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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 potential and self-coupling remain unmeasured, motivating a comprehensive assessment of di-Higgs production and related constraints. This work synthesizes theoretical predictions, experimental searches, collider projections, and effective-field-theory developments, finding that higher-order QCD corrections substantially affect predicted production rates.

  • Problem

    The Higgs potential and self-coupling have not yet been measured, leaving the form of the Higgs field’s energy function experimentally unconstrained.

  • Method

    The paper synthesizes theoretical calculations, effective-field-theory developments, experimental searches, self-coupling constraints, and future-collider projections.

  • Results

    Higher-order QCD corrections increase the total di-Higgs cross section by about a factor of two at leading production order, while NNLO and N3LO corrections reach percent and sub-percent levels for vector-boson fusion.

  • Takeaways & Limitations

    Di-Higgs production provides a central route to studying the Higgs potential and constraining its self-coupling across present and future colliders.

  • Takeaways & Limitations

    For non-SM κλ values, a combined calculation incorporating full top-quark-mass dependence at NLO and NNLO accuracy has not yet been performed.

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.

Authors

The paper is edited by B. Di Micco, M. Gouzevitch, J. Mazzitelli, and C. Vernieri, with contributions from a broad group of listed authors.

  • Authors: B. Di Micco, M. Gouzevitch, J. Mazzitelli, and C. Vernieri are identified as editors among the paper’s contributors.The passage then lists additional contributors, including J. Alison, K. Androsov, J. Baglio, E. Bagnaschi, and many others.

Theoretical status

The theoretical-status section reviews developments needed to extract information from Higgs self-coupling and HH-production measurements, covering SM pair-production predictions and effective-field-theory interpretations of non-resonant deviations.

  • Theoretical status: The theory overview supports extracting the maximum possible information from experimental measurements of the Higgs boson self-coupling and HH production in the Standard Model and beyond.The scope includes both self-coupling measurement and HH-production studies.
  • Theoretical status: It presents updated SM Higgs-pair production cross-section predictions across production modes, emphasizing gluon fusion at the LHC and including fixed-order calculations and Monte Carlo generators.The section identifies gluon fusion as the main production mode at the LHC.
  • Theoretical status: It develops the effective field theory approach to interpret non-resonant deviations from Standard Model expectations.These theory developments are described as crucial for interpreting such deviations.

HH cross section predictions

This section reviews state-of-the-art theoretical predictions for Standard Model Higgs-boson pair production at hadron colliders. It covers production modes, QCD corrections to gluon fusion, Higgs self-coupling dependence, and available Monte Carlo generators.

  • HH cross section predictions: Gluon fusion, g g →HH, is the dominant hadron-collider production mechanism and is mainly mediated by top-quark loops.This mechanism is loop-induced and parallels single-Higgs production.
  • HH cross section predictions: The chapter surveys theoretical predictions for Standard Model Higgs-boson pair production at hadron colliders across all production modes.It frames the discussion as a state-of-the-art summary.
  • HH cross section predictions: Theoretical coverage includes QCD corrections to gluon fusion, dependence on the Higgs self-coupling, and available Monte Carlo generators.The self-coupling can be accessed directly through Higgs pair production, unlike the quartic coupling at the LHC.

1.1 Overview of production modes

Higgs-pair production is dominated by gluon fusion, followed by vector-boson fusion, while double Higgs-strahlung and top-associated channels provide additional mechanisms. QCD corrections substantially enhance gluon-fusion and double-strahlung rates, but are more moderate or process-dependent elsewhere.

  • Gluon fusion: Gluon fusion is the dominant Higgs-pair production mode, mediated mainly by top-quark loops with destructive box–triangle interference involving the trilinear Higgs coupling.Bottom-quark loops provide a smaller contribution.
  • Gluon fusion: QCD corrections increase the gluon-fusion total cross section by about a factor of two relative to the leading-order prediction.Third-order corrections have also been computed in the heavy-top-quark limit.
  • Vector-boson fusion: Vector-boson fusion is the second-largest production mechanism, dominated by t-channel W and Z exchange and involving both continuum and off-shell Higgs-splitting diagrams.Its QCD corrections are known in the structure-function approach.
  • Double Higgs-strahlung: Double Higgs-strahlung has a significantly lower production rate than vector-boson fusion, while its QCD corrections increase total cross sections by about 30%.The main correction component can be translated from the corresponding Drell–Yan calculation.
  • Double Higgs bremsstrahlung off top quarks: At a 100 TeV hadron collider, top-pair-associated Higgs-pair production reaches a cross section close to vector-boson fusion, with NLO corrections reducing the total by 20%.For single-top-associated production, t j HH, NLO corrections are similar in size but positive.

