\Delta)^s]X^*$, as \n$\\sigma_{-1} ((1+\\Delta)^{-s}[D_X,(1+\\Delta)^s]) =  |\\xi|^{-2s}\\sigma_{-1}([D_X,(1+\\Delta)^s])$, etc.\n\nWrite $D_X = \\delta_X + \\Gamma\\cdot X$ in shorthand.  Since $(1+\\Delta)^s$ has scalar leading order symbol, $[\\Gamma\\cdot X,(1+\\Delta)^s]$ has order $2s-1.$  Thus\nwe can compute $\\sigma_{2s-1}([\\Gamma\\cdot X,(1+\\Delta)^s])$ in any coordinate system.  \nIn  Riemannian normal coordinates centered at $\\gamma(\\theta)$, as in the proof of Lemma \\ref{pdo}(ii),  the Christoffel symbols vanish.\nThus $\\sigma_{2s-1}([\\Gamma\\cdot X,(1+\\Delta)^s]) =0.$\n\nBy (\\ref{tsmo}), $\\sigma_{2s-1}([\\delta_X,(1+\\Delta)^s])$ is $s$ times a tensorial expression in $X, \\dot\\gamma, g$,\nsince $\\partial_i\\Gamma_{\\nu j}^{\\ell} = \\frac{1}{3}(R_{i\\nu j}^{\\ \\ \\ \\ell} +\nR_{ij\\nu }^{\\ \\ \\ \\ell})$ in normal coordinates.  The term with $\\Gamma$ vanishes, so \n$\\sigma_{2s-1}( [D_X,(1+\\Delta)^s]) $ is $s$ times this tensorial expression.\n\nThe argument for $\\sigma_{2s-1}([D_\\cdot,(1+\\Delta)^s]X$ is similar.  The term\nwith \n$\\Gamma  $ vanishes. \nBy (\\ref{222}), (\\ref{223a}), \n$$\\sigma_{2s-1}([\\delta_\\cdot,(1+\\Delta)^s]X)^a_b = \ni\\sum_j\\partial_t^j\\partial_\\xi^j|_{t=0, \\xi=0} (p(\\theta, \\xi)^a_e\\Gamma_{bc}^e(\\gamma-t, \\eta +\\xi) X^c(\\theta-t)).$$\nBy (\\ref{tsmo}), the right hand side is linear in $s$ for Re$(s) <0$.  By (\\ref{abc}), this implies \nthe linearity in $s$ for Re$(s)>0.$  \n\nSince $\\sigma_{2s-1}([D_\\cdot,(1+\\Delta)^s]X^*) = (\\sigma_{2s-1}([D_\\cdot, (1+\\Delta)^s]X))^*$, this \nsymbol is also linear in $s$.\n\\end{proof}  \n\n\n\\section{{\\bf The Loop Group Case}}\n\nIn this section, we relate our work to Freed's work on based loop groups\n$\\Omega G$\n\\cite{Freed}.  We find a particular representation of the loop algebra that\ncontrols \nthe order of the curvature of the $H^1$ metric on $\\Omega G.$\n\n\n$\\Omega G\\subset LG$ has tangent space $T_\\gamma\\Omega G\n= \\{X\\in T_\\gamma LG: X(0) = X(2\\pi) = 0\\}$ in some Sobolev topology.  \nInstead of using \n$D^2\/d\\gamma^2$ to define the Sobolev spaces, the usual choice is\n$\\Delta_{S^1} = -d^2\/d\\theta^2$ coupled to the\nidentity operator on the Lie algebra ${\\mathfrak g}$.  Since this operator has\nno kernel on $T_\\gamma\\Omega M$, \n$1 + \\Delta$ is replaced by\n $\\Delta$.  These changes in the $H^s$ inner product\ndo not alter the spaces of Sobolev sections, but the $H^s$ metrics on $\\Omega G$ are \nno longer induced from a metric on $G$ as in the previous sections.\n\n\nThis simplifies the calculations of the Levi-Civita connections.\nIn particular,\\\\\n $[D_Z,\\Delta^s] = 0$, so there is no term $A_XY$ as in (\\ref{axy}).  \nAs a result, one can work directly with the six term formula (\\ref{5one}).\nFor $X, Y, Z$ left\ninvariant vector fields, the first three terms on the right hand side of \n(\\ref{5one}) vanish. Under the standing assumption that $G$ has a left\ninvariant, \nAd-invariant inner product,\none obtains\n$$2\\nabla^{(s)}_XY = [X,Y] + \\Delta^{-s}[X,\\Delta^sY] +\n\\Delta^{-s}[Y,\\Delta^sX]$$\n\\cite{Freed}.  \n\n\nIt is an interesting question to compute the order of the curvature operator\nas a function of $s$.  For based loops, Freed proved that this order is at\nmost $-1$.  In \\cite{andres}, it is\nshown that the order of $\\Omega^s$ is at most $-2$ for all $s\\neq 1\/2, 1$ on\nboth $\\Omega G$ and $LG$, and is exactly $-2$ for $G$ nonabelian.  \n For the case $s=1$, we have a much stronger result.  \n\n\n\\begin{prop} The curvature of the\nLevi-Civita connection for the $H^1$ inner product on $\\Omega\nG$ associated to $-\\frac{d^2}{d\\theta^2}\\otimes {\\rm Id}$ is a $\\Psi{\\rm DO}$ of order $-\\infty.$\n\\end{prop}\n\n\\noindent {\\sc Proof:}\nWe give two quite different proofs.  \n\nBy \\cite{Freed}, the $s=1$ curvature operator $\\Omega = \\Omega^{1}$\nsatisfies\n$$\\left\\langle \\Omega(X,Y)Z,W\\right\\rangle_1 = \\left(\\int_{S^1}[Y,\\dot\nZ],\\int_{S^1}[X,\\dot W]\\right)_{\\mathfrak g} - (X\\leftrightarrow Y),$$\nwhere  the inner product is the Ad-invariant form on the Lie algebra ${\\mathfrak g}$.  We\nwant to write the right hand side of \nthis equation as an $H^1$ inner product with $W$, in\norder to recognize $\\Omega(X,Y)$ as a $\\Psi{\\rm DO}.$\n\n\nLet $\\{e_i\\}$ be an orthonormal basis of ${\\mathfrak g}$, considered\nas a left-invariant frame of $TG$ and as global sections of $\\gamma^*TG.$\n Let $\\cc{i}{j}{k} = ([e_i,e_j],\ne_k)_{\\mathfrak g}$ be the structure constants of ${\\mathfrak g}.$\n(The Levi-Civita connection on left invariant vector fields\nfor the left invariant metric is\ngiven by $\\nabla_XY = \\frac{1}{2}[X,Y]$, so the structure constants\nare twice the Christoffel symbols.)  For $X = X^ie_i =\nX^i(\\theta)e_i, Y = Y^je_j,$ etc., integration by parts \ngives\n$$\\left\\langle\\Omega(X,Y)Z,W\\right\\rangle_1 = \\left(\\int_{S^1} \\dot\nY^iZ^jd\\theta\\right)\\left( \\int_{S^1}\\dot X^\\ell W^m d\\theta\\right)\n\\cc{i}{j}{k}\\cc{\\ell}{m}{n}\\delta_{kn} - (X\\leftrightarrow Y).$$\nSince\n$$\\int_{S^1}\\cc{\\ell}{m}{n}\\dot X^\\ell W^m =\n\\int_{S^1}\\left(\\delta^{mc}\\cc{\\ell}{c}{n}\\dot X^\\ell\ne_m,W^be_b\\right)_{\\mathfrak g} = \\left\n\\langle \\Delta^{-1}(\\delta^{mc}\\cc{\\ell}{c}{n} \\dot X^\\ell e_m),\nW\\right\\rangle_1,$$\nwe get\n\\begin{eqnarray*}\n\\langle\\Omega(X,Y)Z,W\\rangle_1 &=& \\left\\langle\n \\left[\\int_{S^1} \\dot Y^i Z^j\\right]\n\\cc{i}{j}{k}\\delta_{kn}\\delta^{ms}\\cc{\\ell}{s}{n} \\Delta^{-1}(\\dot\nX^\\ell e_m),W\\right\\rangle_1- (X\\leftrightarrow Y)\\\\\n&=&\\left\\langle \\left[ \\int_{S^1}\na_j^k(\\theta,\\theta')Z^j(\\theta')d\\theta'\\right] e_k,W\\right\\rangle_1,\n\\end{eqnarray*}\nwith\n\\begin{equation}\\label{a}a_j^k(\\theta,\\theta') = \\dot Y^i(\\theta')\n\\cc{i}{j}{r}\\delta_{rn}\\delta^{ms}\\cc{\\ell}{s}{n} \n\\left( \\Delta\n^{-1}( \\dot X^\\ell\ne_m)\\right)^k(\\theta) - (X\\leftrightarrow Y).\n\\end{equation}\n\n\n\n\nWe now show that $Z\\mapsto \\left(\\int_{S^1}\na_j^k(\\theta,\\theta')Z^j(\\theta')d\\theta'\\right)e_k$ is a smoothing\noperator.  Applying Fourier transform and Fourier inversion to $Z^j$\nyields\n\\begin{eqnarray*} \\int_{S^1} a_j^k(\\theta,\\theta')Z^j(\\theta')d\\theta'\n&=& \\int_{S^1\\times{\\mathbb R}\\times S^1}\na_j^k(\\theta,\\theta')e^{i(\\theta'\n-\\theta'')\\cdot\\xi}Z^j(\\theta'')d\\theta''d\\xi d\\theta'\\\\\n&=&\n\\int_{S^1\\times{\\mathbb R}\\times S^1} \\left[ a_j^k(\\theta,\\theta')e^{-i(\\theta\n-\\theta')\\cdot\\xi}\\right]e^{i(\\theta\n-\\theta'')\\cdot\\xi}Z^j(\\theta'')d\\theta''d\\xi d\\theta',\n\\end{eqnarray*}\nso $\\Omega(X,Y)$ is a $\\Psi{\\rm DO} $ with symbol \n\\begin{equation}\\label{b} b_j^k(\\theta,\\xi) =\n\\int_{S^1} a_j^k(\\theta,\\theta') e^{i(\\theta-\\theta')\\cdot\\xi} d\\theta',\n\\end{equation}\nwith the usual mixing of local and global notation.\n\nFor fixed $\\theta$,\n(\\ref{b})  contains the Fourier transform of $\\dot Y^i(\\theta')$  and $\\dot X^i(\\theta')$, as\nthese are the only $\\theta'$-dependent terms in (\\ref{a}).\nSince the\nFourier transform is taken in a local chart with respect to a\npartition of unity, and since in each chart $\\dot Y^i$ and $\\dot X^i$ times the\npartition of unity function is compactly supported, the Fourier\ntransform of $a_j^k$ in each chart is rapidly decreasing.  Thus\n$b_j^k(\\theta,\\xi)$ is the product of a rapidly decreasing function\nwith $e^{i\\theta\\cdot\\xi}$, and hence is of order $-\\infty.$\n\n\nWe now give a second proof.  For all $s$,\n$$\\nabla_X Y = \\frac{1}{2}[X,Y] -\\frac{1}{2} \\Delta^{-s}[\\Delta^sX,Y]\n+\\frac{1}{2}\\Delta^{-s}[X,\\Delta^sY].$$\nLabel the terms on the right hand side (1) -- (3).\n As an operator on $Y$ for fixed $X$, the symbol of (1) is\n$\\sigma((1))^a_\\mu = \\frac{1}{2}X^ec_{\\varepsilon\\mu}^a.$\n Abbreviating $\\xii{-s}$ by $\\xi^{-2s}$, we have\n\\begin{eqnarray*} \\sigma((2))^a_\\mu &\\sim & -\\frac{1}{2}c_{\\varepsilon\\mu}^a\n\\left[ \\xi^{-2s}\\Delta^sX^\\varepsilon -\\frac{2s}{i}\\xi^{-2s-1}\n\\partial_\\theta\\Delta^s X^\\varepsilon  \\right.\\\\\n&&\\ \\ \\  \\left. +\\sum_{\\ell=2}^\\infty\\frac{(-2s)(-2s-1)\n\\ldots(-2s-\\ell+1)}{i^\\ell \\ell!}\\xi^{-2s-\\ell}\n\\partial_\\theta^\\ell\\Delta^s X^\\varepsilon \\right]\\\\\n\\sigma((3))^a_\\mu &\\sim &  \\frac{1}{2}c_{\\varepsilon\\mu}^a\n\\left[ X^\\varepsilon+ \\sum_{\\ell=1}^\\infty \\frac{(-2s)(-2s-1)\n\\ldots(-2s-\\ell+1)}{i^\\ell \\ell!} \\xi^{-\\ell}\\partial_\\theta^\\ell X^\\varepsilon\\right].\n\\end{eqnarray*}\nThus\n\\begin{eqnarray}\\label{fourone}\n\\sigma(\\nabla_X)^a_\\mu  &\\sim& \\frac{1}{2}c_{\\varepsilon\\mu}^a\\left[ 2X^\\varepsilon\n   -\\xi^{-2s}\\Delta^sX^\\varepsilon\n+\\frac{2s}{i}\n\\xi^{-2s-1}\\partial_\\theta\\Delta^sX^\\varepsilon\\right. \\nonumber\\\\\n&&\\ \\ \\\n   -\\sum_{ \\ell=2}^\\infty\\frac{(-2s)(-2s-1)\\ldots(-2s-\\ell+1)}{i^\\ell \\ell!}\n\\xi^{-2s-\\ell}\\partial_\\theta^\\ell\\Delta^s X^\\varepsilon \\\\\n&&\\ \\ \\  \\left. + \\sum_{\\ell=1}^\\infty \\frac{(-2s)(-2s-1)\n\\ldots(-2s-\\ell+1)}{i^\\ell \\ell!} \\xi^{-\\ell}\\partial_\\theta^\\ell\n X^\\varepsilon. \\right].\\nonumber\n\\end{eqnarray}\n\nSet $s=1$ in (\\ref{fourone}), and replace $\\ell$\nby\n$\\ell-2$ in the first infinite sum.  Since $\\Delta = -\\partial_\\theta^2$, a\nlittle algebra gives\n\\begin{equation}\\label{fourtwo}\n\\sigma(\\nabla_X)^a_\\mu \\sim c_{\\varepsilon\\mu}^a\\sum_{\\ell=0}^\\infty\n\\frac{(-1)^\\ell}{i^\\ell}\n\\partial_\\theta^\\ell X^\\varepsilon\\xi^{-\\ell}\n=  {\\operatorname{ad\\,}}\\left( \\sum_{\\ell=0}^\\infty\n\\frac{(-1)^\\ell}{i^\\ell}\\partial_\\theta^\\ell\nX\\xi^{-\\ell}\n\\right).\n\\end{equation}\n\nDenote the infinite sum in the last term of (\\ref{fourtwo})\nby $W(X,\\theta,\\xi)$. The map\n$X\\mapsto W(X,\\theta,\\xi)$ takes the  Lie algebra of left invariant vector\nfields on $LG$ to the Lie algebra\n$L{\\mathfrak g}[[\\xi^{-1}]], $\nthe space of formal $\\Psi{\\rm DO}$s of nonpositive integer order on the trivial bundle\n$S^1\\times{\\mathfrak g} \\to S^1$, where the Lie bracket on the\ntarget involves multiplication of power series and bracketing in\n${\\mathfrak g}.$  We claim that this map is a Lie algebra homomorphism.\nAssuming this, we see that\n\\begin{eqnarray*} \\sigma\\left(\\Omega(X,Y)\\right) &=&\n  \\sigma\\left([\\nabla_X,\\nabla_Y] -\\nabla_{[X,Y]}\\right)\n\\sim \\sigma\\left( [{\\operatorname{ad\\,}} W(X), {\\operatorname{ad\\,}} W(Y)] - {\\operatorname{ad\\,}} W([X,Y]) \\right)\\\\\n&=& \\sigma\\left( {\\operatorname{ad\\,}} ( [W(X), W(Y)]) - {\\operatorname{ad\\,}} W([X,Y]) \\right) = 0,\n\\end{eqnarray*}\nwhich proves that $\\Omega(X,Y)$ is a smoothing operator.\n\nTo prove the claim,\nset $X = x^a_n\\eff{n}e_a, Y =y^b_m\\eff{m}e_b$.\nThen\n\\begin{eqnarray*} W([X,Y]) &=&\n W( x^ny^m\\eff{(n+m)}c_{ab}^k e_k) =\\sum_{\\ell=0}^\\infty \\frac{(-1)^\\ell}\n{i^\\ell } c_{ab}^k \\partial_\\theta^\\ell\n \\left(x^a_ny^b_m\\eff{(n+m)}\\right) \\xi^{-\\ell}e_k\\\\\n  {[} W(X)  ,   W(Y)]\n&=& \\sum_{\\ell=0}^\\infty \\sum_{p+q = \\ell}\n\\frac{(-1)^{p+q}}{i^{p+q}} \\partial_\\theta^p \\left(\nx^a_n\\eff{n}\\right) \\partial_\\theta^q\n\\left( y^b_m\\eff{m}\\right)\\xi^{-(p+q)}c_{ab}^k e_k,\n\\end{eqnarray*}\nand these two sums are clearly equal.\n\\hfill $\\Box$\n\n\\bigskip\n\nIt would be interesting to understand how the map $W$ fits into the\nrepresentation theory of the loop algebra $L{{\\mathfrak g}}.$ \n\\bigskip\n\n\\large\n\\noindent {{\\bf Part II. Characteristic Classes on $LM$}}\n\\normalsize\n\\bigskip\n\n\nIn this part, we construct a general theory of Chern-Simons\nclasses on certain infinite rank bundles including the frame\/tangent bundle of \nloop spaces,\nfollowing the construction of primary characteristic classes\nin \\cite{P-R2}. The primary  classes vanish on the tangent bundles of\nloop spaces, which forces the\nconsideration of secondary classes. \nThe key ingredient is to replace the ordinary matrix trace in the Chern-Weil\ntheory of\n invariant polynomials\non  finite dimensional Lie groups with the Wodzicki residue on invertible bounded\n$\\Psi{\\rm DO}$s.  \n\nAs discussed in the Introduction, there are absolute and relative versions of Chern-Simons theory.  We use the relative version, which assigns an odd degree form to a pair \nof connections.\nIn particular, for $TLM$, we can use the $L^2$ (i.e. $s=0$) and\n $s=1$ Levi-Civita connections to form Wodzicki-Chern-Simons (WCS) classes associated to a metric on $M$. \n \n In \\S\\ref{CSCLS}, we develop the general theory of Wodzicki-Chern and WCS classes for \n bundles with structure group $\\Psi{\\rm DO}_0^*$, the group of invertible classical zeroth order pseudodifferential operators.   We show the vanishing of \n  the Wodzicki-Chern classes of $LM$ and more general mapping spaces. \nAs in finite dimensions, we show the existence of  WCS classes in \n  $H^n(LM,{\\mathbb C})$ if dim$(M) = n$ is odd (Definition \\ref{def:WCS})\n  and give the local expression for the WCS classes associated to the Chern character\n  (Theorem \\ref{thm:5.5}).\n   In Theorem \\ref{WCSvan}, we prove that the\n    Chern character WCS class vanishes if dim$(M) \\equiv 3\n   \\ ({\\rm mod}\\ 4)$.\nIn \\S\\ref{dimfive}, we associate to every circle action $a:S^1\\times M^n\\to M^n$\n an $n$-cycle $[a]$\n in $LM$.  