Integrated correlators in a $\mathcal{N}=2$ SYM theory with fundamental flavors: a matrix-model perspective

The D theory is a $\mathcal{N}=2$ conformal SYM theory in four dimensions with gauge group SU($N$), four matter hypermultiplets in the fundamental and two in the anti-symmetric representation, and a flavor symmetry that contains a U(4) factor. Its holographic dual is obtained via a combination of orbifold and orientifold projections from Type II B string theory on $\mathrm{AdS}_5 \times S^5$ and possesses a sector of open strings attached to D7 branes with U(4) Chan-Paton factors. The AdS Veneziano amplitude of the U(4) gluons is dual to four-point correlators of moment-map operators of the U(4) flavor symmetry in the D theory. An integrated version of these correlators is captured by a deformation of the D theory in which the fundamental hypermultiplets acquire a mass. The partition function of this massive theory can be evaluated with matrix-model techniques using localization. In this paper we analyze the matrix model of the mass-deformed D theory arising from localization using the so-called "full Lie algebra" approach in a $1/N$ expansion. In particular, we study the partition function and its mass derivatives up to $O(1/N^2)$ corrections obtaining exact expressions that are valid for all values of the 't Hooft coupling. We also analyze their behavior at strong coupling where our results produce useful constraints on the dual open string scattering amplitudes in AdS.