1.2 QCD corrections for gluon fusion

QCD predictions for gluon-fusion Higgs-pair production progress from heavy-top-limit approximations to full-mass NLO calculations and NNLO-improved Monte Carlo predictions. Finite-top-mass treatments and independent numerical methods achieve controlled agreement and quantify the accuracy of successive approximations.

  • Approximation schemes: The Born-improved NLO prediction approximates the full NLO total cross section within about 15% while remaining reliable mainly at lower Higgs-pair invariant masses.It retains the full mass dependence at leading order but treats virtual and real corrections in the heavy top quark limit.
  • Higher-order corrections: 20−30% is the increase in the total cross section from NNLO QCD corrections when the full leading-order mass dependence is included.These NNLO corrections build on additional virtual and real–virtual contributions beyond the NLO treatment.
  • Approximation schemes: Including full one-loop real-emission mass effects reduces the Born-improved prediction by about 10%, defining the FTapprox treatment.The correction includes the real channels gg → HHg, gq → HHq, and q¯q → HHg.
  • Full-mass NLO calculation: The two independent full-NLO numerical methods agree within their integration errors, with residual differences attributable to the chosen top-quark masses.The analyses use mt = 173 GeV and mt = 172.5 GeV, respectively, and employ numerical integration of the two-loop contributions.
  • NNLO predictions: The FTapprox method has better than 10% accuracy at NLO, supporting NNLOFTapprox predictions that are expected to be more reliable than the remaining uncertainties.A full NNLO Monte Carlo program combines full NLO corrections with NNLO QCD corrections in the heavy top quark limit and reweights partial finite-top-mass effects.

1.3 Cross section as a function of κλ

The section presents full-NLO QCD predictions for Higgs-pair production across trilinear couplings, including total and differential results with full top-quark-mass dependence. It shows substantial κλ-dependent rate and shape variations and provides a publicly available POWHEG-BOX-V2 implementation, while noting that non-SM NNLO matching remains incomplete.

  • Cross sections: κλ = −1 produces the largest total cross section among the considered trilinear-coupling values.The K-factor varies substantially with κλ, unlike in the mt →∞ limit.
  • Cross sections: 1.57–2.16 is the observed range of the K-factor as the trilinear coupling varies.For the 14 TeV results, the κλ = 2 K-factor is 1.57, and these K-factors also apply at 13 TeV except for that case.
  • Differential distributions: κλ ≈ 2.4 generates a characteristic dip in the Higgs-pair invariant-mass distribution from maximal destructive interference between triangle and box contributions.The dip is also visible in the three-dimensional heat map of the mHH distribution.
  • NLO predictions: Full-NLO QCD predictions with full top-quark-mass dependence cover total cross sections at 13, 14, and 27 TeV and differential results at 14 TeV.The calculation includes scale uncertainties and is implemented in the POWHEG-BOX-V2 Monte Carlo framework.
  • Higher-order accuracy: NNLO matching of the full-top-mass NLO result with the improved-HTL calculation has not yet been completed for non-SM κλ values.Such a combination is described as desirable, with work in progress.

1.4 Differential predictions and MC generators for gluon fusion

Non-resonant di-Higgs production is modeled with several public Monte Carlo programs, but parton-shower and matching uncertainties can be large, especially for key differential observables. These effects arise from sub-leading matching contributions and depend strongly on shower settings and implementation.

  • Monte Carlo predictions: Public Monte Carlo predictions for gluon-fusion di-Higgs production include NLO matrix elements with finite top-quark mass and parton showers.The fixed-order calculation was also extended to allow a running top-quark mass, with implementations in POWHEG-BOX and MG5_aMC@NLO.
  • Differential uncertainties: Most NLO-accurate distributions show matching-scheme uncertainties within scale uncertainties, but pT_HH, ΔΦ_HH, and ΔR_HH are strongly affected by parton showers.These distributions are sensitive because their tails are predicted only at the first non-trivial fixed-order level, and matching uncertainties can exceed scale uncertainties.
  • Differential uncertainties: POWHEG-BOX can overshoot the fixed-order pT_HH prediction by about a factor of 2 at large pT_HH, while Dire can also overshoot for its largest shower starting scale.The POWHEG-BOX behavior is connected to shower-generated sub-leading emissions and depends on parameters such as the shower starting scale and hdamp.
  • Origin of uncertainties: The large matching uncertainty is attributed partly to sub-leading terms governed by the large K-factor, large splitting kernel, and shower starting scale.For sufficiently large shower starting scales, shower emissions can affect regions where the fixed-order result should otherwise be reproduced.
  • Parton-shower dependence: In POWHEG-BOX, PYTHIA 8.2 produces a harder sub-leading jet and a harder pT_HH spectrum than PYTHIA 6, whose prediction tends toward the fixed-order result at large pT_HH.The differing recoil against the di-Higgs system explains the shower dependence observed in the tail.

