A new probe of the quartic Higgs self-coupling

Aparna Sankar; Giulia Zanderighi; Ulrich Haisch

REVIEW 2 major objections 4 minor 2 cited by

A new probe of the quartic Higgs self-coupling

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Two-loop wave-function renormalization makes the quartic Higgs self-coupling visible in single-Higgs production rates.

desk verdict The two-loop WFR calculation is serious and well-documented, but the κ4^3 term in Eq. (2.5) is topologically impossible at two loops, which undermines the FCC-ee projection until corrected. read the letter →

arxiv 2505.20463 v1 pith:O7DN7CXT submitted 2025-05-26 hep-ph hep-ex

Ulrich Haisch , Aparna Sankar , Giulia Zanderighi This is my paper

classification hep-phhep-ex

keywords Higgsself-couplingquarticcouplingtwo-loopwave-functionrenormalizationsingle-HiggsproductionFCC-eeHL-LHCSMEFTprecision

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Measuring the quartic Higgs self-coupling is one of the hardest tasks in Higgs physics because it usually requires rare multi-Higgs events. This paper shows a cheaper indirect route: at two loops, the quartic-coupling modifier $\kappa_4$ enters the universal wave-function renormalization of external Higgs lines, and it does so quadratically. Since that rescaling applies to every on-shell Higgs process, precision single-Higgs measurements inherit a $\kappa_4^2$ dependence. Using projected HL-LHC and FCC-ee uncertainties, the paper finds that FCC-ee could constrain $\kappa_4$ to roughly $-7 < \kappa_4 < 15$ at 68% CL when $\kappa_3 = 1$, a reach comparable to double-Higgs production. The result opens single-Higgs rates as an independent window on the quartic self-interaction.

What carries the argument

The central object is the on-shell Higgs wave-function renormalization constant $Z_h$. Its two-loop correction $\delta Z_h^{(2)}$ is obtained by reducing the pure-Higgs two-loop self-energy diagrams with integration-by-parts identities to a small set of scalar master integrals, whose on-shell analytic values are given in closed form. The mechanism that carries the physical argument is the universal rescaling factor $(Z_h^{\rm OS}/Z_h^{\rm MS})^{n/2}$ in Eq. (2.9): expanded to $O(\lambda^2)$, it multiplies every $n$-Higgs amplitude by a factor containing the finite one-loop and two-loop WFR corrections, so the $\kappa_4^2$, $\kappa_3\kappa_5$, $\kappa_3^3$, $\kappa_3^2\kappa_4$, and $\kappa_4^3$ terms enter all on-shell single-Higgs rates through the same coefficient.

What would settle it

Compute the full two-loop $e^+e^-\to Zh$ amplitude keeping the process-dependent $\kappa_3^3-1$ corrections and evaluate the shift at $\mu=m_h$; if the resulting change in the signal strength exceeds the 0.20–0.28% FCC-ee uncertainties used in the fit, the projected bound $-7 < \kappa_4 < 15$ would not survive.

Watch

Extended reading notes

Core claim

The main result is the two-loop correction to the on-shell Higgs wave-function renormalization constant in a general modified-Higgs-potential parameterization (Eq. (2.5)). Its finite part contains terms proportional to $\kappa_4^2$, $\kappa_3\kappa_5$, $\kappa_3^3$, $\kappa_3^2\kappa_4$, and $\kappa_4^3$, with coefficients built from $\zeta(2)$, $\zeta(3)$, and $\mathrm{Cl}_2(\pi/3)$. In the Standard-Model limit $\kappa_3=\kappa_4=1$, $\kappa_5=0$, the expression reproduces the known two-loop SM Higgs anomalous dimension, which the paper uses as a cross-check. The physical claim is that this universal two-loop WFR term makes the quartic Higgs coupling visible in single-Higgs production and decay, playing the same role for $\kappa_4$ that the one-loop WFR term plays for $\kappa_3$.

Load-bearing premise

The projections assume that the universal wave-function rescaling captures all relevant $O(\lambda^2)$ dependence of single-Higgs observables, with the omitted process-dependent two-loop corrections, especially the $\kappa_3^3$ terms, genuinely small.

