Complete one-loop QED corrections to $D_s^+$ leptonic decays and impact on the CKM unitarity test

TL;DR

This work provides the first complete one-loop EW and QED corrections to $D_s^+ \to \ell^+ \nu_\ell$ decays, including both short-distance ($Z W$-box, $\gamma W$-box, and vertex effects) and long-distance soft-photon corrections with resummation. By carefully matching the full theory to the effective weak Hamiltonian and treating the Fermi constant consistently via the muon lifetime, the authors show that universal renormalization cancels in these decays, yielding scheme-dependent but finite results that refine the Sirlin-factor corrections. The long-distance sector is handled with scalar QED, soft-photon emission analysis, and a resummed factor $\Omega_B(E_{\max})$, with PHOTOS corrections accounted for to match experimental analyses; together, these corrections yield a corrected $|V_{cs}|_{D_s} = 0.991 \pm 0.007$, reducing tension in the CKM unitarity tests. The study underscores that the precision of QED corrections presently limits CKM unitarity tests and calls for lattice QCD+QED treatments and investigations of structure-dependent QED effects to further solidify the SM consistency checks.

Abstract

Recently, a violation of the CKM unitarity condition has been reported in the latest charm-meson data and the latest lattice results, once the universal electroweak correction is taken into account. In this article, we analytically derive for the first time the complete one-loop electroweak (EW) and QED corrections to the $D_{s}^+ \to \ell^+ ν_\ell$ decays for $\ell = μ, τ$. Our analysis incorporates both short-distance EW-QED corrections, which are beyond the leading-logarithmic approximation (the so-called Sirlin factor), and long-distance soft-photon corrections depending on the maximum total energy of undetected photons with their resummation. Although the inclusive photon QED corrections to the meson leptonic decays are well known, they do not match the actual measurement circumstances in $D_s^+ \to μ^+ ν_μ$. We find $ |V_{cs}|_{D_s} = 0.991 \pm 0.007 $ from the latest data on $D_s^+$ leptonic decays. We show that properly including these radiative corrections is essential to bring the second-column CKM unitarity tests into agreement with the Standard Model expectation. The study emphasizes that the current limiting factor in confirming CKM unitarity is the precision of QED corrections, and it points out that improving lattice simulations, taking the QED corrections into account, would be desirable for a more robust confirmation.

Figures (13)

• Figure 1: Feynman diagrams for the box contributions.
• Figure 2: Feynman diagrams for the one-loop matchings onto the $c s \ell \nu$ operator. The vertex "$\otimes~\otimes$" denotes the insertion of the four-fermion operator $(\bar{s}\gamma^{\mu}P_L c)(\bar{\nu}_\ell \gamma_\mu P_L \ell)$.
• Figure 3: Vertex corrections with virtual gauge bosons attached between fermions.
• Figure 4: Vertex corrections involving the three-gauge-boson interactions.
• Figure 5: Self-energy diagrams.