{"id":"bd9afd00-1b99-475a-8350-128db4c193f2","arxiv_id":"2502.09246","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The rare decay W to Bc plus photon gets NNLO QCD corrections that reduce the width by 31 percent, giving a branching fraction of 1.522 times 10 to the minus 10 after NLL resummation.","lead":"This paper predicts the rate of an extremely rare decay of the W boson into a Bc meson and a photon, including next-to-next-to-leading-order QCD corrections and all-order resummation of large logarithms. The result, a branching fraction around 10 to the minus 10, is a testing ground for quantum chromodynamics at future colliders.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The NLL prediction rests on a single 50x50 Abel-Pade summation of a formally divergent Gegenbauer series with no stability check; a small change in the resummation scheme could move the NNLO+NLL branching fraction by more than the quoted scale uncertainty.","rationale":"The reader's weakest assumption is exactly the one I consider load-bearing: the formally divergent Gegenbauer series in eq. (4.23) is summed with a single Abel-Pade method, and no stability check against Pade order or summation scheme is presented. The paper's own text flags the divergence, and the numerical tables and figures that support the headline NLL effects depend on this summation. The strongest claim is the NNLO+NLL branching fraction 1.522e-10 and the 31% NNLO reduction; the NNLO reduction itself is not in question, but the NLL modification that yields the final quoted central value is. A 35% shift in |C1|^2 from NNLO to NNLO+NLL in table 2 is large enough that sensitivity tests are essential. I also note the wave-function normalization caveat explicitly stated by the paper (factor-of-two sensitivity), but since that is an acknowledged parametric uncertainty rather than an internal inconsistency, it is secondary. The absence of a machine-checked proof or independent numerical cross-check of the finite NNLO coefficients makes the stability check all the more important. Therefore the verdict should remain CONDITIONAL: the calculation is plausible and the fixed-order part has substantial internal consistency, but the NLL claim needs a concrete resummation-stability test before the paper can be accepted without qualification.","tokens_in":28147,"tokens_out":1762,"duration_ms":15695,"concrete_test":"Recompute the n=0 to n=100 partial sums and the Abel-Pade results for C^(NLL)_i in eqs. (4.22)-(4.23) with, e.g., a 30x30, 50x50, 70x70, and 80x80 Pade approximant, and also with a Borel-based summation or an alternative conformal mapping. If the NNLO+NLL branching fraction in table 3 changes by more than the quoted scale uncertainty (~0.028e-10) or if the NLL correction to |C1|^2 moves by more than ~5%, the resummation is not stable and the NLL claims should be downgraded; if the results stabilize, the concern is settled.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central phenomenological claim, that NLL resummation considerably alters fixed-order NRQCD predictions and significantly reduces scale dependence, is carried by the all-order resummed SDCs in eqs. (4.22) and (4.23). The paper itself states (section 4.3) that the series in eq. (4.23) is formally divergent, and the numerical results in table 2 and figures 2-3 use a single 50x50 Abel-Pade approximant from ref. [101], without any stability or convergence study. This is load-bearing because the NNLO+NLL branching fraction (1.522e-10) is obtained by subtracting a truncated eC(N)LL from the fixed-order result and adding the fully resummed C(N)LL; if the Abel-Pade summation is order-dependent or scheme sensitive, the combination rule in eqs. (4.24a)-(4.24c) can produce spurious shifts. The paper reports that NLL changes |C1|^2 from 2.89 (NNLO) to 1.89 (NNLO+NLL), a 35% shift in the leading form factor contribution, so the claim of a significant NLL effect rests essentially on this one resummation procedure. There is no demonstrated consistency between the truncated NLL expansion in eq. (4.16), the all-order Gegenbauer-space result, and the divergent-series summation, and no discussion of Pade order dependence. The reader's weakest assumption correctly identifies this as the main risk; the concern is about internal stability of the method, not just disagreement with external expectations.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28504,"tokens_out":17098,"duration_ms":145582,"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":[{"comment":"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.","section":"§4.3, eqs. (4.22)–(4.23), tables 2–3"},{"comment":"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.","section":"§3.2, table 1; §4.3, eq. (4.24)"},{"comment":"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.","section":"§5, eq. (4.22), figs. 2–3"},{"comment":"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.","section":"§5, eq. (5.1), table 3"}],"minor_comments":[{"comment":"§2: 'W-bsoson rest frame' is a typo for 'W-boson rest frame'.","section":"§2"},{"comment":"The caption of table 1 begins with the stray text 'T able 1.', and §5 contains 'Combing figure 4' instead of 'Combining figure 4'.","section":"Table 1 caption; §5"},{"comment":"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.","section":"§3.2, after eqs. (3.11)–(3.14)"},{"comment":"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.","section":"Eqs. (3.13)–(3.14)"},{"comment":"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.","section":"§4.1, eq. (4.1)"},{"comment":"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.","section":"§3.2, ref. [25]"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits JHEP's scope well as a hard QCD calculation, and the NNLO NRQCD SDC computation is a substantial new technical contribution. The required revision is well-scoped: the requested stability and internal-consistency checks do not require new physics input beyond what the authors already have in hand. One editorial note: ref. [25], on which the NLO cross-check anchors, is a 2019 arXiv preprint (arXiv:1902.11288) that appears never to have been journal-published; the authors should clarify its status, as the reproducibility of the benchmark depends on its availability. The resummation framework is largely inherited from the same group's Z-decay papers (refs. [51–53]) and from Bodwin et al. [101], but the NNLO computation for this channel is new; I do not see a novelty-disclosure problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"New here is the NNLO QCD correction to the two NRQCD short-distance coefficients for W -> Bc + gamma, plus the NLL resummation of the m_W^2/m_Bc^2 logarithms. That is real, first-time work, extending the NLO calculation of ref. [25]. The NLO cross-check against ref. [25] and the verified IR pole coefficient (gamma_Bc/4) give me reasonable confidence that the framework is sound. The analytical NLO SDCs and the convolution results in appendix B are useful and clearly presented. The numerical master integrals via AMFlow and the clean tables for the branching fractions are also to the authors' credit.