Lattice Calculation of Light Meson Radiative Leptonic Decays

TL;DR

The paper tackles radiative leptonic decays of light pseudoscalars by performing first-principles lattice QCD calculations with $N_f=2+1$ domain-wall fermions at the physical pion mass. It applies the infinite-volume reconstruction (IVR) technique to obtain infinite-volume hadronic matrix elements from finite-volume data, enabling precise determination of the form factors $F_V$ and $F_A$ across the full photon-kinematic range and the associated branching ratios. A key finding is that $O(\alpha^2)$ collinear radiative corrections are essential for electron channels, significantly affecting the comparison with experimental data and resolving discrepancies in $\pi\to e\nu_e\gamma$ while clarifying tensions in $K\to e\nu_e\gamma$ and $K\to \mu\nu_\mu\gamma$. The results are consistent with previous lattice calculations in the no-RC limit and align with KLOE data once RCs are applied, though some tensions with E36 and ISTRA/OKA persist, underscoring the need for the complete $O(\alpha)$ treatment and cross-checks via IVR. Overall, the work provides a robust framework for precise hadronic input to radiative corrections and CKM analyses, with implications for $V_{ud}$ and $V_{us}$ determinations and unitarity tests.

Abstract

In this work, we perform a lattice QCD calculation of the branching ratios and the form factors of radiative leptonic decays $P \to \ell ν_\ell γ$ ($P = π, K$) using $N_f=2+1$ domain wall fermion ensembles generated by the RBC and UKQCD collaborations at the physical pion mass. We adopt the infinite volume reconstruction (IVR) method, which extends lattice data to infinite volume and effectively controls the finite volume effects. This study represents a first step toward a complete calculation of radiative corrections to leptonic decays using the IVR method, including both real photon emissions and virtual photon loops. For decays involving a final state electron, collinear radiative corrections, enhanced by the large logarithmic factors such as $\ln(m_π^2/m_e^2)$ and $\ln(m_K^2/m_e^2)$, can reach the level of $O(10\%)$ and are essential at the current level of theoretical and experimental precision. After including these corrections, our result for $π\to eν_eγ$ agrees with the PIBETA measurement; for \(K \to eν_eγ\), our results are consistent with the KLOE data and exhibit a $1.7σ$ tension with E36; and for $K \to μν_μγ$, where radiative corrections are negligible, our results confirm the previously observed discrepancies between lattice results and the ISTRA/OKA measurements at large photon energies, and with the E787 results at large muon photon angles.

Figures (14)

• Figure 1: For the radiative decay $P^+\to \ell^+\nu_\ell\gamma$, the photon is emitted either from the initial-state meson (Diagram A) or from the final-state lepton (Diagram B). The diagrams correspond to the two terms in $\mathcal{M}^\mu$ in Eq. \ref{['amp']}.
• Figure 2: Radiative corrections evaluated for (i) an inclusive treatment of the second photon (denoted as "inclusive w.r.t $\gamma_2$") and (ii) an angle-independent, laboratory-frame photon-energy cutoff $E_{\gamma_2}^{\text{lab}} < E_{\gamma_2,\text{cut}}^{\text{lab}}$. Left: Dependence of the correction for case (ii) on the kaon momentum $\vec{p}^{~\text{lab}}$ at fixed $E_{\gamma_2,\text{cut}}^{\text{lab}} = 20$ MeV; the KLOE momentum range is indicated in orange. Right: The blue solid line shows the correction for case (ii) as a function of $E_{\gamma_2,\text{cut}}^{\text{lab}}$ at fixed ${p}^{\text{lab}} = 100$ MeV, while the red dotted line shows the correction for case (i). The left and right vertical axes display, respectively, the value of the correction and its percentage relative to the PDG branching ratio $R_\gamma^{\text{PDG}} = 1.62(22)\times 10^{-6}$ParticleDataGroup:2024cfk.
• Figure 3: Idea of the IVR method: Reconstruct the infinite-volume hadronic function from finite-volume lattice data via (i) a temporal reconstruction (IVR) and (ii) a spatial reconstruction ($\delta^{\mathrm{IVR}}$).
• Figure 4: To estimate finite-volume effects, we consider two intermediate states in the decay $P\to e\nu_e\gamma$. In the first case (diagram A), the process $P\to P\gamma \to e\nu_e\gamma$ occurs with time ordering $t < 0$. In the second case (diagram B), the process $P\to 2Pe\nu_e\to e\nu_e\gamma$ occurs with time ordering $t > 0$.
• Figure 5: Results for $R_\gamma^{(\pi)}$ and $R_\gamma^{(K)}$ on the 48I ensemble. The time integral is calculated in the range $t\geq -t_s$. The green points correspond to the results without finite-volume correction, whereas the red and blue points represent results with correction $\delta^{\text{IVR}}_{\text{pt}}$ and $\delta^{\text{IVR}}_{\text{SD}}$, respectively. The blue bands indicate the fits to the plateau region $t_s\in[2.2~\text{fm},\,2.8~\text{fm}]$ of the results with $\delta^{\text{IVR}}_{\text{SD}}$ correction.