Coulomb Corrections to Three-Nucleon Moments

The Helium-3 (${}^3\mathrm{He}$) magnetic moment and Gamow-Teller (GT) matrix element in triton (${}^3\mathrm{H}$) $\beta$-decay are calculated in pionless effective field theory ($\mathrm{EFT}(/\!\!\!\pi)$) to next-to-leading order (NLO). Coulomb corrections are included perturbatively to $\mathcal{O}(\alpha)$ in this framework and should naively be $\alpha M_n/p^*\!\!\sim\!8\%$ corrections, where $p^*\!\!\sim\!88.5$ MeV is related to the three-nucleon binding momentum. Fitting the two-nucleon iso-vector magnetic current low-energy constant (LEC), $L_1$, to the ${}^3\mathrm{H}$ magnetic moment and the two-nucleon iso-scalar magnetic current LEC, $L_2$, to the deuteron magnetic moment we find the NLO ${}^3\mathrm{He}$ magnetic moment in units of nuclear magnetons is -2.130 and the surprisingly small $\mathcal{O}(\alpha)$ correction is 0.00335, $\approx\!0.18\%$ of the LO $\mathrm{EFT}(/\!\!\!\pi)$ prediction. The leading-order (LO) GT matrix element for ${}^3\mathrm{H}$ $\beta$-decay is 0.9806 while again it has a surprisingly small $\mathcal{O}(\alpha)$ Coulomb correction of $-0.000740$, $\approx\!0.08\%$ of the LO $\mathrm{EFT}(/\!\!\!\pi)$ prediction. At NLO we calculate the GT matrix element of ${}^3\mathrm{H}$ $\beta$-decay, including the $\mathcal{O}(\alpha)$ Coulomb correction, in terms of the two-nucleon axial current LEC $l_{1,A}$. Fitting $l_{1,A}$ to the ${}^3\mathrm{H}$ half-life we make a prediction for the proton-proton fusion reduced matrix element of $\Lambda(0)=2.776(331)$. Finally, we attempt to explain the unusually small size of the $\mathcal{O}(\alpha)$ corrections by investigating the Wigner-SU(4) expansion of these observables.