Precise predictions for double Higgs production in association with a vector boson in Effective Field Theory

Gioia Sacchi; Micha{\l} Ryczkowski; Ramona Gr\"ober

REVIEW 2 major objections 4 minor 47 references

NNLO QCD corrections to Vhh production factorize through Drell–Yan coefficient functions, making W±hh K-factors nearly flat while Zhh K-factors vary from 1.25 to 1.6.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · deepseek-v4-flash

2026-08-01 09:11 UTC pith:R5L26OTD

load-bearing objection First NNLO QCD Vhh predictions in SMEFT/HEFT, with solid W±hh results, but the Zhh physics rests on a gg→Zhh amplitude that is never shown – that needs to be fixed before the central claim is usable. the 2 major comments →

arxiv 2607.27334 v1 pith:R5L26OTD submitted 2026-07-29 hep-ph hep-ex

Precise predictions for double Higgs production in association with a vector boson in Effective Field Theory

Ramona Gr\"ober , Micha{\l} Ryczkowski , Gioia Sacchi This is my paper

classification hep-ph hep-ex

keywords NNLO QCDHiggs pair productionVhh associated productionSMEFTHEFTDrell-Yan factorizationK-factorgluon fusion Zhh

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.

Double Higgs production in association with a W or Z boson, pp→Vhh, is a rare but direct probe of the Higgs self-coupling and of whether the Higgs sector behaves as the Standard Model predicts. This paper presents NNLO QCD cross sections for Vhh in both SMEFT (dimension-six, truncated at 1/Λ²) and HEFT, built on the observation that the quark-initiated process factorizes into Drell–Yan production of an off-shell vector boson followed by its decay into Vhh. The central finding is that QCD corrections largely factorize from the EFT dependence for W±hh, so the K-factors are almost flat over the scanned parameter ranges, while Zhh production shows a stronger K-factor dependence on EFT coefficients, driven by the loop-induced gg→Zhh channel. Results are cast as numerical coefficient tables (A_i in HEFT, B_i in SMEFT), letting users evaluate cross sections for arbitrary EFT parameters at 13.6 and 14 TeV, including scale, PDF, and αs uncertainties.

Core claim

The paper's central claim is that the NNLO QCD corrections to Vhh production factorize from the electroweak EFT structure for quark-initiated channels: the process can be viewed as Drell–Yan production of an off-shell vector boson followed by V*→Vhh decay, with universal coefficient functions Δ(z) convoluted with the LO EFT amplitude. This factorization is applied to SMEFT operators that create new Lorentz structures and contact topologies, justified because those interactions are still built from color-singlet quark currents. For W±hh this yields nearly flat K-factors across the EFT parameter ranges; for Zhh, the additional one-loop gg→Zhh subprocess entering at NNLO introduces sensitivity

What carries the argument

The key object is the Drell–Yan factorization structure: the inclusive cross section is written as a convolution of the LO partonic cross section for V*→Vhh with the universal NNLO Drell–Yan coefficient functions Δ_qq, Δ_qg, Δ_gg, supplemented for Zhh by the heavy-quark loop-induced gg→Zhh amplitude. In SMEFT, the LO cross section is modified through rescaled couplings A–D and f_i structures, and genuine contact-topology terms are encoded in new form factors F_Vq and F_X2φ2; the same Drell–Yan coefficient functions are applied to the full SMEFT hard function, which is the load-bearing assumption.

Load-bearing premise

The NNLO QCD corrections to the quark-initiated channels are assumed to be given exactly by the universal Drell–Yan coefficient functions applied to the full SMEFT and HEFT hard function, including the new contact and derivative topologies that do not exist in the Standard Model.

What would settle it

Compute the partonic NLO QCD corrections to the genuine SMEFT contributions—the F_Vq and F_X2φ2 form factors of Eq. 57—directly from their Feynman diagrams; if the resulting NLO-to-LO ratios differ from the Drell–Yan coefficient functions Δ(1), the factorization premise and the flat W±hh K-factor conclusion are refuted. A full NNLO computation of one benchmark point would settle the question definitively.

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

If this is right

For W±hh, NNLO QCD corrections can be treated as an almost universal factor over the scanned EFT space, with scale and PDF+αs uncertainties staying below about 4%.
For Zhh, the K-factor varies from roughly 1.25 to 1.6 as κλ ranges over its allowed interval, so EFT interpretations must use point-by-point NNLO corrections rather than a global K-factor.
The published A_i and B_i coefficient tables allow fast reconstruction of inclusive cross sections for arbitrary EFT parameters at 13.6 and 14 TeV, including uncertainty estimates.
The gg→Zhh component both enlarges the theoretical uncertainty (above 5% in parts of the HEFT parameter space) and adds sensitivity to top-quark operators that the quark-induced channel alone would not provide.
Current experimental bounds (183×σSM and 294×σSM) are an order of magnitude looser than the maximal EFT enhancements found here, so the improved NNLO predictions are ready for HL-LHC sensitivity.

