# Universal behavior of $p$-wave proton-proton fusion near threshold

## TL;DR

This study calculates the p-wave contribution to proton-proton fusion's S factor near threshold using pionless EFT, finding it negligible compared to s-wave contributions, thus simplifying high-precision astrophysical models.

## Contribution

It provides the first precise calculation showing p-wave effects are negligible, contrasting previous chiral EFT results, and refines the total S factor at threshold.

## Key findings

- p-wave S factor is approximately 2.5 x 10^{-28} MeV fm^2
- p-wave contribution is smaller than previous estimates by several orders of magnitude
- total S(0) factor is approximately 4.05 x 10^{-23} MeV fm^2

## Abstract

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 $S(0) = (4.047^{+0.024}_{-0.032}) \times 10^{-23}~{\rm MeV~fm}^2$.