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REVIEW 4 major objections 6 minor 110 references

Rare $W \to B_c + \gamma$ decay up to the NNLO and NLL accuracy in QCD

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that the rare W→Bc+γ decay rate is reduced by about 19% at NLO and 31% at NNLO, and that NLL resummation brings the branching fraction to about 1.5×10^{-10} while taming scale dependence.

desk verdict A solid first-time NNLO NRQCD calculation with a plausible but under-tested NLL resummation; the Abel-Pade load-bearing step needs a stability check before the phenomenology is taken at face value. read the letter →

arxiv 2502.09246 v2 pith:K5WNKJGA submitted 2025-02-13 hep-ph

Xin-Qiang Li , Ya-Dong Yang , Yu-Dong Zhang , Dong-Hui Zheng This is my paper · ORCID
classification hep-ph
keywords WbosonradiativedecayB_cmesonNNLOQCDcorrectionsNRQCDfactorizationlight-coneNLLresummationERBLevolutionbranchingfraction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper computes the extremely rare radiative decay of a $W$ boson into a $B_c$ meson plus a photon through next-to-next-to-leading order in QCD, and finds that the higher-order corrections are large and negative: about $-19\%$ at next-to-leading order and about $-31\%$ at NNLO relative to leading order. Because the $W$ mass and $B_c$ mass are far apart, the calculation also resums the large logarithms of their ratio to next-to-leading logarithmic accuracy, which makes the prediction much less sensitive to the renormalization scale. The final branching fraction is around $1.5\times10^{-10}$. This matters because no radiative hadronic $W$ decay has been observed yet, so a precise Standard Model benchmark is what future collider searches will compare against.

What carries the argument

The amplitude is decomposed into two form factors via $\mathcal{A}_\lambda = F_1\, \varepsilon_W\cdot\varepsilon_\gamma^* + (F_2/m_W^2)\, i\epsilon^{\mu\nu\alpha\beta}\varepsilon_{W,\mu}\varepsilon_{\gamma,\nu}^* p_\alpha k_\beta$. The hard part is computed in NRQCD factorization, whose short-distance coefficients are then refactorized as convolutions of hard-scattering kernels with the $B_c$ light-cone distribution amplitude; the ERBL evolution equation, diagonalized in Gegenbauer moment space, resums the large logarithms, and because the infinite Gegenbauer sum is formally divergent, the paper uses a $50\times50$ Abel-Padé approximant to evaluate it. This combination is what turns fixed-order NNLO coefficients into NLL-resummed predictions.

What would settle it

Recompute the NLL-resummed short-distance coefficients with Abel-Padé approximants of different sizes, such as $25\times25$ and $75\times75$, and check whether the NNLO+NLL branching fraction stays inside the paper's quoted scale and CKM uncertainties; if it moves outside them, the resummation is not stable. An independent analytic or high-precision numerical evaluation of the $O(\alpha_s^2)$ coefficients would also settle whether the $31\%$ NNLO reduction is correct.

Watch

Extended reading notes

Core claim

Within nonrelativistic QCD factorization, the two form factors $F_1$ and $F_2$ of $W^+\to B_c^+\gamma$ are computed through $O(\alpha_s^2)$, and the resulting NRQCD short-distance coefficients are shown to contain large logarithms of $m_W^2/m_{B_c}^2$. Refactorizing those coefficients in light-cone factorization and evolving the $B_c$ light-cone distribution amplitude with the ERBL equation lets the paper resum $\alpha_s^n\ln^n$ and $\alpha_s^{n+1}\ln^n$ contributions to all orders. The central quantitative result is that the NLO and NNLO corrections each reduce the decay width, by roughly $19\%$ and $31\%$ relative to LO, and that the NLL-resummed branching fraction is about $1.522\times10^{-10}$ at the default inputs, with the NLL resummation substantially reducing the renormalization-scale dependence compared with fixed order. The branching fraction also decreases monotonically as $m_c$ increases and increases monotonically as $m_b$ increases.

Load-bearing premise

The NLL numbers depend on a particular mathematical summation trick applied to an infinite series that does not converge; if the trick's answer changes when the series is truncated differently, the NLL-resummed predictions change.

