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arxiv: 2607.06099 · v1 · pith:PLOQPGTV · submitted 2026-07-07 · hep-ph · hep-ex

ggxy: Fast and flexible NLO QCD corrections to gluon-initiated processes

Daniel Stremmer This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 reserved 2026-07-08 16:30 UTCglm-5.2pith:PLOQPGTVrecord.jsonopen to challenge →

Figure 1
Figure 1. Figure 1: Real and imaginary parts of 𝐹 +(1) 3 for 𝑔𝑔 → 𝑍𝐻 as a function of 𝑝𝑇 for √ 𝑠 = 1000 GeV. High-energy and 𝑡 → 0 expansions are shown including mass corrections up to 𝑚 {0,2,4} 𝑍,𝐻 . Also shown are higher mass corrections up to 𝑚 8 𝑍,𝐻 for the 𝑡 → 0 expansion with fewer expansion terms in 𝑡… reproduced from arXiv: 2607.06099
classification hep-ph hep-ex
keywords NLO QCDgluon fusionZH productiontop-quark masstwo-loop amplitudeshigh-energy expansionforward limitparton shower matching
0
0 comments X

The pith

First public code for gg→ZH at NLO QCD with full top-mass effects

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

The paper presents the C++ library ggxy, which computes next-to-leading-order (NLO) QCD corrections for gluon-initiated processes — notably gg→HH, gg→ZH, and gg→ZZ — retaining full dependence on the top-quark mass throughout. The central technical achievement is the evaluation of the two-loop virtual amplitudes, which are the computational bottleneck, by combining two analytical expansion methods: a high-energy expansion (valid when the top mass is small compared to the collision energy and momentum transfer) and a forward-limit expansion (valid when the momentum transfer is small). The paper claims that these two expansions, taken together, cover the entire physically relevant phase space, with all other parts of the NLO calculation (one-loop real corrections, subtraction, phase-space integration) performed exactly. The gg→ZH implementation is described as the first publicly available one with full top-mass dependence at NLO QCD, validated against an independent calculation at the integrated cross-section level across multiple collider energies, and interfaced to POWHEG for parton-shower matching. The library allows free variation of input parameters including the top-quark mass, its renormalization scheme, and external boson masses.

Core claim

The key mechanism is the dual-expansion strategy for two-loop amplitudes: the high-energy expansion (asymptotic, valid at large transverse momentum) and the forward-limit or t→0 expansion (Taylor-type, valid at small transverse momentum) are computed analytically to sufficient depth in mass corrections, and their regions of validity overlap around pT ~ 200 GeV for gg→ZH, allowing a seamless switch between them. A Padé approximation extends the convergence region of the high-energy expansion. This combination replaces a full exact two-loop calculation — which is computationally prohibitive — with a fast, analytically grounded approximation that the paper validates against an independent NLO结果

What carries the argument

High-energy expansion (mt² ≪ s, |t|, asymptotic) and forward-limit expansion (|t| ≪ s, mt², Taylor-type) of two-loop master integrals, combined with a Padé approximation for the high-energy branch; Catani-Seymour dipole subtraction for real corrections; POWHEG interface for parton-shower matching; CRunDec for top-mass scheme conversion.

If this is right

  • Experimental analyses of ZH production at the HL-LHC can now use NLO QCD predictions with full top-mass dependence and parton-shower matching from a public tool, reducing a source of theoretical systematic uncertainty.
  • The 30–40% spread between on-shell and MS-bar top-mass renormalization schemes at NLO — dominated by longitudinal Z polarization — quantifies a leading parametric uncertainty that must be addressed when extracting top-mass-sensitive observables from ZH data.
  • The two-loop amplitude results for gg→Z*Z* can be directly reused for gg→Z*γ* and gg→γ*γ*, extending the library's physics reach without new two-loop calculations.
  • The dual-expansion strategy, validated here for ZH and ZZ, provides a template for other gluon-initiated processes where exact two-loop amplitudes remain out of reach.

