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Higgs boson potential at colliders: status and perspectives

B. Di Micco, M. Gouzevitch, J. Mazzitelli, C. Vernieri, J. Alison, K. Androsov, J. Baglio, E. Bagnaschi, S. Banerjee, P. Basler, A. Bethani, A. Betti, M. Blanke, A. Blondel, L. Borgonovi, E. Brost, P. Bryant, G. Buchalla, T. J. Burch, V. M. M. Cairo, F. Campanario, M. Carena, A. Carvalho, N. Chernyavskaya, V. D'Amico, S. Dawson, N. De Filippis, L. Di Luzio, S. Di Vita, B. Dillon, C. Englert, A. Ferrari, E. Fontanesi, H. Fox, M. Gallinaro, P. P. Giardino, S. Glaus, F. Goertz, S. Gori, R. Gröber, C. Grojean, D. F. Guerrero Ibarra, R. Gupta, U. Haisch, G. Heinrich, P. Huang, P. Janot, S. P. Jones, M. A. Kagan, S. Kast, M. Kerner, J. H. Kim, K. Kong, J. Kozaczuk, F. Krauss, S. Kuttimalai, H. M. Lee, K. Leney, I. M. Lewis, S. Liebler, Z. Liu, H. E. Logan, A. Long, F. Maltoni, S. Manzoni, L. Mastrolorenzo, K. Matchev, F. Micheli, M. Mühlleitner, M. S. Neubauer, G. Ortona, M. Osherson, D. Pagani, G. Panico, A. Papaefstathiou, M. Park, M. E. Peskin, J. Quevillon, M. Riembau, T. Robens, P. Roloff, H. Rzehak, J. Schaarschmidt, U. Schnoor, L. Scyboz, M. Selvaggi, N. R. Shah, A. Shivaji, S. Shrestha, K. Sinha, M. Spannowsky, M. Spira, T. Stefaniak, J. Streicher, M. Sullivan, M. Swiatlowski, R. Teixeira de Lima, J. Thomson, J. Tian, T. Vantalon, C. Veelken, T. Vickey, E. Vryonidou, J. Wells, S. Westhoff, X. Zhao, J. Zurita

arXiv:1910.00012v4hep-phhep-ex

TL;DR

The Higgs-field potential has not yet been measured, leaving its form an open research question. This review synthesizes di-Higgs predictions, searches and self-coupling constraints; it reports that QCD corrections increase the total cross section by about a factor of two over the leading-order prediction.

  • Problem

    The Higgs-field potential has not been measured experimentally, leaving its form an open research question.

    Source: Preface · p. 7, §2.1 · p. 37

  • Method

    The paper summarizes theoretical and experimental di-Higgs results and constraints on the self-coupling from double- and single-Higgs measurements at hadron and lepton colliders.

    Source: Preface · p. 8

  • Results

    QCD corrections increase the total cross section by about a factor of two relative to the leading-order prediction.

    Source: §1.1 · p. 18

  • Takeaways & Limitations

    Studying Higgs-pair production is relevant to understanding the Higgs-field potential, whose properties the Standard Model treats as inputs.

    Source: §2.1 · p. 37

    For non-SM κλ values, the NLO result with full top-quark-mass dependence has not yet been combined with NNLO corrections in the improved heavy-top limit.

    Source: §1.3 · p. 30

Abstract

from arXiv · show

This document summarises the current theoretical and experimental status of the di-Higgs boson production searches, and of the direct and indirect constraints on the Higgs boson self-coupling, with the wish to serve as a useful guide for the next years. The document discusses the theoretical status, including state-of-the-art predictions for di-Higgs cross sections, developments on the effective field theory approach, and studies on specific new physics scenarios that can show up in the di-Higgs final state. The status of di-Higgs searches and the direct and indirect constraints on the Higgs self-coupling at the LHC are presented, with an overview of the relevant experimental techniques, and covering all the variety of relevant signatures. Finally, the capabilities of future colliders in determining the Higgs self-coupling are addressed, comparing the projected precision that can be obtained in such facilities. The work has started as the proceedings of the Di-Higgs workshop at Colliders, held at Fermilab from the 4th to the 9th of September 2018, but it went beyond the topics discussed at that workshop and included further developments.

Theoretical status

The theoretical overview covers developments needed to interpret Higgs self-coupling measurements and study Higgs-pair production in the Standard Model and beyond.

Theoretical status

  • The review presents predictions for Standard Model Higgs-pair production across production modes, emphasizing gluon fusion and including fixed-order results and Monte Carlo generators.These predictions aim to help extract information from experimental measurements.Source: Theoretical status · p. 15
  • It examines effective field theory developments for interpreting non-resonant deviations and their impact on self-coupling determination in double- and single-Higgs final states.Source: Theoretical status · p. 15
  • It studies beyond-Standard-Model scenarios affecting di-Higgs production, focusing mainly on new resonances decaying to Higgs pairs and also considering loop effects.Source: Theoretical status · p. 15

HH cross section predictions

The chapter reviews theoretical predictions for Higgs-pair production at hadron colliders, emphasizing gluon fusion and the QCD precision needed to constrain the trilinear Higgs coupling.

HH cross section predictions

  • The trilinear Higgs coupling can be probed directly through Higgs-pair production, whereas the quartic coupling is inaccessible at the LHC because triple-Higgs production is too small [9–12].Near the Standard Model value, the cross-section dependence is estimated as ∆σ/σ ∼−∆λ/λ, making control of theoretical cross-section uncertainties and higher-order QCD corrections indispensable.Source: HH cross section predictions · p. 17
  • At hadron colliders, Higgs pairs are produced dominantly through loop-induced gluon fusion, mainly mediated by top-quark loops.This mechanism is similar to single-Higgs production.Source: HH cross section predictions · p. 17
  • The chapter covers production modes, QCD corrections to gluon fusion, its dependence on the Higgs self-coupling, and available Monte Carlo generators.Source: HH cross section predictions · p. 17

1.1 Overview of production modes

Higgs-pair production at hadron colliders is dominated by gluon fusion, with vector-boson fusion the second-largest mode. The section also reviews double Higgs-strahlung and top-associated production, whose rates and QCD corrections differ.

Gluon fusion

  • Gluon fusion dominates: destructive interference between box and trilinear-coupling triangle diagrams reduces the LO total cross section by about 50% versus box-only, while QCD corrections raise it by about a factor of two.Predictions include NLO and heavy-top-limit NNLO corrections with partial finite-top-mass effects, and third-order corrections in the heavy-top limit [17] [18] [19] [20] [21] [22] [23] [24] [25].Source: §1.1 · pp. 17–18

Vector-boson fusion

  • VBF is the second-largest mode; its structure-function approximation neglects same-flavor external-quark interference at percent-level accuracy, and NNLO and N3LO corrections are per-cent and sub-per-cent [26] [27] [28] [29].The approximation includes only t-channel W and Z exchange; total-cross-section corrections are known through N3LO, while exclusive calculations are available at NNLO.Source: §1.1 · pp. 18, 20

Double Higgs-strahlung

  • Double Higgs-strahlung has a lower rate than VBF; NLO and NNLO QCD corrections increase its cross section by about 30%, while gg → ZHH adds 20–30% to ZHH production [26] [30] [31].The gluon-induced ZHH contribution at LO is included in the full NNLO QCD corrections [26].Source: §1.1 · p. 20

Double Higgs bremsstrahlung off top quarks

  • At 100 TeV, ttHH approaches VBF in cross section; its NLO QCD corrections reduce the rate by about 20%, whereas tjHH corrections are similarly sized but positive [32].For tjHH, scale uncertainties increase relative to LO, whose uncertainty estimate is not reliable [32].Source: §1.1 · pp. 20–21

1.2 QCD corrections for gluon fusion

The section reviews QCD corrections to gluon-fusion Higgs-pair production, from heavy-top approximations to full-mass NLO calculations and NNLO predictions. It describes how finite top-mass effects, resummation, and alternative expansion methods affect the predictions and their reliability.

1.2 QCD corrections for gluon fusion

  • Born-improved NLO, which retains full LO mass dependence but treats virtual and real corrections in the heavy-top limit, approximates the full total cross section within about 15% [17] [18] [19].NLO corrections in the heavy-top limit increase the cross section by 80–90%.Source: §1.2 · p. 22
  • NNLO QCD corrections increase the total cross section by 20–30%, while NNLL resummation adds O(10%) at μR = μF = mHH and has much smaller effects at mHH/2 [21] [46] [47].The NNLO calculation includes additional one-particle-reducible contributions to virtual corrections and their interference with NLO real corrections.Source: §1.2 · pp. 22–23
  • FTapprox incorporates full one-loop real-emission matrix elements and reduces the Born-improved prediction by about 10%; large-top-mass expansion estimates additional NNLO mass effects of 5% [12] [32] [49] [50].The sizeable virtual two-loop mass effects motivated the full NLO mass calculation.Source: §1.2 · p. 23
  • Full NLO virtual corrections require numerical two-loop box calculations; one method reduces form factors to master integrals, while another parameterizes individual box diagrams [17] [18] [19].Both approaches address ultraviolet and infrared singularities and thresholds numerically, with substantial computational effort.Source: §1.2 · pp. 23–25
  • Full top-mass effects reduce the total NLO cross section by about 15% and are more sizeable at large invariant masses; two numerical methods agree within integration errors [17–19].Both calculations use the narrow-width approximation for the top quark; finite-width effects at LO reduce the total cross section by about 2% [12].Source: §1.2 · pp. 24–25
  • Padé approximants agree with full numerical integration up to mHH ≈ 700–800 GeV, and a Lorentz-invariant expansion agrees up to 800–900 GeV [56] [57].Analytical expansions simplify calculations and can check numerical results, but do not retain full dependence on the expanded parameters; the large-mHH expansion also reveals logarithmic structure and a first approach to NLO bottom-loop contributions [58] [59] [60].Source: §1.2 · pp. 25–27
  • The NNLOFTapprox Monte Carlo improves on Born-improved and FTapprox predictions through subprocess reweighting and full-mass double-real corrections; FTapprox is accurate to better than 10% at NLO [24].At μR = μF = mHH/2, NNLL soft-gluon resummation effects are very small, indicating perturbative stability at this order [61].Source: §1.2 · pp. 27–28

1.3 Cross section as a function of κλ

The section presents full-top-mass NLO QCD predictions for Higgs-pair production across trilinear-coupling values, including total and differential cross sections and a publicly available POWHEG-BOX-V2 generator. It shows that both K-factors and differential shapes depend strongly on κλ, while noting that non-SM predictions have not yet been combined with NNLO corrections.

