Radiative corrections to two-neutrino double-beta decay

Abstract

We use heavy-nucleus effective field theory to compute radiative corrections to two-neutrino double-$β$ decay ($2νββ$). Our main result is the first derivation of a universal radiative-correction factor for double-weak decays -- the analogue of the Sirlin function in single-$β$ decay -- independent of nuclear matrix elements and excitation energies. This "double-weak Sirlin function" depends on the individual electron energies as well as their relative angle and differs significantly from the approximation obtained by summing two single-$β$ decay Sirlin functions. In addition, we calculate the nuclear-structure-dependent component of the radiative corrections and find that they can still be neglected at current experimental sensitivities. On the other hand, the double-weak Sirlin function induces distortions of the electron energies and angular spectra that are comparable in size to the leading nuclear-structure corrections parametrized by the ratio of nuclear matrix elements, $ξ_{31}$. Our results indicate that extractions of nuclear structure information and tests of the Standard Model from high-precision $2νββ$ measurements must include double-weak radiative corrections, implying that recent extractions of $ξ_{31}$ should be revisited.

Figures (5)

• Figure 1: Diagrams contributing to $2\nu\beta\beta$ at $\mathcal{O}(\alpha)$. Double lines denote nuclear states in the heavy-particle EFT. Plain and wiggly lines denote leptons and photons, respectively. Black dots are vertices from the leading Lagrangian. Diagrams analogous to $(a)$, $(b)$, $(c)$, and $(l)$, but with emission from the second electron, are not shown. All possible lepton permutations are implied.
• Figure 2: Top: Full calculation (solid red) and single-$\beta$ decay approximation (dashed black) of the $\mathcal{O}(\alpha)$ corrections to $2\nu\beta\beta$ as defined in Eq. \ref{['eq:Ga']}. We highlight the contribution from the double-weak Sirlin function (dashed-dotted red) against a naive summation of two single-$\beta$ Sirlin functions (dotted black), for fixed $\mu = 2 E_0$, see main text. Bottom: Resulting spectral distortion of the $\mathcal{O}(\alpha)$ corrections, see Eq. \ref{['eq:deltaS']}.We only show $^{76}\mathrm{Ge}$, but the behavior is similar for other isotopes.
• Figure 3: Interplay between radiative corrections and $\xi_{31}$ in the NSM for the electron energy distribution. The red solid line is the result of including $\mathcal{O}(\alpha)$ corrections, while the blue dashed line excludes them. Bands correspond to a $\pm10\%$ variation of $\xi_{31}$. We omit $^{100}\mathrm{Mo}$ since the corresponding $\xi_{31}$ is so large that $\mathcal{O}(\alpha)$ correction do not lead to any appreciable change.
• Figure 4: Interplay between radiative corrections and $\xi_{31}$ in the NSM for the angular distribution in $^{100}\mathrm{Mo}$. Color coding as in Fig. \ref{['fig:eps']}.
• Figure 5: Branching ratio of $2\nu\beta\beta + \gamma$ as a function of the photon energy cut $x_\gamma^{\mathrm{cut}}= E_\gamma^{\mathrm{cut}}/\mathcal{Q}$.