Proton–proton weak capture with the one-loop (N3LO) chiral axial current: testing the convergence of the chiral expansion for the solar fusion S-factor

The weak capture reaction $p(p,e^+\nu_e)d$ ignites the proton–proton chain and fixes the normalization of the solar neutrino spectrum, yet its rate is known only theoretically. All state-of-the-art chiral effective field theory (χEFT) calculations of its astrophysical S-factor employ the two-body weak axial current only at tree level — one-pion exchange with low-energy constants $c_3$, $c_4$ plus a contact term calibrated to tritium β decay — and estimate the convergence of the axial-current expansion statistically rather than testing it by explicit computation. The next-order (one-loop, fourth-order in the chiral counting, called N3LO in the source's convention) two-body axial-current operators were derived in 2016–2017, by time-ordered perturbation theory — precisely the derivation whose absence the 2014 source by Marcucci flagged as the open problem — and independently by the unitary-transformation method, but they have never been evaluated in proton–proton weak capture or in any other few-nucleon weak capture process. The open problem is: compute the low-energy pp-capture S-factor with the complete one-loop two-body axial current, implemented consistently with a chiral interaction and regulator and with the short-range axial coupling re-determined as needed, quantify the loop contribution and its cutoff and scheme dependence, and thereby directly test the convergence of the chiral expansion of the weak axial current for the solar hydrogen-ignition rate.

Background

The reaction $p+p\to d+e^+\nu_e$ (proton–proton weak capture, or pp fusion) is the first and slowest step of the proton–proton chain, by which the Sun and other main-sequence stars burn hydrogen. Its rate cannot be measured in the laboratory at solar energies; it determines, together with the solar composition, the computed luminosity and the flux of pp neutrinos, so the entire normalization of the solar neutrino spectrum rests on theory. The cross section is conventionally quoted as the astrophysical S-factor, $S(E)$, which factors out the exponentially small Coulomb barrier penetration so that $S(E)$ is slowly varying at the low energies (below ~100 keV) relevant to stellar burning.

The theoretical framework is chiral effective field theory (χEFT), a systematic expansion of nuclear forces and electroweak current operators in powers of $Q/\Lambda_b$, where $Q$ is a typical nucleon momentum or the pion mass and $\Lambda_b\simeq 500$–600 MeV is the breakdown scale; the constants of the theory (low-energy constants, LECs) are fixed from few-nucleon data. The reaction is dominated by the axial part of the weak current. At leading order (LO) this is the one-body Gamow–Teller operator, in which the emitted positron and neutrino couple to a nucleon spin–isospin flip with strength given by the axial coupling constant $g_A$; relativistic corrections enter one order higher. Two-body exchange currents then contribute: at the next order where they appear, the two-body axial current consists of a one-pion-exchange (OPE) term whose strength is fixed by the pion–nucleon LECs $c_3$ and $c_4$, and a short-range contact term whose LEC (called $d_R$, equivalently $c_D$ through its relation to the leading three-nucleon force, or $z_0$; $L_{1,A}$ in the pionless theory) is calibrated to the precisely measured Gamow–Teller matrix element in tritium β decay. I will call this tree-level two-body content 'N2LO', following the source and Acharya–Marcucci–Platter (2023); note that other groups label the same operators N3LO (the JLab-Pisa convention, which counts relativistic corrections and Δ-isobar — Δ(1232) resonance — OPE as N2LO) and call the one-loop order treated below N4LO, while Krebs–Epelbaum–Meißner call it 'fourth order'. These label differences do not change the physics.

The one-loop two-body axial current — the analogue for the axial current of the one-loop electromagnetic currents derived by Pastore, Schiavilla, and Goity (2008) by time-ordered perturbation theory (TOPT, a formulation of perturbation theory in which intermediate states carry explicit energy denominators, so that 'reducible' diagrams — those containing purely nucleonic intermediate states — must be separated from irreducible ones and their nonstatic corrections resummed into the wave functions) — was for a long time available only in the early heavy-baryon chiral perturbation theory (HBχPT, the systematic low-energy expansion of QCD with nucleons treated nonrelativistically) derivation of Park et al. (2003). That derivation was later shown to depend on the parametrization of the pion field, to violate current conservation in the chiral limit, and to omit several diagram classes. The 2013 calculation of pp capture by Marcucci, Schiavilla, and Viviani therefore used only the tree-level (N2LO) axial current, together with a chiral N3LO nucleon–nucleon potential and the full electromagnetic interaction (two-photon exchange, vacuum polarization), obtaining $S(0)=(4.030\pm 0.006)\times 10^{-23}$ MeV fm$^2$. In the 2014 conference report (reference 1), Marcucci stated the open problem: the N3LO contributions (in that counting) to the axial current 'arise from loop and TPE terms, and they have not been calculated yet', the TOPT approach 'has not yet been used to derive the axial component of the weak current operator', and it is 'more suitable than the HBχPT one of Park et al. to treat reducible diagrams and to derive the contact terms'; the report concludes that 'it is highly desirable to derive the axial currents up to N3LO in TOPT, following the footsteps of Pastore et al. for the EM operators, and to repeat this study for the pp reaction'. Repeating the pp capture calculation with such currents would directly test the convergence of the chiral expansion.

