Reassessing the gallium anomaly using self-consistent electron wave functions

TL;DR

The study reexamines the gallium anomaly by solving the Dirac-Coulomb problem for both bound and continuum electrons to obtain self-consistent electron wave functions, and by averaging over the nuclear volume to refine the Fermi function and EC/Beta-Decay amplitudes. This leads to a revised neutrino capture cross section on $^{71}$Ga and a more robust, self-consistent framework for comparing theory with gallium experiments. The updated analysis strengthens the anomaly and constrains sterile-neutrino 3+1 interpretations, which remain in tension with reactor, solar, and KATRIN data, highlighting the need for dedicated new-source experiments to resolve the discrepancy.

Abstract

The gallium anomaly, a persistent discrepancy exceeding $4σ$ in the $^{71}$Ga neutrino capture rates from $^{51}$Cr and $^{37}$Ar radioactive sources by the GALLEX, SAGE, and recently BEST experiments, has challenged particle physics and nuclear theory for over three decades. We present a new calculation of the neutrino capture cross-section, abandoning the conventional leading-order approximation for electronic wave functions by numerically solving the Dirac-Coulomb equation for both bound and continuum electron states. Finally, we re-evaluate the gallium anomaly, updating its global significance and presenting the most up-to-date status of its interpretation in terms of sterile neutrinos.

Figures (5)

• Figure 1: Electron density at the nucleus as a function of the radial distance from the origin. We compare our calculations based on RADIALradial (labelled "Cadeddu et al.") with that obtained using GRASPGrasp2018 and with those available in the literature MartinBAND1986BAND1979Behrens. The horizontal dashed lines indicate the values obtained after applying the averaging procedure. The vertical dotted and dash-dotted lines represent $R_{\mathrm{ch}}$ and $R_{\mathrm{box}}=\sqrt{5/3}\, R_{\rm ch}$, respectively.
• Figure 2: Fermi function for $^{71}$Ge as a function of the radial distance (blue solid line), as derived in this work in Eq. (\ref{['eq:fermi']}) using the exact DHFS $g_\kappa(r)$ and $f_\kappa(r)$ wave functions for $E_e=1.031\,\mathrm{MeV}$. We also show the charge density of $^{71}$Ge, $\rho_{\rm ch}(r)$ (gold dashed line) in arbitrary units and the electron density for a bound electron in the 1$s$ state of $^{71}\text{Ge}$ (red long dashed line), as obtained from our numerical code for $C_{\rm ex}=1.25$. The vertical dotted and dash-dotted lines represent $R_{\mathrm{ch}}$ and $R_{\mathrm{box}}=\sqrt{5/3}\, R_{\rm ch}$, respectively.
• Figure 3: Ratios between the experimental values and our theoretical estimations of the event rates in gallium experiments, when considering the (p,n) (upper panel) or the ($^{3}\text{He},^{3}\text{H}$) (lower panel) excited state contribution. Also shown are the best fit value of $\overline{R}$ (green line) and its 1$\sigma$ uncertainty (green band).
• Figure 4: $\Delta\chi^2 = \chi^2 - \chi^2_{\text{min}}$ as a function of $\overline{R}$ obtained from the combined analysis of the gallium data when considering the (p,n) (red line) or the ($^{3}\text{He},^{3}\text{H}$) (purple line) excited state contribution.
• Figure 5: Contours of the $2\sigma$ allowed regions in the plane of the effective oscillation parameters of 3+1 active-sterile neutrino mixing obtained from the combined analysis of the gallium data when considering the (p,n) (red line) or the ($^{3}\text{He},^{3}\text{H}$) (purple line) excited state contribution. The crosses indicate the best-fit points. Also shown are the $2\sigma$ solar bound $\sin^2 2\vartheta_{ee} \lesssim 0.17$Gonzalez-Garcia:2024hmf (cyan line) and the 95% C.L. bound of the KATRIN experiment KATRIN:2025lph (green line).