{"text":"\\section{Introduction and Conclusions}\n\n\\subsection{Introduction}\n\nIn the last years several proposals for the non-perturbative glueball propagators of $QCD$-like confining asymptotically-free gauge theories have been advanced, based\non the $AdS$ String\/ large-$N$ Gauge Theory correspondence \\cite{Mal} and more recently on a Topological Field Theory ($TFT$) underlying the large-$N$ limit of the pure Yang-Mills ($YM$) theory \\cite{boch:quasi_pbs} \\cite{MB0} \\cite{boch:crit_points} \\cite{boch:glueball_prop} \\cite{Top}. \\par\nAbove all these proposal aim to elucidate, at least in the large-$N$ limit, the most fundamental feature of the infrared of large-$N$ $QCD$-like confining asymptotically-free gauge theories, i.e. the existence of a mass gap in the pure glue sector, as opposed to the massless spectrum of gluons in perturbation theory. \\par\nHowever, these proposals predict a variety of spectra for large-$N$ $QCD$ different among themselves, asymptotically quadratic for large masses \\cite{Mal}  \\cite{Witten} \\cite{Brower1} \\cite{PS} \\cite{Brower2} or exactly linear \\cite{Softwall} \\cite{MB0} \\cite{boch:crit_points} \\cite{boch:glueball_prop} \\footnote{Exact linearity in the $TFT$ refers to the joint large-$N$ spectrum of scalar and pseudoscalar glueballs. The $TFT$ in its present formulation does not contain information about higher spin glueballs.}\nin the square of the glueball masses and in general do not agree about the qualitative and quantitative details of the low-energy spectrum but for the existence of the mass gap.  \\par\nIn view of the importance of the problem that these proposals aim to answer and in order to discriminate between the various proposals it is worth investigating whether there is any constraint that we know by the fundamental principles of any confining asymptotically-free gauge theory that any supposed answer for the non-perturbative glueball propagators has to satisfy. \\par\nIn fact, we do know with certainty the implications of the asymptotic freedom for the large-momentum asymptotic behavior of any gauge invariant correlation function. \\par\nIn this paper we do not discuss at all the theoretical justification of the various proposal that we examine, leaving it to the original papers.\nWe limit ourselves to check whether or not the constraint that follows by the asymptotic freedom and by the renormalization group in the ultraviolet ($UV$) is satisfied by any given proposal. Indeed, the importance of this constraint has been pointed out since the early days of large-$N$ $QCD$ \\cite{migdal:multicolor}, see also \\cite{polyakov:gauge}. \\par\nIn fact the purpose of this paper is threefold. \\par\n\n\\subsection{Implications of the renormalization group and of the asymptotic freedom}\n\nFirstly, in sect.(2) we point out that perturbation theory in conjunction with the renormalization group ($RG$) severely constraints the asymptotic behavior of glueball propagators in pure $SU(N)$ Yang-Mills,\nin $QCD$ and in $\\mathcal{N}$ $=1$ $SUSY$ $QCD$ with massless quarks, or in any confining asymptotically-free gauge theory massless to every order of perturbation theory. \\par\nIndeed, we show in this paper, on the basis of $RG$ estimates, that the most fundamental object involved in the problem of the mass gap \\footnote{The lightest glueball is believed to be a scalar\nin pure $YM$ and in the 't Hooft large-$N$ limit of $QCD$.}, the scalar ($S$) glueball propagator in any (confining) asymptotically-free gauge theory with no perturbative physical mass scale, up to unphysical contact terms, i.e. distributions supported at coinciding points,\nhas the following universal, i.e. renormalization-scheme independent, large-momentum asymptotic behavior:\n\\begin{align}\\label{eqn:corr_scalare_inizio}\n&\\int\\langle \\frac{\\beta(g)}{gN}tr\\bigl(\\sum_{\\alpha\\beta}{F_{\\alpha\\beta}^2}(x)\\bigr)\\frac{\\beta(g)}{gN} tr\\bigl(\\sum_{\\alpha\\beta}{F}_{\\alpha\\beta}^2(0)\\bigr)\\rangle_{conn}e^{ip\\cdot x}d^4x \\nonumber\\\\\n= & C_S p^4\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)\\Biggr]\n\\end{align} \nAnalogously for the pseudoscalar ($PS$) propagator:\n\\begin{align}\n&\\int\\langle \\frac{g^2}{N}  tr\\bigl(\\sum_{\\alpha\\beta}{F}_{\\alpha\\beta}\\tilde{F}_{\\alpha\\beta}(x)\\bigr) \\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}{F}_{\\alpha\\beta}\\tilde{F}_{\\alpha\\beta}(0)\\bigr)\\rangle_{conn}e^{ip\\cdot x}d^4x \\nonumber\\\\\n=& C_{PS} p^4 \\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)\\Biggr]\n\\end{align}\nand for a certain linear combination as well, the anti-seldual ($ASD$) propagator:\n\\begin{align}\\label{eqn:corr_asd}\n& \\frac{1}{2}\\int\\langle \\frac{g^2}{N}tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{- 2}(x)\\bigr)\\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{- 2}(0)\\bigr)\\rangle_{conn}e^{ip\\cdot x}d^4x \\nonumber\\\\\n= &  C_{ADS} p^4 \\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)\\Biggr]\n\\end{align}\nwhere $F_{\\alpha\\beta}^-=F_{\\alpha\\beta}-\\tilde F_{\\alpha\\beta}$ and $\\tilde F_{\\alpha\\beta}=\\frac{1}{2} \\epsilon_{\\alpha \\beta \\gamma \\delta} F_{\\gamma \\delta}$. \\par\nThe explicit dependence on the particular $\\Lambda_{\\overline{MS}}$ scale in Eq.(\\ref{eqn:corr_scalare_inizio})-Eq.(\\ref{eqn:corr_asd}) is illusory. A change of scheme affects only the $O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)$ terms.\nThe coincidence of the asymptotic behavior, up to the overall normalization constants that are computed in sect.(3), $C_S, C_{PS},C_{ASD}$, is due to the coincidence of the naive dimension in energy, $4$, and of the one-loop  anomalous dimension, $\\gamma(g)=-2 \\beta_0 g^2+\\cdots$, of these operators deprived of the factors of $\\frac{\\beta(g)}{g}$ or of $g^2$. Euclidean signature is always understood in this paper unless otherwise specified. \\par\n\n\\subsection{Perturbative check of the $RG$ estimates}\nSecondly, in sect.(3) we check the correctness of our $RG$ estimate on the basis of an explicit very remarkable three-loop computation \\footnote{The earlier two-loop computation was performed in \\cite{Kataev:1981gr}.}\nperformed by Chetyrkin et al.\\cite{chetyrkin:scalar}\n\\cite{chetyrkin:pseudoscalar}\nin pure $SU(N)$ $YM$ and in $SU(3)$ $QCD$ with $n_f$ massless Dirac fermions in the fundamental representation. \nFor example, we show that in pure $SU(N)$ $YM$ Chetyrkin et al. result \\cite{chetyrkin:scalar}\n\\cite{chetyrkin:pseudoscalar} can be rewritten by elementary methods as:\n\\begin{align}\\label{eqn:prologo:ris_sommato}\n& \\frac{1}{2} \\int\\langle \\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}{F_{\\alpha\\beta}^{-2}}(x)\\bigr)\\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}{F^{-2}_{\\alpha\\beta}}(0)\\bigr)\\rangle_{conn}\\,e^{-ip\\cdot x}d^4x  \\nonumber\\\\\n&=(1-\\frac{1}{N^2})\\frac{p^4}{2\\pi^2\\beta_0} \\bigl(2g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambda_{\\overline{MS}}^2})-2g_{\\overline{MS}}^2(\\frac{\\mu^2}{\\Lambda_{\\overline{MS}}^2}) \\nonumber\\\\\n& +\\bigl(a+\\tilde{a} -\\frac{\\beta_1}{\\beta_0}\\bigr)g_{\\overline{MS}}^4(\\frac{p^2}{\\Lambda_{\\overline{MS}}^2})\n-\\bigl(a+\\tilde{a} -\\frac{\\beta_1}{\\beta_0}\\bigr)g_{\\overline{MS}}^4(\\frac{\\mu^2}{\\Lambda_{\\overline{MS}}^2})\\bigr)+O(g^6)\n\\end{align} \nwhere $a$ and $\\tilde{a}$ are two scheme-dependent constants that are defined in sect.(3.5) and $g_{\\overline{MS}}$ is the 't Hooft coupling constant in the $\\overline{MS}$ scheme.\nIn Eq.(\\ref{eqn:prologo:ris_sommato}) the terms that depend on $g(\\frac{\\mu^2}{\\Lambda_{\\overline{MS}}^2})$ correspond in the coordinate representation to distributions supported at coincident points (contact terms), and therefore they have no physical meaning.\nRemarkably, the correlator without the contact terms does not in fact depend on the arbitrary scale $\\mu$ (within $O(g^6)$ accuracy) as it should be.\nThe running coupling constant $g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})$ occurs in Eq.(\\ref{eqn:prologo:ris_sommato}) with two-loop accuracy and it is given by:\n\\begin{align}\\label{eqn:prologo:g_pert}\n&g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})=g_{\\overline{MS}}^2(\\frac{\\mu^2}{\\Lambdams^2})\\bigl(1-\\beta_0g_{\\overline{MS}}^2(\\frac{\\mu^2}{\\Lambdams^2})\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&-\\beta_1g_{\\overline{MS}}^4(\\frac{\\mu^2}{\\Lambdams^2})\\log\\frac{p^2}{\\mu^2} +\\beta_0^2g_{\\overline{MS}}^4(\\frac{\\mu^2}{\\Lambdams^2}) \\log^2\\frac{p^2}{\\mu^2}\\bigr) + \\cdots\n\\end{align}\nTherefore, the perturbative computation furnishes an expansion of the correlator in powers of $g_{\\overline{MS}}^2(\\mu)$ and of logarithms. This expansion has been rearranged by elementary methods in terms of the two-loop running coupling $g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})$ in Eq.(\\ref{eqn:prologo:ris_sommato}). \\par\nAt this point our basic strategy to check the $RG$ estimates of sect.(2) consists in substituting in Eq(\\ref{eqn:prologo:ris_sommato}) instead of Eq.(\\ref{eqn:prologo:g_pert}) the $RG$-improved expression for $g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})$ given by:\n\\begin{equation}\\label{eqn:prologo:rg_g}\ng_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})=\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\biggl[1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\biggr]\n+O\\biggl(\\frac{\\log^2\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log^3\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\biggr)\n\\end{equation}\nThe $\\overline{MS}$ scheme is indeed defined \\cite{chetyrkin:schema} in such a way to cancel the term of order of $\\frac{1}{\\log^2\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}$ that would occur in Eq.(\\ref{eqn:prologo:rg_g}) in other schemes.\nBy subtracting the unphysical contact terms and by substituting the $RG$-improved two-loop asymptotic expression for $g_{\\overline{MS}}^2(\\frac{p^2}{\\Lambdams^2})$ it follows the actual large-momentum scheme-independent asymptotic behavior of Eq.(\\ref{eqn:prologo:ris_sommato}):\n\\begin{align}\\label{eqn:prologo_comportamento_pert}\n& \\frac{1}{2} \\int\\langle \\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{- 2}(x)\\bigr)\\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{- 2}(0)\\bigr)\\rangle_{conn}\\,e^{-ip\\cdot x}d^4x  \\nonumber\\\\\n= & (1-\\frac{1}{N^2})\\frac{p^4}{\\pi^2\\beta_0}\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}{\\log\\frac{p^2}{\\Lambda_{\\overline{MS}}^2}}\\Biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambdams^2}}\\biggr)\\Biggr]\n\\end{align} \nas opposed to the perturbative behavior that would follow by Eq.(\\ref{eqn:prologo:g_pert}).\nThe asymptotic result in the other cases is checked similarly. \\par\n\n\\subsection{ $AdS$\/Large-$N$ Gauge Theory correspondence and disagreement with the $RG$ estimates}\n\nThirdly, in this subsection and in the next one, we inquire whether the large-$N$ non-perturbative scalar or pseudoscalar propagators actually computed in the literature agree or disagree with the $RG$ estimate. \\par\nWe find, to the best of our knowledge, that all the scalar propagators presently computed in the literature in the framework of the $AdS$ String\/ large-$N$ Gauge Theory correspondence disagree with the universal asymptotic behavior. \\par\nWe should mention that the comparison of the asymptotics of the scalar glueball propagators in the $AdS$ approach with $YM$ or with $QCD$ at the lowest non-trivial order of perturbation theory has been already performed in \\cite{forkel} \\cite{italiani} \\cite{forkel:holograms} \\cite{forkel:ads_qcd}, but with somehow different conclusions.\nThe reasons is that in \\cite{forkel}  \\cite{italiani} \\cite{forkel:holograms} \\cite{forkel:ads_qcd} the comparison has been performed only with the one-loop result for the scalar glueball propagator, i.e. only with the first term in Eq.(\\ref{eqn:corr_pert_scalar_3l}), that is conformal in the $UV$. No higher order of perturbation theory and no $RG$ improvement has been taken into account in the comparison, as instead we do in this paper. \\par\nHere we enumerate the models based on the $AdS$\/Gauge Theory correspondence for which we could find explicit computations of the scalar glueball propagator in the literature.  \\par\nIn the Hard Wall model (Polchinski-Strassler background \\cite{PS} in the so called bottom-up approach):\n\\[\n\\int\\braket{trF^2(x)trF^2(0)}e^{-i p \\cdot x}d^4x \\sim  p^4 \\biggl[2\\frac{K_1(\\frac{p}{ \\mu})}{I_1(\\frac{p}{\\mu})}-\\log\\frac{p^2}{\\mu^2}\\biggr]\n\\]\nwhere $K_1,I_1$ are the modified Bessel functions \\cite{forkel}. The asymptotic behavior \\cite{forkel} is conformal in the $UV$:\n\\begin{align*}\n&p^4 \\biggl[2\\frac{K_1(\\frac{p}{ \\mu})}{I_1(\\frac{p}{ \\mu})}-\\log\\frac{p^2}{\\mu^2}\\biggr]\n \\sim - p^4\\biggl[\\log\\frac{p^2}{\\mu^2}+ O(e^{- 2 \\frac{p}{\\mu}}) \\biggr]\n\\end{align*}\nwith $p=\\sqrt{p^2}$. Indeed, as recalled in appendix A, in the coordinate representation:\n\\[\n- \\int p^4 \\log\\frac{p^2}{\\mu^2} e^{i p \\cdot x}\\frac{d^4p}{(2\\pi)^4} \\sim \\frac{1}{x^8}\n\\]\nand, as observed in \\cite{forkel} \\cite{italiani} \\cite{forkel:holograms} \\cite{forkel:ads_qcd}, it matches the one-loop large-$N$ $QCD$ result for the perturbative glueball propagator displayed in the first term of Eq.(\\ref{eqn:corr_pert_scalar_3l}). Nevertheless, it disagrees by a factor of $(\\log p)^2$ with the correct asymptotic behavior in Eq.(\\ref{eqn:corr_scalare_inizio}). \\par\nThe Soft Wall model (bottom-up approach) \\cite{Softwall} implies the same leading conformal asymptotic behavior \\cite{forkel} \\cite{italiani} \\cite{forkel:holograms} \\cite{forkel:ads_qcd} in the $UV$ for the scalar glueball propagator:\n\\[\n\\int\\braket{trF^2(x)trF^2(0)}e^{-i p \\cdot x}d^4x  \n \\sim - p^4\\biggl[\\log\\frac{p^2}{\\mu^2}+O(\\frac{\\mu^2}{p^2})\\biggr]\n\\] \nthat therefore disagrees in the $UV$ by the same factor of $(\\log p)^2$. \\par\nA more interesting example of the $AdS$ string \/ large-$N$ Gauge Theory correspondence from the point of view of first principles applies to the cascading $\\mathcal{N}=1$ $SUSY$ $YM$ theory (top-down approach) \\cite{KS1} \\cite{KS2}, because in this case the correct asymptotically-free $\\beta$ function of the cascading theory is exactly reproduced in the supergravity approximation in the Klebanov-Strassler background \\cite{KS1} \\cite{KS2}.\nNevertheless, the asymptotic behavior of the scalar correlator is \\cite{krasnitz:cascading2} \\cite{krasnitz:cascading}:\n\\begin{align*}\n\\int\\braket{trF^2(x)trF^2(0)}e^{-i p \\cdot x}d^4x\\sim p^4 \\log^3 \\frac{p^2}{\\mu^2}\n\\end{align*}\nthat disagrees by a factor of $(\\log p)^4$ with the correct asymptotic behavior in Eq.(\\ref{eqn:corr_scalare_inizio}). \\par\n\n\\subsection{Topological Field Theory and agreement with the $RG$ estimates}\n\nFinally, in sect.(4) we prove that in the large-$N$ limit of pure $SU(N)$ $YM$ the $ASD$ glueball propagator computed in \\cite{boch:glueball_prop}\\cite{boch:crit_points}\\cite{MB0} \\footnote{We use here a manifestly covariant notation as opposed to the one of the $TFT$ employed in \\cite{boch:glueball_prop}\\cite{boch:crit_points}\\cite{MB0}.}:\n\\begin{equation}\\label{eqn:formula_prologo}\n\\frac{1}{2} \\int\\langle \\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{-2}(x)\\bigr)\\frac{g^2}{N} tr\\bigl(\\sum_{\\alpha\\beta}F_{\\alpha\\beta}^{-2}(0)\\bigr)\\rangle_{conn}\\,e^{-ip\\cdot x}d^4x =\n \\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^6 k^2}{p^2+k\\Lambda_{\\overline{W}}^2}\n\\end{equation}\nagrees with the universal $RG$ estimate in Eq.(\\ref{eqn:prologo_comportamento_pert}).  \\par\nSince the proposal for the $TFT$ underlying large-$N$ $YM$ is recent and not widely known we add here a few explanations, but for the purposes of this paper the reader can consider Eq(\\ref{eqn:formula_prologo}) just as a phenomenological model\nfactorizing the $ASD$ glueball propagator on a spectrum linear in the masses squared with certain residues. \\par \nYet, to say it in a nutshell, the rationale behind Eq(\\ref{eqn:formula_prologo}) is as follows. In \\cite{boch:quasi_pbs} \\cite{boch:crit_points} \\cite{Top} it is shown that there is a $TFT$ trivial \\cite{boch:crit_points}  \\cite{Top} at $N=\\infty$ underlying the large-$N$ limit of $YM$. At $N=\\infty$ the $TFT$ is localized on critical points \\cite{MB0} \\cite{boch:crit_points}. However, at the first non-trivial $\\frac{1}{N}$ order the $ASD$ propagator of the $TFT$ arises computing non-trivial fluctuations around the critical points of the $TFT$ \\cite{MB0} \\cite{boch:glueball_prop}. \\par\nIn Eq.(\\ref{eqn:formula_prologo}) $F^-_{\\alpha\\beta}$ is the anti-selfdual part of the curvature $F_{\\alpha\\beta}=\\partial_\\alpha A_\\beta-\\partial_\\beta A_\\alpha +i\\frac{g}{\\sqrt{N}}[A_\\alpha,A_\\beta]$ with the canonical normalization defined in Eq.(\\ref{eqn:F_canonical}), $\\Lambda_{\\overline{W}}$ is the renormalization group invariant scale in the scheme in which it coincides with the mass gap and $g_k=g(\\frac{p^2}{\\Lambda^2_{\\overline{W}}}=k)$ is the 't Hooft running coupling constant at the scale of the pole (in Minkowski space-time) in the scheme defined in \\cite{boch:quasi_pbs}, that is recalled in sect.(4). In fact, the analysis of the $UV$ behavior of Eq.(\\ref{eqn:formula_prologo}) has already been performed at the order of the leading logarithm occurring in Eq.(\\ref{eqn:corr_asd}) in \\cite{boch:glueball_prop}.\nHere we go one step further comparing Eq.(\\ref{eqn:formula_prologo}) with Eq.(\\ref{eqn:corr_asd}) at the order of the next-to-leading logarithm.\nOur basic strategy to obtain the large momentum asymptotics of Eq.(\\ref{eqn:formula_prologo}) is as follows. We write the RHS of Eq.(\\ref{eqn:formula_prologo}) as a sum of physical terms and contact terms according to \\cite{boch:glueball_prop}: \n\\begin{align}\n\\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^6 k^2}{p^2+k\\Lambda_{\\overline{W}}^2}=\n\\frac{1}{\\pi^2}p^4 \\sum_{k=1}^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^2}{p^2+k\\Lambda_{\\overline{W}}^2} +\n \\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}g_k^4\\Lambda^2_{\\overline{W}}(k\\Lambda^2_{\\overline{W}}-p^2)\n\\end{align}\nThe first sum contains the physical terms that in Minkowski space-time carry the pole singularities, while the second sum contains the contact terms, that we ignore in the following.\nWe now consider only the physical terms and to find the leading $UV$ behavior we use the Euler-McLaurin formula according to the technique first introduced by Migdal \\cite{migdal:meromorphization} \\footnote{We understand that Migdal technique has been known to him for decades.} and employed in \\cite{boch:glueball_prop} :\n\\begin{equation}\n\\sum_{k=k_1}^{\\infty}G_k(p)=\n\\int_{k_1}^{\\infty}G_k(p)dk - \\sum_{j=1}^{\\infty}\\frac{B_j}{j!}\\left[\\partial_k^{j-1}G_k(p)\\right]_{k=k_1}\n\\end{equation}\nwhere $B_j$ are the Bernoulli numbers.\nIn our case the terms proportional to the Bernoulli numbers involve negative powers of $p$ and they are therefore suppressed with respect to the first term which behaves as the inverse of a logarithm, so that we ignore them as well. \nWe have:\n\\begin{align}\\label{eqn:prologo_hom_intermedio}\n\\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^6 k^2}{p^2+k\\Lambda_{\\overline{W}}^2}\\sim \n& \\frac{1}{\\pi^2}p^4\\int_1^{\\infty}\\frac{g_k^4\\Lambda_{\\overline{W}}^2}{p^2+k\\Lambda_{\\overline{W}}^2}dk \\nonumber\\\\\n\\sim& \\frac{1}{\\pi^2}p^4\\int_1^{\\infty}\\frac{1}{\\beta_0^2\\log^2\\frac{k}{c}}\n\\biggl(1-\\frac{2\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)\n\\frac{dk}{k+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}\n\\end{align} \nwhere we have used the $RG$-improved asymptotic behavior for large $k$ of the running coupling constant $g_k$ at the scale of the $k$-th pole, i.e. on shell (in Minkowski space-time):\n\\begin{equation}\ng^2_k\\sim \\frac{1}{\\beta_0\\log\\frac{k}{c}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)\n\\end{equation}\nThe constant $c$ is related to the scheme that occurs in the non-perturbative calculation \\cite{boch:quasi_pbs}\\cite{MB0}\\cite{boch:crit_points}\\cite{boch:glueball_prop}. The actual value of $c$ is not relevant in this paper since we study only the universal asymptotic behavior.\nIn sect.(4) we compute the universal leading and next-to-leading behavior of the integral in Eq.(\\ref{eqn:prologo_hom_intermedio}) and the result is:\n\\begin{align}\n& \\frac{1}{\\pi^2}p^4\\int_1^{\\infty}\\frac{1}{\\beta_0^2\\log^2\\frac{k}{c}}\n\\biggl(1-\\frac{2\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)\n\\frac{dk}{k+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}\\nonumber\\\\\n&= \\frac{1}{\\pi^2\\beta_0}p^4 \\biggl[\\frac{1}{\\beta_0\\log\\bigl(\\frac{1}{c}+\\frac{p^2}{c\\Lambda_{\\overline{W}}^2}\\bigr)}-\\frac{\\beta_1}{\\beta_0^3}\\frac{\\log\\log\\bigl(\\frac{1}{c}+\\frac{p^2}{c\\Lambda^2_{\\overline{W}}}\\bigr)}{\\log^2\\bigl(\\frac{1}{c}+\\frac{p^2}{c\\Lambda^2_{\\overline{W}}}\\bigr)}\\biggr]\n+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggr)\\nonumber\\\\\n&=\\frac{1}{\\pi^2\\beta_0}p^4\\biggl[\\frac{1}{\\beta_0\\log\\bigl(\\frac{p^2}{\\Lambda^2_{\\overline{W}}}\\bigr)}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\bigl(\\frac{p^2}{\\Lambda^2_{\\overline{W}}}\\bigr)}{\\log^2\\bigl(\\frac{p^2}{\\Lambda^2_{\\overline{W}}}\\bigr)}\\biggr)\n\\biggr]+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggr)\n\\end{align} \n\n\\subsection{Conclusions}\n\nThe preceding result, for the $ASD$ glueball propagator computed in the $TFT$ underlying large-$N$ pure $YM$, agrees perfectly in the large-$N$ limit with the universal part of the renormalization group improved expression of the perturbative result Eq.(\\ref{eqn:prologo_comportamento_pert}). \\par\nThe agreement is due to the conspiracy between the residues of the poles, that are proportional to the fourth power of the coupling constant renormalized on shell times the fourth power of the glueball mass at the pole, and the exact linearity of the joint scalar and pseudoscalar spectrum of the square of the mass of the glueballs in the $ASD$ correlator of the $TFT$. \\par\nTo the best of our knowledge this is the only non-perturbative result for the scalar or pseudoscalar glueball propagator proposed in the literature that agrees with large-$N$ $YM$ perturbation theory and the renormalization group. \\par\nWhile this agreement is not by itself a guarantee of correctness of Eq.(\\ref{eqn:formula_prologo})  it deserves further investigations, both at theoretical level\nand of further checks. \\par\nBesides, our analysis shows that the $AdS$\/Large-$N$ Gauge Theory correspondence in any of its present strong coupling incarnations, the bottom-up or the top-down approach, for which scalar glueball propagators are available in the literature,\ndoes not capture, not even approximatively, the fundamental ultraviolet feature of $YM$ or of $QCD$ or of any large-$N$  confining asymptotically-free gauge theory in the pure glue sector. \\par\nWhile this conclusion is certainly known to some experts (see for just one example \\cite{KS2}), we think that it is not widely recognized that constructing theories that are conformal in the ultraviolet, as the Hard or the Soft Wall models, or even with the correct beta\nfunction but in the strong coupling phase, as the Klebanov-Strassler supergravity background, is not at all a good approximation for the correct result in the ultraviolet. In this paper, for the first time with leading and next-to-leading logarithmic accuracy, we have computed quantitatively the measure of the disagreement. \\par\nFinally, given the disagreement between the propagators of the $TFT$ and the propagators of the $AdS$\/Large-$N$ Gauge Theory correspondence\nin the infrared for the first few lower-mass glueballs, a careful critical analysis of the two approaches at level of numerical lattice data is needed, and also at theoretical level of further constraints arising by the $OPE$ and by the \nlow-energy theorems of Shifman-Vainshtein-Zakharov ($SVZ$).\n\n\\section{Renormalization group estimates on the universal behavior of correlators}\n\n\\subsection{Definitions}\n\nThe  $SU(N)$ pure $YM$ theory is defined by the partition function:\n\\begin{equation}\\label{eqn:Z1}\nZ=\\int \\mathcal{D}A\\, e^{-\\frac{1}{2g_{YM}^2}\\int tr F^2(x) d^4x}\n\\end{equation}\nwhere we use the simplified notation $tr F^2(x)= \\sum_{\\alpha\\beta} tr \\bigl(F_{\\alpha\\beta}^2\\bigr)$.\nIntroducing the 't Hooft coupling constant $g$ \\cite{'t hooft:large_n}:\n\\begin{equation}\ng^2=g^2_{YM}N\n\\end{equation}\nthe partition function reads:\n\\begin{equation}\nZ=\\int \\mathcal{D}A\\, e^{-\\frac{N}{2g^2}\\int tr F^2(x) d^4x}\n\\end{equation} \nAccording to 't Hooft \\cite{'t hooft:large_n} the large-$N$ limit is defined with $g$ fixed when $N\\rightarrow \\infty$. \\par\nFor the structure of large-$N$ glueball propagators see \\cite{migdal:multicolor}\nand for reviews of the large-$N$ limit see \\cite{polyakov:gauge} and \\cite{makeenko:large_n}.\nThe normalization of the action in Eq.(\\ref{eqn:Z1}) corresponds to choosing the gauge field $A_{\\alpha}$ in the fundamental representation of the Lie algebra, with generators normalized as:\n\\begin{equation}\ntr\\, (t^a t^b)=\\frac{1}{2}\\delta^{ab}\n\\end{equation}\nIn Eq.(\\ref{eqn:Z1}) $F_{\\alpha\\beta}$ is defined by:\n\\begin{equation}\\label{eqn:F_wilsonian}\nF_{\\alpha\\beta}(x)=\\partial_\\alpha A_\\beta-\\partial_\\beta A_\\alpha + i[A_\\alpha,A_\\beta]\n\\end{equation}\nWe refer to the normalization of the action in Eq.(\\ref{eqn:Z1}) as the Wilsonian normalization.\nPerturbation theory is formulated with the canonical normalization, obtained rescaling the field $A_\\alpha$ in Eq.(\\ref{eqn:Z1}) by the coupling constant $g_{YM}=\\frac{g}{\\sqrt{N}}$:\n\\begin{align}\nA_\\alpha \\rightarrow g_{YM}A^{can}_\\alpha\n\\end{align}\nin such a way that in the action the kinetic term becomes independent on $g$:\n\\begin{equation}\n\\frac{1}{2}\\int tr (F^2(A^{can})) (x) d^4 x\n\\end{equation}\nwhere:\n\\begin{align}\\label{eqn:F_canonical}\nF_{\\alpha\\beta}= \\partial_\\beta A^{can}_\\alpha - \\partial_\\alpha A^{can}_\\beta +ig_{YM}[A^{can}_\\alpha , A^{can}_\\beta]\n\\end{align}\nFrom now on we will simply write $F_{\\alpha \\beta}$ for the curvature as a function of the canonical field, without displaying the superscript $can$.\n\n\n\\subsection{A short summary of perturbation theory and of the renormalization group}\n\nWe recall the relation between bare and renormalized two-point connected correlators of a multiplicatively renormalizable gauge-invariant scalar operator $\\mathcal{O}$ of naive dimension in energy $D$:\n\\begin{equation}\nG^{(2)}(p,\\mu,g(\\mu))=Z_{\\mathcal{O}}^{2}(\\frac{\\Lambda}{\\mu}, g(\\Lambda)) G_0^{(2)}(p, \\Lambda, g(\\Lambda))\n\\end{equation} \nwhere $G_0^{(2)}$ is the bare connected correlator in momentum space, computed in some regularization scheme with cutoff $\\Lambda$, and $\\mu$ is the renormalization scale:\n\\begin{equation}\nG_0^{(2)}(p, \\Lambda, g(\\Lambda))= \\int \\braket{\\mathcal{O}(x)\\mathcal{O}(0)}_{conn} e^{i p \\cdot x} d^4x  \\equiv\\braket{\\mathcal{O}(p)\\mathcal{O}(-p)}_{conn}\n\\end{equation} \nSince $YM$ or $QCD$ with massless quarks or $\\mathcal{N}$ $=1$ $SUSY$ $YM$ with massless quarks is massless to every order of perturbation theory and since $\\mathcal{O}$ has naive dimension $D$ we can write:\n\\begin{equation}\nG^{(2)}(p,\\mu, g(\\mu))=p^{2D-4}G_{DL}^{(2)}(\\frac{p}{\\mu}, g(\\mu))\n\\end{equation}  \nwhere $G_{DL}^{(2)}$ is a dimensionless function.\nThe Callan-Symanzik equation for the dimensionless two-point renormalized correlator \nexpresses the independence of the bare two-point correlator with respect to the subtraction point $\\mu$:\n\\begin{equation}\n\\frac{\\mathrm{d} G_0^{(2)}}{\\mathrm{d}\\log\\mu}\\Big|_{\\Lambda,g(\\Lambda)}=0\n\\end{equation}\\begin{equation}\\label{eqn:RG_eq}\n\\left(\\frac{\\partial}{\\partial\\log\\mu}+\\beta(g)\\frac{\\partial}{\\partial g(\\mu)}+2\\gamma_{\\mathcal{O}}(g)\\right)G_{DL}^{(2)}(\\frac{p}{\\mu},g(\\mu))=0\n\\end{equation}\nwhere we have defined the beta function with respect to the renormalized coupling $g(\\mu)$: \n\\begin{equation}\n\\beta(g)=\\frac{\\partial g}{\\partial\\log\\mu}\\Big|_{\\Lambda,g(\\Lambda)}\n\\end{equation}\nand the anomalous dimension:\n\\begin{equation}\n\\gamma_{\\mathcal{O}}(g)= - \\frac{\\partial\\log Z_{\\mathcal{O}}}{\\partial\\log\\mu}\\Big|_{\\Lambda,g(\\Lambda)}\n\\end{equation}\nEq.(\\ref{eqn:RG_eq}) can be rewritten taking into account the dependence of $G_{DL}^{(2)}$ on the momentum $p=\\sqrt {p^2}$:\n\\begin{equation}\\label{eqn:RG_eq_p}\n\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g}-2\\gamma_{\\mathcal{O}}(g)\\right)G_{DL}^{(2)}(\\frac{p}{\\mu},g(\\mu))=0\n\\end{equation}\nThe general solution of Eq.(\\ref{eqn:RG_eq_p}) is:\n\\begin{equation}\\label{eqn:solu_rg}\nG_{DL}^{(2)}(\\frac{p}{\\mu},g(\\mu))=\\mathcal{G}(g(\\frac{p}{\\mu},g(\\mu)))\\, e^{2\\int_{g(\\mu)}^{g(p)}\\frac{\\gamma_{\\mathcal{O}}(g)}{\\beta(g)}dg} \\equiv\n\\mathcal{G}_{\\mathcal{O}}(g(p)) \\, Z^2_\\mathcal{O}(\\frac{p}{\\mu},g(\\mu))\n\\end{equation}\nThe running coupling $g(\\frac{p}{\\mu},g(\\mu))$, that we briefly denote by $g(p)$, solves:\n\\begin{equation}\\label{eqn:eq_rg_flow}\n\\frac{\\partial g(p)}{\\partial \\log p}=\\beta(g(p))\n\\end{equation}\nwith the initial condition $g(1,g(\\mu))=g(\\mu)$. \\par \nThe multiplicative renormalized factor $Z_\\mathcal{O}(\\frac{p}{\\mu},g(\\mu))$ satisfies:\n\\begin{equation}\n\\gamma_{\\mathcal{O}}(g)= - \\frac{\\partial\\log Z_{\\mathcal{O}}}{\\partial\\log \\mu}\n\\end{equation}\nand from now on it is thought as a finite dimensionless function of $g(\\mu)$ and $g(p)$ only $Z_\\mathcal{O}(g(p),g(\\mu))$:\n\\begin{equation}\nZ_{\\mathcal{O}}=e^{\\int_{g(\\mu)}^{g(p)}\\frac{\\gamma_{\\mathcal{O}}(g)}{\\beta(g)}dg}\n\\end{equation}\nEq.(\\ref{eqn:solu_rg}) expresses the solution of the $RG$ equation as a product of a $RG$ invariant ($RGI$) function $\\mathcal{G}_{\\mathcal{O}}$ of $g(p)$ only and of a multiplicative factor $Z_{\\mathcal{O}}^2$ that is determined by the anomalous dimension $\\gamma_{\\mathcal{O}}(g)$ and by the beta function $\\beta(g)$. $\\mathcal{G}_{\\mathcal{O}}$ and $Z_{\\mathcal{O}}$ can be computed order by order in renormalized perturbation theory. \\par\nFrom Eq.(\\ref{eqn:eq_rg_flow}), that represents the coupling constant flow as a function of the momentum, we obtain the well-known behavior of the $RG$-improved 't Hooft running coupling constant with one- and two-loop accuracy, starting from the one- and two-loop perturbative beta function:\n\\begin{equation}\n\\beta(g)=-\\beta_0 g^3 - \\beta_1 g^5 + \\cdots\n\\end{equation}\nwhere for pure $YM$:\n\\begin{align}\n&\\beta_0=\\frac{11}{3}\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n&\\beta_1=\\frac{34}{3}\\frac{1}{(4\\pi)^4}\n\\end{align}\nWith two-loop accuracy we get:\n\\begin{align}\\label{eqn:g_2loop}\n&\\frac{d g}{d\\log p}=-\\beta_0 g^3 - \\beta_1 g^5 \\nonumber\\\\\n\\Rightarrow & \\int_{g(\\mu)}^{g(p)}\\frac{1}{\\beta_0 g^3}(1-\\frac{\\beta_1}{\\beta_0}g^2)dg=-\\log\\frac{p}{\\mu} \\nonumber\\\\\n\\Rightarrow & \\frac{1}{\\beta_0}(\\frac{1}{2g(\\mu)^2}-\\frac{1}{2g(p)^2})\n-\\frac{\\beta_1}{\\beta_0^2}\\log\\frac{g(p)}{g(\\mu)}=-\\log\\frac{p}{\\mu}\\nonumber\\\\\n\\Rightarrow & g^2(p)=\\frac{g^2(\\mu)}{1+2\\beta_0 g^2(\\mu)\\log\\frac{p}{\\mu}-2\\frac{\\beta_1}{\\beta_0}g^2(\\mu)\\log\\frac{g(p)}{g(\\mu)}}\\nonumber\\\\\n\\sim & \\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\n\\left(1+\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\frac{g(p)}{g(\\mu)}}{\\log\\frac{p}{\\mu}}\\right) \\sim\n\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\n\\left(1-\\frac{\\beta_1}{2\\beta_0^2}\\frac{\\log\\frac{g^2(\\mu)}{g^2(p)}}{\\log\\frac{p}{\\mu}}\\right)\\nonumber\\\\\n= & \\frac{1}{\\beta_0\\log\\frac{p^2}{\\mu^2}}\n\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\mu^2}}{\\log\\frac{p^2}{\\mu^2}}\\right)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\mu^2}}\\biggr)\n\\end{align}\nThis is the well known actual $UV$ asymptotic behavior of the running coupling constant.\nThe function $\\mathcal{G}_{\\mathcal{O}}$ in Eq.(\\ref{eqn:solu_rg}) is not known from general principles but can be computed in perturbation theory as a function of $g(\\mu)$ and then expressed in terms of $g(p)$, since  $\\mathcal{G}_{\\mathcal{O}}$ is $RGI$. \nSimilarly, we can evaluate $Z_{\\mathcal{O}}$ using again the one-loop or two-loop perturbative expressions for $\\beta(g)$ and $\\gamma_{\\mathcal{O}}(g)$:\n\\begin{equation}\n\\gamma_\\mathcal{O}(g)=-\\gamma_{0(\\mathcal{O})} g^2 -\\gamma_{1(\\mathcal{O})}g^4 + \\cdots\n\\end{equation}\nWith one-loop accuracy:\n\\begin{equation}\nZ^2_{\\mathcal{O}}\\sim\\left(\\frac{g^2(p)}{g^2(\\mu)}\\right)^{\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}}\\sim\n\\left(\\log(\\frac{p}{\\mu})\\right)^{-\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}}\n\\end{equation}\nand with two-loop accuracy we have:\n\\begin{align}\\label{eqn:z_pert}\nZ^2_{\\mathcal{O}} \\sim &\\left[\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\left(1-\\frac{\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^2\\log\\frac{p}{\\mu}}\\right)\\right]^{\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}}\ne^{\\frac{\\gamma_{1(\\mathcal{O})}\\beta_0-\\gamma_{0(\\mathcal{O})}\\beta_1}{\\beta_0^2}\\left[\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\left(1-\\frac{\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^2\\log\\frac{p}{\\mu}}\\right)\\right]}\\nonumber\\\\\n\\sim &\\left(\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\right)^{\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}} \n\\left(1-\\frac{\\gamma_{0(\\mathcal{O})}\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^3\\log\\frac{p}{\\mu}}\\right)\\nonumber\\\\\n\\times &\\left\\{1+\\frac{\\gamma_{1(\\mathcal{O})}\\beta_0-\\gamma_{0(\\mathcal{O})}\\beta_1}{\\beta_0^2}\\left[\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\n\\left(1-\\frac{\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^2\\log\\frac{p}{\\mu}}\\right)\\right]\\right\\}\\nonumber\\\\\n\\sim &\\left(\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\right)^{\\frac{\\gamma_{0(\\mathcal{O})}}{2\\beta_0}}\n\\Biggl(1-\\frac{\\gamma_{0(\\mathcal{O})}\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^3\\log\\frac{p}{\\mu}}+\n\\frac{\\gamma_{1(\\mathcal{O})}\\beta_0-\\gamma_{0(\\mathcal{O})}\\beta_1}{2\\beta_0^3\\log\\frac{p}{\\mu}}\\nonumber\\\\\n&-\\frac{(\\gamma_{0(\\mathcal{O})}\\gamma_{1(\\mathcal{O})}\\beta_0\\beta_1-(\\gamma_{0(\\mathcal{O})}\\beta_1)^2)\\log\\log\\frac{p}{\\mu}}{4\\beta_0^6(\\log\\frac{p}{\\mu})^2}\n\\Biggr) \n\\end{align}\nIn evaluating the last two expressions we have used the two-loop $RG$-improved expression for $g(p)$ given by Eq.(\\ref{eqn:g_2loop}).\nFrom the two-loop $RG$-improved expression in Eq.(\\ref{eqn:z_pert}) it follows that the leading and next-to-leading logarithms for $Z_{\\mathcal{O}}$ are determined only by $\\beta_0$, $\\beta_1$  and by $\\gamma_{0(\\mathcal{O})}$, that are in fact universal, i.e. scheme independent. Indeed, the two-loop coefficient of the anomalous dimension $\\gamma_{1(\\mathcal{O})}$ does not occur in the first $\\log\\log\\frac{p}{\\mu}$ term, but only in terms that have a subleading behavior as powers of  logarithms.\nKeeping only up to the next-to-leading term in $Z^2_{\\mathcal{O}}$, we obtain for the universal logarithmic behavior of the dimensionless two-point correlator:\n\\begin{equation}\\label{eqn:pert_th_next_to_lead}\nG_{DL}^{(2)}(\\frac{p}{\\mu})\\sim\\left[\\left(\\frac{1}{2\\beta_0\\log\\frac{p}{\\mu}}\\right)\n\\Biggl(1-\\frac{\\beta_1\\log\\log\\frac{p}{\\mu}}{2\\beta_0^2\\log\\frac{p}{\\mu}}\\Biggr)\\right]^{\\frac{\\gamma_{0(\\mathcal{O})}}{\\beta_0}}\n\\mathcal{G}_{\\mathcal O}(g(p))\n\\end{equation}\nThus our aim, in order to get asymptotic estimates, is to determine the one-loop coefficient of the anomalous dimension $\\gamma_{0 (\\mathcal{O})}$ and the $RGI$ function $\\mathcal{G}_{\\mathcal O}$ for our operators $\\mathcal{O}$.\n \n\\subsection{Anomalous dimension of $tr F^2$ and of $trF\\tilde{F}$}\n\nThe operator $\\frac{\\beta(g)}{g}tr F^2$\nis proportional to the conformal anomaly, that is the functional derivative with respect to a conformal rescaling of the metric of the renormalized effective action that must be $RGI$.\nTherefore $\\frac{\\beta(g)}{g}tr F^2$ is $RGI$ as well. \nHence its anomalous dimension vanishes and, using the notation of the previous section, the form of its correlator, ignoring possible contact terms that will be taken into account in sect.(3), is:\n\\begin{equation}\nG^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))=\np^4\\mathcal{G}_{\\frac{\\beta(g)}{g}F^2}(g(p)) \n\\end{equation} \nOn the other hand $tr F^2$  is not $RGI$ and therefore its correlator is:\n\\begin{equation}\nG^{(2)}_{F^2}(p,\\mu,g(\\mu))=\np^4\\mathcal{G}_{F^2}(g(p))Z_{F^2}^2(\\frac{p}{\\mu},g(\\mu))\n\\end{equation}  \nSince the relation between $G^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))$ and $G^{(2)}_{F^2}(p,\\mu,g(\\mu))$ is:\n\\begin{equation}\\label{eqn:relationGg}\nG^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))=\\left(\\frac{\\beta(g)}{g}\\right)^2 G^{(2)}_{F^2}(p,\\mu,g(\\mu)) \n\\end{equation}\nit follows that $\\left(\\frac{\\beta(g)}{g}\\right)^2$ and $Z^2(\\frac{p}{\\mu}, g(\\mu))$ must combine in such a way to obtain a function of $g(p)$ only:\n\\begin{equation}\n\\left(\\frac{\\beta(g(\\mu))}{g(\\mu)}\\right)^2\\, Z_{F^2}^2(\\frac{p}{\\mu}, g(\\mu))\\, \\mathcal{G}_{F^2}(g(p)) =\n\\mathcal{G}_{\\frac{\\beta(g)}{g}F^2}(g(p)) \n\\end{equation}\nTo find the anomalous dimension $\\gamma_{ F^2}$ of $tr F^2$ we exploit once again the property of $\\frac{\\beta(g)}{g}tr F^2$ being $RGI$.\nIts two-point correlator must indeed satisfy the equation:\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g} - 4\\right)G^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))=0\n\\end{equation}\nwhere the last term occurs because we are considering the complete correlator and not its dimensionless part.\nUsing Eq.(\\ref{eqn:relationGg}) we find the anomalous dimension of $tr F^2$:\n\\begin{align}\n&\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g}-4\\right)\n\\biggl[\\left(\\frac{\\beta(g)}{g}\\right)^2 G^{(2)}_{F^2}(p,\\mu,g(\\mu))\\biggr]=0 \\nonumber\\\\\n\\Rightarrow &\\biggl[\\left(\\frac{\\beta(g)}{g}\\right)^2\\frac{\\partial}{\\partial\\log p}\n-\\left(\\frac{\\beta(g)}{g}\\right)^2\\beta(g)\\frac{\\partial}{\\partial g}\\nonumber\\\\\n&-2\\beta(g)(\\frac{\\beta(g)}{g})\\frac{\\partial}{\\partial g}\\left(\\frac{\\beta(g)}{g}\\right) - 4 \\left(\\frac{\\beta(g)}{g}\\right)^2 \\biggr]\nG^{(2)}_{F^2}(p,\\mu,g(\\mu))=0 \\nonumber\\\\\n\\Rightarrow & \\biggl[\\frac{\\partial}{\\partial\\log p}\n-\\beta(g)\\frac{\\partial}{\\partial g}\n-2g\\frac{\\partial}{\\partial g}\\left(\\frac{\\beta(g)}{g}\\right) - 4\\biggr]\nG^{(2)}_{F^2}(p,\\mu,g(\\mu))=0\n\\end{align}\nFrom this equation it follows:\n\\begin{equation}\n\\gamma_{F^2}(g)=g \\frac{\\partial}{\\partial g}\\left(\\frac{\\beta(g)}{g}\\right)\n\\end{equation}\nWith two-loop accuracy this expression reads:\n\\begin{equation}\\label{eqn:dim_anomala_scalar}\n\\gamma_{tr F^2}(g)=-2\\beta_0 g^2-4\\beta_1 g^4 +\\cdots\n\\end{equation} \nKeeping only the first term, we can derive the expression for $Z^2(\\frac{p}{\\mu},g(\\mu))$ with one-loop accuracy:\n\\begin{equation}\nZ^2(\\frac{p}{\\mu},g(\\mu))\\sim\\frac{g^4(p)}{g^4(\\mu)}\n\\end{equation}\nFinally, the correlator of $\\frac{\\beta(g)}{g}trF^2$, with one-loop accuracy, is:\n\\begin{equation}\\label{eqn:corr_l2_rg}\nG^{(2)}_{\\frac{\\beta(g)}{g}F^2}(p,\\mu,g(\\mu))=\np^4\\beta_0^2 g^4(\\mu)\\,\\frac{g^4(p)}{g^4(\\mu)} \\, \\mathcal{G}_{F^2}(g(p))=\np^4\\beta_0^2 g^4(p)\\, \\mathcal{G}_{F^2}(g(p))\n\\end{equation}\nWe can repeat similar calculations for the operator $tr F\\tilde{F}$ in order to compute its anomalous dimension, using the property of $g^2 tr F\\tilde{F}$ being $RGI$.\nIndeed $g^2 tr F\\tilde{F}$ is the density of the second Chern class or topological charge.