Modular invariant gluon-graviton scattering in AdS at one loop

We consider mixed gluon-graviton scattering in Type IIB string theory on AdS$_5\times S^5/\mathbb{Z}_2$ in the presence of D7 branes, which is dual to a mixed correlator of the $SO(8)$ and $SU(2)_L$ flavor multiplets of a certain 4d $\mathcal{N}=2$ $USp(2N)$ gauge theory with complexified coupling $\tau$. We compute this holographic correlator in the large $N$ and finite $\tau$ expansion using constraints from derivatives of the mass deformed sphere free energy, which we compute using supersymmetric localization at large $N$ and finite $\tau$ in terms of modular invariant non-holomorphic Eisenstein series. In particular, we combine this constraint with the known flat space limit to fix the $R^2F^2$ higher derivative correction to the correlator in terms of the weight one non-holomorphic Eisenstein series, and also to fix the logarithmic threshold. We also compute the one-loop correction to the correlator, and match it to the expected flat space limit result.