Editorial extensions

If this is right

At the FCC-ee, fitting single-Higgs signal strengths with $\kappa_3=1$ yields a 68% CL bound $-7 < \kappa_4 < 15$, comparable in strength to double-Higgs production projections.
At the HL-LHC the corresponding single-Higgs bound is much weaker ($-21 < \kappa_4 < 28$), so multi-Higgs channels dominate there, and the combined fit leaves two viable regions, one around the SM point and one near $\{\kappa_3,\kappa_4\} \simeq \{3.5,0\}$.
Combining single-, double-, and triple-Higgs measurements at the FCC could exclude the second BSM region, with single-Higgs data playing a decisive role in that exclusion.
The quadratic $\kappa_4$ dependence first appears at two loops and is entirely due to universal Higgs wave-function renormalization, so it affects every single-Higgs production and decay channel.
In the SM limit the two-loop result reproduces the known SM Higgs anomalous dimension, validating the calculation against earlier two-loop results.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the WFR rescaling is universal but the one-loop coefficients are process dependent, comparing two single-Higgs channels could in principle isolate the two-loop $\kappa_4^2$ effect from other new-physics corrections; the paper does not perform such a channel comparison.
The main theoretical loose end is the $\kappa_3^3$ dependence: Eq. (3.1) drops the process-dependent $\kappa_3^3-1$ terms on the grounds that their coefficient in Eq. (2.5) is small, but a full process-dependent two-loop calculation of $e^+e^-\to Zh$ would be needed to confirm that the projected contours are stable.
The same two-loop WFR mechanism should apply to any extended scalar sector with modified self-couplings, so precision single-scalar measurements could serve as a generic probe of quartic self-interactions beyond the Higgs.
Because the quintic modifier $\kappa_5$ enters only through the small-coefficient $\kappa_3\kappa_5$ term, the single-Higgs probe is largely insensitive to $\kappa_5$; this may make the extracted $\kappa_4$ bound robust against quintic-coupling contamination.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper has a genuinely new analytic target: the two-loop on-shell Higgs WFR constant with general κ3, κ4, and κ5 dependence. The calculation is documented unusually well, with master integrals in an appendix, independent IBP reductions, and cross-checks against AMFlow/TSIL and the known SM anomalous dimension. The writing is clear and the phenomenological framework is standard. That side of the work deserves credit.

The problem is Eq. (2.5). It contains a term proportional to κ4^3, and that term cannot arise at two loops. For a connected two-loop, two-point diagram with 3-, 4-, and 5-point vertices, the loop-vertex identity gives V3 + 2V4 + 3V5 = 4. That bounds the number of quartic vertices by V4 ≤ 2. Three quartic vertices are impossible at two loops. The counterterm insertions in Appendix B are at most linear in κ4, so they cannot generate κ4^3 either. The stress-test note has this right, and it is a stronger threat than the reader's concern about the omitted κ3^3 term.

The κ4^3 term is not just a spectator. In Eq. (3.1) its coefficient is -1.726e-5, and it drives the advertised FCC-ee constraint -7 < κ4 < 15. If that term is spurious, the proposed single-Higgs probe of κ4 largely disappears. The SM anomalous-dimension cross-check is insensitive to the term because it is UV-finite, and the integral checks verify integrals, not diagram power counting. So the standard checks do not catch it.

The reader's flagged omission is also real: Eq. (3.1) drops the κ3^3 contribution that appears in (2.5), and the justification that its coefficient is small does not hold when compared with the retained κ3^2κ4 term. I also notice that the square of the one-loop WFR in Eq. (2.9) should produce a κ3^4 contribution to single-Higgs observables, and I do not see it in (3.1). The numerical section needs a careful independent pass.

What is salvageable is the calculational machinery: the master integrals, the reduction, and the SM-limit checks are solid enough that this looks like a manuscript with a load-bearing error, not a nonsensical submission. A referee should be asked to verify the diagram-counting identity and the κ4^3 term explicitly, and to require a corrected Eq. (2.5) and re-derived projections before the paper is accepted. Send it to peer review, but with that instruction.

Referee Report

2 major / 4 minor

Summary. The paper computes the two-loop on-shell Higgs wave-function renormalization (WFR) constant in a Higgs sector with modified cubic, quartic, and quintic self-couplings, parametrized by κ3, κ4, and κ5. It then uses the universal WFR rescaling to predict O(λ^2) corrections to single-Higgs production rates and derives projected constraints in the (κ3, κ4) plane for the HL-LHC and FCC-ee, comparing them with double- and triple-Higgs production prospects. The central analytic result is Eq. (2.5), and the main phenomenological claim is that FCC-ee single-Higgs measurements could constrain κ4 to roughly -7 < κ4 < 15, a sensitivity comparable to that of double-Higgs production.