\nThe Callan-Symanzik equation is:\n\\begin{align}\n&\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g} - 4\\right)G^{(2)}_{g^2 F\\tilde{F}}(p,\\mu,g(\\mu))\\nonumber\\\\\n&=\\left(\\frac{\\partial}{\\partial\\log p}-\\beta(g)\\frac{\\partial}{\\partial g} - 4 \\right)\\biggl[g^4 G^{(2)}_{F\\tilde{F}}(p,\\mu,g(\\mu))\\biggr]=0\n\\end{align}\nfrom which we obtain the anomalous dimension of $trF\\tilde{F}$:\n\\begin{equation}\\label{eqn:dim_anomala_pseudoscalar}\n\\gamma_{F\\tilde{F}}(g)=2\\frac{\\beta(g)}{g}=-2\\beta_0 g^2 -2\\beta_1 g^4 +\\cdots\n\\end{equation}\nWe notice that while the one-loop anomalous dimensions of $trF^2$ and of $tr F\\tilde{F}$ coincide, the two-loop anomalous dimensions are different.\nThis means that the operator $tr {F^-}^2$ has a well defined anomalous dimension only at one loop, in agreement with the fact that it belongs to the large-$N$ one-loop integrable sector of Ferretti-Heise-Zarembo \\cite{ferretti:new_struct}.\nTherefore, only the universal part of its correlator, that is determined by the one-loop anomalous dimension and by the two-loop $\\beta$ function, can be meaningfully compared with the non-perturbative computation in Eq.(\\ref{eqn:intro_formula_L2}).\n\n\n\n\\subsection{Universal behavior of correlators}\\thispagestyle{empty}\n\n\nKnowing the naive dimension $D$ and the anomalous dimension of a (scalar) operator $\\mathcal{O}_D$, we can write the asymptotic form for $p>>\\mu$ of its correlator obtained by the $RG$ theory.\nIndeed, as we recalled in sect.(2.2), assuming multiplicative renormalizability, the $RG$-improved form of the Fourier transform of the correlator is given by:\n\\begin{equation}\\label{eqn:pert_general_behavior}\nG^{(2)}(p^2)=\\int\\braket{\\mathcal{O}_D(x)\\mathcal{O}_D(0)}_{\\mathit{conn}} e^{i p \\cdot x} d^4x=\np^{2D-4}\\,\\mathcal{G}_{\\mathcal{O}_D}(g(p)) \\, Z^2_{\\mathcal{O}_D}(\\frac{p}{\\mu},g(\\mu))\n\\end{equation}\nwhere the power of $p$ is implied by dimensional analysis, $\\mathcal{G}_{\\mathcal{O}_D}$ is a dimensionless function that depends only on the running coupling $g(p)$, and $Z^2_{\\mathcal{O}_D}$ is the contribution from the anomalous dimension.\nBut in fact in general the correlator of $\\mathcal{O}_D$ is not even multiplicatively renormalizable because of the presence of contact terms. These terms would affect the $UV$ asymptotic behavior, but they are non-physical and therefore they must be subtracted. In fact, they spoil the positivity of the correlator in Euclidean space in the momentum representation, that is required by the Kallen-Lehmann representation (see the comment below Eq.(\\ref{eqn:rg_improved_scalar_2l})). \\par\nIn the coordinate representation of the correlator, for $x \\neq 0$, contact terms do not occur. Therefore, a strategy to avoid that contact terms interfere with the $RG$ improvement is to pass to the coordinate scheme \\cite{chetyrkin:TF}, where the correlator is multiplicatively renormalizable, to compute its $RG$-improved expression, to go back to the momentum representation, and eventually to subtract the contact terms. \\par\nIn the coordinate representation for $x \\neq 0$ the solution of the Callan-Symanzik equation reads:\n\\begin{equation}\\label{eqn:pert_general_behavior_x}\nG_{\\mathcal{O}_D}^{(2)}(x)=\\braket{\\mathcal{O}_D(x)\\mathcal{O}_D(0)}_{\\mathit{conn}}=\n{\\bigl(\\frac{1}{x^2}\\bigr)}^{D}\\,\\mathcal{G}_{\\mathcal{O}_D}(g(x))\\, Z^2_{\\mathcal{O}_D}(x \\mu ,g(\\mu))\n\\end{equation}\nwith $x=\\sqrt {x^2}$, where we have denoted by $g(x)$ the running coupling in the coordinate scheme \\cite{chetyrkin:TF} and by an abuse of notation we have used the same names $\\mathcal{G}$ and $Z$ for the $RGI$ function\nand renormalization factor in the coordinate and momentum representation. \\par\nThe function $\\mathcal{G}(g(p))$ can be guessed at the lowest non-trivial order,\nsince the correlator must be conformal at the lowest non-trivial order in perturbation theory, that implies $\\mathcal{G}(g(x)) \\sim const$. Hence:\n\\begin{equation}\n\\mathcal{G}(\\frac{p}{\\mu}) \\sim const \\,\\log\\frac{p}{\\mu}\n\\end{equation}\nIndeed, in appendix A we show that $ \\int p^{2D-4} \\log\\frac{p}{\\mu} e^{i p \\cdot x} d^4p = const (\\frac{1}{x^2})^{D} $ that is conformal in the coordinate representation. The explicit dependence on $\\mu$,\nthat contradicts $RG$ invariance in the momentum representation, is due to the fact that the correlator in the momentum representation, as opposed to the coordinate representation, is not really multiplicatively renormalizable because (scale dependent) contact terms arise. This can be understood observing that in the coordinate representation for $x \\neq 0$  the lowest-order correlator is independent on the scale $\\mu$\nbut it is not an integrable function, in such a way that its Fourier transform needs a regularization, that introduces the arbitrary scale $\\mu$. \\par\nNaively, we can already derive the leading $UV$ asymptotic behavior:\n\\begin{align}\\label{eqn:naive_rg}\n&G^{(2)}_{\\mathcal{O}_D}(p^2)\\sim p^{2D-4}\\log\\frac{p^2}{\\mu^2}\n\\biggl(\\frac{g^2(p)}{g^2(\\mu)}\\biggr)^{\\frac{\\gamma_{0 (\\mathcal{O}_D)}}{\\beta_0}}\\sim \np^{2D-4}\n(g^2(p))^{\\frac{\\gamma_{0 (\\mathcal{O}_D)}}{\\beta_0}-1}\n\\end{align}\nwhere we have used the fact that $g^2(p)\\sim\\frac{1}{\\log(\\frac{p}{\\mu})}$. It easy to check that for $D=4$ and $\\gamma_{0 (\\mathcal{O}_D)}=2 \\beta_0$ this estimate coincides with Eq.(\\ref{eqn:corr_scalare_inizio})-Eq(\\ref{eqn:corr_asd}). \\par However, this estimate assumes multiplicative renormalizability in the momentum representation and it does not take into account the occurrence of contact terms in the momentum representation of the correlators. \\par Nevertheless, in the next section we confirm by direct computation that after subtracting the contact terms the actual behavior of the scalar and of the pseudoscalar correlator agrees with the estimate in Eq.(\\ref{eqn:naive_rg}). \n\n\\section{Perturbative check of the universal behavior of correlators}\n\nIn this section we obtain the explicit form of the three-loop correlators of $tr F^2$ and of $tr F\\tilde{F}$ starting from their imaginary parts that have been computed in \\cite{chetyrkin:scalar} \\cite{chetyrkin:pseudoscalar} in the $\\overline{MS}$ scheme.\nThe $\\overline{MS}$ scheme can be defined as the scheme in which the two-loop $RG$-improved running coupling does not contain\n$\\frac{1}{\\log^2\\frac{p^2}{\\Lambda_s^2}}$ contributions \\cite{chetyrkin:schema}.\nMore precisely, we consider the equation for the running coupling constant that follows from the two-loop $\\beta$ function:\n\\begin{equation}\n\\log\\frac{p}{\\Lambda_s}=\\int_{g(\\Lambda_s)}^{g(p)}\\frac{dg}{\\beta(g)}=\\frac{1}{2\\beta_0 g^2(p)}+\\frac{\\beta_1}{\\beta_0^2}\\log\\bigl(g(p)\\bigr)+ C +\\cdots\n\\end{equation}\nwhere $C$ is an arbitrary integration constant and $\\Lambda_s$ is the $RGI$ scale in a generic scheme $s$.\nThe value of $C$ in the $\\overline{MS}$ scheme is chosen in such a way to cancel the $\\frac{1}{\\log^2\\frac{p^2}{\\Lambda_s^2}}$ term in the solution:\n\\begin{align}\n&g_s^2(p)=\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda_s^2}}\\biggl[1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log(\\beta_0\\log\\frac{p^2}{\\Lambda_s^2})}{\\log\\frac{p^2}{\\Lambda_s^2}}+\\frac{C}{\\log\\frac{p^2}{\\Lambda_s^2}}\\biggr] \\nonumber\\\\\n\\Rightarrow &C=\\frac{\\beta_1}{\\beta_0^2}\\log(\\beta_0)\n\\end{align}\nThe result reported in \\cite{chetyrkin:scalar}, for $tr F^2$ in the $SU(3)$ $YM$ theory, is:\n\\begin{align}\\label{eqn:im_scalare}\n\\Im\\braket{tr F^2(p)tr F^2(-p)}_{conn}&=\n\\frac{8}{4\\pi}p^4\n\\biggl\\{1+\\frac{\\alpha_{\\overline{MS}}(\\mu)}{\\pi}\\biggl[\\frac{73}{4}-\\frac{11}{2}\\log\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&+(\\frac{\\alpha_{\\overline{MS}}(\\mu)}{\\pi})^2\\biggl[\\frac{37631}{96}\n-\\frac{363}{8}\\zeta(2)-\\frac{495}{8}\\zeta(3)\\nonumber\\\\\n&-\\frac{2817}{16}\\log\\frac{p^2}{\\mu^2}+\\frac{363}{16}\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\biggr\\}\n\\end{align}\nwhere $\\alpha_s=\\frac{g^2_{YM}}{4\\pi}$ and  $\\alpha_{\\overline{MS}}$ is $\\alpha_s$ in the $\\overline{MS}$ scheme.\nFirstly, we want to find from Eq.(\\ref{eqn:im_scalare}) the result for the $SU(N)$  $YM$ theory and we want to express the result in terms of the 't Hooft coupling in the $\\overline{MS}$ scheme $g_{\\overline{MS}}$.\nIn fact, this operation is quite easy since it is known, and it can be checked in \\cite{chetyrkin:pseudoscalar}, that at this order of perturbation theory the rank of the gauge group enters the result only through the Casimir factor $C_A=N$.\nTherefore, to obtain the general result it is sufficient to divide by 3 and to multiply by $N$ the coefficient of $\\alpha_{\\overline{MS}}$ and to divide by 9 and to multiply by $N^2$ the coefficient of $\\alpha_{\\overline{MS}}^2$ .\nThe factors of $N$ and of $N^2$ are then absorbed in the definition of the 't Hooft coupling constant.\nWe obtain:\n\\begin{align}\\label{eqn:ris_chet}\n&\\Im\\braket{tr F^2(p)tr F^2(-p)}_{conn}\\nonumber\\\\\n&=\\frac{N^2-1}{4\\pi}p^4\n\\biggl\\{1+g_{\\overline{MS}}^2(\\mu)\\biggl(\\frac{73}{3(4\\pi)^2}-2\\frac{11}{3(4\\pi)^2}\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl[\\frac{37631}{54(4\\pi)^4}\n-\\frac{242}{3(4\\pi)^4}\\zeta(2)-\\frac{110}{(4\\pi)^4}\\zeta(3)\\nonumber\\\\\n&-\\frac{313}{(4\\pi)^4}\\log\\frac{p^2}{\\mu^2}+\\frac{121}{3(4\\pi)^4}\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\biggr\\}\n\\end{align}\nFrom Eq.(\\ref{eqn:ris_chet}) we derive the complete expression of the correlator,\nassuming the correlator in the form:\n\\begin{align}\\label{eqn:ipotesi}\n\\braket{tr F^2(p)tr F^2(-p)}_{conn}&=\n-\\frac{N^2-1}{4\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{\\overline{MS}}^2(\\mu)\\biggl(f_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl(f_1+f_2\\log\\frac{p^2}{\\mu^2}+f_3\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nWe extract the imaginary part of Eq.(\\ref{eqn:ipotesi}) that arises from the imaginary part of the logarithm in Minkowski signature, $\\log(-\\frac{p^2}{\\mu^2})=\\log\\frac{p^2}{\\mu^2}-i\\pi$. We obtain:\n\\begin{align}\\label{eqn:ris_ipotesi}\n&\\Im\\braket{tr F^2(p)tr F^2(-p)}_{conn}\\nonumber\\\\\n&=\\frac{(N^2-1)}{4\\pi} p^4\\biggl[1+f_0g^2(\\mu)+(f_1-f_3\\pi^2)g^4(\\mu)\\nonumber\\\\\n&-2\\beta_0g^2(\\mu)\\log\\frac{p^2}{\\mu^2}+2f_2g^4(\\mu)\\log\\frac{p^2}{\\mu^2}+3f_3g^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align} \nFinally, comparing Eq.(\\ref{eqn:ris_chet}) with Eq.(\\ref{eqn:ris_ipotesi}) we determine the values of the coefficients $f_i$:\n\\begin{align}\nf_0&=\\frac{73}{3(4\\pi)^2}\\nonumber\\\\\nf_1-f_3\\pi^2&=(\\frac{37631}{54}-\\frac{242}{3}\\zeta(2)-110\\zeta(3))\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n-2\\beta_0&=-2\\frac{11}{3(4\\pi)^2}\\nonumber\\\\\n2f_2&=-\\frac{313}{(4\\pi)^4}\\Rightarrow f_2=-\\frac{313}{2(4\\pi)^4}\\nonumber\\\\\n3f_3&=\\frac{121}{3(4\\pi)^4}\\Rightarrow f_3=\\frac{121}{9(4\\pi)^4}\\Rightarrow f_3=\\beta_0^2\\nonumber\\\\\n\\nonumber\\\\\n\\Rightarrow f_1&=(\\frac{37631}{54}-110\\zeta(3))\\frac{1}{(4\\pi)^4}\n\\end{align}\nTherefore, the correlator is:\n\\begin{align}\\label{eqn:corr_pert_scalar_3l}\n\\braket{tr F^2(p)tr F^2(-p)}_{conn}=\n&-\\frac{(N^2-1)}{4\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g^2(\\mu)\\biggl(f_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g^4(\\mu)\\biggl(f_1+f_2\\log\\frac{p^2}{\\mu^2}+\\beta_0^2\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nSimilarly, the imaginary part of the correlator of $tr F\\tilde{F}$, already written in \\cite{chetyrkin:pseudoscalar} for the gauge group $SU(N)$, is:\n\\begin{align}\n\\Im\\braket{tr F\\tilde{F}(p)tr F\\tilde{F}(-p)}_{conn}=\n&\\frac{N^2-1}{4\\pi}p^4\n\\biggl\\{1+\\frac{\\alpha_{\\overline{MS}}(\\mu)}{\\pi}\\biggl[N\\biggl(\\frac{97}{12}-\\frac{11}{6}\\log\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\nonumber\\\\\n&+(\\frac{\\alpha_{\\overline{MS}}(\\mu)}{\\pi})^2\\biggl[N^2\\biggl(\\frac{51959}{864}-\\frac{121}{24}\\zeta(2)\n-\\frac{55}{8}\\zeta(3)\\nonumber\\\\\n&-\\frac{1135}{48}\\log\\frac{p^2}{\\mu^2}+\\frac{121}{48}\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\biggr\\}\n\\end{align} \nWe obtain:\n\\begin{align}\\label{eqn:corr_pert_pseudoscalar_3l}\n\\braket{tr F\\tilde{F}(p)tr F\\tilde{F}(-p)}_{conn}=\n&-\\frac{(N^2-1)}{4\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{\\overline{MS}}^2(\\mu)\\biggl(\\tilde{f}_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl(\\tilde{f}_1+\\tilde{f}_2\\log\\frac{p^2}{\\mu^2}+\\beta_0^2\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nwhere:\n\\begin{align}\n\\tilde{f}_0&=\\frac{97}{3(4\\pi)^2}\\nonumber\\\\\n\\tilde{f}_1&=(\\frac{51959}{54}-110\\zeta(3))\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n-2\\beta_0&=-2\\frac{11}{3(4\\pi)^2}\\nonumber\\\\\n2\\tilde{f}_2&=-\\frac{1135}{3(4\\pi)^4}\\Rightarrow \\tilde{f}_2=-\\frac{1135}{6(4\\pi)^4}\\nonumber\\\\\n\\end{align} \n\n\\subsection{Correlator of $\\frac{\\beta(g)}{gN}tr F^2$ in $SU(N)$ $YM$ (two loops)}\n\nWe now determine the $UV$ asymptotic behavior for the correlators by employing their $RG$-improved expression.\nFirstly, we recall that in every generic scheme labelled by $a$ the relation between the coupling constant at two different scales is, with  one-loop accuracy:\n\\begin{align}\\label{eqn:scale_relation}\n&\\frac{1}{g_{a}^2(\\mu)}=\\frac{1}{g_{a}^2(p)}-\\beta_0\\log\\frac{p^2}{\\mu^2} \n\\end{align}\nThis relation is necessary to express the correlators in their $RG$-improved form.\nAs a starting simplified example we consider the two-loop expression of the correlator of $\\beta_0 \\frac{g^2}{N}trF^2$ (for the moment we skip overall positive numerical factors in the normalization of the correlator):\n\\begin{align}\\label{eqn:2l_perturbative_corr}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&\\sim -\\beta_0^2 p^4 g_{\\overline{MS}}^4(\\frac{\\mu^2}{\\Lambdams^2})\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{\\overline{MS}}^2(\\frac{\\mu^2}{\\Lambdams^2})\\biggl(f_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nThis expression is renormalization group invariant with one-loop accuracy, since the factor $(\\frac{\\beta(g)}{g})^2$  is $\\beta_0^2 g^4$ if we employ the one-loop $\\beta$ function. \nThe finite term $f_0 g_{\\overline{MS}}^2(\\mu)$ can be absorbed in a change of scheme.\nIndeed, defining:\n\\begin{equation}\ng^2_{a}(\\mu)=g_{\\overline{MS}}^2(\\mu)(1+ag_{\\overline{MS}}^2(\\mu))\n\\end{equation}\nit follows:\n\\begin{align}\\label{eqn:cambio_schema}\ng^4_{a}(\\mu)&=g_{\\overline{MS}}^4(1+2ag_{\\overline{MS}}^2(\\mu)+a^2g_{\\overline{MS}}^4(\\mu))+ O(g^{10})\\nonumber\\\\\ng_{\\overline{MS}}^2(\\mu)&=g^2_{a}(\\mu)(1-ag_{\\overline{MS}}^2(\\mu)+a^2g_{\\overline{MS}}^4(\\mu))+O(g^8)\\nonumber\\\\\n&=g^2_{a}(\\mu)(1-a g_{a}^2(\\mu)+2a^2g^4_{a}(\\mu))+O(g^8)\\nonumber\\\\\ng_{\\overline{MS}}^4(\\mu)&=g^4_{a}(\\mu)(1-2a g^2_{a}(\\mu)+5a^2g^4_{a}(\\mu))+O(g^{10})\n\\end{align}\nWe obtain for the correlator:\n\\begin{align}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n& \\sim -\\beta_0^2 p^4\\log\\frac{p^2}{\\mu^2}g^4_{a}(\\mu)\n\\biggl[1+(f_0-2a)g^2_{a}(\\mu)-\\beta_0g^2_{a}(\\mu)\\log\\frac{p^2}{\\mu^2}+O(g^4\\log\\frac{p^2}{\\mu^2})\\biggr]\n\\end{align}\nTo cancel the finite term it is sufficient to put:\n\\begin{equation}\na=\\frac{f_0}{2}\n\\end{equation}\nHence we obtain:\n\\begin{align}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&\\sim-\\beta_0^2 p^4\\log\\frac{p^2}{\\mu^2}g^4_{a}(\\mu)\\biggl[1-\\beta_0g^2_{a}(\\mu)\\log\\frac{p^2}{\\mu^2}+O(g^4\\log\\frac{p^2}{\\mu^2})\\biggr]\n\\end{align} \nAt this order the term in square brackets is precisely the renormalization factor necessary to renormalize two powers of $g_{a}(\\mu)$.\nWe obtain:\n\\begin{align}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&\\sim-\\beta_0^2 p^4g^2_{a}(\\mu)g^2_{a}(p)\\log\\frac{p^2}{\\mu^2}\\bigl(1+O(g^4\\log\\frac{p^2}{\\mu^2})\\bigr)\n\\end{align} \nFrom  Eq.(\\ref{eqn:scale_relation}) we express the logarithm in terms of the coupling constant:\n\\begin{equation}\n\\beta_0\\log\\frac{p^2}{\\mu^2}=\\frac{1}{g^2_{a}(p)}-\\frac{1}{g^2_{a}(\\mu)}\n\\end{equation}\nThe correlator becomes:\n\\begin{align}\\label{eqn:rg_improved_scalar_2l}\n&\\braket{\\beta_0 \\frac{g^2}{N}tr F^2(p) \\beta_0 \\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&\\sim-\\beta_0 p^4g^2_{a}(\\mu)g^2_{a}(p)\\bigl(\\frac{1}{g^2_{a}(p)}-\\frac{1}{g^2_{a}(\\mu)}\\bigr)\\bigl(1+O(g^4\\log\\frac{p^2}{\\mu^2})\\bigr)\\nonumber\\\\\n&=\\beta_0 p^4\\bigl(g_{a}^2(p)-g_{a}^2(\\mu)\\bigr)\\bigl(1+O(g^4\\log\\frac{p^2}{\\mu^2})\\bigr)\n\\end{align}\nThe second term in the last line is in fact a contact term that has no physical meaning, therefore it may depend on the arbitrary scale $\\mu$ since it must be subtracted anyway.\nThe physical term is positive, despite the correlator that we started with was negative. \nThis is an important feature, since a negative physical term would have been in contrast with the Kallen-Lehmann representation, that requires a positive spectral function.\n\\newline\n\n\\subsection{Correlator of $\\frac{\\beta(g)}{gN}tr F^2$ in $SU(N)$ $YM$ (three loops)}\n\nWe now consider the three-loop result Eq.(\\ref{eqn:corr_pert_scalar_3l}), this time including also the correct normalization factors:\n\\begin{align}\n&\\braket{\\frac{g^2}{N} tr F^2(p)\\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)g_{\\overline{MS}}^4(\\mu)\\frac{p^4}{4\\pi^2}\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{\\overline{MS}}^2(\\mu)\\biggl(f_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl(f_1+f_2\\log\\frac{p^2}{\\mu^2}+\\beta_0^2\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nThis correlator is not supposed to be $RGI$, because the factor of $(\\frac{\\beta(g)}{g})^2=g^4\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2\\bigr)^2$ is missing.\nWe can eliminate the finite terms in the correlator by a redefinition of the scheme:\n\\begin{align}\\label{eqn:cambio_schema2}\ng^2_{ab}(\\mu)&=  g_{\\overline{MS}}^2(\\mu) \\bigl(1+ag_{\\overline{MS}}^2(\\mu)+bg_{\\overline{MS}}^4(\\mu)\\bigr)\\nonumber\\\\\n\\Rightarrow g_{\\overline{MS}}^4(\\mu)&=g^4_{ab}(\\mu)(1-2a g^2_{ab}(\\mu)+(2b+5a^2)g^4_{ab}(\\mu))+O(g^{10})\n\\end{align}\nSubstituting we obtain:\n\\begin{align}\n&\\braket{\\frac{g^2}{N} tr F^2(p)\\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr) g^4_{ab}(\\mu)\\bigl(1-2a g^2_{ab}(\\mu)+(2b+5a^2)g^4_{ab}(\\mu)\\bigr)\n\\frac{p^4}{4\\pi^2}\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times\\quad&\\biggl[1+f_0 g^2_{ab}(\\mu)(1-ag^2_{ab}(\\mu))\n-\\beta_0 g^2_{ab}(\\mu)(1-ag^2_{ab}(\\mu))\\log\\frac{p^2}{\\mu^2}+f_1 g_{ab}^4(\\mu)\\nonumber\\\\\n&+ f_2 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr) g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times\\quad&\\biggl[1+(f_0-2a) g^2_{ab}(\\mu)\n-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}+(f_1+5a^2+2b-af_0) g_{ab}^4(\\mu)\\nonumber\\\\\n&+(f_2+3\\beta_0 a) g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align} \nWe eliminate the two finite terms choosing:\n\\begin{align}\n&a=\\frac{f_0}{2}\\nonumber\\\\\n&f_1+5(\\frac{f_0}{2})^2+2b-\\frac{f_0^2}{2}=0 \\nonumber\\\\\n\\Rightarrow & b=\\frac{3}{8}f_0^2-\\frac{f_1}{2}\n\\end{align}  \nWith this choice of $a$ the coefficient of the $g^4\\log\\frac{p^2}{\\mu^2}$ term becomes:\n\\begin{equation}\nf_2+3\\beta_0 a = f_2+\\frac{3}{2}f_0\\beta_0=-\\frac{68}{3(4\\pi)^4}=-2\\beta_1\n\\end{equation}\nTherefore, the correlator reads:\n\\begin{align}\\label{eqn:corr_scalare_intermedio}\n&\\braket{\\frac{g^2}{N} tr F^2(p)\\frac{g^2}{N}tr F^2(-p)}_{conn}=\n-\\bigl(1-\\frac{1}{N^2}\\bigr) g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nWe notice that the expression in square brackets is the two-loop $Z$ factor determined by the anomalous dimension of $tr F^2$ according to Eq.(\\ref{eqn:dim_anomala_scalar}).\nThe coefficient $2\\beta_1$ should become $\\beta_1$ if we multiply the correlator in Eq.(\\ref{eqn:corr_scalare_intermedio}) by the factor of $\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)^2$, in order to make the correlator $RGI$:\n\\begin{align}\n&\\braket{\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(p)\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0^2 g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)^2\\log\\frac{p^2}{\\mu^2}\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0^2 g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\frac{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)}{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)\\bigr)}\\quad\\nonumber\\\\\n\\times\\quad &\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)\\bigr)\\log\\frac{p^2}{\\mu^2}\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nwhere we have multiplied and divided by $\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)$ in order to exploit the two-loop relation:\n\\begin{equation}\n\\beta_0(1+\\frac{\\beta_1}{\\beta_0}g_{ab}^2(p))\\log\\frac{p^2}{\\mu^2} =\\frac{1}{g_{ab}^2(p)}-\\frac{1}{g_{ab}^2(\\mu)}\n\\end{equation}\nWe now evaluate separately:\n\\begin{align}\n&\\frac{(1+\\frac{\\beta_1}{\\beta_0}g_{ab}^2(\\mu))}{(1+\\frac{\\beta_1}{\\beta_0}g_{ab}^2(p))}\\nonumber\\\\\n&=\\bigl(1+\\frac{\\beta_1}{\\beta_0}g_{ab}^2(\\mu)\\bigr)\n\\bigl(1-\\frac{\\beta_1}{\\beta_0}g_{ab}^2(\\mu)+\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\frac{\\beta_1^2}{\\beta_0^2}g_{ab}^4(\\mu)\\bigr)+O(g^6\\log\\frac{p^2}{\\mu^2})\\nonumber\\\\\n&= 1+\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +O(g^6\\log\\frac{p^2}{\\mu^2})\n\\end{align}\nPutting all together  we get:\n\\begin{align}\n&\\braket{\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(p)\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2}\\quad \\nonumber\\\\\n\\times\\quad & \\bigl(1+\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2}\\bigr)\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\n\\bigl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(\\mu)}\\bigr)\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 g^4_{ab}(\\mu)\n\\frac{p^4}{4\\pi^2} \\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\n\\bigl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(\\mu)}\\bigr)\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nThe factor in square brackets in the last line is now precisely the renormalization factor for two powers of $g_{ab}$. Hence the correlator reads:\n\\begin{align}\\label{corr_scalare_rgi_3_loop_improved}\n&\\braket{\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(p)\\frac{\\beta(g_{ab})}{Ng_{ab}}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 \n\\frac{p^4}{4\\pi^2} g^2_{ab}(\\mu)g^2_{ab}(p)\n\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\n\\bigl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(p)}\\bigr)\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 \n\\frac{p^4}{4\\pi^2} \\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\n\\bigl(g_{ab}^2(\\mu)-g_{ab}^2(p)\\bigr)\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\beta_0 \n\\frac{p^4}{4\\pi^2}\n\\biggl[g_{ab}^2(p)\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)-g_{ab}^2(\\mu)\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)\\biggr]\n\\end{align}\nThe second term in the last line is a contact term, but the first term depends on $g_{ab}(\\mu)$, therefore it is not $RGI$.\nHence Eq.(\\ref{corr_scalare_rgi_3_loop_improved}) is not exactly $RGI$ even after subtracting the contact terms.\nThe scale dependence in the physical term is due to the fact that the correlator is not exact but it is computed to a finite order of perturbation theory.\nWe notice that the scale dependence occurs at order of $g^4$ only and in any case it does not affect the structure of the universal $UV$ behavior but only the overall coefficient in the $RG$ estimate.\nYet it is interesting to determine the precise overall coefficient of the asymptotic behavior. This is done for the correlator of $\\frac{\\beta(g)}{gN}tr F^2$ in $SU(3)$ $QCD$ in sect.(3.7) by assuming its $RG$-invariance, instead of checking it to a finite order of perturbation theory as we just did.\n\n\\subsection{Correlator of $\\frac{g^2}{N} trF^2$ in $SU(N)$ $YM$ (three loops)}\n\nWe now present the result for the correlator of $\\frac{g^2}{N} trF^2$. \nWe recall that in this case we do not expect to get a $RGI$ function to all orders in perturbation theory.\nWe start from Eq.(\\ref{eqn:corr_scalare_intermedio}) and we write it as:\n\\begin{align}\\label{eqn:rg_not_improved_scalar_3l}\n&\\braket{\\frac{g^2}{N}tr F^2(p)\\frac{g^2}{N}tr F^2(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^4_{ab}(\\mu)\n\\frac{1}{\\beta_0} \\frac{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)}{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)\\bigr)}\n\\frac{1}{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)}\n\\biggl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(\\mu)}\\biggr)\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-2\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^4_{ab}(\\mu) \n\\frac{1}{\\beta_0}\n\\frac{1}{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)\\bigr)}\n\\biggl(\\frac{1}{g^2_{ab}(p)}-\\frac{1}{g^2_{ab}(\\mu)}\\biggr)\\nonumber\\\\\n\\times\\quad&\\biggl[1-\\beta_0g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2}\n-\\beta_1 g_{ab}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{ab}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0} \n\\bigl(1-\\frac{\\beta_1}{\\beta_0}g^2_{ab}(\\mu)+\\frac{\\beta_1^2}{\\beta_0^2}g^4_{ab}(\\mu)\\bigr)\n\\bigl(g^2_{ab}(\\mu)-g^2_{ab}(p)\\bigr)\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{ab}(p)-g^2_{ab}(\\mu)+\\frac{\\beta_1}{\\beta_0}g^4_{ab}(\\mu)-\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)g^2_{ab}(\\mu)\\bigr)\n\\end{align}\nSurprisingly we notice that the term that depends on the product $g_{ab}(\\mu)g_{ab}(p)$, that is not $RGI$, is of the same order of $g^4$ as the non-$RGI$ terms in the correlator in Eq.(\\ref{corr_scalare_rgi_3_loop_improved}), that must be $RGI$.\n\n\\subsection{Correlator of $\\frac{g^2}{N} trF\\tilde{F}$ in $SU(N)$ $YM$ (three loops)}\n\nWe repeat the same steps to find the $RG$-improved expression for the correlator of $\\frac{g^2}{N}tr F\\tilde{F}$, that is $RGI$.\nThe three-loop correlator reads:\n\\begin{align}\n\\braket{\\frac{g^2}{N}tr F\\tilde{F}(p)\\frac{g^2}{N}tr F\\tilde{F}(-p)}_{conn}=\n&-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2} g_{\\overline{MS}}^4(\\mu)\\log\\frac{p^2}{\\mu^2}\\quad\\nonumber\\\\\n\\times\\quad &\\biggl[1+g_{\\overline{MS}}^2(\\mu)\\biggl(\\tilde{f}_0-\\beta_0\\log\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+g_{\\overline{MS}}^4(\\mu)\\biggl(\\tilde{f}_1+\\tilde{f}_2\\log\\frac{p^2}{\\mu^2}+\\beta_0^2\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\n\\end{align}\nNow we perform a generic change of scheme as in Eq.(\\ref{eqn:cambio_schema2}):\n\\begin{equation}\ng_{\\tilde{ab}}^2=g_{\\overline{MS}}^2(\\mu)\\bigl(1+\\tilde{a}g_{\\overline{MS}}^2(\\mu)+\\tilde{b}g_{\\overline{MS}}^4(\\mu)\\bigr)\n\\end{equation}\nThe correlator becomes:\n\\begin{align}\n&\\braket{\\frac{g^2}{N}tr F\\tilde{F}(p)\\frac{g^2}{N}tr F\\tilde{F}(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^4_{\\tilde{ab}}(\\mu)\n\\log\\frac{p^2}{\\mu^2}\\quad\\nonumber\\\\\n\\times\\quad&\\biggl[1+(\\tilde{f}_0-2\\tilde{a}) g^2_{\\tilde{ab}}(\\mu)\n-\\beta_0g^2_{\\tilde{ab}}(\\mu)\\log\\frac{p^2}{\\mu^2}+(\\tilde{f}_1+5\\tilde{a}^2+2\\tilde{b}-\\tilde{a}\\tilde{f}_0) g_{\\tilde{ab}}^4(\\mu)\\nonumber\\\\\n&+(\\tilde{f}_2+3\\beta_0 \\tilde{a}) g_{\\tilde{ab}}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{\\tilde{ab}}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align} \nAgain we impose the conditions to eliminate the finite terms:\n\\begin{align}\n&\\tilde{a}=\\frac{\\tilde{f}_0}{2}\\nonumber\\\\\n&\\tilde{f}_1+5(\\frac{\\tilde{f}_0}{2})^2+2\\tilde{b}-\\frac{\\tilde{f}_0^2}{2}=0 \\nonumber\\\\\n\\Rightarrow & \\tilde{b}=\\frac{3}{8}\\tilde{f}_0^2-\\frac{\\tilde{f}_1}{2}\n\\end{align}  \nWith this choice of $\\tilde{a}$ the coefficient of the $g^4\\log\\frac{p^2}{\\mu^2}$ term becomes:\n\\begin{equation}\n\\tilde{f}_2+3\\beta_0 \\tilde{a} = \\tilde{f}_2+\\frac{3}{2}\\tilde{f}_0\\beta_0=-\\frac{34}{3(4\\pi)^4}=-\\beta_1\n\\end{equation}\nSubstituting in the correlator we get:\n\\begin{align}\n&\\braket{\\frac{g^2}{N}tr F\\tilde{F}(p)\\frac{g^2}{N}tr F\\tilde{F}(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^4_{\\tilde{ab}}(\\mu)\n\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times \\quad&\\biggl[1-\\beta_0g^2_{\\tilde{ab}}(\\mu)\\log\\frac{p^2}{\\mu^2}-\\beta_1 g_{\\tilde{ab}}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g_{\\tilde{ab}}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nWe notice that the expression in square brackets is the two-loop $Z$ factor implied by the anomalous dimension of $trF\\tilde{F}$ computed in Eq.(\\ref{eqn:dim_anomala_pseudoscalar}). It renormalizes two powers of $g(\\mu)$.\nTherefore, the correlator reads:\n\\begin{align}\\label{eqn:rg_improved_pseudo_3l}\n&\\braket{\\frac{g^2}{N}tr F\\tilde{F}(p)\\frac{g^2}{N}tr F\\tilde{F}(-p)}_{conn}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^2_{\\tilde{ab}}(\\mu) g^2_{\\tilde{ab}}(p)\n\\log\\frac{p^2}{\\mu^2}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}g^2_{\\tilde{ab}}(\\mu) g^2_{\\tilde{ab}}(p)\\frac{1}{\\beta_0}\\biggl(\\frac{1}{g_{\\tilde{ab}}^2(p)}-\\frac{1}{g^2_{\\tilde{ab}}(\\mu)}\\biggr)\n\\frac{1}{1+\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^2(p)}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{\\tilde{ab}}(\\mu)-g^2_{\\tilde{ab}}(p)\\bigr)\\frac{1}{1+\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^2(p)}\\nonumber\\\\\n&=-\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{\\tilde{ab}}(\\mu)-g^2_{\\tilde{ab}}(p)\\bigr)\\bigl(1-\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^2(p)+\\frac{\\beta_1^2}{\\beta_0^2}g^4_{\\tilde{ab}}(p)\\bigr)\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{4\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{\\tilde{ab}}(p)+\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^2(p)g^2_{\\tilde{ab}}(\\mu)-\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^4(p)-g^2_{\\tilde{ab}}(\\mu)\\bigr) \n\\end{align}\nThe second term in the last line is scale dependent to the order of $g^4$ as the term that occurs in the correlator of $\\frac{g^2}{N}tr F^2$ in Eq.(\\ref{eqn:rg_not_improved_scalar_3l}).\n\n\n\\subsection{Correlator of $\\frac{g^2}{N} tr {F^-}^2$ in $SU(N)$ $YM$ (three loops)}\n\nWe now sum the two results for the correlators of $trF^2$ and of $tr F\\tilde{F}$ to obtain the correlator of $tr {F^-}^2$.\nIndeed, we recall that:\n\\begin{equation}\n\\frac{1}{2}\\braket{tr{F^-}^2(p)tr{F^-}^2(-p)}_{conn}=\n2\\braket{trF^2(p)trF^2(-p)}_{conn}+2\\braket{trF\\tilde{F}(p)trF\\tilde{F}(-p)}_{conn}\n\\end{equation}\nSumming the two results in Eq.(\\ref{eqn:rg_improved_pseudo_3l}) and in Eq.(\\ref{eqn:rg_not_improved_scalar_3l}) we obtain:\n\\begin{align}\\label{eqn:corr_asd_pert}\n&\\frac{1}{2}\\braket{\\frac{g^2}{N}tr{F^-}^2(p)\\frac{g^2}{N}tr{F^-}^2(-p)}_{conn}=\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{2 \\pi^2}\\frac{1}{\\beta_0} \\bigl(g^2_{ab}(p) + g^2_{\\tilde{ab}}(p) -g^2_{ab}(\\mu) -g^2_{\\tilde{ab}}(\\mu) +\\frac{\\beta_1}{\\beta_0}g^4_{ab}(\\mu)\\nonumber\\\\\n&-\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^4(p)+\\frac{\\beta_1}{\\beta_0}g^2_{ab}(p)g^2_{ab}(\\mu)-\\frac{\\beta_1}{\\beta_0}g^2_{\\tilde{ab}}(p)g^2_{\\tilde{ab}}(\\mu)\\bigr)\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{2\\pi^2}\\frac{1}{\\beta_0}\\bigl(g^2_{ab}(p) + g^2_{\\tilde{ab}}(p) -g^2_{ab}(\\mu) -g^2_{\\tilde{ab}}(\\mu) +\\frac{\\beta_1}{\\beta_0}g^4_{ab}(\\mu)\\nonumber\\\\\n&-\\frac{\\beta_1}{\\beta_0} g_{\\tilde{ab}}^4(p)+O(g^6)\\bigr)\n\\end{align}\nwhere surprisingly the mixed terms  $g^2(p)g^2(\\mu)$ cancel to the order of $g^6$.\nThere is no perturbative explanation for such cancellation, but conjecturally the cancellation occurs because of the $RG$ invariance of the non-perturbative formula Eq.(\\ref{eqn:formula_prologo}) in the $TFT$ for the $L=2$ ground state \\cite{boch:crit_points} \\cite{boch:glueball_prop} of the large-$N$ one-loop integrable sector of Ferretti-Heise-Zarembo (see sect.(4)).\nWe can express the last result in terms of the coupling constant in the $\\overline{MS}$ scheme:\n\\begin{align}\\label{eqn:corr_asd_ms}\n&\\frac{1}{2}\\braket{\\frac{g^2}{N}tr{F^-}^2(p)\\frac{g^2}{N}tr{F^-}^2(-p)}_{conn}\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{p^4}{2\\pi^2}\\frac{1}{\\beta_0} \\bigl(2g_{\\overline{MS}}^2(p)-2g_{\\overline{MS}}^2(\\mu) +\\bigl(a+\\tilde{a} -\\frac{\\beta_1}{\\beta_0}\\bigr)g_{\\overline{MS}}^4(p)\\nonumber\\\\\n&+\\bigl(\\frac{\\beta_1}{\\beta_0}-a-\\tilde{a}\\bigr) g_{\\overline{MS}}^4(\\mu)\\bigr)+O(g^6)\n\\end{align}\nthat coincides with Eq.(\\ref{eqn:prologo:ris_sommato}).\n\n\\subsection{Scalar correlators in $SU(3)$ $QCD$ with $n_l$ massless Dirac fermions}\n\n\nIn this section we derive the $RG$-improved expression for the correlators of $trF^2$ and of $\\frac{\\beta(g_{YM})}{g_{YM}}trF^2$ in $QCD$. \\par\nThe three-loop perturbative result for the imaginary part of the correlator of $tr F^2$ in $QCD$ with $n_l$ massless Dirac femions is \\cite{chetyrkin:scalar}:\n\\begin{align}\n&\\Im{\\braket{trF^2(p)trF^2(-p)}_{conn}}\\nonumber\\\\\n&=\\frac{2}{\\pi}p^4\\Biggl\\{1+\\frac{\\alpha_s(\\mu)}{\\pi}\\biggl[\\biggl(\\frac{73}{4}-\\frac{11}{2}\\log\\frac{p^2}{\\mu^2}\\biggr)\n-n_l\\biggl(\\frac{7}{6}-\\frac{1}{3}\\log\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\nonumber\\\\\n&+\\Bigl(\\frac{\\alpha_s(\\mu)}{\\pi}\\Bigr)^2\\biggr[\\frac{37631}{96}-\\frac{363}{8}\\zeta(2)\n-\\frac{495}{8}\\zeta(3)-\\frac{2817}{16}\\log\\frac{p^2}{\\mu^2} +\\frac{363}{16}\\log^2\\frac{p^2}{\\mu^2}\\nonumber\\\\\n&+n_l\\biggl(-\\frac{7189}{144}+\\frac{11}{2}\\zeta(2)+\\frac{5}{4}\\zeta(3)+\\frac{263}{12}\\log\\frac{p^2}{\\mu^2}-\\frac{11}{4}\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+n_l^2\\biggl(\\frac{127}{108}-\\frac{1}{6}\\zeta(2)-\\frac{7}{12}\\log\\frac{p^2}{\\mu^2} +\\frac{1}{12}\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\Biggr\\}\n\\end{align}\nWe write the correlator in terms of the coupling $g_{YM}$ in the $\\overline{MS}$ scheme instead of $\\alpha_s$:\n\\begin{align}\\label{eqn:im_pert}\n&\\Im{\\braket{trF^2(p)trF^2(-p)}_{conn}}\\nonumber\\\\\n&=\\frac{2}{\\pi}p^4\\Biggl\\{1+g_{YM}^2(\\mu)\\biggl[\\biggl(73-22\\log\\frac{p^2}{\\mu^2}\\biggr)\n-n_l\\biggl(\\frac{14}{3}-\\frac{4}{3}\\log\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n&+g_{YM}^4(\\mu)\\biggr[\\biggl(\\frac{37361}{6}-726\\zeta(2)-990\\zeta(3)\n-2817\\log\\frac{p^2}{\\mu^2} +363\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+n_l\\biggl(-\\frac{7189}{9}+88\\zeta(2)+20\\zeta(3)+\\frac{1052}{3}\\log\\frac{p^2}{\\mu^2}-44\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\nonumber\\\\\n&+n_l^2\\biggl(\\frac{508}{27}-\\frac{8}{3}\\zeta(2)-\\frac{28}{3}\\log\\frac{p^2}{\\mu^2} +\\frac{4}{3}\\log^2\\frac{p^2}{\\mu^2}\\biggr)\\biggr]\\frac{1}{(4\\pi)^4}\\Biggr\\}\n\\end{align}\nIf we suppose the correlator to be of the form:\n\\begin{align}\n&\\braket{trF^2(p)trF^2(-p)}_{conn}\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{YM}^2(\\mu)\\Bigl(h_0+h_1\\log\\frac{p^2}{\\mu^2}\\Bigr)\\nonumber\\\\\n&+g_{YM}^4(\\mu)\\Bigl(h_2+h_3\\log\\frac{p^2}{\\mu^2} +h_4 \\log^2\\frac{p^2}{\\mu^2}\\Bigr)\\biggr]\n\\end{align}\nits imaginary part is:\n\\begin{align}\\label{eqn:im_ipotesi}\n&\\Im{\\braket{trF^2(p)trF^2(-p)}_{conn}}\\nonumber\\\\\n&=\\frac{2}{\\pi}p^4\\biggl[1+h_0 g_{YM}^2(\\mu) + 2h_1 g_{YM}^2(\\mu)\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&+(h_2-\\pi^2 h_4)g_{YM}^4(\\mu) +2h_3 g_{YM}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +3h_4 g_{YM}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nComparing Eq.(\\ref{eqn:im_ipotesi}) and Eq.(\\ref{eqn:im_pert}) we get:\n\\begin{align}\nh_0&=\\biggl(73-\\frac{14}{3}n_l\\biggr)\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n2h_1&=\\biggl(-22+\\frac{4}{3}n_l\\biggr)\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n\\Rightarrow h_1&=\\biggl(-11+\\frac{2}{3}n_l\\biggr)\\frac{1}{(4\\pi)^2}=-\\tilde{\\beta}_0\\nonumber\\\\\nh_2-\\pi^2 h_4 &=\\biggl[\\frac{37361}{6}-726\\zeta(2)\n-990\\zeta(3)\n+n_l\\Bigl(-\\frac{7189}{9}+88\\zeta(2)+20\\zeta(3)\\Bigr)\\nonumber\\\\\n&+n_l^2\\Bigl(\\frac{508}{27}-\\frac{8}{3}\\zeta(2)\\Bigr)\\biggr]\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n2h_3 &=\\biggl[-2817+\\frac{1052}{3}n_l-\\frac{28}{3}n_l^2\\biggr]\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n\\Rightarrow h_3 &=\n\\biggl[-\\frac{2817}{2}+\\frac{526}{3}n_l-\\frac{14}{3}n_l^2\\biggr]\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n3h_4&=\\biggl[363- 44n_l+\\frac{4}{3}n_l^2\\biggr]\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n\\Rightarrow h_4&=\\biggl[121- \\frac{44}{3}n_l+\\frac{4}{9}n_l^2\\biggr]\\frac{1}{(4\\pi)^4}=\\tilde{\\beta}_0^2\n\\end{align}\nNow we repeat the same steps as in the $n_l=0$ case. \nWe change renormalization scheme in order to cancel the finite parts:\n\\begin{equation}\ng_{uv}^2(\\mu)=g_{YM}^2(\\mu)\\bigl(1+u g_{YM}^2(\\mu)+vg_{YM}^4(\\mu)\\bigr)\n\\end{equation} \nWe use the perturbative expression for the renormalized coupling constant with two-loop accuracy:\n\\begin{align*}\n&g_{YM}^2(p)=g_{YM}^2(\\mu)\\Bigl(1-\\tilde{\\beta}_0 g_{YM}^2(\\mu)\\log\\frac{p^2}{\\mu^2} -\\tilde{\\beta}_1 g_{YM}^4(\\mu)\\log\\frac{p^2}{\\mu^2}\\nonumber\\\\ &+\\tilde{\\beta}_0^2g^4_{YM}(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\Bigr)\n\\end{align*} \nwhere the tilde refers to the $QCD$ coefficients of the $\\beta$ function: \n\\begin{align}\n\\tilde{\\beta}_0&=\\Bigl(11-\\frac{2}{3}n_l\\Bigr)\\frac{1}{(4\\pi)^2}\\nonumber\\\\\n\\tilde{\\beta}_1 &=\\Bigl(102-\\frac{38}{3}n_l\\Bigr)\\frac{1}{(4\\pi)^4}\n\\end{align}\n\\par\nWe consider now the correlator of $g_{YM}^2 tr F^2$:\n\\begin{align}\n&\\braket{g_{YM}^2 trF^2(p)g_{YM}^2trF^2(-p)}_{conn}\\nonumber\\\\\n&=-\\frac{2g_{YM}^4(\\mu)}{\\pi^2}p^4\\log\\frac{p^2}{\\mu^2}\n\\biggl[1+g_{YM}^2(\\mu)\\Bigl(h_0-\\tilde{\\beta}_0\\log\\frac{p^2}{\\mu^2}\\Bigr)\\nonumber\\\\\n&+g_{YM}^4(\\mu)\\Bigl(h_2+h_3\\log\\frac{p^2}{\\mu^2} +\\tilde{\\beta}_0^2 \\log^2\\frac{p^2}{\\mu^2}\\Bigr)\\biggr]\\nonumber\\\\\n&=-\\frac{2g^4_{uv}(\\mu)}{\\pi^2}p^4\n\\log\\frac{p^2}{\\mu^2}\\quad \\nonumber\\\\\n\\times\\quad&\\biggl[1+(h_0-2u) g^2_{uv}(\\mu)-\n\\tilde{\\beta}_0 g^2_{uv}(\\mu)\\log\\frac{p^2}{\\mu^2}+(h_2+5u^2+2v-uh_0) g_{uv}^4(\\mu)\\nonumber\\\\\n&+(h_3+3\\tilde{\\beta}_0 u) g_{uv}^4(\\mu)\\log\\frac{p^2}{\\mu^2} +\\tilde{\\beta}_0^2 g_{uv}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nChoosing $u=\\frac{h_0}{2}$ to cancel the finite term of order of $g^2$ in the square brackets we get for the coefficient of the term of order of $g^4 \\log\\frac{p^2}{\\mu^2}$:\n\\begin{align}\n&h_3+3\\tilde{\\beta}_0 u\\nonumber\\\\\n&=h_3+\\frac{3}{2}\\tilde{\\beta}_0 h_0\\nonumber\\\\\n&=\\biggl(-\\frac{2817}{2}+\\frac{526}{3}n_l-\\frac{14}{3}n_l^2\\biggr)\\frac{1}{(4\\pi)^4}\n+\\frac{3}{2}\\biggl(73-\\frac{14}{3}n_l\\biggr)\\biggl(11-\\frac{2}{3}n_l\\biggr)\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n&=\\biggl(-\\frac{2817}{2}+\\frac{526}{3}n_l-\\frac{14}{3}n_l^2+\\frac{2409}{2}-73n_l-77n_l+\\frac{14}{3}n_l^2\\biggr)\\frac{1}{(4\\pi)^4}\\nonumber\\\\\n&=-204+\\frac{76}{3}n_l\\nonumber\\\\\n&=-2\\tilde{\\beta}_1\n\\end{align}\nas predicted by Eq.(\\ref{eqn:dim_anomala_scalar}) and by the computational experience gained in the pure $YM$ case. \nTo cancel the finite term of order of $g^4$ we put:\n\\begin{equation}\nh_2+\\frac{5}{2}h_0^2+2v-\\frac{h_0^2}{2}=0\n\\end{equation} \nTherefore, the correlator reads:\n\\begin{align}\\label{eqn:corr_scalare_qcd}\n&\\braket{g_{YM}^2trF^2(p)g_{YM}^2trF^2(-p)}_{conn}\\nonumber\\\\\n&=-\\frac{2g^4_{uv}(\\mu)}{\\pi^2}p^4\n\\log\\frac{p^2}{\\mu^2}\\biggl[1-\\tilde{\\beta}_0 g^2_{uv}(\\mu)\\log\\frac{p^2}{\\mu^2}-2\\tilde{\\beta}_1 g_{uv}^4(\\mu)\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&+\\tilde{\\beta}_0^2 g_{uv}^4(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]\n\\end{align}\nNow we follow the same steps as in the $n_l=0$ case. The only differences are the coefficients of the $\\beta$ function and the parameters $u,v$ that define the new renormalization scheme.