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Complete two-loop Yukawa-induced running of the Higgs-gluon coupling in SMEFT

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper derives the previously missing two-loop contributions to the renormalization group running of the Higgs-gluon coupling in the Standard Model Effective Field Theory, completing the two-loop Yukawa-induced program for tree-level ge

desk verdict Solid, likely-correct completion of a real gap in the two-loop SMEFT RGE program, but the unflagged-by-choice discrepancy with the authors' own earlier paper and the missing fit code make this 'accept after minor revision' rather than 'ready to use.' read the letter →

arxiv 2510.14680 v2 pith:CNKICHHD submitted 2025-10-16 hep-ph

Stefano Di Noi , Barbara Anna Erdelyi , Ramona Gröber This is my paper · ORCID
classification hep-ph
keywords SMEFTtwo-loopRGEHiggs-gluoncouplingYukawa-inducedrunningWarsawbasisclass37vector-likequarksHiggsdatafits
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 completes the missing piece of the two-loop renormalization group running of the effective Higgs-gluon coupling in the Standard Model Effective Field Theory. It presents new contributions from the D^2 H^4 and D H^2 ψ^2 operator classes (Warsaw basis classes 3 and 7), which are generated at tree level in weakly coupled UV completions, joining previously known four-quark and Yukawa-like operator contributions. The result matters because the Higgs-gluon operator enters gluon-fusion Higgs production at tree level in the SMEFT while the SM contribution is loop-induced, so a consistent loop counting requires including the two-loop running from these tree-level generated operators. The authors also fit to Higgs data and find that two-loop running can sizably tighten bounds on some class 7 operators, though the effect is diluted in vector-like quark models where one-loop matching generates many additional operators.

What carries the argument

The kinematic cancellation argument: for a single insertion of a class 7 operator (e.g., O_Ht), the s-channel triangle diagrams contributing to gg→H*H contain a scalar current that contracts with (p1-p2), the difference of the outgoing Higgs momenta, making the diagrams vanish. This justifies discarding all two-loop diagrams in which both external Higgs doublets attach to the effective operator vertex, leaving only diagrams with one external and one internal Higgs leg. The paper also relies on two independent automated computation pipelines based on dimensional regularization and extraction of 1/ε poles to obtain the anomalous dimension, and on the evolution-matrix formalism to convert high-

What would settle it

Compute explicitly the two-loop s-channel triangle diagrams with a single O_Ht insertion and a QCD correction, including the diagram with one Yukawa and one QCD vertex (Fig. 1c), and check whether the sum vanishes; alternatively, compare Eq. (3.1) with the independent two-loop running computation that the authors note is in preparation once it appears, since the paper reports agreement with it.

Watch

Extended reading notes

Core claim

This paper computes the two-loop renormalization group equation for the SMEFT operator O_HG = (H†H) G^A_{μν} G^{A,μν} at order g_s^2/(16π^2)^2, presenting contributions from the class 7 operators O_HQ^(1), O_HQ^(3), O_Ht, O_Hb, O_Htb and the class 3 operators O_H□, O_HD. These contributions complete the two-loop Yukawa-induced running proportional to the third-generation Yukawa couplings for potentially tree-level generated operators. The computation exploits a kinematic cancellation: s-channel triangle diagrams with a single class 7 insertion vanish because the scalar current contracts with (p1-p2), so all diagrams where both external Higgs doublets attach to the effective operator vertex a

Load-bearing premise

The paper relies on the claim that all two-loop diagrams in which both external Higgs doublets attach to the effective class-7 vertex vanish identically because the scalar current contracts with (p1-p2); if any mixed Yukawa-QCD diagram of this type survives, the retained 24 diagrams would be incomplete and Eq. (3.1) would miss contributions.