Significance. If correct, the paper would introduce a qualitatively new indirect probe of the quartic Higgs self-coupling. The technical apparatus is substantial: analytic master integrals are listed in Appendix A, cross-checked against AMFlow, TSIL, and LiteRed, and the SM limit reproduces the known two-loop anomalous dimension. These are real strengths. However, the central numerical claim rests on a term in Eq. (2.5) that is inconsistent with two-loop topology, so the significance of the paper as it stands is not established.

major comments (2)

[Sec. 2, Eq. (2.5)] The last term in Eq. (2.5), [c^(2)_4,0 + 18 c^(1)_2,0 L] κ4^3, cannot arise from any legitimate two-loop contribution to the Higgs self-energy. For a connected two-loop self-energy graph with two external legs, the topological identity is L = V3/2 + V4 + 3V5/2, where Vi is the number of i-point vertices. With L=2 this gives V4 = 2 - V3/2 - 3V5/2 ≤ 2; three quartic vertices require L=3. The one-loop counterterm insertions in Appendix B are at most linear in κ4 (δm_h^2, δλ) or quadratic in κ3 (δZ_h), so they cannot supply the third power of κ4. The notation c^(2)_4,0 suggests the term may have been intended as κ3^4 rather than κ4^3. This is load-bearing because Eq. (3.1) uses the coefficient -1.726×10^-5 (1+1.307L)(κ4^3-1) to derive the FCC-ee bound -7<κ4<15; if the term is removed or reassigned to κ3^4, the projected sensitivity to κ4 is reduced by roughly an order of magnitude, and the paper's central phenomenological claim is not supported.
[Sec. 3, Eq. (3.1)] The omission of all κ3^3-1 terms from Eq. (3.1) is not justified by the argument given in the text. Equation (2.5) contains a universal WFR term -6 c^(1)_2,0 (1+L) κ3^3, whose finite coefficient is about 11.3 at L=0, comparable to the retained κ3^2 κ4 coefficient c^(2)_2,1 ≈ -12.8. Through Eq. (2.9) this universal term contributes roughly +3.8×10^-6 (1+L)(κ3^3-1) to δσ_i, which is not negligible relative to the -8.5×10^-6 (1-1.767L)(κ3^2 κ4 -1) term. The text states that the omission is due to the process-dependent nature of such corrections, but the term from Eq. (2.5) is universal; the process-dependent κ3^3 1PI diagrams discussed in Section 2 are a separate object. The authors should either include the universal κ3^3 WFR term in Eq. (3.1) or demonstrate a cancellation, and should quantify the numerical impact of its omission on the contours in Figure 4.

minor comments (4)

[Abstract and Sec. 4] The abstract and conclusions quote the FCC constraint as '-5 ≲ κ4 ≲ 15', while Section 3 states '-7 < κ4 < 15'; these numbers should be made consistent.
[Table 1] The entries C^{Γf}_1 for b bbar and tau+ tau- are 0.67×10^-5 and 0.33×10^-5, which are several orders of magnitude smaller than the other entries; a footnote explaining whether these are total-width or partial-width coefficients would help the reader.
[Eq. (3.1)] Equation (3.1) mixes universal WFR terms with process-dependent coefficients C^{σ_i}_1 without an explicit derivation of the O(λ) part; a short derivation or a precise reference for each term would improve reproducibility.
[Appendix C] The statement that the scale dependence 'does not alter the overall picture' would be more convincing if the figure quantified the shift of the contours in the negative-κ4 region, where the constraints are less stringent.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the two-loop WFR result (2.5) is an explicit diagram computation cross-checked against AMFlow, TSIL, and LiteRed and against the known SM two-loop anomalous dimension, and the FCC-ee kappa4 reach is a derived projection, not a fitted or self-cited input.

full rationale

Eq. (2.5), the paper's central object, is presented as a direct calculation: 'The generation and computation of the amplitudes were carried out using the Mathematica packages FeynArts, FeynCalc, and FormCalc... reduced using the Tarasov algorithm... cross-checked with LiteRed, resulting in the same final outcome.' The master integrals in Appendix A 'agree with known results in the literature' and 'have also been cross-checked against high-precision numerical results obtained from both the AMFlow and TSIL packages.' The SM limit '(2.7) matches the known two-loop SM value calculated, for instance, in [44, 45], serving as a cross-check of our calculation.' These are independent, externally specified benchmarks, so the central claim does not rest on the authors' own prior conclusions. Eq. (3.1) is then a direct translation of (2.5) into signal-strength shifts via the WFR rescaling (2.9); the process-dependent coefficients C^sigma_i_1, 'directly taken or obtained from [9, 10, 20, 25]', are explicit literature inputs (several self-authored but numerically tabulated and independently available), not quantities fitted to data or renamed as predictions. No parameter in the analysis is adjusted to measurements; the quoted HL-LHC and FCC-ee reaches are projections based on assumed uncertainties from [51] and [50]. The assertion that the kappa4^2, kappa3*kappa5, kappa3^2*kappa4, and kappa4^3 corrections to single-Higgs processes 'arise entirely from Higgs WFR' is justified by the diagram-scaling completeness argument around Figure 3, which is a power-counting claim rather than a self-referential definition. Self-citations ([8], [10], [18], [25], [53]) occur but are not load-bearing for the new kappa4-dependent term, whose master integrals, counterterms, and reduction are all exhibited in this paper. The substantive concerns raised elsewhere — the topological plausibility of the kappa4^3 term in (2.5) and the parenthetical omission of the comparable kappa3^3 term in (3.1) — are physics-correctness risks (and potential errata), not instances of prediction-by-construction. Verdict: no significant circularity; the score is kept at 1 only to reflect that several non-load-bearing input coefficients come from the authors' own earlier papers.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