\nThe result is:\n\\begin{align}\n&\\braket{g_{YM}^2trF^2(p)g_{YM}^2trF^2(-p)}\\nonumber\\\\\n&=\\frac{2}{\\tilde{\\beta}_0\\pi^2}p^4\\Bigl(g^2_{uv}(p)-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^2_{uv}(p)g^2_{uv}(\\mu)-g^2_{uv}(\\mu)+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^4_{uv}(\\mu)\\Bigr)\n\\end{align}\n\\newline\nHence:\n\\begin{align}\n\\biggl(\\frac{\\beta(g_{uv})}{g_{uv}}\\biggr)^2 \\Pi(\\frac{p}{\\mu})\n=\\frac{2\\tilde{\\beta}_0}{\\pi^2}\n\\biggl(g_{uv}^2(p)+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^2_{uv}(\\mu)g_{uv}^2(p)-g_{uv}^2(\\mu)\n-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^4_{uv}(\\mu)\\biggr)\n\\end{align}\nWe recall that $u=\\frac{h_0}{2}$, therefore:\n\\begin{equation}\ng_{uv}^2(\\mu)=g_{YM}^2(\\mu)\\bigl(1+\\frac{h_0}{2}g_{YM}^2(\\mu)+vg_{YM}^4(\\mu)\\bigr)\n\\end{equation}\nHence we get:\n\\begin{align}\\label{eqn:ris_vecchio}\n&\\biggl(\\frac{\\beta(g_{uv})}{g_{uv}}\\biggr)^2 \\Pi(\\frac{p}{\\mu})\\nonumber\\\\\n&=\\frac{2\\tilde{\\beta}_0}{\\pi^2}\n\\biggl(g_{YM}^2(p)+\\frac{h_0}{2}g_{YM}^4(p)+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^2_{YM}(\\mu)g_{YM}^2(p)+\\nonumber\\\\\n&-g_{YM}^2(\\mu)\n-\\frac{h_0}{2}g_{YM}^4(\\mu)-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^4_{YM}(\\mu)+O(g^6)\\biggr)\n\\end{align}\nNow we  multiply the $RHS$ of Eq.(\\ref{eqn:ris_vecchio}) by \n$\\frac{\\bigl(\\frac{\\beta(g_{YM})}{g_{YM}}\\bigr)^2}{\\bigl(\\frac{\\beta(g_{uv})}{g_{uv}}\\bigr)^2}$.\nIndeed, this is necessary to take into account the change of scheme performed to compute Eq.(\\ref{eqn:ris_vecchio}).\nThe additional factor is:\n\\begin{align}\n&\\frac{\\bigl(\\frac{\\beta(g_{YM})}{g_{YM}}\\bigr)^2}{\\bigl(\\frac{\\beta(g_{uv})}{g_{uv}}\\bigr)^2}=\n\\frac{\\bigl(1+\\frac{\\beta_1}{\\beta_0}g_{YM}^2(\\mu)\\bigr)^2}{(1+\\frac{\\beta_1}{\\beta_0}g_{uv}^2(\\mu)\\bigr)^2}\\nonumber\\\\\n&=\\bigl(1+2\\frac{\\beta_1}{\\beta_0}g_{YM}^2(\\mu)+\\frac{\\beta_1^2}{\\beta_0^2}g_{YM}^4(\\mu)\\bigr)\n\\bigl(1-2\\frac{\\beta_1}{\\beta_0}g_{uv}^2(\\mu)+3\\frac{\\beta_1^2}{\\beta_0^2}g_{YM}^4(\\mu)\\bigr)\\nonumber\\\\\n&=1-h_0\\frac{\\beta_1}{\\beta_0}g^4_{YM}(\\mu) +O(g^6)\n\\end{align}\nTherefore, the correlator in Eq.(\\ref{eqn:ris_vecchio}) becomes:\n\\begin{align}\n&\\biggl(\\frac{\\beta(g_{YM})}{g_{YM}}\\biggr)^2 \\Pi(\\frac{p}{\\mu})\\nonumber\\\\\n&=\\frac{2\\tilde{\\beta}_0}{\\pi^2}\n\\biggl(g_{YM}^2(p)+\\frac{h_0}{2}g_{YM}^4(p)+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^2_{YM}(\\mu)g_{YM}^2(p)+\\nonumber\\\\\n&-g_{YM}^2(\\mu)\n-\\frac{h_0}{2}g_{YM}^4(\\mu)-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g^4_{YM}(\\mu)-h_0\\frac{\\beta_1}{\\beta_0}g^4_{YM}(\\mu)+O(g^6)\\biggr)\n\\end{align}\nthat has some dependence on the scale $\\mu$ even after subtracting the contact terms. In the next section we get rid of this dependence \nby using a different method, that assumes the $RG$ invariance of the correlator instead of checking it. \\par\nIn any case the universal $UV$ asymptotic behavior is in agreement with the $RG$ estimate, i.e.:\n\\begin{align}\n&\\braket{\\frac{\\beta(g_{YM})}{g_{YM}}tr F^2(p)\\frac{\\beta(g_{YM})}{g_{YM}}tr F^2(-p)}_{conn}\n\\sim \\frac{p^4}{\\tilde{\\beta}_0 \\log\\frac{p^2}{\\Lambdams^2}}\\Biggl(1-\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambdams^2}}{\\log\\frac{p^2}{\\Lambdams^2}}\\Biggr)\n\\end{align}\n\n\\subsection{$RG$-invariant scalar correlator in $SU(3)$ $QCD$ with $n_l$ massless Dirac fermions}\n\nFirstly, we check the correctness of the finite parts of the scalar correlator in $QCD$, reconstructed in sect.(3.6) from its imaginary part, thanks to another result reported in \\cite{chet:tensore}:\n\\begin{align}\\label{eqn:der_chet}\np^2\\frac{d}{d p^2} \\Pi(p)\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}=\n\\frac{1}{\\pi^2}\\biggl[-2+\\frac{\\alpha_s}{\\pi}\\biggl(-\\frac{73}{2}+\\frac{7}{3}n_l\\biggr)+\\nonumber\\\\\n+\\frac{\\alpha_s^2}{\\pi^2}\\biggl(-\\frac{37631}{48}+\\frac{495}{4}\\zeta(3)+n_l\\Bigl(\\frac{7189}{72}-\\frac{5}{2}\\zeta(3)\\Bigr)-\\frac{127}{54}n_l^2\\biggr)\\biggr]\n\\end{align}\nwith:\n\\begin{equation}\np^4 \\Pi(p)=\\braket{trF^2(p)trF^2(-p)}\n\\end{equation}\nwhere we have changed the overall normalization factor of the correlator with respect to \\cite{chet:tensore} to be coherent with the one used in this paper.\nWe perform the derivative at $p^2=\\mu^2$ of the correlator obtained in sect.(3.6):\n\\begin{align}\\label{eqn:correlator}\n&p^2\\frac{d}{d p^2} \\Pi(\\frac{p}{\\mu})\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}\\nonumber\\\\\n&=-\\frac{d}{d\\log p^2}\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}\\biggl[ \\frac{2}{\\pi^2}\\log\\frac{p^2}{\\mu^2}\n\\biggl(1+g_{YM}^2(\\mu)\\Bigl(h_0+h_1\\log\\frac{p^2}{\\mu^2}\\Bigr)\\nonumber\\\\\n&+g_{YM}^4(\\mu)\\Bigl(h_2+h_3\\log\\frac{p^2}{\\mu^2} +h_4 \\log^2\\frac{p^2}{\\mu^2}\\Bigr)\\biggr)\\biggr] \\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\biggl(1+g_{YM}^2(\\mu) h_0+g_{YM}^4(\\mu) h_2\\biggr) \n\\end{align}\nWe recall that:\n\\begin{align}\\label{eqn:der_mia}\nh_0&=\\biggl(73-\\frac{14}{3}n_l\\biggr)\\frac{1}{(4\\pi)^2}\\nonumber\\\\\nh_2&=\\frac{37361}{6}-990\\zeta(3) + n_l\\Bigl(-\\frac{7189}{9}+20\\zeta(3)\\Bigr)+\\frac{508}{27}n_l^2\n\\end{align}\nIt is easy to verify that Eq.(\\ref{eqn:der_chet}) and Eq.(\\ref{eqn:der_mia}) are in agreement.\nIndeed:\n\\begin{align}\n&\\frac{1}{\\pi^2}\\biggl[-2+\\frac{\\alpha_s}{\\pi}\\biggl(-\\frac{73}{2}+\\frac{7}{3}n_l\\biggr)\\nonumber\\\\\n&+\\frac{\\alpha_s^2}{\\pi^2}\\biggl(-\\frac{37631}{48}+\\frac{495}{4}\\zeta(3)+n_l\\Bigl(\\frac{7189}{72}-\\frac{5}{2}\\zeta(3)\\Bigr)-\\frac{127}{54}n_l^2\\biggr)\\biggr]\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\biggl[1+\\frac{g_{YM}^2}{4\\pi^2}\\biggl(+\\frac{73}{4}-\\frac{7}{6}n_l\\biggr)\\nonumber\\\\\n&+\\frac{g_{YM}^4}{(4\\pi^2)^2}\\biggl(+\\frac{37631}{96}-\\frac{495}{8}\\zeta(3)+n_l\\Bigl(-\\frac{7189}{144}+\\frac{5}{4}\\zeta(3)\\Bigr)+\\frac{127}{108}n_l^2\\biggr)\\biggr]\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\biggl[1+g_{YM}^2h_0+g_{YM}^4 h_2\\biggr]\n\\end{align}\nFrom Eq.(\\ref{eqn:correlator}) it follows the derivative of the correlator of $\\frac{\\beta(g_{YM})}{g_{YM}} trF^2$ with two-loop accuracy:\n\\begin{align}\\label{eqn:der_corr_rgi}\n&p^2\\frac{d}{d p^2} \\biggl(\\frac{\\beta(g_{YM})}{g_{YM}}\\biggr)^2\\Pi(\\frac{p}{\\mu})\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\tilde{\\beta}_0^2 g^4_{YM}(\\mu)\\bigl(1+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g_{YM}^2(\\mu)\\bigr)^2\\biggl[1+g_{YM}^2(\\mu)h_0+g_{YM}^4(\\mu) h_2\\biggr]\n\\end{align}\nSecondly, we write $g_{YM}(p)$ instead of $g_{YM}(\\mu)$  in Eq.(\\ref{eqn:der_corr_rgi}) since $\\log\\frac{p^2}{\\mu^2}=0\\Rightarrow p^2=\\mu^2$ in order to get \nthe large-momentum correlator in a manifestly $RGI$ form.\nExploiting the definition of the $\\beta$ function we can express $d\\log p^2$ in terms of $dg(p)$:\n\\begin{align}\n&\\frac{dg}{d\\log p}=\\beta(g)\n\\Rightarrow  d\\log(p^2)=2\\frac{dg}{\\beta(g)} = \\frac{d(g^2)}{g\\beta(g)}\n\\end{align} \nWe integrate Eq.(\\ref{eqn:der_corr_rgi}) to obtain:\n\\begin{align}\\label{eqn:ris_nuovo}\n&\\frac{d}{d \\log p^2} \\biggl(\\frac{\\beta(g)}{g}\\biggr)^2\\Pi(\\frac{p}{\\mu})\\biggl|_{\\log\\frac{p^2}{\\mu^2}=0}\\nonumber\\\\\n&=-\\frac{2}{\\pi^2}\\tilde{\\beta}_0^2 g^4_{YM}(p)\\bigl(1+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g_{YM}^2(p)\\bigr)^2\\biggl[1+g_{YM}^2(p)h_0+g_{YM}^4(p) h_2\\biggr] \\nonumber\\\\\n\\Rightarrow & \\biggl(\\frac{\\beta(g_{YM})}{g_{YM}}(p)\\biggr)^2 \\Pi(\\frac{p}{\\mu}) - \\biggl(\\frac{\\beta(g_{YM})}{g_{YM}}(\\mu)\\biggr)^2\\Pi(1) \\nonumber\\\\\n&=\\frac{2}{\\pi^2}\\tilde{\\beta}_0^2 \\int_{g_{YM}^2(\\mu)}^{g^2_{YM}(p)}g_{YM}^4\\bigl(1+\\frac{\\tilde{\\beta}_1}{\\tilde{\\beta}_0}g_{YM}^2\\bigr)^2\\biggl[1+g_{YM}^2h_0+g_{YM}^4 h_2\\biggr]\\frac{d(g_{YM}^2)}{\\tilde{\\beta_0}g_{YM}^4(1+\\frac{\\tilde{\\beta_1}}{\\tilde{\\beta_0}}g_{YM}^2)}\\nonumber\\\\\n&=\\frac{2}{\\pi^2}\\tilde{\\beta}_0\\biggl[g_{YM}^2(p)-g_{YM}^2(\\mu)+\\biggl(\\frac{\\tilde{\\beta}_1}{2\\tilde{\\beta}_0}+\\frac{h_0}{2}\\biggr)\\biggl(g_{YM}^4(p)-g_{YM}^4(\\mu)\\biggr)+ O(g^6)\\biggr]\n\\end{align}\nEq.(\\ref{eqn:ris_nuovo}) gives the manifestly $RGI$ form of the correlator after subtracting the $\\mu$-dependent contact terms.\n\n\n\\subsection{Correlators in the coordinate representation}\n\nIn this section we find the $RG$-improved expression for the perturbative correlators in the coordinate representation.\nThis procedure has the main advantage that in the coordinate representation the contact terms do not occur, since they are eliminated by the Fourier transform.\nIndeed, the Fourier transform of $p^4$ is:\n\\begin{equation}\n\\int p^4  e^{ip\\cdot x} \\frac{d^4 p}{(2\\pi)^4}=\n\\Delta^2 \\delta (x) \n\\end{equation}\nthat is supported only at $x=0$. This implies that at points different from zero the contact terms do not occur. \nThe $RG$ improvement and the Fourier transform must commute up to perhaps finite scheme-dependent terms. Therefore, in this way we get another check of the asymptotic behavior.\nIn appendix A we compute the Fourier transforms necessary to pass from the momentum to the coordinate representation.\nIn particular we use the following results:\n\\begin{align}\n\\int {(p^2)}^2\\log\\frac{p^2}{\\mu^2} e^{ip \\cdot x}  \\frac{d^4p}{(2\\pi)^4} &=\n-\\frac{2^6\\cdot 3}{\\pi^2 x^8}\\nonumber\\\\\n\\int {(p^2)}^2\\biggl(\\log\\frac{p^2}{\\mu^2}\\biggr)^2 e^{ip \\cdot x}  \\frac{d^4p}{(2\\pi)^4} &=\n\\frac{2^7\\cdot 3}{\\pi^2 x^8}\\bigl(-\\frac{10}{3}+2\\gamma_E -\\log\\frac{4}{x^2\\mu^2}\\bigr)\\nonumber\\\\\n\\int {(p^2)}^2\\biggl(\\log\\frac{p^2}{\\mu^2}\\biggr)^3 e^{ip \\cdot x}  \\frac{d^4p}{(2\\pi)^4} &=\n\\frac{2^6\\cdot 3}{\\pi^2 x^8}\\bigl(-\\frac{51}{2}+40\\gamma_E-12\\gamma_E^2\\nonumber\\\\\n&-(20-12\\gamma_E)\\log\\frac{4}{x^2\\mu^2} -3\\log^2\\frac{4}{x^2\\mu^2}\\bigr)\\nonumber\\\\\n\\end{align}\nUsing these formulae to compute the Fourier transform of the two-loop perturbative result in Eq.(\\ref{eqn:2l_perturbative_corr}) we get, disregarding the finite parts in Eq.(\\ref{eqn:2l_perturbative_corr}):\n\\begin{align}\\label{eqn:int_corr_2l}\n&-\\int g_{\\overline{MS}}^4(\\mu)p^4\\log\\frac{p^2}{\\mu^2}\\biggl[1-\\beta_0 g_{\\overline{MS}}^2(\\mu)\\log\\frac{p^2}{\\mu^2}\\biggr]e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{3\\cdot 2^6}{\\pi^2 x^8} g_{\\overline{MS}}^4(\\mu)\n+\\beta_0 g_{\\overline{MS}}^6(\\mu)\\frac{3\\cdot 2^6}{\\pi^2 x^8}\\bigl[-2\\log\\frac{4}{x^2\\mu^2} +4\\gamma_E - \\frac{20}{3}\\bigr]\\nonumber\\\\\n&= \\frac{3\\cdot 2^6}{\\pi^2 x^8}g_{\\overline{MS}}^4(\\mu)\\biggl[1+\\bigl(-\\beta_0\\frac{20}{3}+4\\beta_0\\gamma_E\\bigr) g_{\\overline{MS}}^2(\\mu)-2\\beta_0 g_{\\overline{MS}}^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\biggr]\n\\end{align}\nFirstly, the Fourier transform has produced a new finite part. \nSecondly, the coefficient of the logarithm in the square brackets is multiplied by two after the Fourier transform. This implies that the factor in the square brackets renormalizes four powers of $g_{\\overline{MS}}(\\mu)$, as opposed to the momentum representation, where only two powers of the coupling constant were renormalized. \nThis is as expected, since in the coordinate representation the correlator is multiplicatively renormalizable as implied by Eq.(\\ref{eqn:pert_general_behavior_x}). \\par\nTo eliminate the finite term arising from the Fourier transform we change scheme defining:\n\\begin{equation}\ng^2_s(\\mu)= g^2(\\mu)\\bigl[1+\\frac{1}{2}\\bigl(-\\beta_0\\frac{20}{3}+4\\beta_0\\gamma_E\\bigr)g^2(\\mu)\\bigr]\n\\end{equation} \nThe integral in Eq.(\\ref{eqn:int_corr_2l}) reads:\n\\begin{align}\n&-\\int g_{\\overline{MS}}^4(\\mu)p^4\\log\\frac{p^2}{\\mu^2}\\biggl[1-\\beta_0 g_{\\overline{MS}}^2(\\mu)\\log\\frac{p^2}{\\mu^2}\\biggr]e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}=\n\\frac{3\\cdot 2^6}{\\pi^2 x^8} g^4(x)\\nonumber\\\\\n\\end{align} \nwhere $g(x)$ is the one-loop running coupling in the coordinate scheme \\cite{chetyrkin:TF}:\n\\begin{align}\ng^2(x)=g^2(\\mu)\\biggl[1-\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\biggr]\n\\end{align}\nTherefore, the renormalization group improved one-loop asymptotic expression for the correlator is:\n\\begin{equation}\\label{eqn:rg_improved_2l_{a}}\n\\braket{\\frac{g^2}{N} trF^2(x) \\frac{g^2}{N} tr F^2(0)}_{conn}\\sim\n \\bigl(1-\\frac{1}{N^2}\\bigr) \\frac{3\\cdot 2^6}{\\pi^2 x^8} \\frac{1}{\\log^2\\frac{4}{x^2\\mu^2}}\n\\end{equation}\nThe Fourier transform provides automatically the change in sign necessary to obtain a positive expression.\nThis is due to the fact that in the coordinate representation contact terms do not occur.\nWe now go one step further performing the Fourier transform of the three-loop propagators in Eq.(\\ref{eqn:corr_pert_scalar_3l}) and in Eq.(\\ref{eqn:corr_pert_pseudoscalar_3l}). \nWe start with the scalar correlator up to the overall normalization:\n \\begin{align}\\label{eqn:int_corr_3l}\n&-\\int g_{ab}^4(\\mu)p^4\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&\\times\\biggl[1-\\beta_0 (\\mu)g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2} -2\\beta_1 g^4_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g^4_{ab}(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{ab}^4(\\mu)\\biggl[1+\\bigl(-\\beta_0\\frac{20}{3}+4\\beta_0\\gamma_E\\bigr)g^2_{ab}(\\mu)\n-2\\beta_0 g^2_{ab}(\\mu)\\log\\frac{4}{x^2\\mu^2} \\nonumber\\\\\n&+\\bigl(8\\beta_1\\gamma_E-\\frac{40}{3}\\beta_1+\\frac{51}{2}\\beta_0^2 -40\\beta_0^2\\gamma_E+12\\beta_0^2\\gamma_E^2\\bigr)g_{ab}^4(\\mu)\n-4\\beta_1 g_{ab}^4(\\mu)\\log\\frac{4}{x^2\\mu^2} \\nonumber\\\\\n&-\\beta_0^2\\bigl(12\\gamma_E-20\\bigr)g^4_{ab}(\\mu)\\log\\frac{4}{x^2\\mu^2}\n+3\\beta_0^2 g^4_{ab}(\\mu)\\log^2\\frac{4}{x^2\\mu^2}\\biggr]\n\\end{align}\nThe following scheme redefinition:\n\\begin{equation}\ng^2_{st}(\\mu)= g^2_{ab}(\\mu)\\bigl(1+(2\\beta_0\\gamma_E-\\frac{10}{3}\\beta_0)g^2_{ab}+t g_{ab}^4(\\mu)\\bigr)\n\\end{equation}\ncancels the finite term of order of $g^2$ in the square brackets and some terms of order of $g^4\\log\\frac{4}{x^2\\mu^2}$, leaving only the term proportional to $-4\\beta_1$.\nMoreover, the finite term of order of $g^4$ in the square brackets is cancelled by a suitable choice of $t$, as in the previous section.\nEq.(\\ref{eqn:int_corr_3l}) now reads:\n\\begin{align}\\label{eqn:corr_scalare_3l_revisited}\n&-\\int g_{ab}^4(\\mu)p^4\\log\\frac{p^2}{\\mu^2} \\nonumber\\\\\n&\\times\\biggl[1-\\beta_0 (\\mu)g^2_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2} -2\\beta_1 g^4_{ab}(\\mu)\\log\\frac{p^2}{\\mu^2} +\\beta_0^2 g^4_{ab}(\\mu)\\log^2\\frac{p^2}{\\mu^2}\\biggr]e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(\\mu)\n\\frac{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)}{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)}\n\\frac{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)}{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)}\\quad \\nonumber\\\\\n\\times\\quad &\\biggl[1-2\\beta_0 g^2_{st}(\\mu)\\log\\frac{4}{x^2\\mu^2}\n-4\\beta_1 g_{st}^4(\\mu)\\log\\frac{4}{x^2\\mu^2}+3\\beta_0^2 g^4_{st}(\\mu)\\log^2\\frac{4}{x^2\\mu^2}\\biggr]\\nonumber\\\\\n&=\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(\\mu)\n\\frac{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)}{1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)}\\quad \\nonumber\\\\\n\\times\\quad & \\biggl[1-2\\beta_0 g^2_{st}(\\mu)\\log\\frac{4}{x^2\\mu^2}\n-2\\beta_1 g_{st}^4(\\mu)\\log\\frac{4}{x^2\\mu^2}+3\\beta_0^2 g^4_{st}(\\mu)\\log^2\\frac{4}{x^2\\mu^2}\\biggr]\\nonumber\\\\\n&=\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(x)\\Bigl(1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)\n-2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)\\Bigr)\n\\end{align}\nThe scale dependent term in Eq.(\\ref{eqn:corr_scalare_3l_revisited}) occurs now at the order of $g^6$, while in the  momentum representation occurred at the order of $g^4$.\nNow we multiply both sides of Eq.(\\ref{eqn:corr_scalare_3l_revisited}) by $\\bigl(1+\\frac{\\beta_1}{\\beta_0}g_{st}^2(\\mu)\\bigr)^2$, i.e. by the factor necessary to make the correlator $RGI$.\nReinserting the overall normalization, we obtain:\n\\begin{align}\n&\\int \\braket{\\frac{\\beta(g)}{Ng} trF^2(p)\\frac{\\beta(g)}{Ng}trF^2(-p)}_{conn}e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\ \n&= \\frac{1}{4 \\pi^2} \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(x)\\Bigl(1+\\frac{\\beta_1}{\\beta_0}g_{st}^2(\\mu)\\Bigr)^2\\Bigl(1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x)\n-2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)\\Bigr)\\nonumber\\\\\n&= \\frac{1}{4 \\pi^2}\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{st}^4(x)\\Bigl(1+2\\frac{\\beta_1}{\\beta_0}g^2_{st}(x) +O(g^4)\\Bigr)\n\\end{align}\nAs a result the possible scale dependence is of order of $g^8$. \\par\nPerforming the same steps for the pseudoscalar correlator in Eq.(\\ref{eqn:corr_pert_pseudoscalar_3l}) we get:\n\\begin{align}\\label{eqn:corr_pseudoscalare_3l_revisited}\n&\\int \\braket{\\frac{g^2}{N}trF\\tilde{F}(p) \\frac{g^2}{N}trF\\tilde{F}(-p)}_{conn}e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{1}{4 \\pi^2}\\bigl(1-\\frac{1}{N^2}\\bigr) \\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{\\tilde{st}}^4(\\mu)\\biggl[1-2\\beta_0 g^2_{\\tilde{st}}(\\mu)\\log\\frac{4}{x^2\\mu^2}\n-2\\beta_1 g^4_{\\tilde{st}}(\\mu)\\log\\frac{4}{x^2\\mu^2}+3\\beta_1^2g^4_{\\tilde{st}}\\log^2\\frac{4}{x^2\\mu^2}\\biggr] \\nonumber\\\\\n&= \\frac{1}{4 \\pi^2}\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{2^6\\cdot 3}{\\pi^2 x^8}g_{\\tilde{st}}^4(x) \n\\end{align}\nThe correlator in Eq.(\\ref{eqn:corr_pseudoscalare_3l_revisited}) is $RGI$ in the coordinate representation with three-loop accuracy, while in the momentum representation scale-dependent terms of the order of $g^4$ occurred in Eq.(\\ref{eqn:rg_improved_pseudo_3l}). \nAs in the momentum representation we find the correlator of $tr{F^-}^2$ summing the double of the scalar Eq.(\\ref{eqn:corr_scalare_3l_revisited}) and pseudoscalar Eq.(\\ref{eqn:corr_pseudoscalare_3l_revisited}) correlators.\nWe obtain:\n\\begin{align}\\label{eqn:corr_asd_{a}}\n&\\frac{1}{2}\\int \\braket{\\frac{g^2}{N}tr{F^-}^2(p)\\frac{g^2}{N}tr{F^-}^2(-p)}_{conn}e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&=\\frac{1}{4 \\pi^2} \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{2^7\\cdot 3}{\\pi^2 x^8}\\bigl(g^4_{st}(x)+g^4_{\\tilde{st}}(x)+2\\frac{\\beta_1}{\\beta_0}g^6_{st}(x)-2\\frac{\\beta_1}{\\beta_0}g^2_{st}(\\mu)g^4_{st}(x)\\bigr)\\nonumber\\\\\n\\end{align}\nThe scale dependence enters the term of order of $g^6$ as in the momentum representation in Eq.(\\ref{eqn:corr_asd_pert}). \\par\nWe check the correctness of the separation of the contact terms performed in the momentum representation.\nWe verify to the order of the leading logarithm that the Fourier transform of the $RG$-improved expression in the momentum representation in Eq.(\\ref{eqn:corr_asd_ms}) is equal to Eq.(\\ref{eqn:corr_asd_{a}}) in the coordinate representation.\nWithin the leading logarithmic accuracy it is sufficient to put $g^2(p)$:\n\\begin{equation}\ng^2(p)=\\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}}\n\\end{equation}\nTherefore, the Fourier transform of the correlator in Eq.(\\ref{eqn:corr_asd_ms}) can be computed reducing it to a series of positive powers of logarithms:\n\\begin{align}\\label{eqn:serie_p_da_trasformare}\n& \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2\\beta_0}\\int p^4 \\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}} e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&= \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\sum_{l=0}^{\\infty}(-1)^l\\int p^4 g^2(\\mu)\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}\\bigr)^l e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\n\\end{align}\nWe extract the leading logarithms of this Fourier transform.\nBy leading we mean terms that have the highest power of logarithm with the power of $g$ fixed.\nWe use  Eq.(\\ref{eqn:ft_leading}) that furnishes the leading logarithm of the Fourier transform:\n\\begin{equation}\n\\int p^4\\biggl(\\log\\frac{p^2}{\\mu^2}\\biggr)^l e^{ip \\cdot x}  \\frac{d^4p}{(2\\pi)^4} =\n-\\frac{l\\Gamma(4) 2^{5}}{\\pi^2}\n\\frac{1}{x^8}\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{l-1}+\\cdots\n\\end{equation}\nInserting it in Eq.(\\ref{eqn:serie_p_da_trasformare}) we obtain for the leading logarithms: \n\\begin{align}\\label{eqn:x_series_da_tf}\n& \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\int p^4 \\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}} e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4}\\nonumber\\\\\n&= \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\sum_{l=0}^{\\infty}(-1)^{l-1}g^2(\\mu)\\beta_0^l g^{2l}(\\mu)\\frac{l\\cdot\\Gamma(4) 2^5}{\\pi^2}\\frac{1}{x^8}\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{l-1}\n\\end{align}  \nWe compare it with the $ASD$ correlator in the coordinate representation Eq.(\\ref{eqn:corr_asd_{a}}):\n\\begin{align}\\label{eqn:rg_improved_2l_{a}_series}\n&\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4) g^4(x)\\sim\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4) \\biggl(\\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}}\\biggr)^2\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n=0}^{\\infty}\\sum_{l=0}^{\\infty}(-1)^{n+l}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{x^2\\mu^2}{4}\\bigr)^{n+l}\n\\end{align} \nWe want to prove that the two series in Eq.(\\ref{eqn:x_series_da_tf}) and in Eq.(\\ref{eqn:rg_improved_2l_{a}_series}) are equal. \nThe proof is by induction.\nWe prove it for the first non trivial term, i.e. for $l=1$ in Eq.(\\ref{eqn:x_series_da_tf}):\n\\begin{equation}\n\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)\ng^{4}(\\mu)\n\\end{equation} \nthat is equal to the term obtained from Eq.(\\ref{eqn:rg_improved_2l_{a}_series}) putting $n=l=0$.\nAssuming that the equality is valid up to the order of $\\biggl(\\log\\frac{x^2\\mu^2}{4} \\biggr)^{m-1}$, we show that it holds at the order of $\\biggl(\\log\\frac{x^2\\mu^2}{4} \\biggr)^{m}$.\nIndeed, the $m$-power of the logarithm occurs in Eq.(\\ref{eqn:x_series_da_tf}) for $l=m+1$:\n\\begin{align}\\label{eqn:dim_induzione_intermedio}\n&\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\n\\int p^4 \\frac{g^2(\\mu)}{1+\\beta_0 g^2(\\mu)\\log\\frac{p^2}{\\mu^2}} e^{ip\\cdot x}\\frac{d^4p}{(2\\pi)^4} \\nonumber\\\\\n&\\sim \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0} \\bigg[\n\\sum_{l=0}^{m}(-1)^{l-1}g^2(\\mu)\\beta_0^l g^{2l}(\\mu)\\frac{l\\cdot\\Gamma(4) 2^5}{\\pi^2}\\frac{1}{x^8}\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{l-1}+\\nonumber\\\\\n&+(-1)^m\\frac{(m+1)\\cdot 2^5}{\\pi^2 x^8}\\beta_0^{m} \\Gamma(4)\ng^{2m+4}(\\mu)\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{m} \\bigg]\n\\end{align} \nThe $m$-th power of the logarithm in Eq.(\\ref{eqn:rg_improved_2l_{a}_series}) occurs for the $m+1$ couples $(n,l)$ such that $l+n=m$:\n\\begin{align}\n&\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(x)\\nonumber\\\\\n&\\sim \\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0} \\bigg[\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n,l=0}^{\\substack{\\\\n+l\\leq m-1}}(-1)^{n+l}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\bigr)^{n+l}+\\nonumber\\\\\n&+\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n,l=0}^{\\substack{\\\\n+l=m}}(-1)^{m}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\bigr)^{m} \\bigg]\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0}\\bigg[\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n,l=0}^{\\substack{\\\\n+l\\leq m-1}}(-1)^{n+l}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{4}{x^2\\mu^2}\\bigr)^{n+l}+\\nonumber\\\\\n&+(-1)^m\\frac{(m+1)\\cdot 2^5}{\\pi^2 x^8}\\beta_0^{m} \\Gamma(4)\ng^{2m+4}(\\mu)\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{m} \\bigg]\n\\end{align}\nFor the inductive hypothesis the first term in the last expression is equal to the first one in Eq.(\\ref{eqn:dim_induzione_intermedio}), i.e.:\n\\begin{align}\n&\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0} \\bigg[\n\\frac{ 2^5}{\\pi^2 x^8}\\Gamma(4)g^4(\\mu)\\sum_{n,l=0}^{\\substack{\\\\n+l\\leq m-1}}(-1)^{n+l}\\bigl(\\beta_0 g^2(\\mu)\\log\\frac{x^2\\mu^2}{4}\\bigr)^{n+l}  \\bigg]\\nonumber\\\\\n&=\\bigl(1-\\frac{1}{N^2}\\bigr)\\frac{1}{\\pi^2 \\beta_0} \\bigg[\n\\sum_{l=0}^{m}(-1)^{l-1}g^2(\\mu)\\beta_0^l g^{2l}(\\mu)\\frac{l\\cdot\\Gamma(4) 2^5}{\\pi^2}\\frac{1}{x^8}\\biggl(\\log\\frac{4}{x^2\\mu^2}\\biggr)^{l-1} \\bigg]\n\\end{align}\nThe remaining terms, i.e. the terms of order of $\\log^m\\frac{4}{x^2\\mu^2}$, are equal and therefore\nthe proof by induction is complete.\n\n\n\n\n\\section{ $ASD$ correlator in the Topological Field Theory}\\thispagestyle{empty} \n\nWe briefly summarize the results for the glueball propagators in the $TFT$ underlying large-$N$ $YM$ \\cite{boch:quasi_pbs} \\cite{MB0} \\cite{boch:crit_points} \\cite{boch:glueball_prop} \\cite{Top}. \nFor the $ASD$ glueball propagator \\cite{boch:glueball_prop}\\cite{boch:crit_points} \\footnote{We use here a manifestly covariant notation\nas opposed to the one in the $TFT$ \\cite{boch:glueball_prop}\\cite{boch:crit_points}.}:\n\\begin{equation}\\label{eqn:intro_formula_L2}\n\\frac{1}{2}\\braket{\\frac{g^2}{N}tr\\bigl(F^{-2}(p)\\bigr) \\frac{g^2}{N}tr \\bigl(F^{-2}(-p)\\bigr)}_{conn} =\n\\frac{1}{\\pi^2}\\sum_{k=1}^{\\infty}\\frac{k^2 g_k^4\\Lambda_{\\overline{W}}^6 }{p^2+k\\Lambda_{\\overline{W}}^2} + ...\n\\end{equation}\nBesides, in the $TFT$ the two-point correlators of certain scalar operators $\\mathcal{O}_{2L}$ of naive dimension $D=2L$ that are homogeneous polynomials of degree $L$\nin the $ASD$ curvature $F^-$\\cite{boch:glueball_prop}\\cite{boch:crit_points} can be computed asymptotically for large $L$:\n\\begin{align}\\label{eqn:formula}\n&\\braket{\\mathcal{O}_{2L}(p)\\mathcal{O}_{2L}(-p)}_{conn}\n=const\\sum_{k=1}^{\\infty}\\frac{k^{2L-2} Z_k^{-L}\\Lambda_{\\overline{W}}^2 \\Lambda_{\\overline{W}}^{4L-4}}{p^2+k\\Lambda_{\\overline{W}}^2} \n\\end{align}\nThe operators $\\mathcal{O}_{2L}$ occur as the ground state in the integrable sector of large-$N$ $YM$ of Ferreti-Heise-Zarembo \\cite{ferretti:new_struct} asymptotically for large $L$.\nFerreti-Heise-Zarembo have computed their one-loop anomalous dimension for large $L$ \\cite{ferretti:new_struct}:\n\\begin{align}\n\\gamma_{0 (\\mathcal{O}_{2L})}= \\frac{1}{(4\\pi)^2}\\frac{5}{3} L+O(\\frac{1}{L})\n\\end{align}\nThe ground state for $L=2$ is the $ASD$ operator that occurs in Eq.(\\ref{eqn:intro_formula_L2}) for which $\\gamma_{0 (\\mathcal{O}_{4})}=2 \\beta_0$ exactly. \\par\nIn Eq.(\\ref{eqn:intro_formula_L2}) and in Eq.(\\ref{eqn:formula}) $\\Lambda_{\\overline{W}}$ is the $RG$ invariant scale in the scheme in which it coincides with the mass gap. The functions $g^2(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})$ and $Z(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})$ are the solutions of the differential equations:\n\\begin{align} \\label{eqn:eq_def_gk}\n\\frac{\\partial g}{\\partial \\log p}\n&=\\frac{-\\beta_0 g^3+\\frac{1}{(4\\pi)^2}g^3\\frac{\\partial \\log Z}{\\partial \\log p}}{1-\\frac{4}{(4\\pi)^2}g^2} \\nonumber \\\\\n\\frac{\\partial\\log Z}{\\partial\\log p}\n&=2\\gamma_0 g^2 +\\cdots \\nonumber \\\\\n\\gamma_{0}&=\\frac{1}{(4\\pi)^2}\\frac{5}{3}\n\\end{align}\nwhere $p$ is equal to the square root of $p^2$.\nThe definitions of $g_k$ and $Z_k$ are:\n\\begin{align}\n&g_k=g(k)\\\\\n&Z_k=Z(k)\n\\end{align}\nIn \\cite{boch:quasi_pbs} it is shown that Eq.(\\ref{eqn:eq_def_gk}) reproduces the correct universal one-loop and two-loop coefficients of the perturbative $\\beta$ function of pure $YM$. Indeed, substituting in Eq.(\\ref{eqn:eq_def_gk}) we get:\n\\begin{align}\\label{eqn:matching_beta_pert}\n\\frac{\\partial g}{\\partial \\log p}\n&=\\frac{-\\beta_0 g^3+\\frac{2\\gamma_0 }{(4\\pi)^2}g^5}{1-\\frac{4}{(4\\pi)^2}g^2}+\\cdots \\nonumber \\\\\n&=\\bigl(-\\beta_0 g^3+\\frac{2\\gamma_0}{(4\\pi)^2}g^5\\bigr)\\bigl(1+\\frac{4}{(4\\pi)^2}g^2\\bigr)+\\cdots \\nonumber\\\\\n&=-\\beta_0 g^3+\\frac{2\\gamma_0}{(4\\pi)^2}  g^5-\\frac{4\\beta_0}{(4\\pi)^2}g^5+\\cdots \\nonumber\\\\\n&=-\\beta_0 g^3+\\frac{1}{(4\\pi)^4}\\frac{10}{3}g^5 - \\frac{44}{3}\\frac{1}{(4\\pi)^4}g^5+\\cdots\\nonumber\\\\\n&=-\\beta_0 g^3-\\beta_1 g^5+\\cdots\n\\end{align}\nwhere:\n\\begin{align}\n&\\beta_0=\\frac{1}{(4\\pi)^2}\\frac{11}{3}\\\\\n&\\beta_1=\\frac{1}{(4\\pi)^4}\\frac{34}{3}\n\\end{align}\nThese are the correct one- and two-loop coefficients that arise in perturbation theory of pure $YM$ for the 't Hooft coupling.\nTherefore, the renormalization-group improved universal asymptotic behavior of $g_k$ is:\n\\begin{equation}\\label{eqn:gk_as_behav}\ng^2_k\\sim\\frac{1}{\\beta_0\\log\\frac{k}{c}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)\n+O\\biggl(\\frac{1}{\\log^2\\frac{k}{c}}\\biggr)\n\\end{equation}\nand the renormalization group improved universal asymptotic behavior of $Z_k^{-1}$ is:\n\\begin{equation}\\label{eqn:zk_as_behav}\nZ_k^{-1}\\sim (g^2_k)^{\\frac{\\gamma_0}{\\beta_0}}\\sim\n\\Biggl(\\frac{1}{\\beta_0\\log\\frac{k}{c}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{k}{c}}{\\log\\frac{k}{c}}\\biggr)+O\\biggl(\\frac{1}{\\log^2\\frac{k}{c}} \\biggr)  \\Biggr)^{\\frac{\\gamma_0}{\\beta_0}}\n\\end{equation}\nIn this section we find the asymptotics of the $ASD$ propagator in Eqs.(\\ref{eqn:intro_formula_L2}) and of the large-$L$ propagator in Eq.(\\ref{eqn:formula}) at the order of the leading and of the next-to-leading logarithms following the technique employed in \\cite{boch:glueball_prop} at the order of the leading logarithm. \\par\nTo find the asymptotics of the glueball propagator for large $L$ in Eq.(\\ref{eqn:formula}) we follow the strategy explained in sect.(1.4) for the\n$ASD$ correlator. Firstly, we highlight the physical terms contained in Eq.(\\ref{eqn:formula}) neglecting the non-physical contact terms. Secondly, we extract the asymptotic behavior writing the sum in Eq.(\\ref{eqn:formula}) as an integral \\cite{boch:glueball_prop}. Finally, we use the leading and next-to-leading expression for $Z_k^{-1}$ in Eq.(\\ref{eqn:zk_as_behav}) to compare Eq.(\\ref{eqn:formula}) with $RG$-improved perturbation theory.\n\nWe write Eq.(\\ref{eqn:formula}) as \\cite{boch:glueball_prop}\\cite{boch:crit_points}:\n\\begin{align}\n&\\sum_{k=1}^{\\infty}\n\\frac{k^{2(L-1)}Z_k^{-L}\\Lambda_{\\overline{W}}^2 \\Lambda_{\\overline{W}}^{4(L-1)}}{p^2+k\\Lambda_{\\overline{W}}^2}\\nonumber\\\\\n&=\\sum_{k=1}^{\\infty}\n\\frac{((k\\Lambda_{\\overline{W}}^2+p^2)(k\\Lambda_{\\overline{W}}^2-p^2)+p^4)^{L-1} Z_k^{-L}\\Lambda_{\\overline{W}}^2}{p^2+k\\Lambda_{\\overline{W}}^2}\\nonumber\\\\\n&=p^{4L-4}\\sum_{k=1}^{\\infty}\\frac{Z_k^{-L}\\Lambda_{\\overline{W}}^2 }{p^2+k\\Lambda_{\\overline{W}}^2} \\nonumber\\\\\n&+\\sum_{k=1}^{\\infty}\\sum_{m=1}^{L-1}\\binom{L-1}{m}p^{4(L-1-m)}(k\\Lambda_{\\overline{W}}^2+p^2)^{m-1}(k\\Lambda_{\\overline{W}}^2-p^2)^m  Z_k^{-L}\\Lambda_{\\overline{W}}^2\\nonumber\\\\ \n&\\sim p^{4L-4}\\sum_{k=1}^{\\infty}\\frac{ Z_k^{-L}\\Lambda_{\\overline{W}}^2 }{p^2+k\\Lambda_{\\overline{W}}^2}+\\dots\n\\end{align}\nwhere the dots stand for contact terms.\n\nAs in sect.(1.4) we use the Euler-McLaurin formula to approximate the sum to an integral \\cite{boch:glueball_prop}\\cite{boch:crit_points}:\n\\begin{equation}\n\\sum_{k=k_1}^{\\infty}G_k(p)=\n\\int_{k_1}^{\\infty}G_k(p)dk - \\sum_{j=1}^{\\infty}\\frac{B_j}{j!} \\left[\\partial_k^{j-1}G_k(p)\\right]_{k=k_1}\n\\end{equation}\nIn our case the terms proportional to the Bernoulli numbers involve negative powers of $p$ and they are therefore subleading with respect to the first term, hence we ignore them. \n\nWe obtain: \n\\begin{equation}\\label{eqn:int_fond}\n\\sum_{k=1}^{\\infty}\\frac{Z_k^{-L}\\Lambda_{\\overline{W}}^2 }{p^2+k\\Lambda_{\\overline{W}}^2}\\sim\n\\int_{1}^\\infty\\frac{Z_k^{-L}}{k+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}dk\n\\end{equation}\nIn order to compare Eq.(\\ref{eqn:int_fond}) to the $RG$-improved perturbation theory, we substitute for $Z_k^{-1}$ its leading and next-to-leading logarithmic behavior given by Eq.(\\ref{eqn:zk_as_behav}).\nWe define:\n\\begin{equation}\n\\gamma'=\\frac{\\gamma_0}{\\beta_0}L\n\\end{equation}\nand:\n\\begin{equation}\n\\nu=\\frac{p^2}{\\Lambda_{\\overline{W}}^2}\n\\end{equation}\nThe integral that determines the leading asymptotic behavior is:\n\\begin{equation}\\label{eqn:int_fond2}\nI_c^{1}(\\nu)=\\int_{1}^\\infty\\biggl(\\frac{1}{\\beta_0\\log(\\frac{k}{c})}\\biggr)^{\\gamma'}\\frac{dk}{k+\\nu}\n\\end{equation}\nThe next-to-leading logarithmic behavior is determined by:\n\\begin{equation}\\label{eqn:int_fond3}\nI^{2}_c(\\nu)=\\int_{1}^\\infty\\left(\\frac{1}{\\beta_0\\log(\\frac{k}{c})}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k}{c})}{\\log(\\frac{k}{c})}\\right)\\right)^{\\gamma'}\\frac{dk}{k+\\nu}\n\\end{equation}\n$\\gamma'=2$ for the $ASD$ correlator and $\\gamma'=\\frac{\\gamma_0}{\\beta_0}L$ for the large-$L$ correlator.\nWe show in the following that the leading and next-to-leading behavior of $I^{2}_c(\\nu)$ is: \n\\begin{align}\n&I^{2}_c(\\nu)\\sim\\frac{1}{\\gamma_0 L-\\beta_0}\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambdawb^2}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}{\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggr)\\Biggr]^{\\frac{\\gamma_0}{\\beta_0}L-1} \n\\end{align}\nTherefore, the asymptotic behavior of the correlator of the $TFT$ for large-$L$ is:\n\\begin{equation}\n\\braket{\\mathcal{O}_{2L}(p)\\mathcal{O}_{2L}(-p)}_{conn}\\sim p^{4L-4}\\frac{1}{\\gamma_0 L-\\beta_0}\\bigl(g^2(p)\\bigr)^{\\frac{\\gamma_0}{\\beta_0}L-1}\n\\end{equation}\nIt agrees with the naive $RG$ estimate Eq.(\\ref{eqn:naive_rg}).\n\n\\subsection{Asymptotic series to the order of the leading logarithm}\n\nWe now perform an explicit expansion in series of $I_c^{1}(\\nu)$.\nFirstly, we change variables from $k$ to $k+\\nu$:\n\\begin{equation}\nI_c^{1}(\\nu)=\\int_{1+\\nu}^\\infty\\biggl(\\frac{1}{\\beta_0\\log(\\frac{k-\\nu}{c})}\\biggr)^{\\gamma'}\\frac{dk}{k}\n\\end{equation}\nWe have that: \n\\begin{equation}\n[\\log(\\frac{k'-\\nu}{c})]^{-\\gamma'}=\n[\\log(\\frac{k'}{c})]^{-\\gamma'}\\biggl[1+\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\\biggr]^{-\\gamma'}\n\\end{equation}\nIt is easy to see that if $c<1$:\n\\begin{equation}\n\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}<1\n\\end{equation}\nWe define:\n\\begin{equation}\n\\epsilon=\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\n\\end{equation}\nand we exploit the binomial formula \\cite{BIN}:\n\\begin{align}\\label{eqn:espansione_bin}\n(1+\\epsilon)^{-\\gamma'}&=\\sum_{r=0}^{\\infty}\\binom{\\gamma'+r-1}{r}(-1)^r\\epsilon^r\n\\end{align}\nto obtain a series expansion.\nWe proceed order by order in $\\epsilon$.\nAt the order of $\\epsilon^1$ the only contribution is:\n\\begin{equation}\n-\\gamma'\\epsilon\n\\end{equation}\n$\\epsilon$ can be further expanded in powers of $\\eta=\\frac{\\nu}{k'}$, since in the integration domain $\\eta<1$:\n\\begin{align}\\label{eqn:espansione_log}\n\\log(1-\\eta)&=\\sum_{m=1}^{\\infty}\\frac{(-1)^{2m+1}}{m}\\eta^m\n\\end{align}\nUp to the order of $\\eta^1$ this expansion reads: \n\\begin{equation}\n-\\gamma'\\epsilon\\sim\\gamma'\\frac{\\nu}{k'\\log(\\frac{k'}{c})} \n\\end{equation} \nSubstituting in $I_c^{1}(\\nu)$ we get:\n\\begin{align}\\label{eqn: int1}\n&\\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n\\biggl[1+\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\\biggr]^{-\\gamma'}dk'\\nonumber\\\\\n&\\sim \\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'} \n\\biggl[1+\\gamma'\\frac{\\nu}{k'\\log(\\frac{k'}{c})}\\biggr]dk'\\nonumber\\\\\n&=\\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}dk'+\n\\gamma' \\nu\\int_{1+\\nu}^\\infty\\frac{1}{k'^2}\\beta_0^{-\\gamma'}[\\log(\\frac{k'}{c})]^{-\\gamma'-1}dk' \n\\end{align}\nFrom the first integral it follows the leading asymptotic behavior \\cite{boch:glueball_prop}:\n\\begin{equation}\\label{eqn:sol_int_leading}\n\\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}dk'=\n\\frac{1}{\\gamma'-1} \\beta_0^{-\\gamma'}\\left[\\log\\left(\\frac{1+\\nu}{c}\\right)\\right]^{-\\gamma'+1}\n\\end{equation}\nSince for large $\\nu$:\n\\begin{equation}\n\\biggl(\\log\\left(\\frac{1+\\nu}{c}\\right)\\biggr)^{-1}\\sim\n\\bigl(\\log\\nu\\bigr)^{-1} \n\\end{equation}\nit follows the leading asymptotic behavior of Eq.(\\ref{eqn:formula}) \\cite{boch:glueball_prop}:\n\\begin{align}\n\\braket{\\mathcal{O}_{2L}(p)\\mathcal{O}_{2L}(-p)}_{conn}\\sim\\frac{p^{4L-4}}{\\gamma_0 L-\\beta_0}\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambdawb^2}}\\Biggr]^{\\frac{\\gamma_0}{\\beta_0}L-1}\n\\end{align} \nPerforming the same steps for the $ASD$ correlator, i.e. for $L=2$ and $\\gamma'=2$, we get:\n\\begin{equation}\n\\frac{1}{\\pi^2}\\int_1^{\\infty}\\frac{\\bigl(\\beta_0\\log\\frac{k}{c}\\bigr)^{-2}}{k+\\nu}\\sim\n\\frac{1}{\\pi^2\\beta_0}\\Biggl(\\beta_0\\log\\biggl(\\frac{1+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}{c}\\biggr)\\Biggr)^{-1}\\sim\n\\frac{1}{\\pi^2\\beta_0}\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\n\\end{equation}\nthat agrees with the leading logarithm of the asymptotic behavior in Eq.(\\ref{eqn:corr_asd}). \\par\nWe now compute the second term in the last line of Eq.(\\ref{eqn: int1}), that is the first subleading term. \nWe write it as:\n\\begin{equation}\\label{eqn:int_ord1}\n\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\int_{\\frac{1+\\nu}{c}}^\\infty\\frac{1}{k^2}[\\log(k)]^{-\\gamma'-1}dk \n\\end{equation}\nand we integrate by parts:\n\\begin{align}\n&\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\int_{\\frac{1+\\nu}{c}}^{\\infty}\\frac{1}{k^2}[\\log(k)]^{-\\gamma'-1}dk \\nonumber\\\\\n&=\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\biggl[-\\frac{[\\log(k)]^{-\\gamma'-1}}{k}\\bigg|_{\\frac{1+\\nu}{c}}^{\\infty}\n-(\\gamma'+1)\\int_{\\frac{1+\\nu}{c}}^{\\infty}\\frac{dk}{k^2}[\\log(k)]^{-\\gamma'-2}\\biggr]\\nonumber\\\\\n&=\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\biggl[\\frac{c}{1+\\nu}[\\log(\\frac{1+\\nu}{c})]^{-\\gamma'-1}\n-(\\gamma'+1)\\int_{\\frac{1+\\nu}{c}}^{\\infty}\\frac{dk}{k^2}[\\log(k)]^{-\\gamma'-2}\\biggr]\n\\end{align}\nWe notice that the second term in the last line has the same structure as the original integral but with a more negative power of the logarithm.\nThis implies that it is a less relevant term.\nFurthermore, since performing integration by parts repeatedly we always obtain integrals with the same structure, we can derive a possibly asymptotic series expansion for Eq.(\\ref{eqn:int_ord1}): \n\\begin{align}\n&\\frac{\\gamma'\\nu \\beta_0^{-\\gamma'}}{c}\\int_{\\frac{1+\\nu}{c}}^{\\infty}\\frac{dk}{k^2}[\\log(k)]^{-\\gamma'-1}\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\frac{\\nu}{1+\\nu}\\sum_{s=0}^{\\infty}(-1)^s\\left(\\prod_{t=0}^{s}(\\gamma'+t)\\right)[\\log(\\frac{1+\\nu}{c})]^{-\\gamma'-1-s}\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\frac{p^2}{p^2+\\Lambda_{\\overline{W}}^2}\\sum_{s=0}^{\\infty}(-1)^s\\left(\\prod_{t=0}^{s}(\\gamma'+t)\\right)\n\\biggl[\\log\\biggl(\\frac{1+\\frac{p^2}{\\Lambda_{\\overline{W}}^2}}{c}\\biggr)\\biggr]^{-\\gamma'-1-s}\n\\end{align}\n\nNow that we have understood the technique, we derive a complete expression taking into account all the terms coming from the expansion of the logarithm in Eq.