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The paper derives the two-loop virtual amplitudes that are the missing ingredient for NNLO QCD predictions of the top-Yukawa-induced component of bottom-quark pair production with a Higgs boson.

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2026-08-02 17:01 UTC pith:Y5G2ITVZ

load-bearing objection The two-loop y_t-induced b bbar H amplitudes in the heavy-top limit are a genuine first, technically well supported by internal cross-checks and shipped code; the main caveat is that the HTL and leading-colour approximations are not quantitatively validated over the LHC phase space that matters for the claimed NNLO sigma_t prediction. the 2 major comments →

arxiv 2603.29480 v1 pith:Y5G2ITVZ submitted 2026-03-31 hep-ph

Top-Yukawa contributions to ppto bbar{b}H: two-loop leading-colour amplitudes

Heribertus Bayu Hartanto , Rene Poncelet This is my paper
classification hep-ph
keywords top-YukawabbH productiontwo-loop amplitudesheavy-top limitleading colourone-mass pentagon functionsNNLO QCDLHC phenomenology
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.

Bottom-quark pair production in association with a Higgs boson (bbH) has three Yukawa components; the top-Yukawa term dominates but its most accurate prediction still carries about 45% scale uncertainty because the two-loop virtual amplitudes were missing. This paper supplies those amplitudes: analytic two-loop leading-colour finite remainders for the b bbar g g H and b bbar q qbar H channels, computed with a massless bottom quark in the heavy-top limit. The results are expressed in one-mass pentagon functions with rational coefficients reconstructed from finite-field evaluations, and are released as working numerical code for hard functions. If correct, they provide the missing ingredient for NNLO QCD predictions of the top-Yukawa part of bbH production, potentially bringing its theoretical uncertainty under control.

Core claim

The finite remainders of the two-loop amplitudes for 0 -> b bbar g g H and 0 -> b bbar q qbar H, together with the related b bbar b bbar final states, admit analytic expressions as linear combinations of one-mass pentagon-function monomials with rational coefficients in momentum-twistor variables. The paper obtains these expressions in the leading-colour approximation and heavy-top limit, validates them against the universal UV/IR pole structure, a Ward identity, and an independent one-loop comparison, and packages them together with the one-loop full-colour amplitudes in a numerical library that evaluates hard functions for all partonic channels. This closes the missing two-loop virtual pie

What carries the argument

The central device is the heavy-top-limit effective operator L = -(1/4) C1 H G^a_{mu nu} G^{a mu nu}, which replaces the top-quark loop by a local Higgs-gluon coupling, together with the leading-colour approximation (keeping only the dominant powers of the colour and light-flavour counts). The calculation is carried by expressing all master integrals in the one-mass pentagon-function basis — the special functions describing five-point integrals with one external massive leg — and reconstructing the rational coefficients from evaluations over finite fields, using momentum-twistor variables for a rational parametrisation of the external kinematics. This combination turns an otherwise intractab

Load-bearing premise

The computation assumes that the heavy-top-limit effective operator and the leading-colour truncation describe the top-Yukawa-induced bbH amplitude well enough over the phase space that matters at the LHC; the paper does not quantitatively demonstrate the heavy-top limit's accuracy for this process.

What would settle it

Evaluate the two-loop hard function for gg -> b bbar H at a few phase-space points with full top-quark mass dependence, or with subleading-colour terms included, and compare against the heavy-top leading-colour result; a difference comparable to the current 45% scale uncertainty in the relevant phase-space region would falsify the claim that these amplitudes are sufficient for controlled NNLO predictions.

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

If this is right

  • The two-loop amplitudes are the missing ingredient for NNLO QCD corrections to the top-Yukawa component sigma_t in the four-flavour scheme; combined with the massification procedure, NNLO-accurate phenomenological studies of pp -> b bbar H become possible.
  • The current NLO-based predictions for sigma_t carry scale uncertainties around 45%; completing the NNLO program with these amplitudes is expected to bring that uncertainty under much better control, assuming the approximations hold.
  • The same amplitudes are building blocks for NNLO QCD predictions of H plus two jets in the heavy-top limit, with the H + 4-gluon channel remaining as the further ingredient.
  • The one-loop amplitudes are provided in full colour, so the two-loop hard function can be assembled with the |R^(1)|^2 term treated exactly while only the R^(2) term uses the leading-colour approximation.
  • The accompanying numerical implementation evaluates hard functions for all partonic channels and includes a rescaling-based precision check, with higher-precision arithmetic available for numerically difficult phase-space regions.

