Local chiral interactions, the tritium Gamow-Teller matrix element, and the three-nucleon contact term

Abstract

The Gamow-Teller (GT) matrix element contributing to tritium $\ensuremath{\beta}$ decay is calculated with trinucleon wave functions obtained from hyperspherical-harmonics solutions of the Schr\"odinger equation with the chiral two- and three-nucleon interactions including $\mathrm{\ensuremath{\Delta}}$ intermediate states that have recently been constructed in configuration space. Predictions up to next-to-next-to-next-to-leading order (N3LO) in the chiral expansion of the axial current (with $\mathrm{\ensuremath{\Delta}}$'s) overestimate the empirical value by 1--4%. By exploiting the relation between the low-energy constant (LEC) in the contact three-nucleon interaction and two-body axial current, we provide new determinations of the LECs ${c}_{D}$ and ${c}_{E}$ that characterize this interaction by fitting the trinucleon binding energy and tritium GT matrix element. Some of the implications that the resulting models of three-nucleon interactions have on the spectra of light nuclei and the equation of state of neutron matter are briefly discussed. We also provide a partial analysis, which ignores $\mathrm{\ensuremath{\Delta}}$'s, of the contributions due to loop corrections in the axial current at next-to-next-to-next-to-next-to-leading order (N4LO). Finally, explicit expressions for the axial current up to N4LO have been derived in configuration space, which other researchers in the field may find useful.