We calculate the $p$-wave contribution to the proton-proton fusion $S$ factor and its energy derivative in pionless effective field theory (EFT) up to next-to-leading order. The leading contributions are given by a recoil piece from the Gamow-Teller and Fermi operators, and from relativistic $1/m$ suppressed weak interaction operators. We obtain the value of $(2.5\pm0.3 )\times 10^{ -28}~\mathrm{ MeV\ fm^2}$ for the $S$ factor and $(2.2\pm0.2) \times 10^{ -26}~\mathrm{ fm^2}$ for its energy derivative at threshold. These are smaller than the results of a prior study that employed chiral EFT by several orders of magnitude. We conclude that, contrary to what has been previously reported, the $p$-wave contribution does not need to be considered in a high-precision determination of the $S$ factor at astrophysical energies. Combined with the chiral EFT calculation of Acharya { \it et al.} [Phys. Lett. B \bf{ 760}, 584 (2016)] for the $s$-wave channel, this gives a total threshold $S$ factor of $S(0) = (4.047^{ +0.024}_{ -0.032}) \times 10^{ -23}~{ \rm MeV~fm}^2$.
Categories: nucl-th astro-ph.SR hep-ph