Where Pith is reading between the lines

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

  • If the heavy-top limit is not uniformly accurate over the phase space that dominates at the LHC, the NNLO predictions built on these amplitudes may still need mass-dependent correction factors; a direct comparison with a full top-mass two-loop evaluation at benchmark points would calibrate this.
  • The subleading-colour pieces, estimated by the paper at around 10% at the matrix-element level, could become the dominant theoretical uncertainty once the heavy-top NNLO calculation is completed, making a full-colour two-loop computation the natural next step.
  • The same finite-field rational-reconstruction pipeline could in principle be pushed to the H + 4-gluon channel needed for H + two-jet NNLO, though the paper notes that channel will be considerably more demanding.
  • The released hard functions can be used immediately to estimate the phenomenological impact of the omitted subleading-colour terms on differential distributions, using the order-N_c^-2 estimate the paper cites.

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 derives two-loop leading-colour helicity amplitudes for the top-Yukawa-induced contributions to pp -> bbbar b H in the heavy-top limit, for the partonic channels 0 -> bbbar b g g H and 0 -> bbbar b qbar q H, with the bottom quark treated as massless. The finite remainders are expressed in a one-mass pentagon-function basis and the rational coefficients are reconstructed analytically from finite-field evaluations. One-loop amplitudes are provided in full colour and validated against OpenLoops; the two-loop result is checked through IR-pole structure, renormalisation-scale dependence, and a Ward identity. A C++/Mathematica library implementing the amplitudes and hard functions is supplied, together with benchmark numbers.

Significance. If correct, this is a substantial technical achievement. The computation involves non-planar two-loop integral families and provides a necessary ingredient for NNLO predictions of the top-Yukawa component of bbbar H production. The paper ships machine-readable analytic expressions, reproducible benchmark results, and a stability-tested numerical library, and the internal consistency checks (OpenLoops comparison at one loop, IR pole structure, scale-dependence verification, Ward identity) are strong and appropriate for this type of computation. The main caveat is that the phenomenological payoff depends on the heavy-top-limit and leading-colour approximations being sufficiently accurate over the phase space relevant to the LHC; this is not quantitatively assessed in the manuscript, and the authors acknowledge that the leading-colour truncation alone may introduce O(10%) matrix-element corrections.

major comments (2)
  1. [§1, §5, Eq. (2) and Eqs. (15)/(25)] The stated goal is to enable NNLO predictions for sigma_t and reduce its ~45% theoretical uncertainty. The computation relies on the heavy-top-limit operator of Eq. (2) and the leading-colour truncation of Eqs. (15)/(25), but the range of validity of these approximations is not quantified. The leading-colour error is acknowledged as O(N_c^-2), about 10% at the matrix-element level, with unknown cross-section impact; the HTL condition (all kinematic scales much smaller than m_t) is not obviously satisfied for hard b-jets or large m_bb regions that contribute to the cross section. Since the NNLO corrections the amplitudes are meant to provide are of similar size, this is load-bearing for the paper's stated motivation. I request that the authors either provide a quantitative assessment (e.g., a one-loop comparison of the HTL amplitude against the full massive-top amplitude over relevant pha
  2. [§1, §5] The manuscript states that the two-loop amplitude is 'the missing ingredient' to obtain NNLO QCD predictions for sigma_t. A complete NNLO cross section also requires the double-real and real-virtual contributions for the y_t component. The paper does not discuss whether these amplitudes are already available or how they would be obtained. Please clarify which NNLO ingredients exist and which remain, and soften the wording if the two-loop amplitude is only one of the required missing pieces. This is important for accurately representing the path from this work to a full NNLO phenomenological prediction.
minor comments (5)
  1. [Eq. (15b), Table 3] The decomposition in Eq. (15b) includes an n_f^2 term, which is not a leading-colour contribution in the large-N_c expansion. The authors should define their colour-counting convention explicitly and state whether the 'strict leading colour' benchmarks in Table 3 include the n_f^2 terms. This will help readers interpret the size of the neglected subleading-colour terms.
  2. [§5 / Conclusion] The wording '4FS NNLO-accurate phenomenological studies can be performed' is stronger than what is demonstrated, given the caveats above and the need for additional real-emission contributions. A sentence acknowledging the remaining steps would be helpful.
  3. [Appendix D / ancillary files] The documentation lists the file structure but does not include a minimal 'hello world' example or a step-by-step guide to reproducing Table 3 from the Mathematica files. A short worked example would substantially improve usability.
  4. [Table 1] The layout of Table 1 is visually dense and hard to parse because of the repeated integral-family diagrams. Consider splitting it or presenting a simplified summary in the main text, with the full version in an appendix.
  5. [References] Reference [61] appears to contain a malformed DOI ('10.1103/zt4w-c1jk'). Please check and correct it.

