Electroweak Higgs boson pair production: Updated inclusive cross sections

Alexander Karlberg; Gioia Sacchi; Mathieu Pellen; Michael Spira; Ramona Gr\"ober

REVIEW 2 major objections 4 minor 1 cited by

This paper supplies the inclusive cross sections that experiments should use as the reference for electroweak Higgs boson pair production at the LHC and HL-LHC.

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

2026-08-02 19:14 UTC pith:37BZ3UFL

load-bearing objection Useful reference numbers for EW HH production, but the EW uncertainty claim in Table 1 is not valid; fix it before quoting. the 2 major comments →

arxiv 2603.02764 v2 pith:37BZ3UFL submitted 2026-03-03 hep-ph hep-ex

Electroweak Higgs boson pair production: Updated inclusive cross sections

Ramona Gr\"ober , Alexander Karlberg , Mathieu Pellen , Gioia Sacchi , Michael Spira This is my paper

classification hep-ph hep-ex

keywords Higgs boson pair productionvector-boson fusionassociated productionN3LO QCDNNLO QCDNLO electroweak correctionstrilinear Higgs self-couplingLHC cross sections

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.

The paper gives updated inclusive cross sections for the two electroweak Higgs-pair production channels at the LHC and HL-LHC. For vector-boson fusion (VBF) it reports values at third-order QCD combined with next-to-leading-order electroweak corrections, about 1.87 fb at 13.6 TeV; for associated W/Z + hh production it reports second-order QCD values around 0.36-0.40 fb. The numbers are provided at three beam energies, three Higgs masses, and four values of the trilinear Higgs self-coupling. The practical claim is that these tables are the current reference for experimental searches and for translating limits into constraints on the Higgs self-coupling.

Core claim

The central claim is that the inclusive cross sections in Tables 1-5 are the most up-to-date theoretical predictions for electroweak Higgs boson pair production. For VBF hh, the paper combines N3LO QCD with NLO electroweak corrections multiplicatively, obtaining sigma = 1.870 fb at a center-of-mass energy of 13.6 TeV for mh = 125.09 GeV, with scale uncertainties below 0.1% and a 2.7% PDF uncertainty. For W+hh, W-hh, and Zhh, the NNLO QCD values at 13.6 TeV are 0.357, 0.187, and 0.396 fb respectively; the Zhh channel carries larger scale uncertainties (about 3%) because of gluon-initiated contributions. The kappa_lambda dependence, spanning kappa_lambda = 0, 2, 3, is provided so that anomalou

What carries the argument

For VBF hh, the machinery is the factorised VBF approximation: the two quark lines are treated as independent, s-channel diagrams are excluded, t/u interference is dropped, and the NLO electroweak correction is applied as a multiplicative factor to the N3LO QCD cross section, with the renormalisation/factorisation scale set by the di-Higgs transverse momentum. For Vhh, the machinery is off-shell vector-boson production followed by the splitting V* -> V hh, computed at NNLO QCD; for Zhh, gluon-initiated triangle, box, and pentagon diagrams with a finite bottom mass are included, and the scale is the invariant mass of the V hh system.

Load-bearing premise

The load-bearing premise is that the factorised VBF approximation — no s-channel diagrams, no t/u interference, and QCD and EW corrections multiplying independently — is accurate for the inclusive cross section at the sub-percent level; if it is not, the VBF numbers in Table 1 shift.

What would settle it

A calculation of inclusive VBF hh at NLO QCD that keeps s-channel diagrams and t/u interference, compared with the factorised N3LO result, would settle the matter: if the difference exceeds the quoted ~0.05% scale uncertainty, the central values are not robust. A second test is to compute photon-induced VBF contributions with a photon PDF; if they exceed 1%, the stated uncertainty budget is incomplete.

