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arxiv: 2510.26993 · v2 · submitted 2025-10-30 · ✦ hep-lat · hep-ex· hep-ph

Lattice Calculation of Light Meson Radiative Leptonic Decays

Peter Boyle , Norman H. Christ , Xu Feng , Taku Izubuchi , Luchang Jin , Christopher T. Sachrajda , Xin-Yu Tuo This is my paper

Pith reviewed 2026-05-18 03:09 UTC · model grok-4.3

classification ✦ hep-lat hep-exhep-ph
keywords lattice QCDradiative leptonic decayspionkaonform factorsbranching ratiosinfinite volume reconstruction
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The pith

Lattice QCD with infinite-volume reconstruction gives branching ratios for radiative pion and kaon leptonic decays that match some experiments after collinear corrections.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the branching ratios and form factors of the decays π → e ν_e γ, K → e ν_e γ and K → μ ν_μ γ directly from lattice QCD at the physical pion mass. It uses the infinite-volume reconstruction method to remove finite-volume effects from the real-photon emission matrix elements. For the electron modes the authors add the large collinear radiative corrections that arise from the small lepton mass; once these are included the pion result agrees with the PIBETA measurement, the kaon-electron result is consistent with KLOE but shows a 1.7 σ tension with E36, and the muon mode confirms earlier lattice-experiment discrepancies at large photon energy.

Core claim

Using N_f = 2 + 1 domain-wall fermions at the physical pion mass and the infinite-volume reconstruction technique, the calculation yields branching ratios and form factors for P → ℓ ν_ℓ γ (P = π, K) whose comparison with experiment, after collinear corrections for the electron channels, shows agreement with PIBETA for the pion, consistency with KLOE plus 1.7 σ tension with E36 for the kaon-electron mode, and confirmation of the ISTRA/OKA discrepancy at large photon energies for the kaon-muon mode.

What carries the argument

Infinite-volume reconstruction (IVR) method that extends finite-volume lattice data for the radiative matrix elements to infinite volume and thereby controls the dominant finite-volume effects.

If this is right

  • The results provide first-principles inputs for extracting |V_us| and |V_ud| from radiative leptonic decays once virtual-photon corrections are also included.
  • The same IVR framework can be extended to compute the virtual-photon loop contributions that complete the O(α) radiative corrections to the non-radiative leptonic decays.
  • The confirmed discrepancy at large photon energies in the muon channel constrains possible beyond-Standard-Model contributions to the form factors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the calculation is performed at the physical pion mass, the remaining systematic errors are dominated by discretization and the treatment of the photon, opening a direct path to sub-percent precision once finer lattices are used.
  • The method supplies a template for analogous calculations in heavier mesons or baryons where experimental data on radiative decays are sparse.

Load-bearing premise

The infinite-volume reconstruction method fully controls finite-volume effects for the radiative matrix elements at the quoted precision.

What would settle it

A new experimental measurement of the K → e ν_e γ branching ratio in the kinematic region where the 1.7 σ tension appears, with total uncertainty smaller than the current discrepancy, would confirm or refute the lattice prediction.

Figures

Figures reproduced from arXiv: 2510.26993 by Christopher T. Sachrajda, Luchang Jin, Norman H. Christ, Peter Boyle, Taku Izubuchi, Xin-Yu Tuo, Xu Feng.

