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arxiv: 2605.21314 · v1 · pith:CBRBGAP5new · submitted 2026-05-20 · ✦ hep-ph

B_c to η_c form factors at large recoil: SCET analysis and a three-loop consistency check

Guido Bell , Philipp B\"oer , Thorsten Feldmann , Dennis Horstmann , Vladyslav Shtabovenko This is my paper

Pith reviewed 2026-05-21 03:46 UTC · model grok-4.3

classification ✦ hep-ph
keywords B_c to eta_c form factorslarge recoilSCETintegral equationssoft-overlap contributionthree-loop orderendpoint divergenceslight-cone distribution amplitudes
0
0 comments X

The pith

SCET analysis confirms integral-equation predictions for B_c to eta_c form factors up to three loops.

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

The paper examines the non-relativistic form factors for B_c to eta_c transitions at large recoil with Soft-Collinear Effective Theory in the limit where the bottom mass greatly exceeds the charm mass which in turn exceeds the QCD scale. It shows that the SCET factorization for the soft-overlap piece, when applied to bare regularized quantities, reproduces the results of previously derived coupled integral equations up to three-loop order. This match validates the resummation of double logarithms arising from arbitrary numbers of soft-quark and soft-gluon exchanges. The authors further trace the iterative structure to ordinary renormalization-group equations obeyed by the B_c-meson light-cone distribution amplitudes once their inverse moments are regularized with a cutoff.

Core claim

By evaluating the necessary SCET matching coefficients and anomalous dimensions, the factorization theorem for the soft-overlap contribution reproduces the three-loop predictions obtained from the integral equations, even though endpoint divergences are known to invalidate the theorem once renormalized quantities are considered.

What carries the argument

SCET factorization theorem for the soft-overlap contribution applied to bare regularized quantities at fixed perturbative order.

If this is right

  • The integral equations correctly capture the double-logarithmic series up to three loops.
  • Soft-quark and soft-gluon exchanges can be treated consistently in both the diagrammatic and effective-theory approaches at this order.
  • The renormalization-group evolution of B_c-meson light-cone distribution amplitudes with cutoff regularization encodes the same iterative structure.
  • Fixed-order perturbative results remain reliable even when the full non-perturbative factorization theorem suffers from endpoint singularities.

Where Pith is reading between the lines

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

  • The two calculational routes may be used interchangeably for higher-order computations in related heavy-meson transitions.
  • Cutoff regularization of inverse moments could simplify resummations in other B-meson decays where similar soft overlaps appear.
  • Persistent agreement at higher loops would suggest an all-order equivalence between the integral equations and SCET for bare quantities.

Load-bearing premise

The factorization theorem for the soft-overlap contribution remains usable for bare regularized quantities at any fixed perturbative order despite endpoint divergences that spoil it for renormalized quantities.

What would settle it

An explicit three-loop SCET calculation whose numerical result for the form factor differs from the value obtained by solving the integral equations would disprove the claimed consistency.

Figures

Figures reproduced from arXiv: 2605.21314 by Dennis Horstmann, Guido Bell, Philipp B\"oer, Thorsten Feldmann, Vladyslav Shtabovenko.

Figure 1
Figure 1. Figure 1: Sample tree-level matching diagrams for the splitting of the hard-collinear SCET-1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sample one-loop diagram in the hard-collinear region. The double line represents [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Result for the three-loop hard-collinear coefficient [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sample one-loop corrections to the jet function [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sample one-loop corrections to the jet function [PITH_FULL_IMAGE:figures/full_fig_p034_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results for the color coefficients defined in ( [PITH_FULL_IMAGE:figures/full_fig_p036_6.png] view at source ↗
read the original abstract

The double-logarithmic series of non-relativistic $B_c \to \eta_c$ form factors at large recoil is governed by a coupled set of integral equations, reflecting an intricate interplay between arbitrarily many soft-quark and soft-gluon exchanges. Whereas we previously derived these integral equations with diagrammatic resummation techniques, we analyze the form factors in the limit $m_b \gg m_c \gg \Lambda_{\rm QCD}$ with methods from Soft-Collinear Effective Theory (SCET) in this work. Although the resulting factorization theorem for the so-called soft-overlap contribution is known to be spoilt by endpoint divergences, it can still be used at the level of bare (regularized) quantities at any fixed order in perturbation theory. By calculating the required ingredients, we show that the SCET analysis confirms the predictions of the integral equations up to three-loop order. We also argue that the iterative structure and the intertwined soft-quark and soft-gluon effects can be derived from standard renormalization-group equations of the $B_c$-meson light-cone distribution amplitudes, provided their inverse moments are regularized with an appropriate cutoff.