Where Pith is reading between the lines

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

If the Drell–Yan factorization holds, the known N3LO Drell–Yan coefficient functions could be substituted for the NNLO ones to upgrade Vhh predictions to approximate N3LO accuracy without a new multi-loop Vhh calculation.
A direct test of the assumption would be to compute NLO QCD corrections to the genuine SMEFT contact-topology amplitudes (F_Vq and F_X2φ2 of Eq. 57); if their K-factors differ from the Drell–Yan coefficients, the flat-K conclusion for W±hh would need revision.
The strong Ctφ dependence of gg→Zhh suggests that Zhh measurements at the HL-LHC could serve as a complementary handle on top-quark-Z couplings, alongside tt̄Z production.
The large PDF-driven uncertainty found for C(1)φq hints that improved high-x parton densities would sharpen SMEFT constraints extracted from Vhh rates.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

First NNLO QCD Vhh predictions in SMEFT/HEFT, with solid W±hh results, but the Zhh physics rests on a gg→Zhh amplitude that is never shown – that needs to be fixed before the central claim is usable.

read the letter

You should know two things about this paper. It is genuinely the first to give NNLO QCD cross sections for pp→Vhh in SMEFT and HEFT, with a clean coefficient parametrisation (A_i and B_i) that lets other people evaluate cross sections at arbitrary EFT points. That is a real service to the subfield. And the W±hh part of the paper is probably fine: the Drell-Yan factorisation claim there is standard, the uncertainties look carefully treated, and the flat-K-factor conclusion is physically plausible.

The soft spot is exactly where the stress-test points. The Zhh channel's distinctive behaviour – the κλ-dependent K-factor between 1.25 and 1.6, the top-operator sensitivity – comes entirely from the loop-induced gg→Zhh piece. But the text never writes down that amplitude. Section 3.1 says the heavy-quark loop topologies "have to be included"; Sections 3.3 and 3.4 jump straight to the polynomial coefficient fits (Eqs. 46 and 75) and the tables. No matrix element, no phase-space integral, no loop evaluation, no demonstration that the EFT insertions in the box and pentagon diagrams are handled correctly. The triangle cancellation is deferred to Refs. [50,51]. So the single most physics-loaded ingredient of the paper is a black box.

That is disproportionate to the rest of the work, which is transparent about method and inputs. It is also easy to fix in principle: the authors can put the gg→Zhh amplitude (as a function of x1, x2, µh, µV and the EFT coefficients) in the paper or in an ancillary file, and release the code. They mention a GitHub repository but give no link in the text, and only central coefficient values appear in the tables; the promised uncertainties live somewhere unnamed. That is a reproducibility gap, not a fatal one.

One more point, agreeing with the stress-test over the reader's weaker worry: the concern that new SMEFT Lorentz structures might break DY factorisation in the quark-induced channels is unlikely to bite. The new operators are still colour-singlet, vector-like quark currents, so soft/collinear universality should hold. I would not hold the paper hostage to that concern.

So: worth refereeing seriously. A referee should ask for the gg→Zhh derivation or a public code, plus a working repository link, before this goes in. With those, the paper is a solid, modestly useful contribution for Higgs EFT phenomenology at the HL-LHC. Without them, the Zhh numbers are unverifiable.

Class: maybe bring to reading group; I would not cite the Zhh predictions until the amplitude is visible, but I might cite the W±hh coefficient tables.

Referee Report

2 major / 4 minor

Summary. The paper presents NNLO QCD predictions for pp→Vhh (V=W±,Z) in both SMEFT (dim-6, truncated at 1/Λ²) and HEFT (LO chiral expansion). The quark-induced contribution is obtained by convoluting the LO EFT hard function with standard Drell-Yan coefficient functions, while the Zhh channel is supplemented by the loop-induced gg→Zhh contribution that enters at NNLO. Cross sections are parametrized by numerical coefficients A_i (HEFT) and B_i (SMEFT), evaluated at √s=13.6 and 14 TeV, with scale, PDF, and αs uncertainties. The main physics claims are that W±hh K-factors are nearly flat in EFT parameters because QCD corrections factorize from the EFT dependence, while Zhh K-factors vary from ~1.25 to 1.6 with κλ and are additionally sensitive to top-quark operators through the gg→Zhh component.