1.3 Cross section as a function of κλ

  • For the SM coupling, the scale-variation envelope coincides with the 7-point scale-variation band.Scale uncertainties are estimated by varying the common factorisation and renormalization scale by a factor of two around μ0 = mHH/2.Source: §1.3 · p. 29
  • For κλ = −1 the total cross section is largest among the considered values, while K-factors range from 1.57 to 2.16 over −5 ≤ κλ ≤ 12 and vary substantially with κλ [64].This coupling dependence differs from the mt → ∞ limit; the reported K-factors for 14 TeV also apply at 13 TeV, except κλ = 2, whose K-factor is 1.57.Source: §1.3 · p. 29, Table 1.4 · p. 30, Figure 1.9 · p. 30
  • Around κλ = 2.4, maximal destructive interference between triangle and box contributions produces a characteristic dip in the mHH distribution; for κλ < −1 or κλ > 5, triangle contributions increasingly enhance low mHH [64].The same interference is visible but less pronounced in the Higgs-boson transverse-momentum distribution.Source: §1.3 · pp. 29–30, Figure 1.10 · p. 31, Figure 1.11 · p. 31
  • The full-top-mass NLO predictions cover total cross sections at 13, 14, and 27 TeV and differential distributions at 14 TeV, and are implemented in a publicly available POWHEG-BOX-V2 generator [64].The calculation extends the SM result to generic κλ values and includes scale uncertainties.Source: §1.3 · pp. 29–30
  • The full-top-mass NLO result has not yet been combined with the NNLO result in the improved HTL for non-SM κλ; meanwhile, the suggested approximation is to multiply Table 1.4 results by the SM K-factor.The NNLO combination was described as desirable and work in progress.Source: §1.3 · p. 30, Table 1.4 · p. 30

1.4 Differential predictions and MC generators for gluon fusion

The section reviews gluon-fusion di-Higgs predictions matched to parton showers and finds substantial matching and shower uncertainties in key distributions, especially in the di-Higgs transverse-momentum tail. It traces these effects to hard shower emissions and discusses choices that can control them.

1.4 Differential predictions and MC generators for gluon fusion

  • The NLO finite-top-mass calculation is interfaced to POWHEG-BOX, MG5_aMC@NLO, and SHERPA, with fixed-order predictions also extended to a running top-quark mass.The POWHEG and MG5_aMC@NLO implementations use PYTHIA 8.2; SHERPA uses its Catani-Seymour or Dire showers.Source: §1.4 · p. 31, Parton Shower Results · p. 33

Parton Shower Matching

  • MC@NLO uses μPS to divide shower-populated soft emissions from fixed-order real-emission contributions; POWHEG assigns all real emissions to showered Bbar events, which can exponentiate hard contributions.Varying μPS assesses uncertainty from the separation, while POWHEG damping limits hard radiation being exponentiated.Source: Parton Shower Matching · p. 33

Parton Shower Results

  • Matching uncertainties for NLO-accurate observables lie within scale uncertainties, but shower effects and matching uncertainties are large for pHH_T, ΔΦHH, and ΔRHH, sometimes exceeding scale uncertainties.The tails of these distributions are predicted only at the first non-trivial fixed-order contribution.Source: Parton Shower Results · pp. 33–34
  • At large pHH_T, POWHEG-BOX overshoots fixed-order by about a factor of 2, while MG5_aMC@NLO and SHERPA reproduce it at central shower scales; Dire can overshoot at its largest scale.The comparison uses the central shower-scale choices; the fixed-order prediction is expected to be reliable to leading order in this hard region.Source: Parton Shower Results · p. 34, Figure 1.14 · p. 34
  • At low pHH_T, analytic NLO+NLL resummation is marginally compatible with SHERPA and lies between SHERPA and the other implementations.The comparison concerns the region where analytic resummation can be trusted.Source: Parton Shower Results · p. 35, Figure 1.15 · p. 35
  • Hard shower emissions from Bbar events cause tail overshoots when μPS is large; smaller μPS choices, or POWHEG damping with hdamp ≲250 GeV, suppress the effect.The issue arises from sub-leading matching terms, with the large K-factor, splitting kernel, and shower starting scale contributing to the uncertainty.Source: Parton Shower Results · pp. 35–36
  • POWHEG-BOX predictions differ substantially between PYTHIA 8.2 and PYTHIA 6: PYTHIA 8.2 produces harder sub-leading jets and a harder di-Higgs transverse-momentum spectrum.The harder sub-leading jets recoil against the di-Higgs system; PYTHIA 6 yields a softer spectrum that tends toward fixed-order at large pHH_T.Source: Parton Shower Results · p. 36, Figure 1.16 · p. 36

2.1 Introduction to the EFT formalism

Di-Higgs production can test the Standard Model (SM) Higgs potential, while effective field theory (EFT) provides a systematic way to describe broader new-physics effects and constrain them with Higgs-pair and other measurements. The section distinguishes HEFT and SMEFT and explains how SMEFT assumptions and operator effects shape interpretations of possible Higgs self-coupling deviations.

2.1 Introduction to the EFT formalism

  • The SM makes precise predictions for the Higgs potential, but cannot explain its form; Higgs-pair production can test those predictions and probe alternatives to electroweak symmetry breaking.EFT provides a less model-specific framework for describing how new physics could modify pair-production cross sections.Source: §2.1 · p. 37
  • EFT represents short-distance interactions through a systematic Lagrangian of local operators, enabling precise calculations and a finite set of parameters at each expansion order.Its parameters can be constrained using Higgs-pair production alongside single-Higgs, W/Z, and precision electroweak measurements.Source: §2.1 · pp. 37–38
  • Changing the triple-Higgs coupling requires a gauge-invariant formulation; the added terms also generate multi-Higgs vertices needed to cancel potentially troublesome ultraviolet divergences.Treating the Higgs as a singlet or as part of the SM doublet gives different ways to maintain gauge invariance.Source: §2.1 · p. 38

2.1.1 Two EFT extensions of the SM

  • HEFT treats the Higgs as a gauge singlet, whereas SMEFT places it in a gauge doublet; both are gauge-invariant EFTs, but SMEFT is more restrictive and predictive at a given order.They enumerate the same operators using different power-counting schemes, so an operator can enter at different orders.Source: §2.1.1 · pp. 38–39

2.1.2 SMEFT

  • In SMEFT, heavy fields at scale M are integrated out to yield gauge-invariant higher-dimensional operators; for E near mH and M near 1 TeV, dimension-6 effects are estimated at a few percent versus 10^-4 for dimension 8.This estimate assumes coefficients of order 1 and motivates truncating at dimension 6, though the argument is naive and some coefficients can be larger.Source: §2.1.2 · pp. 39–40

How large are the SMEFT parameters?

  • A 10% shift in the top Yukawa coupling can change the extracted Higgs self-coupling by 50%, so other operators must be constrained through measurements, phase-space information, or global fits.Even after basis reduction, 59 independent baryon-number-conserving dimension-6 operators remain for one fermion generation or strongest flavor universality.Source: How large are the SMEFT parameters? · p. 41
  • Precision electroweak data constrain the S and T operator coefficients below 10^-3, while current Higgs-coupling constraints and few-percent SMEFT estimates suggest self-coupling deviations may be unobservable in some cases.Some models nevertheless allow order-1 self-coupling deviations with smaller shifts in other couplings; a strongly first-order electroweak phase transition would require an h3 coefficient about twice the SM value [113,114,115–119].Source: How large are the SMEFT parameters? · pp. 41–42
  • Models with a new boson decaying to HH can produce a high-mass resonance peak alongside a 400–500 GeV peak, where most HH production occurs.The EFT applies only to the lower-energy spectrum; the high-mass resonance requires an explicit new-particle description.Source: How large are the SMEFT parameters? · pp. 41–42

g g →HH in the SMEFT … Perturbativity

This section examines how SMEFT and HEFT parameterize di-Higgs production and Higgs self-coupling measurements, alongside theoretical constraints from vacuum stability and perturbativity. It discusses how perturbativity bounds constrain the trilinear and quartic couplings, informing the interpretation of di-Higgs and single-Higgs constraints and the energy target for searches if the trilinear coupling deviates from the Standard Model.

g g →HH in the SMEFT

  • The g g →HH rate depends strongly on all five relevant SMEFT operators; existing Run 1 constraints leave c6 as the only source of order-10 deviations, while order-1 constraints on c6 require a global fit.Other LHC measurements constrain cH, cu, ctG and cΦG; differential di-Higgs shapes can help disentangle operator effects.Source: Figure 2.2 · p. 44, g g →HH in the SMEFT · p. 44

Single Higgs production in the SMEFT

  • In single-Higgs processes, the trilinear coupling enters only through loop-suppressed NLO corrections, so model-independent constraints require a global fit that reduces the sensitivity found in exclusive fits.The fit must include possible leading-order deformations of single-Higgs interactions; the minimal operator set can depend on collider and precision-electroweak sensitivity.Source: Single Higgs production in the SMEFT · p. 45

2.1.3 HEFT

  • HEFT treats anomalous Higgs couplings as independent parameters, unlike SMEFT relations; extracting chhh therefore requires a global analysis because ctt and cgghh occur only in processes with two Higgs bosons.The framework supplies a field-theory basis for the empirical κ-framework; observed Higgs couplings agree with their SM values to 10–20%, a strong constraint on HEFT parameters.Source: §2.1.3 · pp. 48–49

2.2 EFT vs. complete models: theoretical constraints on κλ

  • ATLAS’s strongest cited double-Higgs constraint is −5.0 < κλ < 12.1, and no experimental constraints on the quartic Higgs self-coupling are available.The bound combines three analyses [62] [144] [145] [146].Source: §2.2 · p. 49

Vacuum stability

  • EFT vacuum-stability arguments cannot provide a model-independent bound on the trilinear Higgs self-coupling [151], because the relevant instabilities lie outside reliable EFT use.In low-scale inverse-seesaw models, deviations up to 30% can destabilize the potential; requiring no instability within an order of magnitude of the neutrino mass scale bounds modifications to κλ = 0.1% [151].Source: Vacuum stability · pp. 49–50

Perturbativity

  • Excessively large Higgs self-couplings can enter the non-perturbative regime, where LO predictions and higher-order corrections cannot be trusted and new phenomena or massive particles may be needed to restore perturbativity.The discussion treats these as complementary implications of perturbativity violation.Source: Perturbativity · p. 50
  • Partial-wave unitarity gives |κλ| ≲6.5 and |λH4/λSM H4| ≲65, with small-ps sensitivity to the trilinear and large-ps sensitivity to the quartic coupling.This kinematic distinction permits model-independent bounds on the trilinear and quartic self-couplings separately.Source: Perturbativity · p. 50
  • Requiring loop corrections to the trilinear vertex to remain below its tree-level value, or |βλH4/λH4| < 1 for the quartic, gives bounds similar to the partial-wave criteria [155] [156].Source: Perturbativity · pp. 50–51
  • The bounds apply to an on-shell HHH vertex across the spectrum above 2mH and are strongest near mHH ≃2mH; |κλ| ≳6 makes perturbative di-Higgs total-cross-section predictions meaningless.They do not directly apply to single-Higgs studies, where the vertex does not involve two on-shell Higgs bosons; |κλ| ∼10 can still yield reliable predictions, albeit with large higher-order corrections [107] [108] [122] [123] [124] [125] [126] [127] [128].Source: Perturbativity · p. 51
  • For HL-LHC di-Higgs cross-section constraints, LO and NLO fits become incompatible at |c6| ≳5, and only |c6| = |κλ−1| ≲5 identifies the Lagrangian parameter of interest [157].Beyond that range, c6 fitted with LO predictions is a parameter of ignorance, and higher-order corrections do not resolve the issue.Source: Perturbativity · p. 51, Figure 2.5 · p. 52
  • One-loop HHH-vertex bounds on c8, accounting for |c6| ≲5, imply |λH4/λSM H4| ≲31 for κλ ≃1, consistent with the separate partial-wave bound [128].Source: Perturbativity · pp. 51–52
  • Current trilinear self-coupling limits remain above the perturbative regime; if a deviation is observed, perturbation theory breaks down at Emax, making that scale a target for future colliders [158].The Emax breakdown result is stated to hold regardless of the Higgs-potential shape.Source: Perturbativity · p. 52

UV-complete models · New Physics in Higgs pair production · Constraints

This section surveys constraints on the Higgs self-coupling and Higgs-pair production in UV-complete models and EFT fits, including information from single-Higgs measurements and di-Higgs production. It also covers kinematic-shape benchmarks and resonant and other new-physics signatures, including singlet-scalar benchmarks.

