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We present a comprehensive study of the $\mathcal{O}(α)$ radiative corrections to two-neutrino and neutrinoless double-$β$ decay within a heavy-nucleus effective field theory. We compute all virtual and real-emission corrections for intermediate nuclear states of arbitrary excitation energy, including new closed-form expressions for the required one-loop integrals with heavy-nucleus propagators.
Let $K/k$ be a finite abelian extension of number fields with Galois group $G$ and let $n\geq 2$. We prove, assuming the relevant $p$-part of the equivariant Tamagawa number conjecture, first-derivative analogues of the Deligne-Ribet integrality theorem and of the Coates-Sinnott conjecture.
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