Editorial extensions

If this is right

  • Equation (3.1) completes the two-loop Yukawa-induced RGE for C_HG from potentially tree-level generated operators, so global SMEFT analyses can consistently include these running effects for gluon-fusion Higgs production.
  • In one-parameter fits to inclusive Higgs data, two-loop running tightens the 2σ bound on C_HQ^(3) sizably, while bounds on class 3 operators C_H□ and C_HD barely change because they enter observables directly.
  • The bound on the top-bottom operator C_Htb becomes looser at two-loop level, indicating cancellations with one-loop running contributions.
  • For UV models with vector-like quarks, the inclusion of one-loop matching to O_HG, O_HW, O_HB, and O_HWB reduces the impact of the two-loop running on the fitted parameter regions.
  • The new contributions are independent of the γ5 scheme, confirming that the scheme dependence in the two-loop RGE of C_HG resides only in the four-top operator contribution.

Reading between the lines

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

  • If the kinematic cancellation is generic, it suggests that class 7 operators mix into C_HG only through diagrams where one Higgs doublet is internal; this could be used to organize higher-loop calculations for similar operators, e.g., in di-Higgs production.
  • The scheme independence of the new terms hints that the γ5 scheme sensitivity in two-loop SMEFT RGEs is confined to specific operator classes (like four-top), which could motivate scheme conventions in future automated two-loop RGE codes.
  • The sizable tightening of the C_HQ^(3) bound implies that fits which omit two-loop running may underestimate the reach of Higgs data on third-generation quark-Higgs couplings; including top-quark and electroweak precision observables could amplify this effect.
  • A direct extension would be computing the corresponding two-loop contributions to O_HW, O_HB, and O_HWB; if the pattern persists, the numerical impact on h→γγ and h→Zγ may be smaller than in the gluon case, as the authors expect from the W-loop dominance in the SM.
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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

2 major / 4 minor

Summary. This paper computes the two-loop renormalization-group contribution to the Wilson coefficient C_HG of the Higgs-gluon operator O_HG in SMEFT, at order g_s^2/(16π^2)^2 and proportional to third-generation Yukawa couplings, induced by Warsaw-basis class 7 (ψ^2 H^2 D) and class 3 (H^4 D^2) operators. The final expression, Eq. (3.1), combines these new terms with earlier four-quark [24] and Yukawa-like [27] contributions, and is claimed to complete the two-loop Yukawa-induced running of C_HG for operators that can be generated at tree level. The calculation is performed in the unbroken phase from the UV poles of gg→H*H, using two independent automated pipelines (FeynRules/FeynArts/FeynCalc/AMFlow+Blade and qgraf/FeynCalc/LiteRed2/FiniteFlow), with an NDR/BMHV scheme check and an agreement statement with independent work [96]. The phenomenological part fits inclusive Higgs signal strengths in both a bottom-up WC fit and top-down vector-like-quark models.

Significance. If correct, Eq. (3.1) completes an important block of the two-loop SMEFT RGE program for the Higgs-gluon coupling, with direct consequences for consistent Higgs fits. The paper's strengths are substantial: two independent diagram-generation and integration pipelines, an explicit γ5-scheme cross-check via the map of Ref. [80], and a statement of agreement with independent work [96]. The RGE result is not circular: no fitted constants or normalization choices enter the computation. However, the completeness of the diagram basis used for Eq. (3.1) rests on a kinematic cancellation for mixed Yukawa-QCD graphs that is asserted but not demonstrated, and this is the main technical concern.