UV-complete models

  • In the singlet extension scan, vacuum instability does not constrain the trilinear-coupling modification, while Higgs-coupling and W-boson-mass measurements exclude parts of parameter space.The scan also classifies points as unstable, metastable, or stable; the text does not provide the maximal allowed deviation value.Source: UV-complete models · pp. 53–54
  • In the MSSM, maximal triple-Higgs deviations are about 15% at tanβ = 5 and MA ∼200 GeV, about 2% for tanβ ≥10, and ≲4% for MA ≳350 GeV [164].The larger deviations occur at low MA, which current heavy-Higgs searches and Higgs-property measurements disfavor; consistent approximations for the Higgs mass and coupling are crucial to the estimate.Source: UV-complete models · pp. 54–55

2.3.1 EFT fit for HH production

  • At the HL-LHC, the projected c6 constraint reaches the 60% level with 10% single-Higgs coupling measurements, versus 40% when only c6 varies [103] [169].With present Higgs-coupling uncertainties, the projected constraint is very weak and leaves c6 unbounded from above; other EFT coefficients can compensate for c6 effects in production or branching fractions.Source: §2.3.1 · pp. 56–57, Figure 2.8 · p. 57

2.3.2 Impact of single Higgs production

  • Single-Higgs measurements probe the trilinear coupling through one-loop corrections in VBF, WH, ZH, ttH, and tHj, and through two-loop corrections in gluon fusion and H→gg,γγ [107] [108] [122] [123] [124] [125] [126] [127] [128].Production channels have process- and kinematics-dependent linear contributions plus a universal quadratic wave-function term, whereas decays have only linear dependence.Source: §2.3.2 · pp. 57–58
  • The single-Higgs-derived bounds are competitive with current double-Higgs bounds [124], but assume deviations only in the trilinear coupling and depend on fit parameters, differential data, and uncertainty assumptions [107, 127].The linear dependence is especially pronounced for WH, ZH, and ttH, with stronger kinematic effects near threshold; VBF dependence is comparatively flat [124] [125] [127].Source: §2.3.2 · pp. 59–60, Figure 2.11 · p. 61
  • Single-Higgs inclusive data alone leave the trilinear coupling unconstrained in a nine-parameter EFT fit, while differential information helps lift degeneracies and improve di-Higgs constraints.With all nine EFT deviations profiled, di-Higgs production gives much stronger constraints; single-Higgs data can help lift degenerate minima around δκλ ∼5, and the combined fit can improve bounds.Source: §2.3.2 · pp. 61–62

2.4 EFT shape benchmarks

  • Twelve shape benchmarks [177] were selected from 1507 simulated EFT samples by clustering their mHH and cosθ∗ distributions, balancing cluster homogeneity against redundancy.The cluster representative is the sample most similar to the other members under a binned two-sample test.Source: §2.4 · pp. 62–63, Table 2.2 · p. 63
  • Observed CMS cross-section upper limits vary by up to two orders of magnitude across shape benchmarks, which can also support coarse reinterpretation of points beyond the EFT domain [170].The mapping to the most similar benchmark uses kinematic information; benchmark-based simulation can also support Monte Carlo reweighting, but cannot cover every EFT correlation.Source: §2.4 · pp. 63–64
  • NLO corrections largely preserve peak, dip, and high-mass-tail features but produce coupling- and mHH-dependent K-factors, so NLO benchmark studies remain desirable [65].The current LO benchmarks may suffice for an initial assessment of analysis sensitivity to shape modifications, but further NLO studies are needed to confirm this.Source: §2.4 · pp. 64–65

New Physics in Higgs pair production

  • New-physics scenarios include light resonances from extended scalar sectors, colored particles in gluon loops, and Higgs-pair plus missing-energy signatures that could reveal a dark sector.Di-Higgs production can break the single-Higgs degeneracy between a shifted top Yukawa coupling and new physics in the gluon-fusion loop [104] [181] [183]; the discussion also considers baryogenesis implications of the self-coupling.Source: New Physics in Higgs pair production · p. 67

3.1 Spin-0 models

Constraints

  • Singlet-scalar benchmarks are assessed against vacuum stability, perturbative unitarity and coupling perturbativity, electroweak precision and W-mass data, and LHC Higgs-search and rate constraints.The allowed maximal pp → h2 → h1h1 signal rate at 13 TeV is compared with the CMS upper limit; the presentation distinguishes all constraints from EW-scale-only constraints.Source: Constraints · p. 69

Non-Z2 … Status of the measurements at LHC

The section surveys non-SM Higgs-sector scenarios that can affect di-Higgs production and reviews their collider signatures, constraints, and associated probes. It also covers LHC Higgs-pair searches and indirect self-coupling measurements, including recommendations for future measurements.

Non-Z2

  • Even after requiring the observed electroweak vacuum to be global, the non-Z2 singlet model allows h2 → h1h1 branching ratios above 80% and di-Higgs rates an order of magnitude above the SM prediction.The additional free parameters a2 and b3 complicate the vacuum structure and affect the h2–h1–h1 coupling; the quoted maxima use b4 at its perturbative-unitarity upper limit and do not impose high-scale RGE perturbativity.Source: Non-Z2 · pp. 70–71

3.2 Interference Effects

  • The interference discussion covers effects of the scalar resonance, on-shell phase effects, and a general parameterization of the overall signal–continuum interference.The continuum can itself differ from the SM because Higgs couplings are modified.Source: §3.2 · pp. 71–72

3.2.1 Off-shell Interference

  • For m2 around 600 GeV, off-shell interference can contribute over 20% of the total cross section and grows as m2 increases.At m_h1h1 much less than m2, the distribution becomes independent of m2; mass-proportional scalar couplings cancel the propagator’s mass dependence, making the interference increasingly important.Source: §3.2.1 · p. 73

3.2.2 On-shell Interference

  • In the singlet extension, constructive on-shell interference between the heavy-scalar resonance and SM box diagram can reach 40% of the Breit–Wigner contribution and enhance the total signal [208].A large relative phase between loop functions drives this effect; at the 900 GeV benchmark, the resonance peak is enhanced by about 25%, with additional enhancement off peak.Source: §3.2.2 · pp. 74–75

3.2.3 Overall Interference

  • In the model-independent parameterization, interference effects depend mostly on the signal-to-background cross-section ratio; at a ratio of about 10, they can increase the cross section by a factor of 1.5.For mH > 2mt, the effect has little dependence on ΓH; the parameterization measures rate changes and peak distortion but not peak shifts, and assumes a constant Wilson coefficient.Source: §3.2.3 · pp. 77–78

3.3 Spin-2 models

  • Warped extra-dimensional models motivate spin-2 di-Higgs resonance searches through Kaluza–Klein gravitons and radions, with KK-graviton masses typically above 1 TeV and potentially lighter radions.The discussion assumes only the lightest KK graviton is experimentally accessible; its couplings to SM particles arise through the energy-momentum tensor.Source: §3.3 · pp. 78–79
  • For bulk Randall–Sundrum models, KK gravitons are produced mainly by gluon fusion and can have di-Higgs branching ratios around 10%, while di-radion branching is about 0.1%.VBF production is typically an order of magnitude lower than gluon fusion, though better background rejection can sometimes improve sensitivity; cited direct KK-graviton bounds are 3.4 TeV in all-hadronic tt̄ and 3.8 TeV for a 15% width in lepton-plus-jets searches.Source: §3.3 · pp. 80–81
  • In the dark-brane scenario, KK gravitons can decay invisibly to dark matter or to radion pairs that yield multiple-Higgs signatures, while radion decays can produce massive SM states at comparable rates.The cited KK-graviton searches also motivate spin-2 resonance searches in di-Higgs production, including composite Higgs models.Source: §3.3 · p. 81

3.4.1 Complex 2-Higgs-Doublet Model

  • In C2HDM benchmarks, the maximum hh rate can exceed the SM rate by about 40 in T1 and 3.2 in T2, through resonant heavy-Higgs decays.The T2 enhancement is smaller because its Higgs spectrum is heavier.Source: §3.4.1 · p. 82, Table 3.3 · p. 83
  • In C2HDM T1, SM-like hh production to bbγγ can reach 68 times the SM rate, while H↓H↓ rates reach about 200 times in bb̄bb̄ or bb̄τ+τ− and 40 times in bb̄γγ.The smaller photon-channel enhancement reflects the smaller branching ratio into photons.Source: §3.4.1 · p. 83, Figure 3.9 · p. 83
  • Other C2HDM channels are less promising: hH↓ enhancements are below 3 in bb̄bb̄ and bb̄τ+τ−, while T2 hh enhancements are around 4.5 in the listed final states.The authors identify T1 as promising for SM-like and non-SM-like Higgs pairs, whereas sufficiently large observed rates could exclude T2.Source: §3.4.1 · p. 83

3.4.2 Singlet extensions of 2HDM

  • In the 2HDM+S alignment limit, non-SM-like CP-even states cannot couple to h125 pairs or vector bosons, and CP-odd states cannot couple to h125 and Z, allowing other Higgs cascade modes.Singlet states couple to SM particles through mixing or may decay to dark matter, so cascades can yield visible or invisible signatures.Source: §3.4.2 · pp. 85–86
  • At 3000 fb−1, projected 2HDM+S cascade searches reach large parameter regions that are difficult for direct additional-Higgs searches, under the stated 750 GeV parent-mass and perfect-alignment benchmark.The reach summary omits existing bounds on additional Higgs bosons; misalignment has negligible quantitative effect on the cascades studied, though it permits other suppressed decays.Source: §3.4.2 · pp. 86–87, Figure 3.10 · p. 86