(\\ref{eqn:espansione_log}),\nsimply substituting it in $I_c^{1}(\\nu)$: \n\\begin{align}\n&\\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n\\biggl[1-\\gamma'\\frac{\\sum_{m=1}^{\\infty}\\frac{(-1)^{2m+1}\\nu^m}{m k^m}}{\\log(\\frac{k'}{c})}\\biggr]\\nonumber\\\\\n&=\\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\\nonumber\\\\\n&-\\gamma'\\beta_0^{-\\gamma'}\n\\sum_{m=1}^{\\infty}\\frac{(-1)^{2m+1}\\nu^m}{m}\\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'^{m+1}}\\left[\\log(\\frac{k'}{c})\\right]^{-\\gamma'-1}\n\\end{align}\nFocusing on the second term:\n\\begin{align}\\label{eqn:ord1_eps}\n&\\gamma'\\beta_0^{-\\gamma'}\n\\sum_{m=1}^{\\infty}\\frac{(-1)^{2m+1}\\nu^m}{m}\\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'^{m+1}}\\left[\\log(\\frac{k'}{c})\\right]^{-\\gamma'-1}\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\gamma'\\sum_{m=1}^{\\infty}\\frac{(-1)\\nu^m}{m c^m}\\left[-\\frac{\\left[\\log(k)\\right]^{-\\gamma'-1}}{m k^{m}}\n\\bigg|_{\\frac{1+\\nu}{c}}^{\\infty}-\n(\\gamma'+1)\\int_{\\frac{1+\\nu}{c}}^{\\infty}dk\\frac{\\left[\\log(k)\\right]^{-\\gamma'-2}}{mk^{m+1}}\\right]\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\sum_{m=1}^{\\infty}\\frac{(-1)\\nu^m}{m c^m}\\left[\\sum_{s=0}^{\\infty}\\frac{(-1)^{s}}{m^{s+1}}\n\\frac{c^m\\prod_{t=0}^{s}(\\gamma'+t)}{(1+\\nu)^m}\\left[\\log\\left(\\frac{1+\\nu}{c}\\right)\\right]^{-\\gamma'-1-s}\\right]\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\sum_{m=1}^{\\infty}\\sum_{s=0}^{\\infty}(-1)^{s+1}\\left(\\frac{\\nu}{1+\\nu}\\right)^m\\frac{\\prod_{t=0}^{s}(\\gamma'+t)}{m^{s+2}}\n\\left[\\log\\left(\\frac{1+\\nu}{c}\\right)\\right]^{-\\gamma'-s-1}\t\n\\end{align}\nTherefore, at the first order in $\\epsilon$ we get:\n\\begin{align}\\label{eqn:int_lead_temp}\n&\\int_{1+\\nu}^\\infty dk'\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n\\biggl[1+\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\\biggr]^{-\\gamma'}\\nonumber\\\\\n&\\sim \\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'} \\nonumber\\\\\n&+\\beta_0^{-\\gamma'}\\sum_{m=1}^{\\infty}\\sum_{s=0}^{\\infty}(-1)^{s}\\left(\\frac{\\nu}{1+\\nu}\\right)^m\\frac{\\prod_{t=0}^{s}(\\gamma'+t)}{m^{s+2}}\n\\left[\\log\\left(\\frac{1+\\nu}{c}\\right)\\right]^{-\\gamma'-s-1}\t \n\\end{align}\nWe find the subleading behavior keeping only the terms with $s=0$ in Eq.(\\ref{eqn:int_lead_temp}).\nWe obtain in the large $\\nu$ limit: \n \\begin{align}\\label{eqn:correzione_serie_as}\n&\\int_{1+\\nu}^\\infty dk'\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n\\biggl[1+\\frac{\\log(1-\\frac{\\nu}{k'})}{\\log(\\frac{k'}{c})}\\biggr]^{-\\gamma'}\\nonumber\\\\\n& \\sim \\int_{1+\\nu}^{\\infty}\\frac{dk'}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-\\gamma'}\n+\\gamma'\\beta_0^{-\\gamma'}\\left[\\log \\nu \\right]^{-\\gamma'-1}\\sum_{m=0}^{\\infty}\\frac{1}{m^2}\\nonumber\\\\\n& \\sim \\frac{1}{\\gamma'-1}\\beta_0^{-\\gamma'}\\left[\\log\\nu\\right]^{-\\gamma'+1}\n+\\gamma'\\beta_0^{-\\gamma'}\\zeta(2)[\\log \\nu]^{-\\gamma'-1}\n\\end{align} \nIt is interesting to notice that the transcendental function $\\zeta(2)=\\frac{\\pi^2}{6}$ occurs, as it often does in Feynman-graph computations.\n\n\n\\subsection{Asymptotic series to the order of the next-to-leading logarithm}\n\nWe now perform a series expansion of $I_c^{2}(\\nu)$:\n\\begin{align}\nI_c^{2}(\\nu)&=\\int_1^{\\infty}\\beta_0^{-\\gamma'}\\left(\\frac{1}{\\log(\\frac{k}{c})}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k}{c})}{\\log(\\frac{k}{c})}\\right)\\right)^{\\gamma'}\\frac{dk}{k+\\nu}\\nonumber\\\\\n&=\\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left(\\frac{1}{\\log(\\frac{k-\\nu}{c})}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k-\\nu}{c})}{\\log(\\frac{k-\\nu}{c})}\\right)\\right)^{\\gamma'}\\frac{dk}{k}\\nonumber\\\\\n&\\sim \\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-\\gamma'}\n\\left(1-\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k-\\nu}{c})}{\\log(\\frac{k-\\nu}{c})}\\right)\\frac{dk}{k}\\nonumber\\\\\n&\\sim  \\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-\\gamma'}\\frac{dk}{k}+\\nonumber\\\\\n&-\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-\\gamma'-1}\\log\\log(\\frac{k-\\nu}{c})\n\\frac{dk}{k}\n\\end{align}\nThe first integral has been evaluated in the previous section and the second term is the new contribution. We evaluate it at the leading order by changing variables and integrating by parts:\n\\begin{align}\\label{eqn:int_next-to-leading}\n&\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-\\gamma'-1}\\log\\log(\\frac{k-\\nu}{c})\n\\frac{dk}{k}\\nonumber\\\\\n&\\sim  \\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k}{c})\\right]^{-\\gamma'-1}\\log\\log(\\frac{k}{c})\n\\frac{dk}{k}\\nonumber\\\\\n&=\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\int_{\\log\\frac{1+\\nu}{c}}^{\\infty}t^{-\\gamma'-1}\\log(t)dt\\nonumber\\\\\n&=\\gamma'\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\left[\\frac{1}{\\gamma'}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-\\gamma'}\\log\\log(\\frac{1+\\nu}{c})+\n\\frac{1}{\\gamma'^2}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-\\gamma'}\\right]\n\\end{align}\nThe second term in brackets is subleading with respect to the first one. \nPutting together  Eq.(\\ref{eqn:int_next-to-leading}) and Eq.(\\ref{eqn:sol_int_leading}) we get for $I_c^{2}(\\nu)$ :\n\\begin{align}\\label{eqn:esp_ntl}\n&\\beta_0^{-\\gamma'}\\int_1^{\\infty}\\left(\\frac{1}{\\log(\\frac{k}{c})}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log(\\frac{k}{c})}{\\log(\\frac{k}{c})}\\right)\\right)^{\\gamma'}\\frac{dk}{k+\\nu}\\nonumber\\\\\n& \\sim \\frac{1}{\\gamma'-1}\\beta_0^{-\\gamma'}\\left(\\log\\frac{1+\\nu}{c}\\right)^{-\\gamma'+1}-\\frac{\\beta_1}{\\beta_0^2}\\beta_0^{-\\gamma'}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-\\gamma'}\\log\\log(\\frac{1+\\nu}{c})\\nonumber\\\\\n&=\\frac{\\beta_0^{-\\gamma'}}{\\gamma'-1}\\biggl(\\log\\frac{1+\\nu}{c}\\biggr)^{-\\gamma'+1}\\left[1-\\frac{\\beta_1(\\gamma'-1)}{\\beta_0^2}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-1}\\log\\log(\\frac{1+\\nu}{c})\\right]\\nonumber\\\\\n&\\sim \\frac{1}{\\beta_0(\\gamma'-1)}\\left(\\beta_0\\log\\frac{1+\\nu}{c}\\right)^{-\\gamma'+1}\\left[1-\\frac{\\beta_1}{\\beta_0^2}\\left(\\log(\\frac{1+\\nu}{c})\\right)^{-1}\\log\\log(\\frac{1+\\nu}{c})\\right]^{\\gamma'-1}\\nonumber\\\\\n&\\sim \\frac{1}{\\beta_0(\\gamma'-1)}(g^2(p))^{\\gamma'-1}+ O\\biggl(\\Bigl(\\frac{1}{\\log\\frac{p^2}{\\Lambda_{\\overline{W}}}}\\Bigr)^{\\gamma'}\\biggr)\n\\end{align}\nThis result agrees with the $RGI$ perturbative estimate in Eq.(\\ref{eqn:naive_rg}).\nRepeating the same steps for the $ASD$ correlator we get: \n\\begin{align}\n&\\frac{1}{\\pi^2}\\int_{1+\\nu}^\\infty\\frac{1}{k'}[\\beta_0\\log(\\frac{k'}{c})]^{-2}dk'\n-\\frac{\\beta_1}{\\pi^2\\beta_0^4}\\int_{1+\\nu}^{\\infty}\\left[\\log(\\frac{k-\\nu}{c})\\right]^{-3}\\log\\log(\\frac{k-\\nu}{c})\n\\frac{dk}{k}\\nonumber\\\\\n& \\sim \\frac{1}{\\pi^2\\beta_0} g^2(p)+O\\biggl(\\frac{1}{\\log^2\\frac{p^2}{\\Lambda_{\\overline{W}}}}\\biggr)\n\\end{align}\nAgain this result agrees with the universal behavior of the $RG$-improved perturbation theory in Eq.(\\ref{eqn:rg_improved_scalar_2l}).\n\n\\newpage\n\\thispagestyle{empty}\n\n\\subsection{Link with the Lerch transcendent and the polylogarithmic function}\n\nWe may obtain the asymptotic behavior by a different method as an independent check, relating the relevant integrals\nto special functions and employing the known asymptotic behavior of the special functions. \\par\n\nWe briefly recall the definition of the Lerch Zeta function \\cite{handbook, wikilerch}:\n\\begin{equation}\nL(\\lambda,s,a)=\\sum_{n=0}^{\\infty}\\frac{e^{2\\pi i\\lambda n}}{(n+a)^s}\n\\end{equation}\nSetting $z=e^{2\\pi i \\lambda}$, we obtain the Lerch transcendent \\cite{handbook, wikilerch}:\n\\begin{equation}\n\\Phi(z,s,a)=\\sum_{n=0}^{\\infty}\\frac{z^n}{(n+a)^s}\n\\end{equation}\nThe Lerch transcendent admits the integral representation:\n\\begin{equation}\\label{eqn:lerch_int_repr}\n\\Phi(z,s,a)=\\frac{1}{\\Gamma(s)}\\int_0^{\\infty}\\frac{t^{s-1}e^{-at}}{1-ze^{-t}}dt\n\\end{equation}\nwhich is valid for $\\Re (a)>0 \\,\\wedge \\, \\Re (s)>0 \\,\\wedge \\,|z|<1$ or $\\Re (a)>0 \\,\\wedge \\, \\Re (s)>1 \\,\\wedge \\,|z|=1$.\nThe Lerch transcendent can be analytically continued to the region \\cite{analytic}:\n\\begin{equation}\n\\mathcal{M}=\\{(z,s,a)\\in (\\mathbb{C}\\setminus \\{0\\})\\times\\mathbb{C}\\times (\\mathbb{C}\\setminus\\mathbb{Z})\\}\n\\end{equation} \nMoreover, we exploit the following recursive formula:\n\\begin{equation}\\label{eqn:lerch_recursive}\n\\Phi(z,s,a)=z^l \\Phi(z,s,a+l)+\\sum_{k=0}^{l-1}\\frac{z^k}{(a+k)^s}\n\\end{equation}\nFinally, we use the relationship between the Lerch transcendent and the polylogarithmic function \\cite{handbook, wikipoly}:\n\\begin{equation}\n\\mathrm{Li}_s(z)=z\\Phi(z,s,1)\n\\end{equation}\nwhere the polylogarithmic function is defined by:\n\\begin{equation}\n\\mathrm{Li}_s(z)=\\sum_{k=1}^{\\infty}\\frac{z^k}{k^s}\n\\end{equation}\n\n\\subsection{Asymptotic behavior and polylogarithmic function}\n\nWe start performing the change of variables $t=\\log\\frac{k}{c}$ in the integral in Eq.(\\ref{eqn:int_fond2}):\n\\begin{equation}\nI^1_c(\\nu)=\\int_{1}^\\infty\\frac{[\\beta_0\\log(\\frac{k}{c})]^{-\\gamma'}}{k+\\nu}dk=\nc\\beta_0^{-\\gamma'}\\int_{\\log\\frac{1}{c}}^{\\infty}\\frac{t^{-\\gamma'}}{c+\\nu e^{-t}}dt\n\\end{equation}\nSetting $c=e^{-\\epsilon}$ in the limit $\\epsilon\\rightarrow 0$ we get the upper bound:\n\\begin{equation}\nI^1_c(\\nu)=\\beta_0^{-\\gamma'}\\int_{\\log\\frac{1}{c}}^{\\infty}\\frac{t^{-\\gamma'}}{1+\\frac{\\nu}{c} e^{-t}}dt\\leq\n\\beta_0^{-\\gamma'}\\int_{\\epsilon}^{\\infty}\\frac{t^{-\\gamma'}}{1+\\frac{\\nu}{e^{-\\epsilon}} e^{-t}}dt = I^1_{1-\\epsilon}(\\nu)\n\\end{equation}\nbut the upper bound is in fact asymptotic since varying $c$  is equivalent to a change of scheme.\nTherefore, we take the limit $\\epsilon\\rightarrow 0$ in order to express $I^1_1$ in terms of the integral representation of the Lerch transcendent in Eq.(\\ref{eqn:lerch_int_repr}). We get:\n\\begin{equation}\nI^1_1(\\nu)=\\beta_0^{-\\gamma'}\\Gamma(-\\gamma'+1)\\Phi(-\\nu,-\\gamma'+1,0)\n\\end{equation}\nWe now exploit the relation in Eq.(\\ref{eqn:lerch_recursive}) with\n$n=1$, $a=0$, $z=-\\nu$ and $s=-\\gamma'+1$: \n\\begin{equation}\n\\Phi(-\\nu,-\\gamma'+1,0)=z \\Phi(-\\nu,-\\gamma'+1,1)\n\\end{equation}\nFinally, we find the relation with the polylogarithmic function:\n\\begin{equation}\nI^1_1(\\nu)=\\beta_0^{-\\gamma'}\\Gamma(-\\gamma'+1)\\mathrm{Li}_{-\\gamma'+1}(-\\nu)\n\\end{equation}\nNow we use the following asymptotic expansion of $\\mathrm{Li}_s$ \\cite{wikipoly}:\n\\begin{equation}\n\\mathrm{Li}_s(z)=\\sum_{j=0}^{\\infty}(-1)^j(1-2^{1-2j})(2\\pi)^{2j}\\frac{B_{2j}}{(2j)!}\\frac{[\\log(-z)]^{s-2j}}{\\Gamma(s+1-2k)}\n\\end{equation}\nto find an asymptotic expansion for $I^1_1(\\nu)$:\n\\begin{equation}\\label{eqn:esp_polylog}\nI^1_1(\\nu)=\\beta_0^{-\\gamma'}\\Gamma(-\\gamma'+1)\\sum_{j=0}^{\\infty}(-1)^j(1-2^{1-2j})(2\\pi)^{2j}\\frac{B_{2j}}{(2j)!}\n\\frac{[\\log\\nu]^{-\\gamma'+1-2j}}{\\Gamma(-\\gamma'+2-2j)}\n\\end{equation}\nWe get the leading behavior of $I_1^1(\\nu)$ from the $j=0$ term in Eq.(\\ref{eqn:esp_polylog}):\n\\begin{equation}\\label{eqn:leading_polylog}\nI_1^1(\\nu)\\sim -\\beta_0^{-\\gamma'}\\Gamma(-\\gamma'+1)\\frac{[\\log \\nu]^{-\\gamma'+1}}{\\Gamma(-\\gamma'+2)}=\n\\frac{[\\beta_0\\log \\nu]^{-\\gamma'+1}}{\\beta_0(\\gamma'-1)}\n\\end{equation}\nKeeping also the $j=1$ term we obtain: \n\\begin{equation}\\label{eqn:subleading_polylog}\nI_1^1(\\nu)\\sim \\frac{[\\beta_0\\log \\nu]^{-\\gamma'+1}}{\\beta_0(\\gamma'-1)}+\n\\gamma'\\beta_0^{-\\gamma'}\\frac{\\pi^2}{6}[\\log \\nu]^{-\\gamma'-1} \n\\end{equation}\nin perfect agreement with Eq.(\\ref{eqn:correzione_serie_as}) since $\\zeta(2)=\\frac{\\pi^2}{6}$.\nReinserting the momentum $p$ in the definition of $\\nu$ the asymptotic result is:\n\\begin{equation}\nI^1_c\\left(\\frac{p^2}{\\Lambda_{\\overline{W}}^2}\\right)\\sim \\frac{[\\beta_0\\log(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})]^{-\\frac{\\gamma_0}{\\beta_0}L+1}}{\\gamma_0 L-\\beta_0}+\n\\gamma_0 L\\frac{\\pi^2}{6}[\\beta_0\\log(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})]^{-\\frac{\\gamma_0}{\\beta_0}L-1}  \n\\end{equation}\nUsing the same technique we find the next-to-leading logarithmic behavior of $I^2_c$. \nIndeed, also in this case we obtain an upper bound putting $c=e^{-\\epsilon}$ and taking the limit $\\epsilon\\rightarrow 0$:\n\\begin{align}\nI^{2}_c(\\nu)&=c\\beta_0^{-\\gamma'}\\int_{\\log\\frac{1}{c}}^\\infty\\left(\\frac{1}{t}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log t}{t}\\right)\\right)^{\\gamma'}\\frac{dt}{c+\\nu e^{-t}}\\\\\n\\leq & \\beta_0^{-\\gamma'}\\int_{\\epsilon}^\\infty\\left(\\frac{1}{t}\\left(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log t}{t}\\right)\\right)^{\\gamma'}\\frac{dt}{1+\\frac{\\nu}{e^{-\\epsilon}} e^{-t}}=I^{2}_{1-\\epsilon}(\\nu)\n\\end{align}\nbut the upper bound is in fact asymptotic since varying $c$  is equivalent to a change of scheme.\nWe now expand $I^{2}_{1-\\epsilon}(\\nu)$:\n\\begin{equation}\nI^{2}_{1-\\epsilon}(\\nu)\\sim \\beta_0^{-\\gamma'}\\int_{\\epsilon}^{\\infty}\\frac{1}{t^{\\gamma'}}\\left(1-\\frac{\\beta_1\\gamma'}{\\beta_0^2}\\frac{\\log t}{t}\\right)\\frac{dt}{1+\\frac{\\nu}{e^{-\\epsilon}}e^{-t} }\n\\end{equation}\nThe first term is equal to $I^1_{1-\\epsilon}(\\nu)$, while the second one is the new contribution.\nThis new term can be linked again to the polylogarithmic function using the relation:\n\\begin{equation}\nt^{-\\gamma'-1}\\log t=-\\frac{\\partial}{\\partial \\alpha} t^{-\\alpha}\\biggl|_{\\alpha=\\gamma'+1}\n\\end{equation} \nWe find :\n\\begin{equation}\nI_{1-\\epsilon}^{2}(\\nu,-\\gamma')\\sim I_{1-\\epsilon}^1(\\nu,-\\gamma')+\\frac{\\beta_1\\gamma'}{\\beta_0^2}\\frac{\\partial}{\\partial\\alpha}I^1_{1-\\epsilon}(\\nu,-\\alpha)\\biggl|_{\\alpha=\\gamma'+1}\n\\end{equation}\nWe take the limit $\\epsilon\\rightarrow 0$ and we perform the derivative in the asymptotic expression of $I_1^1(\\nu,-\\alpha)$ in Eq.(\\ref{eqn:esp_polylog}). Keeping only the leading contribution we obtain:\n\\begin{align}\n\\frac{\\partial}{\\partial\\alpha}I_1^1(\\nu,-\\alpha)\\biggl|_{\\alpha=\\gamma'+1}&=\n\\beta_0^{-\\gamma'}\\frac{\\Gamma(-\\gamma')}{\\Gamma(-\\gamma'+1)}(\\log \\nu)^{-\\gamma'}\\log\\log \\nu=\\\\\n&=-\\frac{\\beta_0^{-\\gamma'}}{\\gamma'}(\\log \\nu)^{-\\gamma'}\\log\\log \\nu\n\\end{align}\nThus the asymptotic behavior to the next-to-leading logarithmic order is:\n\\begin{align}\nI^{2}_c(\\nu)&\\sim  \\frac{[\\beta_0\\log(\\frac{p^2}{\\Lambda_{\\overline{W}}^2})]^{-\\frac{\\gamma_0}{\\beta_0}L+1}}\n{\\gamma_0 L-\\beta_0}\n-\\frac{\\beta_1}{\\beta_0^2}(\\beta_0\\log\\frac{p^2}{\\Lambda_{\\overline{W}}^2})^{-\\frac{\\gamma_0}{\\beta_0}L}\\log\\log\\frac{p^2}{\\Lambda_{\\overline{W}}^2}\\nonumber\\\\\n& \\sim \\frac{1}{\\gamma_0 L-\\beta_0}\\Biggl[\\frac{1}{\\beta_0\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggl(1-\\frac{\\beta_1}{\\beta_0^2}\\frac{\\log\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}{\\log\\frac{p^2}{\\Lambda^2_{\\overline{W}}}}\\biggr)\\Biggr]^{\\frac{\\gamma_0}{\\beta_0}L-1}\n\\end{align}\nthat agrees perfectly with the $RG$ estimate Eq.(\\ref{eqn:esp_ntl}).\n\n\\thispagestyle{empty}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe structure of nuclei involving $\\alpha$ clusters continues to be a subject of very active studies (see \\cite{Freer:2017gip} \nfor a recent review, \\cite{brink2008history} for a historical perspective, \\cite{Arriola:2014lxa} for a discussion of clustering mass formulas and form factors as \nmanifestations of the geometric structure,\nand \\cite{brink1965alpha,freer2007clustered,ikeda2010clusters,Okolowicz:2012kv,beck2012clusters,Beck:2014fja} for additional \ninformation), exploring the ideas dating back to Gamow's original clusterization proposal~\\cite{gamow1931constitution} \nwith modern theoretical~\\cite{Funaki:2006gt,Chernykh:2007zz,KanadaEn'yo:2006ze} and\ncomputational~\\cite{PhysRevLett.109.052501,Barrett:2013nh,Epelbaum:2012qn,Pieper:2002ne,Wiringa:2013ala,Lonardoni:2017egu} methods, as well as with \nanticipated new experimental prospects~\\cite{Yamaguchi:2012sz,Zarubin,Fritsch:2017rxc,Guo:2017tco}. \n\nA few years ago a possible approach of investigating $\\alpha$ clustering in light nuclei via studies of ultra-relativistic nuclear \ncollisions was proposed in Ref.~\\cite{Broniowski:2013dia} and explored in further \ndetail for the ${}^{12}$C nucleus in~Ref.~\\cite{Bozek:2014cva}. Quite remarkably, the experimental application of the method could reveal information on \nthe {\\em ground state} of a light clustered nucleus, i.e. on the lowest possible energy state, via the highest-energy nuclear collisions, such as those carried out at \nultra-relativistic accelerators: the CERN Super Proton Synchrotron (SPS), BNL Relativistic Heavy-Ion Collider (RHIC), or the CERN Large Hadron Collider (LHC).\nIn the first part of this paper we extend the results of Refs.~\\cite{Broniowski:2013dia,Bozek:2014cva} obtained for  ${}^{12}$C  to other light nuclei, \nnamely ${}^{7}$Be, ${}^{9}$Be, and ${}^{16}$O, which are believed to have a prominent cluster structure in their ground states, see Fig.~\\ref{fig:structure}. \n\n\n\\begin{figure}[tb]\n\\centering\n\\vspace{-2mm}\n\\includegraphics[angle=0,width=0.40 \\textwidth]{structures.pdf}\n\\vspace{-4mm}\n\\caption{Schematic view of the cluster structure of light nuclei. The dark blobs indicate $\\alpha$ clusters (in the case of ${}^7$Be, also the ${}^3$He cluster). The \nadditional dot in ${}^9$Be indicates the extra neutron.\n\\label{fig:structure}}\n\\end{figure}\n\nWe recall the basic concepts of Refs.~\\cite{Broniowski:2013dia,Bozek:2014cva}: Spatial correlations in the ground state of a light nucleus, such as the \npresence of clusters, lead to an intrinsic deformation. When colliding with a heavy nucleus (${}^{208}$Pb,  ${}^{197}$Au) at a very high energy, \nwhere due to the Lorentz contraction the collision time is much shorter than any characteristic nuclear time scale, \na reduction of the wave function occurs and a correlated spatial distribution of participant nucleons is formed.\nThis, via individual nucleon-nucleon collisions between the colliding nuclei in the applied Glauber \npicture~\\cite{Glauber:1959aa,Czyz:1968zop,Bialas:1976ed}, leads to an initial distribution of entropy in the \ntransverse plane, whose {\\em eccentricity} reflects the deformation of the ground-state due to correlations. \nIn short, the deformed intrinsic shape of the light nucleus, when hitting a ``wall'' of a heavy target, yields a \ndeformed fireball in the transverse plane.\n\nAs an example, \nif the intrinsic state of the ${}^{12}$C nucleus is a triangle made of three $\\alpha$ particles, then the shape of the initial \nfireball in the transverse plane reflects this triangular geometry. Next, the {\\em shape-flow transmutation} mechanism (cf. Fig.~\\ref{fig:concept}), \na key geometric concept in the phenomenology of ultra-relativistic heavy-ion collisions~\\cite{Ollitrault:1992bk}, generates a large collective \ntriangular flow through the dynamics in the later stages of the evolution, modeled via hydrodynamics (for recent reviews see~\\cite{Heinz:2013th,Gale:2013da,Jeon:2016uym}), \nor transport~\\cite{Lin:2004en}.  As a result, one observes the azimuthal asymmetry of the transverse momentum distributions\nof produced hadrons. Similarly, the dumbbell \nintrinsic shape of the ground states of the ${}^{7,9}$Be nuclei, which occurs when  these nuclei are clustered, leads to a large elliptic flow. \n\nWe remark that the methodology applied in Refs.~\\cite{Broniowski:2013dia,Bozek:2014cva} and in the present work, \nwas used successfully to describe harmonic flow in d+Au collisions~\\cite{Bozek:2011if} (small dumbbells)  \nand in $^3$He+Au collisions~\\cite{Nagle:2013lja,Bozek:2014cya} (small triangles), \nand the predictions later experimentally confirmed in~\\cite{Adare:2013piz,Adare:2015ctn}.\n\n\n\\begin{figure}[tb]\n\\centering\n\\vspace{-2mm}\n\\includegraphics[angle=0,width=0.45 \\textwidth]{concept.pdf}\n\\vspace{-1mm}\n\\caption{Cartoon of ultra-relativistic $^{7,9}$Be+$^{208}$Pb collisions.\nThe clustered beryllium creates a fireball whose initial transverse shape reflects the deformed intrinsic shape of the projectile (left panel). \nSubsequent collective evolution leads to faster expansion along the direction perpendicular to the symmetry axis of the beryllium, and slower expansion \nalong this axis, as indicated by the arrows (right panel). The effect generates specific signatures in the harmonic flow patterns in spectra of the produced hadrons in the final state.\n\\label{fig:concept}}\n\\end{figure}\n\nAs the positions of the nucleons in the colliding nuclei fluctuate, being distributed according to their wave functions, the initial eccentricity, \nand in consequence the harmonic flow, always receives an additional contribution from \nthese random fluctuations~\\cite{Miller:2003kd,Alver:2006wh,Voloshin:2006gz,Broniowski:2007ft,Hama:2007dq,Luzum:2011mm,Bhalerao:2014xra} (the shape fluctuations are indicated \nwith a warped surface of the fireball in Fig.~\\ref{fig:concept}). For that reason the applied measures of the harmonic flow should be able to \ndiscriminate between these two components.\n\nTo a good  approximation, the measured harmonic flow coefficients $v_n$ in the spectra of produced hadrons are linear in the corresponding initial eccentricities \n$\\epsilon_n$ (see, e.g., ~\\cite{Gardim:2011xv,Niemi:2012aj,Bzdak:2013rya}).  This allows for a construction of flow measures given \nin Sec.~\\ref{sec:signatures}, which are independent the of details of the dynamics of the later stages of the collision, and thus carry \ninformation pertaining to the initial eccentricities. We describe such measures in Sect.~\\ref{sec:signatures}.\nWe note that another measure, involving the ratio of the triangular and elliptic flow coefficients, \nhas been recently proposed in Ref.~\\cite{Zhang:2017xda} \nfor the case of ${}^{12}$C, and tested within the AMPT~\\cite{Lin:2004en} transport model.\n\nTo have realistic nuclear distributions with clusters, yet simple enough to be implemented in a Monte Carlo simulation,\nwe apply a procedure explained in Sec.~\\ref{sec:making}, where positions of nucleons are determined within clusters of a given size, whereas \nthe clusters themselves are arranged in an appropriate shape (for instance, triangular for ${}^{12}$C.  \nThe parameters, determining the separation distance between the clusters and their sizes,  \nare fixed in such a way that the resulting one-body nucleon densities compare well to the state-of-the-art Variational Monte Carlo \n(VMC)~\\cite{Wiringa:2013ala,Lonardoni:2017egu} simulations. \nThe simulations for clustered nuclei are compared to the base-line case, where no clustering is present.\n\nOur basic findings, presented in Sec.~\\ref{sec:signatures}, are that clusterization in light nuclei leads to sizable effects in the harmonic flow \npattern in collisions with heavy nuclei. The effect is most manifest for the highest-multiplicity collisions, where additional fluctuations \nfrom the random distribution of nucleons are reduced. For the dumbbell shaped ${}^{7,9}$Be, the measures of \nthe elliptic flow are affected, whereas for the triangular ${}^{12}$C and tetrahedral ${}^{16}$O there are significant imprints of clusterization in \nthe triangular flow. These effects, when observed experimentally, could be promptly \nused to assess the degree of clusterization in light nuclei.\n\nIn the second part of this paper we examine a novel possibility of observing the intrinsic deformation  resulting from clusterization \n of light nuclei with spin, such as ${}^{7,9}$Be, when these are collided with ultra-relativistic {\\em protons}. \nThis interesting but exploratory proposal would require a magnetically \n{\\em polarized} ${}^{7,9}$Be nuclei, which in the ground state have $J^P=3\/2^-$. \n\nIn this case the geometric mechanism is as follows:\nWhen the dumbbell shaped nucleus in $m=1\/2$ ground state is polarized along the proton beam direction, there is a much higher chance \nfor the proton to collide with more nucleons (as it can pass through both clusters) than in the case where it is polarized perpendicular to the beam axis \n(where it would pass through a single cluster only). Thus more participants are formed in the former case. \nThe effect is opposite for the $m=3\/2$ state, as explained in Sect.~\\ref{sec:pA}.\n\nOne could thus investigate the distribution of participant nucleons, $N_w$, for various magnetic numbers $m$ and geometric orientations.\nWe find from our simulations a factor of two effects for $N_w = 4 $ and an order of magnitude effect for $N_w \\ge 6$, when \ncomparing the cases of \\mbox{$m=3\/2$} and \\mbox{$m=1\/2$}, or changing of the direction of the beam relative \nto the polarization axis. We discuss the mechanism and the relevant issues in Sec.~\\ref{sec:pA}.\n\n\n\n\\section{Nucleon distributions in clustered light nuclei}\n\\label{sec:making}\n\n\nTo model the collision process in the applied Glauber framework~\\cite{Glauber:1959aa,Czyz:1968zop,Bialas:1976ed}, we \nfirst need the distributions of centers of nucleons in the considered nuclei. We have adopted a simple and practical procedure \nwhere these distributions are generated randomly in clusters placed at preassigned positions in such a way that \nthe one-body density reproduces the shapes obtained from state-of-the-art \nVariational Monte Carlo (VMC)~\\cite{Buendia:2004yt,Wiringa:2013ala,Lonardoni:2017egu} studies.\n\nExplicitly, our steps are as follows: We set the positions of clusters according to the geometry of Fig.~\\ref{fig:structure}, separating their centers\nfrom each other with the distance $l$. The distribution of the nucleons in each cluster is randomly generated according to the Gaussian function\n\\begin{eqnarray}\nf_i(\\vec{r})=A \\exp \\left (- \\frac{3}{2} \\, \\frac{(\\vec{r}-\\vec{c_i})^2}{r_c^2} \\right ),   \n\\end{eqnarray}\nwhere $\\vec{r}$ is the 3D coordinate of the nucleon, $\\vec{c_i}$ is the position of the center of the cluster $i$, and $r_c$ is the rms radius \nof the cluster, which equals $r_\\alpha$ or  $r_{{}^3{\\rm He}}$ depending on the cluster type. \nWe generate the positions of the nucleons in sequence, alternating the number of the cluster: 1, 2,\\dots, 1, 2,\\dots, until all the nucleons are \nplaced.\n\n\n\n\\begin{table}[tb]\n\\caption{\\label{tab:param} Parameters used in the GLISSANDO simulations to obtain the nuclear distribution: $l$ is the distance between the centers \nof clusters, arranged according to the geometry shown in Fig.~\\ref{fig:structure}, $r_\\alpha$ is the size of the $\\alpha$ cluster, $r_{{}^3{\\rm He}}$ \nis the size of the  ${}^3{\\rm He}$ cluster in \\ensuremath{{}^{7}{\\rm Be}}, and $r_n$ determines the distribution of the extra neutron in \\ensuremath{{}^{9}{\\rm Be}}.}\n\\vspace{3mm}\n\\begin{tabular}{|c|cccc|}\n\\hline\n Nucleus & $l$ [fm] & $r_\\alpha$ [fm] &  $r_{{}^3{\\rm He}}$ [fm]  & $r_n$ [fm]\n\\\\\n\\hline \n\\ensuremath{{}^{7}{\\rm Be}}   & 3.2 & 1.2 & 1.4 & - \\\\\n\\ensuremath{{}^{9}{\\rm Be}}   & 3.6 & 1.1 & -  & 1.9  \\\\\n\\ensuremath{{}^{12}{\\rm C}}     & 2.8 & 1.1 & - & - \\\\\n\\ensuremath{{}^{16}{\\rm O}}    & 3.2 & 1.1 & - & - \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[angle=0,width=0.48\\textwidth]{density_points.pdf}\n\\caption{(Color online) Nuclear density profiles of the considered light nuclei. The points correspond to our Monte Carlo generation of the nuclear distributions \nin GLISSANDO, with parameters listed of Table~\\ref{tab:param} adjusted in such a way that the results from  \nVariational Monte Carlo (VMC)~\\cite{Wiringa:2013ala,Lonardoni:2017egu}(dashed lines)  are properly reproduced. \nWe use the normalization $4\\pi \\int_0^\\infty r^2dr \\, \\rho(r)=1$. \\label{fig:density}}\n\\end{figure}\n\n\nFor ${}^{9}$Be, we add the extra neutron on top of the two $\\alpha$ clusters according to a distribution with a hole in the middle, \n\\begin{eqnarray}\nf_n(\\vec{r})=A' r^2 \\,\\exp \\left (- \\frac{3}{2} \\frac{r^2}{r_n^2} \\right ).\n\\end{eqnarray}\n\nThe short-distance\nnucleon-nucleon repulsion is incorporated by precluding the centers of each pair of nucleons to be closer than the expulsion distance of \n0.9~fm, which is a customary prescription~\\cite{Broniowski:2010jd} in preparing nuclei for the Glauber model in ultra-relativistic nuclear collisions. \nAt the end of the procedure the distributions are shifted such that their center of mass is \nplaced at the origin of the coordinate frame.\nAs a result, we get the Monte Carlo distributions with the built-in cluster correlations.\n\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_7Be.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_7Be.pdf}\n\\caption{(Color online) Scaled standard deviations of rank-$n$ flow coefficients (panel (a)) and ratios of the four-particle to two-particle \ncumulants (panel (b)), plotted as functions of the total number of the wounded nucleons. Clustered nuclei (thick lines) are compared with \nthe case where the nucleons are distributed uniformly with the same one-body radial distributions (thin lines). \\ensuremath{{}^{7}{\\rm Be}+{}^{208}{\\rm Pb}}\\ collisions.\nThe vertical lines indicate the multiplicity percentiles (centralities) corresponding to the indicated values of $N_w$. \n}\n\\label{fig:7Be}\n\\end{figure*}\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_9Be.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_9Be.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:7Be} but for \\ensuremath{{}^{9}{\\rm Be}+{}^{208}{\\rm Pb}}\\ collisions.}\n\\label{fig:9Be}\n\\end{figure*}\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_12C.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_12C.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:7Be} but for \\ensuremath{{}^{12}{\\rm C}+{}^{208}{\\rm Pb}}\\ collisions.}\n\\label{fig:12C}\n\\end{figure*}\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_16O.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_16O.pdf}\n\\caption{(Color online) The same as in Fig.~\\ref{fig:7Be} but for \\ensuremath{{}^{16}{\\rm O}+{}^{208}{\\rm Pb}}\\ collisions.}\n\\label{fig:16O}\n\\end{figure*}\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[angle=0,width=0.495 \\textwidth]{sig_epsn_7Be_rap.pdf}\n\\includegraphics[angle=0,width=0.495 \\textwidth]{ratio_epsn_7Be_rap.pdf}\n\\caption{(Color online) Scaled standard deviations of rank-$n$ flow coefficients (panel (a)) and ratios of the four-particle to two-particle \ncumulants (panel (b)) simulated for the backward, central, and forward rapidity regions, plotted as functions of the total number of the \nwounded nucleons. \\ensuremath{{}^{7}{\\rm Be}+{}^{208}{\\rm Pb}}\\ collisions, clustered nuclei case. Thick lines correspond to the clustered case, and thin lines to the \nuniform distributions.}\n\\label{fig:7Berap}\n\\end{figure*}\n\n\nTo fix the parameters listed in Table~\\ref{tab:param}, we use specific reference radial distribution obtained from \nVMC simulations, which use the Argonne~v18 two-nucleon and Urbana~X three-nucleon potentials, as provided in \n{\\small \\url{http:\/\/www.phy.anl.gov\/theory\/research\/density}}~\\cite{Wiringa:2013ala,Lonardoni:2017egu}. Our distribution parameters \nare then optimized such that the one particle densities $\\rho(r)$ from VMC  \nare properly reproduced.  Thus the radial density of the centers on nucleons serves as a constraint for building our clustered \ndistributions. Figure~\\ref{fig:density} shows the quality of our fit to the one-body densities, which is satisfactory in the \ncontext of modeling ultra-relativistic nuclear collisions.  We note from  Fig.~\\ref{fig:density} that\nthe distributions (except for $^{7}$Be nucleus) develop a dip in the center.\nThe parameters used in our simulations are collected in Table~\\ref{tab:param}.\n\nAs we are interested in specific effects of clusterization, as a ``null result'' we use the {\\em uniform} distributions, i.e., \nwith no  clusters.  We prepare such distributions with exactly the same radial density as the clustered ones. \nThis is achieved easily with a trick, where we randomly re-generate the spherical angles of the nucleons from \nthe clustered distributions, while leaving the radial coordinates intact.\n\n\n\\section{Harmonic flow in relativistic light-heavy collisions}\n\\label{sec:signatures}\n\nAs already mentioned in the Introduction, we use the so-called Glauber approach to model the early stage of the collision. The Glauber model~\\cite{Glauber:1959aa}\nformulated almost sixty years ago to model the elastic scattering amplitude in high-energy collisions, was later \nextended to inelastic collisions~\\cite{Czyz:1968zop}, and subsequently led to the widely used wounded-nucleon model~\\cite{Bialas:1976ed}.\nThe model assumes that \nthe trajectories of nucleons are straight lines and the individual nucleons at impact parameter $b$ interact with a probability $P(b)$, where \n$\\int d^2b \\, P(b) =\\sigma_{\\rm inel}$ is the total inelastic nucleon-nucleon cross section. We use a Gaussian form of $P(b)$, \nwhich for the studied heavy-ion observables is of sufficient accuracy~\\cite{Rybczynski:2011wv}.\n\nGenerally, in the Glauber framework, at the initial stage of the collision the interacting nucleons deposit \nentropy (or energy) in the transverse plane. Such deposition occurs from wounded nucleons, but also from binary collisions.\nSuch an admixture of binary collisions is necessary to obtain proper multiplicity distributions~\\cite{Kharzeev:2000ph,Back:2001xy}.\nIn this model the transverse distribution of entropy takes the form \n\\begin{eqnarray}\n\\rho(x,y)&=&\\frac{1-\\alpha}{2}\\rho_W(x,y) + \\alpha \\rho_{\\rm bin}(x,y), \\label{eq:rho0}\n\\end{eqnarray}\nwhere $\\rho_W(x,y)$ is the distribution of the wounded nucleons, \n$\\rho_{\\rm bin}(x,y)$ is the distribution of the binary collisions, and  $\\alpha$ is the parameter controlling the relative weight of the wounded to binary sources. \nIn our simulations we use $\\alpha=0.12$ (the value fitting the multiplicity distributions at the SPS collision energies). \nThe sources forming the distributions are smeared with a Gaussian of a width of 0.4~fm.\n\nIn the following we show the numerical results of our GLISSANDO~\\cite{Broniowski:2007nz,Rybczynski:2013yba} \nsimulations of collisions of the described above nuclei composed \nof $\\alpha$-clusters with ${}^{208}$Pb nucleus at $\\sqrt{s_{NN}}=17$~GeV, where the corresponding inelastic nucleon-nucleon \ncross section is $\\sigma_{\\rm inel}=32$~mb. Such collision energies are available at SPS and the considered reactions are possible to study in the on-going \nNA61\/SHINE experiment with ${}^{208}$Pb or proton beams. A variety of targets and secondary beams are available in this experiment~\\cite{Abgrall:2014xwa}.\nTherefore the present study may be thought of as a case study for possible NA61\/SHINE investigations.\n\nTo analyze the effects of clusterization in the considered light nuclei on the harmonic flow coefficients\nin the reactions with  ${}^{208}$Pb nuclei, one needs to use appropriate flow measures. The eccentricity coefficients, $\\epsilon_n$, are designed\nas measures of the harmonic deformation in the initial state. They are defined for each collision event as\n\\begin{eqnarray} \n\\epsilon_n e^{i n \\Phi_n} = - \\frac{\\int \\rho(x,y) e^{i n\\phi}  (x^2+y^2)^{n\/2} dx dy}{\\int \\rho(x,y) (x^2+y^2)^{n\/2}  dx dy}, \\label{eq:eps}\n\\end{eqnarray}\nfor $n=2,3,\\dots$, with $\\phi=\\arctan(y\/x)$ and $\\Phi_n$ denoting the angle of the principal axes in the transverse plane $(x,y)$.\n\nThe subsequent collective evolution with hydrodynamic~\\cite{Heinz:2013th,Gale:2013da,Jeon:2016uym} \nor transport~\\cite{Lin:2004en} has a shape-flow transmutation feature: The deformation of shape in the initial stage leads to harmonic \nflow of the hadrons produced in the late stage. The effect is manifest in an approximate proportionality of the flow \ncoefficients $v_n$ to the eccentricities  $\\epsilon_n$, which holds for $n=2$ and $3$ (for higher rank non-linear coupling effects may be present):\n\\begin{equation}\nv_n = \\kappa_n \\epsilon_n \\ , \n\\label{eq:linear}\n\\end{equation} \nThe cumulant coefficients follow an analogous relation: \n\\begin{eqnarray}\nv_n\\{m\\}=\\kappa_n \\epsilon_n\\{m\\}.\n\\end{eqnarray}\nThe proportionality coefficients $\\kappa_n$ depend on various features of the colliding system (centrality, collision energy), \nbut are to a good approximation independent of the eccentricity itself, hence the above relations are linear.\nTo get rid of the influence of the (generally) unknown $\\kappa_n$ coefficients on the results, \none may consider the ratios of cumulants of different order $m$ for a given rank-$n$ flow coefficient $v_n$, e.g., \n\\begin{eqnarray}\n\\frac{v_n\\{m\\}}{v_n\\{2\\}}=\n\\frac{\\epsilon_n\\{m\\}}{\\epsilon_n\\{2\\}} \\ . \\label{eq:ratios} \n\\end{eqnarray}\nTherefore the ratios of the flow cumulants can be directly compared to the corresponding ratios of the eccentricity cumulants.\nIn our work we also use the scaled event-by event standard deviation, ${\\sigma(\\epsilon_n)}\/{\\langle \\epsilon_n \\rangle}$, where\n\\begin{eqnarray}\n \\frac{\\sigma(\\epsilon_n)}{\\langle \\epsilon_n \\rangle} \\simeq \\frac{\\sigma(v_n)}{\\langle v_n \\rangle}. \\label{eq:ev}\n\\end{eqnarray}\n\nIn order to find the specific effects of clusterization, we always compare the obtained results to those corresponding to the  ``uniform'' case, where \nthe nucleons are distributed without clusterization (see Sect.~\\ref{sec:making}).\n\nIn Figs.~\\ref{fig:7Be} and~\\ref{fig:9Be} we show the event-by-event scaled standard deviations of the elliptic ($n=2$), \ntriangular ($n=3$), and quadrangular ($n=4$) flow coefficients, as well as the ratios of the four-particle to \ntwo-particle cumulants, plotted as functions of the total number of wounded nucleons. Since clusters in ${}^{7,9}$Be nuclei form a dumbbell \nshape, the influence of clusterization is, as expected, visible in the $n=2$ (elliptic) coefficients. \nThe behavior seen in panels (a) is easy to explain qualitatively: At large numbers of wounded nucleons the beryllium is oriented in such a way that it \nhits the wall of ${}^{208}$Pb side-wise, as drawn in Fig.~\\ref{fig:concept}. Then the eccentricity of the created fireball, which is an imprint of the \nintrinsic shape of beryllium, is largest. Hence the scaled variance decreases (note division with $\\langle \\epsilon_n \\rangle$ in Eq.~(\\ref{eq:ev})) with $N_w$.\nThe feature is clearly seen from Figs.~\\ref{fig:7Be} and~\\ref{fig:9Be}. Of course, this is not the case for the uniform distributions, where at large \n$N_w$ the scaled standard deviations for all $n$ acquire similar values. A detailed quantitative \nunderstanding of the dependence on $N_w$ requires simulations, as one needs to assess the influence of the random fluctuations on eccentricities, or account for effects \nwhen the beryllium hits the edge of ${}^{208}$Pb. The size of the effect in panels (a) starts to be significant for the 10\\% of the highest-multiplicity events.\n\nThe results for the $v_n\\{4\\}\/v_n\\{2\\}$ (panels (b) of Figs.~\\ref{fig:7Be} and~\\ref{fig:9Be}) are complementary.\nWe note that for high multiplicity collisions\nthe ratio is significantly larger for the clustered case compared to the uniform distributions. This is because the \ntwo-particle cumulants are more sensitive to the random fluctuations than the four-particle cumulants.\n\nFor the case of ${}^{12}$C+${}^{208}$Pb and ${}^{16}$O+${}^{208}$Pb collisions, the significant influence of clusters \nas compared to ``uniform'' case is visible for the rank-3 (triangular) coefficients, see Figs.~\\ref{fig:12C} and \\ref{fig:16O}. \nThis is mainly caused by the triangular and tetrahedral arrangements of clusters in ${}^{12}$C and ${}^{16}$O, respectively.\nThe qualitative understanding is as for the beryllium case, with the replacement of $n=2$ with $n=3$. The case of ${}^{12}$C \nhas also been thoroughly discussed in Ref.~\\cite{Bozek:2014cva}.\n\nAll previously shown simulations were carried out at the mid-rapidity, $y\\sim 0$, region. To study the dependence on rapidity, \nwe apply a model with rapidity-dependent emission functions of the entropy sources. Such an approach is necessary,\nsince in most fixed-target experiments the detectors measure particles produced in rapidity regions which are away from the mid-rapidity\ndomain. \nTaking this into account, we apply the model described in Refs.~\\cite{Bialas:2004su,Bozek:2010bi}. There, the initial density of the fireball \nin the space-time rapidity $\\eta_\\parallel=\\frac{1}{2} \\log (t+z)(t-z)$ and the transverse coordinates ($x, y$) is described by the function:\n\\begin{eqnarray}\n\\rho(\\eta_\\parallel,x,y)&=&(1-\\alpha)[\\rho_A(x,y) f_+(\\eta_\\parallel)\n+ \\rho_B(x,y) f_-(\\eta_\\parallel)] \\nonumber \\\\\n&+& \\alpha\n\\rho_{\\rm bin}(x,y) \\left [ f_+(\\eta_\\parallel) + f_-(\\eta_\\parallel) \\right ].