Where Pith is reading between the lines

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

  • The claim of full phase-space coverage rests on the overlap region being well-controlled; if future differential measurements probe kinematic corners where both expansions simultaneously degrade (e.g., intermediate pT with specific rapidity configurations not shown in the convergence plots), the approximation could develop uncontrolled errors not visible in integrated cross-section validation.
  • The 30–40% mass-scheme dependence, while large, could itself serve as a diagnostic: if the NLO corrections were dominated by terms that resum cleanly, the scheme dependence would be smaller. The persistence of large scheme dependence suggests substantial higher-order terms that are not yet captured, implying that NNLO corrections (if computed) could shift predictions by amounts comparable to the c
  • The factor-of-three runtime increase of ZH over HH suggests the ZH amplitude structure carries additional complexity (extra form factors, polarization channels) that scales unfavorably; this could limit the practicality of parameter scans in data-analysis pipelines unless further optimization is done.

Load-bearing premise

The two-loop amplitudes are approximations, not exact results. The claim that the high-energy and forward-limit expansions together cover the full phase space depends on their overlap region being numerically reliable and on no kinematic configuration existing where both expansions fail at once — a claim supported by convergence plots in specific projections and by agreement of integrated cross sections with one external reference, but not by exhaustive point-by-point testing

What would settle it

A differential observable (a specific bin in pT, rapidity, or invariant mass) at NLO where the ggxy prediction, using the dual-expansion two-loop amplitudes, disagrees with an independent exact two-loop calculation beyond the stated integration uncertainty — particularly in a kinematic region where the overlap between the two expansions is thin or where neither expansion is individually convergent.

Figures

Figures reproduced from arXiv: 2607.06099 by Daniel Stremmer.

Figure 2
Figure 2. Figure 2: Same as in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dependence on the top quark mass renormalization scheme for the distributions 𝑀𝑍𝐻 and 𝑝𝑇,𝐻 . Results are shown in the on-shell and MS top-quark mass renormalization schemes for different values of the top-quark mass renormalization scale 𝜇𝑡 . Lower panels show the ratio to the on-shell scheme. Figures taken from Ref. [2]. 10−4 10−3 10−2 10−1 d σ/d MZH [fb/GeV] transversal Z on-shell µt = mt(mt) µt = mZH µt… view at source ↗
Figure 4
Figure 4. Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

We present the C++ library ggxy, which can be used for the fast and flexible calculation of partonic and hadronic cross sections to gluon-initiated top-mediated processes such as $gg\to HH$, as well as the newly derived processes $gg\to ZH$ and $gg\to ZZ$, at NLO QCD including full top-quark mass dependence. The two-loop virtual amplitudes are implemented using analytical approximations in different kinematic regions, while all other parts of the calculation are exact. This implementation allows to freely modify all input parameters, such as the top-quark mass and its renormalization scheme and the masses of the external $Z$/Higgs bosons. In addition, ggxy has been interfaced to Powheg, which allows the matching to parton showers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 5 minor

Summary. These proceedings present the C++ library ggxy for computing NLO QCD corrections to gluon-initiated processes, focusing on the newly implemented gg→ZH process with full top-quark mass dependence. The two-loop virtual amplitudes are approximated using high-energy and forward-limit expansions that, combined, are claimed to cover the full phase space. All other parts of the calculation (one-loop amplitudes, real corrections, subtraction) are treated exactly. The implementation is validated against Ref. [35] at the integrated cross-section level (Table 1) and interfaced to POWHEG for parton-shower matching. The paper also presents results for gg→Z*Z* and studies the top-quark mass scheme dependence for polarized Z bosons.