major comments (2)
  1. [Sec. 3, Fig. 1c and footnote 2] The exclusion of all two-loop diagrams with both external Higgs legs attached to the O_Ht vertex is load-bearing: it reduces the class-7 set to the 24 diagrams that produce Eq. (3.1). The text explicitly cross-checks the cancellation only for the pure-QCD case, i.e. a gluon-propagator correction between the two QCD vertices (Fig. 1b). For the mixed Yukawa-QCD case (Fig. 1c), where an internal Higgs propagator and chiral projectors are present, the remaining amplitude is not shown to be symmetric under p1↔p2; the effective vertex connects H and H*, which are distinct legs, and the Yukawa vertex introduces a preferred fermion direction. Please provide either an explicit two-loop computation of the Fig. 1c family or a rigorous symmetry proof. As written, the completeness of Eq. (3.1) is not fully established, and the stronger 'all loop orders' claim of footnote 2 inherits the same gap.
  2. [Eq. (3.1), footnote 4] Footnote 4 states that the coefficient 3/2 multiplying the O_tH/O_bH terms in Eq. (3.1) differs from the published result of Ref. [27] by an extra 1/2 and that Ref. [27] 'will be updated'. Since Eq. (3.1) is presented as the complete final RGE, the reader cannot reproduce this term from the cited literature as it stands. Please provide the derivation of this coefficient in the present paper or attach the update/erratum to Ref. [27] before publication.
minor comments (4)
  1. [Eq. (3.1)] There is a stray 'h' immediately after the bracket in the displayed equation: 'g_s^2/(16π^2)^2 h +Yb...' appears to be a typographical artifact.
  2. [Sec. 3] The text states that 'the leading divergence is of O(1/ϵ)' to justify the γ5-scheme comparison; it would be useful to state explicitly whether this is the leading 1/ϵ^2 behavior or only the 1/ϵ pole relevant to the RGE, since two-loop poles can have different scheme dependence.
  3. [Sec. 4.2] The phenomenological comparison is made only against single-operator or paired-operator fits with inclusive Higgs data. A sentence clarifying that no systematic uncertainties from the Higgs signal-strength covariance matrix are included would help the reader judge the strength of the quantitative claims.
  4. [References] The agreement claim with Ref. [96] refers to a manuscript 'in preparation'; if possible, give more details of the cross-checked terms or make the comparison available as an ancillary file.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (3.1) is obtained from two-loop UV divergences with no fitted inputs; self-citations are prior independent calculations and are explicitly corrected where needed.

full rationale

The central result, Eq. (3.1), is derived from the divergent parts of explicitly computed two-loop Feynman diagrams for gg -> H*H, using two independent computational pipelines (FeynRules/FeynArts/FeynCalc/AMFlow and qgraf/FeynCalc/AMFlow). No parameter is fitted to data or to a known result, and no quantity in the claimed prediction is defined in terms of the output. The terms taken from earlier work are clearly identified: the four-quark and Yukawa-like operator contributions come from Refs. [24] and [27], respectively; these are separate prior calculations, and footnote 4 explicitly corrects an overall factor in [27], showing critical reuse rather than blind import. The paper also reports independent agreement with the forthcoming results of Ref. [96]. The computation of class-7 operators uses 'the same pipeline of Ref. [27]' only as a technical tool, not as an input to the physics result. The main potential weakness, the kinematic cancellation used to discard diagrams where both external Higgs legs attach to the effective operator vertex, is a completeness/correctness concern: the paper explicitly cross-checks only the pure-QCD case and extends the argument to mixed Yukawa-QCD diagrams. Even if that extension were incomplete, it would not make the derivation circular, because the cancellation argument does not presuppose the final RGE coefficients. No self-definition, fitted-input-as-prediction, uniqueness-imported-from-authors, ansatz-smuggled-via-citation, or renaming pattern is present. The result is self-contained against external and internal cross-checks.

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

The central RGE result has no fitted free parameters and introduces no new entities. It rests on the Warsaw-basis conventions, the third-generation-only truncation, and the weakly-interacting loop-counting assumption that exclude chromomagnetic and other loop-generated operators at this order.