Editorial extensions

If this is right

  • If the NNLO and NLL results are correct, fixed-order LO or NLO predictions overestimate the $W\to B_c+\gamma$ rate by up to about a third, so future searches should compare against the resummed number.
  • At a future collider producing $O(10^{12})$ $W$ bosons, the predicted branching fraction of roughly $1.5\times10^{-10}$ would translate into hundreds of such decays before reconstruction efficiencies.
  • The NLL resummation reduces the renormalization-scale dependence, making the central value more stable than any fixed-order NRQCD prediction alone.
  • The branching fraction falls monotonically as $m_c$ grows and rises monotonically as $m_b$ grows, so heavy-quark mass inputs matter at the tens-of-percent level.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: the same combined NRQCD-plus-light-cone refactorization could be applied to other exclusive radiative $W$ and $Z$ decays with a large mass hierarchy, such as $W\to D_s+\gamma$, to resum their logarithms to NLL as well.
  • Extension: the formally divergent Gegenbauer series makes the NLL prediction sensitive to the Abel-Padé order; a Borel-type or direct moment-space resummation would provide a quantitative cross-check that the paper does not include.
  • Extension: because the branching fraction depends monotonically on the heavy-quark masses, a future measurement could in principle constrain $m_c$ or $m_b$ once the wave-function normalization is controlled.
  • Extension: the paper leaves the $O(v^2)$ relativistic corrections and $O(m_{B_c}^2/m_W^2)$ power corrections unquantified; those corrections set the floor for how reliable the nominal branching fraction is.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This manuscript computes the rare radiative decay W+ → Bc+ + γ in QCD at NNLO in the NRQCD factorization formalism, then refactorizes the NRQCD short-distance coefficients (SDCs) in light-cone factorization to resum large logarithms ln(mW^2/mBc^2) to all orders in αs at LL and NLL accuracy using the ERBL evolution kernel. The two form factors F1 and F2 are converted into dimensionless SDCs C1 and C2; the NLO coefficients are checked against ref. [25], the NNLO coefficients are obtained numerically from roughly 380 two-loop master integrals with AMFlow, and the remaining IR pole is verified against the NRQCD anomalous dimension. The central phenomenological results are a branching fraction of 1.522×10^-10 at NNLO+NLL (table 3), an approximately 31% reduction of the LO width from NNLO corrections, and the claim that NLL resummation significantly alters the fixed-order predictions and reduces the renormalization-scale dependence. The paper also provides closed analytical results for the ERBL convolutions in appendices A and B.

Significance. If the NLL resummation claim survives scrutiny, this is the most complete prediction available for an exclusive radiative W decay and a useful demonstration that the LC+NRQCD factorization combination works beyond NLO. The paper has genuine strengths: the NLO results are cross-checked against ref. [25]; the IR pole coefficient is numerically verified to equal γBc/4 of eq. (3.6); the convolution results in appendix B are given in closed form; and the LDME is taken from a potential model rather than fitted to the target observable, so the prediction is falsifiable at FCC-hh-scale event samples. The main risk is precisely localized: the all-order NLL results that carry the headline scale-dependence and NLL-impact claims are obtained by Abel-Padé summation of a formally divergent series with a single parameter choice and no stability analysis, while the NNLO finite parts lack an independent validation. Both issues are fixable within the paper's scope.