3.4.3 hMSSM

  • The hMSSM approximates the MSSM for large SUSY particle masses and small or moderate Higgsino mass parameter μ relative to stop masses, encoding radiative corrections through the known 125 GeV Higgs mass.Neglecting diagonal and off-diagonal radiative-correction entries excludes large μ; the remaining Higgs-sector parameters include MA and tanβ.Source: §3.4.3 · p. 87

3.4.4 NMSSM

  • The NMSSM also permits different-Higgs and pseudoscalar-pair final states, with resonant heavy-Higgs decays enhancing rates and possible decays into non-SM states such as neutralinos.Supersymmetry relates Higgs self-couplings to gauge couplings, limiting large deviations from the SM self-coupling.Source: §3.4.4 · p. 89
  • NMSSM scans find bbbb rates up to 9.3 times the SM value for A↓A↓ at 287 fb−1 exclusion luminosity, while hh rates reach at most twice the SM rate.The hH↓ rate reaches about 4.6 times the SM rate at 449 fb−1 exclusion luminosity; the A↓A↓ enhancement is associated with a light pseudoscalar and resonant H↓ production.Source: §3.4.4 · pp. 90–91, Figure 3.11 · p. 90, Figure 3.12 · p. 90

3.4.5 Georgi-Machacek Model

  • The current ATLAS hh→bbbb search excludes Georgi–Machacek model parameter space for mH between about 300 GeV and 1 TeV beyond that constrained by previous searches.The analysis considers resonant H→hh production and uses the model as a benchmark for searches with the full LHC Run 2 dataset.Source: §3.4.5 · p. 92, Figure 3.13 · p. 92

3.5 New Particles in the Loop

  • New colored scalars or fermions in loop amplitudes can change di-Higgs rates and kinematic distributions while leaving single-Higgs production near the SM prediction if their masses have other sources.New heavy quarks consistent with single-Higgs rates must be vector-like rather than obtaining all their mass from the Higgs mechanism.Source: §3.5 · pp. 92–93, Heavy VLQs · p. 93

Singlet VLQ

  • In singlet VLQ models, single-Higgs production stays nearly SM-like, while double-Higgs production can be suppressed; electroweak constraints force it within about 15% of the SM value.For higher top-partner masses, mixing constraints tighten, effects decouple, and invariant-mass distributions remain nearly SM-like.Source: Singlet VLQ · pp. 93–94, Figure 3.14 · p. 93

Full VLQ Generation

  • Full-generation VLQ distributions become very similar to the SM after electroweak precision and single-Higgs constraints are imposed, and the SM EFT matches the model well for Mhh below partner masses.In the EFT, single- and double-Higgs rates are tightly related: increasing the single-Higgs rate decreases the double-Higgs rate, so larger deviations require light particles outside EFT validity.Source: Full VLQ Generation · pp. 94–95, Figure 3.15 · p. 94

Colored scalars

  • A color-triplet scalar needs κ ≳2 to reproduce the SM single-Higgs rate, while a color-octet intermediate state tuned to that rate predicts a highly suppressed double-Higgs rate.Even when total rates match the SM, scalar-loop kinematic distributions differ; reproducing SM rates requires a light scalar, with distributions peaking at lower mhh.Source: Scalar Top Partners · p. 95, Figure 3.16 · p. 95, Figure 3.17 · p. 96

Scalar Top Partners

  • Scalars receiving all their mass from electroweak symmetry breaking produce an unacceptably large single-Higgs rate, while heavy scalars can decouple there and affect the high-mhh double-Higgs tail.For an 800 GeV color-triplet scalar receiving half its mass squared from electroweak symmetry breaking, the single-Higgs rate is in roughly 2σ tension with measurement and the double-Higgs distribution deviates from the SM starting near 2ms.Source: Scalar Top Partners · p. 96, Figure 3.18 · p. 96

MSSM P. Huang

  • Light stops can alter di-Higgs rates and distributions; crossing the 2m_stop threshold produces a second m_hh peak, while the largest MSSM branching-ratio shift is about ±20% in b b̄γγ.Large stop mixing can generate relevant cross-section contributions but is constrained by vacuum stability as well as direct stop searches.Source: MSSM P. Huang · pp. 97–98

3.6 Connection to Cosmology

  • New physics can turn the Standard Model’s smooth electroweak crossover into a first-order transition, with baryogenesis among its possible consequences; di-Higgs measurements probe the Higgs-potential couplings that can strengthen it.Resonant di-Higgs production can probe strong first-order transitions generated by new scalars up to the TeV scale; additional scalar-pair production offers a complementary probe.Source: §3.6 · pp. 98–99, Figure 3.21 · p. 100
  • For a singlet-scalar benchmark, HL-LHC resonant di-Higgs studies and LISA gravitational-wave signals can play complementary roles in exploring the model.The benchmark has a 455 GeV singlet; LISA-detectable points are defined by signal-to-noise ratio above 10, and the collider projections rescale current ATLAS and CMS limits to 3000 fb−1.Source: §3.6 · p. 100, Figure 3.22 · p. 101

3.7 HH and Dark Matter (Missing Energy)

  • At 3 ab−1, nearly all symmetric benchmarks are within HL-LHC discovery reach, whereas planned luminosity cannot exclude or discover resonant benchmarks; resonant 4b2χ sensitivity reaches fb and sub-fb cross sections and improves for compressed spectra.The resonant model’s lower production rates limit benchmark reach; its sensitivity does not depend on mχ provided 2mχ < mA.Source: §3.7 · pp. 102–103, Figure 3.24 · p. 102, Figure 3.26 · p. 103
  • Di-Higgs plus missing energy remains the most sensitive tested channel for both topologies, while dark-matter direct detection or long-lived-particle searches can complement it in probing χ.The authors encourage ATLAS and CMS to add topology-based Higgs-pair plus missing-transverse-energy searches.Source: §3.7 · p. 103

3.8 Summary: precision goals for the measurement of the Higgs pair production process in the light of new physics

  • The precision-goal ladder is Bronze 100% for the largest effects, Silver 25–50% for heavy-scalar mixing and typical electroweak-baryogenesis deviations, Gold 5–10% for loop effects, and Platinum 1% for typical quantum corrections.Gold-level sensitivity could complement searches for new particles at the HL-LHC.Source: §3.8 · pp. 104–105

Status of the measurements at LHC

  • The LHC overview covers direct di-Higgs and indirect single-Higgs self-coupling tests, resonant-state searches, and HH triggers, reconstruction, and calibration; it reviews ATLAS and CMS searches using 2015–2018 data up to about 126 fb−1 and discusses result reinterpretation.The search overview includes non-resonant and resonant gluon-fusion production across Higgs decay signatures.Source: Status of the measurements at LHC · p. 109
  • Chapter 7 presents LHC HH searches across beyond-SM hypotheses, an initial statistical ATLAS–CMS combination, recommendations for improving measurements and interpretation, and first indirect κλ determinations from precision single-Higgs measurements.The indirect determinations use precision measurements of single-Higgs processes described in Chapter 2.Source: Status of the measurements at LHC · p. 109

Detector objects, triggers and analysis techniques

This section reviews detector-object reconstruction and identification techniques relevant to Higgs-pair searches, covering b jets, hadronic τ leptons and photons. It also discusses pileup mitigation, machine-learning methods, b-jet energy corrections and challenges for object performance under HL-LHC conditions.

Detector objects, triggers and analysis techniques

  • ATLAS and CMS combine information from multiple detector subsystems to reconstruct Higgs-decay particles and develop dedicated triggers and calibration strategies for HH final states.The detectors use different technologies and designs but have similar capabilities; identification algorithms commonly use neural networks or boosted decision trees calibrated with data from well-understood SM processes.Source: Detector objects, triggers and… · p. 111

4.1 Jet reconstruction

  • For high-mass HH searches, boosted Higgs decays use large-radius jets, while grooming improves H-jet mass resolution by filtering soft, wide-angle radiation.ATLAS track-calorimeter-cluster jets combine inner-detector angular information with calorimeter energy measurements to improve high-pT resolution. Pileup mitigation is especially important at HL-LHC, where pileup is expected to exceed current levels by more than threefold.Source: §4.1 · pp. 112–113

4.2 Identification of b-jets

  • At 70% b-tagging efficiency, ATLAS reports 0.3% light-jet and 11% charm-jet mis-tag rates; CMS reports 1.1% and 12% at 68% efficiency.ATLAS and CMS combine complementary vertexing, impact-parameter and soft-lepton information using multivariate taggers. At HL-LHC, pileup, tracking degradation and denser track environments challenge b-tagging.Source: §4.2 · pp. 113–115

4.2.1 Boosted H →b¯b taggers

  • For boosted b¯b tagging at 70% efficiency, CMS Deep-Double-b reduces the inclusive mis-tag rate from 4% to 1.2–1.5% versus the BDT tagger.The dedicated tagger uses tracks and secondary vertices, including correlations between the two B-hadron directions; it outperforms tagging two separate jets for dijet pT above 350–400 GeV. CMS and ATLAS also observed Z→b¯b in a single-jet topology, consistent with the SM expectation.Source: §4.2.1 · pp. 115–116

4.3 Specific Corrections for b-jet transverse momentum

  • CMS DNN regression improves b-jet resolution by about 13% and dijet invariant-mass resolution by roughly 20%, while ATLAS corrections improve the latter by about 18%.The CMS DNN also predicts intrinsic jet resolution to better than 20% accuracy across a pT range spanning more than one order of magnitude; its improvement was validated in data. ATLAS obtains a similar mass-resolution improvement with µ-in-jet correction or BDT regression.Source: §4.3 · p. 117

4.4 Hadronic τ object identification

  • ATLAS rejects quark- and gluon-initiated jets using tracker-based τ-vertex and track selections, while CMS HPS combines calorimeter deposits and charged tracks to classify hadronic decay modes [411].Hadronic τ objects are seeded by anti-kT jets with R = 0.4; CMS searches within particle-flow jets and identifies 1-prong, 3-prong, and 1-prong-plus-π0 modes.Source: §4.4 · p. 118
  • Both experiments use multivariate τ identification; BDTs were used in 2015–2017, and ATLAS introduced an RNN in 2018, with jet rejection of O(10^2) to O(10^3).Electron- and muon-to-τ misidentification probabilities are significantly smaller than jet-to-τ probabilities, and jet fake rates have fallen by more than an order of magnitude since 2010 for similar efficiencies. Efficiency-correction uncertainties are approximately 5–6% for visible transverse momenta of 20–60 GeV; τ energy-scale precision is approximately 1.2–2% for ATLAS and less than 1.2% for CMS.Source: §4.4 · p. 119

4.4.1 Boosted H →τ+τ−

  • For boosted τ pairs, R = 0.4 jets can merge the candidates; ATLAS and CMS instead use large-radius jet substructure to identify the pair, with boosted-channel efficiencies of about 80% and 60%.The quoted efficiencies correspond to boosted H→μτh and τhτh decays, respectively, with misidentification rates of 10^-3 and 10^-4; CMS seeds standard τ reconstruction from two subjets in an R = 0.8 jet.Source: §4.4.1 · pp. 119–120

4.5 Photon reconstruction

  • ATLAS calibrates sliding-window photon clusters with standard candles, achieving 2.5% (1%) relative energy resolution for central unconverted photons at ET = 20 (200) GeV [419].CMS uses flexible-window clustering, response corrections, Z→e+e− scale calibration, and multivariate pileup mitigation; its resolution is 1.5% for central electrons with low bremsstrahlung reconstructed as photons [421].Source: §4.5 · pp. 120–121
  • ATLAS photon shower-shape cuts achieve 85–90% (85–95%) efficiency for unconverted (converted) photons, while CMS rectangular cuts or a BDT yield 75–95% efficiency for ET > 20 GeV [423].ATLAS photon isolation has approximately 98% signal efficiency in SM HH→bb̄γγ events; CMS BDT inputs include nearby detector activity and endcap preshower information to identify π0 decays.Source: §4.5 · p. 121

J. Alison

This section reviews online b-jet and τ triggering, then describes HH-specific reconstruction techniques that use kinematic constraints and vertex identification.