All ingredients are standard SMEFT or Higgs-physics inputs. The only ad hoc to paper axiom is the neglect of process-dependent two-loop corrections in the single-Higgs projections, which is also flagged as a red flag.

free parameters (3)

kappa3 and kappa4
Dimensionless modifiers of the cubic and quartic Higgs self-couplings. Treated as free parameters scanned over the kappa3-kappa4 plane; no fit to real data.
kappa5 = 7/4 - 9/4 kappa3 + 1/2 kappa4
Quintic coupling fixed through the SMEFT tree-level relation (1.3) in the projections, not varied independently.
Assumed signal-strength uncertainties Delta_i^f = See Table 2 and Table 3, e.g., 3.6% for ggF h -> gamma gamma at HL-LHC
Projected relative total uncertainties taken from ATLAS S2 scenario [51] and FCC studies [50]; central values assumed to equal SM predictions.

assumptions (5)

standard math Dimensional regularization in d=4-2*epsilon with on-shell and MS renormalization schemes defines the calculation.
Used throughout Section 2 and Appendices A and B.
domain assumption The new physics affecting the Higgs potential can be described by modifying only the cubic, quartic, and quintic self-couplings; higher-dimensional operators beyond Q10 are neglected.
Eq. (1.1) parameterization and the discussion after Eq. (1.3).
ad hoc to paper For single-Higgs observables at O(lambda^2), the universal WFR rescaling captures the kappa4^2, kappa3*kappa5, kappa3^2*kappa4, and kappa4^3 dependence, while process-dependent two-loop corrections (including kappa3^3 terms) can be dropped.
Invoked in Section 3 around Eq. (3.1); this is the load-bearing assumption for the projections and is not fully proven.
domain assumption Future HL-LHC and FCC measurements will return central values equal to SM predictions with the projected uncertainties of the S2 scenario.
Used in the chi^2 definition (3.4) and in Tables 2 and 3.
domain assumption The SMEFT tree-level relations (1.3) connect kappa5 to kappa3 and kappa4 for the two-dimensional projections.
Used after Eqs. (3.5) and (3.6) to fix kappa5.

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Cite this review

Pith. "Pith review of A new probe of the quartic Higgs self-coupling." pith.science (2026). https://pith.science/paper/O7DN7CXT

@misc{pith2026250520463,
  author       = {Pith},
  title        = {Pith review of: A new probe of the quartic Higgs self-coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O7DN7CXT}},
  note         = {Machine review of arXiv:2505.20463}
}

read the original abstract

We calculate the corrections to the Higgs wave-function renormalization constant arising from modified cubic, quartic, and quintic Higgs self-couplings up to the two-loop level. Using our analytic results, we derive two-dimensional constraints on the modifications of the considered Higgs self-interactions that could potentially be set from precision measurements of single-Higgs production processes at the high-luminosity Large Hadron Collider (LHC) and a Future Circular Collider. Our novel constraints are compared to those that might be set by searches for multi-Higgs production at the same facilities. In view of the first LHC results on triple-Higgs production, we also review the current status of Higgs self-coupling determinations after LHC Run 2.

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Forward citations

Cited by 2 Pith papers

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

Search for nonresonant triple Higgs boson production in the final state with six bottom quarks in proton-proton collisions at $\sqrt{s}$ = 13 TeV
hep-ex 2026-07 accept novelty 6.0 of 10

No excess is observed; the 95% CL upper limit on nonresonant HHH→6b is 44 fb (588×SM), with κ3 constrained to −7.4 < κ3 < 12.4 (κ4=1) and κ4 to −177 < κ4 < 185 (κ3=1).
Precision tests of third-generation four-quark operators: $gg \to h$ and $h \to \gamma \gamma$
hep-ph 2025-07 conditional novelty 6.0 of 10

Two-loop SMEFT corrections from third-generation four-quark operators to gg to h and h to gamma gamma are computed with full mass dependence, including new two-loop anomalous dimensions.

