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REVIEW 2 major objections 5 minor 134 references

Heavy jet mass in H→gg and H→qqbar decays is computed to N3LL′ in the dijet limit and NNLL in the trijet limit, matched to NNLO, with Sudakov shoulder resummation that yields sizable corrections and improves perturbative stability.

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2026-08-01 03:11 UTC pith:IOKBBBF2

load-bearing objection A technically serious and mostly honest N3LL′+NNLL calculation of heavy jet mass in Higgs decays, but the N3LL′ label is conditional on an uncomputed three-loop non-global soft function. the 2 major comments →

arxiv 2607.25187 v1 pith:IOKBBBF2 submitted 2026-07-28 hep-ph hep-ex

Heavy Jet Mass in Hadronic Higgs Decays

Luen Clingerman , Xiaoyuan Zhang This is my paper
classification hep-ph hep-ex
keywords heavy jet masshadronic Higgs decaysSudakov shoulderresummationsoft-collinear effective theoryNNLO matchingH→ggH→qqbar
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to put the heavy jet mass distribution in hadronic Higgs decays on a much firmer theoretical footing. It claims that, using effective-field-theory factorization, the distribution in H→gg and H→qqbar can be predicted at next-to-next-to-next-to-leading logarithmic order in the dijet limit and at next-to-next-to-leading logarithmic order in the trijet limit, and matched to next-to-next-to-leading-order fixed order. The crucial new element is a resummation of Sudakov shoulder logarithms near ρ=1/3, where the distribution develops a kink because of incomplete cancellation of infrared singularities; the paper derives the needed one-loop shoulder ingredients and shows the resummation removes the kink and improves stability. If correct, this becomes the reference prediction for extracting α_s, the effective Hgg coupling, and the bottom Yukawa from future lepton-collider Higgs data.

Core claim

The central claim is that the heavy jet mass distribution in both gluon- and quark-initiated hadronic Higgs decays, H→gg and H→qqbar, is computable to N3LL′ accuracy in the dijet limit and to NNLL accuracy in the trijet limit, matched to NNLO fixed order. In the dijet limit the paper assembles the hard, jet, and hemisphere soft functions in soft-collinear effective theory and verifies the singular expansion against fixed-order calculations. In the trijet limit, it derives the leading-power NLO shoulder logarithms analytically from the full H→4 parton amplitudes and four-particle phase space, verifies the shoulder factorization theorem against them and against numerical NLO data, and then res

What carries the argument

The machinery is factorized resummation in soft-collinear effective theory. For the dijet limit, a factorization theorem splits the cross section into a hard function for the back-to-back pair, inclusive jet functions for each hemisphere, and a hemisphere soft function for wide-angle radiation. For the Sudakov shoulder, a separate factorization theorem is constructed around the symmetric three-jet point: it involves the trijet hard function, three jet functions, and a trijet hemisphere soft function that sums the color correlations among three Wilson lines. The paper's new ingredients are the analytic NLO shoulder logarithms, the two-loop trijet hard functions matched from H→3 parton amplitu

Load-bearing premise

The load-bearing premise is that the uncomputed three-loop soft-radiation terms in the dijet hemisphere soft function are small enough to ignore, and that the gluon version of the trijet soft function equals the quark version with color charges substituted; the claimed accuracy fails if either is wrong.

