Influence of the recoil-order and radiative correction on the $\beta$ decay correlation coefficients in mirror decays

Measurements of the $\beta$ decay correlation coefficients in nuclear decay aim for a precision below $1\%$ and theoretical predictions should follow this trend. In this work, the influence of the two dominant standard model correction terms, i.e., the recoil-order and the radiative correction, are studied for the most commonly measured $\beta$ correlations, i.e., the $\beta$-asymmetry parameter ($A_{\beta }$) and the $\beta \text{−}\nu$ angular correlation ($a_{\beta \nu }$). The recoil-order correction is calculated with the well-known Holstein formalism using the impulse approximation to evaluate experimentally undetermined form factors. For the $\beta \text{−}\nu$ angular correlation previously unpublished, semianalytical radiative correction values are tabulated. Results are presented for the mirror $\beta$ decays up to $A=45$. We examine the effect of both corrections and provide a comparison between different isotopes. This comparison will help planning, analyzing, and comparing future experimental efforts.

Figure 1

Size (full line) and sensitivity (dashed line; note the different $y$ axis) of the beta-asymmetry parameter $A_{\beta }$ with respect to $\rho$ for different nuclear spins. The discussed mirror transitions are highlighted. For $J=1/2$, the cancellation of the first-order expression in Eq. (4) at $\rho =\pm \sqrt{3}$ is clearly visible.

Figure 2

Size (full line) and sensitivity (dashed line; note the different $y$ axis) of the beta-neutrino correlation $a_{\beta \nu }$ with respect to $\rho$, with colors corresponding to the nuclear spin. Sensitivity enhancements are obtained for $\rho =\pm \sqrt{3}$.

Figure 3

The influence of the recoil correction terms on $A_{\beta }$ for mirror nuclei with different endpoint energies. No clear trend is observed in the size of the correction terms with increasing endpoint energy. Energies are given as the ratio between the decay energy, $W$, and the endpoint energy, $W_0$. Error bars include the uncertainty from Ref. [23] on the endpoint energy, the weak magnetism form factor $b_{WM}$, and the Gamow-Teller to Fermi strength ratio $\rho$. The latter are the dominant contribution.

Figure 4

The influence of the recoil correction terms for ${}^{19}\mathrm{Ne}$ for both $A_{\beta }$ and $a_{\beta \nu }$. For this isotope, the leading order expressions are suppressed; therefore, the relative importance of the higher-order corrections increases drastically.

Figure 5

The correction ratio $R_A$ when including different parts of the recoil corrections for five different isotopes, i.e., ${}^{13}\mathrm{N}$, ${}^{25}\mathrm{Al}$, ${}^{31}\mathrm{S}$, ${}^{37}\mathrm{K}$, and ${}^{43}\mathrm{Ti}$. Note the different scales on the $y$ axis for the different isotopes.

Figure 6

Radiative correction to the beta-asymmetry parameter as described in Eq. (15) for some selected mirror nuclei. Energies are given as the ratio between the decay energy $W$ and the endpoint energy $W_0$. The influence is seen to be small throughout the energy range and stays well below ${10}^{-3}$ except for the lowest energies. Up to endpoint energies $W_0=3.5$ MeV the size of the correction decreases, but for higher endpoint energies the sign of the correction flips and its influence increases again.

Figure 7

Size of the correction function $g$ (solid) and $h$ (dashed) for different endpoint energies. For all endpoints the difference between both is most pronounced at lower energy ratios $W/W_0$, but their relative size changes with respect to each other. This behavior is the cause of the sign flip in the radiative correction as illustrated in Fig. 6.

Figure 8

The influence of the recoil correction terms for mirror nuclei with different endpoint energies. The size of the correction shows no trend with respect to the endpoint energy. Error bars include the uncertainty from Ref. [23] on the endpoint energy, the weak magnetism form factor $b_{WM}$, and the Gamow-Teller to Fermi strength ratio $\rho$. The latter are the dominant contribution.

Figure 9

The correction ratio $R_a$ when including either only the weak magnetism form factor or the full recoil corrections for five different isotopes, ${}^{13}\mathrm{N}$, ${}^{25}\mathrm{Al}$, ${}^{31}\mathrm{S}$, ${}^{37}\mathrm{K}$, and ${}^{43}\mathrm{Ti}$. Note the different scales on the $y$ axis for the different isotopes.

Figure 10

The radiative correction $r\left(T\right)$ to the recoil spectrum in the beta-neutrino correlation $a_{\beta \nu ,RC}$ as calculated using semianalytical integrals [48, 64] for several mirror decays. The correction is most important for high energy ratios $T/T_0$. The correction is very similar for all decays as is also visible in the inset, with the exception of ${}^3\mathrm{H}$ due to its low endpoint energy. Numerical values are tabulated in Table 4.