We use heavy-nucleus effective field theory to compute radiative corrections to two-neutrino double-$\beta$ decay ($2\nu \beta \beta$). Our main result is the first derivation of a universal radiative-correction factor for double-weak decays—the analog of the Sirlin function in single-$\beta$ 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-$\beta$ 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 correction parametrized by the ratio of nuclear matrix elements, ${\xi }_{31}$. Our results indicate that extractions of nuclear-structure information and tests of the standard model from high-precision $2\nu \beta \beta$ measurements must include double-weak radiative corrections, implying that recent extractions of ${\xi }_{31}$ should be revisited.

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• Beta decay
• Double beta decay