Significance. The paper provides the first publicly available implementation of gg→ZH at NLO QCD with full top-quark mass dependence, which is a valuable contribution to the phenomenology program for the HL-LHC. The code is publicly available (gitlab.com/ggxy/ggxy-release) and interfaced to POWHEG, enabling parton-shower matching. The study of top-quark mass scheme dependence for polarized Z bosons (Figures 3–4) provides useful physics insight. The two-loop amplitudes rely on expansion methods rather than exact numerical evaluation, which is a legitimate and well-established approach in this field.

major comments (2)
  1. §3 and §4, Table 1: The validation against Ref. [35] shows agreement at the 0.1% level for integrated cross sections, but no differential-level comparison against an independent calculation is presented. The paper states that 'several cross-checks at the integrated and differential level' have been performed (§4), but only the integrated-level Table 1 is shown. A differential comparison (e.g., dσ/dpT,H or dσ/dMZH) against an exact calculation such as those of Refs. [20–23] would substantially strengthen the claim of full phase-space coverage, since integrated cross sections can mask compensating region-specific errors. This is particularly relevant because the central claim—that the HE and forward-limit expansions together cover the full phase space—depends on no kinematic region where both expansions fail simultaneously. Including at least one differential-level validation plot or table
  2. §3, Figures 1–2: The convergence of the two expansion methods is demonstrated at √s = 1000 GeV (Fig. 1) and pT = 150 GeV (Fig. 2), but no amplitude-level validation is shown in the near-threshold region 211 < √s < 345 GeV, where only the forward expansion is available (the HE expansion is limited to √s > 2mt ≈ 345 GeV, as acknowledged in §3). In this region, the mass-expansion parameters mZ²/s ~ 0.19 and mH²/s ~ 0.35 are not negligibly small, and |t| can be comparable to s. While the paper notes that 'such small values are never used for the high-energy expansion due to the lower limit on pT,' it would strengthen the manuscript to briefly address whether the forward expansion remains reliable in this near-threshold region, or to cite where this is documented.
minor comments (5)
  1. §4, Table 1: It is not stated whether Ref. [35] uses exact numerical two-loop amplitudes (as in Refs. [20–23]) or similar expansion methods. This distinction is relevant for assessing how independent the validation is; a brief clarification would help.
  2. §3: The paper does not quantify a systematic uncertainty from expansion truncation. A brief statement of the expected size of truncation errors, or a reference to where this is quantified in the companion papers, would be useful.
  3. §1: 'continuos' should be 'continuous'. §4: 'longituidinal' should be 'longitudinal'. §4: 'agree well within in the statistical uncertainties' has a duplicated 'in'.
  4. Figures 1–2 are taken from Ref. [3] and Figures 3–4 from Ref. [2]. While this is noted, the proceedings would benefit from at least one figure showing original validation results (e.g., a differential comparison) not already available in the companion papers.
  5. §2: The notation for the expansion orders (m{0,2,4}, t0, HE, etc.) in Figures 1–2 is compact but not fully self-explanatory; a one-line clarification of the labeling convention would improve self-containedness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Both points are well-taken and we will address them in the revised manuscript.

read point-by-point responses

  1. Referee: §3 and §4, Table 1: No differential-level comparison against an independent calculation is presented, despite the claim that 'several cross-checks at the integrated and differential level' have been performed. A differential comparison would strengthen the claim of full phase-space coverage.

    Authors: The referee is correct that the proceedings only show the integrated-level comparison in Table 1, while the text in §4 states that differential-level cross-checks have been performed. We agree that this is a gap in the presentation. The differential-level validation against the exact numerical results of Ref. [22] (Chen et al., JHEP 08 (2022) 056) is documented in detail in our companion paper Ref. [2] (arXiv:2603.15762), where dσ/dpT,H and dσ/dMZH distributions are compared and agreement at the per-mille level is found across the full kinematic range. However, we acknowledge that this is not visible in the proceedings themselves. In the revised version, we will (i) add a sentence in §4 explicitly referencing where the differential validation can be found, and (ii) if space permits within the proceedings format, include a brief differential comparison plot or table. We note that the concern about compensating region-specific errors is directly addressed by these differential comparisons, which show no such cancellations. revision: yes

  2. Referee: §3, Figures 1–2: No amplitude-level validation is shown in the near-threshold region 211 < √s < 345 GeV, where only the forward expansion is available. The mass-expansion parameters mZ²/s ~ 0.19 and mH²/s ~ 0.35 are not negligibly small, and |t| can be comparable to s. The reliability of the forward expansion in this region should be addressed.