What would settle it

Compute the three-loop non-global-logarithm coefficients s_3,2 and s_3,4 in the HJM hemisphere soft function (currently set to zero in Eq. (2.24)); if at ln ρ ≈ −3.5 they shift the N3LL′ central curve by more than the quoted uncertainty band, the paper's accuracy claim is falsified. Alternatively, measure the HJM spectrum in e+e−→ZH at a future Higgs factory and compare the H→gg/H→qqbar ratio in the shoulder bins ρ≈0.28–0.34; disagreement beyond the predicted band would rule out the shoulder resummation.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A precision measurement of the heavy jet mass spectrum in H→gg and H→qqbar at a future lepton-collider Higgs factory can be compared with a prediction that has all-order logarithms in both the dijet and trijet regions, reducing one of the dominant theoretical uncertainties in α_s extractions.
  • In the H→gg channel the Sudakov shoulder correction reaches tens of percent and becomes order one very close to ρ=1/3, so any interpretation of the trijet region that omits the shoulder resummation is systematically biased.
  • The shoulder resummation removes the unphysical kink at ρ=1/3 and improves perturbative convergence, making the combined N3LL′_dij+NNLL_sh+NNLO curve the reference for future extractions of the Hgg effective coupling and the bottom Yukawa.
  • The analytic NLO shoulder logarithms in both channels provide a direct cross-check for independent resummation or fixed-order calculations at the symmetric trijet point.
  • The same position-space resummation technique also yields a fully resummed thrust distribution in Higgs decays, which is currently the best available handle on the same soft-collinear dynamics.

Where Pith is reading between the lines

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

  • If the missing three-loop non-global-logarithm coefficients in the dijet hemisphere soft function turn out to be numerically significant at the peak (ln ρ ≈ −3.5), the N3LL′ central curve would shift by an amount not covered by the current uncertainty bands; computing those coefficients is a direct test of the paper's accuracy claim.
  • The gluon-trijet soft function is obtained by replacing color factors in the quark-trijet result; a direct two-loop calculation of the gluon case would test whether Casimir scaling holds at the required accuracy for the H→gg shoulder.
  • One could use the matched spectrum to make a data-driven measurement of the non-perturbative parameter Ω_1g from e+e−→ZH events and compare it with the Casimir-scaled expectation Ω_1g/Ω_1q = C_A/C_F; the paper's predictions are a necessary input for that measurement.
  • The order-one shoulder correction in H→gg suggests the same Sudakov-shoulder mechanism could be relevant for other observables with kinematic endpoints, such as C-parameter, where fixed-order predictions would otherwise be unreliable near the boundary.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper computes heavy jet mass (HJM) distributions in H→gg and H→q̄q decays at a future e⁺e⁻ Higgs factory. In the dijet limit ρ→0 it uses SCET factorization with known hard, jet, and soft functions to claim N³LL′ accuracy, matched to NNLO fixed order. In the trijet region ρ≈1/3 it derives the analytic NLO Sudakov-shoulder logarithms from a leading-power expansion of the four-parton matrix elements and phase space, verifies them against EERAD3, and constructs an NNLL shoulder resummation using the factorization theorem of Ref. [112]. The two resummations are combined through the additive matching formula Eq. (4.42). The paper also provides numerical predictions for both channels, including profile-scale uncertainty estimates.

Significance. If the central claims hold, this is the first N³LL′ dijet + NNLL shoulder prediction for HJM in hadronic Higgs decays, and it is directly relevant for α_s, λ(µ), and y_b determinations at future lepton colliders. The paper has genuine strengths: the shoulder-logarithm coefficients are obtained by an independent parameter-free expansion of matrix elements and phase space, the fixed-order singular expansions are checked against EERAD3, the resummed formulas are RG-invariant, and the matching methodology is standard and carefully documented. The main caveat is that the advertised N³LL′ accuracy is conditional on an uncomputed, L-dependent part of the three-loop non-global soft function, which the authors set to zero without a numerical bound.

