New physics searches in nuclear and neutron $\beta$ decay

The status of tests of the standard electroweak model and of searches for new physics in allowed nuclear $\beta$ decay and neutron decay is reviewed including both theoretical and experimental developments. The sensitivity and complementarity of recent and ongoing experiments are discussed with emphasis on their potential to look for new physics. Measurements are interpreted using a model-independent effective field theory approach enabling to recast the outcome of the analysis in many specific new physics models. Special attention is given to the connection that this approach establishes with high-energy physics. A new global fit of available $\beta$-decay data is performed incorporating, for the first time in a consistent way, superallowed $0^+\to 0^+$ transitions, neutron decay and nuclear decays. The constraints on exotic scalar and tensor couplings involving left- or right-handed neutrinos are determined while a constraint on the pseudoscalar coupling from neutron decay data is obtained for the first time as well. The values of the vector and axial-vector couplings, which are associated within the standard model to $V_{ud}$ and $g_A$ respectively, are also updated. The ratio between the axial and vector couplings obtained from the fit under standard model assumptions is $C_A/C_V = -1.27510(66)$. The relevance of the various experimental inputs and error sources is critically discussed and the impact of ongoing measurements is studied. The complementarity of the obtained bounds with other low- and high-energy probes is presented including ongoing searches at the Large Hadron Collider.