4.6.1 b-jet trigger

  • Raising the jet threshold from 60 to 100 GeV halves HH → b ¯b b ¯b sensitivity, with the greatest loss at low mHH, most sensitive to λ [427].Efficient trigger-level b-jet identification is therefore critical for preserving signal acceptance.Source: Figure 4.3 · p. 122, §4.6.1 · p. 122
  • At background rejection 100, online/offline b-tag efficiency differs by ∼2% in ATLAS [428] and ∼6% in CMS [386].CMS online tracking efficiency is 10% below offline for 1–10 GeV tracks, versus better than 98% in ATLAS; inclusive L1 rates, HLT tracking CPU, and online/offline consistency also constrain b-jet triggers.Source: §4.6.1 · pp. 122–124, Figure 4.4 · p. 124

4.6.2 τ trigger

  • CMS’s dynamic-clustering L1 τ trigger combines calorimeter information, calibration, and pile-up mitigation, achieving efficiency close to 100% above threshold for the considered hadronic τ decays [429].The algorithm identifies hadronic decay modes and enables unprecedentedly low online thresholds.Source: §4.6.2 · p. 124

4.7.1 Kinematic fit procedure

  • The CMS kinematic fit improves resonant HH → b ¯b b ¯b mass resolution by 20–40% and sensitivity by 10–20% [405].The fit constrains both dijet masses to the Higgs mass; similar resolution improvements are reported for HH → b ¯bτ+τ− and HH → b ¯bγγ searches [438] [439].Source: §4.7.1 · p. 126
  • ATLAS’s simplified dijet-mass constraint improves mHH resolution by 30–60% across the resonance range, while sculpting non-resonant background at low mHH [144] [145].The CMS reduced-mass estimator improves mHH reconstruction by 25–30 GeV; for a 300 GeV resonance, resolution falls from roughly 50 to 20 GeV [442, 443].Source: §4.7.1 · p. 126
  • CMS’s f MX estimator improves dijet mass resolution by about 10% in boosted HH → b ¯b b ¯b searches [444] [445].The estimator corrects the dijet mass to address out-of-cone and resolution effects.Source: §4.7.1 · p. 127

4.7.2 HH vertex reconstruction with H →γγ decay

  • ATLAS and CMS identify the primary vertex correctly in up to 99.9% of simulated HH → b ¯bγγ events [145] [442].The H → b ¯b decay products help identify the event vertex beyond the diphoton vertex information alone.Source: §4.7.2 · p. 127

Overview of HH searches at the LHC

The section surveys LHC searches across Higgs-pair decay modes and describes the challenges and analysis strategies of the b¯bb¯b channel. It highlights the need for improved low-mHH background modelling and the complementary strengths of other final states.

Overview of HH searches at the LHC

  • The leading search modes are b¯bb¯b, whose large branching fraction is offset by multijet backgrounds; b¯bγγ, which benefits from photon reconstruction and mass resolution; and b¯bτ+τ−, a rate-background compromise with the most stringent individual-channel limit.The b¯bτ+τ− result uses multivariate analysis techniques.Source: Overview of HH searches at the LHC · pp. 129–130
  • Other explored modes include b¯bVV*, WW*γγ, WW*WW*, and τ+τ−τ+τ−; non-b-jet searches have smaller yields but lower backgrounds and are mainly statistics-limited, so their sensitivity should scale better with luminosity.The non-resonant b¯bVV* outlook is described as challenging, with room for improvement.Source: Overview of HH searches at the LHC · p. 130

5.1 HH →b ¯bb ¯b: status and perspectives

  • In b¯bb¯b searches, multijet backgrounds exceed signal rates by several orders of magnitude, and after selection multi-b-jet production contributes 85–90% of background versus 10–15% from top-quark pairs.The Z + jets contribution is estimated at no more than 0.2–0.5%; precise multijet modelling is difficult because reliable simulation requires very large event samples.Source: §5.1 · pp. 130, 132
  • In the latest ATLAS b¯bb¯b search, loosened selections increased background acceptance by a factor of 20, while statistical uncertainty at the mHH peak fell by an order of magnitude to percent level.By the end of HL-LHC data-taking, a sub-percent-level background model will be required, motivating novel data-driven techniques.Source: §5.1 · p. 131, Figure 5.2 · p. 131
  • Resonant searches cover 260 < mHH < 3000 GeV using resolved four-b-jet selections up to 1500 GeV and boosted large-radius jets up to 3 TeV; CMS semi-resolved categorization moderately improves sensitivity for 750–2000 GeV.The resolved regime dominates sensitivity to SM non-resonant HH production.Source: §5.1 · p. 131

5.1.1 Analysis strategies

  • CMS trigger efficiency ranges from 10% at the 2mH threshold to 60% above 800 GeV and is 34% for the non-resonant hypothesis; ATLAS efficiency ranges from 65% at threshold to ⪆99% above 600 GeV.Both experiments use multijet triggers with online b-tagging; the four highest-scoring b-tagged jets are paired to reconstruct Higgs candidates, with ATLAS and CMS selecting the correct pairing at least 90% and 70% of the time, respectively, for non-resonant signal.Source: §5.1.1 · pp. 132–133
  • For resonant b¯bb¯b searches, CMS mass corrections improve reconstructed resonance resolution by 20–40% and sensitivity by 10–20%.The corrections use per-jet regression and constrain Higgs-candidate masses to 125 GeV.Source: §5.1.1 · p. 133

Background modelling

This section reviews background modelling and analysis strategies across Higgs-pair searches, including b-tagged and non-b-jet channels and vector-boson-fusion production. It covers methods for separating signal from background, current limitations and results, and opportunities to improve sensitivity.

Background modelling

  • ATLAS extrapolates low- to high-b-jet-multiplicity data with weighting factors validated in a control region, while CMS uses hemisphere mixing to create artificial background events [461].CMS’s method mixes similar hemispheres from different events to retain background kinematics while disrupting signal correlations; signal-injection tests verify robustness to small signal contamination. Its shape and normalization uncertainties affect the final result by about 9% and 30%, respectively.Source: Background modelling · pp. 133–135

5.1.2 Limitations of current analysis strategies

  • Trigger constraints, statistically limited control regions, and uncertain extrapolations challenge both experiments; ATLAS’s iterative one-dimensional reweighting may miss variable correlations, while CMS hemisphere mixing may dilute t¯t and electroweak correlations.L1 efficiencies are limited for mHH ⪅500 GeV, and online b-tagging CPU demands grow non-linearly with pileup. ATLAS’s attempt to estimate uncertainty by deriving models in orthogonal kinematic regions also faces validation-statistics limits.Source: §5.1.2 · pp. 135–137
  • 33% lower SM HH sensitivity results when the ATLAS background model’s per-bin statistical uncertainty is set equal to that of data, illustrating the analysis’s statistical limitation [144].The hemisphere-mixing model also uses each source event only once, limiting its statistical precision to that of the true background.Source: §5.1.2 · p. 137

5.1.3 Potential improvements

  • CMS b-jet energy regression improved Higgs-mass resolution by 6–12% and search sensitivity by 5–20%; proposed improvements include multivariate reweighting and background validation in higher-statistics three-b-jet samples [405].Three-b-jet validation is most useful at low mHH, where signal contamination is negligible; at mHH ⪆500 GeV, such data could be sensitive to new physics. Measurements of SM ZZ and ZH in b¯bb¯b could also validate background models and reduce systematic impacts.Source: §5.1.3 · pp. 137–138

5.2.1 Overview

  • HH → b¯bγγ has a 0.3% branching fraction but high signal-to-background; with about 36 fb−1, ATLAS and CMS set 95% CL cross-section limits of 0.73 pb and 0.79 pb, respectively, and constrain κλ to −8.2 to 13.2 and −11 to 17 [145] [442].Two high-quality photons suppress multijet backgrounds and provide a trigger advantage for mHH < 400 GeV.Source: §5.2.1 · p. 139

5.2.2 Signal modelling

  • The fully reconstructable b¯bγγ final state has no Higgs-candidate combinatorics; photon mass resolution is σγγ/Mγγ = 1.3%(1.5) for CMS (ATLAS), while CMS quotes Mjj/σjj = 15% after b-jet energy regression.The analyses model Higgs mass peaks with double-sided Crystal-Ball functions, whose Gaussian-core mean and width can encode scale and resolution uncertainties.Source: §5.2.2 · pp. 139–140

5.2.3 Event selection and reconstruction

  • Both experiments use diphoton triggers that are nearly fully efficient for selected photons; ATLAS categorizes events by b-tag multiplicity, whereas CMS uses b-tagging scores and helicity angles for multivariate categorization.ATLAS reconstructs the b¯b candidate from tagged jets or a tagged jet plus a BDT-selected jet; CMS selects jets with the highest b-tagging scores.Source: §5.2.3 · pp. 140–141
  • Three Collins–Soper helicity angles probe Higgs spin and CP properties while reducing dependence on HH transverse momentum; only the two highest-purity categories enter signal extraction.The fully reconstructed mHH spectrum is particularly important for resonant searches and is used in signal extraction.Source: §5.2.3 · p. 142, Figure 5.9 · p. 142

5.2.4 Background modelling

  • Continuum γ+jets backgrounds are modeled from data because QCD corrections and jet-to-photon fragmentation are challenging; function choice can bias signal estimates, motivating Gaussian-process and envelope alternatives.Single-Higgs backgrounds contribute 5–14% of the total and contain an mγγ peak, complicating signal discrimination; Gaussian processes encode fit flexibility through covariance kernels, while envelope fits penalize more complex parameterisations.Source: §5.2.4 · pp. 142–143