For a specific metric on $S^2\\times S^3$ and a specific circle action $a,$\n we prove via exact computer calculations that the WCS class is nonzero by integrating it over $[a].$\n Since the corresponding integral for the cycle associated to the trivial action \n is zero, $a$ cannot be homotoped to the trivial action. \nWe use this result to prove that $\\pi_1({\\rm Diff}\n (S^2\\times S^3))$ is infinite.\n\n\nThroughout this part, $H^*$ always refers to de Rham cohomology for complex valued forms.  By \\cite{beggs}, $H^*(LM)\\simeq H^*_{\\rm sing}(LM,{\\mathbb C}).$\n\n\n\\section{{\\bf Chern-Simons Classes on Loop Spaces}}\\label{CSCLS}\n\nWe begin in \\S5.1 with a review of Chern-Weil and Chern-Simons theory in\nfinite dimensions, following  \\cite{C-S}.  \n\nIn\n\\S5.2, we discuss Chern-Weil and Chern-Simons theory on a class of infinite rank bundles\nincluding the frame bundles of loop spaces.  As in \\S2.7,  the geometric\nstructure group of these bundles\n is $\\Psi{\\rm DO}_0^*$, so we need a trace on the Lie algebra\n$\\Psi{\\rm DO}_{\\leq 0}$ to define invariant polynomials.  There are two\ntypes of  traces, one given by taking the zeroth order symbol and one given by\nthe Wodzicki residue \\cite{paycha-lescure}, \\cite{ponge}.  Here we only consider the \nWodzicki residue trace.  \n\n\nUsing this trace, we generalize the usual definitions of Chern and Chern-Simons classes in\nde Rham cohomology.  In particular,\ngiven a $U(n)$-invariant polynomial $P$ of degree $k$, we define a corresponding \nWCS class $CS^W_P\\in H^{2k-1}(LM)$ if dim$(M) = 2k-1.$    We are forced to consider these secondary classes, because the Wodzicki-Chern classes of mapping spaces\n${\\rm Maps} (N,M)$ vanish.  In Theorem \\ref{thm:5.5}, we give an exact expression for the WCS\nclasses associated to the Chern character.\nIn Theorem \\ref{WCSvan}, we show that these WCS classes in $H^{4k+3}(LM^{4k+3})$\nvanish; in contrast, in finite dimensions, the Chern-Simons classes associated to the Chern character vanish in $H^{4k+1}(M^{4k+1}).$\n\n\n\n\n\\subsection{{\\bf Chern-Weil  and Chern-Simons Theory for Finite Dimensional\n  Bundles} }\n\nWe first review the Chern-Weil construction. \nLet $G$ be a finite dimensional Lie group with Lie algebra ${\\mathfrak  g}$, and let\n $G\\to F\\to M$ be a principal $G$-bundle over a manifold $M$. \nSet $\n {\\mathfrak  g}^k={\\mathfrak  g}^{\\otimes k}$ and let\n\\begin{equation*}I^k(G)\n= \\{P:{\\mathfrak  g}^k\\to {\\mathbb C}\\ | P\\ \\text{symmetric,\n  multilinear, Ad-invariant}\\}\n\\end{equation*}\nbe the degree $k$ Ad-invariant polynomials on ${\\mathfrak  g}.$\n\n\n\\begin{rem}\nFor classical Lie groups $G$, $I^k(G)$ is generated by the polarization of\nthe Newton polynomials $\\operatorname{Tr}(A^\\ell)$, where $\\operatorname{Tr}$ is the usual trace on finite\ndimensional matrices.\n\\end{rem}\n\n\nFor $\\phi\\in\\Lambda^\\ell(F,{\\mathfrak  g}^k)$, $P\\in I^k(G)$, set\n$P(\\phi)=P\\circ \\phi\\in\\Lambda^\\ell(F)$. \n\n\n\\begin{thm}[The Chern-Weil Homomorphism \\cite{K-N}] \\label{previous}\nLet $F\\to M$ have a connection $\\theta$ with curvature $\\Omega_F\\in\n\\Lambda^2(F,{\\mathfrak  g})$. For $P\\in I^k(G)$, $P(\\Omega_F)$ is a closed\n invariant real form on $F$, and so\ndetermines a closed form\n$P(\\Omega_M)\\in \\Lambda^{2k}(M)$.\nThe Chern-Weil map\n\\begin{equation*}\n\\oplus_{k}I^k(G)\\to H^{*}(M), \\ P\\mapsto [P(\\Omega_M)]\n\\end{equation*}\nis a well-defined algebra homomorphism, and in particular is independent of the choice of\nconnection on $F$.\n\\end{thm} \n\nThe proof depends on:\n\\begin{itemize}\n\\item (The {\\it commutativity property}) \nFor $\\phi\\in\\Lambda^{\\ell}(F,{\\mathfrak  g}^k)$, \n\\begin{equation}\\label{eq:deri}\nd(P(\\phi))=P(d\\phi).\n\\end{equation}\n\\item (The {\\it infinitesimal invariance property})\nFor $\\psi_i\\in\\Lambda^{\\ell_i}(F,{\\mathfrak  g})$, $\\phi\\in\\Lambda^{1}(F,{\\mathfrak  g})$ and $P\\in\n  I^k(G)$, \n\\begin{equation}\\label{eq:inva}\n\\sum^k_{i=1}\n(-1)^{\\ell_1+\\dots+\\ell_i}P(\\psi_1\\wedge\\dots\\wedge[\\psi_i,\\phi]\\wedge\\dots\n\\psi_l)=0. \n\\end{equation}\n\\end{itemize}\n$[P(\\Omega_M)]$ is\ncalled the {\\it characteristic class} of $P$. For example, the characteristic class\n associated to  $\\operatorname{Tr}(A^k)$ is the k${}^{\\rm th}$ component of the Chern character of $F$.\n\n\nPart of the theorem's content\nis that for any two connections on $F$,\n$P(\\Omega_1) - P(\\Omega_0) = \ndCS_P(\\theta_1,\\theta_0)$  \nfor some odd form $CS_P(\\nabla_1, \\nabla_0)$.  Explicitly, \n\\begin{equation}\\label{5.1}\nCS_P(\\theta_1,\\theta_0) = \\int_0^1 P(\\theta_1-\\theta_0,\\overbrace{\\Omega_t,...,\\Omega_t}^{k-1})\n\\ dt\n\\end{equation}\nwhere \n$$\\theta_t = t\\theta_0+(1-t)\\theta_1,\\ \\ \\Omega_t = d\\theta_t+\\theta_t\\wedge\\theta_t$$ \\cite[Appendix]{chern}.  \n\n\\begin{rem}\nFor $F\\stackrel{\\pi}{\\to} M$,  \n$\\pi^*F\\to F$ is trivial.\nTake $\\theta_1$ to be the flat connection on $\\pi^*F$\nwith respect to a fixed trivialization.\nLet $\\theta_1$ also\ndenote the connection $\\chi^*\\theta_1$ on $F$, \nwhere $\\chi$ is the global section of $\\pi^*F.$  For any other connection $\\theta_0$ on $F$,  $\\theta_t = t\\theta_0, \\Omega_t = t\\Omega_0 + (t^2-t)\\theta_0\\wedge \\theta_0$. \n Assume an invariant  polynomial $P$ takes values in ${\\mathbb R}.$  Then  we\nobtain the formulas for the transgression form $TP(\\Omega_1)$\non $F$: for \n\\begin{equation}\\label{eq:ChernSimons}\n\\phi_t =t\\Omega_1+\\frac{1}{2}(t^2-t)[\\theta,\\theta],\\ \\ \nTP(\\theta)=l\\int_0^1 P(\\theta\\wedge \\phi^{k-1}_t)dt,\n\\end{equation}\n$dTP(\\theta)=P(\\Omega_1)\\in \\Lambda^{2l}(F)$\n\\cite{C-S}.  $TP(\\Omega_1)$ pushes down to an ${\\mathbb R}\/{\\mathbb Z}$-class on $M$,\nthe absolute Chern-Simons class.\n\\end{rem}\n\nAs usual, these formulas carry over to connections $\\nabla = d+\\omega$\non vector bundles $E\\to M$ in the form\n\\begin{equation}\\label{5.11}\nCS_{P}(\\nabla_1,\\nabla_0) = \\int_0^1 P(\\omega_1-\\omega_0,\\Omega_t,..., \n\\Omega_t)\\ dt,\n\\end{equation}\nsince $\\omega_1-\\omega_0$ and\n$\\Omega_t$ are globally defined forms.  \n\n \n \n \\subsection{{\\bf Chern-Weil and Chern-Simons Theory for $\\Psi{\\rm DO}_0^*$-Bundles}}\n\n \n Let $\\mathcal E\\to\\mathcal M$ be an infinite rank bundle\n over a paracompact Banach manifold\n $\\mathcal M$, with the fiber of $\\mathcal E$ \n modeled on a fixed Sobolev class of sections of \n a finite rank hermitian vector  bundle $E\\to N$, and with structure group $\\pdo_0^*(E)$.  \n For such $\\pdo_0^*$-bundles,\n we can produce\n primary and secondary characteristic classes \n once we choose a trace on $\\Psi{\\rm DO}_{\\leq 0}(E)$.\n Since the adjoint action of $\\pdo_0^*$ on $\\Psi{\\rm DO}_{\\leq 0}$ is by conjugation,  a trace on $\\Psi{\\rm DO}_{\\leq 0}$ will extend to a polynomial on forms\nsatisfying (\\ref{eq:deri}), (\\ref{eq:inva}), so the finite dimensional proofs extend.  \n \nThese traces were classified  in \\cite{lesch-neira, paycha-lescure}, although there are slight variants\nin our special case $N= S^1$ \\cite{ponge}.  Roughly speaking, the traces fall into two classes, the leading order symbol trace \\cite{P-R2} and the Wodzicki residue.  In this paper,\nwe consider only the Wodzicki residue, and refer to \\cite{lrst} for the leading order symbol\ntrace.\n\nFor simplicity, we mainly restrict to the generating invariant polynomials $P_k(A) = A^k$, and \nonly consider $\\mathcal E = TLM$, which we recall is the complexified tangent bundle.  We will work with vector bundles rather than principal bundles.  \n\n\n\n\\begin{defn}  \\label{def:WCS}\n(i) The k${}^{\\rm th}$ {\\it Wodzicki-Chern (WC) form} of a $\\Psi{\\rm DO}_0^*$-connection\n$\\nabla$ on $TLM$ with curvature $\\Omega$ is\n\\begin{equation}\\label{5.1a}\nc_k^W(\\Omega)(\\gamma) =\\frac{1}{k!}\n \\int_{S^*S^1}\\operatorname{tr}\\sigma_{-1}(\\Omega^{k}) \\ d\\xi  dx.\n\\end{equation}\nHere we recall that for each $\\gamma\\in LM$,\n$\\sigma_{-1}(\\Omega^k)$ is a $2k$-form with values in  endomorphisms\n of a trivial bundle\nover $S^*S^1$. \n\n\n\n(ii) The k${}^{\\rm th}$ {\\it Wodzicki-Chern-Simons (WCS) form} of two $\\Psi{\\rm DO}_0^*$-connections \n$\\nabla_0,\\nabla_1$ on $TLM$ is\n\\begin{eqnarray}\\label{5.22}\nCS^W_{2k-1}(\\nabla_1,\\nabla_0) &=&\\frac{1}{k!}\n \\int_0^1 \\int_{S^*S^1}\\operatorname{tr}\\sigma_{-1}((\\omega_1-\\omega_0)\\wedge \n(\\Omega_t)^{k-1})\\ dt\\\\ \n&=&\\frac{1}{k!} \\int_0^1 {\\rm res}^{\\rm w} \n[(\\omega_1-\\omega_0)\\wedge \n(\\Omega_t)^{k-1}]\\ dt.\\nonumber\n\\end{eqnarray}\n\n(iii) The  k${}^{\\rm th}$ {\\it Wodzicki-Chern-Simons form} associated to a Riemannian metric \n$g$ \non $M$, denoted $CS^W_{2k-1}(g)$,  is $CS^W_{2k-1}(\\nabla_1,\\nabla_0)$, where $\\nabla_0, \\nabla_1$ refer to the \n$L^2$ and $s=1$ Levi-Civita connections on $LM$, respectively.\n\n\n(iv) Let $\\Sigma = \\{\\sigma\\}$ be the group of permutations of $\\{1,...,k\\}$. Let $I:\n1\\leq i_1< ...< i_\\ell = k$ be a partition of $k$ (i.e. with $i_0=0$, $\\sum_{j=1}^k\n (i_j-i_{j-1}) = k$) .  For  the symmetric, $U(n)$-invariant, \nmultilinear form on ${\\mathfrak u}(n)$\n\\begin{eqnarray*} P_I(A_1,A_2,...,A_k) &=& \\frac{1}{k!}\n\\sum_\\sigma \\operatorname{tr}(A_{\\sigma(1)}\\cdot...\\cdot A_{\\sigma(i_1)})\n\\operatorname{tr}(A_{\\sigma(i_1+1)}\\cdot...\\cdot A_{\\sigma(i_2)})\\\\\n&&\\qquad \\cdot ...\\cdot \\operatorname{tr}(A_{\\sigma(i_{\\ell-1})}\n\\cdot ...\\cdot A_{\\sigma(k)}),\n\\end{eqnarray*}\ndefine the symmetric, $\\pdo_0^*$-invariant, multilinear form on $\\Psi{\\rm DO}_{\\leq 0}$ by\n\\begin{eqnarray*} P_I^W(B_1,...,B_k) &=&  \\frac{1}{k!}\n \\sum_\\sigma\\left( \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(B_{\\sigma(1)}\\cdot...\\cdot B_{\\sigma(i_1)}) \\right.\\\\\n&&\\qquad \\left. \\cdot\n \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}\n(B_{\\sigma(i_1+1)}\\cdot...\\cdot B_{\\sigma(i_2)})\\right. \\\\\n&&\\qquad \\left. \\cdot ...\\cdot \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(B_{\\sigma(i_{\\ell-1})} \n\\cdot ...\\cdot B_{\\sigma(k)})\\right).\n\\end{eqnarray*}\n  The {\\it Wodzicki-Chern form associated\nto $P_I$} for a $\\Psi{\\rm DO}_0^*$-connection on $TLM$ with curvature $\\Omega$ is \n\\begin{eqnarray}\\label{wcpi} c_{P_I}^W(\\Omega) &=& \nP_I^W(\\Omega,\\Omega,...,\\Omega)\\\\\n &=& \\frac{1}{k!}\n\\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(\\Omega^{k_1}) \\cdot \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(\\Omega^{k_2})\n\\cdot\n...\\cdot \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(\\Omega^{k_\\ell}\\nonumber )\\\\\n&=& \\frac{k_1!k_2!\\cdot...\\cdot k_\\ell !}{k!} \nc_{k_1}^W(\\Omega)c_{k_2}^W(\\Omega)\\cdot ...\\cdot c_{k_\\ell}^W(\\Omega),\\nonumber\n\\end{eqnarray}\nwhere $k_1=i_1-i_0, k_2 = i_2-i_1,...,\nk_\\ell = i_\\ell - i_{\\ell-1}$.\n\nSetting $K = (k_1,...,k_\\ell)$, we also denote $c_{P_I}^W(\\Omega)$ by $c_K^W(\\Omega).$\n\n\n\n(v) \nLet $\\nabla_0,\\nabla_1$ be $\\Psi{\\rm DO}_0^*$-connections on $TLM$ with connection forms\n$\\omega_0, \\omega_1,$ respectively.  The \n {\\it Wodzicki-Chern-Simon form associated to $P_I$ and $\\nabla_0, \\nabla_1$}\nis\n$$\nCS^W_{P_I}(\\nabla_1,\\nabla_0) = \\int_0^1 P_I^W(\\omega_1-\\omega_0,\\Omega_t,...,\n\\Omega_t)dt. $$ \n\\end{defn}\n\nIn (iv) and (v), we do not bother with a normalizing constant, since we do not claim that \nthere is a normalization which gives classes with integral periods.  \nNote that the k${}^{\\rm th}$ WCS class is associated to $P_k(A_1,...,A_k) = \\operatorname{tr}(A_1\\cdot\n...\\cdot A_k)$, i.e. the partition $K = (k)$, or in other words to the polynomial giving the \nk${}^{\\rm th}$ component of the Chern character.\n\nAs in finite dimensions, $c_k^W(\\nabla)$ is a closed $2k$-form, with de Rham cohomology\nclass $c_k(LM)$\n independent of $\\nabla$, as $c_k^W(\\Omega_1)  - c_k^W(\\Omega_0) =\ndCS^W_{2k-1}(\\nabla_1,\\nabla_0).$  \n \n\\begin{rem}  It is an interesting question to determine all the $\\Psi{\\rm DO}_0^*$-invariant polynomials on \n$\\Psi{\\rm DO}_{\\leq 0}.$  As above, $U(n)$-invariant polynomials combine with the Wodzicki residue \n(or the other traces on $\\Psi{\\rm DO}_{\\leq 0}$) to give $\\Psi{\\rm DO}_0^*$-polynomials,\nbut there may be others.  \n\\end{rem}  \n\n\n\nThe tangent space $TLM$, and more generally mapping spaces\nMaps$(N,M)$  with $N$ closed\nhave vanishing Wodzicki-Chern classes.  Here we take a Sobolev topology on Maps$(N,M)$ for some\nlarge Sobolev parameter, so that Maps$(N,M)$ is a paracompact Banach manifold.\nWe denote the de Rham class of $c_{P_I}^W(\\Omega)$ for a connection on $\\mathcal E$ by\n$c_{P_I}(\\mathcal E).$ \n\n\n\n\\begin{prop} \\label{prop:maps} Let $N, M$ be closed manifolds, and let {\\rm Maps}${}_f(N,M)$ denote\nthe component of a fixed $f:N\\to M$.  Then the  cohomology classes\n$c_{P_I}^W({\\rm Maps}_f(N,M)) $ of $T{\\rm Maps}(M,N)$ vanish.\n\\end{prop}\n\n\\begin{proof}\nFor $TLM$, the $L^2$ connection in Lemma \\ref{lem:l2lc}\nhas curvature $\\Omega$ which is a multiplication operator.  Thus $\\sigma_{-1}(\\Omega)$ and hence $\\sigma_{-1}(\\Omega^{i})$ are zero, \nso the WC forms $c_{P_I}(\\Omega)$ also vanish.