Circularity Check

0 steps flagged

No significant circularity: the two-loop amplitudes are derived from Feynman rules, external master integrals and pentagon functions, with no fitted input renamed as a prediction.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The two-loop amplitudes are obtained by generating Feynman diagrams with QGRAF, performing colour decomposition, applying four-dimensional projectors, reducing scalar integrals via IBP to master integrals 'constructed in Refs. [51,53,55-57]' and expressing them in 'one-mass pentagon functions [46,52,54]'; these are external, independently established results, not quantities fitted in this paper. The UV/IR subtraction uses universal pole structures from Refs. [94-97]. The finite-field reconstruction uses FiniteFlow and NeatIBP, which are also external tools. Validation is provided by an independent comparison: 'comparing the full-colour one-loop amplitude against OPENLOOPS [118] through O(eps^0)', as well as a Ward identity check and a rescaling check of the renormalisation-scale dependence. No parameter is fitted to a target amplitude and then renamed a prediction; the benchmark hard functions are evaluations, not fits. The self-citations present are technical or contextual rather than load-bearing: Ref. [40] is cited for the identical tensor/projector decomposition used in the companion y_b calculation (a mathematical construction, not a result depending on the y_t amplitude), and Ref. [91] is cited only to motivate the expected size of subleading-colour corrections, not to define or derive the amplitude. The HTL and leading-colour approximations are stated approximations whose validity is an applicability caveat, not a circularity; the manuscript even acknowledges the subleading-colour impact is 'unknown at this point'. Thus no step reduces, by construction or by self-citation, to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are fitted to data. All numerical choices (mu_R = 173.2 GeV, alpha_s = 0.118, benchmark momenta) are inputs, not fitted values. The paper introduces no new particles or couplings; the effective Hgg operator is from standard heavy-top-limit literature. The result's domain rests on HTL, massless bottom, and leading-colour approximations.

axioms (5)
  • domain assumption Heavy-top-limit effective Lagrangian, Eq. (2), with C1 from Eq. (3)
    The top-quark loop is integrated out into a local H G_mu_nu G^mu_nu operator; the result is exact only in the m_t -> infinity limit.
  • domain assumption Massless bottom-quark kinematics, Eq. (7)
    The computation treats the bottom quark as massless; finite bottom-mass effects are to be added later via massification.
  • domain assumption Leading-colour truncation, Eqs. (11)-(13), (15), (25)
    Only leading terms in N_c and n_f are retained for two-loop amplitudes; subleading colour is estimated at about 10% and not computed.
  • domain assumption Known one-mass master integrals and pentagon function bases from Refs [51,53,55-57] and [46,52,54]
    The amplitudes are expanded in a basis of previously computed master integrals and one-mass pentagon functions; the result inherits their correctness.
  • domain assumption Universal IR pole subtraction Z(L) from Refs [95,97], shown in Appendix A
    The finite remainder is defined by subtracting universal IR poles; if this subtraction is not the correct multi-scale one, the finite remainder would shift.

pith-pipeline@v1.3.0-alltime-deepseek · 30405 in / 12344 out tokens · 119143 ms · 2026-08-02T17:01:34.994972+00:00 · methodology

0 comments
read the original abstract

We derive two-loop scattering amplitudes for bottom-quark pair production in association with a Higgs boson at the LHC, focusing on terms proportional to the top-quark Yukawa coupling. We treat the bottom quark as a massless parton and employ both the leading-colour and heavy-top-quark approximations. The finite remainder of the two-loop amplitude is expressed in terms of one-mass pentagon functions, and the corresponding rational coefficients are reconstructed analytically from evaluations over finite fields. The scattering processes considered in this work also constitute a subset of Higgs+2-jet production at the LHC in the heavy-top-quark approximation.

Figures

Figures reproduced from arXiv: 2603.29480 by Heribertus Bayu Hartanto, Rene Poncelet.

Figure 1
Figure 1. Figure 1: Representative Feynman diagrams illustrating the three [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representative Feynman diagrams illustrating the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sample two-loop Feynman diagrams contributing to leading colour [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sample two-loop Feynman diagrams contributing to leading colour [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The four panels show the distribution of the estimated numerical precision [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

discussion (0)

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Forward citations

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Reference graph

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