major comments (4)
  1. [§4.3, eqs. (4.22)–(4.23), tables 2–3] The central claim of the paper—that NLL resummation considerably alters the fixed-order NRQCD predictions and significantly reduces the scale dependence—rests entirely on summing the formally divergent series of eq. (4.23) with a single 50×50 Abel-Padé approximant taken from ref. [101], and the manuscript reports no stability analysis for this summation. This is load-bearing: table 2 shows that NNLO+NLL changes |C1|^2 from 2.89 to 1.89, a 35% shift comparable in size to the NNLO correction itself, and the NNLO+NLL branching fraction in table 3 is constructed by the subtraction-and-replacement combination of eqs. (4.24a)–(4.24c); if the all-order C(N)LL_i differs from the truncated eC(N)LL_i by terms beyond the intended leading logarithms, the central shift is an artifact. I request three concrete checks: (i) a Padé-order scan (e.g., 20×20, 30×30, 40×40, 60×60) with the spread of |C1|^2, |C2|^2, |Ctot|^2, and Br reported; (ii) for the LL series in eq. (4.22a), which need not be divergent, a comparison of the Abel-Padé result against the truncated partial sums to demonstrate the method's convergence behavior; and (iii) a re-expansion of C^LL_i and C^NLL_i to O(αs^2) to verify that their m_W^2/m_Bc^2 logarithmic terms coincide with eC^LL_i and eC^NLL_i of eq. (4.16), which is the only way to confirm that the subtractions in eqs. (4.24) remove exactly the double-counted logarithms.
  2. [§3.2, table 1; §4.3, eq. (4.24)] The NNLO input to the central prediction is the set of numerical SDCs C(1,2)^(2) in table 1, obtained from roughly 380 two-loop master integrals evaluated with AMFlow, and no independent validation of these finite parts is reported. The verification described in §3.2—that the remaining IR pole equals γBc/4—constrains only the pole coefficient, not the finite terms that produce the quoted 31% reduction of the width. An internal consistency check is available and should be shown: the difference CNNLO_i − eC^NLL_i|_{αs^2} formed in eq. (4.24c) must be independent of ln(m_W^2/m_Bc^2) (apart from the μΛ-dependent pieces absorbed into the LDME), because appendix B gives the full truncated logarithmic content; verifying this simultaneously validates the two-loop finite parts and the resummation subtraction that defines the NNLO+NLL row. Without this or an equivalent cross-check, a systematic error in the two-loop integrals would propagate directly into the headline branching fraction.
  3. [§5, eq. (4.22), figs. 2–3] The mechanism by which the renormalization scale enters the NLL-resummed curves is not defined, which undermines the headline statement that NLL resummation reduces the μR dependence. The resummed SDCs of eq. (4.22) depend on αs(mW) and αs(mBc) and on hard-scattering kernels evaluated at the scale mW, while the terms of eq. (4.16) contain the same fixed hard scale; the manuscript never states what is varied when μR scans from mW/2 to 2mW for the NLO+NLL and NNLO+NLL rows. The paper must specify whether μR replaces mW in the hard-scattering logarithms, in the argument of αs, or in both, and confirm that the renormalization-group structure of eq. (3.5) is preserved order by order after the resummed combination is formed. Figures 2 and 3 and the third error in table 3 are the primary evidence for the scale-reduction claim, so this definition is required for the claim to be meaningful.
  4. [§5, eq. (5.1), table 3] The paper explicitly discloses in §5 that varying |R1S,c̄b(0)|^2 from 1.642 to 3.184 GeV^3 [102] changes the decay rate by roughly a factor of two, but this dominant parametric uncertainty is absent from every error budget in table 3, which propagates only |Vcb|, ΓW, and μR. As the NNLO+NLL branching fraction of 1.522×10^-10 is quoted with four significant digits and is used to estimate FCC-hh event counts, the LDME variation should be reported as a separate systematic uncertainty on the final row, or the headline should be explicitly framed as a prediction at fixed wave-function input. Combining this with the resummation ambiguity of the first major comment would give the phenomenological section an honest total error.
minor comments (6)
  1. [§2] §2: 'W-bsoson rest frame' is a typo for 'W-boson rest frame'.
  2. [Table 1 caption; §5] The caption of table 1 begins with the stray text 'T able 1.', and §5 contains 'Combing figure 4' instead of 'Combining figure 4'.
  3. [§3.2, after eqs. (3.11)–(3.14)] The comparison with ref. [25] uses f_2^(0), f_1^(1), and f_2^(1) without defining these symbols in the present paper; they should be identified as the corresponding SDCs of ref. [25] for the cross-check to be self-contained.
  4. [Eqs. (3.13)–(3.14)] State the branch chosen for ln(−r^2+iϵ) at the physical value r^2 ≈ 0.0061, since the imaginary parts contribute to |Ci|^2 and therefore enter the decay width.
  5. [§4.1, eq. (4.1)] The power corrections O(m_Bc^2/m_W^2) ≈ 6×10^-3 are small relative to the claimed 19% and 31% corrections; adding this numerical estimate would make the leading-twist truncation error explicit.
  6. [§3.2, ref. [25]] Reference [25] is cited in its arXiv form; since the NLO cross-check of §3.2 anchors to it, the publication status of that work should be clarified and the journal version cited if it exists.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the NNLO calculation and branching-fraction prediction are not fitted to the target observable.

full rationale

The central claim (Br around 1.5e-10, NNLO reduces the width by about 31%, and NLL resummation reduces scale dependence) is not obtained by fitting a parameter to W -> Bc + gamma. The NNLO short-distance coefficients C_i^(2) come from a genuine 186 two-loop diagram calculation evaluated with AMFlow, with quark masses, CKM elements, and Gamma_W taken from PDG and the NRQCD matrix element taken from the Buchmuller-Tye potential model. No parameter is tuned to the final branching fraction. The combination rule in eqs. (4.24a)-(4.24c) is a standard double-counting subtraction: the truncated NLL expansion is subtracted from the fixed-order result and the all-order resummed expression is added back, which is not a definitional identification of the prediction with its input. The ERBL kernels and the NLO evolution matrix are taken from the literature, and the Abel-Pade summation method is from ref. [101], not from the overlapping authors. The cited works by overlapping authors (refs. [52,53,96,98]) provide the methodology and the B_c LCDA, but they are separate published calculations; the new NNLO result does not reduce to them. The paper itself notes that the sum in eq. (4.23) is formally divergent and uses a 50x50 Abel-Pade approximant, which is a legitimate robustness concern about the resummation scheme, but it is not circularity because the resummation result is not being used as an input that is then re-derived as the prediction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The prediction relies on two established factorization frameworks, standard evolution kernels from the literature, and several external inputs: a potential-model wave function, pole quark masses, and scale choices. The only genuinely ad hoc element is the Abel-Pade summation of the divergent Gegenbauer series, which is load-bearing for the NLL result. No new particles, forces, or entities are introduced.

