Concurrent tests of Lorentz invariance in $\beta$-decay experiments

Modern experiments on neutron and allowed nuclear $\beta$ decay search for new semileptonic interactions, beyond the ``left-handed'' electroweak force. We show that ongoing and planned $\beta$-decay experiments, with isotopes at rest and in flight, can be exploited as sensitive tests of Lorentz invariance. The variety of correlations that involve the nuclear spin, the direction of the emitted $\beta$ particle, and the recoil direction of the daughter nucleus allow for relatively simple experiments that give direct bounds on Lorentz violation. The pertinent observables are decay-rate asymmetries and their dependence on sidereal time. We discuss the potential of several asymmetries that together cover a large part of the parameter space for Lorentz violation in the gauge sector. High counting statistics is required.