We investigate the contribution of radial excitations to Fermi $\beta$-decay matrix element. To this end, exact no-core shell-model calculations are performed for the mirror $\beta$ decay of tritium, where full convergence can be achieved on an ordinary computer. The differences between the isospin-mixing correction values obtained in the full and in a restricted model spaces are matched to the radial overlap correction term, analogous to that required in the shell-model approach, where the configuration space is extremely limited. We examine this complementary correction term using a nonorthogonal harmonic-oscillator basis, generated by slightly differentiating the oscillator frequencies between the initial and final nuclei, while all desirable properties, including translational invariance, are still preserved. For $N_{\max }\le 8$, we find that the radial excitation contribution is negative, with a typical magnitude of approximately 10% to 20% of the radial diagonal contribution. This effect becomes more pronounced as the model space increases. Therefore, the ${\delta }_{C2}$ values obtained in the shell-model approach, where radial excitations are not explicitly included, are likely overestimated. Based on experimental $ft$ data and the corrective terms adopted in the survey by Hardy and Towner [Phys. Rev. C 102, 045501 (2020)], we show that the incorporation of radial excitations for the superallowed $0^{+}\to 0^{+}$ nuclear $\beta$ decay tends, however, to worsen agreement with the standard model.

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• Beta decay
• Nuclear decay
• Shell model