Framework for maximum likelihood analysis of neutron $\beta$ decay observables to resolve the limits of the $V-A$ law

We assess the ability of future neutron $\beta$ decay measurements of up to $\mathcal{O}\left({10}^{-4}\right)$ precision to falsify the standard model, particularly the $V-A$ law, and to identify the dynamics beyond it. To do this, we employ a maximum likelihood statistical framework which incorporates both experimental and theoretical uncertainties. Using illustrative combined global fits to Monte Carlo pseudodata, we also quantify the importance of experimental measurements of the energy dependence of the angular correlation coefficients as input to such efforts, and we determine the precision to which ill known “second-class” hadronic matrix elements must be determined in order to exact such tests.

Figure 1

Simulated data from the Standard Model data set for $a_{\exp }/\left(\frac{1}{2}\beta \right)$ [panel (a)] and $A_{\exp }/\left(\frac{1}{2}\beta \right)$ [panel (b)] plotted as a function of $T_e$. The solid red line is the result of a simultaneous fit to the $a$ and $A$ data.

Figure 2

Impact of uncertainties in $f_3$ and $g_2$ on the allowed $(\lambda ,b_{\mathrm{BSM}})$ parameter space from a two-parameter simultaneous fit to the $\{a,A\}$ New Physics data set. The bands indicate the 68.3$\%$ CL allowed regions defined by $\Delta {\chi }^2=2.30$ for a joint fit of two free parameters, for the indicated uncertainties in $f_3$ and $g_2$. The input values $\lambda =1.2701$ and $b_{\mathrm{BSM}}=-0.00522$ are indicated by the dashed lines.