\n\nFor $n\\in N$ and $h:N\\to M$,\nlet $\\operatorname{ev}_n: {\\rm Maps}_f(N,M)$ be $\\operatorname{ev}_n(h) = h(n).$ \n Then $D_XY(h)(n) \\stackrel{\\rm def}{=} \n (\\operatorname{ev}_h^*\\nabla^{LC,M})_XY(h)(n)$ is the $L^2$ Levi-Civita connection on \\\\\nMaps$(N,M).$\nAs in Lemma \\ref{lem:l2lc},\nthe curvature of $D$ is a \na multiplication operator.  Details are left to the reader.\n\\end{proof}\n\n\n\\begin{rem}  (i) These mapping spaces fit into the framework of the Families Index Theorem in \nthe case of a trivial fibration\n$Z\\to M\\stackrel{\\pi}{\\to} B$ of closed manifolds.  Given a\nfinite rank bundle $E\\to M$, we get an associated infinite rank bundle ${\\mathcal E} = \\pi_*E\n\\to B$.  For the fibration $N\\to N\\times {\\rm Maps}(N,M)\\to {\\rm Maps}(N,M)$ and $E = {\\rm\n  ev}^*TM$, ${\\mathcal E}$ is $T{\\rm Maps}(N,M).$  A connection $\\nabla$ on $E$ induces a connection $\\nabla^{{\\mathcal E}}$ on ${\\mathcal E}$ defined by\n\\begin{equation*}\n(\\nabla^{{\\mathcal E}}_Z s)(b)\n(z)=\\left( (\\operatorname{ev}^*\\theta^u)_{(Z,0)} u_s\\right)(b,z).\n\\end{equation*}\nHere $u_s(b,z)=s(b)(z)$.\nThe curvature $\\Omega^{{\\mathcal E}}$ satisfies\n\\begin{equation*}\\label{eq:pullback}\n\\Omega^{{\\mathcal E}}(Z,W)s(b)(z)=(\\operatorname{ev}^*\\Omega ) ((Z,0),(W,0)) u_s(b,z).\n\\end{equation*}\nThis follows from\n\\begin{equation*}\n\\Omega^{{\\mathcal E}}(Z,W)s(b)(z)= [\\nabla^{{\\mathcal E}}_Z \\nabla^{{\\mathcal E}}_W\n-\\nabla^{{\\mathcal E}}_W \\nabla^{{\\mathcal E}}_Z -\\nabla^{{\\mathcal E}}_{[Z,W]}]\ns(b)(z).\n\\end{equation*}\nThus the connection and curvature forms take values in multiplication operators, and \nso $c_k^W({\\mathcal E}) = 0.$\n\n\nIf the fibration is nontrivial, the connection on ${\\mathcal E}$ depends on the choice of a horizontal complement to $TZ$ in $TM$, and the corresponding connection and curvature forms take\nvalues in first order differential operators.  \n\n(ii) In finite dimensions, odd Chern forms of complexified real bundles like\\\\\n $T{\\rm Maps}(N,M)$ vanish, because the form involves a composition of an odd number of skew-symmetric matrices.  In contrast, odd WC forms involve terms like\n$\\sigma_{-1}(\\Omega^1)\\wedge\\Omega^M\\wedge...\\wedge\\Omega^M,$ where $\\Omega^1$ is the curvature of the $s=1$ Levi-Civita connection.  By Lemma \n\\ref{old2.2}(ii), $\\sigma_{-1}(\\Omega^1)$ is not skew-symmetric as an endomorphism.  Thus\nit is not obvious that the odd WC forms vanish.\n\nSimilarly, in finite dimensions the Chern-Simons form for the odd Chern classes of complexified real bundles vanish, but this need not be the case for WCS forms.  In fact, we will produce nonvanishing \nWCS classes associated to $c_3^W(TLM^5)$ in \\S\\ref{dimfive}.\n\n\\end{rem}\n\n\nIn finite dimensions, Chern classes are topological obstructions to the\nreduction of the structure group and geometric obstructions to the existence\nof a flat connection.  \nWodzicki-Chern classes for $\\Psi{\\rm DO}_0^*$-bundles \nare also topological and geometric obstructions, but\nthe geometric information is a little more refined due to the grading on the\nLie algebra \n $\\Psi{\\rm DO}_{\\leq 0}$.\n\n\\begin{prop}\n Let ${\\mathcal E}\\to{\\mathcal B}$ be an infinite rank $\\pdo_0^*$-bundle, for\n  $\\pdo_0^*$ acting on\n$E\\to N^n$.  \nIf ${\\mathcal E}$ admits a reduction to the gauge group ${\\mathcal G}(E)$, then\n  $c_k^W({\\mathcal E})  =   0$ for all $k$, and hence $c_{P_I}^W({\\mathcal E}) =0$ for all $P_I$.\nIf ${\\mathcal E}$ admits a \n   $\\pdo_0^*$-connection whose\n  curvature has order $-k$, then $\n  c_{\\ell}({\\mathcal E}) =0$ for $\\ell \\geq [n\/k].$\n \\end{prop}\n\n\\begin{proof}  If the structure group of  ${\\mathcal E}$ reduces to the gauge\n  group, there exists a connection one-form\n  with values in Lie$({\\mathcal G}) = {\\rm End}(E)$, the Lie algebra of multiplication\n  operators.  Thus the Wodzicki residue of powers of the curvature vanishes,\n  so the Wodzicki-Chern classes vanish.\nFor the second statement, the order of the curvature is less than\n$-n$ for $\\ell \\geq [n\/k]$,  so the Wodzicki residue\n  vanishes in this range. \n  \\end{proof}\n  \n  However, we do not have examples of nontrivial WC classes; cf.~\\cite{lrst}, where it is \n  conjectured that these classes always vanish.  \n  \\bigskip\n\n\n\n\nThe relative WCS form is not difficult to compute.  \n\n\\begin{prop}  Let $\\sigma$ be in the group of permutations of $\\{1,\\ldots,2k-1\\}.$ Then\n\\begin{eqnarray}\\label{5.4}\n\\lefteqn{CS^W_{2k-1}(g)(X_1,...,X_{2k-1}) }\\\\\n&=&\n\\frac{2}{(2k-1)!} \\sum_{\\sigma} {\\rm sgn}(\\sigma) \\int_{S^1}\\operatorname{tr} [\n(-2R(X_{\\sigma(1)},\\dot\\gamma)\n-R(\\cdot,\\dot\\gamma)X_{\\sigma(1)} + R(X_{\\sigma(1)},\\cdot)\\dot\\gamma) \\nonumber\\\\\n&&\\qquad \n\\cdot (\\Omega^M)^k(X_{\\sigma(2)},..X_{\\sigma(2k-1)} )].\\nonumber\n\\end{eqnarray}\n\\end{prop}\n\n\n\n\n\\begin{proof}\n$$\\sigma_0((\\omega_1-\\omega_0)_X)^a_b = \\Gamma_{cb}^aX^c -\\Gamma_{cb}^aX^c = 0.$$\nThus \n\\begin{equation}\\label{cswint}\nCS^W_{2k-1}(g) = \\int_0^1 \\int_{S^*S^1}\\operatorname{tr}\\sigma_{-1}(\\omega_1-\\omega_0)\\wedge (\\sigma_0(\\Omega_t))^k\\ dt.\n\\end{equation}\nMoreover,\n\\begin{eqnarray*}\\sigma_0(\\Omega_t) &=& td(\\sigma_0(\\omega_0)) + (1-t)d(\\sigma_0(\\omega_1)) \\\\\n&&\\qquad + \n(t\\sigma_0(\\omega_0) + (1-t)\\sigma_0(\\omega_1))\\wedge (t\\sigma_0(\\omega_0) + (1-t)\\sigma_0(\\omega_1))\\\\\n&=& d\\omega^M + \\omega^M\\wedge \\omega^M\\\\\n&=& \\Omega^M.\n\\end{eqnarray*}\nTherefore\n\\begin{equation}\\label{5.3}\nCS^W_{2k-1}(g) = \\int_0^1 \\int_{S^*S^1}\\operatorname{tr} [\\sigma_{-1}(\\omega_1)\n\\wedge (\\Omega^M)^k]\\ dt,\n\\end{equation}\nsince $\\sigma_{-1}(\\omega_0) = 0.$  We can drop the integral over $t$. \nThe integral over the $\\xi$ variable contributes a factor of $2$: the integrand has\na factor of $|\\xi|^{-2}\\xi$, which equals $\\pm 1$ on the two components of $S^*S^1$.\nSince the fiber of $S^*S^1$ at a fixed $\\theta$ consists of two points \nwith opposite orientation, the ``integral\" over each fiber is $1-(-1) = 2.$ \nThus\n\\begin{eqnarray}\\label{5.4a}  \\lefteqn{\nCS^W_{2k-1}(g)(X_1,...X_{2k-1})    }  \\\\\n&=& =  \\frac{2}{(2k-1)!}   \\sum_\\sigma {\\rm sgn}(\\sigma)  \\int_{S^1}\\operatorname{tr}[\n(-2R(X_{\\sigma(1)},\\dot\\gamma)\n-R(\\cdot,\\dot\\gamma)X_{\\sigma(1)} + R(X_{\\sigma(1)},\\cdot)\\dot\\gamma)\\nonumber\\\\\n&&\\qquad\n\\cdot (\\Omega^M)^k(X_{\\sigma(2)},..X_{\\sigma(2k-1)} )]\\nonumber\n\\end{eqnarray}\nby Lemma \\ref{old2.1}.\n\\end{proof} \n\n\nThis produces odd classes in the de Rham cohomology of the loop space of an odd\ndimensional manifold.\n\n\\begin{thm}\\label{thm:5.5}\n (i)  Let dim$(M) = 2k-1$ and let $P$ be a $U(n)$-invariant polynomial of degree \n$k.$  Then $c^W_P(\\Omega) \\equiv 0$ for any $\\Psi{\\rm DO}_0^*$-connection $\\nabla$ on\n $TLM.$  Thus $CS^W_P(\\nabla_1,\\nabla_0)$ is closed and defines a \n class $[CS^W_P(\\nabla_1,\\nabla_0)]\\in H^{2k-1}(LM).$  In particular, we can\n define $[CS^W_P(g)]\\in H^{2k-1}(LM)$ for a Riemannian metric $g$ on $M$. \n \n \n \n(ii)  For dim$(M) = 2k-1$, the k${}^{\\it th}$ Wodzicki-Chern-Simons form $CS^W_{2k-1}(g)$\nsimplifies to \n \\begin{eqnarray}\\label{csg}\n\\lefteqn{CS^W_{2k-1}(g)(X_1,...,X_{2k-1}) }\\nonumber \\\\\n&=&\n\\frac{2}{(2k-1)!} \\sum_{\\sigma} {\\rm sgn}(\\sigma) \\int_{S^1}\\operatorname{tr}[\n(-R(\\cdot,\\dot\\gamma)X_{\\sigma(1)} + R(X_{\\sigma(1)},\\cdot)\\dot\\gamma)\\\\\n&&\\qquad \n\\cdot (\\Omega^M)^{k-1}(X_{\\sigma(2)},..X_{\\sigma(2k-1)} )].\\nonumber\n\\end{eqnarray}\n\n\n \n\n \\end{thm}\n \n \\begin{proof} (i) Let $\\Omega$ be the curvature of $\\nabla.$\n $c^W_P(\\Omega)(X_1,\\dots, X_{2k})(\\gamma)$ is a sum of monomials of the form\n(\\ref{wcpi}).  This is\na $2k$-form on $M$, and hence vanishes.  \n\n \n (ii) Since\n $$R(X_{1},\\dot\\gamma)\n\\cdot (\\Omega^M)^k(X_{2},..X_{2k-1}) = \n[i_{\\dot\\gamma}\\operatorname{tr}(\\Omega^{k})](X_1,...X_{2k-1}) = \\operatorname{tr}(\\Omega^k)(\\dot\\gamma, X_1,\\ldots,X_{2k-1}),$$\nthe first term on the right hand side of (\\ref{5.4a}) vanishes on a $(2k-1)$-manifold.\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\\begin{rem} There are several variants to the construction of relative WCS classes.\n\n(i) If we define the transgression form $Tc_k(\\nabla)$ with the Wodzicki residue\nreplacing the trace in (\\ref{eq:ChernSimons}), it is easy to check that $Tc_k(\\nabla)$\ninvolves $\\sigma_{-1}(\\Omega).$  For $\\nabla$ the $L^2$ connection, this WCS class vanishes.  For $\\nabla$ the $H^s$ connection, $s>0$, $\\sigma_{-1}(\\Omega)$ involves \nthe covariant derivative of the curvature of $M$ (cf.~Lemma \\ref{old2.2} for $s=1.$)  Thus the\nrelative WCS class is easier for computations than the absolute class $[Tc_k(\\nabla)].$\n\n\n(ii) If we define $CS_k^W(g)$ using the Levi-Civita connection for the $H^s$ \nmetric instead of\nthe $H^1$ metric, the WCS class is simply multiplied by the artificial parameter $s$ by \nLemma \\ref{lem:33}.  Therefore setting $s=1$ is not only computationally convenient, it \nregularizes the WCS, in that it extracts the $s$-independent information.\nThis justifies the following definition:\n\n\n\\begin{defn}  \\label{def:regularized}\nThe {\\it regularized  $k^{th}$ WCS class} associated to a Riemannian metric \n$g$ on $M$ is $CS_k^{W, {\\rm reg}}(g) \\stackrel{\\rm def}{=} \nCS_k^W(\\nabla^{1},\\nabla^0)$, where $\\nabla^{1}$ is the $H^1$ connection\nand $\\nabla^0$ is the $L^2$ Levi-Civita connection.  \n\\end{defn}  \n\n\\end{rem}\n\n\\bigskip\n\nWe conclude this section with a vanishing result that does not have a finite dimensional \nanalogue.\n  \\begin{thm} \\label{WCSvan}\n  The {\\it  k}${}^{\\it  th}$ WCS class $CS_k^W(g)$\n   vanishes if ${\\rm dim}(M) \\equiv 3 \\ ({\\rm mod}\\  4).$\n  \\end{thm}\n\n\\begin{proof}  Let dim$(M) =2 k-1$. Since $\\Omega^M$ takes values in skew-symmetric endomorphisms, \nso does\n$(\\Omega^M)^{k-1}$ if $k$ is even, i.e. if ${\\rm dim}(M) \\equiv 3 \\ ({\\rm mod}\\  4).$\n  The term \n  $-R(\\cdot,\\dot\\gamma)X_{\\sigma(1)} + R(X_{\\sigma(1)},\\cdot)\\dot\\gamma$ in (\\ref{csg}) is a symmetric\n  endomorphism.  For in Riemannian normal coordinates, this term is\n  $(-R_{bdca} +R_{cbda})X^c\\dot\\gamma^d \\equiv A_{ab}$, say, so the curvature terms in \n  $A_{ab} - A_{ba}$ are\n  \\begin{eqnarray*}\n  -R_{bdca} +R_{cbda} + R_{adcb} - R_{cadb} &=& -R_{bdca} +R_{cbda} \n  + R_{cbad} - R_{dbca}\\\\\n  &=& -R_{bdca} +R_{cbda} -R_{cbda} + R_{bdca}=0.\n  \\end{eqnarray*}\nThus the integrand in  (\\ref{csg}) is the trace of a symmetric endomorphism composed with a skew-symmetric endormorphism, and so \nvanishes.\n\\end{proof}\n\n\\begin{exm}  We contrast Theorem \\ref{WCSvan} with the situation in finite dimensions. \nLet dim$(M)=3.$ \nThe only invariant monomials of degree two are $\\operatorname{tr}(A_1A_2)$ and \n$\\operatorname{tr}(A_1)\n\\operatorname{tr}(A_2)$ (corresponding to $c_2$ and $c_1^2$, respectively).  \n\nFor $M$, $\\operatorname{tr}(A_1A_2)$ gives rise \nto the classical Chern-Simons invariant for $M$.  However, the Chern-Simons class associated to \n$\\operatorname{tr}(A_1)\\operatorname{tr}(A_2)$ involves $\\operatorname{tr}(\\omega_1-\\omega_0)\\operatorname{tr}(\\Omega_t)$, \nwhich vanishes since both forms take values in skew-symmetric endomorphisms.\n\nIn contrast, on $LM$ we know that the WCS class $CS^W_3$ associated to \n$\\operatorname{tr}(A_1A_2)$ vanishes.  The WCS associated to $\\operatorname{tr}(A_1)\\operatorname{tr}(A_2)$ involves \n$\\operatorname{tr}\\sigma_{-1}(\\omega_1-\\omega_0) = \\operatorname{tr}\\sigma_{-1}(\\omega_1)$ and $\\operatorname{tr}\\sigma_{-1}(\\Omega_t).$  \nBoth $\\omega_1$ and $ \\Omega_t$ take values in skew-symmetric $\\Psi{\\rm DO}$s, but\nthis does not imply that the terms in their symbol expansions are skew-symmetric.  In fact, a calculation using Lemma \\ref{old2.1} shows that $\\sigma_{-1}(\\omega_1)$ is not skew-symmetric. \nThus the WCS class associated to $\\operatorname{tr}(A_1)\\operatorname{tr}(A_2)$ may be nonzero.\n\n\n\\end{exm}\n\n\n\n\n\n\n\\section{{\\bf An Application of Wodzicki-Chern-Simons Classes to Circle\n    Actions}}\\label{dimfive}\n\n\nIn this section we use WCS classes to distinguish different $S^1$ actions on\n$M=S^2\\times S^3$. We use this to conclude that $\\pi_1({\\rm Diff}(M) , id)$ is infinite.  \n\nRecall that  $H^*(LM)$ denotes de Rham cohomology of complex valued \nforms.   In particular, integration of closed forms over homology cycles gives a pairing of\n$H^*(LM)$ and $H_*(LM,{\\mathbb C})$. \n\n For any closed oriented manifold $M$, let $a_0,a_1:S^1\\times M\\to M$ be two smooth actions.  Thus\n$$a_i(0,m) = m, \\ a_i(\\theta,a(\\psi,m)) = a_i(\\theta + \\psi, m).$$\n\n\\begin{defn} (i)  $a_0$ and $a_1$\n are {\\it smoothly  homotopic} if there exists a smooth map \n$$F:[0,1]\\times S^1\\times M\\to M,\\ F(0,\\theta,m) = a_0(\\theta,m),\\\n F(1,\\theta,m) = a_1(\\theta,m).$$\n\n(ii)  $a_0$ and $a_1$ are {\\it smoothly  homotopic through actions} if\n $F(t,\\cdot,\\cdot):S^1\\times M\\to M$ is an action for all $t$.\n\n\\end{defn}\n\nWe can rewrite an action in two equivalent ways.\n\n\n\\begin{itemize}\n\\item $a$ determines (and is determined by)\n$a^D:S^1\\to {\\rm Diff} (M)$ given by\n$a^D(\\theta)(m) = a(\\theta,m).