Reference graph

Works this paper leans on

86 extracted references · 13 canonical work pages · cited by 2 Pith papers

[1]

Buchmüller and D

W. Buchmüller and D. Wyler, Nucl. Phys. B268, 621 (1986)

1986
[2]

Grzadkowski, M

B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, JHEP10, 085 (2010), arXiv:1008.4884 [hep-ph]

arXiv 2010
[3]

Brivio and M

I. Brivio and M. Trott, Phys. Rept.793, 1 (2019), arXiv:1706.08945 [hep-ph]

arXiv 2019
[4]

Isidori, F

G. Isidori, F. Wilsch, and D. Wyler, Rev. Mod. Phys.96, 015006 (2024), arXiv:2303.16922 [hep-ph]

arXiv 2024
[5]

de Blas, M

J. de Blas, M. Chala, M. Perez-Victoria, and J. Santiago, JHEP04, 078 (2015), arXiv:1412.8480 [hep-ph]

arXiv 2015
[6]

Durieux, M

G. Durieux, M. McCullough, and E. Salvioni, JHEP12, 148 (2022), [Erratum: JHEP02, 165 (2023)], arXiv:2209.00666 [hep-ph]

arXiv 2022
[7]

McCullough, Phys

M. McCullough, Phys. Rev. D90, 015001 (2014), [Erratum: Phys. Rev. D92, 039903 (2015)], arXiv:1312.3322 [hep-ph]. – 18 –

arXiv 2014
[8]

Gorbahn and U

M. Gorbahn and U. Haisch, JHEP10, 094 (2016), arXiv:1607.03773 [hep-ph]

arXiv 2016

Show all 86 references

[9]

Degrassi, P

G. Degrassi, P. P. Giardino, F. Maltoni, and D. Pagani, JHEP12, 080 (2016), arXiv:1607.04251 [hep-ph]

2016 arXiv
[10]

Bizoń, M

W. Bizoń, M. Gorbahn, U. Haisch, and G. Zanderighi, JHEP07, 083 (2017), arXiv:1610.05771 [hep-ph]

2017 arXiv
[11]

Degrassi, M

G. Degrassi, M. Fedele, and P. P. Giardino, JHEP04, 155 (2017), arXiv:1702.01737 [hep-ph]

2017 arXiv
[12]

G. D. Kribs, A. Maier, H. Rzehak, M. Spannowsky, and P. Waite, Phys. Rev. D95, 093004 (2017), arXiv:1702.07678 [hep-ph]

2017 arXiv
[13]

Di Vita, C

S. Di Vita, C. Grojean, G. Panico, M. Riembau, and T. Vantalon, JHEP09, 069 (2017), arXiv:1704.01953 [hep-ph]

2017 arXiv
[14]

Maltoni, D

F. Maltoni, D. Pagani, A. Shivaji, and X. Zhao, Eur. Phys. J. C77, 887 (2017), arXiv:1709.08649 [hep-ph]

2017 arXiv
[15]

Di Vita, G

S. Di Vita, G. Durieux, C. Grojean, J. Gu, Z. Liu, G. Panico, M. Riembau, and T. Vantalon, JHEP02, 178 (2018), arXiv:1711.03978 [hep-ph]

2018 arXiv
[16]

Maltoni, D

F. Maltoni, D. Pagani, and X. Zhao, JHEP07, 087 (2018), arXiv:1802.07616 [hep-ph]

2018 arXiv
[17]

Liu, K.-F

T. Liu, K.-F. Lyu, J. Ren, and H. X. Zhu, Phys. Rev. D98, 093004 (2018), arXiv:1803.04359 [hep-ph]

2018 arXiv
[18]

Bizoń, U

W. Bizoń, U. Haisch, and L. Rottoli, JHEP10, 267 (2019), arXiv:1810.04665 [hep-ph]

2019 arXiv
[19]

Borowka, C

S. Borowka, C. Duhr, F. Maltoni, D. Pagani, A. Shivaji, and X. Zhao, JHEP04, 016 (2019), arXiv:1811.12366 [hep-ph]

2019 arXiv
[20]

Gorbahn and U

M. Gorbahn and U. Haisch, JHEP04, 062 (2019), arXiv:1902.05480 [hep-ph]

2019 arXiv
[21]

Degrassi and M

G. Degrassi and M. Vitti, Eur. Phys. J. C80, 307 (2020), arXiv:1912.06429 [hep-ph]

2020 arXiv
[22]

Haisch and G

U. Haisch and G. Koole, JHEP02, 030 (2022), arXiv:2111.12589 [hep-ph]

2022 arXiv
[23]

Gao, X.-M

J. Gao, X.-M. Shen, G. Wang, L. L. Yang, and B. Zhou, Phys. Rev. D107, 115017 (2023), arXiv:2302.04160 [hep-ph]

2023 arXiv
[24]

H. T. Li, Z.-G. Si, J. Wang, X. Zhang, and D. Zhao, Chin. Phys. C49, 023107 (2025), arXiv:2407.14716 [hep-ph]

2025 arXiv
[25]

