Radiative corrections to proton-proton fusion in pionless effective field theory

We study the leading radiative correction to proton-proton fusion using the pionless effective field theory framework at leading order. We derive the relevant matrix elements and evaluate them using the method of regions. We benchmark the accuracy of our approximations by carrying out numerical computations of the full expressions. We show that the first order radiative corrections due to the exchange of a Coulomb photon between positron and proton-proton systems map onto the Sirlin function and the $\mathcal{O}(\ensuremath{\alpha})$ contribution from the Fermi function. We furthermore find that the nuclear structure dependent radiative correction omitted in the previous analysis by Kurylov et al. [Phys. Rev. C 67, 035502 (2003)] gives an up to $0.2%$ correction to the $pp$ fusion $S$ factor with its size ultimately depending on a two-nucleon counterterm ${L}_{1A}$ that renormalizes the axial two-body current.