The astrophysical $S$ factor for proton-proton weak capture is calculated in chiral effective field theory over the center-of-mass relative-energy range 0–100 keV. The chiral two-nucleon potential derived up to next-to-next-to-next-to leading order is augmented by the full electromagnetic interaction including, beyond Coulomb, two-photon and vacuum-polarization corrections. The low-energy constants entering the weak current operators are fixed so as to reproduce the $A=3$ binding energies and magnetic moments and the Gamow-Teller matrix element in tritium $\beta$ decay. Contributions from $S$ and $P$ partial waves in the incoming two-proton channel are retained. The $S$ factor at zero energy is found to be $S(0)=(4.030\pm 0.006)\times {10}^{-23}\mathrm{MeV}{\mathrm{fm}}^2$, with a $P$-wave contribution of $0.020\times {10}^{-23}\mathrm{MeV}{\mathrm{fm}}^2$. The theoretical uncertainty is due to the fitting procedure of the low-energy constants and to the cutoff dependence.

• Received 13 March 2013

DOI:https://doi.org/10.1103/PhysRevLett.110.192503