  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 Schrodinger 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.