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

If this is right

If these numbers stand, they become the baseline for LHC experiments searching for VBF and Vhh double-Higgs production, replacing older handbook values.
The sub-percent scale uncertainties on VBF hh imply that theory is not the limiting factor for VBF interpretations; PDF uncertainty dominates the error budget.
The separate W+hh and W-hh values allow independent constraints on the couplings of two Higgs bosons to W and Z bosons, something VBF alone cannot provide.
The kappa_lambda rows let any future measurement be mapped directly onto a value or bound for the trilinear self-coupling within the assumed coupling-modifier framework.
The larger scale uncertainty on Zhh means that channel's interpretive power is limited by missing higher-order or top-mass-renormalisation effects, as the paper itself notes.

Where Pith is reading between the lines

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

If full non-factorisable and s-channel contributions were included for inclusive VBF hh, the quoted sub-percent scale uncertainty would be tested; the paper's own numbers give the benchmark for that comparison.
The assertion that electroweak uncertainties are fully contained in the PDF uncertainty is untested; a dedicated estimate would either confirm the 2.7% band or enlarge it.
The tables show the cross section rising steeply for kappa_lambda = 3 and dropping for kappa_lambda = 0 in the associated-production channels, suggesting these modes could act as discriminators for the sign and magnitude of the self-coupling even though their SM rates are small.
The omission of photon-induced contributions is stated to be below 1%, but since the recommended PDF set lacks a photon, an independent check with a photon-containing PDF would settle whether that estimate holds.

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

Useful reference numbers for EW HH production, but the EW uncertainty claim in Table 1 is not valid; fix it before quoting.

read the letter

Punchline: this is a useful update of reference cross sections for EW HH production, with the new content being PDF4LHC21, 13.6 TeV, and κλ=0,2,3. The numbers are credible, but the caption to Table 1 overclaims about EW uncertainties, and that should be fixed before experiments quote these as final.

What's new: the underlying calculations are from public codes like PROVBFHH and RECOLA+MOCANLO, and the original papers are cited properly. The setup is clearly specified: masses, widths, scale choice, PDF set. The tables give state-of-the-art QCD+EW values for VBF HH at N3LO QCD+NLO EW, and NNLO QCD for Vhh. The VBF approximation is defined explicitly, and using it for an inclusive cross section is standard practice in this field. The numbers look internally consistent.

The soft spot: the statement in Table 1's caption that EW uncertainties are 'fully contained in the PDF uncertainty' is not logically sound. PDF uncertainties are determined by the PDF fit; they don't reflect missing higher-order electroweak corrections. The NLO EW correction is included, but the uncertainty due to two-loop EW effects or scheme dependence is not quantified. For a table intended as a reference for ATLAS and CMS, the error budget should either include an estimate of missing EW corrections or explicitly state that they are not included. Right now the caption invites the reader to treat the total uncertainty as complete, which it isn't. This is a local, fixable issue. The cross sections themselves are computed with established codes and are not the concern.

Other notes: the Zhh numbers use mb=4.9 GeV for the gluon-initiated contributions while the rest uses mb=0; the text acknowledges this, so it's not a hidden inconsistency. The κλ tables without EW corrections are clearly labeled.

Who should read this: experimental colleagues who need central values and uncertainties for their HH searches, and anyone who wants an up-to-date reference. It doesn't change physics but it's exactly what a community report should provide.

Recommendation: send to peer review. A referee should ask for a correction to the EW uncertainty statement, but the paper does not require major rework.

Referee Report

2 major / 4 minor

Summary. This short community note presents updated inclusive cross sections for electroweak Higgs boson pair production at the LHC and HL-LHC. For vector-boson fusion (VBF) production of hh, the authors quote cross sections at N3LO QCD+NLO EW accuracy, obtained by combining N3LO QCD results with the NLO EW corrections of Ref. [41] in a multiplicative way (Eq. (4)). For associated V hh production (W+hh, W−hh, Zhh), the paper quotes NNLO QCD cross sections. All numbers are given for √s = 13, 13.6, 14 TeV, for mh = 125, 125.09, 125.38 GeV, and for κλ = 1 (SM) together with κλ = 0, 2, 3. The setup is described in Section 2: PDF4LHC21 PDFs, αs(MZ)=0.118, GF scheme, and a scale choice for VBF (Eq. (3)) different from that used for V hh. The tables in Sections 3 and 4 are intended as reference values for experimental searches.