$  $a^D(\\theta)$ is a diffeomorphism because \n$$a^D(-\\theta)(a^D(\\theta,m)) = a(-\\theta, a(\\theta,m)) = m.$$\nSince $a^D(0) = id,$  we get a class $[a^D]\\in \\pi_1({\\rm Diff} (M), id)$, the\nfundamental group of ${\\rm Diff} (M)$ based at $id.$  Here Diff$(M)$ is a Banach manifold\nas an open subset of the Banach manifold of ${\\rm Maps} (M) = {\\rm Maps} (M,M)$ of some fixed Sobolev class.\n\n\n\\item $a$ determines (and is determined by)\n$a^L:M\\to LM$ given by $ a^L(m)(\\theta) = a(\\theta,m)$. This determines a class\n$[a^L]\\in H_n(LM,{\\mathbb Z})$ with $n = {\\rm dim}(M)$ by setting $[a^L] = a^L_*[M].$\n  In concrete terms, if we triangulate $M$ as the $n$-cycle $\\sum_i n_i\\sigma_i$,\nwith $\\sigma_i:\\Delta^n\\to M$, \nthen $[a^L]$ is the homology class of\nthe cycle $\\sum_i n_i (a^L\\circ \\sigma_i).$  \n\n\\end{itemize}\n\nWe give a series of elementary lemmas comparing these maps.\n\n\n\\begin{lem}\\label{lem:one} $a_0$ is smoothly homotopic to $a_1$ through actions iff $[a^D_0] =\n\t  [a^D_1]\\in \\pi_1({\\rm Diff} (M), id).$\n\\end{lem}\n\n\\begin{proof} ($\\Rightarrow$) Given $F$ as above, set $G:[0,1]\\times S^1\\to\n\t  {\\rm Diff} (M)$ by $G(t,\\theta)(m) = F(t,\\theta,m).$  We have $G(0,\\theta)(m)\n\t  = a_0(\\theta,m) = a^D(\\theta)(m)$, $G(1,\\theta)(m) = a_1(\\theta, m)\n\t  = a^D_1(\\theta)(m)$.\n$G(t,\\theta)\\in{\\rm Diff} (M)$, because \n$$G(t,-\\theta)(G(t,\\theta)(m)) = F(t,-\\theta,F(t,\\theta,m)) = F(t,0,m) = m.$$\n(This uses that $F(t,\\cdot,\\cdot)$ is an action.)\nSince $F$ is\nsmooth, $G$ is a continuous (in fact, smooth) map of ${\\rm Diff} (M)$.\nThus $a^D_0, a^D_1$ \nare homotopic as elements of\\\\\n ${\\rm Maps} (S^1,{\\rm Diff} (M))$, so $[a^D_0] = [a^D_1].$\n\\bigskip\n\n\\noindent ($\\Leftarrow$)  Let $G:[0,1]\\times S^1\\to {\\rm Diff} (M)$ be a continuous\nhomotopy from\n$a^D_0(\\theta) = G(0,\\theta) $ to $a^D_1(\\theta) = G(1,\\theta)$ with $G(t,0) = id$\nfor all $t$. \nIt is possible to approximate\n$G$ arbitrarily well by a smooth map, since $[0,1]\\times S^1$ is compact.  Set\n$F:[0,1]\\times S^1\\times M\\to M$ by\n$F(t,\\theta,m) = G(t,\\theta)(m).$  \n $F$ is smooth.  Note that\n$F(0,\\theta,m) = G(t,\\theta)(m) = a^D_0(\\theta)(m) = a_0(\\theta,m)$, and\n$F(1,\\theta,m) = a_1(\\theta,m).$  Thus $a_0$ and $a_1$ are smoothly homotopic.\n\\end{proof}\n\n\nThere are similar results for $a^L.$\n\n\\begin{lem}\\label{lem:three}  $a_0$ is smoothly homotopic to $a_1$ iff $a^L_0,\na^L_1:M\\to LM$ are smoothly homotopic.\n\\end{lem}\n\n\n\\begin{proof}  Let $F$ be the homotopy from $a_0$ to $a_1$.  Set\n\t  $H:[0,1]\\times M \\to LM $ by $H(t,m)(\\theta) = F(t,\\theta,m).$  Then\n$H(0,m)(\\theta) = F(0,\\theta,m) = a_0(\\theta,m) =  a^L_0(m)(\\theta)$,\n  $H(1,m)(\\theta) =  a^L_1(m)(\\theta),$  so $H$ is a homotopy from $\n  a^L_0$ to $ a^L_1.$ \nIt is  easy to check that $H$ is smooth.\n\n\nConversely, if $H:[0,1]\\times M \\to LM $ is a smooth homotopy from $a^L_0$ to\n$a^L_1$, set $F(t,\\theta, m) = H(t,m)(\\theta).$  \n\\end{proof}\n\n\\begin{cor}\\label{cor:one} If $a_0$ is smoothly homotopic to $a_1$, then\n$[a^L_0] =  [a^L_1]\\in H_n(LM,{\\mathbb Z}).$  \n\\end{cor}\n\n\\begin{proof} By the last Lemma, $a^L_0$ and $a^L_1$ are homotopic. Thus \n$[a^L_0] = a^L_{0,*}[M] = a^L_{1,*}[M] = [a^L_1].$\n\\end{proof}\n\nThis yields a technique to use WCS classes to distinguish actions and to investigate \n$\\pi_1({\\rm Diff}(M) ,id).$  From now on, ``homotopic\" means ``smoothly homotopic.\"\n\n\n\\begin{prop} \\label{prop:two} Let dim$(M)=2k-1.$ Let $a_0, a_1:S^1\\times M\\to M$ be actions.\n\n\n(i)  If $\\int_{[a^L_0]} CS^W_{2k-1 } \\neq \\int_{[a^L_1]} CS^W_{2k-1 }$, then $a_0$ and $a_1$\n  are not homotopic through actions, and $[a^D_0]\\neq [a^D_1]\\in \\pi_1({\\rm Diff} (M),id).$\n\n(ii)  If $\\int_{[a_1^L]} CS^W_{2k-1 } \\neq 0,$ then\n  $\\pi_1({\\rm Diff} (M), id)$  is infinite.\n\n\\end{prop}\n\n\\begin{proof} \n\n(i) By Stokes' Theorem, $[a^L_0]\\neq [a^L_1]\\in H_n(LM,{\\mathbb C})$.\n  By Corollary \\ref{cor:one}, $a_0$ and $a_1$ are not homotopic,\n   and hence not homotopic\n  through actions.  By\nLemma \\ref{lem:one}, $[a^D_0]\\neq [a^D_1]\\in \\pi_1({\\rm Diff} (M),id).$\n\n(ii) Let $a_n$ be the $n^{\\rm th}$ iterate of \n $a_1$, i.e. $a_n(\\theta,m) =\na_1(n\\theta,m).$  \n\nWe claim that \n $\\int_{[a^L_n]}CS^W_{2k-1 } =\nn\\int_{[a^L_1]}CS^W_{2k-1 }$.  By (\\ref{5.4}), every term in $CS^W_{2k-1 }$ is of the\nform $\\int_{S^1}\\dot\\gamma(\\theta) f(\\theta)$, where $f$ is a periodic function on the\ncircle.  Each loop $\\gamma\\in\na^L_1(M)$ corresponds to the loop $\\gamma(n\\cdot)\\in a^L_n(M).$  Therefore the term\n$\\int_{S^1}\\dot\\gamma(\\theta) f(\\theta)$ is replaced by \n$$\\int_{S^1} \\frac{d}{d\\theta}\\gamma(n\\theta) f(n\\theta)d\\theta \n = n\\int_0^{2\\pi} \\dot\\gamma(\\theta)f(\\theta)d\\theta.$$\nThus $\\int_{[a^L_n]}CS^W_{2k-1 } = n\\int_{[a^L_1]}CS^W_{2k-1 }.$\n By (i), the $[a^L_n]\\in \n\\pi_1({\\rm Diff} (M), id)$\nare all distinct.  \n\n\\end{proof}\n\n\n\n\n\\begin{rem}\nIf two actions\nare homotopic through actions,\nthe $S^1$ index of an equivariant operator of the two actions is the same. (Here equivariance\nmeans for each action $a_t, t\\in [0,1].$)\nIn contrast to Proposition \\ref{prop:two}(ii), the $S^1$ index of an equivariant operator\ncannot distinguish actions on odd dimensional manifolds, as the\n$S^1$ index vanishes. This can be seen from the\nlocal version of the\n$S^1$ index theorem \\cite[Thm. 6.16]{BGV}. For the normal bundle to the\nfixed point set is always even dimensional, so the fixed point set consists of\nodd dimensional submanifolds.  The integrand in the fixed point submanifold\ncontribution to the $S^1$-index is the constant term in the short time\nasymptotics of the appropriate heat kernel.  In odd dimensions, this constant\nterm is zero.\n\n  In \\cite{MRT2}, we  interpret the $S^1$ index theorem as\nthe integral of an equivariant characteristic class over $[a^L]$.\n\\end{rem}\n\n\nWe now apply these methods to a Sasaki-Einstein metric on $S^2\\times S^3$\nconstructed in \\cite{gdsw}\n to prove the following:\n\n\\begin{thm} (i) There is an $S^1$ action on $S^2\\times S^3$ that is not smoothly homotopic\nto the trivial action.\n\n(ii) $\\pi_1({\\rm Diff} (S^2\\times S^3), id)$ is infinite.\n\\end{thm}\n\nThe content of  (i) is that although the $S^1$-orbit\n $\\gamma_x$ through $x\\in S^2\\times S^3$\nis contractible to $x$, the contraction cannot be constructed to be \nsmooth in $x$.  \n\n\\begin{proof}\nAccording to  \\cite{gdsw}, the locally defined metric \n\\begin{eqnarray}\\label{metric} g &=&\\frac{1-cy}{6}(d\\theta^2 + \\sin^2\\theta d\\phi^2) + \n\\frac{1}{w(y)q(y)} dy^2 + \\frac{q(y)}{9}[d\\psi^2 -\\cos\\theta d\\phi^2]\\nonumber\\\\\n&&\\qquad + w(y)\\left[d\\alpha + \\frac{ac-2y+y^2c}{6(a-y^2)}[d\\psi -\\cos\\theta d\\phi]\\right]^2,\n\\end{eqnarray}\nwith \n$$w(y) = \\frac{2(a-y^2)}{1-cy}, q(y) = \\frac{a-3y^2+2cy^3}{a-y^2},$$\nis a family of Sasaki-Einstein metrics on a coordinate ball in the variables\n$(\\phi, \\theta, \\psi, y, \\alpha).$  Here $a$ and $c$ are constants, and we can take $a\\in (0,1], c=1$.  \nFor $p,q$ relatively prime, $q<p$, and satisfying $4p^2-3q^2 = n^2$ for some integer $n$,  \nand for  $a = a(p,q)< 1$, the metric extends to a $5$-manifold $Y^{p,q}$ which has\nthe coordinate ball as a dense subset.  \nIn this case, $( \\phi, \\theta, \\psi, y)$\nare  spherical coordinates on $S^2\\times S^2$ with a nonstandard metric, and $\\alpha$ is the fiber coordinate of an \n$S^1$-fibration $S^1\\to Y^{p,q}\\to S^2\\times S^2.$  \n$Y^{p,q}$ is diffeomorphic to $S^2\\times S^3,$ and\n has first Chern class which integrates over the two $S^2$ factors to $p+q$ and $p$ \\cite[\\S2]{gdsw}.\nThe coordinate ranges are $\\phi\\in (0,2\\pi), \\theta \\in (0,\\pi), \\psi\\in (0,2\\pi)$,\n$\\alpha\\in (0,2\\pi\\ell)$,\nwhere $\\ell = \\ell(p,q)$, and \n$y\\in (y_1, y_2)$, with the $y_i$ the two smaller roots of $a-3y^2+2y^3=0$.  $p$ and\n$q$ determine $a, \\ell, y_1, y_2$ explicitly \\cite[(3.1), (3.4), (3.5), (3.6)]{gdsw}.  \n\n\n\n\nFor these choices of $p, q$, we get an $S^1$-action $a_1$ on $Y^{p,q}$ by rotation in the $\\alpha$-fiber.\nWe claim that for e.g. $(p,q) = (7,3)$\n\\begin{equation}\\label{neqo} \\int_{[a_1^L]} CS_5^W(g) \\neq 0.\n\\end{equation}\nBy Proposition \\ref{prop:two}(iii), this implies $\\pi_1({\\rm Diff} (S^2\\times S^3), id)$ is infinite.\nSince the trivial action $a_0$ has $\\int_{[a_0^L]} CS_5^W(g) = 0$ (by the proof of \nProposition \\ref{prop:two}(ii) with $n=0$), $a_0$ and $a_1$ are not smoothly homotopic by\nProposition \\ref{prop:two}(i).   Thus showing (\\ref{neqo}) will prove the theorem.\n\nSet $M = S^2\\times S^3$. Since $a_1^L:M\\to LM$ has degree one on its image,\n\\begin{equation}\\label{cswf} \n\\int_{[a_1^L]} CS_5^W(g) = \\int_M a_1^{L,*} CS_5^W(g).\n\\end{equation}\nFor $m\\in M$, \n$$a_1^{L,*}CS_5^W(g)_m = f(m)d\\phi\\wedge d\\theta\\wedge dy\\wedge d\\psi\\wedge \nd\\alpha$$ \nfor some $f\\in C^\\infty(M)$.  We determine\n$f(m)$ by explicitly computing $a_{1,*}^L(\\partial_\\phi),..., a_{1,*}^L(\\partial_\\alpha),$ (e.g. \n$a_{1,*}^L(\\partial_\\phi)(a^L(m))(t) = \\partial_\\phi|_{a(m,t)}$ ),\nand noting\n\\begin{eqnarray}\\label{f} f(m) &=& f(m)d\\phi\\wedge d\\theta\\wedge dy\\wedge d\\psi\\wedge \nd\\alpha(\\partial_\\phi,\\partial_\\theta,\\partial_y,\\partial_\\psi,\\partial_\\alpha)\\nonumber\\\\\n&=& a_1^{L,*}CS_5^W(g)_m(\\partial_\\phi,...,\\partial_\\alpha)\\\\\n&=& CS_5^W(g)_{a_1^L(m)}(a_{1,*}^L(\\partial_\\phi),...,a_{1,*}^L(\\partial_\\alpha)).\\nonumber\n\\end{eqnarray}\nSince $CS^W_5(g)$ is explicitly computable from the formulas in \\S\\ref{localsymbols}, we can\ncompute $f(m)$ from (\\ref{f}).  Then $\\int_{[a_1^L]} CS_5^W(g) = \\int_{M} f(m)\nd\\phi\\wedge d\\theta\\wedge dy\\wedge d\\psi\\wedge \nd\\alpha$ can be computed as an ordinary integral in the dense coordinate space. \n\n\nVia this method, in the Mathematica file {\\tt ComputationsChernSimonsS2xS3.pdf} at {\\tt http:\/\/math.bu.edu\/people\/sr\/},  $\\int_{[a_1^L]} CS_5^W(g)$ is computed as\n a function of $(p,q).$  For example, \n$(p,q) = (7,3)$,\n$$\\int_{[a_1^L]} CS_5^W(g)  = -\\frac{1849\\pi^4}{22050}.$$\nThis formula is exact; the rationality up to $\\pi^4$ follows from $4p^2-3q^2$ being a perfect\nsquare, as then the various integrals computed in (\\ref{cswf}) with respect to our coordinates\nare rational functions evaluated at rational endpoints. \n In particular, (\\ref{neqo})\nholds.\n  \\end{proof}\n\n\n\\begin{rem}  \nFor $a=1$, the metric extends to the closure of the coordinate chart, but the total space is $S^5$ with the standard metric.  \n$\\pi_1({\\rm Diff} (S^5))$ is torsion \\cite{F-H}.  By Proposition \\ref{prop:two}(ii), $\\int_{[a^L]} CS^W_5 = 0$\nfor any circle action on $S^5.$  In the formulas in the Mathematica file, $\\int_{[a^L]}CS^W_5$ is proportional to\n$(-1+a)^2$, which vanishes at $a=1$.  \nThis gives a check of the validity of the computation.\n\\end{rem}\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSchwinger long ago predicted that a strong, constant electric field will create electron-positron pairs \\cite{schwinger1951gauge}. This has, however, never been experimentally observed as the electric field strengths required are several orders of magnitude larger than has ever been achieved in the laboratory \\cite{danson2015petawatt,Turcu:2016dxm}. The rate of pair production becomes large as the electric field strength approaches an appreciable fraction of the Schwinger critical field, $E_c= m_e^2c^3\/ e \\hbar$, corresponding to an electric field intensity $I_c=\\tfrac{1}{2}\\ensuremath{\\epsilon_0} c E_c^2 \\approx 2\\times 10^{29}~\\mr{Wcm}^{-2}$. It has also long ago been predicted that one can create electron-positron pairs from a thermal bath of photons \\cite{weaver1976reaction}, via the two photon Breit-Wheeler process, $\\gamma\\gamma\\to e^+e^-$ \\cite{breit1934collision}. This too has never been observed due to the unattainably high temperatures required (although an experiment has been proposed that uses a quasi-thermal radiation field for one of the two photons \\cite{pike2014photon}). In this case, the rate of pair production becomes large when $k_B T$ becomes an appreciable fraction of the rest mass energy of an electron and positron, $2m_e c^2\\sim 1~\\mr{MeV}$.\n\nOne can, however, combine these two ingredients. Starting from an initial state containing a thermal bath of photons at a high temperature and adding a strong, constant electric field, one finds the rate of pair production is significantly faster than either the Schwinger process or the purely thermal process alone. The nature of the process depends on the relative magnitudes of the electric field strength and the temperature. At low temperatures, when $E\/E_c\\gg k_B T\/2m_ec^2$, the process is essentially Schwinger pair production, in which virtual electron-positron pairs tunnel quantum mechanically through their energy barrier to existence. In the opposite limit, $E\/E_c\\ll k_B T\/2m_ec^2$, at high temperatures, virtual electron-positron pairs are given sufficient energy from the thermal bath to transition over the barrier classically. At intermediate temperatures the process can be described as thermally enhanced quantum tunnelling, whereby the virtual electron-positron pairs tunnel from an excited state at nonzero separation.