5.2.5 Signal extraction

  • The CMS simultaneous mγγ–m j j fit improves search sensitivity by ≈10% over the ATLAS approach, but its factorisation assumption must be rechecked as datasets grow.The ATLAS approach avoids reliance on modeling mass correlations; the CMS approach constrains non-resonant backgrounds using the full m j j distribution.Source: §5.2.5 · p. 144

5.2.6 Systematic uncertainties

  • Run 2 HH → b ¯bγγ searches remain statistics-limited; experimental uncertainties include 1–5% from photon scale and resolution, 1–5% from jet energy scale and resolution, and 10–20% from flavour tagging.PDF and scale uncertainties on the non-resonant signal model are 3–6%, while single-Higgs background normalisation uncertainties are 1–20%; ATLAS also assigns a 100% uncertainty to ggF and WH with extra heavy flavour.Source: §5.2.6 · p. 145

5.2.7 Machine learning in HH →b ¯bγγ: use and challenges

  • Classifier generalisation is addressed with separate kinematic regimes and, in CMS, an ensemble of signal hypotheses; parameterised neural networks can also interpolate across model parameters.The parameterised network trains effectively distinct classifiers for parameter values such as resonance mass or κλ within one training procedure.Source: §5.2.7 · p. 146

5.3 HH →b ¯bτ+τ−: status and perspectives

  • The HH → b ¯bτ+τ− branching fraction is 7.3%; analyses combine eτh, µτh, and τhτh, which cover 88% of ττ decays, amid heavy-flavour, top-pair, multijet, and increasing single-Higgs backgrounds.Reconstruction must identify b-jets and hadronic τ decays while rejecting misidentified objects.Source: §5.3 · pp. 146–147

5.3.1 Analysis strategies

  • ATLAS uses MMC and CMS uses SVfit to estimate mττ from visible decay products and missing transverse energy [487] [489]; the τhτh category with two b-tagged jets is most sensitive.ATLAS uses lepton-plus-τ triggers in semileptonic channels, gaining 3% final sensitivity; higher future luminosity may require raised thresholds unless track-based techniques are used. CMS also splits resonant events above 700 GeV into resolved and boosted categories [438].Source: §5.3.1 · pp. 147–148

5.3.2 Modelling of background contributions

  • The bbττ background model relies on simulation for top pairs and Z/γ*+jets shapes, with data control regions constraining normalisations and fake contributions.Top pairs are the dominant background; Z+heavy-flavour predictions require large data-derived corrections because they do not match observations, with ATLAS using one normalisation factor and CMS using three generator-level b-jet categories.Source: §5.3.2 · pp. 149–150

5.3.3 Limitations of the current result and perspectives

  • Observed non-resonant HH→bbττ limits are 12.7×SM for ATLAS and 31.4×SM for CMS; CMS sensitivity is weaker by a factor of 1.7.CMS's lower signal efficiency, limited final-selection statistics, and BDT-score cut strategy contribute to the gap; an independent CMS-data study found comparable-to-ATLAS results using multivariate methods in all channels and the continuous BDT score.Source: §5.3.3 · pp. 150–151
  • Although current bbττ searches are statistically dominated, multi-jet and top-pair normalisation uncertainties and τh and b-tagging efficiencies are important targets for future sensitivity gains.Multi-jet normalisation uncertainties are 5–30% and top-pair normalisation uncertainties 10–17%; τh and b-tagging efficiencies affect signal normalisation by up to 10–16% and 6–8%, respectively.Source: §5.3.3 · p. 151

5.4 HH →b ¯bV V ∗: status and perspectives

  • The bbVV channel has the second-largest HH branching fraction and offers hadronic, semileptonic, and dilepton modes, but its prospects are challenging, especially because of top backgrounds.It also covers bbZZ-like final states and can probe scalar resonances decaying to HH or mixed non-resonant HS production when the scalar is lighter than twice the Higgs mass.Source: §5.4 · p. 151

5.4.1 HH →b ¯bW W ∗(ℓνq ¯q)

  • For bbWW→ℓνqq, the observed and expected 95% CL limits are 300 and 300+100/−80 times the SM cross section, respectively.Boosted-analysis expected limits are higher than resolved-analysis limits above 1300 GeV for scalar resonances and 800 GeV for spin-2; resolved searches count events while boosted searches fit the reconstructed mHH shape.Source: §5.4.1 · pp. 152–153

5.4.3 New kinematic observables for HH →b ¯bW W ∗

  • The bbWW dilepton channel can exploit Topness and Higgsness, which encode top-pair and HH kinematics, respectively, to separate dominant top backgrounds from signal.The proposed observables include subsystem MT2 for top production and subsystem ŝmin for HH; Higgsness and Topness occupy different regions in their joint distribution, while M(b)T2 cuts can suppress top and bbττ backgrounds.Source: §5.4.3 · pp. 154–155
  • For HH→bbWW*, the m_min(ℓℓ) distribution ends near mH for signal, while backgrounds extend above it; m_min(bbℓℓ) for ttbar peaks near the 2mt threshold.Both HH and ttbar m̂s distributions peak around 400 GeV; the passage relates this to near-rest top quarks versus relatively boosted Higgs bosons, motivating angular-separation and mass variables.Source: §5.4.3 · pp. 156–157

5.5 HH, other signatures: status and perspectives

  • Channels without b-jets have smaller HH signal yields but are generally less background-contaminated; their sensitivities are statistically limited, and the cited SM branching fractions are 9.85·10^-4, 4.67·10^-2, and 4.00·10^-3.The channels discussed are γγWW*, WWWW*, and τ+τ−τ+τ−, respectively; ATLAS studied the first two and CMS the third.Source: §5.5 · p. 158

5.5.1 HH →γγW W ∗

  • ATLAS set an observed (expected) limit of 230 (160) times the SM cross section for non-resonant HH→γγWW* production, with data yields consistent with SM expectations.In Run 1, the corresponding limit was 1150 (680) times the SM cross section; the Run 2 analysis observed 33 events against 24±5.0 expected background events without the pT cut.Source: §5.5.1 · p. 159, Figure 5.19 · p. 159

5.5.3 HH →τ+τ−τ+τ−

  • The CMS HH→τ+τ−τ+τ− analysis was blinded and expected sensitivity to resonant HH signals with cross sections of order 10 pb.It selected a 2-lepton plus 2-hadronic-tau final state covering 31.2% of the channel signal.Source: §5.5.3 · pp. 160–161

5.5.4 Potential improvements

  • Potential improvements for channels without b-jets include multivariate selections, expanded final states, data-driven background estimates, and HH-mass reconstruction; their sensitivity is expected to improve faster with more LHC data.Earlier analyses of established b-jet channels gained up to a factor of 2–3 in sensitivity at the same luminosity; the section calls for studying new channels for the HL-LHC.Source: §5.5.4 · pp. 161–162
  • The HH→τ+τ−τ+τ− mass-reconstruction algorithm achieves mHH resolutions of 22% (7%) at resonance masses of 300 (500) GeV, improving to 4% (6%) with correct pairing.Incorrect pairing occurs in 13% (2%) of simulated events at those respective masses and degrades the resolution.Source: §5.5.4 · p. 162, Figure 5.21 · p. 162

5.6 HH production in the VBF mode

  • VBF HH probes the otherwise unconstrained HHVV coupling; doubling its SM value can enhance the cross section by up to two orders of magnitude, while high-mHH behavior probes δc.At 300 fb^-1, a phenomenological study estimates about 40% precision on c2V; the channel is less sensitive than ggF to the Higgs self-coupling.Source: §5.6 · p. 163
  • In the CMS bbγγ search, adding VBF HH improved sensitivity by 1.3%, although VBF HH accounts for 5% of the total cross section.The signal model had 13% efficiency times acceptance overall (10% above 350 GeV and 3% below); the search BDT was trained on ggF events, limiting potential gains.Source: §5.6 · p. 163, Figure 5.25 · p. 165
  • VBF-focused categories can improve sensitivity when combined with ggF HH searches, since VBF gains relative to ggF at large mHH [451].Resonant VBF HH searches would complement ggF resonance searches, but remain sparsely studied because of the small cross section; the search tested resonance masses of 260–1000 GeV under narrow- and broad-width hypotheses.Source: §5.6 · pp. 165–166
  • The ATLAS search targets VBF HH in the b¯bb¯b final state using forward jets with |∆η| > 5 and m_jj > 1 TeV.The selection also requires at least four central b-tagged jets and uses multivariate jet energy regression, improving b-jet energy resolution by about 10%.Source: §5.6 · p. 165
  • 1600 fb observed (1000 fb expected) is the 95% CL upper limit on non-resonant VBF HH production in the first search.With cV and κλ fixed to their SM values, the observed (expected) excluded ranges are c2V < -1.00 or c2V > 2.67 (c2V < -1.07 or c2V > 2.78), not reaching the SM value c2V = 1.Source: §5.6 · pp. 165–166

Results presentation

Di-Higgs search results are often published for specific benchmark signals, limiting reinterpretation when new signals differ. The section recommends publishing reusable statistical and analysis information so the results can be tested against a wider range of models.

Results presentation

  • Benchmark-specific CLs limits often cannot be reinterpreted when a new signal differs substantially in width or kinematics, motivating experiments to publish more reusable information.Benchmark signals include resonances and rescaled SM HH production; benchmark results can be directly applied when the signal properties are sufficiently similar.Source: Results presentation · p. 167

6.1 Examples from other BSM searches

  • Reinterpretation methods include cut-and-count yields, Gaussian-shaped resonance limits, and simplified likelihoods, but trade signal-shape information, model coverage, or accuracy.Cut-and-count is less dependent on signal shape but would lose important HH discrimination; Gaussian parameterizations suit resonances, while simplified likelihoods encode background uncertainties through covariance matrices.Source: §6.1 · pp. 168, 170

6.2 Options for the future

  • PYHF represents workspaces in human-readable JSON, while RECAST [527] can run full reinterpretations but remains collaboration-internal and cumbersome for routine external use.RECAST accepts signal LHE files and can produce a CLs value using the archived search simulation and analysis; external interpretations still require the usual ATLAS approval and publication process.Source: §6.2 · pp. 170, 172

6.3 Machine-learning vs interpretability

  • Truth-level and fully simulated inputs agreed within 10% in an ATLAS stop 1-lepton BDT search [530], showing that machine learning need not preclude reinterpretation.Publishing the MVA configuration enables others to rerun the selection; the passage also notes that KERAS models can be saved in JSON or YAML formats.Source: §6.3 · p. 172

6.4 Considerations for non-resonant signatures and EFT

  • Shape benchmarks map EFT parameter space to HH limits, with a rather small difference from direct EFT-shape limits; rare benchmark switches can cause discontinuities near exclusion boundaries.The benchmark method offers stable estimates of excluded EFT regions, but cannot directly cover unique signals absent from the benchmark set.Source: §6.4 · pp. 172, 174, Figure 6.6 · p. 173

6.5 Conclusions and recommendations

  • Publish full editable likelihoods where possible, covariance matrices as a less-preferred fallback, and machine-learning configurations with validation information to extend the impact of searches.The recommendations prioritize PYHF JSON likelihoods; covariance matrices permit likelihood reproduction with lower reinterpretation accuracy. MVA configurations should ideally be accompanied by comparisons of fully simulated and truth-level samples.Source: §6.5 · p. 174

LHC results · Higgs boson potential at future colliders

The section reviews LHC searches for resonant and non-resonant Higgs-pair production and the direct and indirect constraints on the Higgs self-coupling from Higgs-pair and single-Higgs measurements. It projects future LHC constraints, discusses their limitations, and motivates future colliders.

