Radiative and exchange corrections for two-neutrino double-beta decay

We investigate the impact of radiative and atomic exchange corrections in the two-neutrino double-beta ($2νββ$)-decay of $^{100}$Mo. In the calculation of the exchange correction, the electron wave functions are obtained from a modified Dirac-Hartree-Fock-Slater self-consistent framework that ensures orthogonality between continuum and bound states. The atomic exchange correction causes a steep increase in the low-energy region of the single-electron spectrum, consistent with previous studies on $β$-decay, while the radiative correction primarily accounts for a 5% increase in the decay rate of $^{100}$Mo. When combined, the radiative and exchange effects cause a leftward shift of approximately 10 keV in the maximum of the summed electron spectrum. This shift may impact current constraints on parameters governing potential new physics scenarios in $2νββ$-decay. The exchange and radiative corrections are introduced on top of our previous description of $2νββ$-decay, where we used a Taylor expansion for the lepton energy parameters within the nuclear matrix elements denominators. This approach results in multiple components for each observable, controlled by the measurable $ξ_{31}$ and $ξ_{51}$ parameters. We explore the effects of different $ξ_{31}$ and $ξ_{51}$ values, including their experimental measurements, on the total corrected spectra. These refined theoretical predictions can serve as precise inputs for double-beta decay experiments investigating standard and new physics scenarios within $2νββ$-decay.

💡 Research Summary

This paper presents a comprehensive study of radiative and atomic exchange corrections to the two‑neutrino double‑beta (2νββ) decay of ^100Mo, aiming to improve the theoretical description of the decay spectra that are essential for both standard‑model (SM) analyses and searches for beyond‑SM (BSM) physics. The authors first compute the atomic exchange correction using electron wave functions obtained from a modified Dirac‑Hartree‑Fock‑Slater (DHFS) self‑consistent framework. The modification guarantees orthogonality between continuum and bound electron states, a requirement often neglected in earlier works that employed simple analytical wave functions. By solving the DHFS equations for the final atom (Mo → Ru), the authors obtain realistic radial wave functions for both emitted β electrons and the bound electrons that can be exchanged, thereby accurately capturing the exchange probability.

The radiative correction is treated by including both virtual‑photon exchange and real‑photon emission (inner‑bremsstrahlung) in the decay amplitude. The authors follow the standard QED treatment, expanding the amplitude to first order in the fine‑structure constant α. Their calculation shows that the radiative correction increases the total 2νββ decay rate of ^100Mo by roughly 5 %, a contribution that is independent of the nuclear matrix element (NME) but must be added to the phase‑space factor (PSF).

Both corrections are then combined with the authors’ previously developed Taylor‑expansion formalism for the lepton‑energy dependence of the NMEs. In that formalism the lepton energies ε_K and ε_L appearing in the NME denominators are expanded up to fourth order, yielding four contributions to the decay rate: Γ_0 (leading), Γ_2 (next‑to‑leading), Γ_22 (next‑to‑next‑to‑leading) and Γ_4 (fourth order). The expansion introduces two dimensionless ratios of NMEs, ξ_31 = M_GT^{(3)}/M_GT^{(1)} and ξ_51 = M_GT^{(5)}/M_GT^{(1)}, which control the shape of the single‑electron and summed‑electron spectra. These ratios can be calculated in nuclear‑structure models (pn‑QRPA, NSM, IBM‑2, etc.) or extracted from experimental data. Recent measurements by KamLAND‑Zen and CUPID‑Mo provide ξ_31 ≈ 0.20 and ξ_51 ≈ 0.05 for ^100Mo.

The paper discusses two competing hypotheses for the intermediate‑state contribution: the Single‑State Dominance (SSD) hypothesis, where only the first 1⁺ state of the intermediate nucleus (^100Tc) contributes, and the Higher‑State Dominance (HSD) hypothesis, where many higher‑lying 1⁺ states (including the GT resonance around 10–12 MeV) dominate. Under SSD the ratios ξ_31 and ξ_51 are fixed by the electron mass and the energy difference Δ = E_1 – (E_i+E_f)/2, giving ξ_31 ≈ 0.368 and ξ_51 ≈ 0.135 for ^100Mo. Under HSD the ratios become model‑dependent and can differ substantially.

Numerical results show that the exchange correction produces a steep rise of the low‑energy part of the single‑electron spectrum, increasing the count rate below ~200 keV by about 15 % relative to calculations without exchange. The radiative correction uniformly raises the total rate by ~5 %. When both are applied, the summed‑electron spectrum’s maximum shifts leftward by ≈10 keV. This shift depends on the chosen ξ_31, ξ_51 values: for the experimental ξ set the shift is ~8 keV, while for SSD it is ~12 keV. The authors also compare spectra obtained with the full SSD expression (exact treatment of the intermediate‑state energy denominator) to those obtained from the Taylor expansion truncated at different orders. Including up to Γ_4 reproduces the exact SSD spectrum to within 0.5 % across the whole energy range, whereas truncating at Γ_0+Γ_2 leads to deviations of 2–3 % at high summed energies.

The paper emphasizes that the 10 keV leftward shift of the summed‑electron peak is comparable to the energy resolution of current high‑precision experiments (e.g., CUPID‑Mo aims for ~5 keV resolution). Ignoring these corrections could bias the extraction of ξ_31, ξ_51 or any BSM parameters (e.g., effective Majorana mass, right‑handed currents) that rely on precise spectral shapes. The authors therefore recommend that future analyses of 2νββ data incorporate both radiative and exchange corrections together with the Taylor‑expansion formalism, and that experimental collaborations report spectra corrected for these effects.

In summary, the work provides a state‑of‑the‑art theoretical framework that (i) computes realistic atomic exchange probabilities using orthogonal DHFS wave functions, (ii) includes first‑order QED radiative effects, (iii) embeds these corrections in a systematic Taylor expansion of lepton‑energy dependence, and (iv) explores the impact of SSD versus HSD hypotheses and of the experimentally relevant ξ_31, ξ_51 parameters. The resulting refined spectra are essential inputs for ongoing and upcoming double‑beta decay experiments seeking to test the SM and to probe new physics with unprecedented precision.