\n\nHere we consider the viability of observing the thermal Schwinger process. If this were realised, it would be the first observation of semiclassical pair production in quantum electrodynamics (QED), a class of nonperturbative phenomena with applications in many branches of physics, including astrophysics, cosmology, heavy ion collisions, and plasma physics (see for example \\cite{Ruffini:2009hg}). It would also open up the controlled study of semiclassical pair production in general, a very basic process in quantum field theory and one that has proved elusive experimentally.\n\nTo put our work in context, we note that several other mechanisms have been proposed to lower the intensities required for Schwinger pair production, by including high frequency fields \\cite{brezin1970pair,Ringwald:2001ib,Popov:2001ak,Piazza:2004sv,dunne2009catalysis,Otto:2015gla,Panferov:2015yda,Torgrimsson:2016ant,Kohlfurst:2012rb,Jansen:2013dea}, or the Coulomb fields of highly charged nuclei \\cite{Muller:2003zzd,PhysRevA.73.053409,DiPiazza:2009py,FillionGourdeau:2012zs}. Further, it has been pointed out that once an initial seed pair has been produced, a cascade of pair production will follow for sufficiently strong field intensities \\cite{Bell:2008zzb,PhysRevLett.105.080402,Nerush:2010fe}. This may dramatically amplify any signal of Schwinger pair production. On the experimental side of strong-field QED, there has been much progress on a variety of fronts \\cite{burke1997positron,mackenroth2011nonlinear,thomas2012strong,blackburn2014quantum,cole2018experimental,poder2018experimental}, and the next generation of high intensity lasers offer exciting possibilities to discover and investigate qualitatively new phenomena in QED \\cite{DiPiazza:2011tq,Turcu:2016dxm}.\n\nIn the regime we consider here, there is an additional significance, in that the formula for the rate of pair production is a rare example of an all-orders and all-loop result in QED \\cite{gould2017thermal,Gould:2018ovk}. Perturbative estimates of the rate of pair production are orders of magnitude slower, so could be distinguished experimentally. Hence, one can probe QED beyond the perturbative loop expansion, and could decide experimentally on the validity of conjectured all-order behaviours of QED \\cite{cvitanovic1977asymptotic,affleck1981pair,lebedev1984virial,huet2017asymptotic,gould2017thermal,Gould:2018ovk}. Further, an experimental verification of the formulae would directly carry over to strongly-coupled physics, and hence provide an important validation of principle in the search for magnetic monopoles \\cite{acharya2016search,gould2017magnetic}. As such, an experimental search for the thermal Schwinger process is well motivated even independently of its connection to the pure Schwinger process.\n\n\\section{Theoretical results}\nAt zero temperature, and for electric field strengths somewhat below $E_c$, the rate of electron-positron pair production per unit volume in a constant electric field is given by Schwinger's result \\cite{schwinger1951gauge},\n\\begin{equation}\n \\Gamma_{\\mr{Schwinger}}(E)\\approx \\frac{(eE)^2}{4\\pi^3 c \\hbar^2}\\mr{e}^{-\\frac{\\pi m_e^2 c^3}{e  E \\hbar}}. \\label{eq:rate_schwinger}\n\\end{equation}\nLoop corrections to this formula have been computed at two loops \\cite{ritus1975lagrange,ritus1977connection,lebedev1984virial,ritus1998effective}, (see also \\cite{Gies:2016yaa}) and have been resummed to all loops (within the quenched approximation) \\cite{affleck1981pair,lebedev1984virial}. However, the loop corrections are small, and only give an $O(1\\%)$ enhancement over Schwinger's one-loop result. At leading loop order, there are also corrections which are exponentially subdominant for small $E\/E_c$ \\cite{schwinger1951gauge}.\n\nIf one adds a thermal bath of photons, the rate is enhanced. Note that in this analysis it is crucial that there are very few charged particles in the initial thermal state as their presence would Debye screen the electric field. The Debye screening length should be longer than the scales relevant for pair production (which we give in Section \\ref{sec:schematic}). Hence, we assume $k_B T\\ll m_e c^2$.\n\nDepending on the relative magnitudes of $E$ and $T$ one finds three different regimes. In the lowest temperature regime, the energy of the thermal bath is less than the energy that an electron-positron pair would gain when accelerated by the electric field over their Compton wavelength. That is, for temperatures lower than\n\\begin{equation}\n    T_{CW}:=e E\\hbar\/(2m_e c k_B). \\label{eq:lowT}\n\\end{equation}\nIn this regime, the thermal bath is negligible and electron-positron pairs are produced by quantum tunnelling from virtuality in vacuum, to reality. Above $T_{CW}$ the thermal bath excites virtual electron-positron pairs significantly above their ground state. Pair production then takes the form of quantum tunnelling from an excited state. At higher temperatures still, virtual electron-positron pairs acquire sufficient energy from thermal fluctuations to go over the energy barrier classically. This process dominates over quantum tunnelling for temperatures greater than\n\\begin{equation}\n    T_{WS}:=\\left(4 e E^3\\ensuremath{\\epsilon_0}\\hbar^4\/\\pi^3 m_e^2 k_B^4\\right)^{1\/4}\n\\end{equation}\nThis temperature can be understood as that when the lowest thermal (Matsubara) frequency $2\\pi k_BT\/\\hbar$ is equal to the exponential decay rate of the unstable electron-positron transition state, $2\\sqrt{2}(eE^3\\ensuremath{\\epsilon_0}\/m_e^2)^{1\/4}$.\\footnote{The labels $C$, $W$ and $S$ follow the notation of Ref.~\\cite{gould2017thermal}. They stand for \\emph{circular}, \\emph{wavy} and \\emph{straight} (or \\emph{sphaleron}) respectively, and refer to the shape of the instanton describing the processes at low, intermediate and high temperatures.}\n\n\\begin{figure}\n  \\includegraphics[width=1.0\\columnwidth]{allApproxQEDRatesExpPlusWriting5.pdf}\n  \\caption[Thermal Schwinger Rate]{Approximate rates of pair production for low, medium and high temperatures. The coloured region is where the approximations leading to Eqs. \\eqref{eq:rate_schwinger}, \\eqref{eq:rate_w} and \\eqref{eq:rate_sphaleron} are well satisfied. All rates are normalised by $\\Gamma_{\\mr{Ref}}=0.1~\\muup\\mr{m}^{-3}\\muup\\mr{s}^{-1}$. For intermediate temperatures we use only the leading two contributions in the fine structure constant whereas in the high temperature regime all orders are included. This leads to the apparent discontinuity which will be smoothed out by contributions at higher orders.}\n \n  \\label{fig:allRatesApprox}\n\\end{figure}\n\n\\emph{Low temperatures,} $T<T_{CW}$:\nFor very low temperatures, $T\\ll T_{CW}$, corrections to the Schwinger rate can be considered perturbatively. The dominant contribution to the rate in this regime is thus given by Fig.~\\ref{fig:feyn}~(a), just as at $T=0$. The leading thermal corrections arise at two loops and are \\cite{gies2000qed}\n\\begin{equation}\n    \\Gamma_C=\\Gamma_{\\mr{Schwinger}}\\left[1+\\frac{\\pi^4 e^2}{1440\\ensuremath{\\epsilon_0}\\hbar c} \\left(\\frac{T}{T_{CW}}\\right)^4+\\dots\\right].\n   \n\\end{equation}\nAs $T$ increases towards $T_{CW}$, higher order perturbative corrections become important. The exponential enhancement due to these corrections has been computed at leading \\cite{monin2010photon} and next-to-leading \\cite{gould2017thermal} order. The corrections to the exponent of the rate are small if $E\\lesssim E_c$, except if one is very close to $T_{CW}$ in which case even the exponent is unknown. Where one can trust the calculations, the thermal enhancement at low temperatures is less that 1\\%.\n\n\\emph{Intermediate temperatures,} $T_{CW}<T<T_{WS}$:\nAt $T_{CW}$, the rate goes through a sharp transition. In this intermediate temperature range, the rate of pair production is significantly (exponentially) enhanced by the presence of the thermal bath. For intermediate temperatures, the exponent of the rate is given by \\cite{selivanov1986tunneling,ivlev1987tunneling,brown2015schwinger,gould2017thermal,Draper:2018lyw}\n\\begin{align}\n  &\\Gamma_{W}(E,T)\\sim \\nonumber \\\\\n  &\\exp\\left[-2\\frac{m_e^2c^3}{eE\\hbar}\\left( \\arcsin\\left(\\frac{T_{CW}}{T}\\right) +  \\frac{T_{CW}}{T} \\sqrt{1 - \\frac{T_{CW}^2}{T^2}}\\right)\\right], \\label{eq:rate_w}\n\\end{align}\nthough there has been some dispute on this \\cite{medina2015schwinger,Korwar:2018euc}. The precise formula, including the prefactor of the exponential has recently been worked out in Ref.~\\cite{Torgrimsson:2019sjn}. There it was also shown that the dominant contribution to the rate in this regime is given by Fig.~\\ref{fig:feyn}~(b).\n\n\\emph{High temperatures,} $T_{WS}<T$:\nAt $T_{WS}$ there is again a sharp transition in the rate. In the high temperature regime, the pair production process can be seen as a thermal process enhanced by the presence of the electric field. Unlike the lower temperature regimes, the rate is not dominated by the diagrams in Fig.~\\ref{fig:feyn}; infinitely more such diagrams contribute to the leading approximation to the rate, leading to a significant nonperturbative enhancement. Some of the present authors have recently calculated the rate \\cite{gould2017thermal,Gould:2018ovk}. It is given by\n\\begin{equation}\n \\Gamma_{S}(E,T)\\approx \\frac{4 k_B T_{WS}\\left(m_e k_B T\\right)^{3\/2}\\mr{e}^{-\\frac{2m_e c^2}{k_B T}+\\frac{\\sqrt{e^3E\/\\pi\\ensuremath{\\epsilon_0}}}{k_B T}}}{(4\\pi)^{3\/2}\\hbar^4\\sin \\left(\\frac{\\pi T_{WS} }{T }\\right)\\sinh^2\\left(\\frac{\\pi T_{WS}}{\\sqrt{2} T }\\right)} .\\label{eq:rate_sphaleron}\n\\end{equation}\nIn deriving this expression we assumed the following three strong inequalities: $e E\\hbar\/m_e^2c^3\\ll 1$, $k_BT\/m_e c^2\\ll 1$ and $e (k_B T)^2 \/E c^2 \\ensuremath{\\epsilon_0} \\hbar^2\\ll 1$. Note that these imply the calculation is only valid when the rate of pair production is not too fast. For temperatures much larger than $T_{WS}$, this equation simplifies to \n\\begin{align}\n\\Gamma_S(E,T) & \\approx \\frac{m_e^{3\/2} T^2 \\left(k_B T\\right)^{5\/2}}{\\pi^{9\/2}\\hbar^4 T_{WS}^2}  \\mr{e}^{-\\frac{2m_e c^2}{k_B T}+\\frac{\\sqrt{e^3E\/\\pi\\ensuremath{\\epsilon_0}}}{k_B T}} \\nonumber \\\\\n&\\approx \\frac{ m_e^{5\/2} (k_B T)^{9\/2}}{ 2 \\pi ^3 \\hbar^{6} \\sqrt{ \\ensuremath{\\epsilon_0} e E^{3}} } e^{-\\frac{2 m_e c^2}{k_BT}+\\frac{\\sqrt{e^3E\/\\pi\\ensuremath{\\epsilon_0}}}{k_B T}}.\\label{eq:rate_sphaleron_high}\n\\end{align}\nAs one can see from this expression, surprisingly the rate is higher for weaker fields. This behaviour cannot be extrapolated to arbitrarily weak fields as the validity of our approximations breaks down for $E\\lesssim e (k_B T)^2 \/c^2 \\ensuremath{\\epsilon_0} \\hbar^2$.\n\nFor a thermal bath of photons in zero electric field, electron-positron pairs can be produced by two photon fusion (the Breit-Wheeler process). Integrating the cross section for this process over the photon thermal distribution one finds \\cite{weaver1976reaction}\\footnote{Note that there are several typos in Ref.~\\cite{weaver1976reaction}. We have repeated the calculation for low temperatures, both fully numerically and in the nonrelativistic approximation, finding agreement with Eq. \\eqref{eq:breit_wheeler}.}\n\\begin{align}\n \\Gamma_{BW}(T)&\\approx 2 \\int \\frac{d^3 p}{(2\\pi )^3}\\frac{d^3 q}{(2\\pi )^3} \\frac{p^\\mu q_\\mu}{p^0 q^0}\\theta\\left((p^\\mu+q^\\mu)^2-4m^2\\right)\\nonumber \\\\\n & \\qquad\\qquad\\qquad\\qquad f(p^0)f(q^0)\\sigma_{BW}\\left((p^\\mu+q^\\mu)^2\\right)\\nonumber \\\\\n &\\approx \\frac{e^4 m_e (k_B T)^3}{2(2\\pi)^4 c^3 \\ensuremath{\\epsilon_0}^2 \\hbar^6}\\mr{e}^{-\\frac{2m_e c^2}{k_B T}},\\label{eq:breit_wheeler}\n\\end{align}\nwhere $f(E)=1\/(\\mr{e}^{E\/k_BT}-1)$ and $\\sigma_{BW}(s)$ is the Breit-Wheeler cross section \\cite{breit1934collision}. This expression is valid for low temperatures, $k_B T\/m_e c^2 \\ll 1$, up to about $O(100~\\mr{keV}\/k_B)$ with an accuracy of a few percent. In the absence of an intense electric field, higher order perturbative effects involving more photons are expected to be subdominant if $e k_B T\/\\sqrt{c \\hbar \\ensuremath{\\epsilon_0}} \\ll m_ec^2$, or $k_BT\\ll 3 m_e c^2$ \\cite{king2012pair}.\n\nWe note that the addition of a constant electric field does not enhance the perturbative Breit-Wheeler process as a constant field consists of zero energy photons. Beyond this idealised limit, corrections to the Breit-Wheeler rate due to the presence of an additional source of photons with wavelength $\\lambda$ are suppressed by $\\exp(-\\lambda m^2 c^3\/(2\\pi \\hbar k_B T))$. For a typical laser source with $\\lambda = 0.8~\\muup \\mr{m}$ and a thermal bath of temperature $T=20~\\mr{keV}\/k_B$, say, this correction is completely negligible $\\sim 10^{-10^6}$.\n\nThe Breit-Wheeler process should not be thought of as a competing process to thermal Schwinger pair production. Fig.