{"id":"334d768a-7121-4e9c-873d-bc66f1eae44a","arxiv_id":"2603.18576","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The O(α_s^4) top-Yukawa-induced correction to H→bb is computed analytically, increasing the width by 0.4% and lowering scale uncertainty to 0.4%.","lead":"This paper computes the fourth-order QCD correction to the Higgs boson decay to bottom quarks that comes from the top-quark Yukawa coupling, including the full bottom-quark mass dependence. The new correction shifts the decay width by +0.4% and reduces the scale uncertainty to 0.4%, a precision needed for future Higgs factories.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's verdict ACCEPT is well-founded. The central claim is about the C1C1 channel at O(α_s^4), a partial result that is clearly labeled. I examined the most plausible soft spots: (1) the asymptotic truncation at O(z^{-1}); (2) possible missing Wilson-coefficient cross-terms (2 c1 c2 Δ1) at O(α_s^4); and (3) the significance of the missing C1C2 channel. (1) The truncation is safe: at z≈900, even a maximally log-enhanced O(z^{-2}) term changes Δ2 by at most a few percent, corresponding to a shift in the total width below 0.02%, far below the 0.4% effect. (2) The O(α_s^4) C1C1 contribution should include both c1^2 Δ2 and 2 c1 c2 Δ1; the numerical table (0.0087 MeV) is consistent with including both, and the known components make omission unlikely. (3) The missing C1C2 term is disclosed and does not invalidate the C1C1-specific claim. The paper's internal checks (leading-log ratios, agreement between two integration methods) support correctness. The only residual weakness is the lack of an independent numerical cross-check of the three-loop integrals, but that is a standard reason for MODERATE confidence, not a reason to reject. Therefore the verdict remains ACCEPT, unchanged.","tokens_in":13412,"tokens_out":30311,"duration_ms":252997,"concrete_test":"Evaluate the exact one-fold integral representations (eqs. 23-24) numerically at z=m_H^2/m_b^2≈900, using the given R_{\\bar b}(x) and R_{4b}(x), and compare the resulting Δ^{C1C1}_{2,\\bar b\\bar b} with the asymptotic expression (eqs. 26-27) truncated at O(z^{-1}). If the difference exceeds 0.1% of the C1C1 O(α_s^4) contribution (i.e., δΓ > 0.00001 MeV), the 0.4% claim would need revision. This directly settles the reader's weakest assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central claim. The reader's weakest assumption — that the O(z^{-1}) asymptotic truncation of the new master integrals is accurate — is quantitatively safe. Even under a worst-case O(z^{-2}) term enhanced by log^6(z) at z≈900, the fractional shift in Δ^{C1C1}_2 is at most a few percent, translating to a shift in Γ_{H→bb} below 0.02% of the total width, far smaller than the claimed 0.4% correction or the 0.4% scale uncertainty. The omitted O(z^{-2}) terms therefore cannot alter the conclusion. The calculation is internally consistent: the leading-log ratios quoted in eqs. (29)-(30) match the explicit expansions, and the two methods for the master integrals agree. The partial nature (missing C1C2 at O(α_s^4)) is clearly stated, and the label N4LO(part) is used in the tables/figures. The only residual risk is the absence of an independent check of the three-loop integrals, which the reader already captures as MODERATE confidence.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents the O(alpha_s^4) correction to the H -> b bbar decay width in the C1C1 channel, i.e., the contribution from squared amplitudes with two insertions of the top-quark-Yukawa-induced H G G effective operator, retaining the full bottom-quark mass dependence. The calculation uses the optical theorem, automated amplitude generation and reduction (FeynArts/FeynCalc/Kira), and a master-integral analysis based on differential equations. Of the 38 master integrals, 35 are taken from previous work; the three new integrals M36-M38 are solved analytically, with one elliptic sector expressed through one- and two-fold integrals of complete elliptic integrals. Numerical results are obtained from the asymptotic expansion in z = m_H^2/m_b^2 truncated at O(z^{-1}). The authors find that the new O(alpha_s^4) C1C1 term increases the N3LO width by about 0.4% and reduces the renormalization-scale uncertainty to about 0.4%, giving Gamma_{H->bb}^{MS} = 2.421 (+0.008/-0.010) +/- 0.005 MeV. The calculation is explicitly partial: the C1C2 channel at O(alpha_s^4) is not computed, and the notation 'N4LO(part)' is used