    Authors: This is a valid concern. We should clarify the situation more explicitly in the manuscript. The forward expansion is a Taylor expansion in the small variables t, q₃², and q₄² simultaneously. In the near-threshold region 211 < √s < 345 GeV, while the mass-expansion parameters mZ²/s and mH²/s are indeed not negligibly small, the key point is that the forward expansion is used in this region only at relatively small pT (since the switch to the HE expansion occurs at larger pT), which corresponds to small |t|/s. The expansion in |t| is the primary expansion parameter, and the mass corrections are included up to sufficiently high order (m⁸ for the external masses, as shown in Figure 1) to ensure convergence. We acknowledge, however, that this reasoning is not spelled out in the proceedings. In the revised version, we will add a brief discussion addressing the reliability of the forward expansion in the near-threshold region, including a reference to Ref. [3] (arXiv:2509.07072) where the convergence properties are studied in more detail. We will also note that the differential-level validation mentioned in our response to the first comment covers this kinematic region and provides an indirect but important check on the amplitude-level accuracy. revision: partial

Circularity Check

0 steps flagged

No significant circularity found; self-citations are to first-principles amplitude calculations, validated against an independent group.

full rationale

The paper's central claim — that ggxy provides NLO QCD corrections to gg→ZH with full top-mass dependence using high-energy and forward-limit expansions — rests on two pillars: (1) the two-loop amplitude calculations from Ref. [3] (co-authored by Stremmer) and (2) validation against Ref. [35] (an independent group, Campillo Aveleira et al.). Ref. [3] derives the amplitudes from first-principles QCD using systematic Taylor and asymptotic expansions in well-defined kinematic limits, not by fitting to data or to the target observables. The expansions are defined in terms of physical kinematic parameters (s, t, m_t, external masses), not in terms of the cross sections they aim to predict. The Padé approximation (following Ref. [16]) is a standard mathematical resummation technique, not an ansatz fitted to results. Table 1 validates integrated cross sections against Ref. [35], which is an independent collaboration, providing genuine external support. Figures 1–2 (taken from Ref. [3]) demonstrate internal consistency between two independently derived expansion methods in their overlap region — a legitimate cross-check, not circular reasoning. The self-citations to Refs. [1, 2, 3] are normal practice for a software/phenomenology paper building on prior technical work, and the cited calculations are parameter-free QCD computations with externally falsifiable predictions. No step in the derivation chain reduces to its own inputs by construction. The minor self-citation load (Ref. [3] for amplitudes, Refs. [1,2] for the code) is not circular because the amplitude calculations are independent first-principles results, not fitted quantities renamed as predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper introduces no new physical entities, particles, or forces. It is a computational implementation of Standard Model processes. The free parameters (top-quark mass, renormalization scale, boson masses) are physical inputs that the user can vary, not parameters fitted to make the derivation work. The axioms are standard NLO QCD methodology assumptions plus the domain-specific claim about expansion coverage.

axioms (3)
  • domain assumption The high-energy and forward-limit expansions of the two-loop amplitudes, when combined, cover the full physically relevant phase space with sufficient accuracy.
    This is the central premise of the method, invoked in Section 2 and demonstrated numerically in Section 3 (Figures 1–2). The expansions are asymptotic/Taylor series whose convergence must be verified kinematic point by kinematic point.
  • standard math The Catani-Seymour dipole subtraction scheme correctly handles the infrared singularities in the real corrections for these processes.
    Standard NLO QCD methodology, invoked in Section 4. Well-established in the literature (Ref. [30]).
  • standard math The one-loop 2→3 amplitudes computed by Recola/Collier/CutTools/OneLOop are correct and stable.
    These are established public tools for one-loop calculations, invoked in Section 4. Their correctness is assumed but not independently verified in this paper.

pith-pipeline@v1.1.0-glm · 11717 in / 2617 out tokens · 489879 ms · 2026-07-08T16:30:36.244825+00:00 · methodology

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Reference graph

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