Haisch and M

U. Haisch and M. Niggetiedt, JHEP10, 236 (2024), arXiv:2408.13186 [hep-ph]

2024 arXiv
[26]

Englert and M

C. Englert and M. McCullough, JHEP07, 168 (2013), arXiv:1303.1526 [hep-ph]

2013 arXiv
[27]

Craig, C

N. Craig, C. Englert, and M. McCullough, Phys. Rev. Lett.111, 121803 (2013), arXiv:1305.5251 [hep-ph]

2013 arXiv
[28]

Aadet al.(ATLAS), Phys

G. Aadet al.(ATLAS), Phys. Lett. B843, 137745 (2023), arXiv:2211.01216 [hep-ex]

2023 arXiv
[29]

Hayrapetyanet al.(CMS), Phys

A. Hayrapetyanet al.(CMS), Phys. Lett. B861, 139210 (2025), arXiv:2407.13554 [hep-ex]

2025 arXiv
[30]

Chiesa, F

M. Chiesa, F. Maltoni, L. Mantani, B. Mele, F. Piccinini, and X. Zhao, JHEP09, 098 (2020), arXiv:2003.13628 [hep-ph]

2020 arXiv
[31]

Gonzalez-Lopez, M

M. Gonzalez-Lopez, M. J. Herrero, and P. Martinez-Suarez, Eur. Phys. J. C81, 260 (2021), arXiv:2011.13915 [hep-ph]

2021 arXiv
[32]

Stylianou and G

P. Stylianou and G. Weiglein, Eur. Phys. J. C84, 366 (2024), arXiv:2312.04646 [hep-ph]. – 19 –

2024 arXiv
[33]

Papaefstathiou and G

A. Papaefstathiou and G. Tetlalmatzi-Xolocotzi, JHEP06, 124 (2024), arXiv:2312.13562 [hep-ph]

2024 arXiv
[34]

Brigljevicet al., Eur

V. Brigljevicet al., Eur. Phys. J. C84, 1183 (2024), arXiv:2407.03015 [hep-ph]

2024 arXiv
[35]

Z. Dong, X. Sun, B. Guo, L. Zhang, Z. Li, J. Wang, Z. Li, Y. Ban, and Y. Mao, (2025), arXiv:2504.04037 [hep-ph]

2025 arXiv
[36]

Aadet al.(ATLAS), Phys

G. Aadet al.(ATLAS), Phys. Rev. D111, 032006 (2025), arXiv:2411.02040 [hep-ex]

2025 arXiv
[37]

Hahn, Comput

T. Hahn, Comput. Phys. Commun.140, 418 (2001), arXiv:hep-ph/0012260

2001 arXiv
[38]

Shtabovenko, R

V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun.256, 107478 (2020), arXiv:2001.04407 [hep-ph]

2020 arXiv
[39]

T. Hahn, S. Paßehr, and C. Schappacher, PoSLL2016, 068 (2016), arXiv:1604.04611 [hep-ph]

2016 arXiv
[40]

O. V. Tarasov, Nucl. Phys. B502, 455 (1997), arXiv:hep-ph/9703319

1997 arXiv
[41]

Mertig and R

R. Mertig and R. Scharf, Comput. Phys. Commun.111, 265 (1998), arXiv:hep-ph/9801383

1998 arXiv
[42]

R. N. Lee, J. Phys. Conf. Ser.523, 012059 (2014), arXiv:1310.1145 [hep-ph]

2014 arXiv
[43]

Denner, Fortsch

A. Denner, Fortsch. Phys.41, 307 (1993), arXiv:0709.1075 [hep-ph]

1993 arXiv
[44]

M. E. Machacek and M. T. Vaughn, Nucl. Phys. B222, 83 (1983)

1983
[45]

Luo, H.-w

M.-x. Luo, H.-w. Wang, and Y. Xiao, Phys. Rev. D67, 065019 (2003), arXiv:hep-ph/0211440

2003 arXiv
[46]

Senaha, Phys

E. Senaha, Phys. Rev. D100, 055034 (2019), arXiv:1811.00336 [hep-ph]

2019 arXiv
[47]

Braathen and S

J. Braathen and S. Kanemura, Phys. Lett. B796, 38 (2019), arXiv:1903.05417 [hep-ph]

2019 arXiv
[48]

Braathen and S

J. Braathen and S. Kanemura, Eur. Phys. J. C80, 227 (2020), arXiv:1911.11507 [hep-ph]

2020 arXiv
[49]

Braathen, S

J. Braathen, S. Kanemura, and M. Shimoda, JHEP03, 297 (2021), arXiv:2011.07580 [hep-ph]

2021 arXiv
[50]