in the tables and figures.","tokens_in":13719,"tokens_out":16159,"duration_ms":147988,"significance":"If correct, the result is a valuable step toward precision predictions for the dominant Higgs decay at future lepton colliders, and it provides a nontrivial application of modern multiloop technology involving elliptic sectors. The calculation is based on standard, well-tested tools and has several concrete strengths: no parameter is fitted to the target observable (the inputs are PDG values; master-integral boundary conditions are fixed by high-precision AMFlow evaluation of the same integrals and PSLQ), a cross-check against an epsilon-factorised differential-equation approach is reported, and the partial nature of the result is honestly labelled. The main limitations are that the final numerical prediction relies on asymptotic expansions whose stated accuracy is not demonstrated in the manuscript, and the independent epsilon-factorised check is deferred to a forthcoming paper; these reduce confidence but do not, in my assessment, undermine the central claim.","major_comments":[],"minor_comments":[{"comment":"The footnote star in the O(alpha_s^4) row is ambiguous. If it is intended to mark the absent C1C2 entry, the star should be placed in the empty C1C2 cell. If it is intended to mark the C2C2 entry, it contradicts Section 2 and Eq. (8), where Delta^{C2C2}_{4,bbar} is quoted as known and is used in Eq. (32). Please clarify the typesetting.","section":"Section 4, Table 1"},{"comment":"Eq. (32) is labelled 'N4LO QCD' even though the computation is partial and the text, Table 1, and Figures 2-3 use 'N4LO(part)'. Because the missing C1C2 O(alpha_s^4) term could in principle be comparable to the computed C1C1 term, the equation label should be changed or an explicit caveat added, e.g., 'N4LO(partial, C1C1 only)'.","section":"Section 4, Eq. (32)"},{"comment":"The sentence 'these asymptotic results are accurate enough for phenomenological studies since the omitted higher-power terms introduce a correction of less than 0.1%' is an assertion. Since the numerical results in Table 1 and Eq. (32) are based on the truncated O(z^{-1}) expansions, please provide a numerical comparison of the full integral representations (23)-(24) with the asymptotic forms at z ~ 897, or give a quantitative bound. I do not regard this as blocking: even a conservative O(z^{-2}) term enhanced by log^6(z) changes the total width by well below 0.02%, far smaller than the quoted 0.4% correction. But the stated 0.1% accuracy should be substantiated in the text.","section":"After Eq. (27)"},{"comment":"The phrase 'ratio of the leading logarithms' is terse and could be misread as the ratio of the full Delta functions. It is correct only if 'leading logarithm' means the coefficient of the highest power of log z at each power in z: at z^0, Delta2's highest log is the log^3 z term from the C_A^2 C_F color structure, giving Eq. (29), and at z^{-1}, the log^4 z / z term divided by Delta1's 12/z constant gives Eq. (30). Please define this terminology explicitly so the reader can verify the quoted ratios without reconstructing the expansions.","section":"Section 3, Eqs. (29)-(30)"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is technically sound and within the scope of the journal. The requested clarifications are local and do not affect my confidence in the main numerical result. I would ask the authors to respond explicitly to the asymptotic-accuracy request and to correct the Table 1/Eq. (32) labelling issues. The reliance on reference [58] for the epsilon-factorised cross-check is acceptable, but the published version should state clearly which parts of the cross-check are documented in the present paper and which are deferred."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real calculation, not a placeholder. The authors compute the previously unknown O(α_s^4) correction in the C1C1 channel to H→bb with full bottom-quark mass dependence. The genuinely new ingredients are the three-loop master integrals M36–M38, and the analytic result for Δ^{C1C1}_{2,b̄b} appears here for the first time. The numerical impact—a +0.4% shift and a scale uncertainty that drops from 0.7% to 0.4%—is credible.