That missing derivation now exists in two forms. Baroni, Girlanda, Pastore, Schiavilla, and Viviani (2016) derived the two-nucleon axial charge and current operators up to one loop in TOPT: the loop corrections to the axial current are finite, conserve the current in the chiral limit, and introduce no new LECs at that order (four new contact LECs $z_1$–$z_4$ enter only the axial charge), and the work documents precisely the reducible-diagram and contact-term issues raised in the source. Independently, Krebs, Epelbaum, and Meißner (2017) derived the complete axial charge, current, and pseudoscalar operators to fourth order in the chiral expansion — one-body to two loops, two-body to one loop, three-body at tree level — using the method of unitary transformations (a scheme for constructing currents and forces in the same Fock-space basis), showing that the ambiguity associated with the freedom of choosing that basis can be eliminated by requiring renormalizability and matching pion-pole terms to the nuclear forces. The two derivations yield operator sets that differ by documented, unitary-ambiguity-related choices, so any application must keep the current consistent with the interaction family and regulator it is paired with. What has never been done is the second half of the source's program: no published pp-capture calculation — nor any calculation of the sister weak reactions — includes these one-loop two-body axial currents. The current state of the art (Acharya, Marcucci, Platter 2023; Barlucchi, Gnech, Degl'Innocenti, Marcucci 2026) computes $S(E)$ order by order for the interaction but stops the current at the tree-level two-body order, and estimates the missing current-truncation error statistically (order-by-order spreads with $(Q/\Lambda_b)^4$ heuristics, and most recently a Bayesian truncation analysis), yielding $S(0)\simeq 4.07$–$4.10\times 10^{-25}$ MeV b. The Solar Fusion III review recommends $S(0)=4.09(1\pm 0.015)\times 10^{-25}$ MeV b, with a chiral-truncation component of about 0.5% in its error budget — a number that only an explicit next-order computation can validate or replace. Notably, a 2026 analysis of tritium β decay at next-to-next-to-leading order found its Gamow–Teller matrix element to be finely tuned in χEFT and 'point[ing] towards large higher-order two-body corrections', so the uncomputed one-loop order is not guaranteed to be small.

Problem Statement

The open problem left by the source (Marcucci's 2014 conference report on proton–proton weak capture in chiral effective field theory) after the 2016–2017 derivation of the operators is the second half of its program: the proton–proton weak capture S-factor has never been computed with the complete next-order chiral axial two-body current, so the convergence of the chiral expansion for the solar hydrogen-ignition rate has never been directly tested. Specifically, all of the following are jointly required to resolve the problem: implement the full one-loop (fourth-order in the chiral expansion; 'N3LO' in the source's counting) two-nucleon axial weak current — as derived by time-ordered perturbation theory (Baroni et al., 2016) or equivalently by the unitary-transformation derivation (Krebs et al., 2017), including the pion-loop corrections, the complete contact-term structure, and the consistent treatment of reducible diagrams — in a form regulated consistently with a chosen chiral nucleon–nucleon interaction and the accompanying three-nucleon force required for the consistency of that scheme; re-determine the short-range axial-current low-energy constant (for example from the tritium Gamow–Teller matrix element computed with the same currents, since the existing calibration ignores the loop corrections); and evaluate the low-energy pp-capture astrophysical S-factor $S(E)$, its threshold value $S(0)$ and energy derivatives, with an order-by-order breakdown separating the leading Gamow–Teller, tree-level two-body, and one-loop two-body contributions, together with their regulator and scheme dependence. The scientific objective is to quantify the one-loop two-body axial contribution and thereby to test directly the convergence of the chiral expansion of the weak axial current for the pp fusion rate — in place of the purely statistical truncation estimates on which all current uncertainty budgets for $S(0)$ rest — and to state whether that contribution lies within or exceeds the roughly 0.5–1.5% theory uncertainty currently quoted for the recommended solar fusion rate.

References

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