7.1 HH results and combination: status and perspectives

  • The HH combination uses a joint likelihood across signal and background regions, correlating systematic uncertainties and interpolating between tested resonance masses where needed.Current searches probe resonant and non-resonant production, including assumptions that BSM effects alter only the total rate or only the Higgs self-coupling.Source: §7.1 · p. 175

7.2.1 SM-like production

  • ATLAS and CMS set observed (expected) non-resonant HH limits of 6.9 (10) and 22.2 (13) times the SM prediction, respectively.Combining channels improved the expected upper limit by 71% for ATLAS and 68% for CMS versus the best single channel; the leading channels were bbτ+τ− for ATLAS and bbγγ for CMS.Source: Table 7.1 · p. 176, §7.2.1 · p. 176

7.2.2 Higgs self-coupling constraint

  • The κλ-dependent signal acceptance peaks near κλ = 2, where destructive box–triangle interference minimizes the cross section; the combined observed limit is slightly worse than ATLAS’s alone.The latter follows from a 2σ excess in CMS; polynomial prediction parameters used by LHC experiments differ by up to 15% from recent finite-mt NLO calculations.Source: §7.2.2 · pp. 177, 179, 180

7.2.3 More general EFT scans

  • HH production alone cannot constrain κλ without assumptions about κt; CMS instead reports limits across EFT benchmarks with distinct rates and kinematics.The HEFT scan groups coupling choices into 12 kinematic clusters plus the SM and κλ = 0 scenarios; combining single-Higgs and HH measurements is expected to yield a more model-independent κλ determination.Source: §7.2.3 · pp. 181–182

7.3 Resonant HH production mode

  • No resonant HH signal was observed; the most sensitive channels were bbbb, bbτ+τ−, and bbγγ, with bbbb dominating at high resonance masses.Searches covered spin-0 and spin-2 resonances; boosted bbτ+τ− contributed above 1 TeV, while CMS bbWW single-lepton searches showed sensitivity below 1.5 TeV [535]. ATLAS also studied bbWW in a boosted regime [462].Source: §7.3 · pp. 182–183

7.4 Interpretation in complete models

  • ATLAS interpreted spin-0 resonance limits in hMSSM and electroweak-singlet models, but the results apply only under channel-dependent narrow-width conditions.The ΓS/mS thresholds are 2% for bbγγ, 5% for bbτ+τ−, and 10% for bbbb; wider regions are removed or shown separately, with fewer channels combined.Source: §7.4 · p. 186

7.5 Impact of systematic errors and concluding remarks

  • The reported Run 2 HH results, based on 27.5–36.1 fb−1, are limited by dataset size rather than systematic uncertainties.Systematic uncertainties nevertheless have a non-negligible impact in individual analyses.Source: §7.2.1 · p. 177, §7.5 · p. 187
  • Non-resonant searches are limited by background-estimation systematics, while high-mass resonance searches are limited by event counts in the current dataset.Multijet modelling is particularly important in bbbb and bbττ channels; data-driven extrapolation can improve with more data, but its extrapolation uncertainty is intrinsic. Object reconstruction, luminosity, and limited simulation statistics also contribute.Source: §7.5 · p. 188
  • Combined ATLAS and CMS projections reach 3.5 times the SM cross section at Run 2 and 2.4 times at Run 3; the Run 3 limit worsens to 5 times if systematics dominate.With reduced systematic effects, the expected Run 3 interval is −1.2 < κλ < 7.5 at 95% CL.Source: §7.5 · p. 188

7.6 Constraints on κλ from single Higgs boson measurements

  • The single-Higgs fit gives −3.2 < κλ < 11.9 observed and −6.2 < κλ < 14.4 expected, comparable to direct HH constraints from 36.1 fb−1.Sensitivity is driven mainly by γγ, ZZ, WW, ggF, and ttH; current STXS binning in VBF, WH, and ZH adds little. The fit assumes other Higgs couplings take their SM values.Source: §7.6 · pp. 189, 191
  • Single-Higgs sensitivity weakens when κV or κF is floated, with the expected lower 95% CL κλ exclusion limit degraded by 50% when κV and κλ are fitted together.Direct HH sensitivity to κλ vanishes if κt floats; accounting for such coupling deformations requires a simultaneous single- and double-Higgs analysis.Source: §7.6 · p. 191

Higgs boson potential at future colliders

  • Beyond-LHC colliders may be needed to establish the Higgs self-coupling if its value is near the Standard Model prediction; projections cover the HL-LHC, lepton colliders, and 27 and 100 TeV hadron colliders.The HL-LHC projection assumes 14 TeV collisions and about 3 ab−1; the document compares collider capabilities at the end of this part.Source: Higgs boson potential at future… · p. 195

Higgs self-coupling at HL-LHC … J. Tian

These parts examine projected constraints on the Higgs self-coupling at the HL-LHC and e+e− colliders using single- and double-Higgs measurements. They cover complementary sensitivities, factors shaping collider projections, and the prospects and limits of extracting the coupling while controlling other SMEFT effects.

Higgs self-coupling at HL-LHC

  • The HL-LHC targets 14 TeV and 3000–4000 fb−1 over about ten years, with roughly 200 collisions per crossing and detector upgrades for radiation and pileup challenges.Planned changes include extended silicon tracking and timing detectors intended to separate nearby-in-space collisions by their time spread.Source: Higgs self-coupling at HL-LHC · p. 197

8.1 Measurement of the Higgs boson self-coupling at HL-LHC

  • HL-LHC sensitivity projections consider ggF HH at average pileup of 200, reduce theoretical uncertainties by half, and scale experimental uncertainties with luminosity to detector-dependent floors.The analyses assume 1% luminosity uncertainty, neglect limited simulated-sample uncertainties, and assume reconstruction improvements offset higher-pileup degradation.Source: §8.1 · p. 198

8.2 Double Higgs boson production measurements

  • HL-LHC HH projections study b b̄b b̄, b b̄τ+τ− and b b̄γγ, with CMS also considering b b̄WW* and b b̄ZZ* [173, 542, 559].For b b̄b b̄, resolved and boosted strategies address different event topologies; boosted events also probe BSM scenarios with enhanced high-mHH production.Source: §8.2 · pp. 198, 200

8.2.1 Results

  • With statistical uncertainties only, ATLAS (CMS) projects 3.5σ (2.8σ) significance and 95% CL κλ intervals of −0.4–7.3 (−0.18–3.6) [173].The b b̄γγ and b b̄τ+τ− channels provide the best sensitivity, followed by b b̄b b̄; b b̄WW* and b b̄ZZ* add sensitivity in combination.Source: Table 8.2 · p. 201, §8.2.1 · p. 201
  • ATLAS and CMS projections combined reach 4σ significance with systematics (4.5σ without) and constrain κλ to 0.1–2.3 at 95% CL (0.5–1.5 at 68% CL) [173].The combined projections exclude κλ = 0 at 95% CL.Source: §8.2.1 · p. 201, Figure 8.1 · p. 202

8.3 Single Higgs measurements

  • At 3 ab−1, single-Higgs differential measurements project 20–40% cross-section uncertainties and a 95% CL κλ interval of −4.1–14.1, assuming κt = 1 [129].The projection uses tt̄H with H → γγ and is less competitive than direct HH constraints; it uses one production mode and one decay channel.Source: §8.3 · pp. 202–203
  • HH measurements are expected to drive the κλ bound, while differential single-Higgs data can lift the degeneracy near δκλ ∼5 and global fits must account for other Higgs couplings.The passage identifies SMEFT with nine free parameters and HEFT with five as frameworks for such fits.Source: §8.3 · p. 203

9.1 Introduction

  • At e+e− colliders, single-Higgs measurements can benefit from sub-percent precision and multiple observables, while double-Higgs production is available only above 500 GeV.Two collision energies can help lift global-fit degeneracies; direct pair production proceeds through e+e− → ZHH and e+e− → νν̄HH.Source: §9.1 · p. 205
  • e+e− measurements can separate κλ deviations from other new-physics effects, but allowing additional effects can prevent a robust quartic self-coupling bound.A two-parameter cubic-and-quartic analysis omits other possible BSM effects, whose inclusion can dramatically change the results.Source: §9.1 · p. 206

9.2 Scope of the proposed e+e−Higgs factories

  • The four proposed e+e− Higgs factories are CEPC, FCC-ee, ILC, and CLIC, with staged programs spanning Higgs-factory energies through higher-energy runs.The proposals include 240 GeV programs for CEPC and FCC-ee, ILC stages at 250 and 500 GeV, and CLIC stages at 380 GeV, 1.5 TeV, and 3 TeV; proposed data sets differ by stage and facility.Source: §9.2 · pp. 206–207

9.3 Determination of the Higgs self-coupling from single Higgs reactions — CEPC, FCC-ee, CLIC, ILC

  • Single-Higgs fits show substantial κλ sensitivity in a one-parameter model, but SMEFT effects weaken it and some collider scenarios yield no closed constraint.The quoted e+e−-only uncertainties remove the assumed independent 50% HL-LHC contribution; some multi-parameter fits do not close.Source: §9.3 · p. 208, Table 9.1 · p. 208

9.4 HH production processes at e+e−colliders

  • The ZHH and ν¯νHH processes have opposite κλ dependence, so one cross-section increases for either sign of deviation from the SM value.Their small cross-sections make the processes difficult to identify; left-handed electron-beam polarization can raise the ν¯νHH cross-section by almost a factor of 2.Source: §9.4 · pp. 208–209, Figure 9.4 · p. 210

J. Tian

  • At the ILC, 4 ab−1 at 500 GeV gives a 27% κλ uncertainty, while 8 ab−1 at 1 TeV gives 10% precision in one-parameter fits.The 500 GeV result combines HH channels with a 16.8% total-cross-section precision; at 1 TeV, vector-boson fusion dominates HH production.Source: J. Tian · p. 211
  • ILC projections favor 500–600 GeV for ZHH and at least 1 TeV for ν¯νHH; ZHH at 500 GeV could measure a doubled self-coupling with 15% precision.Realistic full-simulation precision is a factor of 4–5 worse than the ideal case, with clustering, flavor tagging, and additional channels identified as improvement opportunities.Source: J. Tian · pp. 211–212