\n\nThe methods are standard but demanding: FeynArts/FeynCalc/Kira for amplitude generation and IBP reduction, AMFlow for boundary values, canonical differential equations with two square roots, and one-fold integral representations for the elliptic top sector. They also report agreement with an ε-factorised approach, though the details are deferred to a forthcoming paper. That deferral is the main soft spot: the independent check of the new MIs is not in this paper, so the reader has to take the cross-check on faith. This is a normal situation in multiloop work and not a red flag, but it does mean the result is not yet independently verifiable from this paper alone.\n\nThe asymptotic expansion in z=m_H^2/m_b^2, truncated at O(z^{-1}), is used for all numerics. The authors claim <0.1% accuracy. I did a quick stress-test: even a pessimistic O(z^{-2}) term enhanced by log^6(z) at z≈900 shifts the total width by well under 0.02%, so this is safe. The ratio of leading logarithms in eqs. (29)-(30) matches the explicit expansions, and the color structures are distinct, which makes the log-enhancement story internally consistent.\n\nThe paper is honest about what is missing: the C1C2 channel at O(α_s^4) is not computed, so this is N4LO(part), and the tables/figures say so. The Wilson coefficients are taken from the literature, and the bottom-quark mass dependence in the C2C2 channel is argued to be negligible. I did not find any parameter fitted to the target observable; the inputs are PDG values.\n\nWho is this for? People working on Higgs decay phenomenology and on multiloop elliptic integrals. It is not a new framework or a resolution of a long-standing puzzle; it is a substantial, expected step in an established program. It deserves a serious referee: the calculation is intricate, and the auxiliary-file results should be spot-checked. I would send it to review without hesitation.","headline":"Solid partial-N4LO result for H→bb; new three-loop integrals and a clear 0.4% effect, but the full N4LO still waits on the C1C2 channel.","tokens_in":14112,"tokens_out":2245,"would_cite":true,"duration_ms":22181,"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":"The O(α_s^4) corrections to the Higgs decay into bottom quarks induced by top-quark Yukawa couplings increase the decay width by 0.4% over the previous N3LO result and cut the scale uncertainty from 0.7% to 0.4%.","keywords":["Higgs decay","bottom quark","top-quark Yukawa","alpha_s^4","QCD corrections","effective field theory","master integrals","decay width"],"falsifier":"Numerically integrate the exact integral representations for the master integrals M37 and M38 at $z \\approx (m_H/m_b)^2 \\approx 900$, evaluating the one-fold and two-fold integrals with complete elliptic functions, and compare these values to the $O(z^{-1})$ asymptotic expressions used in the paper; any discrepancy above roughly 0.1% would mean the central correction is not reliable.","tokens_in":13348,"feed_emoji":"⚛️","tokens_out":7335,"duration_ms":61591,"temperature":0.7,"texified_at":"2026-08-05T21:01:05.876301+00:00","pith_summary":"Higgs decay to bottom quarks is the dominant Standard-Model Higgs decay and the cleanest handle on the bottom-quark Yukawa coupling, so its theory prediction must match the few-per-mil precision planned at future lepton colliders. The authors compute the previously missing $O(\\alpha_s^4)$ correction to this width that comes from two insertions of the gluonic effective operator (the C1C1 channel), retaining the full bottom-quark mass dependence. They find that this correction increases the decay width by 0.4% relative to the N3LO result – larger than the 0.21% experimental precision expected at a Higgs factory – and reduces the renormalization-scale uncertainty from 0.7% to 0.4%. The paper argues that the full C1C2 channel remains, but the C1C1 contribution alone improves the prediction to $\\Gamma = 2.421^{+0.008}_{-0.010}(\\text{scale}) \\pm 0.005(\\alpha_s)$ MeV in the MS scheme.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":6668,"prompt_tokens":842,"completion_tokens":5826,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":842,"completion_tokens_details":{"reasoning_tokens":5021}},"feed_headline":"Top-Yukawa terms add 0.4% to the Higgs→bb width","feed_subtitle":"The 0.4% shift exceeds the projected 0.21% precision of a future Higgs factory.","key_machinery":"The argument rests on the effective Lagrangian $\\frac{H}{v} (C_1 O_1 + C_2 O_2)$ with $O_1 = G_{a,\\mu\\nu}^2$ and $O_2 = m_b \\bar{b} b$, where the Wilson coefficients $C_1, C_2$ carry the decoupled top-quark effects. The width is split as $C_2C_2 + C_1C_2 + C_1C_1$, and the $C_1C_1$ piece is computed through the optical theorem. The key new objects are the master integrals M36–M38: M36 