Bayesian analysis of proton-proton fusion in chiral effective field theory

The astrophysical $S$-factor for the proton-proton fusion is calculated in the low-energy regime for a variety of nuclear interactions and consistent nuclear currents, derived within chiral effective field theory. We estimate, for the first time, the theoretical uncertainty on the $S$-factor due to the truncation of the chiral expansion of the currents using a Bayesian analysis. In order to reach an accuracy at the percent level in the calculation, the electromagnetic potential includes contributions beyond the leading Coulomb interaction, such as two-photon exchange and vacuum polarization. The initial proton-proton state is expanded in partial waves and only the ${}^1S_0$ contribution is included, as it is known that the other partial-waves effects are negligible. The low-energy constant entering the contact term in the weak axial current operator is calibrated to reproduce the Gamow-Teller matrix element in Tritium $\beta$-decay. The value $S(0)$ is found to be $S(0)=(4.068 \pm 0.025)\times 10^{-25} \: \text{MeV}\: \text{b}$.