Correlators in non-conformal $\mathcal{N}=2$ gauge theories from localization

We study two-point correlation functions of chiral/anti-chiral operators in SU(N) $\mathcal{N}=2$ gauge theories with massless hyper-multiplets in a representation $\mathcal{R}$ associated with a non-vanishing $\beta$-function. Using supersymmetric localization on the four-sphere $S^4$, these observables can be evaluated by matrix correlators of normal ordered operators. We show that, within a specific regime of validity, standard perturbative calculations based on Feynman diagrams and renormalization procedures in flat space perfectly match the localization predictions up to two loops generalizing and extending previous results. Our analysis highlights that non-trivial interference effects between evanescent terms and the UV poles of the bare coupling are essential for the agreement. On the matrix model side, we employ a direct diagrammatic procedure akin to the Feynman diagram expansion on the field theory description; this simplifies the comparison for generic operators and general matter representation $\mathcal{R}$.