Figure 1
Figure 1. Figure 1: For the radiative decay P + → ℓ +νℓγ, 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 Mµ in Eq. (3). amplitude is iM[P → ℓνℓγ] = − GF eVCKM √ 2 ϵµ(k, λ)Mµ (k, pℓ , pνℓ ), Mµ (k, pℓ , pνℓ ) = fP L µ (k, pℓ , pνℓ ) − H µν M (k, p)lν(pℓ , pνℓ ). (3) Here, ϵµ(k, λ) denotes the photon’s polarization vec… view at source ↗
Figure 2
Figure 2. Figure 2: Radiative corrections evaluated for (i) an inclusive treatment of the second photon [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Idea of the IVR method: Reconstruct the infinite-volume hadronic function from finite [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: To estimate finite-volume effects, we consider two intermediate states in the decay [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Results for R (π) γ and R (K) γ on the 48I ensemble. The time integral is calculated in the range t ≥ −ts. The green points correspond to the results without finite-volume correction, whereas the red and blue points represent results with correction δ IVR pt and δ IVR SD , respectively. The blue bands indicate the fits to the plateau region ts ∈ [2.2 fm, 2.8 fm] of the results with δ IVR SD correction. Reg… view at source ↗
Figure 6
Figure 6. Figure 6: Continuum extrapolation of R (π) γ in four phase space regions. The lattice results are calculated by including the O(α 2Le) radiative correction, as defined in Eq. (14). For comparison, the results from PIBETA experiment are also shown [3]. and the structure-dependent correction δ IVR SD is negligible relative to the statistical errors. In the case of R (K) γ , this finite-volume correction is negligible … view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the branching ratios R (π) γ for region O from lattice calculations and the PIBETA experiment [3]. Lattice results are presented both without radiative corrections (“wo. RC” as defined in Eq. (13)), using data from this work and Ref. [12], and with leading-order collinear radiative corrections (“w. O(α 2Le) RC” as defined in Eq. (14)) from this work. errors, indicating that discretization eff… view at source ↗
Figure 8
Figure 8. Figure 8: Continuum extrapolation of R (K) γ in phase-space region 1–5. Results are shown for (i) inclusive with respect to the second photon (Eq. (14), denoted as “inclusive RC”) and (ii) with a laboratory-frame energy cut on the second photon (Eq. (16) with ⃗p lab = 100 MeV and Elab γ2,cut = 20 MeV, denoted as “RC w. Elab γ2,cut”). For comparison, measurements from the KLOE [16] and E36 [17] experiments are also s… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the branching ratios R (K) γ for region 1-5 from this work and from the lattice calculations of Ref. [12, 14], as well as the KLOE and E36 experimental measurements [16, 17]. Lattice results are presented both without radiative corrections (“wo. RC”, defined in Eq. 13) and with radiative corrections inclusive with respect to the second photon (“inclusive RC”, defined in Eq. 14) or with a labo… view at source ↗
Figure 10
Figure 10. Figure 10: Continuum-extrapolated values of the differential branching ratio [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Continuum extrapolated values of the ratio [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Form factors FA and FV for the pion. The left panels show the results in the 48I and 64I ensembles. The right panels show the continuum-extrapolated results. The dashed lines representing the O(p 4 ) χPT predictions [32]. For comparison, we also show the results of previous lattice calculations in Ref. [11] (denoted as “Rome-Southampton 21”). We choose 128 uniformly spaced values of xγ selected within the… view at source ↗
Figure 13
Figure 13. Figure 13: Form factors FA and FV for the kaon. The left panels show the results in the 48I and 64I ensembles. The right panels show the continuum-extrapolated results. The dashed lines representing the O(p 4 ) χPT predictions [1]. For comparison, we also show the results of previous lattice calculations in Ref. [11] (denoted as “Rome-Southampton 21”) and Ref. [14] (denoted as “Rome-Southampton 25”). We choose 128 u… view at source ↗
Figure 1
Figure 1. Figure 1: In deriving the contribution from diagram B, we restrict our consideration to the [PITH_FULL_IMAGE:figures/full_fig_p049_1.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of lattice data with the point-particle approximation (dashed lines) and [PITH_FULL_IMAGE:figures/full_fig_p052_14.png] view at source ↗
read the original abstract

In this work, we perform a lattice QCD calculation of the branching ratios and the form factors of radiative leptonic decays $P \to \ell \nu_\ell \gamma$ ($P = \pi, K$) 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 $\ln(m_\pi^2/m_e^2)$ 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 $\pi \to e\nu_e\gamma$ agrees with the PIBETA measurement; for \(K \to e\nu_e\gamma\), our results are consistent with the KLOE data and exhibit a $1.7\sigma$ tension with E36; and for $K \to \mu\nu_\mu\gamma$, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript reports a lattice QCD calculation of branching ratios and form factors for the radiative leptonic decays P → ℓ ν_ℓ γ (P = π, K) on N_f=2+1 domain-wall fermion ensembles at the physical pion mass. The authors adopt the infinite-volume reconstruction (IVR) method to control finite-volume effects, compute the relevant hadronic matrix elements, and include collinear radiative corrections (enhanced by large logarithms) for electron final states. After these corrections, the π → e ν_e γ result agrees with the PIBETA measurement; K → e ν_e γ is consistent with KLOE and shows 1.7σ tension with E36; K → μ ν_μ γ confirms prior lattice-experiment discrepancies at large photon energies.

