We analyze the renormalon diagram of gauge theories on

Article funded by SCOAP3

1$ theory on $\R^3\times \S^1$ is continuously connected to the $n_W=0$ theory on $\R^4$ by taking the limits $L\rightarrow \infty$ and $n_W\rightarrow 0$.}} on $\R^4$~\cite{Shifman-ml-2008ja,Unsal-ml-2008ch}. The $L$ dependence kicks in at the scale $L\sim \Lambda_{\mbox{\scriptsize QCD}}^{-1}$, so that there is no reason to expect that the $L$ dependence of the renormalon diagram is canceled by the non-perturbative effects. This is even more so the case because of the fact that the regime of small $L$ is characterized by the hierarchy of scales for which $L\ll \text{(monopole separation)}$, so that the monopole screening of the perturbative propagators happens at the momentum scales much lower than the scale $1/L$, the point at which momentum dependence is cut off in the renormalon diagram. As $L$ is increased, however, these two scales (i.e.\ the monopole diluteness and the IR cutoff scale $L$) become of the order $\Lambda_{\rm QCD}^{-1}$. This happens when the coupling is already strong, and perturbation theory is not justified anyway, so low momenta in Feynman diagrams should be cut off by non-perturbative effects. If complemented by the non-perturbative monopoles, however, the perturbation theory will have a new IR cutoff scale which is now as important as $L$, so appropriate non-perturbative corrections need to be introduced in the perturbative~propagators. \enl It is perhaps important to stress that although the theory on $\R^3\times \S^1$ does not have renormalon poles in the Borel plane because the factorial growth $n!$ is cut off by the presence of the IR scale $L$, for $L\gg\Lambda_{\rm QCD}^{-1}$ there will be some factorial growth which still may have physical meaning.\footnote{Indeed there is a lot of phenomenological work where renormalon behavior can be extracted from a finite box~\cite{Bauer-ml-2011ws,Bali-ml-2013pla,Bali-ml-2014fea,Bali-ml-2014sja}, and this will also be true for a large $L$ theory on $\R^3\times \S^1$.} Indeed at large $L$ the perturbation theory will complain about the fact that one is using it in the regime where it should not be used. But since the unphysical growth of the series is cut off at large $n$, and since formally no singularity in the Borel plane exists due to the renormalon diagram, it is not as straightforward to attach meaning to this growth and to connect it to non-perturbative saddles. It seems clear, however, that whatever physics can be extracted in the regime of large $L\gg\Lambda_{\rm QCD}^{-1}$, it should remain the same as on $\R^4$, due to the presence of the mass gap of order $\Lambda_{\rm QCD}$. As the radius $L$ is reduced and as the threshold $\Lambda_{\rm QCD}^{-1}$ is reached, the semi-classical instanton-monopoles and bions will be the source of mass-gap and condensates, but no renormalon growth will be observed at all. On the other hand there will be factorial growth of diagrams associated with these saddles. Whether for $L \gg \Lambda_{\rm QCD}$ there is a connection between the two, seemingly different and unrelated factorial growths, remains an open and important~question. These arguments are heuristic, however, but they do give hope that the theory on large $L$ can indeed be studied for small $L$, where all effects can be systematically accounted for and the large $L$ (and possibly as $n_W\rightarrow 0$) limit taken. The article is organized as follows. In section~\ref{IR renormalons: the sickness and cure} we review the renormalon problem and how it arises in gauge theories on $\R^4$. We also qualitatively discuss why it is expected that no renormalons appear on $\R^3\times \S^1$. In section~\ref{Strategy and the calculation method} we introduce our computational strategy and obtain the general structure of the vacuum polarization tensors \emph{exactly} using the background field method for arbitrary external momentum. In section~\ref{Calculating the polarization tensor on a compact circle} we carry out systematic and \emph{exact} calculations of the vacuum polarization tensor to one loop. To the best of our knowledge, this is the first exact calculations of the vacuum polarization diagram on $\mathbb R^3 \times \S^1$ and the results are easily applied to the case of thermal QCD and QED. Finally, in section~\ref{The static limit, resummation and absence of IR renormalons} we show explicitly that no renormalons exist on $\mathbb R^3 \times \S^1$. We conclude in section~\ref{sec:conclusion}. Various appendices summarize miscellaneous sums and integrals used in the computations. In particular, in appendix~\ref{eq:integrals_deriv} we use a novel method to obtain the exact result of new untabulated integrals. ]]>

3$ the charged fermions will generally have different masses, and hence one has to restore to the original expressions~(\ref{Pi for fermions all groups}) or~(\ref{Pi fermion in Cartan}). In fact, one can obtain the result~(\ref{final expression Pi in cartan}) directly from~(\ref{Pi for fermions all groups}) for $N=2,3$ by setting $A_3^bT^b=\pm \mu/L$ and using $\mbox{Tr}_{\mbox{\scriptsize{adj}} }[T^eT^d]=f^{adc}f^{cea}=N\delta^{ed}$. This is a huge simplification since one can then use the same background field Feynman rules, as given by Abbott~\cite{Abbott-ml-1980hw}, to compute the non-abelian one-loop corrections on $\mathbb R^3 \times \mathbb S^1$, shown in figure~\ref{fig:vac_pol_gauge_sector}, provided that we substitute the ghosts and gluons propagators on $\mathbb R^4$ with the propagators on $\mathbb R^3 \times \mathbb S^1$ in the presence of a non-trivial holonomy. The diagrams of figure~\ref{fig:vac_pol_gauge_sector} (for $N=2,3$) add up to~\cite{Chaichian-ml-1996yr}: \begin{align}\nn \Pi_{mn}^{{\mbox\scriptsize NA}\,ij}(p, \omega)=&\,\delta^{ij}\frac{g^2 N}{2L}\sum_{q\in \mathbb Z}\int\frac{d^3 k}{(2\pi)^3}\\ &\times\frac{4\delta_{mn}P^2+2\left(P_m K_n+P_n K_m\right)+4K_m K_n-3P_m P_n-2(K+P)^2\delta_{mn}}{\left[k^2+\left(\frac{2\pi q+\mu}{L}\right)^2 \right]\left[(\pmb k+\pmb p)^2+\left(\frac{2\pi q+\mu}{L}+\omega\right)^2 \right]} \nn \\ &+ (\mu \rightarrow -\mu)\,. \label{the nonabelian polarization} \end{align} \begin{figure} \centering \includegraphics[width=.7\textwidth]{vac_pol_gauge_sector.pdf} ]]>