Supersymmetric localization and non-conformal $\mathcal{N}=2$ SYM theories in the perturbative regime

Abstract: We examine the relation between supersymmetric localization on $\mathbb{S}^4$ and standard QFT results for non-conformal theories in flat space. Specifically, we consider 1/2 BPS circular Wilson loops in four-dimensional SU($N$) $\mathcal{N}$= 2 SYM theories with massless hypermultiplets in an arbitrary representation $\mathcal{R}$ such that the $\beta$-function is non-vanishing. On $\mathbb{S}^4$, localization maps this observable into an interacting matrix model. Although conformal symmetry is broken at the quantum level, we show that within a specific regime of validity the matrix model predictions are consistent with perturbation theory in flat space up to order $g^6$. In particular, at this order, localization predicts two classes of corrections proportional to $\zeta(3)$ whose diagrammatic origins in field theory are remarkably different. One class of $\zeta(3)$-like corrections emerges via interference effects between evanescent terms and the ultraviolet (UV) poles associated with the bare coupling constant, while the second one stems from a Feynman integral which retains the same form in flat space and on the sphere.