\n\\label{eq:em}\n\\end{eqnarray}\nwhich straightforwardly generalizes Eq.~(\\ref{eq:rho0}), assuming factorized profiles from a given source.\nHere $\\rho_{A,B}(x,y)$ denotes the transverse density of the wounded sources from the nuclei $A$ and $B$, \nwhich move in the forward and backward directions, respectively. The entropy emission functions  \n$f_{\\pm}(\\eta_\\parallel)$ are given explicitly in~\\cite{Bozek:2010bi}. They are peaked in the forward or backward \ndirections, respectively, reflecting the fact that a wounded nucleon emits preferentially in its own forward hemisphere.\n\nIn the Fig.~\\ref{fig:7Berap} we plot, as functions of $N_w$, the scaled standard deviations of the rank-2 and 3 flow coefficients and ratios \nof the four-particle to two-particle cumulants calculated in backward ($\\eta_\\parallel = -2.5 $), central ($\\eta_\\parallel = 0$), and forward ($\\eta_\\parallel = 2.5 $) \nrapidity regions (at the SPS collision energy of $\\sqrt{s_{NN}}=17$~GeV the rapidity of the beam is $\\sim 2.9$). \nWe focus on results here for \\ensuremath{{}^{7}{\\rm Be}+{}^{208}{\\rm Pb}}\\ collisions, as for the other light clustered nuclei the results\nare qualitatively similar. The centrality dependence \nof the scaled standard deviation of second-rank flow coefficients (panel (a)) is similar for all considered regions of phase-space, however its magnitude \ngrows when we move from the backward (${}^{208}$Pb) to the forward (beryllium) hemisphere. The effect has to do with a \na much larger number of wounded nucleons in the backward compared to the forward hemisphere in the applied model \nof the rapidity dependence. This makes the random fluctuations smaller in the backward compared to the forward hemisphere, giving the effect seen in \nFig.~\\ref{fig:7Berap}.\n\nWe note that the previously discussed difference of the behavior of eccentricities between the clustered and uniform cases holds also \nfor other regions in rapidity, which makes the effect possible to study also in the fixed-target experiments with detectors covering the \nforward rapidity region.\n\n\\section{Proton-polarized light nucleus scattering}\n\\label{sec:pA}\n\nIn this section we present a more exploratory study, as the investigation needs the magnetic field \nto polarize the beryllium nuclei along a chosen direction. Polarized nuclear targets or beams have not yet been\nused in ultra-relativistic collisions. Nevertheless, our novel effect, also geometric in its origin, is worth \npresenting as a possibility for future experiments.\n\nSince the ground states of ${}^{7,9}$Be nuclei have $J^P=3\/2^-$, they can be polarized. Then, due to their\ncluster nature, the intrinsic symmetry axis correlated to the polarization axis in a specific way described in detail below. \nOne can thus control (to a certain degree)  the orientation of the intrinsic dumbbell shape. This, in turn, can be probed in \nultra-relativistic collisions with protons, as more particles are produced when the proton goes along the dumbbell, compared to the case when \nit collides perpendicular to the symmetry axis. \n\nWe wish to consider the beryllium nuclei polarized in magnetic field, therefore the first task is to obtain \nstates of good quantum numbers in our model approach, where we prepare intrinsic states \nwith the method described in Sect.~\\ref{sec:making}. \nWe use the Peierls-Yoccoz projection (see, e.g., ~\\cite{ring}), which is a \nstandard tool in nuclear physics of (heavy) deformed nuclei. The basic formula to pass from an intrinsic wave function $\\Psi^{\\rm intr}_k(\\Omega)$,\nwhere $\\Omega$ is the spherical angle of the symmetry axis and $k$ is the intrinsic spin projection, to the state of good quantum numbers $|j,m\\rangle$ has the form\n\\begin{eqnarray}\n|j,m\\rangle = \\sum_k \\int d\\Omega D^{j}_{m,k}(\\Omega) |\\Psi_k^{\\rm intr}(\\Omega) \\rangle, \\label{eq:PY}\n\\end{eqnarray}\nwhere $ D^{j}_{m,k}(\\Omega)$ is the Wigner $D$ function.\n\n\nThe ${}^7$Be nucleus has the following cluster decomposition and angular momentum decomposition between the spin of the clusters and the orbital angular momentum of the clusters:\n\\begin{eqnarray}\n{}^7{\\rm Be} &=& {}^4{\\rm He} + {}^3{\\rm He}, \\label{eq:7Be} \\\\  \n\\frac{3}{2}^-  &=& 0^+ + \\tfrac{1}{2}^+ + 1^-, \\nonumber\n\\end{eqnarray}\nwhere $0^+$ is the $J^P$ of the $\\alpha$ particle,  $ \\tfrac{1}{2}^+$ of ${}^3{\\rm He}$, and $ 1^-$ is the orbital angular \nmomentum. Similarly, for ${}^9$Be \n\\begin{eqnarray}\n{}^9{\\rm Be} &=& {}^4{\\rm He} + {}^4{\\rm He} + n, \\label{eq:9Be} \\\\  \n\\frac{3}{2}^-  &=& 0^+ + 0^+ +  \\tfrac{1}{2}^+ + 1^-, \\nonumber\n\\end{eqnarray}\nwhere the neutron is assumed to be in an $S$ state, and the $J^P$ of the angular motion of the two $\\alpha$ clusters is $1^-$. The Clebsch-Gordan decomposition is \n\\begin{eqnarray}\n|\\tfrac{3}{2},m &=& \\tfrac{3}{2}\\rangle = |\\tfrac{1}{2},\\tfrac{1}{2}\\rangle \\otimes |1,1\\rangle, \\label{eq:CG} \\\\\n|\\tfrac{3}{2},m &=& \\tfrac{1}{2}\\rangle = \n\\sqrt{\\tfrac{2}{3}}|\\tfrac{1}{2},\\tfrac{1}{2}\\rangle \\otimes |1,0\\rangle + \\sqrt{\\tfrac{1}{3}} |\\tfrac{1}{2},-\\tfrac{1}{2}\\rangle \\otimes |1,1\\rangle. \\nonumber \n\\end{eqnarray}\nIn the intrinsic frame, where the clusters are at rest, the angular momentum comes from the spin of $ {}^3{\\rm He}$ or $n$ in the cases of  ${}^7$Be \nor ${}^9$Be, respectively, hence the available values of $k$ are $\\pm \\tfrac{1}{2}$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.4 \\textwidth]{wig2.pdf} \n\\caption{The distributions of the intrinsic symmetry axis of ${}^{7,9}$Be in the polar angle $\\theta$, Eq.~(\\ref{eq:tilt}), \nfollowing from the Peierls-Yoccoz projection method. \\label{fig:wig}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[angle=0,width=0.23 \\textwidth]{pl32_t.pdf} \\includegraphics[angle=0,width=0.23 \\textwidth]{pl12_t.pdf} \\\\\n\\vspace{-2mm}\n\\includegraphics[angle=0,width=0.23 \\textwidth]{pl32_t_p.pdf} \\includegraphics[angle=0,width=0.23 \\textwidth]{pl12_t_p.pdf} \n\\vspace{-5mm}\n\\caption{Schematic representation of collisions of protons with polarized $^{7,9}$Be. The sphere presents the $\\alpha$ or $^3$He clusters \nand the clouds indicate the quantum washing out of the symmetry axis of the intrinsic states, in accordance to Eq.~(\\ref{eq:PY}). The tube represents \nthe proton beam with the area given by the total inelastic proton-proton cross section. Arrows show the direction of the magnetic field which corresponds \nto the quantization axis of spin. Details in the text. \\label{fig:geom}}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[angle=0,width=0.4 \\textwidth]{7z.pdf} ~~ \\includegraphics[angle=0,width=0.4 \\textwidth]{7y.pdf}\n\\caption{Results of Monte Carlo simulations of p+$^7$Be collisions. We note that for $N_w \\ge 3 $  the probability of wounding \n$N_w$ nucleons is higher for $m=1\/2$ than for $m=3\/2$ in the case when $\\vec{B}$ is parallel to z axis (panel (a)). For situation when $\\vec{B}$ \nis perpendicular to z, we observed more wounded nucleons for $m=3\/2$ than for $m=1\/2$ (panel (b)).   \\label{fig:distr1}}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[angle=0,width=0.4 \\textwidth]{9z.pdf} ~~ \\includegraphics[angle=0,width=0.4 \\textwidth]{9y.pdf} \n\\caption{The same as in Fig.~\\ref{fig:distr1} but for p+$^9$Be collisions. \\label{fig:distr2}}\n\\end{figure*}\n\n\nAccording to Eq.~(\\ref{eq:PY}), we have for both nuclei \n\\begin{eqnarray}\n|\\tfrac{3}{2},m\\rangle = \\sum_{k=\\pm \\tfrac{1}{2}} \\int d\\Omega D^{3\/2}_{m,k}(\\Omega) |\\Psi_k^{\\rm intr}(\\Omega) \\rangle. \\label{eq:PY32}\n\\end{eqnarray}\nUnder the assumptions $\\langle \\Psi_{\\rm intr}(\\Omega') | \\Psi_{\\rm intr}(\\Omega) \\rangle \\simeq \\delta(\\Omega-\\Omega')$, which becomes \nexact in the limit of many nucleons, but still holds to a sufficiently good accuracy for 7 or 9 nucleons, we find \n\\begin{eqnarray}\n\\left | \\langle \\theta, \\phi |\\tfrac{3}{2},m\\rangle \\right |^2 =  [D^{3\/2}_{m,1\/2}(\\theta,\\phi)]^2+[D^{3\/2}_{m,-1\/2}(\\theta,\\phi)]^2. \n\\end{eqnarray}\nExplicitly, \n\\begin{eqnarray}\n&& \\left | \\langle \\theta, \\phi |\\tfrac{3}{2},\\tfrac{3}{2} \\rangle \\right |^2 = |Y_{11}(\\theta,\\phi)|^2= \\frac{3}{8\\pi} \\sin^2 \\theta, \\label{eq:tilt} \\\\\n&& \\left | \\langle \\theta, \\phi |\\tfrac{3}{2},\\tfrac{1}{2} \\rangle \\right |^2 = \\tfrac{2}{3} |Y_{10}(\\theta,\\phi)|^2 + \\tfrac{1}{3} |Y_{11}(\\theta,\\phi)|^2 \\nonumber \\\\\n&& \\hspace{2cm} =  \\frac{1}{8\\pi} \\left ( 1+3\\cos^2 \\theta \\right ), \\nonumber\n\\end{eqnarray}\nin accordance to Eq.~(\\ref{eq:CG}). The distributions (\\ref{eq:tilt}), which depend on the polar angle $\\theta$ and \nnot on the azimuthal angle $\\phi$, are shown in Fig.~\\ref{fig:wig}. \n\n\nThe prescription for the Monte Carlo simulations that follows from the above derivation is that the symmetry axes of ${}^{7,9}$Be should be \nrandomly tilted in each collision event according to the distributions (\\ref{eq:tilt}). We note that the $m=1\/2$ state is approximately aligned \nalong the spin projection axis (the distribution peaks at $\\theta=0$ or $\\theta= \\pi$), \nwhereas the $m=3\/2$ state is distributed near the equatorial plane (with the maximum at $\\theta=\\pi\/2$). \n\nSuppose that the targets of ${}^{7,9}$Be are 100\\% polarized along the direction of the magnetic field $B$ and consider collisions with a proton beam \nparallel or perpendicular to $B$. Then the geometry of the collision is influenced by the distributions of the intrinsic symmetry axis, \nas pictorially displayed in Fig.~\\ref{fig:geom}.\n \nThe figure shows schematically the collisions of protons with a polarized $^{7,9}$Be target, with the spheres representing the $\\alpha$ or $^3$He \nclusters and the clouds indicating the quantum distribution of the symmetry axis of the intrinsic states, in accordance to Eq.~(\\ref{eq:PY}). In the two \nleft panels of the Fig.~\\ref{fig:geom}, corresponding to $m=3\/2$ states the clusters are distributed near the equatorial plane, whereas in the two right \npanels, corresponding to $m=1\/2$ states the distribution of the clusters is approximately align along the quantization axis given by the magnetic field \ndirection B. The tubes represent the proton beam, drawn in such a way that the area of the tube is given by the total inelastic proton-proton cross section.\nWe can distinguish several geometric cases in the top panels of Fig.~\\ref{fig:geom} the proton beam is parallel to the direction of $\\vec{B}$, we notice \nthat for the $m=1\/2$ case the chance of hitting two clusters, thus wounding more nucleons, is higher than for $m=3\/2$ case. The effect is opposite \nwhen the proton beam is perpendicular to $\\vec{B}$ as can be seen from the two bottom panels. \n\nThe above discussed simple geometric mechanism finds its realization in numerical Monte Carlo simulations. Distribution of the number of \nwounded nucleons (in a logarithmic scale) is shown in Fig.~\\ref{fig:distr1} and in Fig.~\\ref{fig:distr2}. We note from panels (a) that in the \ncase of $\\vec{B}$ parallel to z (beam direction) indeed the probability of wounding more nucleons $N_w\\ge 3$ is larger for $m=1\/2$ than \nfor $m=3\/2$. The effect for $N_w=5$ reaches about a factor of 5, and increases for higher $N_w$. Note however, that at higher $N_w$ the collisions \nbecome very rare, thus statistical errors would preclude measurements.\nIn the case when $\\vec{B}$ is perpendicular to z (panels (b)) the effect is opposite \nwith higher probability of wounding more nucleons for $m=3\/2$ than for $m=1\/2$.\n\n\n\\section{Summary and conclusions}\n\\label{sec:summa}\n\nWe have shown that clusterization in light nuclei leads to characteristic signatures which could be studied in ultra-relativistic nuclear collisions. \nThe presence of clusters leads to specific intrinsic geometric deformation, which in collisions with a heavy nucleus \ngenerates hallmark harmonic flow patterns, especially \nfor the  collisions of highest multiplicity of the produced particles. \nAs the phenomenology of flow and the corresponding data analysis methods are standard, we believe that the proposal \nis experimentally feasible, requiring collisions with appropriate beams and then using the well developed and tested data analysis \ntechniques. We note that in the NA61\/SHINE experiment the beryllium beams and targets, studied in this paper, \nhave already been used~\\cite{Abgrall:2014xwa}. \n\nWe have also explored an opportunity following from the fact that the ground states of  ${}^{7,9}$Be have a non-zero spin, which allows for their \npolarization in an external magnetic field. Then, clusterization leads to significant effects in the spectra of participant (or spectator) nucleons in ultra-relativistic \ncollisions with the protons. We have found a factor of two effects for $N_w = 4 $ and an order of magnitude effect for $N_w \\ge 6$, when changing the orientation of the direction of the beam \nrelative to the polarization axis, or when comparing the spin states \\mbox{$m=3\/2$} and \\mbox{$m=1\/2$}. As the polarized nuclei have not, \nup to now, been used in ultra-relativistic nuclear collisions, our proposal is to be considered in future experimental proposals.\n\nFinally, we note that the effects of $\\alpha$ clusterization for heavier nuclei are small in the sense that the resulting intrinsic eccentricities are much \nsmaller than in the light systems considered in this paper. Therefore the investigations with the ${}^{7,9}$Be, ${}^{12}$C, and ${}^{16}$O nuclei \nwould be most promising. \n\n\\begin{acknowledgments}\nThe numerical simulations were carried out in laboratories created under the project\n``Development of research base of specialized laboratories of public universities in Swietokrzyskie region'',\nPOIG 02.2.00-26-023\/08, 19 May 2009.\nMR was supported by the Polish National Science Centre (NCN) grant 2016\/23\/B\/ST2\/00692 and WB by NCN grant 2015\/19\/B\/ST2\/00937.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nSupersymmetry (SUSY) is one of the most promising extensions of the\nStandard Model (SM) \\cite{MSSM} since, among other things, it solves\nthe hierarchy problem, provides a natural candidate for dark matter\n\\textit{etc}. It also introduces many new sources of CP violation\nthat may be needed to explain baryon asymmetry of the universe.\nThese phases, if large $\\mathcal{O}(1)$, can cause problems with\nsatisfying experimental bounds on lepton, neutron and mercury\nEDMs~\\cite{susycp}. This can be overcome by pushing sfermion spectra\nabove a TeV scale or arranging internal cancelations\n\\cite{Ibrahim:2007fb}.\n\nMost unambiguous way to detect the presence of CP-violating phases\nwould be to study CP-odd observables measurable at future\naccelerators --- the LHC and the ILC. Such observables in the\nchargino sector are, for example, triple products of momenta of\ninitial electrons, charginos and their decay products\n\\cite{Kittel:2004kd}. However they require polarized initial\nelectron\/positron beams or measurement of chargino polarization.\n\nIn this talk we present another possibility of detecting\nCP-violating phases in the chargino sector. As it was recently\npointed out \\cite{Osland:2007xw,my}, in non-diagonal chargino pair\nproduction\n\\begin{equation}\ne^+e^-\\to\\tilde{\\chi}_1^\\pm\\tilde{\\chi}_2^\\mp \\label{produkcja}\n\\end{equation}\na CP-odd observable can be constructed beyond tree-level from\nproduction cross section without polarized $e^+e^-$ beams or\nmeasurement of chargino polarization. We show here the results of\nthe full one-loop calculation of this effect. In the\nreaction~(\\ref{produkcja}) the CP violation can be induced by the\ncomplex higgsino mass parameter $\\mu$ or complex trilinear coupling\nin top squark sector $A_t$. Since these asymmetries can reach a few\npercent, they can be detected in simple event-counting experiments\nat future colliders.\n\n\\section{CP-odd asymmetry at one loop}\n\nIn $e^+e^-$ collisions charginos are produced at tree-level via the\n$s$-channel $\\gamma,Z$ exchange and $t$-channel $\\tilde{\\nu}_e$\nexchange. As it was shown in \\cite{Choi:1998ei} no CP violation\neffects can be observed at the tree-level for the production\nprocesses of diagonal $\\tilde{\\chi}_i^+ \\tilde{\\chi}_i^-$ and\nnon-diagonal $\\tilde{\\chi}_i^+ \\tilde{\\chi}_j^-$ chargino pairs\nwithout the measurement of polarization of final chargino. However\nthe situation is different for non-diagonal production if we go\nbeyond tree-level approximation.\n\nRadiative corrections to the chargino pair production include the\nfollowing generic one-loop Feynman diagrams: the virtual vertex\ncorrections, the self-energy corrections to the $\\tilde{\\nu}$, $Z$\nand $\\gamma$ propagators, and the box diagrams contributions. We\nalso have to include corrections on external chargino legs.\n\nOne-loop corrected matrix element squared is given by\n\\begin{eqnarray}\n|\\mathcal{M}_{\\mathrm{loop}}|^2 = |\\mathcal{M}_{\\mathrm{tree}}|^2 +\n2 \\mathrm{Re}(\\mathcal{M}_{\\mathrm{tree}}^*\n\\mathcal{M}_{\\mathrm{loop}} )\\, .\n\\end{eqnarray}\nAccordingly, the one-loop CP asymmetry for the non-diagonal chargino\npair is defined as\n\\begin{eqnarray}\n&& A_{12}=\\frac{\\sigma^{12}_{\\rm loop}-\n  \\sigma^{21}_{\\rm loop}}{\\sigma^{12}_{\\rm tree}+\n  \\sigma^{21}_{\\rm tree}}\\, ,\n  \\label{CPasym}\n\\end{eqnarray}\nwhere $\\sigma^{12}$, $\\sigma^{21}$ denote cross sections for\nproduction of $\\tilde{\\chi}_1^+ \\tilde{\\chi}_2^-$ and\n$\\tilde{\\chi}_2^+ \\tilde{\\chi}_1^-$, respectively. Since the\nasymmetry vanishes at tree-level it has to be finite at one loop,\nhence no renormalization is needed.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.9]{Rolbiecki_fig1a.eps}\\hskip 1cm\n\\includegraphics[scale=0.35]{Rolbiecki_fig1b.eps}\n\\end{center}\n\\caption{Box diagram with selectron exchange and its contribution to\nthe asymmetry $A_{12}$ vs.\\ center of mass energy. The selectron\nmass is 403~GeV.\\label{fig:thrs_scan}}\n\\end{figure}\n\nThe CP asymmetry Eq.~(\\ref{CPasym}) arises due to the interference\nbetween complex couplings, which in our case are due to complex\nmixing matrices of charginos or stops, and non-trivial imaginary\npart from Feynman diagrams --- the absorptive part. Such\ncontributions appear when some of the intermediate state particles\nin loop diagrams go on-shell. This is illustrated in\nFig.~\\ref{fig:thrs_scan} where the contribution to $A_{12}$ from\ndouble selectron exchange appears at the threshold for selectron\npair production at $\\sqrt{s}=806$~GeV.\n\n\n\\section{Numerical results}\n\nFor the numerical results in this section we use two parameter sets\n(A) and (B) with gaugino\/higgsino mass parameters defined as follows\nat the low scale:\n\\begin{eqnarray}\n&\\mbox{A:}&\\quad |M_1| = 100\\mbox{ GeV},\\quad M_2 = 200\\mbox{ GeV},\n\\quad|\\mu| = 400\\mbox{ GeV},\\nonumber \\\\\n&\\mbox{B:}&\\quad |M_1| = 250\\mbox{ GeV}, \\quad M_2 = 200\\mbox{ GeV},\n\\quad |\\mu| = 300\\mbox{ GeV},\\nonumber\n\\end{eqnarray}\nand with $\\tan\\beta=10$. This gives the following chargino masses:\n\\begin{eqnarray*}\n&\\mbox{A:}&\\quad m_{\\tilde{\\chi}^-_1} = 186.7 \\mbox{ GeV},\\quad\nm_{\\tilde{\\chi}^-_2} = 421.8 \\mbox{ GeV},\\\\\n&\\mbox{B:}&\\quad m_{\\tilde{\\chi}^-_1} = 175.6 \\mbox{ GeV},\\quad\nm_{\\tilde{\\chi}^-_2} = 334.5 \\mbox{ GeV}.\n\\end{eqnarray*}\nFor the sfermion mass parameters in scenario (A) we assume\n\\begin{eqnarray*}\n&&m_{\\tilde{q}}\\equiv M_{\\tilde{Q}_{1,2}}=M_{\\tilde{U}_{1,2}}=M_{\\tilde{D}_{1,2}}=450\\mbox{ GeV},\\nonumber\\\\\n&&M_{\\tilde{Q}}\\equiv M_{\\tilde{Q}_{3}}=M_{\\tilde{U}_{3}}=M_{\\tilde{D}_{3}}=300\\mbox{ GeV},\\\\\n&&m_{\\tilde{l}}\\equiv\nM_{\\tilde{L}_{1,2,3}}=M_{\\tilde{E}_{1,2,3}}=150\\mbox{ GeV},\n\\end{eqnarray*}\nand for the sfermion trilinear coupling:\n$|A_{t}|=-A_{b}=-A_{\\tau}=A=400\\mbox{ GeV}$. Scenario (B) is for\ncomparison with Ref.~\\cite{Osland:2007xw} for which we take\n$$M_{S}= M_{\\tilde{Q}}= M_{\\tilde{U}} =M_{\\tilde{D}} =M_{\\tilde{L}}=\nM_{\\tilde{E}}=10\\mbox{ TeV}.$$\n\nIn our numerical analysis we consider the dependence of the\nasymmetry (\\ref{CPasym}) on the phase of the higgsino mass parameter\n$\\mu = |\\mu| e^{i \\Phi_\\mu}$ and soft trilinear top squark coupling\n$A_t = |A_t| e^{i \\Phi_t}$. In Fig.~\\ref{fig2} we show the CP\nasymmetry in scenario (A) as a function of the phase of $\\mu$ and\n$A_t$, left and middle panel, respectively.  Contributions due to\nbox corrections, vertex corrections and self energy corrections have\nbeen plotted in addition to the full result. In this scenario the\nasymmetry can reach $\\sim 1\\%$ for the $\\mu$ parameter and $\\sim\n6\\%$ for $A_t$, respectively. We note that for the asymmetry due to\nthe non-zero phase of the higgsino mass parameter there are\nsignificant cancelations among various contributions. In addition,\nwe also show in the right panel of Fig.~\\ref{fig2} the dependence of\nthe asymmetry due to $A_t$ as a function of $\\tan\\beta$.\n\nFor the asymmetry generated by the $\\mu$ parameter all possible\none-loop diagrams containing absorptive part contribute. The\nsituation is different for the phase of  the trilinear coupling\n$A_t$ --- when chargino mixing matrices remain real. In this case\nonly vertex and self-energy diagrams containing stop lines\ncontribute to the asymmetry~\\cite{my}.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.365]{Rolbiecki_fig2a.eps}\n\\includegraphics[scale=0.35]{Rolbiecki_fig2b.eps}\n\\includegraphics[scale=0.365]{Rolbiecki_fig2c.eps}\n\\end{center}\\vspace{-0.1cm}\n\\caption{Asymmetry $A_{12}$ in scenario (A) as a function of the\nphase of $\\mu$ parameter (left), the phase of $A_t$ (middle), and as\na function of $\\tan\\beta$ with $\\Phi_t=\\pi\/3$ (right). Different\nlines denote full asymmetry (full line) and contributions from box\n(dashed), vertex (dotted) and self energy (dash-dotted) diagrams.\n\\label{fig2}}\n\\end{figure}\n\n\nWe present also the results for the heavy sfermion scenario~(B).\nThis is to compare with~\\cite{Osland:2007xw} where only box diagrams\nwith $\\gamma$, $W$, $Z$ exchanges have been calculated neglecting\nall sfermion contributions. As can be seen in the left panel of\nFig.~\\ref{fig3} these gauge-box diagrams constitute the main part of\nthe asymmetry $A_{12}$, however this is due to partial cancelation\nof vertex and self-energy contributions. For lower values of the\nuniversal scalar mass $M_S$ the discrepancy between full and\napproximate result of~\\cite{Osland:2007xw} increases significantly.\nThis is illustrated in the middle and right panel of Fig.~\\ref{fig3}\nwhere we show two paths of approaching of the full result to the\ngauge-box approximation as the function of $M_S$. As can be seen\nthese paths depend strongly on the center of mass energy.\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.335]{Rolbiecki_fig3a.eps}\n\\includegraphics[scale=0.32]{Rolbiecki_fig3b.eps}\n\\includegraphics[scale=0.32]{Rolbiecki_fig3c.eps}\n\\end{center}\n\\caption{Left: Asymmetry $A_{12}$ in scenario (B) as a function of\nthe phase $\\Phi_\\mu$. Different lines denote full asymmetry (full\nline) and contributions from box (dashed), vertex (dotted) and self\nenergy (dash-dotted) diagrams. Middle and Right: Asymmetry $A_{12}$\nas a function of the universal scalar mass $M_S$ with\n$\\Phi_\\mu=\\pi\/2$ at different cms. The full lines denote full result\nand dashed lines show only the box contributions after neglecting\ndiagrams with slepton exchange.\\label{fig3}}\n\\end{figure}\n\n\n\n\n\\section{Summary}\nIt has been shown that CP-odd asymmetry can be generated in\nnon-diagonal chargino pair production with unpolarized\nelectron\/positron beams. The asymmetry is pure one-loop effect and\nis generated by interference between complex couplings and\nabsorptive parts of one loop integrals. The effect is significant\nfor the phases of the higgsino mass parameter $\\mu$ and the\ntrilinear coupling in stop sector $A_t$. At future linear collider\nit may give information about CP violation in chargino and stop\nsectors.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nThe flow around objects in a bulk medium is a classical problem in fluid dynamics which occurs in numerous natural phenomena and has many practical applications. The majority of research concerns high Reynolds number flows, where vortex shedding plays an important role, and mixing is greatly enhanced~\\cite{Choi2008}. Until recently, and even though it was pointed out more than 60 years ago, much less attention has been given to mixing due to the drift volume, a mechanism described by potential flow known as Darwin's drift~\\cite{Darwin1953}. One of the recent studies has shown that for many marine animals Darwin's drift is the dominant contribution to mixing by swimming~\\cite{Katija2009}. Darwin's original proposition states that the drift volume induced by an object, which was started infinitely far away from an imaginary plane and moves infinitely far to the other side of the plane is equal to the added mass volume of that object~\\cite{Darwin1953}. This prediction is based on potential flow, and has been confirmed two decades ago by Eames \\emph{et al.}~\\cite{Eames1994}, who added specific information concerning the method of evaluating the integrals to calculate the drift volume and of taking into account partial drift for finite systems. An open question is how the drift volume is influenced in the case of a less smooth object, or at higher Reynolds numbers, when flow separation starts to play a role. This problem is related to the drift volume of a vortex ring, which was considered analytically for a continuously expanding vortex~\\cite{Turner1964} and experimentally for a steady vortex ring~\\cite{Dabiri2006}.\n\nWe approach this problem experimentally by impulsively starting a disc from the interface of two immiscible liquids, oil and water, similarly and with the same setup as we did before when studying the dynamics of an air-water interface when a disc would impact it~\\cite{Bergmann2006,Bergmann2009,Gekle2009a,Gekle2010,Gekle2010d}. Besides being a practical solution to studying drift volumes, this method has relevance for, e.g., the mechanism of oil dispersion in water~\\cite{Murphy2015}. In our experiment, when the disc moves down, it drags along the oil, which then obtains a particular funnel-shaped profile (see, e.g., Fig.~\\ref{fig:sequence_thick_layer}). Both gravity and surface tension are deforming the shapes, and we determine how these profiles depend on the velocity of the disc. We observe that there exists a universal profile which becomes more prominent for higher velocities, for which this universal shape will extend deeper into the fluid. Surprisingly, however, despite the observed universal behavior, these profiles do not agree with potential flow simulations we performed using a boundary integral technique. We attribute this difference to the formation of a vortex ring, which distorts the potential flow. We specifically show that the drift volume is \\emph{larger} than that predicted by potential flow.\n\n\\section{Experimental setup}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=70mm]{fig01}\n    \\caption{\\label{fig:setup} Schematic view of the experiment, with disc radius $R_D$ and disc velocity $V_D$. We start with the bottom of the disc at rest at the interface between a deep layer of oil (45 mm) on top of a deep layer of water (25 cm), after which we pull down the disc at constant speed $V_D$. We define $z=0$ at the undisturbed oil-water interface.}\n\\end{figure}\nThe experimental setup (Fig.~\\ref{fig:setup}) consisted of a water reservoir with a cross section of 15 cm by 15 cm and a height of 50 cm. A linear motor that was mounted below the tank pulls a disc with a radius $R_D=20~\\mathrm{mm}$ through the water surface at a constant speed $V_D$, by means of a thin rod (radius 3 mm) connecting the linear motor with the disc. The disc was accelerated with a maximum acceleration of $42~\\mathrm{m\/s^2}$ until the desired velocity ($V_D$) was reached. The events were recorded with a Photron SA2 high-speed color camera at frame rates ranging from $1$ to $8~\\text{kHz}$. The main control parameter of the experiment is an effective Froude number $\\Froude^*$, which is similar to the regular Froude number defined as the disc speed $V_D$, made dimensionless using the disc radius $R_D$ and the gravitational acceleration $g$. Only we replace $g$ by the effective gravitational acceleration $g^*$ of the oil phase inside the water phase, as one would use to determine the wave speed of gravitational waves on a density interface $g^*=g(\\rho_w-\\rho_o)\/(\\rho_w+\\rho_o)$ \\cite{Kundu2004_waves}, yielding\n\\begin{equation}\n    \\Froude^*=\\frac{V_D^2}{gR_D}\\left(\\frac{\\rho_w+\\rho_o}{\\rho_w-\\rho_o}\\right),\n\\end{equation}\nwhere $\\rho_w$ and $\\rho_o$ are the densities of water and oil respectively. In our experiments we used sunflower oil, which has a density $\\rho_o=900~\\mathrm{kg\/m^3}$ and a viscosity $\\nu\\sim50\\cdot10^{-6}\\mathrm{m^2\/s}$. Next to demineralized water we used a solution of table salt in water to increase the density of the water phase. We dissolved $1.0~\\mathrm{kg}$ of table salt in $5000~\\mathrm{ml}$ water, resulting in $\\rho_{sw}=1140~\\mathrm{kg\/m^3}$. The thickness of the oil layer in these experiments was $45~\\mathrm{mm}$, which was thick enough to be considered as infinite. We verified this by performing the same experiment with increased oil layer thicknesses of $90~\\mathrm{mm}$ and $135~\\mathrm{mm}$, which did not influence our results.\n\n\n\\section{Results}\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=86mm]{fig02}\n    \\caption{\\label{fig:sequence_thick_layer}\n    Snapshots from experiments with two different values of the effective Froude number $\\Froude^*=2.7$ ($a$i-$c$i), and $\\Froude^*=43$ ($a$ii-$c$ii). Corresponding pictures have been taken at the same dimensionless times $\\tau=2,~\\tau=3,~\\tau=4$. In the top experiment gravity has a clear influence on the shape of the entrained oil column. The bottom experiment is in the inertial regime, where gravity has negligible influence. A vortex ring is formed above the disc, which grows with the same dimensionless rate, independent of the Froude number. Black dashed line is the result from boundary integral simulations.\n    }\n\\end{figure}\nFigure \\ref{fig:sequence_thick_layer} shows two experiments in which we pulled down the disc from the oil-water interface at different velocities $V_D = 0.25$ and $1.0~\\mathrm{m\/s}$. Initially the disc was at rest and its lower surface was aligned with the oil-water interface. Then, the disc was set into motion and in a short period of time obtained a constant speed $V_D$ \\footnote{With an acceleration of $42~\\mathrm{m\/s^2}$, it takes $0.024~\\mathrm{s}$ to reach $V_D=1.0~\\mathrm{m\/s}$ ($\\Froude^*=43$). The duration of the experiment in that case is $0.08~\\mathrm{s}$. The acceleration does not significantly influence the experiment, as can be appreciated in Fig.~\\ref{fig:all_profiles}}. A vortex ring appeared just above the disc, along with a smooth profile in the center which connects to the thick oil layer at the top. We first focus on this smooth profile, which has similarities with the shapes seen in \\cite{Lian1989}, although that study  concentrated on the formation of the vortex ring.\n\nIn order to compare the experiments for different disc speeds, we define a dimensionless time $\\tau=z(t)\/R_D$, where z(t) is the depth the disc has reached at time $t$ after starting from $z=0$ at $t=0$, i.e., at equal dimensionless times the disc has reached the same vertical position below the undisturbed oil-water interface, measured in units of the disc radius $R_D$. By varying the acceleration we verified that at equal target velocities $V_D$ and dimensionless times $\\tau$ the observed column shapes are independent of the precise value of the acceleration used in experiments~\\footnote{We have used accelerations ranging from $21$ to $42~\\mathrm{m\/s^2}$ and found no discernible differences in the shape of the entrained oil column, up to the point in time that gravity becomes significant, where small differences would be introduced due to the somewhat longer time that is needed to reach a depth $z(t)$ at small accelerations.}. If the disc would be moving at a constant velocity $V_D$ all the time, i.e., if the acceleration phase would be infinitely small, then $\\tau$ and $t$ would be related as $\\tau=tV_D\/R_D$. In the remainder of the article we will for simplicity ignore the existence of the acceleration phase and take $z(t) = V_D t$.\n\nComparing Fig.~\\ref{fig:sequence_thick_layer}(\\emph{a}i) and (\\emph{a}ii), we see that at $\\tau=2$ the shape of the entrained oil is very similar for $\\Froude^*=2.7$ and $\\Froude^*=43$, although the amount of vorticity in the vortex ring appears to be\nmuch larger for the higher speed. At $\\tau=3$ the effect of buoyancy becomes visible for the lower Froude number, where a difference in the shape of the entrained oil between Fig.~\\ref{fig:sequence_thick_layer}(\\emph{b}i) and (\\emph{b}ii) is appreciable. In the last frame, Fig.~\\ref{fig:sequence_thick_layer}(\\emph{c}i) and (\\emph{c}ii) at $\\tau=4$, the oil in the case of $\\Froude^*=2.7$  has clearly moved back up due to buoyancy, leaving only a relatively straight cylinder of oil behind. For $\\Froude^*=43$, the shape is still unaffected by gravity at this point in time.\n\nIn order to see what the effect of the density difference is on the shape that we obtain at high Froude numbers, we performed experiments with demineralized water ($\\rho=998~\\mathrm{kg\/m^3}$) at $V_D=1~\\mathrm{m\/s}$ ($\\Froude^*=99$) and with salt water ($\\rho=1140~\\mathrm{kg\/m^3}$) at $V_D=1.5~\\mathrm{m\/s}$ ($\\Froude^*=97$). Although there is a factor two in the density difference between the oil and the water phases (demineralized vs. salt water), the difference in the profiles is negligible, which leads us to conclude that the shape of the entrained oil column does not strongly depend on the relative density difference between the fluids. The use of salt water does however have an experimental advantage: The oil-water interface became less contaminated with oil and water droplets after the experiment is finished, which reduced the time that we had to wait between two experiments until the surface was smooth enough to clearly observe formation of the profile of the entrained oil. For experimental convenience, we used salt water in all experiments, except noted otherwise.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=85mm]{fig03}\n    \\caption{\\label{fig:all_profiles}\n    Profiles of the entrained oil, for effective Froude numbers ranging from $2.7$ to $97$. We observe universal profiles for $\\Froude^*\\rightarrow\\infty$, each color in the image corresponds to one Froude number. (\\emph{a}) at $\\tau=2$, all shapes collapse. (\\emph{b}) at $\\tau=3$, a difference becomes visible for $\\Froude^*=2.7$ and $\\Froude^*=5.3$ (\\emph{c}) $\\tau=4$, for increasing $\\Froude^*$, the shapes of the entrained oil converge to a single universal profile independent of $\\Froude^*$. ($d$-$f$) shows the same profiles (only retaining shapes fulfilling the universality criterium $\\tau^2 < Fr^*$) together with the boundary integral simulation results (dotted lines). The position and size of the disc is indicated by the black rectangle. The profile of the oil-water interface is not shown in the vicinity of the disc because the profile was not visible due to the vortex ring, see Fig. \\ref{fig:sequence_thick_layer}.\n    }\n\\end{figure}\n\nIn Fig.~\\ref{fig:all_profiles} we compare the profiles of the entrained oil at equal dimensionless times $\\tau$ and for a wide range of Froude numbers. Every experiment in Fig.~\\ref{fig:all_profiles}(\\emph{a-c}) consists of several repetitions of the experiment, sometimes using two different disc accelerations ($21$ and $42~\\mathrm{m\/s^2}$), indicating the excellent reproducibility of the experiment and the irrelevance of the initial startup motion of the disc and the actual acceleration that is used in this startup phase.\n\nThe appearance of differences in the shapes shown in Fig.~\\ref{fig:all_profiles}(\\emph{a-c}) are a result of gravity that is pushing the oil phase upwards. This will only happen if the time is long enough for gravity to become more important than the inertia that is pulling the oil phase down. We can predict the moment that differences appear by comparing the inertial time scale $t_{in}\\equiv R_D\/V_D$ to the gravitational time scale $t_g \\equiv\\sqrt{R_D\/g^*}$. Gravity is expected to play a role only if $t\\gtrsim t_g$, which, after dividing both sides by the inertial time scale can be written as\n\\begin{equation}\n    \\tau^2\\gtrsim\\Froude^*,\n    \\label{eq:OilTimeScales}\n\\end{equation}\nwhere we have used $t\/t_{in}=\\tau$. If we now again look at Fig.~\\ref{fig:all_profiles}(\\emph{a-c}), we expect according to Eq.~(\\ref{eq:OilTimeScales}) to see a difference for $\\Froude^*\\lesssim4$ at $\\tau=2$, for $\\Froude^*\\lesssim9$ at $\\tau=3$, and for $\\Froude^*\\lesssim16$ at $\\tau=4$. These predictions agree well with the moment that we observe differences in the experimental profiles in Fig.~\\ref{fig:all_profiles}(\\emph{a-c}).\n\nWe now proceed to compare our experimental findings to potential flow solutions. For this, we use the boundary integral method as described in~\\cite{Oguz1993,Bergmann2009,Gekle2010d,Gekle2011a,Pozrikidis2011}. We performed boundary integral simulations of an impulsively started disc in an infinite bath of fluid, moving at a constant speed $V_D$. The disc had the same size and thickness as in the experiment. We injected tracers at the position corresponding to the initial oil-water interface, \\textit{i.e.}, aligned with the bottom of the disc. The tracers were then advected with the flow field around the disc. Because we used tracers in an infinite bath of a single liquid, the motion of the tracers corresponds to the case of $Fr^*\\rightarrow\\infty$. To validate the numerical code, we verified our simulations by calculating the displaced volume in the case of a sphere. We found good agreement with the analytical results of Eames~\\textit{et al.}~\\cite{Eames1994} for $\\rho_{max}\/|x_0|\\rightarrow\\infty$, where $\\rho_{max}$ is the radius of the reference plane that is taken into account in the calculation of the drift volume and $x_0$ the initial axial distance of the sphere from this plane (see Eames~\\textit{et al.}~\\cite{Eames1994} for details). We note that in our experiments the initial position of the disc is at the reference plane (the oil-water interface), such that $x_0\\rightarrow0$, and, consequently, $\\rho_{max}\/|x_0|\\rightarrow\\infty$.\n\nFig.~\\ref{fig:all_profiles}\\emph{d-f} shows that the shapes are approximated by our simulation results, but also that there exists a significant difference close to the undisturbed surface. A closer inspection reveals that the discrepancy increases as the disc moves further down: while at $\\tau=2$ ($d$) there is a reasonable agreement between the simulation and experiment, at $\\tau=4$ ($f$) the difference is much larger. The same discrepancy is also illustrated in Fig.~\\ref{fig:sequence_thick_layer}($a$ii-$c$ii).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig04}\n    \\caption{\\label{fig:radiusTime}\n    Radius of the entrained column oil as a function of dimensionless time $\\tau$ at different depths $z$ for the experiment (closed symbols) and the simulation (solid lines). Blue circles: $z\/R_D=-0.5$, green diamonds: $z\/R_D=-1.0$, red triangles: $z\/R_D=-1.5$, black squares: $z\/R_D=-2.0$.}\n\\end{figure}\n\nWe further investigate the development of this difference between the experiments and simulation in Fig.~\\ref{fig:radiusTime}, where we plot the radius of the oil\/water interface at different depths $z$ below the location of the undisturbed interface at $z=0$. First, we observe that the larger the distance to the undisturbed surface is, the smaller the discrepancy becomes. The difference in radius even appears to switch sign at the deepest position plotted here (black squares), although this difference is close to the experimental uncertainty. Second, the figure shows an apparent qualitative difference between the experiments and simulations at $z\/R_D=-0.5$, for which the simulation shows an initial decrease of the radius, which then seemingly asymptotes to a constant value. The experiment on the other hand shows a significant increase in radius after the initial decrease~\\footnote{Actually, also in the simulations we observe such an increase, only much smaller ($<2\\%$ of the minimum radius) than in the experiment}.