assumptions (5)
  • domain assumption Warsaw basis operator definitions and SM Lagrangian conventions of Refs [1,14,16] are assumed for all Wilson coefficients and signs.
    Section 2: 'We follow the Warsaw basis of Ref. [1] ... and Ref. [14] for the definitions of the SM Lagrangian.' The result of Eq. (3.1) is convention-dependent.
  • domain assumption Only third-generation quarks (t_R, b_R, Q_L) are retained; first- and second-generation contributions are ignored.
    Section 2 states 'We consider only the third family of quarks'; this truncation is not quantified.
  • domain assumption Loop-counting: in weakly-interacting renormalizable UV completions, operators generated only at loop level (e.g. chromomagnetic O_G, O_HG itself) cannot contribute to the two-loop RGE of C_HG at O(g_s^2 Y_t^2).
    Sections 1 and 5 and Refs [45,46]; if this counting fails, additional operator classes would enter at the same order.
  • domain assumption Kinematic vanishing of the one-loop triangle scalar current: the gg→H*H amplitude with a single class-7 insertion has only s-channel diagrams and the current contraction yields (p1−p2)·..., which vanishes; this is used to discard all diagrams where both external Higgses attach to the effective verte
    Section 3, discussion around Fig. 1; load-bearing because it restricts the diagram set to 24+12 diagrams.
  • standard math In NDR, traces with an odd number of γ5 are dropped because they only contribute to the CP-odd operator C~_HG; NDR and BMHV are argued to give the same result for the CP-even part via the map of Ref [80].
    Section 3, γ5 scheme paragraph.

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Cite this review

Pith. "Pith review of Complete two-loop Yukawa-induced running of the Higgs-gluon coupling in SMEFT." pith.science (2026). https://pith.science/paper/CNKICHHD

@misc{pith2026251014680,
  author       = {Pith},
  title        = {Pith review of: Complete two-loop Yukawa-induced running of the Higgs-gluon coupling in SMEFT},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CNKICHHD}},
  note         = {Machine review of arXiv:2510.14680}
}
read the original abstract

We compute the two-loop renormalisation group equation for the effective Higgs to gluon coupling in Standard Model Effective Field Theory. Concretely, we present the contributions generated by the operators belonging to class 3 and 7 in the Warsaw basis, completing the two-loop renormalization program of the Higgs-gluon coupling proportional to the top Yukawa coupling for potentially tree-level generated operators. We investigate the phenomenological impact of the contributions in fits to Higgs data both in a bottom-up approach and a top-down approach in terms of UV models with vector-like quarks.

Figures

Figures reproduced from arXiv: 2510.14680 by the authors.

Figure 1
Figure 1. Sample of the diagrams with a single insertion of the operator OHt which vanish because of kinematics, see main text for details. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs doublet, a black square to denote an insertion of OHt and a black dot to denote a SM vertex insertion. g H g H∗ (a) g H g H∗ (b) g H g H∗ (c) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Sample of the two-loop order diagrams with a single insertion of the operator OHt. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs doublet, a black square to denote an insertion of OHt, a black dot to denote a SM vertex insertion and a white star to denote the Yukawa coupling counterterm. g g H H∗ (a) g g H H∗ (b) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Sample of the two-loop order diagrams with a single insertion of the operators OH□ and OHD. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs doublet, a white square to denote an insertion of OH□ and OHD and a black dot to denote a SM vertex insertion. diagrams are purely s-channel, depending on the linear combination q1+q2 = p1+p2 where we have used p1,2 (q1,2) to d… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Allowed ranges at 2σ for the WCs obtained by performing one parameter fits. The yellow (orange) line is obtained by considering one-loop (two-loop) running effects. coefficients are not sizeably affected by the inclusion of the two-loop effects in the RGEs. These opera…
Figure 5
Figure 5. Figure 5: Allowed regions at 2σ obtained via a two parameter fit including one-loop (in yellow) and two-loop (in orange) running effects. We consider in pairs the operators belonging to class 3 and to class 7. For class 7, we further separate between the cases in which the quark…
Figure 6
Figure 6. Figure 6: Allowed regions at 2σ for pairs of couplings for the considered models. The orange (yellow) coloured areas correspond to allowed values derived accounting for two-loop (one-loop) running effects. −L(int) M.5 = [λQ7 ] rp Q¯r 7LHup R + [λT2 ] rp 2 T¯I r 2RH˜ † τ I q p L …
Figure 7
Figure 7. Figure 7: Two fits performed for Model 1, considering either the complete matching up to one loop for all the considered operators (left) or only the tree-level matching results (right). In both cases, the allowed parameter space at 2σ is plotted in yellow (orange) when the one-…

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