Significance. If the results are correct, the paper provides a useful phenomenological tool for HL-LHC studies: the explicit LO structure, the DY convolution setup, the tabulated HEFT/SMEFT coefficients, and the uncertainty prescriptions are all directly usable for fast evaluations over the considered parameter ranges. The authors are also to be credited for making the coefficient tables available in a repository and for clearly stating the scales/PDF sets used. However, the most distinctive physics claim—the Zhh K-factor variation and top-operator sensitivity—rests entirely on the gg→Zhh amplitude, whose derivation is not shown. Until that ingredient is made verifiable, the central conclusions cannot be fully assessed.

major comments (2)

[§3.1, §3.3–3.4; Eqs. (46), (75); Tables 5, 14]
[§4.1 and §3.4 around Eq. (57) and p.16]

minor comments (4)

[§4.2] The sentence 'Tables 12, 13 and 14 show the numerical results for the B_i coefficients in Eqs. (73), (74) and (76)' likely refers to Eq. (75) for the NNLO Zhh parametrisation; Eq. (76) is the input parameter list. Please correct the cross-reference.
[§4] The GitHub repository is mentioned but the URL/DOI is not given. Please provide a direct link or DOI so that the coefficient tables and any supporting code can be located.
[§3.3–3.4] The interpolation procedure is described only as solving a linear system at 'a sufficient number of simulated phase space points.' Please report the number of points, the parameter ranges, and the interpolation residuals for the A_i and B_i coefficients. This is important for judging the accuracy of the tabulated central values.
[§4.1] In the 'one non-SM κ at a time' scans in Figs. 4 and 5, the caption states that all other κ coefficients are set to their SM values, but it does not say whether the scans are restricted to the intervals in Table 2. Please clarify in the captions.

Circularity Check

1 steps flagged

No derivational circularity: SM NNLO baseline is externally benchmarked; EFT predictions are original convolutions. Only the coefficient tables are interpolations of the paper's own calculation, a presentation-level fit.

specific steps

fitted input called prediction [Sec. 3.3 (Eqs. 43-46) and Sec. 4.1 (text after Table 2); analogous SMEFT B_i procedure in Secs. 3.4/4.2]
"The central values of these coefficients are determined by solving a system of linear equations derived from Eqs. (44) and (46), evaluated at a sufficient number of simulated phase space points. ... These tables enable the computation of predictions for any set of LO HEFT effective couplings."

The A_i coefficients are solved from the very equations (44)/(46) that define the cross-section parametrization, using the authors' own matrix-element outputs as 'simulated data points'. Any cross section or K-factor subsequently labeled a 'prediction' is therefore the interpolation surface by construction, not an independent result. This is a mild, presentation-level circularity: the underlying LO matrix-element and DY-convolution calculation is independent, so the physics conclusions do not reduce to the fit; only the coefficient-table deliverable is a self-fit.

full rationale

The central physics derivation is not circular. The SM NNLO baseline, although cited from the authors' Refs. [1] and [9], is built on the external Drell-Yan coefficient functions of Ref. [23] (Hamberg-van Neerven-Matsuura) and is benchmarked against the fully differential NNLO computations of Refs. [10,11] (Li/Wang), so the self-citations are backed by independent, externally falsifiable results. The EFT part is an original calculation: LO HEFT/SMEFT matrix elements are generated with FeynArts/FormCalc/SmeftFR and convoluted with standard DY kernels; the flatness of the W+/-hh K-factor is a computed output, not assumed. The one genuine presentation-level circularity is the coefficient tables: A_i (and B_i) are obtained by solving the same polynomial equations they are later used to evaluate, so the 'predictions for arbitrary EFT parameters' are interpolations of the paper's own calculation. I do not count this as load-bearing for the physics, because the interpolated surface encodes the same matrix-element content. The largest actual risk is not circularity but verification: the gg->Zhh amplitude is never displayed; Section 3.1 only states Eq. (28), and Sections 3.3/3.4 move directly to coefficient parametrizations, with the triangle-diagram cancellation delegated to Refs. [50,51], of which Ref. [51] is an in-preparation self-citation. That is a reproducibility/validation gap, not a circular derivation. The DY factorization assumption for the new SMEFT form factors is likewise an assumption, not a circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

No new particles, forces, or mediators are introduced; the only new objects are analytic form factors and interpolated coefficients, neither of which are physical entities.

free parameters (2)