~\\ref{fig:feyn}~(b) shows the dominant Feynman diagram for thermal Schwinger pair production in the intermediate temperature regime. Applying the Optical Theorem, one can see that a unitarity cut of diagram (b) gives the Breit-Wheeler process~\\cite{Torgrimsson:2019sjn}, except that the effect of the electric field has been accounted for to all orders, giving a nonperturbative enhancement over the pure Breit-Wheeler process. In fact, Eq.~\\eqref{eq:rate_w} reduces to Eq.~\\eqref{eq:breit_wheeler} for temperatures $T\\gg T_{CW}$.\n\nHowever, in the high temperature regime, $T>T_{WS}$, the diagrams of Fig.~\\ref{fig:feyn} cease to dominate the rate of pair production and there is a further nonperturbative enhancement that cannot easily be understood diagramatically. As can be seen from Eqs.~\\eqref{eq:rate_sphaleron} and \\eqref{eq:rate_sphaleron_high}, the dependence of the rate, $\\Gamma_S$, on the fine-structure constant is non-analytic even after one absorbs one power of $e$ into the electric field.\n\\begin{figure}\n  \\includegraphics[width=0.7\\columnwidth]{feyn.pdf}\n  \\caption{Feynman diagrams which dominate the rate of thermal Schwinger pair production in the (a) low and (b) intermediate temperature regimes. Double lines denote the electron propagator including the effect of the external electric field to all orders and the wiggly lines denote photons from the thermal bath. In the high temperature regime, the rate is not dominated by a single such Feynman diagram but infinitely many diagrams contribute to the leading approximation to the rate.}\n  \\label{fig:feyn}\n\\end{figure}\n\nFor completeness, we note that the addition of a single electromagnetic plane wave to a thermal bath of photons also leads to nonperturbatively enhanced electron-positron pair production. This is true even in the long-wavelength limit, $\\lambda m c\/\\hbar \\to \\infty$, showing the collective, nonperturbative nature of the phenomenon. In this case, the Breit-Wheeler rate is additively enhanced by \\cite{king2012pair},\n\\begin{equation}\n\\Gamma_{\\mr{Plane}} \\approx \\frac{3^{3\/4} e^2 (k_B T)^2 m^2 }{16 \\pi ^{5\/2} \\ensuremath{\\epsilon_0} \\hbar^5} \\left(\\frac{e E \\hbar k_B T}{\n   m^3 c^5}\\right)^{1\/4} \n   \\mr{e}^{-\\sqrt{\\frac{16c^5 m^3}{3 e E \\hbar k_B T }}}. \\label{eqn:rate_king} \n\\end{equation}\nThis result is valid for $\\sqrt{k_B T E\/(m c^2 E_c)}\\ll 1$. The crucial difference from that of a constant electric field is that the electromagnetic invariant $E^2-c^2B^2$ of a plane wave vanishes. As we will note later, Eq. \\eqref{eqn:rate_king} is orders of magnitude smaller than the thermal Schwinger rate, showing that the absence of the magnetic field is crucial for pair production.\n\n\\section{Observability}\n\nWe would like to understand exactly how high the temperatures and how strong the electric fields need to be to get a measurable rate of pair production. To answer that, we will make a simple comparison with the experiment of Ref.~\\cite{burke1997positron}, which was the first experiment to observe the (multi-photon) Breit-Wheeler process. They observed $106\\pm14$ positrons produced in this way, from a total spacetime interaction volume of order $10^{-21}~\\mr{m}^3 \\mr{s}$ (when integrated over all laser shots). Hence we take as our observable reference rate $\\Gamma_{\\mr{Ref}}=10^{23} ~\\mr{m}^{-3}\\mr{s}^{-1}=0.1~\\muup\\mr{m}^{-3}\\muup\\mr{s}^{-1}$, which is approximately Avogadro's number of positrons per metre cubed per second. One can therefore reasonably expect that a normalised rate greater than 1 will be required for the rate of pair production to be measurable. In Fig.~\\ref{fig:allRatesApprox} we show the thermal Schwinger rate in all three regimes, normalised by this reference rate.\n\nThe almost perfectly vertical lines of constant rate in the low temperature regime reflect that, in this regime, the thermal enhancements are small. As such, this regime offers no advantages over pure Schwinger pair production for experimentally observing pair production. On the other hand, in the intermediate and high temperature regimes, the thermal enhancements are very significant. Of these two regimes, observing pair production in the high temperature regime is easier, because the electric field intensities required are orders of magnitude smaller, while the temperatures required are very similar.\n\nFrom Figure \\ref{fig:allRatesApprox} one can see that temperatures around $O(20~\\mr{keV}\/k_B)$ or above are needed in order to produce an observable number of positrons. Perhaps the leading method of producing high temperature thermal photons is with a laser and cavity, or holhraum. The aim of achieving inertial confinement fusion (ICF) has been a powerful incentive in developing these technologies. Thermal distributions of $0.3~\\mr{keV}\/k_B$ have been achieved since 1990, though about $0.4~\\mr{keV}\/k_B$ is likely the upper limit of this approach \\cite{lindl2004physics}. Unfortunately at these temperatures, the thermal enhancement of the Schwinger rate is negligible.\n\nWhen ICF is achieved, the burning thermonuclear plasma leads to significantly higher energy densities. Charged particles in the plasma are expected to reach temperatures from $O(20~\\mr{keV}\/k_B)$ to $O(200~\\mr{keV}\/k_B)$, depending on the composition and size of the burning plasma \\cite{tabak1996role,rose2013electron}. Burning deuterium (D) plasmas are expected to be hotter than burning deuterium-tritium (DT) plasmas, as the peak nuclear reaction rate is at higher energies for D-D nuclear reactions. For a fixed composition, larger plasmas reach higher temperatures.\n\nAs the plasma is not optically thick, the effective temperature of the photons is lower than that of the charged particles. For representative examples, of burning deuterium plasma with radii $r=120~\\muup\\mr{m}$ and $r=150~\\muup\\mr{m}$, one finds that the photon energy density is equal to that of a Planck distribution with two degrees of freedom at $T=22~\\mr{keV}\/k_B$ and $T=26~\\mr{keV}\/k_B$ respectively. The photon distribution can be calculated using the approach of Ref.~\\cite{rose2013electron}. However, the result is further from equilibrium than that of the charged particles. For now, we will assume a thermal distribution of photons, though we will return to this point in Section \\ref{sec:distributions}.\n\n\\section{An experimental scheme} \\label{sec:schematic}\n\n\\begin{figure}\n  \\includegraphics[width=1.0\\columnwidth]{experimentSchematic3.pdf}\n  \\caption{Schematic of the experimental set-up. Two counter propagating high energy beams are focused into an X-ray radiation field produced by a burning fusion.}\n  \\label{fig:schematic}\n\\end{figure}\n\nSo, in order to observe the thermal Schwinger process, we would propose combining two lasers with combined intensity $O(10^{23}~\\mr{Wcm}^{-2})$ with a source of thermal photons with temperature $O(20~\\mr{keV}\/k_B)$. A possible schematic for such an experiment is shown in Figure \\ref{fig:schematic}. The region of interest for our purposes is on the left-hand side, outside the ignited thermonuclear plasma. A window is needed to hold up the material expansion long enough to allow the high-intensity lasers to interact with the radiation from the ICF capsule. The wall of the hohlraum would in principle be able to act in this way whilst transmitting the majority of the radiation, though this would require specific design. As long as the distances from the nuclear plasma are small compared with its radius, the geometric reduction of the intensity will not be significant.\n\nThe electric field is provided by a high intensity laser, split into two counter-propagating beams. These are focused so that the magnetic fields cancel in the vicinity of a given point, whereas the electric fields reinforce. Assuming standard parameters for the laser, with wavelength $\\lambda \\sim 0.8~\\muup m$, the field maxima of the two beams are expected to be approximately of size $O(\\lambda^3)$ and of time extent $O(\\lambda\/c)$, with approximately 10-20 field maxima per shot, amounting to a possible pair production region of size $5\\times 10^{-32}~\\mr{m}^3\\mr{s}$. The integrals of the rate over the interaction region can be carried out in the locally constant field approximation (see for example Refs.~\\cite{galtsov1983macroscopic,Gavrilov:2016tuq}), within which the region around the field maxima will dominate the pair production. However, in what follows we simply multiply the rates by the approximate spacetime volume of the field maxima, which is sufficient to get the order of magnitude correct.\n\nTo achieve $\\gtrsim 1$ electron-positron pair produced per shot, requires a rate $5\\times 10^6$ times faster than $\\Gamma_{\\mr{Ref}}$ (see Fig.~\\ref{fig:sphaleron_rate}). Assuming a thermal distribution of photons from the burning nuclear plasma, this could be achieved at\n\\begin{align}\n T_\\star &\\approx 20~\\mr{keV}\/k_B, \\nonumber \\\\\n I_{E\\star} &\\approx 1.3\\times 10^{23}~ \\mr{W cm}^{-2}, \\label{eq:parameters}\n\\end{align}\nwhere $I_E$ refers to the combined intensity of the two beams. This parameter point is shown as a star in Fig.~\\ref{fig:sphaleron_rate}. For comparison, we also consider a second point with a significantly higher production rate, at \n\\begin{align}\n T_\\blacktriangle &\\approx 26~\\mr{keV}\/k_B, \\nonumber \\\\\n I_{E\\blacktriangle} &\\approx 3.7\\times 10^{23}~\\mr{W cm}^{-2}. \\label{eq:parameters_triangle}\n\\end{align}\nFor these two sets of parameters, the numbers of positrons produced per shot via the thermal Schwinger, Breit-Wheeler and pure Schwinger (without thermal enhancement) processes are given in Table \\ref{table:positrons_per_shot}. We also include the nonperturbative enhancement to the number of positrons produced due to only one of the two laser beams, given by Eq. \\eqref{eqn:rate_king}. \n\nNote that the pure Breit-Wheeler process can take place in a larger region than that of the Schwinger pair production, which is not accounted for in Table~\\ref{table:positrons_per_shot}. In order to ensure that the thermal Schwinger process dominates, and taking into account its $O(10^6)$ times higher rate, the volume of the interaction region should be significantly less than about $10^7 \\lambda^3\\approx 5\\times 10^{-3}\\mr{mm}^3$. This can be achieved by modifying the diameter of the hohlraum window and by focusing the laser fairly close to the window. Further, the directionality of emitted electrons and positrons can help distinguish between production mechanisms, with the Breit-Wheeler process producing pairs more or less isotropically and the Schwinger process producing pairs along the electric field of the counterpropagating lasers.\n\nIt is encouraging that a relatively small increase in both radiation temperature and laser intensity\nproduces such a significant increase in the production rate. One can also see that the thermal Schwinger process has a huge nonperturbative enhancement. A simple perturbative estimate of the number of positrons produced by the Breit-Wheeler process underestimates the actual number by a factor of $10^6$. Such large enhancements are a generic feature of the thermal Schwinger process in the high temperature regime \\cite{Gould:2018ovk}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ c c c c c } \n  \\toprule\n   & Thermal Schwinger & Breit-Wheeler & Eq. \\eqref{eqn:rate_king} & Schwinger\\\\\n  \\colrule\n  $\\star$ & 3 & $3\\times 10^{-6}$ & $10^{-167}$ & $10^{-1817}$\\\\\n  $\\blacktriangle$ & $1\\times 10^6$ & 1 & $10^{-105}$ & $10^{-1062}$ \\\\\n\\end{tabular}\n\\caption{Numbers of positrons produced per shot via different mechanisms for the two sets of parameters given by Eqs. \\eqref{eq:parameters} and \\eqref{eq:parameters_triangle}. The column labelled Eq. \\eqref{eqn:rate_king} is that for a thermal bath plus a single laser beam (rather than counter-propagating beams).}\n\\label{table:positrons_per_shot}\n\\end{center}\n\\end{table}\n\nOne can also compare the thermal Schwinger rate to that obtained in a thermal bath plus only a single high intensity laser beam (in which case the magnetic field does not cancel). In this case, the enhancement of the rate of pair production due to the high intensity laser is given by Eq.~\\eqref{eqn:rate_king}. For the parameters of either Eqs. \\eqref{eq:parameters} or \\eqref{eq:parameters_triangle}, one finds that the enhancement is negligible and the rate of this process is smaller than the thermal Schwinger rate by a factor of $\\sim 10^{\u2212100}$ or more, as can be seen in Table \\ref{table:positrons_per_shot}.\n\nFor the validity of the locally constant field approximation, it is important that the electric field, as well as the photon distribution from the plasma, are slowly varying on the time and length scales of the pair creation process, described by an instanton. The time scale of the instanton is $t_{\\mr{inst}}\\sim \\hbar\/k_B T$ and the length scale is $\\sqrt{e\/(4\\pi \\ensuremath{\\epsilon_0} E)}$ \\cite{gould2017thermal,Gould:2018ovk}. Using temperature and electric field strengths determined by Eq. \\eqref{eq:parameters}, this amounts to $3\\times 10^{-20}~s\\approx 10^{-11}~\\mr{m}\/c$ and $5\\times 10^{-12}~\\mr{m}$ respectively. The smallest length scale on which the electric field varies is the wavelength of the laser. Assuming a laser with wavelength of $\\lambda \\sim 0.8~\\muup m$, one can safely treat the electric field as constant. Further, one would expect the photon distribution from the plasma to vary on a length scale of order the size of the hohlraum window. This will likewise be much larger than the length scale of the instanton, $\\sim 5\\times 10^{-12}~\\mr{m}$, and hence the locally constant field approximation is applicable.