9.6 Measurement of HH production — CLIC

  • CLIC studies find 3.6σ evidence for ν¯νHH and 5.9σ observation for ZHH at 1.4 TeV; at 3 TeV, 5 ab−1 yields 7.3% ν¯νHH cross-section precision.The 3 TeV study also reports ν¯νHH observation with 700 fb−1; CLIC uses cross-section and differential information to extract the self-coupling.Source: §9.6 · p. 212, 350 CLICdp · p. 213

350 CLICdp

  • In CLIC ν¯νHH, destructive interference makes the total cross-section ambiguous because κλ = 2.2 gives the SM cross-section; the HH-mass shape or ZHH measurement resolves it.The κλ constraint is extracted by a template fit to the binned reconstructed HH invariant-mass distribution and can be combined with the ZHH cross-section result.Source: 350 CLICdp · p. 213

J. Tian

  • At 500 GeV, non-self-coupling new physics can swamp κλ effects in e+e−→Z HH, while its small cross section and restricted phase space prevent useful differential separation [126].The cross section depends on 17 SMEFT coefficients.Source: J. Tian · pp. 215–216
  • 2.4% is the total uncertainty from SMEFT coefficients other than c6, giving a 5% systematic error on c6 extraction [126] [582].The contributing uncertainties are 2.8% from cH and cWW and 0.9% from cHL, c′HL, and cHE; the 5% systematic is added in quadrature to the larger statistical error.Source: J. Tian · p. 216
  • Controlling other dimension-6 contributions makes the total e+e−→Z HH cross section a model-independent measurement of c6 within the broad SMEFT class [126].The cWW correction can be determined from single-Higgs data and subtracted without loss of accuracy in determining c6.Source: J. Tian · p. 216
  • For ν¯νHH, a similar SMEFT expression is expected, but coefficients cHL, c′HL, and cHE grow as s/mZ^2; improved precision electroweak measurements can constrain them further [583].The passage identifies deriving the corresponding cross-section expression as a prerequisite for the analysis.Source: J. Tian · p. 216
  • Figure 9.11 compares 1-loop κλ-induced shifts in e+e−→ZH and h→WW* with expected FCC-ee and ILC measurement uncertainties [574].For Q > 200 GeV, the uncertainty bars concern σ(e+e−→ZH); near Q ∼40 GeV, they concern Γ(h→WW*) from full-program SMEFT fits.Source: Figure 9.11 · p. 217

C. Grojean · Higgs self-coupling at future hadron colliders · Higgs self-coupling at future colliders: Summary

The section examines projected indirect and direct measurements of the Higgs self-coupling at future colliders, including constraints from di-Higgs channels and related interactions. It discusses projected precision under varying assumptions and analysis maturity, noting that direct comparisons require caution.

C. Grojean

  • Single-Higgs measurements at FCC-ee and ILC can determine c6 with 40–60% uncertainty independently of HH results, but the extraction requires multi-energy data and is sensitive to other SMEFT parameters.The c6 contribution is loop-level and has a coefficient of order 1%, unlike the order-1 coefficients of most other parameters; a full fit involves 17 SMEFT parameters.Source: C. Grojean · pp. 217–219

9.9 The quartic Higgs self-coupling

  • Future e+e− colliders can set first coarse bounds on c8 and λ4, although the SM triple-Higgs rate is unmeasurable; combining HH and HHH, especially W-fusion HHH, improves constraints.The study finds that CLIC at 3 TeV via W-fusion HHH can achieve constraints similar to those at a future 100 TeV hadron collider.Source: §9.9 · p. 221

9.10 Conclusions

  • FCC-ee and ILC single-Higgs measurements project 44% and 58% precision on the self-coupling, respectively, while ILC at 500 GeV projects 27% from Z HH production.Combining the ILC single-Higgs and Z HH determinations would reach 24%; ILC at 1 TeV or CLIC at 1.5 and 3 TeV projects 10% from HH production. These determinations are described as robust within SMEFT, although the 10% projection needs further study of model independence.Source: §9.10 · pp. 222–223
  • The FCC-ee and ILC can provide robust, SMEFT-based self-coupling determinations from single-Higgs data at two centre-of-mass energies; the expected precision improves with increased FCC-ee running or detectors.FCC-ee precision is projected to improve from 44% to 27% with four interaction points or doubled running time; these indirect measurements are statistics-limited and described as essentially free of model-dependent assumptions within SMEFT.Source: C. Grojean · pp. 218–219, §9.10 · p. 222

10.1 The Higgs-self coupling at ps = 27 TeV

  • Without systematic uncertainties, the 68% CL precision on δκλ is approximately 40% in b ¯bγγ and 20% in b ¯bτ+τ−; Fig. 10.2 gives 10–20% under these assumptions.A global fit including all possible SM deviations gives at best 200% precision on the self-coupling; varying only its modifier gives 50% in the best case using differential single-Higgs information. Double-Higgs production gives 10–20% precision, depending on the assumed systematic-uncertainty scenario.Source: §10.1 · pp. 227–228

10.2 The Higgs self-coupling at ps = 100 TeV

  • The 100 TeV study assumes 30 ab−1 and a reference FCC-hh detector, using DELPHES fast simulation and signal samples with full top-mass dependence and an NNLL+NNLO-matched K-factor.These are the supplied setup details for the 100 TeV collider projections.Source: §10.2 · p. 228

10.2.1 HH →b ¯bγγ

  • At 100 TeV, the HH → b ¯bγγ analysis projects δκλ = 5.5% with statistical uncertainty alone and 6% with a 1% signal-normalisation uncertainty.The study varies systematic effects, background normalisations and detector assumptions; the supplied result passage reports these two uncertainty scenarios.Source: §10.2.1 · p. 229
  • δκλ = 5% is projected for HH →b ¯bγγ at the FCC-hh, while background-yield variations by factors of 2 or 0.5 shift the precision by ≈±1%.The study also finds an expected precision of 6.5% under a 1% single-Higgs background normalisation uncertainty; less optimistic detector assumptions degrade precision by 1–2%.Source: §10.2.1 · p. 230

10.2.2 HH →b ¯bZ Z (4ℓ)

  • At 68% CL, the HH →b ¯bZ Z(4ℓ) channel projects 14% precision with statistical uncertainties only, 15% with 1% systematics, and 24% with 3% systematics.The channel has σb ¯b4ℓ = 178 ab at the FCC-hh and is inaccessible at the HL-LHC; a higher lepton pT threshold changes statistical-only precision from 14% to 15%.Source: §10.2.2 · pp. 230, 232, Figure 10.7 · p. 232

10.2.3 HH →b ¯bb ¯b +jet

  • δκλ = 30% is projected in the boosted HH →b ¯bb ¯b +jet analysis, with QCD background-yield variations by factors of 2 or 0.5 shifting precision by ≈±10%.The strategy uses a recoiling Higgs pair at small mHH, where sensitivity to the trilinear coupling is greater; extending to semi-resolved and resolved regions could improve the measurement.Source: §10.2.3 · pp. 232–233

10.2.4 HH →b ¯bW W ∗

  • At L = 30 ab−1, HH →b ¯bW W ∗ is projected to reach 5σ significance, corresponding to δκλ = 40%.The study considers one hadronic and one leptonic W decay and uses a BDT trained to discriminate signal from the dominant t ¯t background.Source: §10.2.4 · p. 234

10.2.5 Summary of 100 TeV studies

  • δκλ = 5% appears achievable at 100 TeV across several techniques and decay modes, including b ¯bγγ [585] [587] [592] and b ¯bτ+τ− [590].The projection assumes L = 30 ab−1 and future theoretical progress in signal and background modelling.Source: §10.2.5 · pp. 234–235

10.3 Other Probes of Multi-Linear Higgs Interactions

  • In the conservative scenario, bounds from double-Higgs production are stronger than those from the conservative triple-Higgs analysis for the Higgs quartic interaction; the two production modes provide complementary constraints.The indirect method extends precision single-Higgs probes of λ3 to λ4 and di-Higgs production. Direct triple-Higgs bounds strongly depend on b-tagging efficiency, with conservative and optimistic scenarios of 60% and 80%, respectively.Source: §10.3 · p. 236
  • FCC-hh di-Higgs production via vector boson fusion can constrain δc2V to better than ±1%, improving on the 40% precision projected at the HL-LHC.The mHH distribution probes the Higgs-sector gauge structure; the projection relies on cV measured to a few per mille at FCC-ee and a suppressed cubic-coupling contribution at multi-TeV masses.Source: §10.3 · p. 237, Figure 10.13 · p. 237
  • The t ¯tHH cross section grows by a factor of ∼75 from 14 TeV to 100 TeV and increases for |ctt| > 0 or |κλ| > 0.The t ¯tHH vertex is accessible through dimension-6 operators; nonlinear EFT allows its coupling to be probed independently of t ¯tH, unlike the correlated linear-realisation couplings.Source: §10.3 · p. 237, Figure 10.15 · p. 238

Higgs self-coupling at future colliders: Summary

  • Run 3 is not expected to observe SM HH production; its best projected cross-section limit is 1–3 times the SM value at 95% CL, leaving self-coupling measurement out of reach.The projection assumes around 600 fb^-1 combined ATLAS and CMS data and current detector performance.Source: Higgs self-coupling at future… · p. 239
  • At HL-LHC, 3 ab^-1 per interaction point enables a 5σ observation of HH production, but self-coupling precision remains about 50%, driven mainly by HH measurements.The observation corresponds to establishing at 95% CL that the self-coupling is nonzero.Source: Higgs self-coupling at future… · p. 239
  • Single-Higgs measurements at e+e− factories can constrain the self-coupling through global fits, but silver-level precision requires a second energy stage; FCC-ee with four interaction points is projected to reach 24%.The stated second-stage energies are 365 GeV for FCC-ee, 500 GeV for ILC, and 1.5 TeV for CLIC; operation above 240 GeV is not in the CEPC baseline plan.Source: Higgs self-coupling at future… · pp. 239–240
  • Projections based on HH production reach 5% precision at FCC-hh and 15% at HE-LHC, and the highest-energy lepton and hadron colliders can attain gold-level precision.These estimates rely on measuring the HH production cross section.Source: Higgs self-coupling at future… · p. 240
  • Future-collider estimates are not directly comparable because they use differing SMEFT assumptions, and single-Higgs projections below HL-LHC combine with its projected 50% error.Direct e+e− estimates use full-simulation analyses based on current methods, while hadron-collider projections assume detector performance under very high pileup.Source: Higgs self-coupling at future… · p. 240

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