Kinematic sensitivity to the Fierz term of $\beta$-decay differential spectra

The current most stringent constraints on exotic scalar or tensor couplings in neutron and nuclear $\beta$ decay, involving left-handed neutrinos, are obtained from the Fierz interference term. The sensitivity to this term in a correlation coefficient is usually driven by an energy-averaged kinematic factor that increases monotonically toward smaller values of the $\beta$ endpoint energies. We first point out here that this property does not hold for certain differential observables that are directly sensitive to the Fierz term, such as the $\beta$ or the recoil energy spectrum. This observation is relevant for the selection of sensitive transitions in searches for exotic couplings through spectrum shape measurements. We then point out previous errors in the exploitation of measurements of the $\beta \text{−}\nu$ angular correlation coefficient and discuss their impact on the extraction of constraints on exotic couplings.

Figure 1

Variation of the sensitivity factor $\langle m/W\rangle$ as a function of the endpoint energy $E_0$.

Figure 2

The solid red line shows the $1\sigma$ statistical uncertainties obtained from fits of simulated $\beta$ energy spectra as a function of the endpoint energy $E_0$. The dashed brown line shows the result obtained with the approximation given by Eq. (7).

Figure 3

Energy dependence of the ratio $m/W$, for two different endpoint energies. Notice that the $E$ axis is rescaled to ease the comparison.

Figure 4

The $1\sigma$ statistical uncertainties on $b$ obtained from fits of simulated recoil momentum spectra as a function of the endpoint energy $E_0$. The solid lines represent $\mathrm{\Delta }b$ and correspond to Fermi (red) and Gamow-Teller (blue) transitions for which the value of $a$ was, respectively, fixed to $a_F$ or $a_{\mathrm{GT}}$. The dotted lines represent $\mathrm{\Delta }{b}_{\tilde{a}}$, that is, the uncertainty on $b$ obtained by incorrectly reinterpreting a fit of $a$ with $b=0$, through the $\tilde{a}$ prescription [cf. Eq. (18)].

Figure 5

The solid red ellipse shows the $1\sigma$ region obtained from a fit of simulated recoil momentum spectra with ${10}^7$ events, for the ${}^6\mathrm{He}$ decay, where both $a$ and $b$ were left as free parameters. The blue shaded band shows the $1\sigma$ bound on the combination $\widehat{a}=a+0.127b$, whereas the black dotted lines represent the $1\sigma$ bound obtained using the $\tilde{a}$ prescription.