Tritium β decay and proton-proton fusion in pionless effective field theory

The Gamow-Teller and Fermi matrix elements, $\langle \mathbf{GT}\rangle$ and $\langle \mathbf{F}\rangle$, respectively, for tritium $\beta$ decay are calculated to next-to-leading order (NLO) in pionless effective field theory in the absence of Coulomb interactions and isospin violation giving the leading order predictions ${\langle \mathbf{GT}\rangle }_0=0.9807$ and ${\langle \mathbf{F}\rangle }_0=1$. Using an experimentally determined value for the tritium $\beta$ decay GT matrix element, the two-body axial current low energy constant is fixed at NLO yielding $L_{1,A}=6.01\pm 2.08{\mathrm{fm}}^3$ at the renormalization scale of the physical pion mass, which agrees with predictions based on naive dimensional analysis. The impact of $L_{1,A}$ on proton-proton fusion is also discussed. Finally, the consequences of Wigner-SU(4) spin-isospin symmetry are considered for the Gamow-Teller matrix element.