  We calculate the $\beta$ spectrum in the decay of $^6$He using Quantum Monte
Carlo methods with nuclear interactions derived from chiral Effective Field
Theory and consistent weak vector and axial currents. We work at second order
in the multipole expansion, retaining terms suppressed by $\mathcal
O(q^2/m_\pi^2)$, where $q$ denotes low-energy scales such as the reaction's
$\mathcal Q$-value or the electron energy, and $m_\pi$ the pion mass. We go
beyond the impulse approximation by including the effects of two-body vector
and axial currents. We estimate the theoretical error on the spectrum by using
four potential models in the Norfolk family of local two- and three-nucleon
interactions, which have different cut-off, fit two-nucleon data up to
different energies and use different observables to determine the couplings in
the three-body force. We find the theoretical uncertainty on the $\beta$
spectrum, normalized by the total rate, to be well below the permille level,
and to receive contributions of comparable size from first and second order
corrections in the multipole expansion. We consider corrections to the $\beta$
decay spectrum induced by beyond-the-Standard Model charged-current
interactions in the Standard Model Effective Field Theory, with and without
sterile neutrinos, and discuss the sensitivity of the next generation of
experiments to these interactions.