HEFT interpolation coefficients A_i (W+, W-, Z; LO and NNLO) = Tabulated: e.g. A^{Z}_{11,NNLO}=0.4028 fb
Determined by solving a linear system at simulated phase-space points generated from the paper's own matrix-element formulas; not physical inputs, but the published parametrization relies on them.
SMEFT interpolation coefficients B_i = Tabulated: e.g. B^{W+}_{4,NNLO}=30.3666 fb·TeV²
Same numerical interpolation procedure; linear order in 1/Λ²; no error bars attached to the coefficients themselves.

axioms (4)

domain assumption Quark-induced NNLO QCD corrections factorize as DY V* production with universal coefficient functions Δ(z) applied to the LO EFT hard function.
Assumed throughout Sec. 3.1/3.4, especially in Eqs (29)–(34) and for contact operators with new Lorentz structures (Eq. 57).
domain assumption SMEFT truncated at dimension six with single insertions; cross sections kept linear in 1/Λ²; light Yukawas and CP-odd operators neglected.
Defines the SMEFT calculation in Secs 2.1/3.4; standard framework, but the truncation means only interference-type contributions are included.
domain assumption HEFT LO chiral Lagrangian with multiplicative κ-factors and power counting Nχ - Ngs (Eq. 9).
Sec 2.2; this separates QCD and chiral orders and justifies the κ-framework substitution in Eq. (42).
domain assumption gg→Zhh EFT amplitudes are evaluated at one loop with SMEFT vertex insertions (SmeftFR) and are not corrected beyond this order.
Secs 3.1/3.4; the gluon-induced term drives Zhh EFT dependence, but its calculation is not shown in detail.

pith-pipeline@v1.3.0-daily-deepseek · 25235 in / 16629 out tokens · 160138 ms · 2026-08-01T09:11:07.479522+00:00 · methodology

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read the original abstract

We present next-to-next-to-leading order (NNLO) QCD predictions for Higgs boson pair production in association with a weak gauge boson, $pp \to Vhh$ with $V=W^\pm,Z$, in effective field theory descriptions of new physics. We consider both the Standard Model Effective Field Theory (SMEFT), truncated at dimension six, and the Higgs Effective Field Theory (HEFT) at leading order in the chiral expansion. The calculation builds on the factorisation of the quark-induced production process into Drell-Yan production of an off-shell vector boson and its subsequent decay into $Vhh$, supplemented in the $Zhh$ channel by the loop-induced $gg \to Zhh$ contribution that first enters at NNLO QCD. We provide inclusive predictions at $\sqrt{s}=13.6$ and $14.0$ TeV, including scale, PDF, and $\alpha_s$ uncertainties, and express the results in terms of numerical coefficients that allow fast evaluations for arbitrary EFT parameters in the considered ranges. We find that, for $W^\pm hh$ production, QCD corrections largely factorise from the EFT dependence, leading to almost flat $K$-factors. In contrast, $Zhh$ production shows a stronger dependence on the EFT coefficients because of the gluon-induced component.

Figures

Figures reproduced from arXiv: 2607.27334 by Gioia Sacchi, Micha{\l} Ryczkowski, Ramona Gr\"ober.

**Figure 2.** Figure 2: Illustrative diagrams for the gluon initiated process. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Example diagrams featuring genuine SMEFT contributions to quark-initiated [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: W+hh (left) and W−hh (right) cross section at NNLO QCD and K-factor as a function of various κ coefficients at √ s = 13.6 TeV. All other κ coefficients are set to their SM values. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

**Figure 5.** Figure 5: Zhh cross section at NNLO QCD and K-factor as a function of various κ coefficients at √ s = 13.6 TeV. All other κ coefficients are set to their SM values. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗

**Figure 6.** Figure 6: Cross section and K-factor of W+hh (left) and W−hh (right) as function of Cφ (top) and CφW (bottom) at √ s = 13.6 TeV. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗

**Figure 7.** Figure 7: Cross section and K-factor of W+hh (left) and W−hh (right) as function of Cφ□ (top) and C (3) φq (bottom) at √ s = 13.6 TeV. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

**Figure 8.** Figure 8: Cross section and K-factor of Zhh as a function of Cφ (left top), Cφ□ (right top), CφW (left bottom) and Ctφ (right bottom) at √ s = 13.6 TeV. behaviour makes W±hh a particularly clean channel for interpreting deviations in terms of modified Higgs self-interactions and modified couplings of one or two Higgs bosons to electroweak gauge bosons. The situation is different for Zhh production. In this channel, … view at source ↗

discussion (0)

Reference graph

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