Vertex correction to nuclear matrix elements of double-$\beta$ decays

The predicted neutrinoless double-$\beta$ ($0\nu\beta\beta$) decay is the crucial phenomenon to prove the existence of the Majorana neutrino, which gives a foundation to leptogenesis to explain the matter prevalence of the universe. The nuclear matrix element (NME) of $0\nu\beta\beta$ decay is an important theoretical quantity to determine the effective neutrino mass and help the detector design for the next generation of the $0\nu\beta\beta$ decay search. Reliable calculation of this NME is a long-standing problem because of the diversity of the predicted values of the NME. The main reason for this difficulty is that the effective strength of the Gamow-Teller transition operator $g_A$ for this decay is unknown. I will show the lowest-order vertex corrections for the $0\nu\beta\beta$ and the $2\nu\beta\beta$ NME of $^{136}$Xe in the framework of the hybrid application of the quantum field theory to the leptons and the Rayleigh-Schr\"{o}dinger perturbation to the nucleus. The unperturbed nuclear states are obtained by the quasiparticle random-phase approximation. These corrections reduce the $0\nu\beta\beta$ NME by 30%. The effective $g_A$ referring to this reduced NME is also obtained, and it is shown for the first time that the effective $g_A$ for the $0\nu\beta\beta$ NME is not quite different from that for the $2\nu\beta\beta$ NME; the difference is only 10%. This indicates the possibility that the phenomenological effective $g_A$ to reproduce the experimental half-life of the $2\nu\beta\beta$ decay can be approximately used for the calculation of the $0\nu\beta\beta$ NME.