free parameters (5)
  • NRQCD LDME from Buchmuller-Tye wave function |R(0)|^2 = 1.642 GeV^3
    Nonperturbative input from a potential model in ref. [102]; the paper notes varying it to 3.184 GeV^3 changes the rate by roughly a factor of two, so this parameter dominates the normalization uncertainty.
  • Charm and bottom pole masses m_c, m_b = m_c = 1.69 GeV, m_b = 4.80 GeV
    Converted from MSbar running masses at two loops; they fix the momentum fraction x0 = m_c/(m_b+m_c) and enter the kinematics.
  • Factorization scale mu_Lambda = 1 GeV
    Scale at which the NRQCD LDME and the Bc LCDA are evaluated; the mu_Lambda dependence cancels between SDCs and LDME at fixed order.
  • Renormalization scale mu_R = m_W = 80.3692 GeV (default)
    Default scale; varied over m_W/2 to 2m_W to estimate uncertainty.
  • Abel-Pade order = 50x50
    Parameter used to sum the formally divergent Gegenbauer series in eq. (4.23); the result depends on this choice, with no stability cross-check.
assumptions (6)
  • domain assumption NRQCD factorization at leading order in the heavy-quark relative velocity v
    Invoked in section 3.1 to factorize the W to Bc plus gamma amplitude into short-distance coefficients and a NRQCD matrix element; the leading-power approximation m_Bc ~ m_b + m_c is used.
  • domain assumption Light-cone factorization at leading twist for the NRQCD SDCs
    Invoked in section 4.1 to refactorize the SDCs as convolutions of hard-scattering kernels with the Bc leading-twist LCDA, dropping power corrections of O(m_Bc^2/m_W^2).
  • standard math ERBL evolution equation for the Bc LCDA controls the resummation
    Used in section 4.2 to evolve the LCDA from m_Bc to m_W; the one- and two-loop kernels are taken from refs. [44-46, 104-110].
  • standard math Strategy of regions isolates the hard contribution in full QCD matching
    Used in section 3.2 to extract NRQCD SDCs by computing only the hard region of loop integrals; soft and collinear regions are assumed to cancel against NRQCD matrix elements.
  • ad hoc to paper The formally divergent Gegenbauer series in eq. (4.23) is summable by Abel-Pade
    Section 4.3 states the sum is formally divergent and applies a 50 by 50 Pade approximant; the NLL predictions depend on this choice, which is not cross-checked.
  • domain assumption The Bc meson mass is approximated by m_b + m_c
    Used throughout at leading order in v, for example in eq. (3.7); sets the momentum fractions x0 and 1 minus x0.

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Cite this review

Pith. "Pith review of Rare $W \to B_c + \gamma$ decay up to the NNLO and NLL accuracy in QCD." pith.science (2026). https://pith.science/paper/K5WNKJGA

@misc{pith2026250209246,
  author       = {Pith},
  title        = {Pith review of: Rare $W \to B_c + \gamma$ decay up to the NNLO and NLL accuracy in QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5WNKJGA}},
  note         = {Machine review of arXiv:2502.09246}
}
abstract

We perform a detailed theoretical study of the rare radiative decay of the $W$ boson into a $B_c$ meson and an on-shell photon. The decay amplitude is described by two independent form factors, which are calculated up to the next-to-next-to-leading order (NNLO) in QCD within the nonrelativistic QCD (NRQCD) factorization formalism. Since the two typical energy scales, the $W$-boson mass $m_W$ and the $B_c$-meson mass $m_{B_c}$, involved in the process are widely separated, large logarithms of $m_W^2/m_{B_c}^2$ present in the NRQCD short-distance coefficients are also resummed to all orders in $\alpha_s$ up to the next-to-leading logarithmic (NLL) accuracy, by employing the light-cone factorization approach. Taking into account all these corrections, we then perform a phenomenological exploration of this rare decay. It is found that, relative to the leading-order result, the decay width of the process is reduced by the next-to-leading-order and NNLO corrections, with a net effect of $\sim19\%$ and of $\sim31\%$, respectively. Furthermore, the NLL resummation can considerably alter the fixed-order NRQCD predictions, especially for the $\mathcal{O}(\alpha_s)$ correction. We also find that the radiative corrections increase the renormalization scale dependence of the branching fraction, which is however significantly reduced by the NLL resummation. The dependence of the branching fraction on the heavy-quark masses $m_{b,c}$ is also investigated, which shows a monotonic decrease (increase) with $m_c$ ($m_b$).

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