Free energy on the sphere for non-abelian gauge theories

We compute the Sd partition function of the fixed point of non-abelian gauge theories in continuous d, using the ϵ-expansion around d = 4. We illustrate in detail the technical aspects of the calculation, including all the factors arising from the gauge-fixing procedure, and the method to deal with the zero-modes of the ghosts. We obtain the result up to NLO, i.e. including two-loop vacuum diagrams. Depending on the sign of the one-loop beta function, there is a fixed point with real gauge coupling in d > 4 or d < 4. In the first case we extrapolate to d = 5 to test a recently proposed construction of the UV fixed point of 5d SU(2) Yang-Mills via a susy-breaking deformation of the E1 SCFT. We find that the F theorem allows the proposed RG flow. In the second case we extrapolate to d = 3 to test whether QCD3 with gauge group SU(nc) and nf fundamental matter fields flows to a CFT or to a symmetry-breaking case. We find that within the regime with a real gauge coupling near d = 4 the CFT phase is always favored. For lower values of nf we compare the average of F between the two complex fixed points with its value at the symmetry-breaking phase to give an upper bound of the critical value nf∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {n}_f^{\ast } $$\end{document} below which the symmetry-breaking phase takes over.