Effective Field Theory

The supplied passage identifies the editors as F. Goertz, D. Pagani, and G. Panico, but provides no substantive content on effective field theory.

  • Effective Field Theory: The passage lists F. Goertz, D. Pagani, and G. Panico as editors.

2.1 Introduction to the EFT formalism

The section presents EFT as a systematic framework for studying how new physics can alter Higgs-pair production beyond the SM. It distinguishes HEFT and SMEFT descriptions and explains how high-scale interactions are encoded through gauge-invariant operators and power counting.

  • 2.1 Introduction to the EFT formalism: Higgs-pair production probes the Higgs potential and the origin of electroweak symmetry breaking, questions the SM itself does not explain.The study aims to understand why the Higgs field acquires a vacuum expectation value, fills the universe, and gives mass to elementary particles.
  • 2.1 Introduction to the EFT formalism: EFT systematically approximates short-distance interactions through a Lagrangian with an enumerable set of parameters, enabling precise calculations even with non-renormalisable terms.It provides an effective description of underlying models whose effects appear at accessible energies.
  • 2.1 Introduction to the EFT formalism: A gauge-invariant treatment of modified triple-Higgs interactions requires additional Lagrangian terms beyond simply rescaling the SM vertex.These terms introduce multi-Higgs vertices, contribute to higher-order electroweak corrections, and cancel potentially troublesome ultraviolet divergences.
  • 2.1.1 Two EFT extensions of the SM: HEFT and SMEFT enumerate the same operators using different power-counting schemes, so an operator’s assigned order can differ between the two formalisms.The SMEFT is described as more restrictive than HEFT.
  • 2.1.2 SMEFT: SMEFT assumes new-physics fields act at short distances or high scales, with a hierarchy between the SM scale mZ and the new-interaction scale M.The absence of discovered new-physics particles at the LHC motivates this hierarchy.
  • 2.1.2 SMEFT: Below M, integrating out high-energy fields yields a theory containing SM fields and potentially higher-dimension operators whose coefficients are fixed by dimensional analysis.Renormalisable terms have dimension 4 or less, while operators with dimension d > 4 carry coefficients proportional to (mass)4−d.
  • How large are the SMEFT parameters?: The largest SMEFT effects arise when a new boson produces an s-channel resonance that decays to HH, altering the di-Higgs mass spectrum.Such resonant models are identified as producing the largest effects in HH production.

g g →HH in the SMEFT … Perturbativity

The paper presents reduced SMEFT descriptions for extracting the Higgs self-coupling from double- and single-Higgs production, while emphasizing operator correlations and limitations. It also derives vacuum-stability and perturbativity constraints, including bounds on self-couplings and the energy scale where new physics should appear.

  • g g →HH in the SMEFT: The operators cH, cu, ctG, cΦG, and c6 affect double-Higgs production, while c6 changes only Higgs pair production and cH universally rescales single-Higgs cross sections.Additional operators become relevant for Higgs decays and QCD or electroweak corrections.
  • g g →HH in the SMEFT: In g g →HH SMEFT analyses, the total cross section depends strongly on all relevant operator coefficients, complicating extraction of the triple Higgs coupling.Analyses commonly restrict the operator set, although the full SMEFT contains 59 or more dimension-6 operators.
  • g g →HH in the SMEFT: Other LHC measurements can constrain several nuisance operators, and double-Higgs invariant-mass shapes can help disentangle their contributions in a global fit.Top, t t̄H, single-Higgs, and gluon-fusion measurements constrain ctG, cu, cΦG, and cH, respectively.
  • Single Higgs production in the SMEFT: Single-Higgs self-coupling extraction requires a more complex operator treatment because even small leading-order operator effects can overshadow a modified Higgs trilinear coupling.A proposed high-luminosity-LHC setup includes 9 effective operators in addition to the trilinear deformation, with simplifying omissions of dipole, fermion-gauge, and four-fermion operators.
  • Vacuum stability: Vacuum-stability constraints do not generally map directly onto trilinear Higgs self-coupling modifications because scalar-potential couplings can decorrelate the two effects.In low-scale inverse-seesaw models, trilinear deviations up to 30% can drive the Higgs potential into an unstable regime.
  • Perturbativity: Partial-wave unitarity yields |κλ| ≲6.5 and |λH4/λSM H4 | ≲65, with distinct kinematic dependence allowing separate trilinear and quartic bounds.The trilinear bound is obtained at small center-of-mass energy, whereas quartic effects become important only at large energy.
  • Perturbativity: For |κλ| ≳6, perturbative predictions for total double-Higgs cross sections become meaningless because the strongest bound occurs near the double-Higgs threshold.The perturbativity bounds concern configurations with two Higgs bosons on-shell and need not directly apply to other kinematics.
  • Perturbativity: A measured trilinear-coupling deviation δλ = κλ−1 implies that perturbation theory breaks down at an Emax scale, providing a target energy for future collider searches for new physics.The breakdown scale is reported to be independent of the specific shape of the Higgs potential.