\n\nThe soft spots, in order of size. The NLL claim rests on a single 50x50 Abel-Pade summation of a formally divergent Gegenbauer series, with no stability check. Because the combination rule in eqs. (4.24a)-(4.24c) subtracts the truncated NLL and adds the fully resummed one, an unreliable summation can shift |C1|^2 by more than the quoted scale uncertainty. This is not cosmetic: the paper's central qualitative point, that NLL resummation significantly alters the fixed-order predictions, depends on that one procedure. A Pade-order scan and a second resummation scheme would settle the point either way. The wave-function normalization uncertainty, which the authors themselves estimate as a factor of two in the decay rate, is stated but not propagated into the quoted branching fractions; the error bars in table 3 therefore understate the real parametric uncertainty. Separately, the NNLO finite parts are numerical only, with no independent validation. That is acceptable for a first calculation, but it would be better to publish the numerical SDCs with enough precision for others to check them. Minor typos in the text do not affect the results.\n\nOverall: the fixed-order NNLO part is solid, the resummation part is plausible but not yet demonstrated stable. This is honest progress within an established program. I would send it to a serious referee, with instructions to push hard on the Abel-Pade stability and on the wave-function uncertainty. If the authors can show the NLL result is stable under resummation-scheme variations, the paper will be a useful reference for NRQCD factorization studies.","headline":"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.","tokens_in":29043,"tokens_out":1669,"would_cite":true,"duration_ms":18864,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["W boson radiative decay","B_c meson","NNLO QCD corrections","NRQCD factorization","light-cone factorization","NLL resummation","ERBL evolution","branching fraction"],"falsifier":"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.","tokens_in":27921,"feed_emoji":"⚛️","tokens_out":12426,"duration_ms":103144,"temperature":0.7,"pith_summary":"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.","feed_headline":"Rare W to Bc photon decay rate drops 31% at NNLO","feed_subtitle":"NNLO+NLL QCD calculation fixes W→Bc+γ at 1.522e-10 and makes the prediction stable against scale choice.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the light-cone QCD factorization framework and hard-scattering kernels for exclusive radiative W and Z decays.","marker":"[24]"},{"why":"Provides the previous NLO combined light-cone and NRQCD prediction for W→Bc+γ that this paper extends to NNLO.","marker":"[25]"},{"why":"Gives the analogous NNLO and NLL resummation treatment for Z-boson radiative decays to quarkonia, used as the comparison for the impact of resummation.","marker":"[52, 53]"},{"why":"Establishes the NRQCD factorization formalism used to compute the short-distance coefficients.","marker":"[54]"},{"why":"Defines the ERBL evolution equation whose kernels drive the LL and NLL resummation.","marker":"[44-46]"},{"why":"Supplies the Abel-Padé summation method used to evaluate the formally divergent Gegenbauer series.","marker":"[101]"},{"why":"Provides the Buchmüller-Tye 1S wave function at the origin used to fix the NRQCD long-distance matrix element.","marker":"[102]"},{"why":"Provides the anomalous dimension of the NRQCD current that enters the factorization-scale dependence of the coefficients.","marker":"[55, 56]"},{"why":"Computes the two-loop master integrals numerically via auxiliary mass flow, enabling the NNLO coefficients.","marker":"[90, 91]"},{"why":"Supplies the experimental inputs (m_W, m_{B_c}, |V_{cb}|, α) used for the numerical branching-fraction predictions.","marker":"[19]"}],"fun_headline_variants":["NNLO cuts W→Bc+γ width 31%, NLL stabilizes scales","W→Bc+γ at NNLO: 31% lower width, NLL removes scale drift","Rare W to Bc photon: NNLO reduces rate 31%, NLL firms prediction","W→Bc+γ with NNLO+NLL: width down 31%, scale uncertainty tamed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["NNLO cuts W→Bc+γ width 31%, NLL stabilizes scales","W→Bc+γ at NNLO: 31% lower width, NLL removes scale drift","Rare W to Bc photon: NNLO reduces rate 31%, NLL firms prediction","W→Bc+γ with NNLO+NLL: width down 31%, scale uncertainty tamed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00056,"raw_usage":{"total_tokens":2749,"prompt_tokens":1124,"completion_tokens":1625,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":740,"completion_tokens_details":{"reasoning_tokens":1532}},"tokens_in":740,"tokens_out":1625,"duration_ms":12230,"temperature":1.0,"reasoning_tokens":1532,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T22:10:22.788564+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Optimized predictions for $W \\to B_c + \\gamma$ by combining light-cone and NRQCD approaches","cited_arxiv_id":"1902.11288","evidence_quote":"Provides the previous NLO combined light-cone and NRQCD prediction for W→Bc+γ that this paper extends to NNLO."},{"cited_title":"New approach to the resummation of logarithms in Higgs-boson decays to a vector quarkonium plus a photon","cited_arxiv_id":"1603.06793","evidence_quote":"Supplies the Abel-Padé summation method used to evaluate the formally divergent Gegenbauer series."}],"review_version":1}