major comments (2)
  1. [§2.3, Eq. (2.24) and Table 1] The N³LL′ dijet claim requires the three-loop HJM soft function, including its L-dependent non-global terms. Eq. (2.24) writes s_NGL,3(L)=2(s_3+s_3,2 L²+s_3,4 L⁴) and sets s_3,2=s_3,4=0; the retained s_3 is the thrust-like constant. These omitted α_s³ ln⁴ρ-type terms enter the exponent of Eq. (2.40) and are not bounded by profile-scale variation, which probes RG-scale dependence rather than this fixed-order boundary uncertainty. In the H→gg peak region lnρ≈−(2.3–3), the omitted terms are parametrically comparable to the NNLL′→N³LL′ shift shown in Fig. 4. Since the abstract's first claim is N³LL′ accuracy, this acknowledged modeling choice needs to be quantified, or the accuracy claim should be qualified. An estimate obtained by varying s_3,2 and s_3,4 over natural-size ranges would be a concrete way to assess the impact.
  2. [§4.2.2, Eqs. (4.14)–(4.15)] The gluon trijet hemisphere soft function, which underpins the new H→gg shoulder resummation, is obtained by C_F→C_A Casimir scaling from the quark-channel result. At the one-loop order needed for the quoted NNLL shoulder accuracy this is plausible, but the all-gluon operator involves three adjoint Wilson lines and a different color flow, and the paper gives no test or estimate of where Casimir scaling breaks down at higher orders. Since H→gg is the main phenomenological target, the validity range of this replacement should be stated explicitly; if a two-loop estimate is not available, the residual uncertainty from this source should at least be acknowledged in the accuracy budget.
minor comments (5)
  1. [§3.4, Eqs. (3.24)–(3.29)] The notation 'ln2 r' is ambiguous in the typeset equations; it should be written as ln²r (and similarly for ln3, ln4, etc.) to avoid confusion with ln(2r).
  2. [§2.1.3, Eq. (2.20)] The phrase 'known fully at two loops and partially at three loops' is followed by a model for s_NGL,3 that sets two of the three polynomial coefficients to zero. Please state in the text which three-loop pieces are actually known and which are being modeled, so that readers can distinguish computed from assumed inputs.
  3. [§4.5 and Fig. 9] The right-shoulder NNLLsh+NLO curve in Fig. 9 is shown to become negative quickly, and the text attributes this to a second shoulder. This limitation should be stated in the abstract or in the introduction's scope statement, since the phrase 'trijet limit' could be read as covering the right shoulder region as well.
  4. [§4.6, Eq. (4.41)] The modified dijet profile parameters t_2=0.2, t_s=1/3 are introduced without a plot showing the hand-off between the dijet and shoulder resummations. The text explains the rationale, but a small figure of g_sh(ρ) and the dijet profile weights together would make the matching more transparent.
  5. [General] The paper is overall carefully written, but the multi-line fixed-order expressions in Eqs. (2.33)–(2.36) are difficult to check. An ancillary file with machine-readable expressions, or at least a numerical validation table against EERAD3, would help reproducibility.

Circularity Check

0 steps flagged

No significant circularity: the heavy-jet-mass predictions are derived from independently computed SCET ingredients and fixed-order matrix elements; the missing three-loop non-global logarithms are a stated limitation, not a circular input.

full rationale

The derivation chain is not circular. The dijet N3LL' resummation is assembled from the standard SCET factorization theorem (Eq. 2.6) using published hard, jet, and soft functions, and the singular expansion in Sec. 2.2 is checked against external EERAD3 data and the NNLO results of Ref. [28] (Fig. 1). The shoulder logarithms are not fitted inputs: Sec. 3.4 computes the NLO r ln^2 r and r ln r coefficients by direct power expansion of the 1 -> 4 matrix elements and phase space (Eq. 3.7, scaling Eq. 3.8), and Sec. 4.3 then expands the shoulder factorization theorem and verifies that it reproduces those independently derived coefficients. This is a genuine consistency check rather than a construction-by-definition. The matching formula (4.42) is the standard additive subtraction of singular limits evaluated at the fixed-order scale, not a renamed fit. The non-singular constants fit in Eq. (4.21) are comparison parameters, not part of the claimed resummed prediction. Self-citations to Refs. [65], [75], and [112] provide the shoulder framework, kernel boundary condition, and profile-function choices, but the current paper tests the framework against its own independent NLO calculation, so these citations are not load-bearing in a circular way. The principal caveat is the acknowledged modeling choice in Eq. (2.24): the three-loop HJM-specific non-global terms s_{3,2}=s_{3,4}=0 are set to zero because the complete three-loop soft function is unknown. This is an accuracy/robustness limitation and not a circular reduction of the prediction to its inputs, since the omitted terms are not fitted from the target distribution. The score is therefore 0 on circularity, with the correctness caveat noted separately.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

