An $\textit{ab initio}$ strategy for taming nuclear-structure dependence of $ V_{ud} $ extractions: the $ {}^{10}\mathrm{C} \rightarrow {}^{10}\mathrm{B} $ superallowed transition

TL;DR

This work addresses the precision determination of the CKM top-row element $|V_{ud}|$ from superallowed decays by computing the nuclear-structure-dependent radiative correction $\,\delta_{NS}$ for the $^{10}$C → $^{10}$B transition with an \\textit{ab initio} approach. The authors perform a no-core shell-model calculation using chiral EFT Hamiltonians (NN-N$^4$LO(500) with 3N forces) and SRG evolution to obtain the nuclear γW-box contribution, including a novel treatment of nuclear shadowing and a detailed error budget via a Lanczos-strength-type method for the resolvent. They report $\,\delta_{NS} = -0.422(14)_{PME}(4)_{\Omega}(9)_{\chi}(24)_{sh}(12)_{n,el} \%$, and a total radiative correction $\Delta_R^V + \delta_{NS} = 0.02057(29)_{nuc}(14)_{n,inel}(10)_{hi}$, indicating a 1.6× reduction in uncertainty relative to prior shell-model estimates. The results establish a systematically improvable, \\textit{ab initio} framework for electroweak radiative corrections in nuclei, enabling a more precise extraction of $V_{ud}$ and guiding future theoretical and experimental efforts for superallowed decays and related processes.

Abstract

We report the first \textit{ab initio} calculation of the nuclear-structure-dependent radiative correction $ δ_{ \mathrm{NS} } $ to the $ {}^{10}\mathrm{C} \rightarrow {}^{10}\mathrm{B} $ superallowed transition, computed with the no-core shell model and chiral effective field theory. We obtain $δ_{ \mathrm{NS} } = - 0.422 (29)_{ \mathrm{nuc} } (12)_{ n,\mathrm{el} } $ with a $1.6$-times reduction in the total uncertainty when compared to the current literature estimate based on the shell model and Fermi gas picture. This work paves the way for a precise determination of $V_{ud}$ from superallowed beta decays within a systematically improvable framework.

Figures (5)

• Figure 1: Example trajectories of the $\nu$-integral poles in Eq. (\ref{['sec:theory:eq:gW_box']}) coming from the (i) nuclear propagators in $T_3$ (ii) photon propagator and (iii) electron propagator, with sets labelled by $\mathcal{N}$, $\gamma$ and $e$, respectively.
• Figure 2: Breakdown of the $\square_{ \gamma W }^{ b, \mathrm{nuc} }$ into (i) different electroweak operator structures in the Compton amplitude and (ii) each moment in the multipole expansion, obtained with (a) the chiral $\text{NN-N}^{4}\text{LO}(500) {+} 3\text{N}_{\text{lnl}}$ and (b) $\text{NN-N}^{4}\text{LO}(500) {+} 3\text{N}_{\text{lnl}}^{*}$ interactions, in the NCSM. Residue contributions and contributions from the electron energy expansion are shown as hatched and solid bars, respectively.
• Figure 3: Slice of $i T^{ \mathrm{mag} }_{ J = 1 } \otimes i T^{ 5, \mathrm{el} }_{ J = 1 }$ contribution to the $\gamma W$-box integrand at $E_e \approx m_e$. The Compton residue (dashed), $\mathcal{O}( E_e )$ expansion (dash-dotted) and total sum (dotted) are as detailed in Eq. \ref{['sec:theory:eq:gW-box_expansion']}. The vertical dashed line corresponds to $\mathbf{ {q} } _{ \mathrm{max} } = \sqrt{ M_{ f }^2 - M_{ k }^2 }$ where $k$ is the lowest-lying ${}^{10} \mathrm{B}$($1^+$) state.
• Figure 4: Sequences of $\delta_{\text{NS}}$ evaluations in the NCSM with $N_{ \mathrm{max} } = 3,\, 5,\, 7$ model space truncations and frequencies $\hbar \Omega = 16 - 20 \, \text{MeV}$. The $E_{7}$ point is obtained with the $\text{NN-N}^{4}\text{LO}(500) + 3\text{N}_{\text{lnl}}^{*}$ interaction; all other points are generated with the $\text{NN-N}^{4}\text{LO}(500) + 3\text{N}_{\text{lnl}}$ interaction. Error bars are as described in the text.
• Figure S 1: The nuclear $\gamma W$-box diagrams.