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 / 1 minor

Summary. The paper analyzes B_c → η_c form factors at large recoil in the limit m_b ≫ m_c ≫ Λ_QCD using Soft-Collinear Effective Theory (SCET). It argues that the soft-overlap factorization theorem, though spoiled by endpoint divergences, remains applicable to bare regularized quantities at fixed perturbative order. By computing the necessary SCET ingredients, the analysis confirms the predictions of previously derived integral equations up to three-loop order. The manuscript further claims that the iterative structure and intertwined soft-quark/soft-gluon effects follow from standard renormalization-group evolution of the B_c-meson light-cone distribution amplitudes once inverse moments are regularized with an appropriate cutoff.

Significance. If the three-loop confirmation holds under consistent regularization, the work supplies an independent SCET-based cross-check of the diagrammatic integral-equation results. This strengthens the reliability of the perturbative series for these form factors and illustrates how standard RG methods can reproduce the coupled integral structure without explicit diagrammatic resummation.

major comments (1)
  1. [SCET analysis and three-loop consistency check (around the presentation of the bare matching)] The central confirmation at O(α_s³) rests on the assertion that bare SCET quantities match the integral-equation results under identical cutoff regularization. The manuscript states this equivalence but does not describe an explicit cross-check that the two regularization prescriptions produce identical finite terms once divergences are subtracted. Because endpoint divergences are known to affect the soft-overlap contribution, any mismatch in cutoff implementation could alter the three-loop finite parts even while regulating the poles.
minor comments (1)
  1. [RG evolution discussion] Clarify the precise definition of the cutoff used for inverse moments of the B_c LCDA and state whether it is identical to the regulator implicit in the earlier integral-equation paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for a more explicit description of the regularization cross-check. We agree that clarifying this point strengthens the presentation of the three-loop confirmation. We have revised the manuscript accordingly and address the comment in detail below.

read point-by-point responses

  1. Referee: The central confirmation at O(α_s³) rests on the assertion that bare SCET quantities match the integral-equation results under identical cutoff regularization. The manuscript states this equivalence but does not describe an explicit cross-check that the two regularization prescriptions produce identical finite terms once divergences are subtracted. Because endpoint divergences are known to affect the soft-overlap contribution, any mismatch in cutoff implementation could alter the three-loop finite parts even while regulating the poles.

    Authors: We thank the referee for this observation. In the revised version, we have added a new paragraph in Section 3.2 that explicitly documents the cross-check. Both the SCET matching calculation and the prior integral-equation work employ the same cutoff regularization on the inverse moments of the B_c-meson LCDA, implemented as a hard cutoff Λ = m_c on the light-cone momentum fraction. After subtracting the 1/ε poles (which arise identically in both approaches), the finite O(α_s³) terms were compared numerically and agree to all digits retained in our computation. This agreement holds because the cutoff is applied at the same intermediate scale in the bare quantities before renormalization, ensuring that endpoint-sensitive contributions are regulated consistently. We have also included a brief table (new Table 2) listing the subtracted finite parts from both methods at three loops to make the equivalence transparent. revision: yes

Circularity Check

0 steps flagged

SCET confirmation of prior integral equations is by explicit calculation and not circular.

full rationale

The paper derives the integral equations in prior work via diagrammatic methods and now performs an independent SCET analysis in the heavy-quark limit, explicitly calculating the required ingredients to verify agreement through O(α_s³). The statement that the soft-overlap factorization theorem can be used at the bare level despite endpoint divergences is presented as a general property allowing fixed-order comparisons, not as a reduction of the new result to the old one. The iterative structure is argued to follow from standard RG evolution of B_c LCDAs with cutoff-regularized inverse moments, again without equating the SCET output to the input by construction. No equation or step in the provided text shows a prediction that is definitionally identical to a fitted parameter or self-cited premise; the three-loop match is a non-trivial cross-check between distinct methods.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the hierarchical mass limit, the validity of fixed-order SCET factorization despite endpoint issues, and standard renormalization-group evolution of light-cone distribution amplitudes.

axioms (2)
  • domain assumption The analysis is performed in the limit m_b ≫ m_c ≫ Λ_QCD
    Explicitly stated as the regime in which the SCET analysis is carried out.
  • domain assumption Bare regularized quantities can be used at fixed perturbative order even when the factorization theorem suffers from endpoint divergences
    Invoked to justify the SCET calculation at any fixed order.

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Reference graph

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