\n\n\\begin{figure}\n  \\includegraphics[width=1.0\\columnwidth]{sphaleronQEDRateExp2.pdf}\n  \\caption[Thermal Schwinger rate at high temperature]{The number of electron-positron pairs produced per shot in the experiment proposed here. The coloured region is the high temperature region, and where the approximations leading to Eq. \\eqref{eq:rate_sphaleron} are valid. The solid black line is the boundary between the intermediate and high temperature regions, defined by $T=T_{WS}$. The dashed black line is defined by $E=0.2e(k_BT)^2\/\\ensuremath{\\epsilon_0} c^2 \\hbar^2$ and the dotted black line by $T=0.2m_ec^2\/k_B$. The star and diamond are the points referred to in Eqs. \\eqref{eq:parameters} and \\eqref{eq:parameters_triangle}.}\n  \\label{fig:sphaleron_rate}\n\\end{figure}\n\nIn the region where the electric field and thermal photons collide, electrons and positrons will be produced with an approximately thermal spectrum of velocities and then accelerated in opposite directions antiparallel and parallel respectively to the electric field. Their thermal velocities are isotropic in the lab frame and are expected to be rather large, $\\tfrac{1}{3}\\bar{v^2}\\sim k_B T\/m_e \\sim (0.2 c)^2$. The field then accelerates the particles over a distance $\\lesssim \\lambda$, giving them a highly relativistic velocity, $v\\approx c$, parallel to the electric field and up to energies of order $eE\\lambda \\sim 1~\\mr{GeV}$. Once produced, the electrons and positrons may be deflected in opposite directions with a magnet, after which their momenta can be measured by a calorimeter, as in Ref. \\cite{burke1997positron}. If the combined intensity of the lasers is greater than around $10^{24}~\\mr{Wcm}^{-1}$, a seed electron-positron pair produced by the thermal Schwinger process will induce a cascade of pair production, so amplifying any positive signal~\\cite{Bell:2008zzb,PhysRevLett.105.080402,Nerush:2010fe}.\n\nIn the absence of charged particles, the thermal Schwinger process is the dominant mechanism of electron-positron pair production. However, if charged particles are not adequately shielded, other pair production processes are possible, such as the trident mechanism ($e^-Z\\to e^- e^+e^- Z$) and the Bethe-Heitler process ($\\gamma Z\\to e^+e^- Z$). Another possibility is for non-linear Compton scattering of charged particles in the laser field, producing high energy photons which then take part in the Breit-Wheeler process. Debye screening by charged particles will also inhibit the thermal Schwinger process if the Debye length is not much longer than the length scale of the pair creation process, $\\sqrt{e\/(4\\pi \\ensuremath{\\epsilon_0} E)}$. For the parameters of Eq. \\eqref{eq:parameters}, one requires the density of charged particles to be much less than one per $\\mr{pm}^3$. In the regime we have considered here, the purely thermal and the purely Schwinger pair production rates are orders of magnitude lower than the combination. Thus by performing null shots, with either only the burning plasma or only the high intensity laser, one can measure the presence of any backgrounds.\n\n\\section{Photon distributions}\\label{sec:distributions}\n\nLet us return to consider the distribution of photons in the burning plasma. This must be close to equilibrium for our approach to be valid. To investigate this we have solved the Boltzmann equation for the distribution of photons for a range of different plasma sizes and compositions. We have followed the method of Ref.~\\cite{rose2013electron}, including the effect of Compton scattering. For our representative example, of a burning deuterium plasma of radius $r=150~\\muup\\mr{m}$, the photon intensity at the surface of this plasma is shown as the full black line in Fig.~\\ref{fig:intensity}.\n\n\\begin{figure}\n  \\includegraphics[width=1.0\\columnwidth]{planckLogIntensities2.pdf}\n  \\caption[Photon Intensity]{Photon intensity in a D-burning target of radius $150~\\muup\\mr{m}$, along with various approximations to it. Note that a purely thermal distribution at $T=148~\\mr{keV}\/k_B$ would lie at much higher intensities.}\n  \\label{fig:intensity}\n\\end{figure}\n\nEquating the photon energy density to that of a thermal distribution, one finds that the effective temperature of the distribution is $26~\\mr{keV}\/k_B$. Doing the same for the photon number density, one instead finds a somewhat lower effective temperature of $16~\\mr{keV}\/k_B$, showing that the distribution is shifted to higher energies with respect to a thermal distribution. Plotting the photon intensity of a Planck distribution at $T=26~\\mr{keV}\/k_B$, the blue dotted line in Fig.~\\ref{fig:intensity}, one can see the shift to higher energies.\n\nThe high energy tail of the distribution, above about $700~\\mr{keV}$, is an exponential fall off and hence fits well a Boltzmann tail with an effective temperature of $T=148~\\mr{keV}\/k_B$, though scaled down by a normalisation, or equivalently a negative photon chemical potential\\footnote{The photon chemical potential must be zero in equilibrium, but not necessarily out of equilibrium. In this context its presence is natural as photon number conserving processes dominate.}, $\\mu=-1097~\\mr{keV}$, plotted as the dot-dashed green line in Fig.~\\ref{fig:intensity}. At the lowest energies, the distribution rises above this, and can be better described by a purely thermal distribution at a much lower temperature, $T=7.9~\\mr{keV}\/k_B$, plotted as the dashed red line in Fig.~\\ref{fig:intensity}. At intermediate energies, the distribution is not well described by a Bose-Einstein distribution. Nevertheless, the overall shape of the distribution is qualitatively similar to a thermal distribution, being smooth and highly occupied with a power-like rise at low energies and an exponential decrease at high energies, though we have used four different effective temperatures to describe different aspects of it, ranging from $7.9~\\mr{keV}\/k_B$ to $148~\\mr{keV}\/k_B$.\n\nIn two counterpropagating laser beams with intensity given by Eq.~\\eqref{eq:parameters} or \\eqref{eq:parameters_triangle}, one finds that the intermediate temperature regime of thermal Schwinger pair production would be reached at temperatures above\n\\begin{align}\n    T_{CW,\\star} &=0.20~\\mr{keV}\/k_B, \\nonumber \\\\\n    T_{CW,\\blacktriangle} &= 0.32~\\mr{keV}\/k_B, \\label{eq:tcw} \n\\end{align}\nand the high temperature regime would be reached at temperatures above\n\\begin{align}\n    T_{WS,\\star} &=2.5~\\mr{keV}\/k_B, \\nonumber \\\\ \n    T_{WS,\\blacktriangle} &= 3.7~\\mr{keV}\/k_B. \\label{eq:tws} \n\\end{align}\nAll four effective temperatures we have used to describe the distribution of photons in the burning plasma are well above these temperatures. We thus expect the high temperature regime to provide a better description of pair production in this setup than either the low or intermediate temperature regimes, which would imply that the diagrams of Fig.~\\ref{fig:feyn} do not dominate pair production and there is a nonperturbative enhancement over both pure Schwinger and pure thermal pair production.\n\nPhysically, it is clear that the process of pair production should not depend on the photon gas being precisely in equilibrium: if we use the picture of tunnelling from an excited state, one would expect that it is the energy and density of the photon distribution, rather than the nearness to equilibrium, that matters.\n\nOn the other hand, the condition of equilibrium is necessary for the calculation because it leads to important simplifications in the calculation of the production rate. The nonperturbative calculation of Eq.~\\eqref{eq:rate_sphaleron} \\cite{gould2017thermal,Gould:2018ovk} relied heavily on the Matsubara formalism~\\cite{Bloch1932,Matsubara:1955ws}, which is only valid in equilibrium. In the high temperature regime a resummation of all-orders of the perturbative loop expansion was required. Generalising the result to any out-of-equilibrium distribution is beyond the scope of this paper.\n\nWe note however that the diagrams of Fig.~\\ref{fig:feyn} can be calculated in an arbitrary photon distribution, following the approach of Ref.~\\cite{Torgrimsson:2019sjn}, though in the high temperature regime these diagrams are not dominant. Considering the calculation in this distribution, it can be seen that these diagrams reproduce the perturbative Breit-Wheeler rate up to very small corrections, essentially because the photon gas is highly occupied at energies much greater than $k_B T_{CW}$ (see Eqs.~\\eqref{eq:lowT} and \\eqref{eq:tcw}). Further, perturbative corrections in this distribution due to the high intensity laser require one photon from the high energy tail of the distribution and hence are suppressed by $\\exp(-\\lambda m^2 c^3\/(2\\pi \\hbar k_B T) + \\mu\/k_B T)\\sim 10^{-10^{5}}$, where $T$ and $\\mu$ here refer to the green dot-dashed line in Fig.~\\ref{fig:intensity}. Thus any enhancement due to the high intensity laser must be a nonperturbative phenomenon which goes beyond the diagrams of Fig.~\\ref{fig:feyn}.\n\nBecause the full nonperturbative calculation of the rate of pair production is beyond the scope of this paper, the possibility of nonperturbative enhancements in our proposed setup is conjectural. However, as all the effective temperatures we have used to describe the photon distribution are larger than $T_{WS}$, Eq.~\\eqref{eq:tws}, we expect the high temperature regime to best describe the photon distribution in question. We thus expect a nonperturbative enhancement over the perturbative prediction, as is the case in equilibrium where the enhancement to the positron yield was $O(10^6)$. The experiment we have proposed here would be able to test this plausible conjecture, by performing null shots without the counterpropagating laser beams, for which only the perturbative process is possible. This would be able to determine which features of a photon distribution are important for the nonperturbative enhancements to pair production which feature in the thermal Schwinger effect, and how generic such enhancements are.\n\n\\section*{Acknowledgements}\nO.G. would like to thank Holger Gies and Greger Torgrimsson for discussions related to this work. O.G. was supported from the Research Funds of the University of Helsinki. S.M. was supported by Engineering and Physical Sciences Research Council grant No. EP\/M018555\/1 and by Horizon 2020 under European Research Council Grant Agreement No. 682399. A.R. was supported by the U.K. Science and Technology Facilities Council grant ST\/P000762\/1. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background}\n\nIn this paper, let $\\Sigma$ be a fixed, compact, smooth, oriented surface of genus \n$g > 1$. We also denote $\\sigma |dz|^2$ as a {hyperbolic metric} on $\\Sigma$, for conformal \ncoordinates $z$. On $(\\Sigma, \\sigma |dz|^2)$, we denote the Laplacian as\n\\begin{center}\n$\\Delta = {\\frac{4}{\\sigma}}{\\frac{\\partial^2}{\\partial z \\partial \\bar{z}}}$,\n\\end{center}\nwith nonpositive eigenvalues.\n\n\n{Teichm\\\"{u}ller space} is a complex manifold with the {mapping class group} acting by biholomorphisms, of {complex dimension} \n$3g-3$, the number of independent closed curves in any pair-of-pants decomposition \nof the surface. \n\n\nThe {Weil-Petersson} metric is invariant under the action of the {mapping class group}, hence it decends to a \nmetric on the {moduli space}. It is shown that the {Weil-Petersson} metric is K\\\"{a}hlerian (\\cite {Ah1}), \nwith negative {sectional curvature} (\\cite {Tr}, \\cite {Wp3}). The {Weil-Petersson} Riemannian curvature \ntensor is given as the Tromba-Wolpert formula\n(\\cite {Tr}, \\cite {Wp3}):\n\\begin{center} \n$R_{\\alpha\\bar{\\beta}\\gamma\\bar{\\delta}} = \n\\int_{\\Sigma}D(\\mu_{\\alpha}\\bar\\mu_{\\beta})\\mu_{\\gamma}\\bar\\mu_{\\delta}dA + \n\\int_{\\Sigma}D(\\mu_{\\alpha}\\bar\\mu_{\\delta})\\mu_{\\gamma}\\bar\\mu_{\\beta}dA$,\n\\end{center}\nwhere $\\Delta$ is the Laplacian for the {hyperbolic metric} $\\sigma$ on the surface, and \n$D = -2(\\Delta -2)^{-1}$ is a self-adjoint, compact operator.\nThose $\\mu$'s in the formula are tangent vectors, i.e., harmonic {Beltrami differential}s. There is a natural \npairing between \n$QD(\\Sigma) =\\{\\phi(z) dz^2\\}$ and $HB(\\Sigma) =\\{\\mu(z) {\\frac{d\\bar{z}}{dz}}\\}$:\n\\begin{center}\n$<\\phi dz^2,\\mu {\\frac{d\\bar{z}}{dz}}> = Re \\int_{\\Sigma}\\phi \\mu dzd\\bar{z}$.