Bayesian analysis of proton-proton fusion in chiral effective field theory

Abstract

The astrophysical $S$-factor for the proton-proton fusion is calculated in the low-energy regime for a variety of nuclear interactions and consistent nuclear currents, derived within chiral effective field theory. We estimate, for the first time, the theoretical uncertainty on the $S$-factor due to the truncation of the chiral expansion of the currents using a Bayesian analysis. In order to reach an accuracy at the percent level in the calculation, the electromagnetic potential includes contributions beyond the leading Coulomb interaction, such as two-photon exchange and vacuum polarization. The initial proton-proton state is expanded in partial waves and only the ${}^1S_0$ contribution is included, as it is known that the other partial-waves effects are negligible. The low-energy constant entering the contact term in the weak axial current operator is calibrated to reproduce the Gamow-Teller matrix element in Tritium $β$-decay. The value $S(0)$ is found to be $S(0)=(4.068 \pm 0.025)\times 10^{-25} \: \text{MeV}\: \text{b}$.

Figures (6)

• Figure 1: The $s$-wave $u_0(r)$ (dashed lines) and $d$-wave $u_2(r)$ (solid lines) reduced radial wave functions calculated using the NVIa potential (red lines) and the EMN potential at N$^3$LO and cutoff value $\Lambda$ = 500 MeV (blue lines).
• Figure 2: The astrophysical $S$-factor expectation value ${\tilde{S}}_n^{({\rm int})}(E)$ calculated using the EMN potential with fixed cutoff $\Lambda$ = 500 MeV and with the current at N$^3$LO with $z_0$ fixed at zero for the LO and NLO potentials, and fixed to reproduce the tritium $\beta$-decay in the other cases. Each band color represents the $\sigma^{(\text{int})}_n(E)$ truncation uncertainty at each interaction order: yellow band for LO ($n = 0$), orange band for NLO ($n = 1$), red band for N$^2$LO ($n = 2$) and purple band for N$^3$LO ($n = 3$).
• Figure 3: The astrophysical $S$-factor expectation value ${\tilde{S}}_n^{({\rm cur})}(E)$ calculated using the EMN potential at N$^3$LO with fixed cutoff $\Lambda$ = 500 MeV. Each band color represents the $\sigma^{(\text{cur})}_n(E)$ truncation uncertainty at each current order: yellow band for LO ($n = 0$), red band for N$^2$LO ($n = 1$) and purple band for N$^3$LO ($n = 2$).
• Figure 4: The average value of the energy-dependent $S$-factor $\langle S(E)\rangle$, as defined in Eq. \ref{['eq: Mean S final results']} (black line) together with the energy-dependent variance $\sigma_S (E)$ as defined in Eq. \ref{['variance S factor total']} (orange band).
• Figure 5: The GP modeling of the interaction chiral expansion coefficients and its diagnostics for the EMN450 interaction. Panel (a): the simulators (solid lines - i.e. our calculation) along with the corresponding GP emulators (dashed lines) and their $2\sigma$ intervals (bands). The data used for training are denoted by dots. Panel (b): the Mahalanobis distances compared to the mean (interior line), $50\%$ (box) and $95\%$ (whiskers) credible intervals of the reference distribution. Panel (c): the pivoted Cholesky diagnostics versus the index along with $95\%$ credible intervals (gray lines). Panel (d): the credible interval diagnostics for the truncation error bands. The $1(2)\sigma$ is represented with the dark(light) gray band.