Durieux, C

G. Durieux, C. Grojean, J. Gu, and K. Wang, JHEP09, 014 (2017), arXiv:1704.02333 [hep-ph]. [51]Projections for measurements of Higgs boson cross sections, branching ratios, coupling parameters and mass with the ATLAS detector at the HL-LHC, Tech. Rep. (CERN, Geneva, 2018)

2017 arXiv
[52]

Alioli, P

S. Alioli, P. Nason, C. Oleari, and E. Re, JHEP06, 043 (2010), arXiv:1002.2581 [hep-ph]

2010 arXiv
[53]

Bizoń, U

W. Bizoń, U. Haisch, L. Rottoli, Z. Gillis, B. Moser, and P. Windischhofer, JHEP02, 170 (2024), arXiv:2402.03463 [hep-ph]

2024 arXiv
[54]

Borowka, N

S. Borowka, N. Greiner, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk, U. Schubert, and T. Zirke, Phys. Rev. Lett.117, 012001 (2016), [Erratum: Phys. Rev. Lett.117, 079901 (2016)], arXiv:1604.06447 [hep-ph]

2016 arXiv
[55]

Borowka, N

S. Borowka, N. Greiner, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, JHEP10, 107 (2016), arXiv:1608.04798 [hep-ph]

2016 arXiv
[56]

Heinrich, S

G. Heinrich, S. P. Jones, M. Kerner, G. Luisoni, and E. Vryonidou, JHEP08, 088 (2017), arXiv:1703.09252 [hep-ph]. – 20 –

2017 arXiv
[57]

Heinrich, S

G. Heinrich, S. P. Jones, M. Kerner, G. Luisoni, and L. Scyboz, JHEP06, 066 (2019), arXiv:1903.08137 [hep-ph]

2019 arXiv
[58]

Butterworthet al., J

J. Butterworthet al., J. Phys. G43, 023001 (2016), arXiv:1510.03865 [hep-ph]

2016 arXiv
[59]

Alwall, R

J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, JHEP07, 079 (2014), arXiv:1405.0301 [hep-ph]

2014 arXiv
[60]

Maltoni, E

F. Maltoni, E. Vryonidou, and M. Zaro, JHEP11, 079 (2014), arXiv:1408.6542 [hep-ph]. [61]Projected sensitivity of Higgs boson pair production combining theb¯bγγandb ¯bτ +τ − final states with the ATLAS detector at the HL-LHC, Tech. Rep. (CERN, Geneva, 2022)

2014 arXiv
[62]

Abadaet al.(FCC), Eur

A. Abadaet al.(FCC), Eur. Phys. J. C79, 474 (2019)

2019
[63]

A. J. Barr, M. J. Dolan, C. Englert, D. E. Ferreira de Lima, and M. Spannowsky, JHEP02, 016 (2015), arXiv:1412.7154 [hep-ph]

2015 arXiv
[64]

Azatov, R

A. Azatov, R. Contino, G. Panico, and M. Son, Phys. Rev. D92, 035001 (2015), arXiv:1502.00539 [hep-ph]

2015 arXiv
[65]

H.-J. He, J. Ren, and W. Yao, Phys. Rev. D93, 015003 (2016), arXiv:1506.03302 [hep-ph]

2016 arXiv
[66]

Continoet al., (2016), 10.23731/CYRM-2017-003.255, arXiv:1606.09408 [hep-ph]

R. Continoet al., (2016), 10.23731/CYRM-2017-003.255, arXiv:1606.09408 [hep-ph]

2016 arXiv
[67]

M. L. Manganoet al., (2016), 10.23731/CYRM-2017-003.1, arXiv:1607.01831 [hep-ph]

2016 arXiv
[68]

Gonçalves, T

D. Gonçalves, T. Han, F. Kling, T. Plehn, and M. Takeuchi, Phys. Rev. D97, 113004 (2018), arXiv:1802.04319 [hep-ph]

2018 arXiv
[69]

Chang, K

J. Chang, K. Cheung, J. S. Lee, C.-T. Lu, and J. Park, Phys. Rev. D100, 096001 (2019), arXiv:1804.07130 [hep-ph]

2019 arXiv
[70]

Papaefstathiou and K

A. Papaefstathiou and K. Sakurai, JHEP02, 006 (2016), arXiv:1508.06524 [hep-ph]

2016 arXiv
[71]

Chen, Q.-S

C.-Y. Chen, Q.-S. Yan, X. Zhao, Y.-M. Zhong, and Z. Zhao, Phys. Rev. D93, 013007 (2016), arXiv:1510.04013 [hep-ph]

2016 arXiv
[72]

B. Fuks, J. H. Kim, and S. J. Lee, Phys. Rev. D93, 035026 (2016), arXiv:1510.07697 [hep-ph]

2016 arXiv
[73]

