Tritium $\beta$-decay and Proton-Proton Fusion in Pionless Effective Field Theory

The Gamow-Teller and Fermi matrix elements, $<\mathbf{G}\mathbf{T}>$ and $<\mathbf{F}>$, respectively, for tritium $\beta$-decay are calculated to next-to-leading order (NLO) in pionless effective field theory in the absence of Coulomb and isospin violation giving the leading order predictions ${<\mathbf{G}\mathbf{T}>}_0=0.9807$ and ${<\mathbf{F}>}_0=1$. 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 $L_{1,A}=6.01\pm 2.08$ fm${}^3$ at the renormalization scale of the physical pion mass, which agrees with predictions based on naive dimensional analysis. Finally, the consequences of Wigner-SU(4) spin-isospin symmetry are considered for the Gamow-Teller matrix element.