Tritium $beta$-decay in chiral effective field theory

We evaluate the Fermi and Gamow-Teller (GT) matrix elements in tritium $beta$-decay by including in the charge-changing weak current the corrections up to one loop recently derived in nuclear chiral effective field theory ($chi$ EFT). The trinucleon wave functions are obtained from hyperspherical-harmonics solutions of the Schru007fodinger equation with two- and three-nucleon potentials corresponding to either $chi$ EFT (the N3LO/N2LO combination) or meson-exchange phenomenology (the AV18/UIX combination). We find that contributions due to loop corrections in the axial current are, in relative terms, as large as (and in some cases, dominate) those from one-pion exchange, which nominally occur at lower order in the power counting. We also provide values for the low-energy constants multiplying the contact axial current and three-nucleon potential, required to reproduce the experimental GT matrix element and trinucleon binding energies in the N3LO/N2LO and AV18/UIX calculations.