Significance. If the quoted numbers are correct, this paper provides a convenient, up-to-date compilation of the highest-order inclusive cross sections for two important Higgs-pair production modes, and the tables would indeed be valuable reference values for ATLAS and CMS. A strength of the manuscript is that the calculations are based on publicly available codes (PROVBFHH v2.1.0, RECOLA+MOCANLO) with a clearly specified setup, and the κλ dependence is included for anomalous coupling interpretations. The main weakness is that the uncertainty treatment, especially the EW uncertainty in the VBF channel, is not fully satisfactory; this is a local but important caveat for a reference document. The VBF approximation itself is standard and clearly stated, so the numerical central values are credible.

major comments (2)

[Section 3, Table 1 caption] The caption states that 'no EW uncertainties are included, but that they are assumed to be fully contained in the PDF uncertainty which dominates.' This is not a valid assumption: PDF uncertainties reflect the knowledge of parton distribution functions and are unrelated to missing higher-order electroweak corrections. The NLO EW correction is included, but the uncertainty from neglected higher-order EW effects (e.g., two-loop EW and scheme dependence) is not quantified. The scale uncertainty is only about ±0.05%, so an unquantified EW uncertainty of order 1% — which is plausible for an NLO EW correction at the LHC — would be comparable to or larger than the scale uncertainty and could be a sizable fraction of the total quoted uncertainty. For a reference value to be used by experiments, this should be either estimated and included or explicitly flagged as a separate missing uncertainty i
[Section 4.2, Eq. (1) vs. mb=4.9 GeV] The general setup in Eq. (1) sets mb = 0 GeV, but the gg→Zhh contributions (triangle, box, pentagon diagrams) are computed with mb = 4.9 GeV. This is mentioned in the text, but the numerical impact of this choice on the Zhh cross sections is not assessed. Since the gg-initiated contribution is non-negligible for Zhh and the scale uncertainty is larger than in W±hh, the paper should either quantify the effect of the bottom-mass treatment or justify why it is negligible at the reported precision. This is relevant for the consistency of the quoted Zhh numbers.

minor comments (4)

[Section 3, Eq. (4)] The multiplicative combination of N3LO QCD and NLO EW corrections is clearly stated, but the paper does not comment on the size of the NLO EW correction δVBF_NLO EW/σVBF_LO. A single sentence with the numerical value at, say, √s=13.6 TeV would help the reader judge the importance of the missing EW uncertainty.
[Section 4.2, formatting] There are minor formatting issues: 'g g' should be 'gg' in several places, and the sentence 'Those contributions have been computed in [35] on which the numbers presented here are based' is awkward. No scientific content is affected.
[Tables 3–5 captions] The captions state 'including scale uncertainties and PDF uncertainties in percent in this order at NNLO QCD'. The phrase 'in this order' is unclear; consider rephrasing to 'at NNLO QCD in QCD' or similar.
[General] The paper relies on public codes, which is good, but it would be helpful for reproducibility to list the exact version of RECOLA/MOCANLO (MoCaNLO is cited as a 2026 preprint) and, if possible, to provide a short table of input parameters or a link to a repository with the numbers.

Circularity Check

0 steps flagged

No significant circularity: the note transparently recomputes published cross sections with updated inputs; self-citations are to independent public codes.

full rationale

This is an LHC Higgs Working Group note that updates inclusive cross sections using existing theoretical calculations, not a derivation that fits parameters to its own target. The paper states this explicitly: 'since we present cross section updates based on existing theoretical calculations, this note is rather short. We therefore refer the reader to the original references for details on the various calculations.' The VBF numbers are produced with the publicly available code PROVBFHH and RECOLA+MOCANLO, citing the original independent calculations; no parameter is fitted to the quoted cross sections. The combination in Eq. (4) is a standard multiplicative VBF approximation, and the paper clearly labels it as such. The only flagged weakness is the Table 1 caption statement that 'no EW uncertainties are included, but that they are assumed to be fully contained in the PDF uncertainty which dominates.' That is an acknowledged limitation and an uncertainty-assumption, not a circular reduction: it does not define the cross section in terms of itself, and it does not make the central numbers equivalent to their inputs. Similarly, Section 5 notes a missing top-mass renormalisation uncertainty for Zhh, which is another stated limitation. Self-citations to prior code papers are appropriate because those codes are publicly available, parameter-free implementations with stated assumptions that do not include the present target cross sections. No load-bearing argument reduces to a self-citation chain. The result is a set of numerical predictions from independent published calculations, so the circularity score is 0.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 0 invented entities