Kilian, S

W. Kilian, S. Sun, Q.-S. Yan, X. Zhao, and Z. Zhao, JHEP06, 145 (2017), arXiv:1702.03554 [hep-ph]

2017 arXiv
[74]

B. Fuks, J. H. Kim, and S. J. Lee, Phys. Lett. B771, 354 (2017), arXiv:1704.04298 [hep-ph]

2017 arXiv
[75]

A. I. Davydychev and J. B. Tausk, Nucl. Phys. B397, 123 (1993)

1993
[76]

Scharf and J

R. Scharf and J. B. Tausk, Nucl. Phys. B412, 523 (1994)

1994
[77]

Fleischer, F

J. Fleischer, F. Jegerlehner, O. V. Tarasov, and O. L. Veretin, Nucl. Phys. B539, 671 (1999), [Erratum: Nucl. Phys.B571, 511 (2000)], arXiv:hep-ph/9803493

1999 arXiv
[78]

Liu and Y.-Q

X. Liu and Y.-Q. Ma, Comput. Phys. Commun.283, 108565 (2023), arXiv:2201.11669 [hep-ph]

2023 arXiv
[79]

S. P. Martin and D. G. Robertson, Comput. Phys. Commun.174, 133 (2006), arXiv:hep-ph/0501132

2006 arXiv
[80]

A. V. Smirnov and F. S. Chuharev, Comput. Phys. Commun.247, 106877 (2020), arXiv:1901.07808 [hep-ph]. – 21 –

2020 arXiv
[81]

Ferguson and D

H. Ferguson and D. Bailey,A Polynomial Time, Numerically Stable Integer Relation Algorithm, Tech. Rep. (1992)

1992
[82]

Aadet al.(ATLAS), JHEP11, 097 (2024), arXiv:2402.05742 [hep-ex]

G. Aadet al.(ATLAS), JHEP11, 097 (2024), arXiv:2402.05742 [hep-ex]

2024 arXiv
[83]

Aadet al.(ATLAS), Phys

G. Aadet al.(ATLAS), Phys. Rev. Lett.133, 101801 (2024), arXiv:2406.09971 [hep-ex]

2024 arXiv
[84]

Aadet al.(ATLAS), Phys

G. Aadet al.(ATLAS), Phys. Rev. D108, 052003 (2023), arXiv:2301.03212 [hep-ex]

2023 arXiv
[85]

Aadet al.(ATLAS), Phys

G. Aadet al.(ATLAS), Phys. Lett. B858, 139007 (2024), arXiv:2404.17193 [hep-ex]

2024 arXiv
[86]

Aadet al.(ATLAS), Phys

G. Aadet al.(ATLAS), Phys. Rev. D110, 032012 (2024), arXiv:2404.12660 [hep-ex]

2024 arXiv
[87]

Aadet al.(ATLAS), JHEP01, 066 (2024), arXiv:2310.12301 [hep-ex]

G. Aadet al.(ATLAS), JHEP01, 066 (2024), arXiv:2310.12301 [hep-ex]

2024 arXiv
[88]

Aadet al.(ATLAS), JHEP02, 037 (2024), arXiv:2310.11286 [hep-ex]

G. Aadet al.(ATLAS), JHEP02, 037 (2024), arXiv:2310.11286 [hep-ex]. – 22 –

2024 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Buchmüller and D

W. Buchmüller and D. Wyler, Nucl. Phys. B268, 621 (1986)

1986

[2] [2]

Grzadkowski, M

B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, JHEP10, 085 (2010), arXiv:1008.4884 [hep-ph]

arXiv 2010

[3] [3]

Brivio and M

I. Brivio and M. Trott, Phys. Rept.793, 1 (2019), arXiv:1706.08945 [hep-ph]

arXiv 2019

[4] [4]

Isidori, F

G. Isidori, F. Wilsch, and D. Wyler, Rev. Mod. Phys.96, 015006 (2024), arXiv:2303.16922 [hep-ph]

arXiv 2024

[5] [5]

de Blas, M

J. de Blas, M. Chala, M. Perez-Victoria, and J. Santiago, JHEP04, 078 (2015), arXiv:1412.8480 [hep-ph]

arXiv 2015

[6] [6]

Durieux, M

G. Durieux, M. McCullough, and E. Salvioni, JHEP12, 148 (2022), [Erratum: JHEP02, 165 (2023)], arXiv:2209.00666 [hep-ph]

arXiv 2022

[7] [7]

McCullough, Phys

M. McCullough, Phys. Rev. D90, 015001 (2014), [Erratum: Phys. Rev. D92, 039903 (2015)], arXiv:1312.3322 [hep-ph]. – 18 –

arXiv 2014

[8] [8]

Gorbahn and U

M. Gorbahn and U. Haisch, JHEP10, 094 (2016), arXiv:1607.03773 [hep-ph]

arXiv 2016