[Paper Review] Real photon emissions in leptonic decays

Giulia Maria de Divitiis, Antonio Desiderio+12 more|arXiv (Cornell University)|Aug 27, 2019

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper presents a non-perturbative, $O(a)$-improved lattice QCD calculation of form factors governing radiative leptonic decays $P \to \ell\bar{\nu}_\ell\gamma$ for pseudoscalar mesons (pion, kaon, $D$, $D_s$). By separating the infrared-divergent point-like contribution from structure-dependent, infrared-safe terms, it enables model-independent $O(\alpha_{\text{em}})$ predictions for leptonic decay rates, significantly improving precision in CKM matrix element determinations and offering the first first-principles access to hard-photon emission in heavy mesons.

ABSTRACT

We present a non-perturbative calculation of the form factors which contribute to the amplitudes for the radiative decays $P o \ell \bar ν_\ell γ$, where $P$ is a pseudoscalar meson and $\ell$ is a charged lepton. Together with the non-perturbative determination of the virtual photon corrections to the processes $P o \ell \bar ν_\ell$, this will allow accurate predictions to be made at $O(α_{em})$ for leptonic decay rates for pseudoscalar mesons ranging from the pion to the $B$ meson. We are able to separate unambiguously the point-like contribution, the square of which leads to the infrared divergence in the decay rate, from the structure dependent, infrared-safe, terms in the amplitude. The fully non-perturbative, $O(a)$ improved calculation of the inclusive leptonic decay rates will lead to significantly improved precision in the determination of the corresponding Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Precise predictions for the emission of a hard photon are also very interesting, especially for the decays of heavy $D$ and $B$ mesons for which currently only model-dependent predictions are available to compare with existing experimental data.

Motivation & Objective

• To provide a non-perturbative, model-independent calculation of form factors for radiative leptonic decays $P \to \ell\bar{\nu}_\ell\gamma$ at $O(\alpha_{\text{em}})$.
• To separate the infrared-divergent point-like contribution from structure-dependent, infrared-safe terms in the amplitude.
• To enable precise predictions of leptonic decay rates for pseudoscalar mesons from pion to $B$ meson, including hard-photon emission.
• To improve the precision of Cabibbo-Kobayashi-Maskawa (CKM) matrix element determinations by including non-perturbative electromagnetic corrections.
• To provide the first first-principles access to structure-dependent corrections in radiative decays of heavy $D$ and $B$ mesons, previously only available via model-dependent estimates.

Proposed method

• Using $2+1+1$ twisted mass fermions on lattice QCD gauge configurations with three lattice spacings ($a \approx 0.0619$, 0.00815, 0.0085 fm) and unphysical light-quark masses ($m_{ud}^{\overline{\text{MS}}}(2\,\text{GeV}) = 11.7\,\text{MeV}$).
• Computing the hadronic matrix elements $\langle P(\bm{p})| \mathtt{T}\{j_W^\alpha(0) j_{\text{em}}^\mu(y)\} |0 \rangle$ via time-correlation functions to extract form factors.
• Employing $O(a)$-improved vector and axial currents and implementing time-reversal symmetry to enhance signal-to-noise in the correlation functions.
• Using effective-mass and residue analyses of pseudoscalar-pseudoscalar and pseudoscalar-axial two-point functions to extract meson masses, decay constants $f_P$, and matrix elements.
• Defining ratios $R_{A,V}(t)$ of three-point to two-point functions to isolate the form factors $F_A$ and $F_V$, with plateaus identified in the region $0 \ll t \ll T/2$.
• Subtracting the point-like contribution $2f_P/(m_P x_\gamma)$ from $R_A(x_\gamma)$ to extract the structure-dependent $F_A(x_\gamma)$, and comparing with chiral perturbation theory predictions.

Experimental results

Research questions

• RQ1 Can non-perturbative lattice QCD accurately compute the form factors for $P \to \ell\bar{\nu}_\ell\gamma$ decays across the full kinematic range?
• RQ2 To what extent can the structure-dependent, infrared-safe contributions to the amplitude be separated from the point-like, infrared-divergent term?
• RQ3 How do the extracted form factors for $K$, $D$, and $D_s$ mesons compare with chiral perturbation theory predictions at $O(p^6)$?
• RQ4 What is the magnitude and momentum dependence of the structure-dependent contributions in radiative decays of heavy mesons, particularly for $E_\gamma \lesssim 400\,\text{MeV}$?
• RQ5 Can this method provide model-independent, first-principles predictions for hard-photon emission in $D$ and $B$ meson decays, previously only accessible via phenomenological models?

Key findings

• The point-like contribution $2f_P/(m_P x_\gamma)$ dominates the form factor $R_A(x_\gamma)$ for both $K$ and $D_s$ mesons across the explored $x_\gamma$ range.
• The structure-dependent form factor $F_A(x_\gamma)$ for the $K$ meson is consistent with the chiral perturbation theory prediction $F_A(x_\gamma) = 8m_K(L_9^r + L_{10}^r)/f_K \approx \text{const.}$, with $L_9^r + L_{10}^r \simeq 0.0017$.
• The $F_V(x_\gamma)$ form factor for the $K$ meson agrees with the chiral perturbation theory prediction $F_V(x_\gamma) = m_K/(4\pi^2 f_K)$, indicating consistency with low-energy theorems.
• For the $D_s$ meson, $F_A(x_\gamma)$ and $F_V(x_\gamma)$ are extracted for $0 \leq x_\gamma \leq 0.4$ ($E_\gamma \lesssim 400\,\text{MeV}$), covering the physical range relevant for experimental comparisons.
• The method successfully isolates structure-dependent contributions from the infrared-divergent point-like term, enabling precise, non-perturbative access to $O(\alpha_{\text{em}})$ corrections.
• Preliminary results show good signal quality and plateau regions in $R_{A,V}(t)$, supporting the feasibility of extending the analysis to the full kinematic range and to bottom mesons in future work.