The paper introduces no new entities. The central claim (cross section values) depends on standard inputs, the VBF approximation, and a multiplicative EW combination. The only new 'parameters' are the scale choices and the specific PDF/energy settings, all external.

free parameters (4)

αs(MZ) = 0.118
Taken from the PDF4LHC21 set; external input, not fitted to the target cross sections.
mt, mb, MZ, MW, GF = mt=172.5 GeV, mb=0 GeV (PDF), mb=4.9 GeV (Zhh loops), MZ=91.1876 GeV, MW=80.379 GeV, GF=1.16638e-5 GeV^-2
Standard PDG inputs, chosen by hand; not fitted. mb=0 in general setup, but mb=4.9 GeV for gg→Zhh triangle/box/pentagon diagrams (Section 4.2).
PDF set = PDF4LHC21_40
External global fit; not a parameter of this paper.
Renormalisation/factorisation scales = VBF: μ = sqrt(mh/2 * sqrt((mh/2)^2 + pT,hh^2)) (Eq 3); Vhh: μR=μF=MVhh
Scale choices are hand-chosen per process; central values affect the quoted uncertainties.

axioms (7)

domain assumption Standard Model validity and perturbative QCD/EW factorisation
Used throughout; standard for cross-section predictions.
domain assumption VBF approximation: s-channel omitted, t/u only, no interference, factorised quark lines
Section 3 defines VBF in this 'usual sense' and applies it consistently at all orders.
domain assumption N3LO QCD corrections for VBF obtained in factorised approximation via DIS structure functions
PROVBFHH uses HOPPET with parametrised 2/3-loop DIS coefficient functions; this is an approximation to full N3LO VBF.
ad hoc to paper Multiplicative combination of N3LO QCD and NLO EW corrections (Eq. 4)
The paper combines N3LO QCD and NLO EW multiplicatively, which is a standard but not rigorously justified ansatz.
domain assumption EW uncertainties are fully contained in PDF uncertainties
Table 1 caption states this without quantitative evidence; a fragile assumption.
domain assumption Photon-induced contributions are negligible for VBF
Section 3 cites Ref. [33] that these are below 1% in MS scheme; the paper nevertheless omits them for consistency.
domain assumption For gg→Zhh, loop contributions with mb=4.9 GeV are relevant and correctly handled
Section 4.2 states that the triangle/box/pentagon diagrams need a non-zero bottom mass; this is an external choice.

pith-pipeline@v1.3.0-alltime-deepseek · 12333 in / 9951 out tokens · 91215 ms · 2026-08-02T19:14:46.268909+00:00 · methodology

0 comments

read the original abstract

We present updated inclusive cross sections for electroweak Higgs boson pair production for energies of relevance to the LHC and High-Luminosity phase of the LHC. The cross sections are presented at N$^3$LO QCD+NLO EW for vector-boson fusion and NNLO QCD for associate production with a vector boson. We compute the cross sections using the most up-to-date theory inputs, both in the Standard Model and for a few anomalous values of the trilinear Higgs self-coupling.

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Precise predictions for double Higgs production in association with a vector boson in Effective Field Theory
hep-ph 2026-07 conditional novelty 6.0

First NNLO QCD cross sections for pp→Vhh in SMEFT and HEFT, provided as interpolated coefficients; W±hh K-factors are almost EFT-independent while Zhh is sensitive to gluon-initiated contributions.

Reference graph

Works this paper leans on

4 extracted references · cited by 1 Pith paper

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