No invented entities. The prediction rests on: standard SCET factorization theorems (dijet, Eq 2.6; shoulder, Eqs 4.3-4.4) adopted from the literature; HEFT operator basis; literature hard functions (Refs 121-127); public fixed-order data (EERAD3, Ref 28); and hand-chosen profile parameters. The only explicitly ad hoc element is the truncated three-loop NGL polynomial (Eq 2.24). The two premises least tested by the text are Casimir scaling of the gluon trijet soft function and the small-hemisphere-mass assumption of the shoulder factorization.

free parameters (4)
  • s_3,2, s_3,4 (three-loop NGL polynomial coefficients) = 0
    Eq (2.24): unknown coefficients of the three-loop non-global dijet soft function, set to zero by hand. Affects the N³LL′ central curve at the α_s³ L⁴ level.
  • dijet profile parameters = µ_min_s=1.1 GeV; n0=2; n1=10; t2=0.25; ts=0.40
    Tab 2: hand-chosen transition/freeze-out scales, default from Ref [75]; varied for the uncertainty band.
  • shoulder profile parameters = ρ_L1=0.10, ρ_L2=0.23, ρ_L3=0.28; ρ_R1=1/3, ρ_R2=0.338, ρ_R3=0.342
    Sec 4.6, Tab 3: chosen from LO singular-vs-nonsingular crossover (Fig 10); right-side values not varied.
  • shoulder scale exponent v_j and soft variation e_sh_s = v_j=1/2; e_sh_s=1
    Sec 4.5: canonical-shoulder scale choices, varied over [0.4,0.6] and [1/2,2] for the band.
axioms (6)
  • domain assumption Dijet SCET factorization (Eq 2.6): double-differential hemisphere-mass distribution = hard × jet × hemisphere-soft convolution at leading power in λ=√ρ
    Sec 2.1; standard SCET result, adopted without re-derivation.
  • domain assumption Shoulder factorization (Eqs 4.3-4.4): near ρ=1/3 the distribution is governed by trijet hard, three jet functions, and a trijet hemisphere soft function, with the assumption that both hemisphere masses are small (O(λ)) and only their difference enters the measurement
    Sec 4.1; includes the assertion that hard O(1)-mass configurations cannot generate ln r terms. Inherited from Refs [112,65].
  • domain assumption Casimir scaling C_F→C_A for the gluonic trijet hemisphere soft function and its anomalous dimensions
    Sec 4.2.2: 'wide-angle radiation only resolves the total color charge'; assumed for the new H→gg channels.
  • domain assumption HEFT operator basis (Eq 1.3) with massless light quarks and no O_g–O_q interference
    Sec 1; follows Ref [31] and standard HEFT.
  • ad hoc to paper Three-loop NGL soft function modeled by an even polynomial with vanishing L² and L⁴ coefficients
    Eq (2.24): explicit model, flagged by the authors as replaceable.
  • domain assumption Boundary kernel K(z,r) = (1 − e^{−izr} + r(1 − e^{iz}))/z² fixes the shoulder double-integration boundaries
    Sec 4.4, Eq (4.35): chosen so the expanded distribution reproduces full-theory leading behavior near r=0; verified against Sec 3.4 analytic logs.

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read the original abstract

The heavy jet mass distributions in hadronic Higgs decays are computed to next-to-next-to-next-to-leading logarithmic order (N${}^3$LL${}^\prime$) in the dijet limit and NNLL in the trijet limit, matched to the next-to-next-to-leading order (NNLO). Both resummation results are obtained from the factorization theorems in Soft-Collinear Effective Theory. In particular, we study the Sudakov shoulders in the trijet region, originating from the incomplete cancellation of infrared singularities between final states with different parton multiplicities, and resum the induced large logarithms to all orders. The shoulder resummation yields sizable corrections and improves the perturbative stability of the distribution. Our results provide state-of-the-art predictions for heavy jet mass in $H \to gg$ and $H\to q\bar{q}$ and can be applied to precision Higgs measurements at future $e^+e^-$ colliders.

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