\n\nA qualitative explanation for both these observations originates from the formation and growth of the vortex which introduces an additional downward velocity (and velocity gradient) inside the entrained volume of oil, since oil will continuously flow into the vortex. There where the shape of the entrained oil volume $r(z,t)$ is column-like (i.e., $\\partial r\/\\partial z$ is small), this will lead to simple stretching and thus a decreasing radius of the column~\\cite{Eggers2008}. However, where there exists a large gradient in the radius, the overall downward translation may introduce an increase in radius in the lab frame. This can be seen as follows. Because the volume $\\propto r^2v_z$ of the entrained oil is conserved, the interface $r(z,t)$ obeys the PDE\n\\begin{equation}\n\t\\frac{\\partial r^2}{\\partial t} + \\frac{\\partial}{\\partial z}(r^2 v_z)=0,\n\\end{equation}\nwhere $v_z(z,t)$ ($\\geq 0$) is the downward velocity in the vertical direction inside the column, which for simplicity is assumed to be independent of the radial coordinate. This can immediately be rewritten as\n\\begin{equation}\n\t\\frac{\\partial r}{\\partial t} + v_z\\frac{\\partial r}{\\partial z} + \\frac{1}{2}r\\frac{\\partial v_z}{\\partial z}=0.\n\\end{equation}\nNote that for such a stretching flow the second term in the above expression is always negative, since $\\partial r\/\\partial z < 0$ for the measured profiles, whereas the third (or stretching) term is always positive because $\\partial v_z\/\\partial z \\geq 0$. In case of a purely columnar shape, the second term is zero, and therefore $\\partial r\/\\partial t$ is negative. In case of a funnel-shaped profile with a strong radial gradient ($\\partial r\/\\partial z \\ll 0$) and sufficient downward velocity, the magnitude of the second term may well\nbe larger than the stretching term, which will result in a positive $\\partial r\/\\partial t$.\n\nThe remaining question is why we still observe universal shapes even though there is a clear influence from the vortex. The reason is that for early times ($\\tau\\lesssim8$) the\ndimensionless size and strength of a vortex behind a impulsively started disc is independent of the disc speed, as was shown in~\\cite{Yang2012}. That the same holds for our experiments can clearly be seen in Fig.~\\ref{fig:sequence_thick_layer}, where the vortex for $Fr^*=2.7$ has the same size as that for $Fr^*=43$, when they are compared at the same dimensionless time.\n\nWe now proceed to providing an estimate of the magnitude of the downward velocity $v_z$ due to the presence of the vortex. Given the circulation $\\Gamma$ of a vortex line in a closed loop $C$, the velocity $\\bf v$ at position $\\bf r$ can be obtained from the Biot-Savart law\n\\begin{equation}\n\t{\\bf v}({\\bf r}) = \\frac{\\Gamma}{4\\pi}\\oint_C\\frac{d{\\bf l}\\times({\\bf l} - {\\bf r})}{|{\\bf l} - {\\bf r}|^3},\n\t\\label{eq:BiotSavart}\n\\end{equation}\nwhere $\\bf l$ is the core of the vortex line. For a ring-shaped vortex with a radius equal to the disc radius $R_D$ we can write for points $\\bf r$ on the axis of symmetry that, taking into account that the velocity is purely vertical, $|{\\bf l} - {\\bf r}|=\\sqrt{R_D^2+(z')^2}$ and ${\\bf \\hat{e}_z} \\cdot (d{\\bf l}\\times({\\bf l} - {\\bf r})) = R_D dl$, with $\\bf \\hat{e}_z$ the unit vector in the $z$-direction, $dl=R_Dd\\theta$, and $z'=z-z_c$, where $z_c$ is the vertical position of the core of the vortex. The $z$-component of the integral of Eq.~(\\ref{eq:BiotSavart}) can now be evaluated straightforwardly as\n\\begin{equation}\n\tv_z(z') = \\frac{\\Gamma}{4\\pi}\\int_0^{2\\pi}\\frac{R_D^2}{(R_D^2 + z'^2)^{3\/2}}d\\theta\n\\end{equation}\nand gives the vertical component of the velocity $v_z$ as\n\\begin{equation}\n\tv_z = \\frac{R_D^2\\Gamma}{2(R_D^2+z'^2)^{3\/2}},\n\\end{equation}\nor, in dimensionless form,\n\\begin{equation}\n\t\\tilde v_z = \\frac{\\tilde\\Gamma}{2(1+\\tilde z'^2)^{3\/2}},\n\\end{equation}\nwhere $\\tilde v_z = v_z\/V_D$, $\\tilde z = z\/R_D$, and the dimensionless, time-dependent circulation $\\tilde\\Gamma=\\Gamma\/(R_DV_D)$ is independent of the disc speed~\\cite{Yang2012}. In Fig.~\\ref{fig:vzVortex} we compare the vertical velocity on the axis of symmetry resulting from our potential flow calculation to the one induced by the vortex ring at different instances of time. We have used the empirical relation from Yang~\\emph{et al.}~\\cite{Yang2012} for $\\tilde\\Gamma(\\tau)$~\\footnote{$\\tilde\\Gamma\\approx-4.67(1-0.75e^{-0.416\\tau} - (1-0.75)e^{-24.927\\tau})$} and approximated the velocity of the vortex core to be linear (i.e., $\\tilde z'=\\tilde z_D + 0.25\\tau$). The latter approximately matches the position of the vortex core in the experiments, where we observe that the core is about one disc radius above the disc at $\\tau=4$ (see Fig.~\\ref{fig:sequence_thick_layer}). Clearly, the velocities and velocity gradients are greatly enhanced by the starting vortex, and an influence on the shape of the entrained oil column is therefore expected. Most importantly note that the velocity profiles of Fig.~\\ref{fig:vzVortex} are independent of the disc velocity, resulting in a mechanism through which the universality of the shapes shown in Fig.~\\ref{fig:all_profiles} are preserved.\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig05}\n    \\caption{\\label{fig:vzVortex}\n    Comparison of the estimated non-dimensional velocity $\\tilde v_z = v_z\/V_D$ on the axis of symmetry as a function of the non-dimensional distance $\\tilde z -\\tilde z_D = (z-z_D)\/R_D$ to the center of the disc for the potential flow solution (solid line) and the time-dependent contribution from the vortex ring (dotted lines) at four different instances in dimensionless time $\\tau = tR_D\/V_D$.}\n\\end{figure}\n\nTo further quantify the influence of the vortex on the drift volume, we calculate the entrained volume of oil in both the simulations and the experiment. Because a part of the interface is masked by the vortex, we do not calculate the complete displaced volume~\\cite{Eames1994}, but only compare the part that is accessible in both the experiment and the simulation. In the simulations we find, at $\\tau=4$, between the depths $z\/R_D=-0.2$ and $z\/R_D=-1.0$ (with $z=0$ at the unperturbed oil\/water interface), an entrained volume of 4.24 ml, while in the experiment 7.25 ml of oil is entrained. Extending the range to $z\/R_D=-2.0$ gives 5.57 ml and 8.66 ml for the simulation and experiment respectively. Clearly, the experimental volume of entrained oil is significantly larger than the volume predicted by the potential flow simulations.\n\n\n\\section{Conclusions and outlook}\nWe have performed experiments where we started a disc at an oil-water interface and pulled it down at a constant speed. We have shown that at high speeds, gravity and surface tension can be neglected, and the  entrained oil obtains a universal funnel shape, independent of the Froude number. However, a vortex ring is formed at the disc edge, which influences the shape of the entrained oil resulting in a qualitative and quantitative difference compared to the potential flow solution. Nonetheless, the universality of the funnel shape is conserved. The shape also appears insensitive to the relative density difference, at least for the density differences studied.\n\nThe effect might be investigated further for flows at lower Reynolds numbers, exploring the limit where the vortex disappears. In our current setup this regime is inaccessible due to the influence of gravity at low disc speeds. A consequence of our finding is that the displaced volume as predicted by \\citep{Darwin1953} is underestimated in cases where flow separation is of importance. This introduces a non-trivial shape-effect on the entrainment in the wake of, for example, ocean life \\cite{Katija2009}. The preserved universality in such cases, however, may help in simplifying and generalizing analysis in these situations.\n\n\\begin{acknowledgments}\nWe acknowledge financial support from FOM and the NWO-Spinoza program.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nA number of fundamental problems in algebraic topology can be\ndescribed as measuring the extent to which a given functor \\ \n$T:{\\mathcal C}\\to{\\mathcal D}$ \\ between model categories induces an equivalence of\nhomotopy categories: more specifically, which objects (or maps) from\n${\\mathcal D}$ are in the image of $T$, and in how many different ways. For example\\vspace{2 mm}:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\alph{enumi})\\ }\n\\item How does one distinguish between different topological spaces\nwith the same homology groups, or with chain-homotopy\nequivalent chain complexes?  How can one realize a given map of\nchain complexes up to homotopy\\vspace{2 mm}?\n\\item When do two simply-connected topological spaces have the same\nrational homotopy type\\vspace{2 mm}?  \n\\item When is a given topological space a suspension, up to homotopy?\nDually, how many distinct loop space structures, if any, can a given topological\nspace carry\\vspace{2 mm}?\n\\item Is a given $\\Pi$-algebra\\ (that is, a graded group with an action\nof the primary homotopy operations) realizable as the homotopy\ngroups of a topological space, and if so, in how many ways\\vspace{2 mm}? \n\\end{enumerate}\n\nOur goal is to describe a unified approach to such problems that\nworks for functors between \\emph{spherical} model categories, for which\nseveral familiar concepts and constructions are available. These\ninclude a set ${\\mathcal A}$ of \\emph{models} (to play the role of spheres, in\nparticular determining the corresponding homotopy groups \\ $\\pinC{\\ast}$), \\ \nPostnikov systems, and $k$-invariants. If a functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\\nrespects this additional structure, we obtain a natural long exact\nsequence of the form: \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eles}\n\\dotsc\\to \\Gamma_{n}X~\\xra{s}~\\pinC{n} X~\\xra{h}~\n\\pinD{n} TX~\\xra{\\partial}~\\Gamma_{n-1}X\\dotsc~,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent which generalizes the EHP sequence, J.H.C. Whitehead's ``certain exact\nsequence'', and the spiral exact sequence of Dwyer, Kan, and\nStover. See \\ \\eqref{efive} \\ below.\n\nUnder these hypotheses, given an object $G$ in ${\\mathcal D}$, we want to find\nan object $X$ in ${\\mathcal C}$ with \\ $TX\\simeq G$. \\ The key step is to choose \\ \n$\\pinC{\\ast} X$ \\ which fits into \\ \\eqref{eles}. \\ We describe an inductive \nprocedure for doing this, using the Postnikov systems in both\ncategories, together with an obstruction theory for lifting $G$ to\n${\\mathcal C}$, along the following lines: \n\n\\begin{tha}\nGiven \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ and \\ $G\\in{\\mathcal D}$ \\ as above, for each \\ $X\\in{\\mathcal C}$ \\ \nwith \\ $TX\\simeq G$, \\ there is a tower of fibrations in ${\\mathcal C}$:\n$$\n\\dotsb \\xra{p\\q{n+1}} \\Xpn{n+1} \\xra{p\\q{n}} \\Xpn{n} \\xra{p\\q{n-1}} \\dotsb\n\\xra{p\\q{0}} \\Xpn{0}~,\n$$\ncalled the \\emph{modified Postikov tower for} $X$ (Def.\\ \\ref{dmps}), \nwith $G$ mapping compatibly to \\ $T\\Xpn{n}$ \\ for each $n$, and \\\n$X\\simeq\\operatorname{holim}_{n} \\Xpn{n}$.\n   \nConversely, given such a tower up to level $n$, the obstruction to\nextending it to level \\ $n+1$ \\ lies in \\\n$\\HL{n+3}{G}{\\Gamma_{n+1}\\Xpn{n}}$, \\ and the choices for \\ $\\Xpn{n+1}$ \\ \nare classified by:\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{$\\bullet$~}\n\\item a class in \\ $\\HL{n+2}{G}{\\Gamma_{n+1}{\\Xpn{n}}}$;  \n\\item a class in \\ $\\HL{n+2}{\\Xpn{n}}{K_{n+1}}$, \\ where \\ \n$K_{n+1}:=\\operatorname{Coker}\\,\\pi_{n+2}\\rho\\q{n}$, \\ for \\ \n$\\rho\\q{n}:P_{n+2}G\\to P_{n+2}T\\Xpn{n}$.\n\\end{enumerate}\n\\end{tha}\n\nSee Theorem \\ref{tfour}.\n\n\\subsection{Related work}\n\\label{srw}\\stepcounter{thm}\n\nThe comparison problems discussed above are familiar ones in \nalgebraic topology:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\alph{enumi})\\ }\n\\item The question of the realizability of a graded algebra as a\n  cohomology ring was first raised explicitly by Steenrod in\n  \\cite{SteCA}, but it goes back to Hopf (in \\cite{HopfT}) in the\n  rational case. The ``Steenrod problem'' of realizing a given \\\n  $\\pi_{1}$-action in homology has been studied, for example, \n  in \\cite{ThP,JRSmiT1}. \n\\item The comparison between integral and rational homotopy type \n  was implicit in the notion of a Serre class (cf.\\ \\cite{SerG,ACuH}),\n  although an explicit formulation was only possible after the \n  construction of the rationalization functors of Quillen and Sullivan\n  in \\cite{QuiR,SulG}.\n\\item Possible loop space structures on a given $H$-space were\n  analyzed extensively, starting with the work of Sugawara and\n  Stasheff (cf.\\ \\cite{SugG,StaH}). The dual question on\n  identifying suspensions has also been studied (see, e.g., \\cite{BHilS}).\n\\item The question of the realizability of homotopy groups goes back\n  to J.H.C.~Whitehead, in \\cite{JWhR} (see also \\cite{JWhSH}), and has\n  reappeared in recent years in the context of $\\Pi$-algebra s (cf.\\\n  \\cite{DKStE,DKStB}). The relationship between homology and homotopy\n  groups, which is relevant to the realization problem for both, was\n  studied in  \\cite{JWhSB,JWhC} (in which the ``certain exact sequence'' \n  was introduced)\\vspace{2 mm}.\n\\end{enumerate}\n\nIn \\cite{BauCF}, H.-J.\\ Baues gave what appears to be the first \ngeneral theory covering a wide spectrum of such realization problems.\nThis was an outgrowth of his earlier work on classifying\nhomotopy types of finite dimensional CW complexes in\n\\cite{BauCHF,BauHH} (which in turn builds on \\cite{JWhSF}). \n\nHis initial setting consists of a homological cofibration \ncategory ${\\mathcal C}$ (corresponding to, and extending, the notion of a\nresolution model category) under a theory  of coactions $\\mathbb T$\n(corresponding to the category \\ $\\Pi_{\\A}$ \\ of \\S \\ref{ssmod}). \nBaues then constructs a generalized ``certain exact sequence'' \nsimilar to \\ \\eqref{eles}, \\ and\nprovides an inductive obstruction theory for realizing a chain complex\n(or a chain map) by a $\\mathbb T$-complex (corresponding to a CW complex, or\nmore generally a cofibrant object in ${\\mathcal C}$) \\ -- \\ see \n\\cite[VI, (2.2-2.3)]{BauCF}). \n\nThese results apply inter alia to the problem of realizing \na chain complex by a topological space (the motivating example \nfor Baues's approach), as well as to the realization of a $\\Pi$-algebra\\ \n(cf.\\ \\cite[D, (7.9)]{BauCF}).  However, here we consider \nfunctors between two different model categories that \nare not covered by \\cite{BauCF}. In particular, our original\nmotivating example \\ -- \\ the realization of a \\emph{simplicial} $\\Pi$-algebra\\\n(by a simplicial space) \\ -- \\ shows that in the relative context a\nmore refined obstruction theory may be necessary:  \ncompare Theorem (2.3) of \\cite[VI]{BauCF} with Theorem \\ref{tfour} below.\n\n\\begin{remark}\\stepcounter{subsection}\nAnother set of closely related questions  \\ -- \\ which do not quite fit\ninto the framework described here, though they can also be stated as\nrealization problems \\ -- \\  arise in categories of structured ring\nspectra; see for example \\cite{RobO} and \\cite[Cor.\\ 5.9]{GHopkM}\\vspace{2 mm} . \n\\end{remark}\n\n\\subsection{Notation and conventions}\n\\label{snac}\\stepcounter{thm}\n\n$\\TT_{\\ast}$ \\ denotes the category of pointed connected topological\nspaces; \\ $\\Set_{\\ast}$ \\ that of pointed sets, and \\ ${\\EuScript Gp}$ \\ that of\ngroups. \\ For any category ${\\mathcal C}$, \\ $\\operatorname{gr}{\\mathcal C}$ \\ denotes the category of \nnon-negatively graded objects over ${\\mathcal C}$, and \\ $s{\\mathcal C}$ \\ the category of\nsimplicial objects over ${\\mathcal C}$. \\ $s{\\EuScript Set}$ \\ is denoted by ${\\mathcal S}$, \\\n$s\\Set_{\\ast}$ \\ by \\ $\\Ss_{\\ast}$, \\ and \\ $s{\\EuScript Gp}$ \\ by \\ ${\\mathcal G}$. \\ The constant\nsimplicial object an an object \\ $X\\in{\\mathcal C}$ \\ is written \\ $\\co{X}\\in s{\\mathcal C}$. \n\nIf ${\\mathcal C}$ has all coproducts, then given \\ $A\\in{\\mathcal S}$ \\ and \\ $X\\in{\\mathcal C}$, \\\nwe define \\ $X\\hat{\\otimes} A\\in s{\\mathcal C}$ \\ by \\ \n$(X\\hat{\\otimes} A)_{n}:=\\coprod_{a\\in A_{n}} X$, \\ \nwith face and degeneracy maps induced from those of $A$. \nFor \\ $Y\\in s{\\mathcal C}$, \\ define \\ $Y\\otimes A\\in s{\\mathcal C}$ \\ by \\ \n$(Y\\otimes A)_{n}:= \\coprod_{a\\in A_{n}} Y_{n}$ \\ (the diagonal of the\nbisimplicial object \\ $Y\\hat{\\otimes} A$) \\ -- \\ so that for \\ $X\\in{\\mathcal C}$ \\\nwe have \\ $X\\hat{\\otimes} A=\\co{X}\\otimes A$.\n\nThe category of chain complexes of $R$-modules is denoted by \\\n${\\EuScript Chain}_{R}$ \\ (or simply \\ ${\\EuScript Chain}$, \\ for $R=\\mathbb Z$).\n\n\\subsection{Organization:}\n\\label{sorg}\\stepcounter{thm}\n\nIn Section \\ref{csmc} we define \\emph{spherical} model categories,\nhaving the additional structure mentioned above. \nMost examples of such categories are in particular \\emph{resolution} \nmodel categories, which are described in Section \\ref{crmc}; we\nexplain how to produce the needed structure for them in Section \\ref{csp}.\nWe define \\emph{spherical functors} between such categories, and\nconstruct the comparison exact sequence for them, in\nSection \\ref{csf}. This is applied in Section \\ref{ccps} to study the\neffect of a spherical functor on Postnikov systems. Finally, in\nSection \\ref{cfib} we construct an obstruction theory as above for the\nfiber of a spherical functor. In Section \\ref{cat} we indicate how the\ntheory works for the above examples.\n\n\\subsection{Acknowledgements}\n\\label{sack}\\stepcounter{thm}\n\nI would like to thank Paul Goerss for many hours of discussion on various\nissues connected with this paper, and especially for his essential\nhelp with Sections \\ref{ccps}-\\ref{cfib}, the technical core of this\nnote. I would also like to thank Hans Baues for explaining the\nrelevance of his work in \\cite{BauCF} to me.\n\n\\setcounter{thm}{0}\\section{Spherical model categories}\n\\label{csmc}\n\nBefore defining the additional structure we shall need, we briefly\nrecapitulate the relevant homotopical algebra:\n\n\\subsection{Model categories}\n\\label{smc}\\stepcounter{thm}\n\nRecall that a \\emph{model category} is a bicomplete category ${\\mathcal C}$\nequipped with three classes of maps: weak equivalences,\nfibrations, and cofibrations, related by appropriate lifting\nproperties. By inverting the weak equivalences we obtain the\nassociated homotopy category \\ $\\operatorname{ho}{\\mathcal C}$, \\ with morphism set \\ \n$[X,Y]=[X,Y]_{{\\mathcal C}}$. \\ We shall concentrate on \\emph{pointed} model\ncategories (with null object $\\ast$). \\ See \\cite{QuiH} or \\cite{PHirM}.   \n\n\\subsection{The set of models}\n\\label{ssmod}\\stepcounter{thm}\n\nThe additional initial data that we shall require for our model\ncategory consists of a set ${\\mathcal A}$ of cofibrant homotopy cogroup objects\nin ${\\mathcal C}$, called \\emph{models} (playing the role of the spheres in $\\TT_{\\ast}$).\nGiven such a set ${\\mathcal A}$, let \\ $\\Pi_{\\A}$ \\ denote the smallest subcategory of ${\\mathcal C}$\ncontaining ${\\mathcal A}$ and closed under weak equivalences, arbitrary\ncoproducts, and suspensions. Note that every object in \\ $\\Pi_{\\A}$ \\ is a\nhomotopy cogroup object, too.  \n\n\\begin{example}\\label{esmod}\\stepcounter{subsection}\nLet \\ ${\\mathcal C}={\\mathcal G}$ \\ be the category of simplicial groups, \\\n$S^{k}=\\Delta[k]\/\\partial\\Delta[k]$ \\ the standard simplicial $k$-sphere\nin\\ $\\Ss_{\\ast}$, \\ $G:\\Ss_{\\ast}\\to{\\mathcal G}$ \\ the Kan's loop functor (cf.\\ \\cite[\\S 26.3]{MayS}), \nand \\ $F:\\Ss_{\\ast}\\to{\\mathcal G}$ \\ the free group functor. For each \\ $n\\geq 1$,  \\  \n$\\gS{n}:= GS^{n}\\in{\\mathcal G}\\cong FS^{n-1}$ \\ will be called the\n\\textit{$n$-dimensional ${\\mathcal G}$-sphere}, \\ with \\\n$\\Sigma^{k}\\gS{n}\\simeq\\gS{n+k}$. \\  These, and their coproducts, are\ncofibrant strict cogroup objects for ${\\mathcal G}$. \\ Here \\\n${\\mathcal A}:=\\{\\gS{1}=\\co{\\mathbb Z}\\}$; \\ in fact, throughout this paper ${\\mathcal A}$ will\nbe either a singleton, or countable. \n\\end{example}\n\n\\begin{remark}\\label{ssg}\\stepcounter{subsection}\nThe adjoint pairs of functors:\n$$\n\\TT_{\\ast}\\ \\ \\substack{S\\\\ \\rightleftharpoons\\\\ \\|-\\|}\\ \\ \\Ss_{\\ast}\\ \\ \n\\substack{G\\\\ \\rightleftharpoons\\\\ \\bar{W}}\\ \\ {\\mathcal G}\n$$\n\\noindent induce equivalences of the corresponding homotopy categories \\ -- \\ \nwhere \\ $\\bar{W}:{\\mathcal G}\\to\\Ss_{\\ast}$ \\ is the \nEilenberg-Mac~Lane classifying space functor, \\ $S:\\TT_{\\ast}\\to\\Ss_{\\ast}$ \\ is the \nsingular set functor, and \\ $\\|-\\|:\\Ss_{\\ast}\\to\\TT_{\\ast}$ \\ is the geometric realization \nfunctor (cf.\\ \\cite[\\S 14,23]{MayS}). Thus to study the usual homotopy category\nof (pointed connected) topological spaces, we can work in ${\\mathcal G}$ (or \\ $\\Ss_{\\ast}$), \\ \nrather than \\ $\\TT_{\\ast}$. \n\\end{remark}\n\n\\begin{defn}\\label{dpis}\\stepcounter{subsection}\nIf ${\\mathcal A}$ is a set of models for ${\\mathcal C}$, then given \\ $X\\in{\\mathcal C}$, \\  for\neach \\ $A\\in{\\mathcal A}$ \\ let \\ $\\pia{k}(X):=[\\Sigma^{k}A,X']_{{\\mathcal C}}$, \\\nwhere \\ $X'\\to X$ \\ is a (functorial) fibrant replacement. We write \\ \n$\\pinC{k} X$ \\ for \\ $(\\pia{k}X)_{A\\in{\\mathcal A}}$, \\ and \\ \n$\\pinC{\\ast} X:=(\\pinC{k}X)_{k=0}^{\\infty}$.\n\\end{defn}\n\n\\subsection{Theories and algebras}\n\\label{staa}\\stepcounter{thm}\n\nRecall that a \\emph{theory} is a small category  $\\Theta$ with \nfinite products (so in particular, an FP-sketch \\ -- \\ cf.\\ \n\\cite[\\S 5.6]{BorcH2}), and a $\\Theta$-\\emph{algebra} (or\n\\emph{model}) is a product-preserving functor \\ $\\Theta\\to{\\EuScript Set}$. \\ \nThink of $\\Theta$ as encoding the operations and relations for \na ``variety of universal algebras'', the category \\ $\\Alg{\\Theta}$ \\ \nof $\\Theta$-algebras (which is \\emph{sketched} by $\\Theta$).\n\nFor example, the obvious category $\\mathfrak{G}$, which sketches groups, is\nequivalent to the opposite of the homotopy category of (finite) wedges\nof circles. An $\\mathfrak{G}$-\\emph{theory} $\\Theta$\n(cf.\\ \\cite[\\S 2]{BPescF}) is one equipped with a map of theories \\ \n$\\coprod_{S}\\,\\mathfrak{G}\\to\\Theta$ \\ (coproduct taken in the category of\ntheories, over some index set $S$) which is bijective on objects. This\nimplies that each $\\Theta$-algebra has the underlying \nstructure of an $S$-graded group, so that \\ $\\Alg{\\Theta}$ \\  \ncan be thought of as a ``variety of (graded) groups with operators'' \n(cf.\\ \\cite[I, (2.5)]{BauCF}).\n\n\\begin{remark}\\label{rpis}\\stepcounter{subsection}\nWe will assume that all the functors \\ $\\pinC{n}$ \\ ($n\\geq 0$) \\ take\nvalue in a category \\ $\\Alg{\\PiC}$ \\ sketched by a $\\mathfrak{G}$-theory $\\Theta$, \nand thus equipped with a faithful forgetful\nfunctor \\ $U_{{\\mathcal C}}:\\Alg{\\PiC}\\to{\\EuScript Gp}^{{\\mathcal A}}$ \\ into the category of\n${\\mathcal A}$-graded groups. \nThe objects of \\ $\\Alg{\\PiC}$ \\ are called \\ \\emph{$\\PiC$-algebra s}.\n\nFor topological spaces, with \\ ${\\mathcal A}=\\{\\bS{1}\\}$, \\ the $\\PiC$-algebra s are simply\ngroups. If we use rational spheres as the models, then \\ $\\Alg{\\PiC}$ \\ \nis the category of $\\mathbb Q$-vector spaces. A more interesting example\nappears in \\S \\ref{dpa} below. \n\\end{remark}\n\n\\subsection{Constructions based on models}\n\\label{scbm}\\stepcounter{thm}\n\nThere are a number of familiar constructions for topological spaces which we\nrequire for our purposes.  We can \\emph{define} them once we are given\na set of models ${\\mathcal A}$ as above, although they do not always exist (see\n\\S \\ref{snsrmc} below).  \n\n\\begin{defn}\\label{dfpt}\\stepcounter{subsection}\nA \\emph{Postnikov tower} (with respect to ${\\mathcal A}$) is a functor that\nassigns to each \\ $Y\\in{\\mathcal C}$ \\ a tower of fibrations: \n$$\n\\dotsc \\to P^{{\\mathcal A}}_{n}Y\\xra{p\\q{n}}P^{{\\mathcal A}}_{n-1}Y\\xra{p\\q{n-1}}\\dots \n\\to P^{{\\mathcal A}}_{0}Y~,\n$$\nas well as a weak equivalence \\ $r:Y\\to P^{{\\mathcal A}}_{\\infty}Y:=\\lim_{n}P^{{\\mathcal A}}_{n}Y$ \\ \nand fibrations \\ $P^{{\\mathcal A}}_{\\infty}Y\\xra{r\\q{n}}P^{{\\mathcal A}}_{n}Y$ \\ such that \\\n$r\\q{n-1}=p\\q{n}\\circ r\\q{n}$ \\ for all $n$. \\ Finally, \\ \n$(r\\q{n}\\circ r)_{\\#}:\\pinC{k}Y\\to\\pinC{k}(P^{{\\mathcal A}}_{n}Y)$ \\ \nis an isomorphism for \\ $k\\leq n$, \\ and \\ $\\pinC{k}(P^{{\\mathcal A}}_{n}Y)$ \\\nis zero for \\ $k>n$. \\ \n\nWhen ${\\mathcal A}$ is clear from the context, we denote \\ $P_{n}^{{\\mathcal A}}$ \\\nsimply by \\ $P_{n}$.\n\\end{defn}\n\n\\begin{example}\\label{epost}\\stepcounter{subsection}\nFor a free chain complex \\ $C_{\\ast}\\in{\\EuScript Chain}_{R}$ \\ of modules over a\nring $R$, we may take \\ $C'_{\\ast}:=P_{n}C_{\\ast}$ \\ where \\\n$C'_{i}=C_{i}$ \\ for \\ $i\\leq n+1$, \\ $C'_{n+2}=Z_{n+1}C_{\\ast})$, \\ and \\ \n$C'_{i}=0$ \\ for \\ $i\\geq n+3$. \\ The map \\ $r\\q{n}:C_{\\ast}\\to C'_{\\ast}$ \\ \nis defined by \\ $r\\q{n}_{n+2}:=\\partial_{n+2}:C_{n+2}\\to Z_{n+1}C_{\\ast}$. \n\\end{example}\n\n\\begin{defn}\\label{drem}\\stepcounter{subsection}\nGiven an $\\PiC$-algebra\\ \\ $\\Lambda$, \\ a \\emph{classifying object} \\ $B_{\\C}\\Lambda$ \\\n(or simply \\ $B\\Lambda$) \\ for $\\Lambda$ is any \\ $B\\in s{\\mathcal C}$ \\ such that \\\n$B\\simeq P_{0}K$ \\ and \\ $\\pinC{0}B\\cong\\Lambda$.\n\\end{defn}\n\nThe name is used by analogy with the classifying space of a group,\nwhich classifies $G$-bundles. One can interpret \\ $B_{\\C}\\Lambda$ \\ similarly,\nthough perhaps less naturally (see, e.g., \\cite[\\S 4.6]{BJTurR}).\n\n\\begin{defn}\\label{dmod}\\stepcounter{subsection}\nA  \\emph{module over a $\\PiC$-algebra} $\\Lambda$ is an abelian group object \nin \\ $\\Alg{\\PiC}\/\\Lambda$ (cf.\\ \\cite[\\S 2]{QuiC}), and the\ncategory of such is denoted by \\ $\\RM{\\Lambda}$. \n\\end{defn}\n\n\\begin{remark}\\label{rmodule}\\stepcounter{subsection}\nSince any $\\PiC$-algebra\\ is in particular a (graded) group, if \\\n$p:Y\\to\\Lambda$ \\ is a module, then \\ $Y=K\\times\\Lambda$  \\ \n(as sets!) for \\ $K:=\\operatorname{Ker}\\,(p)$, \\ with an appropriate $\\PiC$-algebra\\ structure \n(cf.\\ \\cite[\\S 3]{BlaG}). \\ We may call $K$ itself a\n$\\Lambda$-\\emph{module} (which corresponds to the traditional\ndescription of an $R$-module, for a ring $R$). \n\\end{remark}\n\n\\begin{example}\\label{emodule}\\stepcounter{subsection}\nFor any object \\ $X\\in{\\mathcal C}$ \\ as above, the \\ ${\\mathcal A}\\times\\mathbb N$-graded group \\ \n$\\pinC{\\ast} X$ \\ has an action of the ${\\mathcal A}$-primary homotopy\noperations, corepresented by the maps in \\ $\\operatorname{ho}\\Pi_{\\A}$ \\ (see \n\\S \\ref{dpa} below). In particular, one of these operations,\ncorresponding to the action of the fundamental group on the higher\nhomotopy groups, makes each \\ $\\pinC{n}X$ \\ ($n\\geq 1$) \\ into a\nmodule over $\\pinC{0}X$ \\ (see Fact \\ref{ffour} below).\n\\end{example}\n\n\\begin{defn}\\label{dem}\\stepcounter{subsection}\nGiven an abelian $\\PiC$-algebra\\ $M$ and an integer \\ \n$n\\geq 1$, \\ an \\emph{$n$-dimensional $M$-Eilenberg-Mac~Lane object} \\ \n$\\EC{M}{n}$ \\ (or simply \\ $\\EK{}{}{M}{n}$) \\ is any \\ $E\\in s{\\mathcal C}$ \\ such that \\ \n$\\pinC{n}E\\cong M$ \\ and \\ $\\pinC{k}E=0$ \\ for \\ $k\\neq n$.\n\\end{defn}\n\n\\begin{defn}\\label{deaem}\\stepcounter{subsection}\nGiven a $\\PiC$-algebra\\ $\\Lambda$, \\ a module $M$ over $\\Lambda$, and an integer \\ \n$n\\geq 1$, \\ an \\emph{$n$-dimensional extended $M$-Eilenberg-Mac~Lane object} \\ \n$\\ECL{M}{n}$ \\ (or simply \\ $\\EL{M}{n}$) \\ is any homotopy abelian group object \\ \n$E\\in s{\\mathcal C}\/\\Lambda$, \\ equipped with a section $s$ for \\ \n$p\\q{0}:E\\to P_{0}E\\simeqB\\Lambda$, \\ such that \\ \n$\\pinC{n}E\\cong M$ \\ as modules over $\\Lambda$; \\ and \\ $\\pinC{k}E=0$ \\ \nfor \\ $k\\neq 0,n$.\n\\end{defn}\n\n\\begin{defn}\\label{dki}\\stepcounter{subsection}\nGiven a Postnikov tower functor as in \\S \\ref{dfpt}, an $n$-th \n\\emph{$k$-invariant square} (with respect to ${\\mathcal A}$)  \nis a functor that assigns to each \\ $Y\\in{\\mathcal C}$ \\ a homotopy pull-back square:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{ezero}\n\\xymatrix@R=25pt{\\ar @{} [dr] |<<<{\\framebox{\\scriptsize{PB}}}\nP^{{\\mathcal A}}_{n+1}Y \\ar[r]^{p\\q{n+1}} \\ar[d] &\n  P^{{\\mathcal A}}_{n}Y \\ar[d]^{k_{n}}\\\\ B\\Lambda \\ar[r] & \\EL{M}{n+2}\n}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent for \\ $\\Lambda:=\\pinC{0}Y$ \\ and \\ $M:=\\pinC{n+1}Y$, \\ where\n\\ $p\\q{n+1}:P_{n+1}Y\\to P_{n}Y$ \\ is the given fibration of the\nPostnikov tower. \n\nThe map \\ $k_{n}:P_{n}Y\\to\\EL{M}{n+2}$ \\ is the $n$-th\n(functorial) $k$-\\emph{invariant} for $Y$. \n\\end{defn}\n\n\\begin{example}\\label{ekinv}\\stepcounter{subsection}\nIf \\ $C_{\\ast}$ \\ is a chain complex of $R$-modules, and \\ \n$P_{n}C_{\\ast}=C'_{\\ast}$ \\ as in \\S \\ref{epost}, we may take \\\n$\\EK{}{}{H_{n+1}C_{\\ast}}{n+2}=E_{\\ast}$, \\ where \\ $E_{i}=0$ \\ for \\\n$i<n+2$, \\ $E_{n+2}=Z_{n+1}C_{\\ast}$, \\ and \\  \n$E_{n+3}=B_{n+1}C_{\\ast}$. \\ Then \\ $k_{n}:C'_{\\ast}\\to E_{\\ast}$ \\ is\ndefined by \\ $\\operatorname{Id}:C'_{n+1}\\to E_{n+1}$.\n\nOf course, if $R$ is a principle ideal domain (or a hereditary ring),\nsuch as $\\mathbb Z$, then the $k$-invariants for \\ $C_{\\ast}$ \\ are trivial,\nsince in that case any two free (or projective) chain complexes with\nthe same homology are homotopy equivalent, by \\cite[Prop.\\ 3.5]{DolH}. \nBut this need not hold for an arbitrary ring $R$.\n\\end{example}\n\n\\subsection{Spherical models}\n\\label{dsmod}\\stepcounter{thm}\n\nA set of objects \\ ${\\mathcal A}:=\\{A\\}_{A\\in{\\mathcal A}}$ \\ in a model category\n${\\mathcal C}$ is called a collection of \\emph{spherical models} if the\nfollowing axioms hold\\vspace{2 mm} : \n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{Ax~\\arabic{enumi}.~}\n\\item Each \\ $\\Sigma^{n}A$ \\ ($A\\in{\\mathcal A}$, \\ $n\\in\\mathbb N$) \\ is a\n  cofibrant homotopy cogroup object in ${\\mathcal C}$\\vspace{2 mm}.\n\\item For any \\ $X\\in{\\mathcal C}$ \\ and \\ $n\\geq 1$, \\ $\\pinC{n}X$ \\ has a\n  natural structure of a module over \\ $\\pinC{0}X$\\vspace{2 mm}.\n\\item A map \\ $f:X\\to Y$ is a weak equivalence if and only if \\ $\\pia{n}f$ \\ is a \nweak equivalence for each \\ $A\\in{\\mathcal A}$ \\ and \\ $n\\in\\mathbb N$\\vspace{2 mm}.\n\\item ${\\mathcal C}$ has Postnikov towers with respect to ${\\mathcal A}$\\vspace{2 mm}.\n\\item For every $\\PiC$-algebra\\ $\\Lambda$ \\ and module $M$ over $\\Lambda$, the\n  classifying object \\ $B\\Lambda$ \\ and extended $M$-Eilenberg-Mac~Lane\n  object \\ $\\EL{M}{n}$ \\ exist (and are unique up to homotopy) for\n  each \\ $n\\geq 1$\\vspace{2 mm}.\n\\item ${\\mathcal C}$ has $k$-invariant squares with respect to ${\\mathcal A}$ for\n  each \\ $n\\geq 0$\\vspace{2 mm}. \n\\end{enumerate}\n\nIf each model \\ $\\Sigma^{k}A$ \\ ($A\\in{\\mathcal A}$, \\ $k\\in\\mathbb N$) \\ is a \ncofibrant \\emph{strict} cogroup object \\ -- \\ which implies that \nevery object in \\ $\\Pi_{\\A}$ \\ is such, up to weak equivalence \\ -- \\ we\ncall ${\\mathcal A}$ a collection of \\emph{strict} spherical models.\n\nA pointed simplicial model category \\ ${\\mathcal C}$ equipped with\na collection \\ ${\\mathcal A}:=\\{A\\}_{A\\in{\\mathcal A}}$ \\ of spherical models\nis called a \\emph{spherical model category}, \\ and we denote it\nby \\ $\\lra{{\\mathcal C};{\\mathcal A}}$. \\ Such a category is \\emph{stratified} in the\nsense of Spali\\'{n}ski (cf.\\ \\cite{SpalSM}). \n\n\\begin{example}\\label{esmc}\\stepcounter{subsection}\nThe category \\ $\\Ss_{\\ast}$ \\ of pointed simplicial sets, as well as the\ncategory \\ $\\TT_{\\ast}$ \\ of pointed connected topological spaces, have\nspherical model category structures with \\ ${\\mathcal A}=\\{\\bS{1}\\}$. \\ \n(Functorial $k$-invariants in these categories are provided by the\nconstruction of \\cite[\\S 5]{BDGoeR}; \\ in both cases \\\n$\\Alg{\\PiC}\\approx{\\EuScript Gp}$).  \\ Similarly for the category \\ ${\\EuScript Chain}_{R}$ \\ of\nchain complexes over $R$, with the constructions indicated in \\S \\ref{epost} and\n\\S \\ref{ekinv}.\n\\end{example}\n\nIn the examples we have in mind, our model categories enjoy\nadditional useful properties, which we can summarize in the following:\n\n\\begin{defn}\\label{dssmc}\\stepcounter{subsection}\nA spherical model category \\ $\\lra{{\\mathcal C};{\\mathcal A}}$ \\ as above is called\n\\emph{strict} if the following axioms hold\\vspace{2 mm} :  \n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{Ax~\\arabic{enumi}.~}\n\\item ${\\mathcal C}$ \\ is a pointed right-proper cofibrantly generated simplicial \nmodel category (cf.\\ \\cite[11.1, 13.1]{PHirM}), in which every object is fibrant.\n\\item ${\\mathcal C}$ is equipped with a faithful forgetful functor \\\n$\\hat{U}:{\\mathcal C}\\to{\\mathcal D}$, \\ with left adjoint $\\hat{F}$ \\ -- \\ where \\ ${\\mathcal D}$\n  is one of the ``categories of groups'' \\ ${\\mathcal D}={\\EuScript Gp}$, \\ $gr{\\EuScript Gp}$, \\\n  ${\\mathcal G}$, \\ $\\RM{R}$, \\ or \\ $s\\RM{R}$, \\ for some ring $R$.\n\\item The adjoint pair \\ $(\\hat{U},\\hat{F})$ \\ \\emph{create} the \nmodel category structure on ${\\mathcal C}$ in the sense of \\cite[\\S 4.13]{BlaN} \\ -- \\  \nso in particular $\\hat{U}$ creates all limits in ${\\mathcal C}$.\n\\item ${\\mathcal A}$ is a collection of strict spherical models, each of which lies\nin the image of the composite \\ $\\hat{F}\\circ F':{\\mathcal S}\\to {\\mathcal C}$, \\ \nwhere \\ $F':{\\mathcal S}\\to{\\mathcal D}$ \\ is adjoint to the forgetful functor \\\n$U':{\\mathcal D}\\to{\\mathcal S}$, \\ with the group structure on \\ $\\operatorname{Hom}_{{\\mathcal C}}(A,X)$ \\\ninduced from that of \\ $\\hat{U}(X)$.\n\\end{enumerate}\n\\end{defn}\n\n\\setcounter{thm}{0}\\section{Resolution model categories}\n\\label{crmc}\n\nMany examples of spherical model categories fit into the\nframework originally conceived by Dwyer, Kan and Stover in\n\\cite{DKStB} under the name of ``$E^{2}$ model categories,'' and later \ngeneralized by Bousfield (see \\cite{BousCR,JardBE}. A slightly\ndifferent generalization is given by Baues in \n\\cite[Ch.~D, \\S 2]{BauCF} under the name of \\emph{spiral} model categories.\n\nFirst, some preliminary concepts:  \n \n\\begin{defn}\\label{dmat}\\stepcounter{subsection}\nThe $n$-th \\emph{matching object} for a simplicial object $X$ over\n${\\mathcal C}$ is defined by\n$$\nM_{n}X=\\{(x_{0},\\dotsc,x_{n})\\in(X_{n-1})^{n+1}~|\\ \nd_{i}x_{j}=d_{j-1}x_{i} \\text{\\ \\ for all\\ }0\\leq i<j\\leq n\\}\n$$\n(see \\cite[X,\\S 4.5]{BKaH}). Note that each face map \\\n$d_{k}:X_{n}\\to X_{n-1}$ \\ factors through the obvious map \\ \n$\\delta_{n}:X_{n}\\to M_{n}X$.\n\\end{defn}\n\n\\begin{defn}\\label{dlat}\\stepcounter{subsection}\nThe $n$-th \\emph{latching object} of a simplicial object $X$ over ${\\mathcal C}$\nis defined \\ $L_{n}X:=\\coprod_{0\\leq i\\leq n-1} X_{n-1}\/\\sim$, \\ where for any \\ \n$x\\in X_{n-k-1}$ \\ and \\ $0\\leq i\\leq j \\leq n-1$ \\ we set \\ \n$s_{j_{1}}s_{j_{2}}\\dotsc s_{j_{k}}x$ \\ in the $i$-th copy of \\ $X_{n-1}$ \\ \nequivalent to \\ $s_{i_{1}}s_{i_{2}}\\dotsc s_{i_{k}}x$ \\ in the $j$-th copy \nof \\ $X_{n-1}$ \\ whenever the simplicial identity \\ \n$s_{i}s_{j_{1}}s_{j_{2}}\\dotsc s_{j_{k}}=s_{j}s_{i_{1}}s_{i_{2}}\\dotsc\ns_{i_{k}}$ \\ holds. The map \\ $\\sigma_{n}:L_{n}X\\to X_{n}$ \\ is defined\n\\ $\\sigma_{n}(x)_{i}=s_{i}x$, \\ where \\ $(x)_{i}\\in (X_{n-1})_{i}$.\n\\end{defn}\n\nThere are two canonical ways to extend a given model category structure on \n$\\hat{\\C}$ to \\ ${\\mathcal C}:=s\\hat{\\C}$:\n\n\\subsection{The Reedy model structure}\n\\label{sreed}\\stepcounter{thm}\n\nThis is defined by letting a simplicial map \\ $f:X\\to Y$ \\ in \\\n${\\mathcal C}:=s\\hat{\\C}$ \\ be:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\roman{enumi})}\n\\item a weak equivalence if \\ $f_{n}:X_{n}\\to Y_{n}$ \\ is a weak equivalence \nin $\\hat{\\C}$ \\ for each \\ $n\\geq 0$;\n\\item a (trivial) cofibration if \\ \n$f_{n}\\amalg \\sigma_{n}:X_{n}\\amalg_{L_{n}X}L_{n}Y\\to Y_{n}$ \\ is a \n(trivial) cofibration in $\\hat{\\C}$ \\ for each \\ $n\\geq 0$;\n\\item a (trivial) fibration if \\ \n$f_{n}\\times \\delta_{n}:X_{n}\\to Y_{n}\\times_{M_{n}Y}M_{n}X$ \\ is a \n(trivial) fibration in $\\hat{\\C}$ \\ for each \\ $n\\geq 0$.\n\\end{enumerate}\n\nSee \\cite[15.3]{PHirM}.\n\n\\subsection{The resolution model category}\n\\label{srmc}\\stepcounter{thm}\n\nLet $\\hat{\\C}$ be a pointed cofibrantly generated right proper model category\n(in our cases, every object will be fibrant, though this is not needed\nin general  \\ -- \\ cf.\\ \\cite{JardBE}). Given a \nset $\\hat{\\A}$ of models for $\\hat{\\C}$ \\ (\\S \\ref{ssmod}), we let \\ \n${\\mathcal A}:=\\{\\co{\\Sigma^{k}\\hat{A}}\\}_{k\\in\\mathbb N,\\hat{A}\\in\\hat{\\A}}$ \\ \n(the constant simplicial objects on \\ $\\Sigma^{k}\\hat{A}\\in\\hat{\\C}$) \\ be\nthe set of models for ${\\mathcal C}$. \\ Note that \\ \n$\\Sigma^{n}\\co{\\Sigma^{k}\\hat{A}}:=\\co{\\Sigma^{k}\\hat{A}}\\hat{\\otimes} S^{n}$ \\ \n(\\S \\ref{ssmod}), \\ so we shall generally reserve the notation \\\n$\\Sigma^{k}$ \\ for (internal) suspension in $\\hat{\\C}$, and \\\n$-\\otimes S^{n}$ \\ for the (simplicial) suspension in \\ ${\\mathcal C}=s\\hat{\\C}$. \n\n\\begin{remark}\\label{rmods}\\stepcounter{subsection}\nIf we do not assume that each \\ $\\hat{A}\\in\\hat{\\A}$ \\ is a homotopy\ncogroup object in $\\hat{\\C}$, we take \\\n${\\mathcal A}:=\\{(\\Sigma^{k}\\hat{A})\\otimes S^{1}\\}_{A\\in{\\mathcal A}}$ \\  as our\ncollection of models for ${\\mathcal C}$.  \n\\end{remark}\n\n\\begin{defn}\\label{dffm}\\stepcounter{subsection}\nA map \\ $f:V\\to Y$ \\ in  \\ ${\\mathcal C}=s\\hat{\\C}$ \\ is called \\emph{homotopically\n$\\hat{\\A}$-free} if for each \\ $n\\geq 0$, \\ there is \n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\alph{enumi})\\ }\n\\item a cofibrant object \\ $W_{n}$ \\ in \\ $\\Pi_{\\hat{\\A}}\\subset\\hat{\\C}$, \\ and\n\\item a map \\ $\\varphi_{n}:W_{n}\\to Y_{n}$ \\ in ${\\mathcal C}$ inducing a trivial \ncofibration \\ $(V_{n}\\amalg_{L_{n}V}L_{n}Y)\\amalg W_{n}\\to Y_{n}$.\n\\end{enumerate}\n\\end{defn}\n\nWe define the \\emph{resolution model category structure} on \\ $s\\hat{\\C}$ \\ \ndetermined by $\\hat{\\A}$, by letting a simplicial map \\ $f:X\\to Y$ \\ be:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\roman{enumi})}\n\\item a \\emph{weak equivalence} if \\ $\\pia{n}f$ \\ is a weak equivalence \nof simplicial groups for each \\ $A\\in{\\mathcal A}$ \\ and \\ $n\\geq 0$.\n\\item a \\emph{cofibration} if it is a retract of a homotopically $\\hat{\\A}$-free map;\n\\item a \\emph{fibration} if it is a Reedy fibration (\\S \\ref{sreed}(iii)) \nand \\ $\\pia{n}f$ \\ is a fibration of simplicial groups for each \\ \n$A\\in{\\mathcal A}$ \\ and \\ $n\\geq 0$\n\\end{enumerate}\n\n\\begin{defn}\\label{dnc}\\stepcounter{subsection}\nGiven a fibrant \\ $X\\in s\\hat{\\C}$, \\ define its $n$-\\emph{cycles} object \\ \n$Z_{n}X$ \\ to be \\ $\\{ x\\in X_{n}\\,|\\ d_{i}x=0 \\ \\text{for}\\\ni=0,\\dotsc,n\\}$ \\ (the fiber of \\ $\\delta_{n}:X_{n}\\to M_{n}X$ \\ of \n\\S \\ref{dmat}). \\ Similarly, the $n$-\\emph{chains} object for $X$ is \\ \n$C_{n}X=\\{ x\\in X_{n}\\,|\\ d_{i}x=0 \\ \\text{for}\\ i=1,\\dotsc,n\\}$.