Free energy on the sphere for non-abelian gauge theories

We compute the $S^d$ partition function of fixed point non-abelian gauge theories in continuous $d$, using $\epsilon$-expansion around $d=4$. illustrate detail technical aspects calculation, including all factors arising from gauge-fixing procedure, and method to deal with zero-modes ghosts. obtain result up NLO, i.e. two-loop vacuum diagrams. Depending on sign one-loop beta function, there is a real coupling $d>4$ or $d<4$. In first case we extrapolate $d=5$ test recently proposed construction UV $5d$ $SU(2)$ Yang-Mills via susy-breaking deformation $E_1$ SCFT. find that $F$ theorem allows RG flow. second $d=3$ whether QCD$_3$ group $SU(n_c)$ $n_f$ fundamental matter fields flows CFT symmetry-breaking phase. within regime near $d=4$ phase always favored. For lower values compare average between two complex points its value at give an upper bound critical $n_f^*$ below which takes over.