First-Principles Determination of the Proton-Proton Fusion Matrix Element from Lattice QCD

Abstract

Proton-proton fusion is the fundamental weak reaction initiating stellar energy production, and a first-principles determination of its matrix element remains a long-standing goal of nuclear theory. We present a lattice QCD calculation of the pp fusion matrix element at m_pi~432 MeV. We implement Lellouch-Luscher (LL) finite-volume (FV) corrections within a 2+J->2 framework, accounting for two-nucleon (2N) rescattering, to relate FV matrix elements to infinite-volume counterparts. Excited-state contamination is suppressed using bi-local nucleon-nucleon interpolating operators and a variational analysis with three lowest momenta. This enables determination of 2N energy spectrum and scattering parameters via Luscher's FV formalism. Before including rescattering effects in the LL factor, we obtain <d|J|pp>/g_A = 0.984(10), where g_A is the axial charge. The deviation from unity indicates a small nonvanishing 2-body current contribution. Our analysis shows that rescattering effects in LL factors substantially modify the 2-body contribution, while large uncertainties in 2N scattering parameters propagate strongly into FV corrections. Thus, precise determination of the 2-body low-energy constant L_{1,A} remains highly challenging with current lattice inputs. Despite the large uncertainty, L_{1,A}=6.0(7.1) fm^3 is compatible, at the level of naturalness, with phenomenological extractions. This work demonstrates feasibility and intrinsic challenges of ab initio lattice QCD calculations of weak 2N reactions, and establishes a foundation for future studies at or near the physical pion mass.

Figures (7)

• Figure 1: One-body contribution$(a)$ and two-body contribution $(b)$.
• Figure 2: Effective mass curves obtained using different dinucleon interpolators. PP denotes hexaquark interpolators at both the source and sink, while PSPS corresponds to bi-local interpolators at both source and sink. Data points for the $^1\mathrm{S}_0$ and $^3\mathrm{S}_1$ channels are slightly offset horizontally for clarity.
• Figure 3: Effective mass curves (upper panels) and coupling factors (lower panels) from the variational analysis for the $^1\mathrm{S}_0$ and $^3\mathrm{S}_1$ channels. Black points denote the raw data, while colored points indicate results after applying the variational method. The bar charts display the overlap factors $\left| Z^i_n \right|^2\equiv \left| \Braket{\Omega|O_i(t)|n} \right|^2$, as defined in Eq. (\ref{['eq:overlapFactor']}).
• Figure 4: Upper panels: Two-state fits to the effective masses of the proton, diproton ($^1\mathrm{S}_0$), and deuteron ($^3\mathrm{S}_1$). Lower panels: Energy shifts extracted using the ratio method (blue), subtracted correlators (orange), and the variational method (yellow band).
• Figure 5: Scattering phase shift analysis for $^1\mathrm{S}_0$ (upper) and $^3\mathrm{S}_1$ (lower) channels.