This paper calculates the low-energy rate for proton–proton fusion and, for the first time in this problem, uses a Bayesian method to estimate a key theoretical uncertainty. Proton–proton fusion (p + p → d + e⁻ + ν̄e) sets the pace of the proton–proton chain that powers Sun-like stars. Nuclear physicists usually report that rate as an astrophysical S-factor, S(E), which isolates the nuclear physics from the effect of the Coulomb barrier. The authors give a new zero-energy value S(0) = (4.068 ± 0.025) × 10⁻²⁵ MeV·barn (MeV·b) and provide controlled error estimates for that number and its low-energy derivatives.

The team worked within chiral effective field theory (χEFT), a framework that builds nuclear forces and weak currents order by order in a controlled expansion. They tested many modern χEFT interactions. One group of forces is non-local (the EMN family) available from leading order up to next-to-next-to-next-to-leading order (N3LO) with cutoffs 450, 500 and 550 MeV. The other group is a set of local Norfolk (NV) N3LO potentials with different regulator choices and fit ranges. The weak axial current that drives the fusion reaction is included up to N3LO. A short-range contact term in that current contains a low-energy constant (LEC), labelled z0 (related to another LEC cD), and this was fixed so the theory reproduces the Gamow–Teller matrix element measured in tritium beta decay. That calibration ties the weak current used here to other observable nuclear decays.

To reach percent-level accuracy the authors also included electromagnetic effects beyond the simple Coulomb force. These include two-photon exchange and vacuum polarization in the proton–proton initial state. The initial pp state was expanded in partial waves and only the 1S0 contribution was kept, because other partial waves are known to be negligible at the energies of interest. They computed S(E) in the range 3–30 keV at 1 keV steps and fitted the results with a third-degree Taylor polynomial to extract S(0) and its first two derivatives. The statistical quality of that fit is very high: the relative fitting error per data point is about 0.013% and is negligible compared to other uncertainties.