Compute Z‑independent O(α) and O(α²) ultrasoft anomalous dimensions for subleading two‑ and three‑body electroweak potentials

Determine the Z‑independent O(α) and O(α²) anomalous dimensions governing the renormalization‑group evolution of the electroweak potentials V^0, V_{m_e}, V_+, and V_+^{3b} that appear in the EFT description of superallowed 0^+→0^+ nuclear β decays, in order to complete the evolution of the corresponding effective couplings C_eff^{(i)}(μ) between the nuclear scale and μ≈m_e.

Background

In the EFT framework developed in the paper, the dominant ultrasoft photon effects are captured by renormalization‑group equations for effective couplings C_eff^{(i)}(μ). For the leading vector coupling these anomalous dimensions are known to the required accuracy, but for several subleading two‑ and three‑body potentials only the Z‑enhanced pieces are controlled.

The authors note that the Z‑independent O(α) and O(α²) parts of the anomalous dimensions for V^0, V_{m_e}, V_+, and V_+^{3b} are not yet determined within their analysis. Although these terms are expected to affect contributions that enter at O(α εχ) or O(α επ̸), establishing them would complete the RG evolution and reduce residual scale dependence in precision extractions of V_ud.