Significance. If the numerical results and error budgets hold, the work supplies the first lattice results for these radiative decays that incorporate real-photon emission via IVR and addresses O(10%) collinear corrections at the precision needed for current experiments. It constitutes a concrete step toward a complete lattice treatment of radiative corrections to leptonic decays and provides falsifiable predictions that can be tested against existing and forthcoming data from PIBETA, KLOE, E36, and ISTRA/OKA.

major comments (1)
  1. Abstract and the paragraph introducing the IVR method: the central numerical claims (post-correction agreement with PIBETA at the level required to resolve O(10%) collinear logs, plus the quoted 1.7σ tension statements) rest on the assertion that IVR fully removes finite-volume contamination from the hadronic tensor with real-photon emission. No explicit multi-volume convergence test or direct comparison of IVR-reconstructed versus large-volume results for the same radiative form-factor combination is reported. Because radiative decays introduce additional infrared structure and momentum-dependent photon propagators, the volume dependence may differ from the non-radiative case; this must be demonstrated at the target precision before the experimental comparisons can be considered robust.
minor comments (2)
  1. The abstract states that the study is a 'first step toward a complete calculation … including both real photon emissions and virtual photon loops,' yet the manuscript does not outline the planned extension to virtual loops or quantify the remaining systematic uncertainty from their omission.
  2. Error budgets and fit details for the form factors and branching ratios are referenced but not shown in the provided text; a dedicated table or section summarizing statistical, systematic, and discretization uncertainties would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that requires clarification regarding the robustness of the IVR method in the radiative case. We respond to the single major comment below and indicate the changes we will make.

read point-by-point responses

  1. Referee: [—] Abstract and the paragraph introducing the IVR method: the central numerical claims (post-correction agreement with PIBETA at the level required to resolve O(10%) collinear logs, plus the quoted 1.7σ tension statements) rest on the assertion that IVR fully removes finite-volume contamination from the hadronic tensor with real-photon emission. No explicit multi-volume convergence test or direct comparison of IVR-reconstructed versus large-volume results for the same radiative form-factor combination is reported. Because radiative decays introduce additional infrared structure and momentum-dependent photon propagators, the volume dependence may differ from the non-radiative case; this must be demonstrated at the target precision before the experimental comparisons can be considered robust.

    Authors: We agree that an explicit demonstration of IVR convergence for the radiative matrix elements would strengthen the manuscript. The IVR procedure reconstructs the infinite-volume hadronic tensor from the finite-volume lattice data by subtracting the known long-distance contribution (computed from the physical form factors and dispersion relations) and adding back the infinite-volume counterpart; the real photon is treated as an external leg whose propagator is evaluated in infinite volume. This structure is formally independent of the non-radiative case, yet the additional infrared sensitivity introduced by the photon momentum could in principle alter the residual volume dependence. Because our ensembles are at a single physical volume, a direct multi-volume comparison at the physical pion mass is not available. In the revised manuscript we will (i) add a dedicated paragraph in the IVR section explaining why the reconstruction remains valid for the radiative tensor, (ii) present a numerical test on a smaller-volume ensemble (where both direct and IVR results can be compared) for a representative set of kinematics, and (iii) quantify the residual finite-volume uncertainty that remains after IVR. These additions will be placed before the experimental comparisons so that the quoted agreement and tension statements rest on a more explicitly validated error budget. revision: partial

Circularity Check

0 steps flagged

No circularity: direct lattice matrix elements with external IVR extension

full rationale

The derivation computes branching ratios and form factors from explicit lattice QCD matrix elements on RBC/UKQCD physical-mass ensembles. The IVR method is adopted to extend data to infinite volume and control finite-volume effects, but the paper presents this as a technical reconstruction step applied to computed correlators rather than a definition that presupposes the target observables. Collinear radiative corrections are included after the lattice computation and compared to external experimental data (PIBETA, KLOE, etc.); these comparisons are post-hoc validations, not inputs to the lattice pipeline. No self-definitional loops, fitted-input predictions, or load-bearing self-citations that reduce the central claims to the paper's own fitted parameters appear in the provided derivation chain. The calculation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on standard lattice QCD assumptions plus the effectiveness of the IVR method for controlling finite-volume effects in radiative matrix elements. No new particles or forces are introduced.

axioms (2)
  • domain assumption Domain-wall fermion ensembles generated by RBC/UKQCD at physical pion mass accurately represent QCD in the continuum limit after extrapolation.
    Invoked when stating use of Nf=2+1 physical-mass ensembles.
  • domain assumption The infinite-volume reconstruction method extends lattice data to infinite volume without introducing uncontrolled systematic errors at the target precision.
    Central to the claim that finite-volume effects are effectively controlled.

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Forward citations

Cited by 1 Pith paper

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