forms a canonical basis of multiple polylogarithms, while M37 and M38 satisfy first-order differential equations whose solutions are one-fold integrals of complete elliptic integrals (and two-fold for the $b\\bar{b} b\\bar{b}$ cut). The asymptotic expansion in $z = m_H^2/m_b^2$, truncated at $O(z^{-1})$, exposes double logarithms such as $\\log^2(z)$ and $\\log^4(z)$ t","core_discovery":"The central claim is that at fourth order in the strong coupling, the top-Yukawa-induced C1C1 channel contributes a +0.4% correction to $\\Gamma(H \\to bb)$ over the N3LO prediction, and that including it reduces the scale uncertainty to 0.4%. The calculation is analytic: the decay width is decomposed via effective operators, the forward-scattering amplitude is expanded in master integrals, and the three new integrals M36–M38 are solved, two of them expressed as one-fold integrals of complete elliptic integrals (and two-fold for the four-bottom final state). Using the asymptotic expansion in $z = m_H^2/m_b^2$, the paper obtains a closed expression for the correction and evaluates it numerically, giving the M","pith_inferences":["If the missing C1C2 channel at O(α_s^4) turns out to be comparable in size to the C1C1 one, the complete N4LO correction could be close to a full percent, strengthening the case for resumming the subleading-power logarithms before comparing to Higgs-factory data.","The success of solving the elliptic top-sector master integrals via ϵ-factorised differential equations suggests the same approach could be applied to other multiscale QCD processes where elliptic and polylogarithmic sectors mix.","The paper's reliance on the O(z^{-1}) asymptotic expansion could be checked by comparing it with the exact numerical evaluation of the one-fold/two-fold integrals, offering a direct test of the 0.4% claim without waiting for a full next-order calculation."],"forward_implications":["The 0.4% correction exceeds the 0.21% experimental precision projected at a future lepton collider, so it must be included when extracting the bottom-quark Yukawa coupling from a width measurement.","The renormalization-scale uncertainty of the partial N4LO width drops from 0.7% to 0.4%, making the prediction competitive with the planned measurement accuracy.","A measured width with 0.21% uncertainty would determine the bottom-quark mass to about 0.36% precision when combined with this prediction.","The sizeable logarithms in z reveal a slowly convergent series for the top-induced part, implying that a full N4LO result (including the missing C1C2 channel) is needed to reach sub-percent theoretical accuracy.","The analytic structure of the correction, including the elliptic master integrals, provides a benchmark for developing all-order resummation at subleading power."],"fun_headline_variants":["Top-Yukawa shifts Higgs→bb width by 0.4%","0.4% top-quark effect beats Higgs factory precision","New α_s^4 term cuts Higgs width uncertainty to 0.4%","Top-Yukawa effect at α_s^4 adds 0.4% to Higgs→bb","0.4% from top-Yukawa narrows Higgs→bb width uncertainty"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The numerical results assume that the expansion in $m_H^2/m_b^2$ truncated at $O(z^{-1})$ is accurate to better than 0.1%; if the omitted higher-power terms are actually larger, the claimed 0.4% correction could shift significantly.","fun_headline_variants_meta":{"raw":{"variants":["Top-Yukawa shifts Higgs→bb width by 0.4%","0.4% top-quark effect beats Higgs factory precision","New α_s^4 term cuts Higgs width uncertainty to 0.4%","Top-Yukawa effect at α_s^4 adds 0.4% to Higgs→bb","0.4% from top-Yukawa narrows Higgs→bb width uncertainty"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00081,"raw_usage":{"total_tokens":3387,"prompt_tokens":736,"completion_tokens":2651,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":2546}},"tokens_in":480,"tokens_out":2651,"duration_ms":19400,"temperature":1.0,"reasoning_tokens":2546,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T17:52:56.861627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically integrate the exact integral representations for the master integrals M37 and M38 at $z \\approx (m_H/m_b)^2 \\approx 900$, evaluating the one-fold and two-fold integrals with complete elliptic functions, and compare these values to the $O(z^{-1})$ asymptotic expressions used in the paper; any discrepancy above roughly 0.1% would mean the central correction is not reliable.","supporting_citations":[],"review_version":1}