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The Gamow-Teller (GT) matrix element contributing to tritium $\beta$ decay iscalculated with trinucleon wave functions obtained fromhyperspherical-harmonics solutions of the Schr\"odinger equation with thechiral two- and three-nucleon interactions including $\Delta$ intermediatestates that have recently been constructed in configuration space. Predictionsup to N3LO in the chiral expansion of the axial current (with $\Delta$'s)overestimate the empirical value by 1--4 \%. By exploiting the relation betweenthe low-energy constant (LEC) in the contact three-nucleon interaction andtwo-body axial current, we provide new determinations of the LECs $c_D$ and$c_E$ that characterize this interaction by fitting the trinucleon bindingenergy and tritium GT matrix element. Some of the implications that theresulting models of three-nucleon interactions have on the spectra of lightnuclei and the equation of state of neutron matter are briefly discussed. Wealso provide a partial analysis, which ignores $\Delta$'s, of the contributionsdue to loop corrections in the axial current at N4LO. Finally, explicitexpressions for the axial current up to N4LO have been derived in configurationspace, which other researchers in the field may find useful.

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