\n\\end{center}\n\n\nA remarkable property of the {Weil-Petersson} metric is that it is incomplete (\\cite {Ma}, \n\\cite {Wp1}). This is caused by pinching off at least one short closed geodesic on the \nsurface. The {Weil-Petersson} completion modulo the {mapping class group} is topologically the Deligne-Mumford \ncompactification of the {moduli space}. The compactification divisor, thus \nconsists of a union of lower dimensional {Teichm\\\"{u}ller space}s, each such space consists of noded \n{Riemann surface}s, obtained by pinching nontrivial short closed geodesics on the surface \n(\\cite {B}, \\cite {Ma}). Therefore the compactification divisor can be described \nvia the systole $l_{0}(\\sigma)$ as the set $\\{l_{0}(\\sigma) = 0\\}$.\n\n\nSince we will analyze {harmonic map}s between compact hyperbolic surfaces, we recall some \nfundamental facts here. For a Lipschitz map $w:(\\Sigma, \\sigma |dz|^2) \\rightarrow \n(\\Sigma, \\rho |dw|^2)$, where $\\sigma |dz|^2$ and $\\rho |dw|^2$ are {hyperbolic metric}s on \n$\\Sigma$, and $z$ and $w$ are conformal coordinates on $\\Sigma$, \none follows Sampson (\\cite {S}) to define \n\\begin{center}\n${\\mathcal{H}}(z) = {\\frac{\\rho(w(z))}{\\sigma(z)}}|w_z|^2, {\\mathcal{L}}(z) = \n{\\frac{\\rho(w(z))}{\\sigma(z)}}|w_{\\bar{z}}|^2$.\n\\end{center}\nWe call ${\\mathcal{H}}(z)$ the {\\it holomorphic energy density}, and ${\\mathcal{L}}(z)$ \nthe {\\it anti-holomorphic energy density}. Then the energy density function of $w$ is \nsimply $e(w)= {\\mathcal{H}} + {\\mathcal{L}}$, and the total energy is then given by\n\\begin{center}\n$E(w,\\sigma,\\rho) = \\int_{\\Sigma}e\\sigma|dz|^2$.\n\\end{center}\n\n\nWe also note that the {Jacobian determinant} relative to the $\\sigma$ metric is therefore given by \n$J(z) = {\\mathcal{H}}(z) - {\\mathcal{L}}(z)$.\n\n\nThe map $w$ is called {\\it harmonic} if it is a critical point of this energy \nfunctional, i.e., it satisfies Euler-Lagrange equation:\n\\begin{center}\n$w_{z\\bar{z}}+ {\\frac{\\rho_w}{\\rho}}w_z w_{z\\bar{z}} = 0$.\n\\end{center}\n\n\nThe $(2,0)$ part of the pullback $w^{*}\\rho$ is the so-called {\\it {Hopf differential}}:\n\\begin{center}\n$\\phi(z)dz^2 = (w^{*}\\rho)^{(2,0)} = \\rho(w(z)) w_z {\\bar{w}}_zdz^2$.\n\\end{center}\nIt is routine to check that $w$ is harmonic if and only if $\\phi dz^2 \\in QD(\\Sigma)$, \nand $w$ is conformal if and only if $\\phi=0$.\n\n\nIn our situation, there is a unique \n{harmonic map} $w:(\\Sigma, \\sigma) \\rightarrow (\\Sigma, \\rho)$ in the homotopy class of the \nidentity, moreover, this map $w$ is a diffeoemorphism with positive Jacobian $J$, and \n${\\mathcal{H}}>0$ (\\cite {ES}, \\cite {Hr}, \\cite {S}, \\cite {SY}). \n\n\nA key observation to link the harmonic maps to Teichm\\\"{u}ller theory is that one \nobtains a map from {Teichm\\\"{u}ller space} to $QD(\\Sigma)$, for some fixed {hyperbolic metric} $\\sigma$. More \nspecifically, this map sends any {hyperbolic metric} on $\\Sigma$ to a holomorphic quadratic \ndifferential associated to the unique {harmonic map} in the homotopy \nclass of the identity. This map is a diffeomorphism (\\cite {S}, \\cite{Wf1}).\n\n\\section{Proof of Main Theorems} \n\\noindent \nLet $\\mu = \\mu (z){\\frac{d\\bar{z}}{dz}}$ be a unit {Weil-Petersson} normed harmonic {Beltrami differential}, thus \n$\\int_{\\Sigma}|\\mu|^2 dA = 1$. The {Weil-Petersson} {holomorphic sectional curvature} in the direction of $\\mu$ is given by\n\\begin{center}\n$K_h = -2\\int_{\\Sigma}D(|\\mu|^2)|\\mu|^2 dA$.\n\\end{center} \nIts upper bound, in terms of the genus, is known, proved by Wolpert:\n\\begin{lem}(\\cite {Wp4})\n$K_h < -{\\frac{1}{2\\pi(g-1)}}$.\n\\end{lem}\n\n\n\\begin{rem}\nThis upper bound holds without restriction on the systole. Since all \n{sectional curvature}s are negative, so the {Ricci curvature} and scalar curvature are bounded from \nabove by $-{\\frac{1}{2\\pi(g-1)}}$, and $-{\\frac{3(3g-2)}{4\\pi}}$, respectively (\\cite {Wp4}).  \n\\end{rem}\n\n\nWe now show the following pointwise estimate on $|\\mu(z)|$, where the tangent \nvector $\\mu(z){\\frac{d\\bar{z}}{dz}}$ is normalized to have unit {Weil-Petersson} \nnorm, and we shall apply this estimate to prove our main theorems.\n\\begin{theorem}\nFor $\\mu(z){\\frac{d\\bar{z}}{dz}} \\in HB(\\Sigma)$ with $||\\mu||_{WP} =1$, there \nexists a positive constant $h_0$, independent of $g$, such that \n$|\\mu (z)| \\le h_0$, for all $z \\in \\Sigma$, where the surface $\\Sigma$ is in \nthe thick part of the {moduli space}.\n\\end{theorem}\n\\begin{proof}\nRecall that $\\mu(z){\\frac{d\\bar{z}}{dz}}$ is a harmonic {Beltrami differential}, hence is a symmetric \ntensor given as $\\bar{\\phi}(ds^2)^{-1}$ for $\\phi$ a {holomorphic quadratic differential} with at most simpole poles \nat the cusps and $ds^2$ the {hyperbolic metric} tensor (\\cite{Wp2}). Since the surface $\\Sigma$ lies in the \nthick part of {moduli space}, hence no cusps and this {holomorphic quadratic differential} $\\phi = \\phi(z)dz^{2}$ has no poles.\n\n\nNote that $\\phi \\in QD(\\Sigma)$, as stated in the previous section, by a theorem of \nWolf (\\cite{Wf1}), there exists a {hyperbolic metric} $\\rho$ on surface $\\Sigma$, and a unique \n{harmonic map} $w: (\\Sigma,\\sigma) \\rightarrow (\\Sigma,\\rho)$, such that $\\phi$ is the \nHopf differential associated to this {harmonic map} $w$, i.e., \n$\\phi(z)dz^{2} = \\rho(w(z)) w_z {\\bar{w}}_zdz^{2}$.\n\n\nMuch of our study will be analyzing this {harmonic map} $w$. Note that even though \n$inj_{\\sigma}(\\Sigma) > r_{0} >0$, the metric $\\rho$ might not lie in the thick part \nof the {moduli space}. We recall that the holomorphic and anti-holomorphic energy density \nfunctions of $w$ are defined as \n${\\mathcal{H}}(z) = {\\frac{\\rho(w(z))}{\\sigma(z)}}|w_z|^2$, and ${\\mathcal{L}}(z) = \n{\\frac{\\rho(w(z))}{\\sigma(z)}}|w_{\\bar{z}}|^2$, respectively. \n\n\nThe energy density function of $w$ is $e(w)= {\\mathcal{H}} + {\\mathcal{L}}$, while the {Jacobian determinant} \nbetween {hyperbolic metric}s $\\sigma$ and $\\rho$ is therefore \n$J(z) = {\\mathcal{H}}(z) - {\\mathcal{L}}(z)$. Since the map $w$ is a diffeomorphism \nwith positive {Jacobian determinant}, we have ${\\mathcal{H}}(z) > {\\mathcal{L}}(z) \\ge 0$.\n\n\nWe also find that \n${\\mathcal{H}}{\\mathcal{L}} = {\\frac{|\\phi|^{2}}{\\sigma^{2}}} = |\\mu|^{2}$, so \nthe zeros of ${\\mathcal{L}}$ are the zeros of $|\\mu|$, or equivalently, the zeros of $\\phi$.\n\n\nLet $\\nu$ be the {Beltrami differential} of the map $w$, defined by \n$\\nu = {\\frac{w_{\\bar z}d\\bar z}{w_{z}dz}}$. It measures the failure of $w$ to be \nconformal, and since $J > 0$, we have $|\\nu|<1$. \n\n\nOne easily finds that\n $|\\nu|^{2} = {\\frac{\\mathcal{L}} {\\mathcal{H}}}$, therefore,\n \\begin{center}\n $|\\mu|= {\\sqrt{{\\mathcal{H}}{\\mathcal{L}}}} = {\\mathcal{H}}|\\nu| < {\\mathcal{H}}$.\n \\end{center}\nThus it suffices to estimate ${\\mathcal{H}}(z)$ to bound $|\\mu|$ pointwisely.\n\n\nLet $z_{0} \\in \\Sigma$ such that ${\\mathcal{H}}(z_{0}) = max_{z \\in \\Sigma}{\\mathcal{H}}(z)$. \nWe follow a calculation of Schoen-Yau to define a local one-form $\\theta = \\sqrt{\\sigma(z)}dz$, \nand find (\\cite {SY}):\n \\begin{center}\n $|w_{\\theta}|^{2}= {\\frac{\\rho}{\\sigma}}|w_{z}|^{2} = {\\mathcal{H}}(z)$,\n \\end{center}\n and \n \\begin{eqnarray}\n \\Delta |w_{\\theta}|^{2}= 4|w_{\\theta \\theta}|^{2}+2J|w_{\\theta}|^{2}-2|w_{\\theta}|^{2}.\n \\end{eqnarray}\n We rewrite this as \n \\begin{eqnarray}\n \\Delta{\\mathcal{H}}= 4|w_{\\theta \\theta}|^{2}+2J{\\mathcal{H}}-2{\\mathcal{H}} > -2{\\mathcal{H}}.\n \\end{eqnarray}\n \n \n Therefore ${\\mathcal{H}}$ is a subsolution to an elliptic equation $(\\Delta+2)f=0$.\n \n \n Recalling that $inj_{\\sigma}(\\Sigma)>r_{0}>0$, we embed a hyperbolic ball $B_{z_{0}}({\\frac{r_{0}}{2}})$ \n into $\\Sigma$, centered at $z_{0}$ with radius ${\\frac{r_{0}}{2}}$. Morrey's theorem (\\cite{Mo}, theorem 5.3.1) \n on subsolutions of elliptic differential equations guarantees that there is a constant $C(r_{0})$, such that, \n \\begin{center}\n ${\\mathcal{H}}(z_{0}) = sup_{B_{z_{0}}({\\frac{r_{0}}{4}})}{\\mathcal{H}}(z) \n \\le C(r_{0})\\int_{B_{z_{0}}({\\frac{r_{0}}{2}})}{\\mathcal{H}}(z)\\sigma dzd\\bar{z}$.\n \\end{center}\n \n \n Another consequence of formula $(3)$ is the Bochner identity (see \\cite {SY}), as now $log{\\mathcal{H}}$ \n is well defined:\n \\begin{eqnarray}\n \\Delta log{\\mathcal{H}}= 2{\\mathcal{H}}-2{\\mathcal{L}} -2.\n \\end{eqnarray} \n The minimal principle implies ${\\mathcal{H}}(z) \\ge 1$ for all $z \\in \\Sigma$, and we find\n \\begin{center}\n $\\int_{\\Sigma}{\\mathcal{L}}dA \\le \\int_{\\Sigma}{\\mathcal{H}}{\\mathcal{L}}dA = \n \\int_{\\Sigma}|\\mu|^{2}dA = ||\\mu||_{WP}= 1$. \n \\end{center}\n \n \nIt is not hard to see that we can actually bound the total energy of this {harmonic map} $w$ from above, \nin terms of the genus. More precisely, we recall that $w$ is a diffeomorphism, so\n\\begin{center}\n$ \\int_{\\Sigma}({\\mathcal{H}}-{\\mathcal{L}})dA =  \\int_{\\Sigma}JdA = Area(w(\\Sigma)) = 4\\pi (g-1)$,\n\\end{center}\ntherefore the total energy satisfies\n\\begin{eqnarray*}\nE(w) & = & \\int_{\\Sigma}e(w)dA =  \\int_{\\Sigma}({\\mathcal{H}}+{\\mathcal{L}})dA \\\\\n& = & \\int_{\\Sigma}({\\mathcal{H}}-{\\mathcal{L}})dA + 2\\int_{\\Sigma}{\\mathcal{L}}dA \\\\\n&\\le& Area(\\Sigma,\\sigma) + 2 \\\\\n&= & 4\\pi(g-1) +2.\n\\end{eqnarray*}\nTherefore\n \\begin{eqnarray*}\n\\int_{B_{z_{0}}({\\frac{r_{0}}{2}})}{\\mathcal{H}}(z)\\sigma dzd\\bar{z} \n& < & \\int_{B_{z_{0}}({\\frac{r_{0}}{2}})}({\\mathcal{H}}(z) + {\\mathcal{L}}(z))\\sigma dzd\\bar{z} \\\\\n& = & E(w) - \\int_{\\Sigma \\backslash B_{z_{0}}({\\frac{r_{0}}{2}})}e(z) \\sigma dzd\\bar{z} \\\\\n& \\le & 4\\pi(g-1) +2 - (4\\pi(g-1)-A_{1}({B_{z_{0}}}({\\frac{r_{0}}{2}}))) \\\\\n&=& A_{1}({B_{z_{0}}}({\\frac{r_{0}}{2}})) + 2.\n\\end{eqnarray*}\nwhere $A_{1}({B_{z_{0}}}({\\frac{r_{0}}{2}}))$ is the hyperbolic area of the ball \n$B_{z_{0}}({\\frac{r_{0}}{2}})$.  \n\n\nWe set $h_0 = C(r_0)(A_{1}({B_{z_{0}}}({\\frac{r_{0}}{2}})) + 2)$, then $h_0$ \nis independent of the genus $g$, as it is obtained from a local estimate in a \ngeodesic ball. Therefore, \n \\begin{center}\n $|\\mu(z)| < {\\mathcal{H}}(z) \\le {\\mathcal{H}}(z_0) < h_0$,\n \\end{center}\nfor all $z \\in \\Sigma$.\n\\end{proof}\n\\begin{rem}\nThe assumption of the surface lying in the thick part of the {moduli space} is essential to this \nargument, since we used an estimate in an embedded geodesic ball.\n\\end{rem}\n\\noindent\nAs an application of this estimate, we can prove theorem 1.1 easily. \n\n\\begin{proof}(of theorem 1.1) Recall that the operator $D = -2(\\Delta -2)^{-1}$ is self-adjoint, and $||\\mu||_{WP} = 1$:\n\\begin{center}\n$|K_h| = 2\\int_{\\Sigma}D(|\\mu|^2)|\\mu|^2 dA < 2h_0^2 \\int_{\\Sigma}D(|\\mu|^2)dA = 2h_0^2$.\n\\end{center}\n\\end{proof}.\n\n\nWe now shift our attention to general {Weil-Petersson} {sectional curvature}s and to prove theorem 1.2.\n\\begin{proof}(of theorem 1.2)\nWe recall from the previous section the Riemannian curvature tensor of the {Weil-Petersson} metric \nis given by (\\cite {Tr}, \\cite{Wp3}):\n\\begin{center} \n$R_{\\alpha\\bar{\\beta}\\gamma\\bar{\\delta}} = \n\\int_{\\Sigma}D(\\mu_{\\alpha}\\bar\\mu_{\\beta})\\mu_{\\gamma}\\bar\\mu_{\\delta}dA + \n\\int_{\\Sigma}D(\\mu_{\\alpha}\\bar\\mu_{\\delta})\\mu_{\\gamma}\\bar\\mu_{\\beta}dA$,\n\\end{center}\nwhere $\\mu$'s in the formula are harmonic {Beltrami differential}s, and $D$ again is the operator \n$-2(\\Delta -2)^{-1}$.\n\n\nTo calculate the {sectional curvature}, we choose two arbitrary orthonormal harmonic {Beltrami differential}s \n$\\mu_0$ and $\\mu_1$. In other words, we have \n\\begin{center}\n$\\int_{\\Sigma}|\\mu_0|^2 dA = \\int_{\\Sigma}|\\mu_1|^2 dA = 1$ and \n$\\int_{\\Sigma}\\mu_0 {\\bar{\\mu}}_1 dA = 0$.\n\\end{center}\nThen the Gaussian curvature of the plane spanned by $\\mu_0$ and $\\mu_1$ is \n(\\cite {Wp3})\n\\begin{eqnarray*}\nK(\\mu_0,\\mu_1) &=& {\\frac{1}{4}}(R_{0\\bar{1}0\\bar{1}} - R_{0\\bar{1}1\\bar{0}} - \nR_{1\\bar{0}0\\bar{1}}+R_{1\\bar{0}1\\bar{0}}) \\nonumber \\\\\n&=& Re(\\int_{\\Sigma}D(\\mu_{0}\\bar\\mu_{1})\\mu_{0}\\bar\\mu_{1}dA) - \n{\\frac{1}{2}}Re(\\int_{\\Sigma}D(\\mu_{0}\\bar\\mu_{1})\\mu_{1}\\bar\\mu_{0}dA) \\nonumber \\\\\n&-& {\\frac{1}{2}}\\int_{\\Sigma}D(|\\mu_{1}|^2)|\\mu_{0}|^2dA. \n\\end{eqnarray*}\nFrom (\\cite {Wp3}, lemma 4.3) and H\\\"{o}lder inequality, we have\n\\begin{eqnarray*}\n|Re(\\int_{\\Sigma}D(\\mu_{0}\\bar\\mu_{1})\\mu_{0}\\bar\\mu_{1}dA)| &\\le& \n\\int_{\\Sigma}|D(\\mu_{0}\\bar\\mu_{1})||\\mu_{0}\\bar\\mu_{1}|dA \\nonumber \\\\\n&\\le& \\int_{\\Sigma}\\sqrt{D(|\\mu_{0}|^{2})}\\sqrt{D(|\\mu_{0}|^{2})}|\\mu_{0}\\bar\\mu_{1}|dA \\nonumber \\\\\n&\\le&  \\int_{\\Sigma}D(|\\mu_{0}|^2)|\\mu_{1}|^2dA = \\int_{\\Sigma}D(|\\mu_{1}|^2)|\\mu_{0}|^2dA, \n\\end{eqnarray*}\nand similarly\n\\begin{center}\n$|\\int_{\\Sigma}D(\\mu_{0}\\bar\\mu_{1})\\mu_{1}\\bar\\mu_{0}dA| \\le \\int_{\\Sigma}D(|\\mu_{1}|^2)|\\mu_{0}|^2dA$. \n\\end{center}\nTherefore \n\\begin{center}\n$|K(\\mu_0,\\mu_1)| \\le 2\\int_{\\Sigma}D(|\\mu_{1}|^2)|\\mu_{0}|^2dA$.  \n\\end{center}\nWe apply theorem 3.3 to find that there is a $h_0 > 0$, independent of $g$, such that $|\\mu_{0}| < h_0$. \nSo $|K| < 2h_0^2 \\int_{\\Sigma}D(|\\mu_{1}|^2)dA = 2h_0^2$.\n\\end{proof}\n\n\nIt is now straightforward to see that theorem 1.3 holds since all {sectional curvature}s are negative. \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}