\n\\end{defn}\n\nIf $X$ is fibrant, the map \\ \n$\\mathbf{d}_{0}=\\mathbf{d}_{0}^{n}:=d_{0}\\rest{C_{n}X}:C_{n}X\\to Z_{n-1}X$ \\ fits into a \nfibration sequence:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwo}\n\\dotsb \\Omega Z_{n}X\\to Z_{n+1}X\\xra{j^{X}_{n+1}}C_{n+1}X \\xra{\\mathbf{d}_{0}^{n+1}} Z_{n}X\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent (see \\cite[Prop.\\ 5.7]{DKStB}). \n\n\\begin{defn}\\label{dpa}\\stepcounter{subsection}\nA \\emph{$\\PiA$-algebra} \\ is a product-preserving functor from \\\n$(\\operatorname{ho}\\Pi_{\\A})^{\\operatorname{op}}$ \\ to sets. The category of $\\PiA$-algebra s is denoted by \\ $\\Alg{\\PiA}$.\n\nEquivalently, we can think of an $\\PiA$-algebra\\ $\\Lambda$ as an \\ \n$\\mathbb N\\times{\\mathcal A}$-graded group equipped with an action of the ${\\mathcal A}$-primary\nhomotopy operations (corepresented by the maps in \\ $\\operatorname{ho}\\Pi_{\\A}$). \n\nThus we can think of the functor \\ $\\pinC{\\ast}$ \\ as taking value in \\ $\\Alg{\\PiA}$.\nThis explains the additional $\\PiC$-algebra\\ structure on the ${\\mathcal A}$-graded groups \\ \n$\\pinC{n}X$, \\ mentioned in \\S \\ref{rpis}: when \\ ${\\mathcal C}=s\\hat{\\C}$, \\ we\nhave \\ $\\Alg{\\PiC}:=\\Alg{\\Pi_{\\hA}}$.\n\\end{defn}\n\n\\begin{example}\\label{epa}\\stepcounter{subsection}\nWhen \\ ${\\mathcal C}={\\mathcal G}$, \\ and \\ ${\\mathcal A}=\\{\\gS{1}\\}$ \\ -- \\ so \\ $\\Pi_{\\A}$ \\ is the\ncategory of wedges of ${\\mathcal G}$-spheres (\\S \\ref{esmod}) \\ -- \\  then (up \nto indexing) \\ $\\Alg{\\PiA}$ \\ is the usual category of $\\Pi$-algebra s (see\n\\cite[\\S 2]{StoV}): \\ graded groups equipped with an action of the\nprimary homotopy operations (Whitehead products and compositions). \n\\end{example}\n\n\\subsection{Examples of resolution model categories}\n\\label{ermc}\\stepcounter{thm}\n\nIn this paper we shall be interested mainly in the following instances\nof resolution model categories:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\alph{enumi})\\ }\n\\item Let \\ $\\hat{\\C}={\\EuScript Gp}$ \\ with the \\emph{trivial} model category structure: \\ \ni.e., only isomorphisms are weak equivalences, and every map is both a fibration\nand a cofibration. Let \\ $\\hat{\\A}=\\{\\mathbb Z\\}$ \\ consist of the free cyclic group \n(whose coproducts are the cogroup objects in \\ ${\\EuScript Gp}$). \\ \nThe resulting resolution model category\nstructure on \\ ${\\mathcal G}:= s\\hat{\\C}$ \\ is the usual one (cf.\\ \\cite[II,\\S 3]{QuiH}. \\ \nHere \\ $\\Alg{\\PiC}\\approx {\\EuScript Gp}$ \\ -- \\ there is no extra structure on the\nindividual homotopy groups of a simplicial group.\n\nNote that if we tried to do the same for \\ $\\hat{\\C}={\\EuScript Set}$, \\ there are no \nnontrivial cogroup objects, while in \\ ${\\mathcal S}$ \\ not all objects are fibrant.\nNote also that the category \\ $\\TT_{\\ast}$ \\ of pointed topological spaces, \nwhich is one of the main examples we have in mind, has a spherical\nmodel category structure which is not strict (\\S \\ref{dssmc}). This\nexplains the significance of Remark \\ref{ssg} in our context\\vspace{2 mm}.\n\n\\item The previous example extends to any category $\\hat{\\C}$ of (possibly graded) \nuniversal algebras with an underlying group structure \\ -- \\ such as\nrings, $R$-modules,  associative algebras, Lie algebras, and so on \\ -- \\ \nso that ${\\mathcal C}$ is corepresented by a $\\mathfrak{G}$-theory $\\Theta$, in the\nlanguage of \\cite[\\S 4]{BPescF}. Here ${\\mathcal A}$ consists\nof free monogenic algebras (one for each isomorphism class), and\nthus once more \\ $\\Alg{\\PiC}\\approx{\\mathcal C}$. \n\\item We can iterate the process by taking ${\\mathcal G}$ for $\\hat{\\C}$, and letting \\ \n$\\hat{\\A}:=\\{\\gS{n}\\}_{n=1}^{\\infty}$ \\ (\\S \\ref{esmod}). We thus obtain\na resolution model category structure on \\ $s{\\mathcal G}$ \\ (or equivalently, on\nthe category of simplicial spaces). \n\nIn this case the homotopy groups \\ $\\pi^{s{\\mathcal G}}_{k,n}X$, \\ denoted\nbriefly by \\ $\\pi^{\\natural}_{n}X$, \\ are the ``bigraded groups'' of\n\\cite{DKStB}, and Proposition 5.8 there shows that, for a fibrant\nsimplicial space \\ $X\\in s{\\mathcal G}$, \\ we have \\ \n$\\pia{n}{X}\\cong\\pi_{0}\\,\\operatorname{map}(A\\otimes S^{n},X)$\\vspace{2 mm}.\n\\item If ${\\mathcal C}$ is a resolution model category and $I$ is some small\ncategory, the category \\ ${\\mathcal C}^{I}$ \\ of $I$-diagrams in ${\\mathcal C}$ also has\na resolution model category structure, in which the models consists of\nall free $I$-diagrams \\ $F[A,i]$ \\ for $i\\in\\operatorname{Obj} I$ \\ and \\ \n$A\\in{\\mathcal A}$, \\ where \\ $F[A,i](j):=\\coprod_{\\operatorname{Hom}_{I}(i,j)}~A$. \\ \nSee \\cite[\\S 1]{BJTurR}).\n\\end{enumerate}\n\n\\begin{remark}\\label{rbgg}\\stepcounter{subsection}\nIn all these examples, if \\ $Y\\in {\\mathcal C}=s\\hat{\\C}$, \\ is fibrant, then for\neach \\ $n\\geq 0$ \\ we have an exact sequence: \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{ethree}\n\\pi_{\\ast}^{\\hat{\\C}}C_{n+1}X~\\xra{(\\mathbf{d}_{0}^{n+1})_{\\#}}~\n\\pi_{\\ast}^{\\hat{\\C}}Z_{n}X~\\xra{\\hat{\\vartheta}_{n}}~ \n\\pinC{n}{Y}\\to 0.\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\end{remark}\n\n\\setcounter{thm}{0}\\section{Constructions in resolution model categories}\n\\label{csp}\n\nNot all spherical model categories are resolution model categories \n(see \\S \\ref{esmc}), but all known examples appear to be Quillen\nequivalent to such. Conversely, the examples of resolution model categories \\ \n$\\lra{{\\mathcal C}=s\\hat{\\C};{\\mathcal A}}$ \\ we are interested in are spherical (though this\ndoes not hold in general \\ -- \\ see \\S \\ref{snsrmc} below). We briefly\nindicate why this is so. \n\n\\subsection{Postnikov sections}\n\\label{sps}\\stepcounter{thm}\n\nGiven \\ $Y\\in s\\hat{\\C}$, for each \\ $n\\geq 0$ \\ define \\ $Y\\q{n}\\in s\\hat{\\C}$ \\ \nby setting \\ $Y\\q{n}_{k}:= Y_{k}$ \\ for \\ $k\\leq n+1$ \\ and \\ \n$Y\\q{n}_{k}:= M_{k}(Y\\q{n})$ \\ (\\S \\ref{dmat}) for \\ $k\\geq n+2$. \\ \nNote that for any \\ $X\\in s{\\mathcal C}$, \\ $M_{k}X$ \\ depends only on $X$\nthrough dimension \\ $(k-1)$, \\ so this definition is valid inductively.\nDenote the obvious maps by \\ $r\\q{n}:Y\\to Y\\q{n}$ \\ and \\ \n$p\\q{n}:Y\\q{n+1}\\to Y\\q{n}$ \\ (see \\cite[\\S 1.2]{DKanO}).\n\nNow for any \\ $X\\in s\\hat{\\C}$, \\ choose a functorial fibrant replacement\n$Y$, and set \\ $P_{n}X := Y\\q{n}$, \\ with \\ \n$\\varphi\\q{n}:X\\to P_{n}X$ \\ defined to be the composite of \\ $r\\q{n}$ \\ \nwith the trivial cofibration \\ $i:X\\to Y$, \\ and \\ \n$p\\q{n}:P_{n+1}X\\to P_{n}X$ \\ defined as above. \n\n\\begin{remark}\\label{rcsk}\\stepcounter{subsection}\nThe functor \\ $-\\q{n}:{\\mathcal C}\\to{\\mathcal C}$ \\ is right adjoint to the \\\n$(n+1)$-skeleton functor \\ $\\sk{n+1}$, \\ so \\ $P_{n}X$ \\ depends only\non \\ $\\sk{n+1}X$, \\ even if $X$ is not fibrant. If $X$ is fibrant, we can find \\ \n$Y\\simeq P_{n}X$ \\ with \\ $\\sk{n+1}Y=\\sk{n+1}X$.\n\\end{remark}\n\n\\begin{fact}\\label{fone}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), the tower:\n$$\nX\\to\\dotsc \\to P_{n+1}X\\xra{p\\q{n}} P_{n}X\\to\\dotsc \\to P_{0}X\n$$\nis a functorial Postnikov tower for \\ ${\\mathcal C}=s\\hat{\\C}$ \\ with respect to ${\\mathcal A}$ \\ \n(\\S \\ref{dfpt}).  \n\\end{fact}\n\n\\begin{proof}\nFrom \\S \\ref{sreed} and \\S \\ref{sps} it follows that if  \\ $Y\\in\ns\\hat{\\C}$ \\ is fibrant, then \\ so is each \\ $Y\\q{n}$, \\ and  \nfor each $n$, \\ $Y\\q{n+1}\\to Y\\q{n}$ \\ is a fibration, \\ \n$Z_{k}Y\\q{n}=0$ \\ and \\ $C_{k}Y\\q{n}\\xra{\\mathbf{d}_{0}}Z_{k-1}Y$ \\ is an\nisomorphism for \\ $k\\geq n+2$. \\ The claim then follows from \\eqref{ethree}.\n\\end{proof}\n\n\\begin{fact}\\label{ftwo}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), \\ there is a\nclassifying object \\ $B\\Lambda$ \\ for any $\\PiA$-algebra\\ \\ $\\Lambda$, \\ and it is\nunique up to homotopy. \n\\end{fact}\n\n\\begin{proof}\nIn the algebraic cases of \\S \\ref{ermc}(a-b), \\ we may take \\ $B\\Lambda$ \\\nto be (a cofibrant model for) the constant simplicial object on\n$\\Lambda$. For simplicial spaces, \\ $B\\Lambda$ \\ may be constructed as for\ntopological spaces, using generators and relations (see \\cite[\\S 8.9]{BDGoeR}). \nThe extension to the diagram case of \\S \\ref{ermc}(d) is objectwise.\n\\end{proof}\n\n\\begin{fact}\\label{fthree}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), \\ for each \\ $n\\geq 1$ \\ \nthere is an $n$-dimensional $M$-Eilenberg-Mac~Lane object \\ \n$\\EK{}{}{M}{n}$ \\ for any abelian $\\PiA$-algebra\\ $M$, \\ and there is an\n$n$-dimensional extended $M$-Eilenberg-Mac~Lane object \\ \n$\\EL{M}{n}$ \\ for any $\\PiA$-algebra\\ $\\Lambda$ and module $M$ over\n$\\Lambda$. \\ Each of these is unique up to homotopy. \n\\end{fact}\n\n\\begin{proof}\nIn the algebraic cases of \\S \\ref{ermc}(a-b), \\ we may take \\\n$\\EK{}{}{M}{n}$ \\ to be the iterated Eilenberg-Mac Lane construction\n$\\bar{W}$ on \\ $BM$, \\ while \\ $\\EL{M}{n}$ \\ is a semi-direct product \\ \n$\\EK{}{}{M}{n}\\ltimesB\\Lambda$ (see \\cite[Prop.\\ 2.2]{BDGoeR}).\nFor simplicial spaces, use the explicit construction of \\cite[\\S 8.9]{BDGoeR} \\\nThe extension to the diagram case is again objectwise.\n\\end{proof}\n\n\\begin{fact}\\label{ffour}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), \\ for each \\ $n\\geq 1$ \\ \nand \\ $X\\in{\\mathcal C}$, \\ $\\pinC{n}X$ \\ has a natural structure of a \nmodule over \\ $\\pinC{0}X$.\n\\end{fact}\n\n\\begin{proof}\nNote that by \\cite[II,1,(6)]{QuiH} we have \\ \n$\\operatorname{map}(A\\otimes S^{n},X)\\cong\\operatorname{map}_{{\\mathcal S}}(S^{n},\\operatorname{map}(A,X))$ \\ \n(unpointed maps), \\ so \\ $\\pinC{n}X\\to\\pinC{0}X$ \\ associates to each \\ \n$f:A\\otimes S^{n}\\to X$ \\ its component in \\ $\\operatorname{map}(A,X)$. \\ This defines \nan \\emph{abelian} algebra over \\ $\\pinC{0}X$ \\ by \\cite[Prop.\\ 6.26]{BPescF}).\n\\end{proof}\n\n\\begin{fact}\\label{ffive}\\stepcounter{subsection}\nIn each of the examples of \\S \\ref{ermc}(a-d), \\ for each \\ $X\\in s{\\mathcal C}$, \\ \n$\\Lambda:=\\pinC{0}X$ \\ and \\ $n\\geq 1$, \\ the commutative square\nobtained by applying the functor \\ $P_{n+2}$ \\ to the pushout diagram:\n$$\n\\xymatrix@C=40pt{\\ar @{} [dr] |>>>>>{\\framebox{\\scriptsize{PO}}}\nP_{n+1}X  \\ar[r]^-{p\\q{n+1}} \n\\ar[d] & P_{n}X \\ar[d]^{k_{n}}\\\\ B\\Lambda \\ar[r] & Y\n}\n$$\nis an $n$-th $k$-invariant square (Def.\\ \\ref{dki}) \\ -- \\ that is, \\ \n$P_{n+2}Y\\simeq\\EL{\\pia{n+1}X}{n+2}$.\n\\end{fact}\n\n\\begin{proof}\nSee \\cite[\\S 5]{BDGoeR}.\n\\end{proof}\n\nWe may summarize these facts in the following:\n\n\\begin{thm}\\label{tone}\\stepcounter{subsection}\nThe following resolution model categories (cf.\\ \\S \\ref{ermc})\nare strict spherical model categories:\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\roman{enumi}.\\ }\n\\item The category \\ ${\\mathcal C}=s\\hy{\\Theta}{\\Set_{\\ast}}$ \\ of simplicial\n  $\\Theta$-algebras for any $\\mathfrak{G}$-theory $\\Theta$,\n  with $\\hat{\\A}$ consisting of monogenic free $\\Theta$-algebras;\n\\item In particular, the category \\ ${\\mathcal C}={\\mathcal G}$ \\ of simplicial groups, with \\\n  $\\hat{\\A}=\\{\\mathbb Z\\}$; \n\\item The category \\ $s{\\mathcal G}$ \\ of bisimplicial groups (``simplicial\n  spaces''), with \\ $\\hat{\\A}=\\{\\gS{1}\\otimes S^{k}\\}_{k=0}^{\\infty}$.\n\\item The category \\ ${\\mathcal C}^{I}$ \\ of $I$-diagrams in a strict spherical\n  model category ${\\mathcal C}$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{thm}\\label{ttwo}\\stepcounter{subsection}\nThe following are spherical model categories (which are not strict):\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\roman{enumi}.\\ }\n\\item The category \\ $\\Ss_{\\ast}$ \\ of pointed simplicial sets, with \\ ${\\mathcal A}=\\{S^{1}\\}$; \n\\item The category \\ $\\TT_{\\ast}$ \\ of pointed topological spaces, with \\ \n      ${\\mathcal A}=\\{\\bS{1}\\}$; \n\\item The category \\ $s\\TT_{\\ast}$ \\ of simplicial pointed topological spaces, with \\ \n      $\\hat{\\A}=\\{\\bS{1}\\otimes S^{k}\\}_{k=1}^{\\infty}$.\n\\end{enumerate}\n\\end{thm}\n\n\\subsection{Non-spherical model categories}\n\\label{snsrmc}\\stepcounter{thm}\n\nConsider the trivial model category structure on \\ $\\hat{\\C}={\\EuScript Gp}$, \\ with \\ \n$\\hat{\\A}:=\\{A=\\mathbb Z\/p\\}$ \\ (for $p$ an odd prime). \\ This defines a\nresolution model category structure on ${\\mathcal G}$ \\ -- \\ or equivalently, on \\ \n$\\TT_{\\ast}$ \\ (see Remark \\ref{rmods}). \\ Note that \\ $-\\otimes S^{n}$ \\ \ncorresponds to suspension of simplicial sets, not\nsimplicial abelian group, so the model \\ $A\\otimes S^{n}\\in{\\mathcal G}$ \\\ncorresponds to the $n$-dimensional mod $p$ Moore space \\ \n$\\bS{n-1}\\cup_{p}\\be{n}$. \n\nThus \\ $\\pia{k}X:=[A\\otimes S^{k},X]$ \\ is by definition the $k$-th\n\\emph{mod $p$ homotopy group} of $X$ \\ -- \\  denoted by \\ $\\pi_{k}(X;\\mathbb Z\/p)$ \\ \nin \\cite[Def.\\ 1.2]{NeiP} \\ -- \\  which fits into a short exact sequence:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{efour}\n0\\to\\pi_{k}X\\otimes\\mathbb Z\/p \\to\n        \\pi_{k}(X;\\mathbb Z\/p)\\to\\operatorname{Tor}^{\\mathbb Z}_{1}(\\pi_{k-1}X,\\mathbb Z\/p)\\to 0\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent for \\ $k\\geq 2$ \\ (see \\cite[Prop.\\ 1.4]{NeiP}). \\ In particular, \nfor \\ $Y:=A\\otimes S^{n}$ \\ ($n\\geq 4$) \\ we have\n$$\n\\pi_{i}(Y;\\mathbb Z\/p)=\\begin{cases}\\mathbb Z\/p & \\text{for \\ }i=n-1,n,\\\\\n             0     & \\text{for \\ } 2\\leq i<n-1 \\text{\\ or \\ } i=n+1,\n\\end{cases}\n$$\n\\noindent with the two non-trivial groups connected by a Bockstein \n(cf.\\ \\cite[\\S 1]{NeiP})\\vspace{2 mm}.\n\nHowever, the resolution model category structure on ${\\mathcal G}$ determined by ${\\mathcal A}$ is \nnot spherical: \\ if it were, in particular there would be Postnikov functors \\\n$P_{k}=P_{k}^{{\\mathcal A}}$ \\ for all \\ $k\\geq 1$ \\ (Def.\\ \\ref{dfpt}). \\ \nFrom \\eqref{efour} we see that, disregarding torsion prime to $p$,  \nbecause of the Bockstein we must have \\ \n$P_{n-1}Y\\simeq E(\\mathbb Z,n-1)$ \\ and \\ $P_{n}Y\\simeq E(\\mathbb Z\/p,n-1)$ \\ \n(for \\ $Y=S^{n-1}\\cup_{p} e^{n}$). \\ But then there is no\nnon-trivial map \\ $P_{n}Y\\to P_{n-1}Y$. \n\n\\subsection{Cohomology in spherical model categories}\n\\label{scpa}\\stepcounter{thm}\n\nNote that the  $k$-invariants of a simplicial object\nactually take value in cohomology groups, as expected:\n\n\\begin{prop}\\label{ptwo}\\stepcounter{subsection}\nFor each $\\PiA$-algebra\\ $\\Lambda$ and module $M$ over $\\Lambda$, the functors \\\n$D^{n}:{\\mathcal C}\/B\\Lambda\\to{\\Ab{\\EuScript Gp}}$ \\ \\textup{($n>0$)}, \\ defined \\ \n$D^{n}(X):=[X,\\EL{M}{n}]_{B\\Lambda}$, \\ \nare \\emph{cohomology functors} on ${\\mathcal C}$ \\ -- \\ that is, they \nare homotopy invariant, take arbitrary coproducts to products, vanish \non the spherical models \\ $\\Sigma^{n}A$, \\ except in degree $n$, \nand have Mayer-Vietoris sequences for homotopy pushouts.\n\\end{prop}\n\nWe therefore denote \\ $[X,\\EL{M}{n}]_{B\\Lambda}$ \\ by \\ $\\HL{n}{X}{M}$.\n\n\\begin{proof}\nSee \\cite[Thm.\\ 7.14]{BPescF}.\n\\end{proof}\n\n\nFact \\ref{fthree} then follows from Brown Representability, since \\\n$\\EL{M}{n}$ \\ represents the $n$-th Andr\\'{e}-Quillen cohomology\ngroup in ${\\mathcal C}$; see  \\cite[\\S 6.7]{BDGoeR} and \\cite[\\S 4]{BlaG}.  \n\n\\setcounter{thm}{0}\\section{Spherical functors}\n\\label{csf}\n\nOur objective is to study functors between model categories, and\ninvestigate the extent to which they induce an equivalence of homotopy\ncategories. Our methods work only for functors between spherical model\ncategories which take models to models, in the following sense:\n\n\\begin{defn}\\label{dsfunc}\\stepcounter{subsection}\nLet \\ $\\lra{{\\mathcal C};{\\mathcal A}}$ \\ and \\ $\\lra{{\\mathcal D};{\\mathcal B}}$ \\ be two spherical model\ncategories. A functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is called \\emph{spherical} if \n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\roman{enumi}.\\ }\n\\item $T$ defines a bijection \\ ${\\mathcal A}\\to{\\mathcal B}$;\n\\item $T\\rest{\\Pi_{\\A}}$ \\ preserves coproducts and suspensions;\n\\item $T$ induces an equivalence of categories \\ $\\Alg{\\PiC}\\approx\\Alg{\\PiD}$ \\ \n(in fact, it suffices that \\ $\\Alg{\\PiD}$ \\ be a full subcategory of \\ $\\Alg{\\PiC}$).\n\\end{enumerate}\n\\end{defn}\n\n\\subsection{Examples of spherical functors}\n\\label{ssfunc}\\stepcounter{thm}\n\nIn the cases we shall be considering (those mentioned in the\nintroduction), ${\\mathcal C}$ and ${\\mathcal D}$ will be strict spherical resolution model\ncategories, with \\ ${\\mathcal C}=s\\hat{\\C}$ \\ and \\ ${\\mathcal D}=s\\hat{\\D}$, \\ and $T$ will \nbe prolonged from a functor \\ $\\hat{T}:\\hat{\\C}\\to\\hat{\\D}$\\vspace{2 mm} . \n\nThe four examples:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\alph{enumi})\\ }\n\\item For \\ $\\lra{\\hat{\\C};\\hat{\\A}}=\\lra{{\\EuScript Gp};\\{\\mathbb Z\\}}$ \\ and \\  \n$\\lra{\\hat{\\D},\\hat{\\B}}=\\lra{{\\Ab{\\EuScript Gp}};\\{\\mathbb Z\\}}$, \\ let \\ $\\hat{T}={\\EuScript Ab}:{\\EuScript Gp}\\to{\\Ab{\\EuScript Gp}}$ \\\nbe the abelianization functor.\n\nHere \\ ${\\mathcal C}=s\\hat{\\C}={\\mathcal G}$, \\ so \\ $\\operatorname{ho}{\\mathcal C}$ \\ is equivalent to the homotopy\ncategory of pointed connected topological spaces (\\S \\ref{ssg}), \nwhile \\ ${\\mathcal D}=s\\hat{\\D}$, \\ the category of simplicial abelian groups,\nis equivalent to the category of chain complexes under the\nDold-Kan correspondence (see \\cite[\\S 1]{DolH}). Thus \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\\nrepresents the singular chain complex functor \\ $C_{\\ast}:\\TT_{\\ast}\\to{\\EuScript Chain}$.\n\nNote that \\ $\\Alg{\\PiC}={\\EuScript Gp}$, \\ while \\ $\\Alg{\\PiD}={\\Ab{\\EuScript Gp}}$, \\ in this case, so \nstrictly speaking $T$ does not induce an equivalence of categories. \\\nBut since \\ ${\\Ab{\\EuScript Gp}}$ \\ is a full subcategory of \\ ${\\EuScript Gp}$, \\ we can in\nfact think of \\ $\\pi^{\\natural}$ \\ as taking values in groups\\vspace{2 mm} .\n\\item For \\ $\\lra{\\hat{\\C};\\hat{\\A}}=\\lra{{\\EuScript Gp};\\{\\mathbb Z\\}}$ \\ and \\  \n$\\lra{\\hat{\\D},\\hat{\\B}}=\\lra{{\\EuScript Hopf};\\{H\\}}$, \\ where \\ ${\\EuScript Hopf}$ \\ is the\ncategory of complete Hopf algebras over $\\mathbb Q$, \\ $H$ is the\nmonogenic free object in this category, let \\ $\\hat{Q}:{\\EuScript Gp}\\to{\\EuScript Hopf}$ \\ \nbe the functor which associates to a group $G$ the completion of the\ngroup ring \\ $\\mathbb Q[G]$ \\ by powers of the augmentation ideal. \n\nAgain, \\ ${\\mathcal C}=s\\hat{\\C}$ \\ is a model category for connected \ntopological spaces, while \\ ${\\mathcal D}=s\\hat{\\D}$ \\ is a model category for the\nrational simply-connected spaces (see \\cite{QuiR}); \\ $Q$ (when\nrestricted to connected simplicial groups) represents the\nrationalization functor. \\ Once more, \\ $\\Alg{\\PiC}={\\EuScript Gp}$, \\ while \\\n$\\Alg{\\PiD}$ \\ is the subcategory of vector spaces over $\\mathbb Q$\\vspace{2 mm}. \n\\item For \\ $\\lra{\\hat{\\C},\\hat{\\A}}=\\lra{\\Set_{\\ast};\\{S^{0}\\}}$ \\ \n(so that \\ $\\lra{{\\mathcal C},{\\mathcal A}}=\\lra{{\\mathcal S};\\{S^{1}\\}}$, \\ by Remark\n\\ref{rmods}), and \\ $\\lra{\\hat{\\C};\\hat{\\A}}=\\lra{{\\EuScript Gp};\\{\\mathbb Z\\}}$, \\ let \\\n$\\hat{F}:\\Set_{\\ast}\\to{\\EuScript Gp}$ \\ be the free group functor.\n\nAgain, we think of both \\ ${\\mathcal C}=s\\hat{\\C}={\\mathcal G}$ \\ and \\ ${\\mathcal D}=s\\hat{\\D}=\\Ss_{\\ast}$ \\ as\nmodel categories for pointed topological spaces, (under the respective\nequivalences of \\S \\ref{ssg}) \\ -- \\ so $F$ here represents \nthe suspension functor \\ $\\Sigma:\\TT_{\\ast}\\to\\TT_{\\ast}$ \\ (rather than \\\n$\\Omega\\Sigma$, \\ as one might think at first glance)\\vspace{2 mm}.\n\\item For \\ $\\lra{\\hat{\\C};\\hat{\\A}}=\\lra{{\\mathcal G};\\{\\gS{k}\\}_{k=0}^{\\infty}}$ \\ and \\  \n$\\lra{\\hat{\\D},\\hat{\\B}}=\\lra{\\Alg{\\Pi};\\{\\pi_{\\ast}\\gS{k}\\}_{k=0}^{\\infty}}$, \\ let \\ \n$\\widehat{\\pi_{\\ast}}:{\\mathcal G}\\to\\Alg{\\Pi}$ \\ be the graded homotopy group functor \\ $X\\mapsto\\pi_{\\ast} X$. \\ \nHere \\ ${\\mathcal C}=s{\\mathcal G}$ \\ is a model category for simplicial spaces\\vspace{2 mm}.\n\\end{enumerate}\n\n\\begin{thm}\\label{tthree}\\stepcounter{subsection}\nLet \\ $\\lra{{\\mathcal C};{\\mathcal A}}$ \\ and \\ $\\lra{{\\mathcal D};{\\mathcal B}}$ \\ be spherical model\ncategories,  and let \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ be a spherical functor. Then for\neach \\ $X\\in{\\mathcal C}$ \\ and \\ $A\\in{\\mathcal A}$ \\ there is a natural long exact\nsequence of $\\PiC$-algebra s: \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{efive}\n\\dotsc\\to \\Gamma^{X}_{\\alpha,n}X~\\xra{s_{n}^{X}}~\\pia{n}\nX~\\xra{h_{n}^{X}}~\\pinC{T_{\\ast}(\\alpha),n}\nTX~\\xra{\\partial_{n}^{X}}~\\Gamma^{T}_{\\alpha,n-1}X\\dotsc~. \n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\end{thm}\n\nWe call \\eqref{efive} the \\emph{comparison exact sequence} for\n$T$. Compare \\cite[V, (5.4)]{BauCF}.\n\n\\begin{proof}\nIf \\ $\\tilde{X}\\to X$ \\ is a functorial fibrant replacement, the functor \n$T$ induces a natural transformation \\ \n$\\tau:\\operatorname{map}_{{\\mathcal C}}(A,\\tilde{X})\\to \\operatorname{map}_{{\\mathcal D}}(TA,\\widehat{T\\tilde{X}})$, \\ which\nwe may functorially change to a fibration of simplicial sets, \nwith fiber \\ $F(X)$. \\ Setting \\ $\\Gamma^{T}_{\\alpha,n}:=\\pi_{n}F(X)$, \\ \nthe corresponding long exact sequence in homotopy is\n\\eqref{efive}. \n\nNote that the map \\ $h^{X}_{n}=h^{X}$ \\ is also natural in the\nvariable \\ $A$, \\ so the graded map \\ $h^{X}_{\\ast}:\\pinC{n}X\\to\\pinD{n}TX$ \\ \nis a morphism of $\\PiC$-algebra s (i.e., \\ $\\Pi_{\\hA}$-algebras).\n\\end{proof}\n\n\\subsection{Applications of Theorem \\protect{\\ref{tthree}}}\n\\label{satt}\\stepcounter{thm}\n\nThe Theorem is not very useful in this generality. However, in all \nthe examples of \\S \\ref{ssfunc}, we obtain\ninteresting (though mostly known) exact sequences\\vspace{2 mm} :\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\alph{enumi})\\ }\n\\item For \\ $\\hat{T}={\\EuScript Ab}:{\\EuScript Gp}\\to{\\Ab{\\EuScript Gp}}$ \\ the abelianization functor, where \\ \n$T:{\\mathcal G}\\to s{\\Ab{\\EuScript Gp}}$ \\ represents the singular chain complex functor \\ \n$C_{\\ast}:\\TT_{\\ast}\\to{\\EuScript Chain}$ \\ (cf.\\ \\S \\ref{ssfunc}(a)), the sequence \n\\eqref{efive} is the ``certain exact sequence'' of J.H.C. Whitehead:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentyfive}\n\\dotsc\\to \\Gamma_{n} X \\to\\pi_{n} X~\\xra{h_{n}}~ H_{n}(X;\\mathbb Z)\\to\n\\Gamma_{n-1} X \\dotsc\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\n(See \\cite{JWhC}).  In particular, the third term in this sequence, \\\n$\\Gamma^{{\\mathcal A}}_{n}(X)$, \\ is simply the $n$-th homotopy group of the\ncommutator subgroup of \\ $GX$\\vspace{2 mm}. \n\\item For \\ $Q:{\\mathcal G}\\to s{\\EuScript Hopf}$ \\ of \\S \\ref{ssfunc}(b), representing \nthe rationalization functor, \\ we obtain a long exact sequence\nrelating the integral and rational homotopy groups of a\nsimply-connected space $X$. The third term in \\eqref{efive} may be\ndescribed in terms of the torsion subgroup of \\ $\\pi_{\\ast} X$ \\ together\nwith \\ $\\pi_{\\ast} X\\otimes\\mathbb Q\/\\mathbb Z$\\vspace{2 mm}. \n\n\\item The free group functor \\ $\\hat{F}:\\Set_{\\ast}\\to{\\EuScript Gp}$ \\ of \\S\n\\ref{ssfunc}(c) represents the suspension \\ $\\Sigma:\\TT_{\\ast}\\to\\TT_{\\ast}$, \\ \nand indeed for \\ $K\\in\\Ss_{\\ast}$ \\ the map \\ $h^{K}$, \\ which is the composite:\n\n\\begin{equation*}\n\\begin{split}\n\\pi_{n}K~=~\\pi_{0}\\operatorname{map}_{\\Ss_{\\ast}}(S^{n},K)~&~\n\\longrightarrow~\\pi_{0}\\,\\operatorname{map}_{{\\mathcal G}}(FS^{n},FK)\\\\\n\\xra{\\cong}~& \\pi_{0}\\,\\operatorname{map}_{\\Ss_{\\ast}}(\\Sigma S^{n},\\Sigma K)~=~\\pi_{n+1}\\Sigma K~,\n\\end{split}\n\\end{equation*}\n\\noindent is the suspension homomorphism, so \\eqref{efive} is a\ngeneralized EHP sequence (cf.\\ \\cite{BauR,GanG,NomE})\\vspace{2 mm}. \n\\item For \\ $\\pi_{\\ast}:s{\\mathcal G}\\to s\\Alg{\\Pi}$ \\ as in \\S \\ref{ssfunc}(d), it\nturns out that for any simplicial space \\ $X\\in s{\\mathcal G}$, \\ the induced map \\\n$h^{X}_{n}$ \\ is the ``Hurewicz homomorphism'' \\ \n$h_{n}:\\pi^{\\natural}_{n}X\\to\\pi_{n}\\pi_{\\ast} X$ \\ of \\cite[7.1]{DKStB}, \\ \nwhile \\ $\\Gamma^{T}_{n}X$ \\ is just \\ $\\Omega\\pi^{\\natural}_{n-1}X$ \\ -- \\ that is, \\ \n$\\Gamma^{T}_{i,n}X=\\pi^{\\natural}_{i+1,n-1}X$ \\ for each $i$. Thus \\ \n\\eqref{efive} \\ is the \\emph{spiral long exact sequence}:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{efifteen}\n\\dotsc \\pi_{n+1}\\pi_{\\ast} X \\xra{\\partial^{\\star}_{n+1}} \\Omega\\pi^{\\natural}_{n-1}X\n\\xra{s_{n}} \\pi^{\\natural}_{n}X \\xra{h_{n}} \\pi_{n}\\pi_{\\ast} X\\to\n\\dotsb \\pi^{\\natural}_{0}X\\xra{h_{0}}\\pi_{0}\\pi_{\\ast} X \\to 0\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent  of \\cite[8.1]{DKStB}. Of course, \\ $\\pi^{\\natural}_{-1}{X}=0$, \\ \nso \\ $h_{0}$ \\ is an isomorphism\\vspace{2 mm} .\n\\end{enumerate}\n\nNote that for \\ $T:{\\mathcal C}\\to{\\mathcal D}$ as above, the homotopy groups \\ $\\pinD{n}TX$ \\ for\nany \\ $X\\in{\\mathcal C}=s\\hat{\\C}$ \\ may be computed using the Moore chains \\\n$C_{\\ast}TX$ \\ as in \\S \\ref{dnc}; each  \\ $\\pinD{n}TX$ \\ is a \\ $\\Pi_{\\D}$-algebra,\nabelian for \\ $n\\geq 1$.  \n\n\\subsection{Explicit construction of the spiral exact sequence}\n\\label{sses}\\stepcounter{thm}\n\nIt may be helpful to inspect in detail the construction of last long\nexact sequence, since it is perhaps the least familar of the four.\nSpecificializing to \\ $\\hat{\\C}={\\mathcal G}$ \\ and \\ $T=\\pi_{\\ast}$, \\ we have:\n\n\\begin{lemma}\\label{lone}\\stepcounter{subsection}\nFor fibrant \\ $X\\in{\\mathcal C}$, \\ the inclusion \\ $\\iota:C_{n}X\\hookrightarrow X_{n}$ \\ \ninduces an isomorphism \\ $\\iota_{\\star}:\\pi_{\\ast} C_{n}X\\cong C_{n}(\\pi_{\\ast} X)$ \\ \nfor each \\ $n\\geq 0$. \n\\end{lemma}\n\n\\begin{proof}\nSee \\cite[Prop.\\ 2.11]{BlaCW}.\n\\end{proof}\n\nTogether with \\eqref{ethree}, this yields a commuting diagram:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentyseven}\n\\xymatrix{\n\\pi_{\\ast} C_{n+1}X \\rto^{(\\mathbf{d}_{0})_{\\#}} \\dto_{\\iota_{\\star}}^{\\cong} & \n\\pi_{\\ast} Z_{n}X \\ar@{->>}[r]^{\\hat{\\vartheta}_{n}} \\dto^{\\hat{\\iota}_{\\star}} & \n\\pi^{\\natural}_{n}X \\ar@{.>}[d]^{h_{n}} \\\\\nC_{n+1}(\\pi_{\\ast} X) \\rto^{d_{0}^{\\pi_{\\ast} X}} & \nZ_{n}(\\pi_{\\ast} X) \\ar@{->>}[r]^{\\vartheta_{n}} & \\pi^{\\natural}_{n}\\pi_{\\ast} X\n}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent which defines the dotted morphism of $\\Pi$-algebra s \\ \n$h_{n}:\\pi^{\\natural}_{n}X\\to\\pi_{n}(\\pi_{\\ast} X)$. \\ Note that for \\ $n=0$ \\ the map \\ \n$\\hat{\\iota}_{\\star}$ \\ is an isomorphism, so $h$ is, too.\n\nIf  \\ $X\\in s{\\mathcal G}$ \\ is fibrant, applying \\ $\\pi_{\\ast}$ \\ to the fibration sequence \n\\eqref{etwo} yields a long exact sequence, with connecting homomorphism \\ \n$\\partial_{n}:\\Omega\\pi_{\\ast} Z_{n} X=\\pi_{\\ast}\\Omega Z_{n} X \\to\\pi_{\\ast} Z_{n+1} X $; \\ \n\\eqref{ethree} then implies that\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{efourteen}\n\\Omega\\pi^{\\natural}_{n}X=\\Omega\\operatorname{Coker}\\,(\\mathbf{d}_{0}^{n+1})_{\\#}\\cong\\operatorname{Im}\\,\\partial_{n}\n\\cong \\operatorname{Ker}\\,(j^{X}_{n+1})_{\\#}\\subseteq\\pi_{\\ast} Z_{n+1}X,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent and the map \\ $s_{n+1}:\\Omega\\pi^{\\natural}_{n}X\\to\\pi^{\\natural}_{n+1}X$ \\ in\n\\eqref{efour} is then obtained by composing the inclusion \\ \n$\\operatorname{Ker}\\,(j^{X}_{n+1})_{\\#}\\hookrightarrow\\pi_{\\ast} Z_{n+1}X$ \\ with the quotient map \\ \n$\\hat{\\vartheta}_{n+1}:\\pi_{\\ast} Z_{n+1}X\\to\\pi^{\\natural}_{n+1}X$ \\ of \\eqref{ethree}.\n\nSimilarly, \\ $h_{n}:\\pi^{\\natural}_{n}X\\to\\pi^{\\natural}_{n}\\pi_{\\ast} X$ \\ is induced by the \ninclusion \\ \n$(j^{X}_{n})_{\\#}:\\pi_{\\ast} Z_{n}X\\to Z_{n}\\pi_{\\ast} X\\subseteq C_{n}\\pi_{\\ast} X$, \\ and \\ \n$\\partial^{\\star}_{n+2}:\\pi^{\\natural}_{n+2}\\pi_{\\ast} X\\to\\Omega\\pi^{\\natural}_{n}X$ \\ is induced by \\ \nthe composite \\ \n$$\nZ_{n+2}\\pi_{\\ast} X\\subseteq C_{n+2}\\pi_{\\ast} X\\cong \\pi_{\\ast} C_{n+2}X\n\\xra{(\\mathbf{d}_{0}^{n+2})_{\\#}}Z_{n+1}\\pi_{\\ast} X,\n$$\n\\noindent which actually lands in \\ \n$\\operatorname{Ker}\\,(j^{X}_{n+1})_{\\#}\\cong \\Omega\\pi^{\\natural}_{n}X$ \\ by the exactness of the \nlong exact sequence for the fibration.\n\nMoreover, for each \\ $n\\geq 0$, \\ \\eqref{etwentyseven} may be extended\n(after rotating by $90^{\\circ}$) \\ to a commutative diagram with exact\nrows and columns: \n$$\n\\xymatrix{\n& 0 \\dto & 0\\dto & 0\\dto & & \\\\ \n0 \\rto & \\operatorname{Ker}\\, s_{n} \\ar@{|->}[r] \\dto & B_{n+1}X \\dto\n\\ar@{->>}[r]^>>>>>{(j_{n})_{\\ast}} & B_{n+1}\\pi_{\\ast} X_{n+2}\\rto\\dto & 0\\dto & \\\\\n0 \\rto & \\Omega\\pi^{\\natural}_{n-1}X \\ar@{|->}[r] \\dto & \n\\pi_{\\ast} Z_{n}X \\dto^{\\hat{\\vartheta}_{n}} \\rto^{(j_{n})_{\\ast}} & \nZ_{n}\\pi_{\\ast} X \\ar@{->>}[r] \\dto^{\\vartheta_{n}} & \\operatorname{Coker}\\, h_{n} \\rto \\dto^{=} & 0 \\\\\n0 \\rto & \\operatorname{Ker}\\, h_{n} \\ar@{|->}[r] \\dto & \\pi^{\\natural}_{n}X \\dto \\rto^{h_{n}} & \n\\pi_{n}\\pi_{\\ast} X \\ar@{->>}[r] \\dto & \\operatorname{Coker}\\, h_{n}\\rto\\dto & 0 \\\\\n & 0 & 0 & 0 & 0 &\n}\n$$\n\\noindent in which \\ \n$B_{n+1} X:=\\operatorname{Im}\\,(\\mathbf{d}_{0}^{X_{n+2}})_{\\#}\\subseteq\\pi_{\\ast} Z_{n} X$ \\ and \\  \n$B_{n+1}\\pi_{\\ast} X_{n+2}:=\\operatorname{Im}\\,\\mathbf{d}_{0}^{\\pi_{\\ast} X_{n+2}}$ \\ are the respective\nboundary objects.\n\nThe maps \\ $\\partial^{\\star}_{n+1}$, \\ $s_{n}$, \\ and \\ $h_{n}$, \\ \nas defined above, form the spiral long exact sequence.\n\n\\subsection{Inverse spherical functors}\n\\label{sisf}\\stepcounter{thm}\n\nWe may sometimes be interested in functors between spherical model\ncategories which are not quite spherical. Thus, if \\ \n$T:\\lra{{\\mathcal C};{\\mathcal A}}\\to\\lra{{\\mathcal D};{\\mathcal B}}$ \\ is a spherical functor as in \\S\n\\ref{dsfunc}, a functor \\ $V:{\\mathcal D}\\to{\\mathcal C}$ \\ equipped with a natural\ntransformation \\ $\\vartheta:\\operatorname{Id}_{{\\mathcal C}}\\to VT$ \\ is called an\n\\emph{inverse spherical functor} to $T$. \n\n\\begin{example}\\label{eisf}\\stepcounter{subsection}\nFor the free group functor \\ $F:\\Set_{\\ast}\\to{\\EuScript Gp}$ \\ of \\S \\ref{ssfunc}(c),\nthe forgetful functor \\ $\\hat{U}:{\\EuScript Gp}\\to\\Set_{\\ast}$ \\ (right adjoint to $F$)\nwith the adjunction counit \\ $\\eta:\\operatorname{Id}\\to UF$ \\ as the natural\ntransformation $\\vartheta$, yields the inverse spherical functor \\ \n$U:{\\mathcal G}\\to\\Ss_{\\ast}$. \\ Here we do not think of ${\\mathcal G}$ as a model for $\\TT_{\\ast}$ \\ -- \\ \nrather, $U$ represents the forgetful functor from loop spaces\n(topological groups) to spaces.\n\nSimilarly, the adjoint to the abelianization functor \\ ${\\EuScript Ab}:{\\EuScript Gp}\\to{\\Ab{\\EuScript Gp}}$ \\ \nis the inclusion \\ $\\hat{I}:{\\Ab{\\EuScript Gp}}\\to{\\EuScript Gp}$, \\ and the corresponding functor \\ \n$I:s{\\Ab{\\EuScript Gp}}\\to{\\mathcal G}$ \\ represents the factorization of the Dold-Thom infinite\nsymmetric product functor \\ $SP^{\\infty}:\\TT_{\\ast}\\to\\TT_{\\ast}$ \\ \nthrough \\ ${\\EuScript Chain}$.\n\\end{example}\n\n\\begin{prop}\\label{pfour}\\stepcounter{subsection}\nIf \\ $V:{\\mathcal D}\\to{\\mathcal C}$ \\ is an inverse spherical functor to $T$,\nthen for each \\ $Y\\in{\\mathcal D}$ \\ and \\ $B\\in{\\mathcal B}$ \\ there is a natural \nlong exact sequence:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{esix}\n\\dotsc\\to \\Delta^{V}_{B,n}Y\\to\\pinD{B,n} Y\\xra{V_{\\#}} \n\\pinC{V_{\\ast}(B),n} VY\\to \\Delta^{V}_{B,n-1}Y\\dotsc\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\end{prop}\n\n\\begin{proof}\nIf $V$ is an inverse spherical functor, because \\ $T\\rest{{\\mathcal A}}$ \\ is a\nbijection onto ${\\mathcal B}$, there is an \\ $A\\in{\\mathcal A}$ \\ such that \\ \n$B=TA$. \\ As before, $V$ induces a natural transformation \\ \n$\\nu:\\operatorname{map}_{{\\mathcal D}}(B,\\tilde{Y})\\to \\operatorname{map}_{{\\mathcal D}}(VB,\\widehat{V\\tilde{Y}})$ \\ and the\nnatural transformation \\ $\\vartheta:A\\to VTA$ \\ yields \\ \n$\\vartheta^{\\#}:\\operatorname{map}_{{\\mathcal D}}(VTA,\\widehat{V\\tilde{Y}})\\to\n\\operatorname{map}_{{\\mathcal D}}(A,\\widehat{V\\tilde{Y}})$ \\ so we get a composite map \\ \n$\\operatorname{map}_{{\\mathcal D}}(B,\\tilde{Y})\\to\\operatorname{map}_{{\\mathcal D}}(A,\\widehat{V\\tilde{Y}})$, \\ with homotopy\nfiber \\ $E(Y)$. \\ If we let \\ $\\Delta^{V}_{\\beta,n}Y:=\\pi_{n} E(Y)$, \\ \nthe fibration long exact sequence is \\eqref{esix}. \n\\end{proof}\n\n\\begin{remark}\\label{rlesisf}\\stepcounter{subsection}\nNote that in contradistinction to Theorem \\ref{tthree}, \\ \n$V_{\\#}$ \\ of \\eqref{esix} need not respect any operations, since we\nonly have a bijection \\ $T\\rest{{\\mathcal A}}:{\\mathcal A}\\to{\\mathcal B}$, \\ not a functor. \n\nFor \\ $U:{\\mathcal G}\\to\\Ss_{\\ast}$ \\ as in \\S \\ref{eisf}, we may assume \\ $X\\in{\\mathcal G}$ \\\nis of the form \\ $X\\simeq GK$ \\ for \\ $K\\in\\Ss_{\\ast}$, \\ and then \\ \n$V_{\\#}$ \\ is the identity:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentythree}\n\\begin{split}\n\\pi_{n}K~=~\\pi^{\\natural}_{n}X&=~\\pi_{0}\\,\\operatorname{map}_{{\\mathcal G}}(FS^{n-1},GK)~\\to~\n\\pi_{0}\\,\\operatorname{map}_{\\Ss_{\\ast}}(UFS^{n-1},UGK)\\\\\n& \\xra{\\eta^{\\#}}~\\pi_{0}\\,\\operatorname{map}_{\\Ss_{\\ast}}(S^{n-1},UGK)~=~\\pi_{n}K~,\n\\end{split}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent so \\eqref{esix} is not interesting in this case. \n\\end{remark}\n\n\\setcounter{thm}{0}\\section{Comparing Postnikov systems}\n\\label{ccps}\n\nThe basic problem under consideration in this paper may be formulated as\nfollows\\vspace{2 mm} : \n\n\\subsection*{Question}\nGiven a spherical functor \\ $T:\\lra{{\\mathcal C};{\\mathcal A}}\\to\\lra{{\\mathcal D};{\\mathcal B}}$ \\\nand an object \\ $G\\in{\\mathcal D}$, \\ what are the different objects \\ $X\\in{\\mathcal C}$ \\ \n(up to homotopy) such that \\ $TX\\simeq G$\\vspace{2 mm} ?\n\nAs shown in the previous section, such a pair \\ \n$\\lra{X,G}$ \\ must be connected by a comparison exact sequence. Thus,\nin order to reconstruct $X$ from $G$, we first try to determine \\ \n$\\pinC{\\ast}X$, \\ and its relation to \\ $\\pinD{\\ast}G$. \n\nIn order to proceed further, we must make an additional assumption on\n$T$, contained in the following:\n\n\\begin{defn}\\label{dspecial}\\stepcounter{subsection}\nA spherical (or inverse spherical) functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is called\n\\emph{special} if:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\roman{enumi}.\\ }\n\\item ${\\mathcal C}=s\\hat{\\C}$ \\ and \\ ${\\mathcal D}=s\\hat{\\D}$ \\ are spherical resolution model\n  categories, and $T$ is prolonged from a functor \\ $\\hat{T}:\\hat{\\C}\\to\\hat{\\D}$. \\ \n\\item For any $\\PiA$-algebra\\ $\\Lambda$ and module $M$ over $\\Lambda$, \\ $T$ induces a \nhomomorphism of (graded) groups \\ $\\phi_{T}:\\Lambda\\to\\pinD{0}TB_{\\C}\\Lambda$. \\ \n\\item This \\ $\\phi_{T}$ \\ induces a functor \\ \n$\\hat{T}:\\RM{\\Lambda}\\to\\RM{\\phi_{T}\\Lambda}$ \\ which is an\nisomorphism on $\\Lambda$-modules (see Remark \\ref{rmodule}).\n\\item For each \\ $n\\geq 1$ \\ and $n$-dimensional extended\n$M$-Eilenberg-Mac~Lane object \\ $E=\\ECL{M}{n}$, \\ there is a natural\nisomorphism \\ $\\pinD{n}TE\\cong M$ \\ which respects $\\hat{T}$ in the\nobvious sense.\n\\item The natural map\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentyfour}\n[X,\\ECL{M}{n}]_{B_{\\C}\\Lambda}\\to [TX,E\\sp{\\hat{T}\\Lambda}\\sb{{\\mathcal D}}(M,n)]_{B_{{\\mathcal D}}\\hat{T}L}~,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent defined by composition with the projection\n$$\n\\rho:T\\ECL{M}{n}\\to P_{n}T\\ECL{M}{n}~=~E\\sp{\\hat{T}\\Lambda}\\sb{{\\mathcal D}}(M,n)~, \n$$ \nis an isomorphism.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{example}\\label{especial}\\stepcounter{subsection}\nAll the functors we have considered hitherto, except for the rationalization\nfunctor \\ $Q:{\\mathcal G}\\to s{\\EuScript Hopf}$ \\ of \\S \\ref{ssfunc}(b), are special\\vspace{2 mm}:\n\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{(\\alph{enumi})\\ }\n\\item For the singular chain functor \\ $T:{\\mathcal G}\\to s{\\Ab{\\EuScript Gp}}$, induced by \nabelianization, this follows from the Hurewicz Theorem (recall that \\\n$\\pinC{0}X$ \\ is the fundamental group, in our indexing for \\ $X\\in{\\mathcal G}$).\n\\item For the suspension \\ $\\Sigma:\\TT_{\\ast}\\to\\TT_{\\ast}$, \\ induced by \nthe free group functor \\ $F:\\Set_{\\ast}\\to{\\EuScript Gp}$, \\ this follows (in the\nsimply connected case) from the Freudenthal Suspension Theorem.\n\\item For the homotopy groups functor \\ $\\pi_{\\ast}:s{\\mathcal G}\\to s\\Alg{\\Pi}$, \\ (i)-(iii)\nfollow by inspecting the spiral long exact sequence \\eqref{efifteen},\nwhile (iv) is \\cite[Prop.\\ 8.7]{BDGoeR}.\n\\item For the inverse spherical functor \\ $U:{\\mathcal G}\\to\\Ss_{\\ast}$ \\ of \\S \\ref{eisf},\ninduced by the forgetful functor \\ $\\hat{U}:{\\EuScript Gp}\\to\\Set_{\\ast}$, \\ this is\nimmediate from \\eqref{etwentythree}\\vspace{2 mm}.\n\\end{enumerate}\n\\end{example}\n\n\\begin{lemma}\\label{ltwo}\\stepcounter{subsection}\nAny special spherical functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ as above \\emph{respects\nPostnikov systems} \\ -- \\ that is, for any \\ $X\\in{\\mathcal C}$ \\ \nand \\ $n\\geq 0$ \\ we have:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eeight}\nP^{{\\mathcal D}}_{n}T P^{{\\mathcal C}}_{n} X\\cong P^{{\\mathcal D}}_{n}T X \\ - \\ \n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\nso that \\ $\\pinC{k}T X\\cong \\pinD{k}TP_{n}X$ \\ and \\ \n$\\Gamma_{k}X\\cong \\Gamma_{k}P_{n}X$ \\ for \\ $k\\leq n$. \n\\end{lemma}\n\n\\begin{proof}\nThis follows from the constructions in \\S \\ref{sps} and the proof of\nTheorem \\ref{tthree}.\n\\end{proof}\n\n\\subsection{Postnikov systems and spherical functors}\n\\label{spssf}\\stepcounter{thm}\n\nFrom now on, assume \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is a special spherical functor.\nUltimately, for each object \\ $G\\in{\\mathcal D}$, \\ we would like find any and all \\\n$X\\in{\\mathcal C}$ \\ such that \\ $TX\\simeq G$. \\ First, however, we try to\ndiscover what can be said about \\ $TX$ \\ and its Postnikov systems\nfor a given \\ $X\\in{\\mathcal C}$. \\ Using the comparison exact sequence for $T$\nand Lemma \\ref{ltwo}, we see that:  \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{enine}\n\\pinD{k}TP_{n}X\\cong\\begin{cases}\n        \\pinD{k}TX & \\text{for \\ } k\\leq n,\\\\\n        \\operatorname{Coker}\\,\\{h^{X}_{n+1}:\\pinC{n+1}X\\to\\pinD{n+1}T X\\} & \\text{for \\ }\n          k=n+1,\\\\\n        \\Gamma_{k-1}P_{n}X & \\text{for \\ }k\\geq n+2~.\n\\end{cases}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\n\\begin{fact}\\label{fsix}\\stepcounter{subsection}\nIf \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is a special spherical functor, applying \\\n$\\pinC{n+2}$ \\ to the $n$-th $k$-invariant \\\n$k_{n}:P_{n}X\\to \\ECL{\\pinC{n+1}X}{n+2}$ \\  \nyields the homomorphism \\ $s^{X}_{n+1}:\\Gamma_{n+1}X\\to\\pinC{n+1}X$.\n\\end{fact}\n\n\\begin{proof}\nSince $T$ is special, \\ $\\pinD{n+2}T\\ECL{\\pi^{\\natural}_{n+1}X}{n+2}\\cong\\pinC{n+1}X$, \\ \nand \\ $\\pinD{n+2}TP_{n}X\\cong\\Gamma_{n+1}X$ \\ from \n\\eqref{enine}, so this follows from the naturality of the comparison\nexact sequence, applied to the maps in \\eqref{ezero}.\n\\end{proof}\n\n\\begin{lemma}\\label{lthree}\\stepcounter{subsection}\nIf \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ is a special spherical functor, for any \\ $X\\in{\\mathcal C}$, \n$$\n\\xymatrix{\nP_{n+1}TP_{n}X \\rto \\dto & P_{n+1}TP_{n-1}X \\dto \\\\ \nP_{n+1}TB_{\\C}\\Lambda \\rto^-{\\protect{Tk_{n}}} & P_{n+1}T\\ECL{\\pinC{n}X}{n+2}\n}\n$$\n\\noindent is a homotopy pullback square in \\ ${\\mathcal D}\/TB_{\\C}\\Lambda$, \\ where \\ \n$\\Lambda:=\\pinC{0}X$. \n\\end{lemma}\n\n\\begin{proof}\nSet \\ $E:=T\\ECL{J}{n+1}$, \\ $M^{n-1}:=TP_{n-1}X$, \\ and \\ $M^{n}:=TP_{n}X$. \\ \nThe naturality of the comparison exact sequence, applied to the maps\nin \\eqref{ezero}, \\ combined with Fact \\ref{fsix}, imply that the\nvertical maps in the following commutative diagram are isomorphisms:\n$$\n\\xymatrix{\n\\pinD{n+2}E \\rto \\dto^{\\cong} & \\pinD{n+1}M^{n} \\rto \\dto^{\\cong} &\n\\pinD{n+1}M^{n-1} \\rto^>>>>{Tk_{n-1}} \\dto^{\\cong} & \n\\pinD{n+1}E \\rto \\dto^{\\cong} &\n\\pinD{n}M^{n} \\rto \\dto^{\\cong} & \\pinD{n}M^{n-1} \\dto^{\\cong} \\\\\n0\\rto & \\operatorname{Coker}\\, h^{X}_{n+1} \\ar@{|->}[r] & \\Gamma_{n}X \\rto^{s^{X}_{n}} &\n\\pinC{n}X \\rto^{h^{X}_{n}} & \\pinD{n}T X \\ar@{->>}[r] & \\operatorname{Coker}\\, h_{n}^{T}\n}\n$$\n\\noindent and since the bottom row is part of the comparison long\nexact sequence, and the rest of the top sequence to the right \nis exact for by \\eqref{eeight}, the $k$-invariant square \\eqref{ezero} \ninduces a long exact sequence after applying \\ $\\pi^{\\natural}$ \\ (except in the\nbottom dimensions). \\ The obvious map from \\ $M^{n}$ \\ to the fiber of \\\n$Tk_{n-1}$ \\ is thus a weak equivalence in \\ ${\\mathcal D}\/TB_{\\C}\\Lambda$ \\ through\ndimension \\ $n+1$.\n\\end{proof}\n\n\\begin{cor}\\label{czero}\\stepcounter{subsection}\nFor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ as above, \\ for any \\ $X\\in{\\mathcal C}$ \\ and \\ $n\\geq 1$ \\ \nthe natural map \\ $r\\q{n}:X\\to P_{n}X$ \\ of \\S \\ref{sps} induces an \\ \nisomorphism \\ $\\Gamma_{k}X\\cong\\Gamma_{k}P_{n}X$ \\ for \\ $k\\leq n+1$.\n\\end{cor}\n\n\\begin{proof}\nFor each \\ $A\\in{\\mathcal A}$, \\ take fibers vertically and horizontally of\nthe commutative square:\n$$\n\\xymatrix{\n\\operatorname{map}_{B_{\\C}\\Lambda}(A,P_{n}X) \\rto^-{\\protect{h^{P_{n}X}}} \\dto^{(k_{n})_{\\ast}} & \n\\operatorname{map}_{B_{\\D}\\Lambda}(TA,TP_{n}X) \\dto^{(Tk_{n})_{\\ast}} \\\\ \n\\operatorname{map}_{B_{\\C}\\Lambda}(A,\\ECL{\\pinC{n+1}X}{n+2}) \\rto^-{\\protect{h^{E}}} & \n\\operatorname{map}_{B_{\\D}\\Lambda}(TA,T\\ECL{\\pinC{n+1}X}{n+2})~,\\\\\n}\n$$\nand use Lemma \\ref{lthree} and \\S \\ref{dspecial}(iv).\n\\end{proof}\n\n\\begin{remark}\\label{rhem}\\stepcounter{subsection}\nFor \\ ${\\mathcal C}=s{\\mathcal G}$ \\ this follows from the fact that \\\n$\\Gamma_{n}X\\cong\\Omega\\pi^{\\natural}_{n-1}X$, \\ while for the algebraic cases\nof \\S \\ref{ermc}(i-ii), this follows from the fact that \\\n$H_{n+1}(K(\\pi,n);\\mathbb Z)=0$ \\ for \\ $n\\geq 1$.\n\\end{remark}\n\n\\subsection{The extension}\n\\label{sext}\\stepcounter{thm}\n\nThe map \\ $r\\q{n}:X\\to P_{n}X$ \\ induces a map of comparison exact\nsequences: \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentysix}\n\\xymatrix{\n\\pinC{n+2}X \\dto \\rto^{h^{X}_{n+2}} & \n\\pinC{n+2}TX \\dto^{\\pinD{n+2}Tr\\q{n}} \\rto^{\\partial^{\\star}_{n+2}} & \n\\Gamma_{n+1}X \\dto^{=} \\rto^{s_{n+1}} & \\pinC{n+1}X \\dto \\rto^{h^{X}_{n+1}} & \n\\pinD{n+1}TX \\dto^{\\pinD{n+1}Tr\\q{n}} \\rto^{\\partial^{\\star}_{n+1}} & \n\\Gamma_{n}X \\dto^{=} \\\\\n0 \\rto & \\pinD{n+2}M^{n} \\rto^{\\cong} & \\Gamma_{n+1}P_{n}X \\rto & 0 \\rto &\n\\pinD{n+1}M^{n} \\rto & \\Gamma_{n}P_{n}X\n}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent so that \\ $\\pinC{n+1}X$ \\ fits into a short exact sequence \nof $\\PiA$-algebra s:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eten}\n0\\to \\operatorname{Coker}\\,\\pinD{n+2}Tr\\q{n} \\to \\pinC{n+1}X \\to\n\\operatorname{Ker}\\,\\pinD{n+1}Tr\\q{n} \\to 0,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent where \n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwentyone}\n\\operatorname{Coker}\\,\\pinD{n+2}Tr\\q{n}\\cong \\operatorname{Ker}\\, h^{X}_{n+1}\\hspace{2 mm} \\text{and}\\hspace{2 mm} \n\\operatorname{Ker}\\,\\pinD{n+1}Tr\\q{n} \\cong \\operatorname{Im}\\, h^{X}_{n+1}.\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nSince \\ $h^{X}_{n+1}$ \\ is a map of modules over \\ $\\Lambda:=\\pinC{0}X$, \\ \nby Theorem \\ref{tone}, \\eqref{eten} is actually a short exact sequence\nof modules over $\\Lambda$, and we can classify the possible values of \\ \n$J\\in\\RM{\\Lambda}$ \\ (the candidates for \\ $\\pinC{n+1}X$) \\ using the following:\n\\begin{prop}\\label{peleven}\\stepcounter{subsection}\nGiven \\ $Tr\\q{n}:T X\\to T P_{n}X$, \\ a choice for the \nisomorphism class of \\ $\\pinC{n+1}X$ \\ uniquely determines an element of \\ \n$$\n\\operatorname{Ext}_{\\RM{\\Lambda}}(\\operatorname{Ker}\\,(Tr\\q{n})_{n+1}, \\operatorname{Coker}\\,(Tr\\q{n})_{n+2}).\n$$\n\\end{prop}\n\n\\begin{proof}\nSince \\ $\\RM{\\Lambda}$ \\ is an abelian category, with a set  \\ \n$\\{A_{\\operatorname{ab}}\\otimes S^{n}\\amalgB_{\\D}\\Lambda\\}_{A\\in{\\mathcal A},n\\in\\mathbb N}$ \\ \nof projective generators, the argument of \\cite[III]{MacH} carries\nover to our setting. \n\\end{proof}\n\n\\begin{remark}\\label{rinfo}\\stepcounter{subsection}\nObserve that given \\ $P_{n}X$, \\ we know the comparison exact sequence \n\\eqref{efive} for \\ $X$ \\ only from \\ \n$s_{n}:\\Gamma_{n-1}X\\to\\pinC{n}X$ \\ down. However, if \\ \n$\\pinD{i}Tr\\q{n}:\\pinD{i}TX\\to\\pinD{i}M^{n}$ \\ (for \\ $i\\geq 0$) \\ and the \nextension \\eqref{eten} are also known,  all we need in order to\ndetermine \\eqref{efive} for $X$ from \\ \n$\\partial_{n+3}^{\\star}:\\pinD{n+3}TX\\to\\Gamma_{n+1}X$ \\ down is the \nhomomorphism\n$$\n\\pinD{n+3}Tr\\q{n+1}:\\pinD{n+3}TX\\to\\pinD{n+3}TP_{n+1}X~,\n$$\nwhich is just \\ $\\partial_{n+3}^{\\star}$, \\ as one can see \nfrom \\eqref{etwentysix}.\n\\end{remark}\n\n\\begin{prop}\\label{ptwelve}\\stepcounter{subsection}\nFor any \\ $\\Lambda\\in{\\mathcal D}$, \\ $J',J''\\in\\RM{\\Lambda}$, \\ and \\ \n$n\\geq 2$, \\ there is a natural isomorphism \\ \n$\\operatorname{Ext}_{\\RM{\\Lambda}}(J'',J')\\cong\\HL{n+1}{\\EDL{J''}{n}}{J'}$.\n\\end{prop}\n\nIn particular, this implies that \\ $\\HL{n+1}{\\EDL{-}{n}}{-}$ \\ is \n\\emph{stable} \\ -- \\ i.e., independent of $n$.\n\n\\begin{proof}\nBy Proposition \\ref{ptwo}\\emph{ff.} there is a natural isomorphism \\ \n$$\n\\HL{n+1}{\\EDL{J''}{n}}{J'}\n\\cong[\\EDL{J''}{n},\\EDL{J'}{n+1}]_{s{\\mathcal D}\/B_{\\D}\\Lambda},\n$$\n\\noindent and given a map \\ $\\psi:\\EDL{J''}{n}\\to\\EDL{J'}{n+1}$, \\ we can \nform the fibration sequence over \\ $B_{\\D}\\Lambda$ \\ (that is, pullback square\nas in \\eqref{ezero}):\n$$\n\\Omega\\EDL{J''}{n}\\xra{\\Omega\\psi}\\Omega\\EDL{J'}{n+1}\\simeq\\EDL{J'}{n}\n\\to F\\to\\EDL{J''}{n}\\xra{\\psi}\\EDL{J'}{n+1}.\n$$\n\\noindent From the corresponding long exact sequence in homotopy \nfor this sequence in ${\\mathcal D}$, we obtain a short exact sequence of \nmodules over $\\Lambda$:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eeleven}\n0\\to J'\\to J\\to J''\\to 0.\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nOn the other hand, given a short exact sequence \\eqref{eeleven} in \\ \n$\\RM{\\Lambda}$, \\ we can construct a map \\ \n$\\psi:\\EDL{J''}{n}\\to\\EDL{J'}{n+1}$ \\ over \\ $B_{\\D}\\Lambda$ \\ as follows:\n\nAssume \\ $E:=\\EDL{J''}{n}$ \\ is constructed  starting with \\ \n$\\sk{n-1}\\EDL{J''}:=\\sk{n-1}B_{\\D}\\Lambda$, \\ and \\ \n$E_{n}\\simeq W\\amalg L_{n}B_{\\D}\\Lambda$ \\ (cf.\\ \\S \\ref{dlat}), \\ where $W$ is free, \nequipped with a surjection \\ $\\phi:W\\to J''$. \\ Because \\ $J\\to\\hspace{-5 mm}\\to J''$ \\ is \na surjection, and $W$ is free, we can lift $\\phi$ to \\ \n$\\phi':W\\to J$, \\ defining a map \\ $\\tilde{\\phi}':Z_{n}\\EDL{J''}{n}\\to J$. \\ \nSince \\  $\\pinD{n}\\EDL{J''}{n}=J''$, \\ the restriction of \\ \n$\\tilde{\\phi}'$ \\ to \\ $B_{n}\\EDL{J''}{n}=\\operatorname{Ker}\\,\\{Z_{n}\\EDL{J''}{n}\\to J''\\}$ \\ \nfactors through \\ $\\psi:B_{n}\\EDL{J''}{n}\\to J'=\\operatorname{Ker}\\,\\{J\\to J''\\}$. \\\nPrecomposing with \\ $\\mathbf{d}_{0}:C_{n+1}\\EDL{J''}{n}\\to B_{n}\\EDL{J''}{n}$ \\ defines \\ \n$\\psi:\\EDL{J''}{n}\\to\\EDL{J'}{n+1}$, \\ which classifies \\eqref{eeleven} \nas before.\n\\end{proof}\n\n\\begin{cor}\\label{cone}\\stepcounter{subsection}\nFor $\\Lambda$, $J'$, and $J''$ as above, there is a natural isomorphism:\n$$\n\\operatorname{Ext}_{\\RM{\\Lambda}}(J'',J')\\cong\\HL{n+1}{\\ECL{J''}{n}}{J'}.\n$$\n\\end{cor}\n\n\\begin{proof}\nThis follows from \\eqref{etwentyfour}-\\eqref{enine} and the \nnaturality of \\ $P^{{\\mathcal D}}_{n+1}$.\n\\end{proof}\n\n\\begin{defn}\\label{dmps}\\stepcounter{subsection}\nGiven \\ $X\\in{\\mathcal C}$, \\ its \\emph{$n$-th modified Postnikov section}, \ndenoted by \\ $\\bPa{n}X$, \\ is defined as follows\\vspace{2 mm} :\n\nLet \\ ${\\EuScript K}:=\\{f:A\\otimes S^{n+1}\\to X~|\\ \nA\\in{\\mathcal A},~[f]\\in\\operatorname{Ker}\\, h_{n+1}^{T}\\subset\\pinC{n+1}X\\}$, \\ and let\n$C$ be the cofiber of the obvious map \\ \n$\\Phi:\\bigvee_{f\\in{\\EuScript K}}~A\\otimes S^{n+1}\\to X$ \\  \n(so that \\ $\\pinC{n+1}C\\cong\\operatorname{Coker}\\,\\Phi$), \\ with \\ $\\bPa{n}X:= P_{n+1}C$. \\ \nThere are then natural maps \\ $\\bpa{n+1}:P_{n+1}X\\to\\bPa{n}X$ \\ \n(induced by \\ $X\\to C$), \\ as well\nas \\ $\\bpc{n}:\\bPa{n}X\\to P_{n}X$ \\ (which is just \\ \n$p\\q{n}_{C}:P_{n+1}C\\to P_{n}C\\cong P_{n}X$), \\ with \\ \n$\\bpc{n}\\circ\\bpa{n}=p\\q{n}_{X}:P_{n+1}X\\to P_{n}X$. \\ \nNote that \\ $\\pinC{n+1}\\bPa{n}X\\cong\\operatorname{Im}\\, h^{X}_{n+1}$, \\ and \\ \n$P_{n}\\bPa{n}X\\cong P_{n}X$.\n\\end{defn}\n\nThe map \\ $\\brp{n}:=\\bpa{n}\\circ r\\q{n}:X\\to\\bPa{n}X$ \\ induces a map of \ncomparison exact sequences:\n$$\n\\xymatrix{\n\\pinC{n+2}X \\rto^{h^{X}_{n+2}} \\dto & \n\\pinD{n+2}T X \\rto^{\\partial^{\\star}_{n+2}} \\dto^{\\pinD{n+2}T\\brp{n}} &\n\\Gamma_{n+1}X \\rto^{s_{n+1}} \\dto^{=} &\n\\pinC{n+1}X \\rto^{h^{X}_{n+1}} \\dto &\n\\pinD{n+1}T X \\rto^{\\partial_{n+1}} \\dto_{=}^{\\pinD{n+1}T\\brp{n}}\n& \\Gamma_{n}X \\dto^{=} \\\\\n0 \\rto & \\pinD{n+2}T\\bPa{n}X \\rto^{\\cong} & \\Gamma_{n+1}\\bPa{n}X \\rto^{0} & \n\\pinC{n+1}\\bPa{n}X \\ar@{|->}[r] & \\pinD{n+1}T\\bPa{n}X \\rto & \\Gamma_{n}\\bPa{n}X\n}\n$$\n\\noindent so that:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{etwenty}\n\\pinD{k}T\\bPa{n}X\\cong\\begin{cases}\n        \\pinD{k}TX& \\text{for \\ } k\\leq n+1,\\\\\n        \\Gamma_{n+1}X & \\text{for \\ }k=n+2,\\\\\n        \\Gamma_{k-1}\\bPa{n}X & \\text{for \\ }k\\geq n+3~.\n\\end{cases}\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nThus \\ $\\brp{n}$ \\ induces a weak equivalence \\ $P_{n+1}TX\\simeq\nP_{n+1}T\\bPa{n}X$, \\ which, together with the existence of the\nappropriate maps \\ $P_{n+1}X\\xra{\\bpa{n}}\\bPa{n}X\\xra{\\bpc{n}}P_{n}X$, \\ \ndetermines \\ $\\bPa{n}X$ \\ up to homotopy. In fact we have:\n\n\\begin{prop}\\label{pfive}\\stepcounter{subsection}\n$\\bPa{n}X$ \\ is determined uniquely (up to weak equivalence) by \\ \n$P_{n}X$ \\ and the map \\ $\\rho:=P_{n+1}Tr\\q{n}:P_{n+1}TX\\to P_{n+1}TP_{n}X$.\n\\end{prop}\n\n\\begin{proof}\nNote that \\ $I_{n+1}:=\\operatorname{Ker}\\, \\pinD{n+1}\\rho$ \\ is isomorphic to \\ \n$\\operatorname{Im}\\, h^{X}_{n+1}$ \\ and \\ $C_{n+1}:=\\operatorname{Im}\\, \\pinD{n+1}\\rho$ \\ \nis isomorphic to \\ $\\operatorname{Coker}\\, h^{X}_{n+1}$ \\ by \\eqref{etwentyone}. \\ \n\nWe construct \\ $Y\\simeq\\bPa{n}X$ \\ as follows, starting with \\ \n$\\sk{n+1}Y:=\\sk{n+1}P_{n}X$; \\ by Remark \\ref{rcsk}, we may \nassume \\ $\\sk{n+1}TX=\\sk{n+1}TP_{n}X$, \\ so that \\ \n$P_{n}TX\\cong P_{n}TP_{n}X$. \\ By Fact \\ref{ffive}), the lower right \nhand square in Figure \\ref{fig6} commutes in ${\\mathcal D}$, thus inducing the\nrest of the diagram, in which the rows and columns are fibration\nsequences over \\ $B_{\\D}\\Lambda$. \n\n\\begin{center}\n\\setcounter{figure}{\\value{thm}}\\stepcounter{subsection}\n\\begin{figure}[hbtp]\n\\begin{picture}(300,110)(25,-5)\n\\put(15,100){$F$}\n\\multiput(35,105)(3,0){34}{\\circle*{.5}}\n\\put(137,105){\\vector(1,0){2}}\n\\put(85,110){$\\hat{\\rho}$}\n\\put(145,100){$P_{n+1}TP_{n}X$}\n\\multiput(205,105)(3,0){13}{\\circle*{.5}}\n\\put(245,105){\\vector(1,0){2}}\n\\put(217,110){$\\tkp{n}$}\n\\put(255,100){$\\EDL{I_{n+1}}{n+2}$}\n\\put(20,92){\\vector(0,-1){30}}\n\\put(8,75){$\\simeq$}\n\\put(25,75){$\\lambda$}\n\\put(170,92){\\vector(0,-1){30}}\n\\put(175,75){$p\\q{n}_{TP_{n}X}$}\n\\put(290,92){\\vector(0,-1){30}}\n\\put(295,75){$i_{\\ast}$}\n\\put(0,50){$P_{n+1}TX$}\n\\put(45,55){\\vector(1,0){55}}\n\\put(65,62){$p\\q{n}_{TX}$}\n\\put(105,50){$P_{n}TX\\cong P_{n}TP_{n}X$}\n\\put(205,55){\\vector(1,0){45}}\n\\put(220,61){$k_{n}^{TX}$}\n\\put(255,50){$\\EDL{\\pinD{n+1}TX}{n+2}$}\n\\put(20,42){\\vector(0,-1){30}}\n\\put(170,42){\\vector(0,-1){30}}\n\\put(175,28){$k_{n}^{TP_{n}X}$}\n\\put(290,42){\\vector(0,-1){30}}\n\\put(295,28){$q_{\\ast}$}\n\\put(10,0){$B_{\\D}\\Lambda$}\n\\put(55,5){\\vector(1,0){65}}\n\\put(125,0){$\\EDL{C_{n+1}}{n+2}$}\n\\put(215,5){\\vector(1,0){35}}\n\\put(230,10){$=$}\n\\put(255,0){$\\EDL{C_{n+1}}{n+2}$}\n\\end{picture}\n\\caption[fig6]{}\n\\label{fig6}\n\\end{figure}\n\\setcounter{thm}{\\value{figure}}\n\\end{center}\n\nIn particular, the induced map \\ \n$\\tkp{n}:P_{n+1}TP_{n}X\\to\\EL{I_{n+1}}{n+2}$ \\ provides a canonical lifting of:\n$$\nk_{n}^{TX}\\circ p\\q{n}_{TP_{n}X}:P_{n+1}TP_{n}X\\to\\EDL{\\pinD{n+1}TX}{n+2}\n$$\n\\noindent to \\ $\\EDL{I_{n+1}}{n+2}$. \\ Composing it with the natural map \\ \n$r\\q{n+1}:TP_{n}X\\to P_{n+1}TP_{n}X$ \\ defines an element in: \n$$\n[TP_{n}X,\\EDL{I_{n+1}}{n+2}]\\cong\\HL{n+2}{P_{n}X}{I_{n+1}}~,\n$$\n\\noindent which we call the \\emph{$n$-th modified $k$-invariant} for $X$\\vspace{2 mm}.\n\nIf \\ $\\hk{n}:P_{n}X\\to\\ECL{I_{n+1}}{n+2}$ \\ is the map corresponding to\n$\\tkp{n}$ \\ under \\eqref{etwentyfour}), then its homotopy fiber $Y$ is \n(weakly equivalent to) \\ $\\bPa{n}X$, \\ as one can verify by calculating \\ \n$\\pinC{\\ast}Y$. \\ Note that Lemma \\ref{lthree} implies that \\ \n$F\\simeq P_{n+1}TP_{n}X$, \\ so that $\\lambda$ is the homotopy inverse\nof the weak equivalence \\ $P_{n+1}\\rho:TX\\to TP_{n}X$, \\ which\ncompletes the construction. \n\\end{proof}\n\n\\begin{remark}\\label{rmin}\\stepcounter{subsection}\nNote that there is a certain indeterminacy in our description of \\ \n$\\tkp{n}$, \\ and thus of \\ $\\hk{n}$, \\ since we must make the lower right\ncorner of Figure \\ref{fig6} into a strict commuting diagram of fibrations,\nrather than one which commutes only up to homotopy. However,\n\\end{remark}\n\n\\begin{fact}\\label{fseven}\\stepcounter{subsection}\nThe indeterminacy for \\ $\\tkp{n}$ \\ as an induced map is contained in \nthe indeterminacy for \\ $\\tkp{n}$ \\ as a $k$-invariant\nfor \\ $P_{n+1}TX=P_{n+1}TY$.\n\\end{fact}\n\n\\begin{proof}\nLet \\ $M:=TP_{n}X$. \\ Making the lower right corner of Figure\n\\ref{fig6} commute on the nose (assuming \\ $q_{\\ast}$ \\ is already a\nfibration) requires the choice of a homotopy\n$$\nH:P_{n}TX\\to\\Omega\\EDL{C_{n+1}}{n+2}=\\EDL{C_{n+1}}{n+1}~,\n$$\n\\noindent so the indeterminacy for \\ $\\tkp{n}$ \\ as defined above is \\ \n$\\psi_{\\ast}p^{\\ast}[P_{n}TX,\\EDL{C_{n+1}}{n+1}]$, \\ where \\ \n$\\psi:\\EDL{C_{n+1}}{n+1}\\to\\EDL{I_{n+1}}{n+2}$ \\ classifies the extension \\ \n$$\n0\\to I_{n+1}\\to\\pinD{n+1}TX\\to C_{n+1}\\to 0\n$$\n\\noindent (Proposition \\ref{ptwelve}), and \\ \n$p=p\\q{n}_{M}:P_{n+1}M\\to P_{n}M=P_{n}TX$. \n\nOn the other hand, the $k$-invariant \\ \n$\\tkp{n}^{M}:P_{n+1}M\\to\\EDL{I_{n+1}}{n+2}$ \\ \nfor \\ $P_{n+1}TP_{n}X$ \\ (which is \\ $P_{n+1}TX$) \\ is \ndetermined only up to the actions of the group \\ $\\operatorname{haut}_{\\Lambda}(P_{n+1}M)$ \\\nof homotopy self-equivalences of \\ $P_{n+1}M$ \\ over \\\n$B_{\\D}\\Lambda$, \\ and of \\ $\\Aut_{\\Lambda}(I_{n+1})$, \\ the group of automorphisms of\nmodules over $\\Lambda$ of \\ $I_{n+1}$, \\ in \\\n$[P_{n+1}M,\\EDL{I_{n+1}}{n+2}]$. \nThus given a map \\ $f:P_{n}M\\to\\EDL{C_{n+1}}{n+1}$, \\ we obtain a \nself-map \\ $g:P_{n+1}M\\to P_{n+1}M$ \\ such that \\ \n$P_{n}g=\\operatorname{Id}_{P_{n}M}$ \\ and \\ $\\pinD{n+1}g=\\operatorname{Id}$, \\ by letting \\ \n$g=\\operatorname{Id}+i_{\\ast}p^{\\ast}(f)$, \\ for \\ $i:\\EDL{C_{n+1}}{n+1}\\to P_{n+1}M$ \\ \nthe inclusion of the fiber. It is readily verified that \n$g$ induces the identity on \\ $\\pinD{\\ast}P_{n+1}M$, \\ so \\ \n$[g]\\in \\operatorname{haut}_{\\Lambda}(P_{n+1}M)$, \\ and that \\ \n$\\tkp{n}+\\psi_{\\ast}p^{\\ast}(f)$ \\ is obtained from \\ $\\tkp{n}$ \\ under\nthe action of \\ $[g]$ \\ on \\ $\\HL{n+2}{P_{n+1}M}{I_{n+1}}$.\n\\end{proof}\n\n\\begin{notation}\\label{nbpa}\\stepcounter{subsection}\nGiven \\ $W\\simeq P_{n}X$ \\ and \\ $\\rho:P_{n+1}TX\\to P_{n+1}TW$, \\ \nProposition \\ref{pfive} allows us to write \\ $\\bPa{n}(W,\\rho)$, \\ or\nsimply \\ $\\bPa{n}W$ \\ for \\ $\\bPa{n}X\\in{\\mathcal C}$, \\ which\nthey determine up to homotopy. This comes equipped with a weak equivalence \\ \n$\\rho:P_{n+1}TX\\to P_{n+1}T\\bPa{n}W$ \\ lifting $\\rho$.\n\\end{notation}\n\n\\begin{cor}\\label{ctwo}\\stepcounter{subsection}\nThe weak equivalence \\ $\\rho:P_{n+1}TX\\to P_{n+1}T\\bPa{n}W$ \\ is\nwell-defined up to homotopy.\n\\end{cor}\n\n\\begin{proof}\nThe map $\\rho$ is inverse to $\\lambda$ in Figure \\ref{fig6}, which is induced\nby the upper right hand square, which is determined by \\ $\\tkp{n}$ \\ \nand thus up to a self-equivalence \\ $g:P_{n+1}TW\\to P_{n+1}TW$, \\ \naccording to Fact \\ref{fseven}. \\ But such a $g$ induces a canonical\nself-equivalence \\ $g':F'\\to F$, \\ where \\ \n$F':=\\operatorname{Fib}\\,(\\tkp{n}\\circ g)$, \\ and the resulting \\ \n$\\lambda':F'\\simeq P_{n+1}TX$ \\ satisfies \\ \n$\\lambda\\circ g'\\simeq\\lambda'$.\n\\end{proof}\n\n\\begin{defn}\\label{dae}\\stepcounter{subsection}\nFor $W\\simeq P_{n}X$ \\ and \\ $\\rho:P_{n+1}TX\\to P_{n+1}TW$ \\ as above, \nan extension\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{ethirteen}\n0\\to \\operatorname{Coker}\\, \\pinD{n+2}\\rho\\hookrightarrow J\\to\\hspace{-5 mm}\\to\\operatorname{Ker}\\,\\pinD{n+1}\\rho \\to 0\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent is called \\emph{allowable} if its classifying cohomology class \\ \n$$\n[\\psi]\\in\\HL{n+3}{\\EDL{\\operatorname{Coker}\\, \\pi_{n+2}\\rho}{n+2}}{\\operatorname{Ker}\\, \\pi_{n+1}\\rho }\n$$\n(cf.\\ Proposition \\ref{ptwelve}) satisfies \\ $[\\psi]\\circ\\hk{n}=0$.\n\\end{defn}\n\n\\begin{prop}\\label{pthirteen}\\stepcounter{subsection}\nFor any \\ $X\\in{\\mathcal C}$, \\ the extension \\eqref{eten} is\nallowable.\n\\end{prop}\n\n\\begin{proof}\nWriting \\ $V\\simeq P_{n+1}X$ \\ and \\ $Y\\simeq\\bPa{n}X$, \\ by\nnaturality we have a commutative square:\n$$\n\\xymatrix{\nP_{n}V \\rto^-{\\protect{k_{n}}} \\dto^{=} & \\ECL{\\pinC{n+1}V}{n+2}\n\\dto^{q_{\\ast}} \\\\ \nP_{n}Y \\rto^-{\\protect{k_{n}}} & \\ECL{\\operatorname{Ker}\\,\\pi_{n+1}Tr\\q{n}}{n+2}. \\\\\n}\n$$\n\nLemma \\ref{lthree} and \\eqref{etwentyfour} then yield the following\ncommuting diagram in ${\\mathcal D}$ in which the rows and columns are all\nfibration sequences over \\ $B_{\\D}\\Lambda$: \n\n\\begin{center}\n\\begin{picture}(380,170)(0,-50)\n\\put(0,100){$\\EDL{\\operatorname{Ker}\\, h_{n}}{n+1}$}\n\\put(100,105){\\vector(1,0){45}}\n\\put(150,100){$B_{\\D}\\Lambda$}\n\\put(180,105){\\vector(1,0){40}}\n\\put(225,100){$\\EDL{\\operatorname{Ker}\\, h_{n+1}}{n+2}$}\n\\put(50,95){\\vector(0,-1){33}}\n\\put(160,95){\\vector(0,-1){30}}\n\\put(270,95){\\vector(0,-1){30}}\n\\put(30,50){$TP_{n+1}X$}\n\\put(79,55){\\vector(1,0){60}}\n\\put(95,62){$Tr\\q{n}$}\n\\put(143,50){$TP_{n}X$}\n\\put(182,55){\\vector(1,0){40}}\n\\put(195,61){$k$}\n\\put(225,50){$\\EDL{\\pinC{n+1}X}{n+2}$}\n\\put(50,45){\\vector(0,-1){33}}\n\\put(55,28){$r\\q{n+2}$}\n\\put(160,45){\\vector(0,-1){30}}\n\\put(165,28){$=$}\n\\put(270,45){\\vector(0,-1){30}}\n\\put(255,28){$q_{\\ast}$}\n\\put(40,0){$TY$}\n\\put(70,5){\\vector(1,0){70}}\n\\put(143,0){$TP_{n}X$}\n\\put(177,5){\\vector(1,0){40}}\n\\put(200,11){$\\tkp{}$}\n\\put(225,0){$\\EDL{\\operatorname{Im}\\, h_{n+1}}{n+2}$}\n\\put(270,-7){\\vector(0,-1){23}}\n\\put(257,-18){$\\psi$}\n\\put(225,-43){$\\EDL{\\operatorname{Ker}\\, h_{n+1}}{n+3}$}\n\\end{picture}\n\\end{center}\n\n\\noindent The map $k$ is induced by \\ $k_{n}$, \\ and \\ $\\tkp{}$ \\ is\ninduced by \\ $\\hk{n}$. \\ The claim then follows from the\nuniversal property for fibrations. \n\\end{proof}\n\n\\setcounter{thm}{0}\\section{The fiber of a special spherical functor}\n\\label{cfib}\n\nLet  \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ be a special spherical functor.\nWe would like to use the results of Section \\ref{ccps}  in order to\ndetermine whether a given \\ $G\\in{\\mathcal D}$ \\ is (up to homotopy) of the form \\ \n$TX$ \\ for some \\ $X\\in{\\mathcal C}$ \\ -- \\  and if so, how we can distinguish \nbetween such \\emph{realizations}, or liftings. \n\n\\subsection{Lifting objects of ${\\mathcal D}$}\n\\label{srsp}\\stepcounter{thm}\n\nLet us assume for simplicity that \\ $\\Lambda:=\\pinD{0}G$ \\ is a $\\PiC$-algebra,\nand that the map \\ $\\phi_{T}:\\Lambda\\to\\pinD{0}TB_{\\C}\\Lambda$ \\ of \\S\n\\ref{dspecial}(i) is an isomorphism. [In the general case, we are\nfaced with an additional, purely algebraic, problem of determining the\nfiber of the functor \\ $T_{\\ast}:\\Alg{\\PiC}\\to\\Alg{\\PiD}$ \\ \n(compare \\cite{BPescF}); \\ we bypassed this question in \\S \\ref{dsfunc}(iv).  \n   \nWe want a map \\ $\\varphi:TX\\to G$ \\ inducing isomorphisms \\\n$\\pinD{i}TX\\to\\pinD{i}G$ \\ for \\ $i\\geq 0$. \\ \nOur approach is inductive: we are trying to define a tower in ${\\mathcal C}$:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eseven}\n\\dotsb \\xra{p\\q{n+1}} \\Xpn{n+1} \\xra{p\\q{n}} \\Xpn{n} \\xra{p\\q{n-1}} \\dotsb\n\\xra{p\\q{0}} \\Xpn{0}\\simeqB_{\\C}\\Lambda\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent which are to serve as the modified Postnikov tower of the\n(putative) \\ $X\\in{\\mathcal C}$ -- \\ so that in the end we will have \\ \n$X:=\\operatorname{holim}_{n}\\Xpn{n}$.\n\nAt the $n$-th stage \\ ($n\\geq 0$), we have constructed \\ $\\Xpn{n}$ \\ as our \ncandidate for \\ $\\bPa{n}X$ \\ -- \\ so in particular if we let \\ \n$\\Xn{n}:=P_{n}\\Xpn{n}$, \\ (our candidate for the ordinary $n$-th Postnikov \nsection of \\ $X$), \\ then \\ $T\\Xn{n}$ \\ satisfies \\eqref{enine}, \\ \n$T\\Xpn{n}$ \\ satisfies \\eqref{etwenty}, \\ and of course \\ \n$\\Xpn{n}=P_{n+1}\\Xpn{n}$.\n\nAssume also, as part of our inductive hypothesis, a given weak equivalence:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{eseventeen}\n\\hr{n}:P_{n+1}G\\xra{\\simeq}P_{n+1}T\\Xpn{n}.\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nWe start the induction with \\ $\\Xn{0}:=B_{\\C}\\Lambda$. \\ The natural\nmap \\ $r\\q{1}:G\\to P_{1} TB_{\\C}\\Lambda=B_{\\D}\\Lambda$ \\ allows us to define \\\n$\\Xpn{0}$, \\ together with \\ $\\hr{0}:P_{1}G\\xra{\\simeq}P_{1}T\\Xpn{0}$, \\ \nas in Definition \\ref{dmps} (see \\S \\ref{rmin}).\n\n\\subsection{Lifting $\\rho\\q{n}$}\n\\label{slrpn}\\stepcounter{thm}\n\nThe first stage in the inductive step occurs in ${\\mathcal D}$: we must lift \\ \n$\\hr{n}$ \\ to \\ $\\rho\\q{n}:P_{n+2}G\\to P_{n+2}T\\Xpn{n}$. \\ \nNote that by Remark \\ref{rinfo} and Fact \\ref{fsix}, we already know the \ncomparison exact sequence \\eqref{efive} for the putative $X$ from \\ \n$h_{n+1}$ \\ down; the lifting \\ $\\rho:=\\rho\\q{n}$ \\ will determine \\ \n$\\partial_{n+2}:\\pinD{n+2}G\\to\\Gamma_{n+1}\\Xpn{n}$ \\ in addition, \nsince this is just \\ $\\pi_{n+2}\\rho$, \\ so that \\ \n$C_{n+2}:=\\operatorname{Im}\\, \\pinD{n+2}\\rho$ \\ is our candidate for \\ \n$\\operatorname{Coker}\\, h_{n+2}^{X}$, \\ while \\ $K_{n+1}:=\\operatorname{Coker}\\, \\pinD{n+2}\\rho$ \\ is our \ncandidate for \\ $\\operatorname{Ker}\\, h_{n+1}^{X}$.\n\nFrom \\eqref{etwenty} we see that the obstruction is the class:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{enineteen}\n\\chi_{n}:=k_{n+1}^{T\\Xpn{n}}\\circ\\rho\\q{n}\\in\n\\HL{n+3}{G}{\\Gamma_{n+1}\\Xpn{n}}~,\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\\noindent and the different liftings are classified by \\ \n$\\HL{n+2}{G}{\\Gamma_{n+1}\\Xpn{n}}$.\n\n\\subsection{Constructing \\ $\\Xn{n+1}$}\n\\label{scpx}\\stepcounter{thm}\n\nThe next step is to choose a cohomology class \\ \n$\\hk{n}$ \\ in \\ $\\HL{n+2}{\\Xpn{n}}{K_{n+1}}$. \\ This fits into a\ncommutative diagram with rows and fibers all fibration sequences over \\ $B_{\\C}\\Lambda$:\n$$\n\\xymatrix{\nB_{\\C}\\Lambda \\rto \\dto & \\ECL{I_{n+1}}{n+1} \\rto^{=} \\dto^{i} &\n\\ECL{I_{n+1}}{n+1} \\dto^{\\psi} \\\\\n\\Xn{n+1} \\rto^{\\bpa{n}} \\dto & \\Xpn{n} \\rto^>>>>>>>{\\hk{n}}\\dto & \n\\ECL{K_{n+1}}{n+2} \\dto^{j_{\\ast}} \\\\\n\\Xn{n+1}\\rto & \\Xn{n} \\ar@{.>}[r]^>>>>>>>>>>{k_{n}} & \\ECL{J }{n+2}\n}\n$$\nfor the bottom fibration sequence \\ $\\Xn{n+1}\\to\\Xn{n}\\to\\ECL{J }{n+2}$ \\ \nas indicated (though we shall not need this).\n\nNote that $J$, our candidate for \\ $\\pinC{n+1}X$, \\ fits into the short \nexact sequence of modules over $\\Lambda$:\n$$\n0\\to K_{n+1}\\hookrightarrow J\\to\\hspace{-5 mm}\\to I_{n+1}\\to 0,\n$$\n\\noindent as in \\eqref{eten}, and is classified by \\ \n$\\psi:=\\hk{n}\\circ i\\in\\HL{n+2}{\\ECL{I_{n+1}}{n+1}}{K_{n+1}}$, \\ \nas in Corollary \\ref{cone}. Moreover, this extension is\nobviously allowable in the sense of \\S \\ref{dae}.\n\n\\subsection{Lifting $\\rho$}\n\\label{slr}\\stepcounter{thm}\n\nTo complete the induction on \\eqref{eseventeen}, we must lift \\ \n$\\rho:G\\to P_{n+2}T\\Xpn{n}$. \\ This will be done in two steps\\vspace{2 mm} :\n\nFirst, note that we obtain a commuting diagram:\n\n\\begin{center}\n$$\n\\begin{picture}(350,180)(0,0)\n\\put(15,170){$P_{n+2}G$}\n\\put(60,175){\\vector(1,0){205}}\n\\put(140,180){$\\rho$}\n\\put(270,170){$P_{n+2}T\\Xpn{n}$}\n\\multiput(60,168)(2,-1){25}{\\circle*{.5}}\n\\put(110,143){\\vector(2,-1){2}}\n\\put(78,143){$\\rho$}\n\\put(115,135){$P_{n+2}T\\Xn{n+1}$}\n\\put(205,142){\\vector(2,1){55}}\n\\put(230,143){$\\tilde{i}_{\\ast}$}\n\\put(35,162){\\vector(0,-1){62}}\n\\put(3,130){$p\\q{n+1}_{G}$}\n\\put(150,128){\\vector(0,-1){30}}\n\\put(155,112){$p\\q{n+1}_{T\\Xn{n+1}}$}\n\\put(290,162){\\vector(0,-1){62}}\n\\put(295,130){$p\\q{n+1}_{T\\Xpn{n}}$}\n\\put(15,85){$P_{n+1}G$}\n\\put(60,90){\\vector(1,0){50}}\n\\put(80,95){$f$}\n\\put(80,82){$\\simeq$}\n\\put(115,85){$P_{n+1}T\\Xn{n+1}$}\n\\put(208,90){\\vector(1,0){55}}\n\\put(230,95){$g$}\n\\put(230,82){$\\simeq$}\n\\put(270,85){$P_{n+1}T\\Xpn{n}$}\n\\put(35,77){\\vector(0,-1){62}}\n\\put(15,40){$k_{n+1}^{G}$}\n\\put(150,77){\\vector(0,-1){28}}\n\\put(155,60){$k_{n+1}^{T\\Xn{n+1}}$}\n\\put(290,77){\\vector(0,-1){62}}\n\\put(295,40){$k_{n+1}^{T\\Xpn{n}}$}\n\\put(-30,0){$\\EDL{\\pi^{\\natural}_{n+2}G}{n+3}$}\n\\put(75,5){\\vector(1,0){190}}\n\\put(145,11){$(\\pi^{\\natural}_{n+2}\\rho)_{\\ast}$}\n\\multiput(70,10)(2,1){23}{\\circle*{.5}}\n\\put(116,33){\\vector(2,1){2}}\n\\put(90,30){$q_{\\ast}$}\n\\put(123,35){$\\EDL{C_{n+2}}{n+3}$}\n\\put(213,37){\\vector(2,-1){53}}\n\\put(235,30){$i_{\\ast}$}\n\\put(270,0){$\\EDL{\\pinD{n+1}TX}{n+3}$}\n\\end{picture}\n$$\n\\end{center}\n\n\\noindent in which the columns are fibration sequences over \\\n$B_{\\C}\\Lambda$, \\ since by definition \\ \n$$\n\\pinD{n+2}\\rho:\\pinD{n+2}G \\to\\pinD{n+1}T\\Xpn{n}=\\pinD{n+1}TX\n$$\nfactors through \\ $C_{n+2}:=\\operatorname{Im}\\, \\pinD{n+2}\\rho$, \\ so that the\nbottom triangle  commutes. \n\nSince the natural $K$-invariant \\ \\ $k_{n+1}^{G}$ \\ is given,\nthe other two $k$-invariants in the diagram above are determined by\ninverting the given homotopy equivalences \\ \n$f:P_{n+1}G\\to P_{n+1}T\\Xn{n+1}$ \\ and \\ \n$g:P_{n+1}G\\to P_{n+1}T\\Xpn{n}$ \\ (assuming all objects in ${\\mathcal D}$ are\nfibrant and cofibrant), \\ and letting \\ \n$k_{n+1}^{T\\Xn{n+1}}:=q_{\\ast}\\circ k_{n+1}^{G}\\circ f^{-1}$ \\ and \\ \n$k_{n+1}^{T\\Xpn{n}}:=i_{\\ast}\\circ k_{n+1}^{G}\\circ g^{-1}$, \\ \nusing Fact \\ref{ffive}.\n\nTherefore, the map \\ $\\rho:G\\to P_{n+2}T\\Xpn{n}$ \\ \nlifts to \\ $\\rho:P_{n+2}G\\to P_{n+2}T\\Xn{n+1}$ \\ (which is induced by \\ \n$q_{\\ast}$). \\ In fact, the lifting $\\rho$ is unique up to homotopy.\nMoreover, from the proof of Proposition \\ref{pfive} we see that this suffices \nto define \\ $\\Xpn{n+1}$, \\ as well as determining a lifting of $\\rho$ to \na weak equivalence \\ $\\hr{n+1}:P_{n+2}G\\to P_{n+2}T\\Xpn{n+1}$\\vspace{2 mm} . \n\nWe may summarize our results in:\n\n\\begin{thm}\\label{tfour}\\stepcounter{subsection}\nGiven \\ $G\\in{\\mathcal D}$, \\ there is an object \\ $X\\in{\\mathcal C}$ \\ \nsuch that \\ $TX\\simeq G$ \\ if and only if there is a tower as in \n\\eqref{eseven}, serving as the modified Postnikov tower for $X$. \\ If we \nhave constructed \\ $\\Xpn{n}$ \\ satisfying \\eqref{eseventeen} for \\ $n$, \\\na necessary and sufficient condition for the existence of an \\ $\\Xpn{n+1}$ \\ \nsatisfying \\eqref{eseventeen} for \\ $n+1$ \\ is the vanishing of \\ \n$\\chi_{n}\\in\\HL{n+3}{G}{\\Gamma_{n+1}\\Xpn{n}}$. \\ The choices are classified by: \n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{$\\bullet$~}\n\\item $\\HL{n+2}{G}{\\Gamma_{n+1}{\\Xpn{n}}}$ \\ (distinguishing the\n  liftings of \\ $\\hr{n}$ \\ to \\ $P_{n+2}T\\Xpn{n}$); \\ and \n\\item $\\hk{n}\\in\\HL{n+2}{\\Xpn{n}}{K_{n+1}}$, \\ where \\ \n$K_{n+1}:=\\operatorname{Coker}\\, \\pi_{n+2}\\rho\\q{n}$, \\ \nup to self-homotopy equivalences of \\ $\\Xpn{n}$ \\ over \\ $B_{\\C}\\Lambda$ \\ and \\ \n$\\Aut_{\\Lambda}(K_{n+1})$. \\ In particular, this distinguishes the class of \\ \n$\\pinC{n+1}X$ \\ in \\ \n$\\operatorname{Ext}_{\\RM{\\Lambda}}(\\operatorname{Ker}\\,(Tr\\q{n})_{n+1}, \\operatorname{Coker}\\,(Tr\\q{n})_{n+2})$.\n\\end{enumerate}\n\\end{thm}\n\nNote that \\ $\\Gamma_{n+1}\\Xpn{n}=\\Gamma_{n+1}\\Xpn{n+1}=\\Gamma_{n+1}X$, \\ \nby Corollary \\ref{czero}.\n\n\\subsection{Moduli spaces}\n\\label{sreal}\\stepcounter{thm}\n\nIt is possible to refine the statement of our fundamental problem of\nlifting \\ $G\\in{\\mathcal D}$ \\ to ${\\mathcal C}$ in terms of \\emph{moduli} spaces\\vspace{2 mm} : \n\nGiven a model category ${\\mathcal C}$, let $\\mathfrak{W}$ be a homotopically small \nsubcategory of ${\\mathcal C}$,  such that all maps in $\\mathfrak{W}$ are weak equivalences,\nand if \\ $f:X\\to Y$ \\ is a weak equivalence in ${\\mathcal C}$ with either $X$ or\n$Y$ in $\\mathfrak{W}$, then \\ $f\\in\\mathfrak{W}$. \\  \nRecall from \\cite[\\S 2.1]{DKanCD} that the nerve \\ $B\\mathfrak{W}$ \\ of such a\ncategory is called a \\emph{classification complex}. Its components are\nin one-to-one correspondence with the weak homotopy types (in ${\\mathcal C}$) \nof the objects of $\\mathfrak{W}$, and the component containing \\ $X\\in{\\mathcal C}$ \\ is\nweakly equivalent to the classifying space \\ $B\\operatorname{haut} X$ \\ of the\nmonoid of self-weak equivalences of $X$. \n\n\\begin{defn}\\label{drealsp}\\stepcounter{subsection}\nGiven a spherical functor \\ $T:{\\mathcal C}\\to{\\mathcal D}$ \\ and \\ $G\\in{\\mathcal D}$, \\ we denote \nby \\ ${\\EuScript M}(G)$ \\ the category of objects in ${\\mathcal D}$ weakly equivalent to \n$G$ (with weak equivalences as morphisms), and by \\ ${\\EuScript TM}(G)$ \\ \nthe category of objects \\ $X\\in{\\mathcal C}$ \\ such that \\ \n$TX\\in{\\EuScript M}(G)$ \\ (again, with weak equivalences in ${\\mathcal C}$ as \nmorphisms). The ``pointed'' version is denoted\nby \\ $\\R{}(G)$ \\ -- \\ the category of pairs \\ $(X,\\rho)$, \\ \nwhere \\ $X\\in{\\mathcal C}$ \\ and \\ $\\rho:G\\to TX$ \\ is a specified weak equivalence.\n\\end{defn}\n\nIn all our examples the obvious functors \\ $\\R{}(G)\\xra{F}{\\EuScript TM}(G)\\xra{T} {\\EuScript M}(G)$ \\  \npreserve fibrant and cofibrant objects, and thus induce a homotopy \npullback diagram:\n$$\n\\xymatrix{\nB\\R{}(G) \\rto^{BF} \\dto &\n\tB{\\EuScript TM}(G) \\dto^{BT} \\\\\n\\{\\operatorname{Id}_{G}\\} \\rto & B{\\EuScript M}(G)\\\\\n}\n$$\n\\noindent and there are weak equivalences \\ \n$B{\\EuScript TM}(G) \\simeq \\coprod_{X\\in \\pi_{0}{\\EuScript TM}(G)} B\\operatorname{haut} X$, \\ where \\ \n$B{\\EuScript M}(G) \\simeq B\\operatorname{Aut}(G)$ \\ for \\ $\\operatorname{Aut}(G)$ \\ the monoid \nof self weak equivalences of $G$.\n\n\\subsection{Towers of moduli spaces}\n\\label{streal}\\stepcounter{thm}\n\nAlthough \\ $B{\\EuScript TM}(G)$ \\ is the more natural object of interest in \nour context, it is more convenient to study \\ $B\\R{}(G)$ \\ by means of a \ntower of fibrations, corresponding to the Postnikov system of \\ $X\\in\\R{}(G)$:\n\nLet \\ $\\R{n}(G)$ \\ denote the category whose objects are pairs \\ \n$(\\Xpn{n},\\rho')$, \\ where \\ $\\Xpn{n}\\in{\\mathcal C}$ \\ has \\ \n$P_{n+1}\\Xpn{n}\\simeq\\Xpn{n}$ \\ and \\ \n$\\rho':P_{n+1}G\\to P_{n+1}T\\Xpn{n}$ \\ is a weak equivalence.\nThe maps of \\ $\\R{n}(G)$ \\ are weak equivalences compatible \nwith the maps \\ $p\\q{n}$.\t\n\nAs in \\cite[Thm.\\ 9.4]{BDGoeR}, one can show that \\ \n$B\\R{}(G)\\simeq \\operatorname{holim}_{n} B\\R{n}(G)$, \\ so we may try to obtain \ninformation about the moduli space \\ ${\\EuScript TM}(G)$ \\ by studying the \nsuccessive stages in the tower:\n\\setcounter{equation}{\\value{thm}}\\stepcounter{subsection}\n\\begin{equation}\\label{esixteen}\n\\dotsc B\\R{n+1}(G) \\xra{BF_{n}}B\\R{n}(G) \\xra{BF_{n-1}}\\dotsc\n\\to B\\R{1}(G).\n\\end{equation}\n\\setcounter{thm}{\\value{equation}}\n\nHowever, from the discussion above we see that we need several \nintermediate steps in the study of \\ $B\\R{n+1}(G)\\to B\\R{n}(G)$, \\ \ncorresponding to the additional choices made in obtaining \\ \n$\\bPa{n+1}X$ \\ and \\ \n$p\\q{n+1}:P_{n+2}G\\xra{\\simeq}P_{n+2}T\\bPa{n+1}X$ \\ from \\ \n$\\bPa{n}X$ \\ and \\ $p\\q{n}:P_{n+1}G\\xra{\\simeq}P_{n+1}T\\bPa{n}X$. \\ \nAs a result one obtains a refinement of the tower \\ \\eqref{esixteen}, \\ \nwhere the successive fibers $F$ are either empty, or else generalized\nEilenerg-Mac Lane spaces, whose homotopy groups may be described in\nterms of appropriate Quillen cohomology groups. We leave the details\nto the reader; compare \\cite[Thm.\\ 9.6]{BDGoeR}.\n\n\\setcounter{thm}{0}\\section{Applying the theory}\n\\label{cat}\n\nThe approach to the lifting problem for a spherical functor \\ \n$T:{\\mathcal C}\\to{\\mathcal D}$ \\ described in the previous section is somewhat\nunwieldy. However, in specific applications it may simplify in various\nways.  We illustrate this by a number of examples:\n\n\\subsection{Singular chains}\n\\label{ssc}\\stepcounter{thm}\n\nConsider the singular chain functor \\ $C_{\\ast}:\\TT_{\\ast}\\to{\\EuScript Chain}$, \\ which\nin the form \\ $T:{\\mathcal G}\\to s{\\Ab{\\EuScript Gp}}$ \\ is induced by abelianization \n(see \\S \\ref{ssfunc}(a)). Thus, given a chain complex \\ $G_{\\ast}$, \\\nwe would like to find all topological spaces $X$ (if any) with \\ \n$C_{\\ast}X\\simeq G_{\\ast}$. \\ Over $\\mathbb Z$, this is equivalent to the\nquestion of realizing a given sequence of homology groups. \n\nOur approach uses  Whitehead's exact sequence \\\n\\eqref{etwentyfive} \\ to relate the (trivial) Postnikov system for the\nchain complex \\ $G_{\\ast}$ \\ to the modified Postnikov system for the\nspace $X$,  in which we attach at each stage not a single new homotopy\ngroup, but a pair of groups in adjacent dimensions, corresponding to\nthe image and kernel respectively of the Hurewicz homomorphism.\n\nIt should be observed that the functor $T$ involves only \n``algebraic'' categories \\ ${\\mathcal C}=s\\hat{\\C}$, \\ where $\\hat{\\C}$ \\ -- \\ in our\ncase, \\ ${\\EuScript Gp}$ \\ or \\ ${\\Ab{\\EuScript Gp}}$ \\ -- \\ has a trivial model category\nstructure, as in \\S \\ref{ermc}(a-b). \\ The analysis \nin Section \\ref{cfib} then simplifies considerably, in as much as the \ncategories of $\\PiC$-algebra s and $\\PiD$-algebra s are simply \\ ${\\EuScript Gp}$ \\ and \\ ${\\Ab{\\EuScript Gp}}$, \\\nrespectively. \n\nAs noted in the Introduction, Baues's \\cite[VI, (2.3)]{BauCF} is\nactually a generalization the obstruction theory described here for\nthis case. His earlier approach in \\cite{BauHH} (as well as that of \nBenkhalifa in \\cite{BenkT} is parallel to this, though not framed in the\nsame cohomological language. See \\cite{MandE} for another viewpoint. \n\n\\subsection{Rationalization}\n\\label{srat}\\stepcounter{thm}\n\nOn the other hand, the rationalization functor \\\n$(-)_{\\mathbb Q}:{\\mathcal T}\\to{\\mathcal T}_{\\mathbb Q}$, \\ induced by the completed group ring\nfunctor \\ $\\hat{Q}:{\\EuScript Gp}\\to{\\EuScript Hopf}$ \\ (cf.\\ \\S \\ref{ssfunc}(b)), is spherical\nbut not special (Def.\\ \\ref{dspecial}), and so the theory described\nhere does not apply as is. In fact, one can see why if one\nconsiders the comparison exact sequence for $\\hat{Q}$ \\ (\\S \\ref{satt}(b)):\ngiven a (simply-connected) rational space \\ $G\\in{\\mathcal T}_{\\mathbb Q}$, \\ for\neach $\\mathbb Q$-vector space \\ $\\pi_{n}G$, \\ we need an abelian group \\  \n$A=\\pi_{n}X$ \\ such that \\ $A\\otimes\\mathbb Q\\cong\\pi_{n}G$, \\ and then lift\nthe rational $k$-invariants for $X$ to integral ones. \\ Thus, much of\nthe indeterminacy for $X$ is algebraic.\n\n\\subsection{Suspension}\n\\label{ssusp}\\stepcounter{thm}\n\nThe suspension functor \\ $\\Sigma:\\TT_{\\ast}\\to\\TT_{\\ast}$, \\ induced by the free\ngroup functor \\ $\\hat{F}:\\Set_{\\ast}\\to{\\EuScript Gp}$ \\ as in \\S \\ref{ssfunc}(c), is\nsimilar to singular chains, with the generalized EHP sequence\nreplacing the ``certain long exact sequence'', and the modified\nPostniov systems involve the kernel and image of the suspension\nhomomorphism \\ $E:\\pi_{n}X\\to\\pi_{n+1}\\Sigma X$.\n\n\\subsection{Homotopy groups}\n\\label{shg}\\stepcounter{thm}\n\nThe motivating example for the treatment in this paper \\ -- \\ and the\nonly one which requires the full force of Section \\ref{cfib} \\ -- \\ \nis the functor \\ $\\pi_{\\ast}:\\TT_{\\ast}\\to\\Alg{\\Pi}$, \\ prolonged to simplicial spaces\n(as in as in \\S \\ref{ssfunc}(d)). \\  However, even this case\nsimplifies greatly if we want to realize a single $\\Pi$-algebra\\ $\\Lambda$ \\ -- \\\nthat is, we take \\ $G\\in s\\Alg{\\Pi}$ \\ to be the constant simplicial $\\Pi$-algebra \\ $B\\Lambda$.\n\nIndeed, given a simplicial space $X$ with \\ $\\pi_{\\ast} X\\simeqB\\Lambda$ \\ (which\nimplies that \\ $\\pi_{\\ast} \\|X\\|\\cong G$), from the spiral exact\nsequence \\eqref{efifteen} we find that \\\n$\\pi^{\\natural}_{n}X\\cong\\Omega^{n}\\Lambda$ \\ for all \\ $n\\geq 0$, \\ so that \\ \n$h_{n}:\\pi^{\\natural}_{n}X\\to\\pi^{\\natural}_{n}\\pi_{\\ast} X$ \\ is trivial for \\ $n>0$. \\ We do\nnot need the modified Postnikov system in this case: the\nobstructions to realizing $\\Lambda$ (or $G$) are just the classes \\ \n$\\chi_{n}\\in H^{n+3}(\\Lambda;\\Omega^{n+1}\\Lambda)$, \\ and the \ndifference obstructions distinguishing between the different\nrealizations are \\ $\\delta_{n}\\in H^{n+2}(\\Lambda;\\Omega^{n+1}\\Lambda)$ \\ \n($n\\geq 1$). \\ See \\cite{BDGoeR} and \\cite[\\S 5]{BJTurR} for two\ndescriptions of this case.\n\n\\begin{remark}\\label{risf}\\stepcounter{subsection}\nOur obstruction theory is irrelevant, of course, for the inverse\nspherical functor \\ $U:{\\mathcal G}\\to\\Ss_{\\ast}$ \\ (see \\S \\ref{eisf}) \\ -- \\ that is, \nin determining loop structures on a given topological space.\nNevertheless, from \\eqref{etwentythree} we can easily recover the\nwell-known fact that \\ $X\\simeq\\Omega Y$ \\ is a loop space if and only if its\n$k$-invariants are suspensions of those of $Y$ (cf.\\ \\cite{AHKaneN}).\n\\end{remark}\n\n\\subsection{Lifting morphisms}\n\\label{slm}\\stepcounter{thm}\n\nIn all of the above examples, one can ask the analogous question\nregarding the lifting of \\emph{maps}, or more complicated diagrams,\nfrom ${\\mathcal D}$ to ${\\mathcal C}$. This can be addresses via Theorem \\ref{tfour} by\ntransfering the spherical structure from ${\\mathcal C}$ and ${\\mathcal D}$ to the diagram\ncategories \\ ${\\mathcal C}^{I}$ \\ and \\ ${\\mathcal D}^{I}$ \\ (cf.\\ \\S \\ref{ermc}(d)).\nSee \\cite[\\S 8]{BJTurR} for a detailed example. \n\nNote that the $k$-invariants for a map of chain complexes are not\ntrivial (cf.\\ \\cite[(3.8)]{DolH}), so the theory for realizing chain maps in \\ \n$\\TT_{\\ast}$ \\ is correspondingly more complicated.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}