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

The section examines UV-complete and EFT descriptions of Higgs pair production, emphasizing theoretical constraints, operator correlations, and kinematic effects. It also surveys new-resonance signatures and direct or indirect constraints on the Higgs self-coupling.

  • UV-complete models: In the singlet model scan, Higgs coupling and W-boson-mass measurements exclude distinct parameter regions, while vacuum instability does not constrain trilinear-coupling modifications.The maximal allowed deviations are determined within the viable parameter space rather than by vacuum instability.
  • UV-complete models: 15% deviation of the SM Higgs triple coupling arises in the MSSM for tanβ = 5 and low MA values, with MA ∼200 GeV.The estimate requires consistent approximations for the Higgs mass and triple-Higgs coupling, including higher-order corrections.
  • 2.3.1 EFT fit for HH production: Dimension-six operators modify Higgs pair production through Yukawa, gluonic, and contact-interaction effects beyond the SM-like triangle and box diagrams.The negative triangle–box interference decreases the cross section for positive c6, while Yukawa-like and gluonic operators have particularly pronounced effects.
  • 2.3.1 EFT fit for HH production: 60% level projected determination of c6 follows from 10% single-Higgs coupling information, compared with an unbounded constraint using present uncertainties and 40% with only c6 varied.These projections assume the SM and an HL-LHC integrated luminosity of 3000 fb−1.
  • 2.3.2 Impact of single Higgs production: Single-Higgs production constrains the trilinear coupling indirectly through electroweak loop corrections, with channel-dependent linear and universal quadratic κλ contributions.Vector-boson-associated and top-associated production have larger linear dependence than several other processes, while gluon fusion and H→gg,γγ enter at two-loop electroweak order.
  • 2.3.2 Impact of single Higgs production: Inclusive single-Higgs fits can leave λH3 unconstrained when additional EFT deviations are included, whereas differential information can improve combined constraints with double-Higgs production.Indirect bounds are competitive under exclusive κλ variations but weaken when relevant EFT operators are fitted simultaneously.
  • 2.4 EFT shape benchmarks: Shape benchmarks expose local-minimum regions and typical distribution distortions, with observed CMS di-Higgs limits varying up to two orders of magnitude across benchmarks.NLO corrections preserve main qualitative features but can produce strongly varying K-factors across mHH and with anomalous couplings.
  • New Physics in Higgs pair production · Constraints: Light new degrees of freedom, especially resonances below EFT validity, can strongly enhance Higgs pair production through decays to Higgs pairs.The singlet extension provides the simplest example, while interference with box and SM-Higgs-triangle backgrounds is emphasized; allowed rates are compared with CMS limits.

Non-Z2 … Status of the measurements at LHC

The section surveys non-Z2 singlet phenomenology, vector-like-quark and colored-scalar effects, MSSM modifications, and the experimental program probing the Higgs self-coupling through single- and double-Higgs production. It also organizes future precision goals by the classes of new-physics effects they can test.

  • Non-Z2: In the non-Z2 singlet model, the physical inputs include v = 246 GeV, vS = 0, m1 = 125 GeV, m2 > 2m1, and the mixing angle θ.The remaining free parameters are a2, b3, and b4.
  • Singlet VLQ: Singlet VLQ effects can substantially modify double-Higgs production while leaving single-Higgs production nearly SM-like, but electroweak precision data restrict the double-Higgs rate to within ∼15% of the SM value.The deviations are suppressions, and the kinematic distributions remain nearly SM-like.
  • Full VLQ Generation: In full VLQ generations, constrained SM and VLQ invariant-mass distributions are very similar, while sizable double-Higgs deviations require light new particles in a region where the EFT is not valid.Single- and double-Higgs rates are bound together in the constrained parameter space.
  • Colored scalars: A color-triplet scalar reproducing the SM single-Higgs rate requires κ ≳2, while Higgs-generated heavy scalars can change the single-Higgs rate by 54% and are incompatible with current gluon-fusion limits.Heavy scalars decouple quickly in the single-Higgs rate but may affect the high-mhh tail of double-Higgs production.
  • Scalar Top Partners: For scalar top partners with m0 = m_s^2/2, the single-Higgs rate is in roughly 2σ tension with current measurements, while double-Higgs deviations begin around 2m_s.The benchmark is constrained by the relation between single- and double-Higgs rates.
  • MSSM P. Huang: In the MSSM, Higgs decay branching-ratio modifications are generally small, with the largest change about ±20% in the b b̄γγ channel; light stops can also alter the double-Higgs invariant-mass distribution.The double-Higgs final-state rate depends on both production and decay branching ratios.
  • Status of the measurements at LHC: Precision targets range from 100% sensitivity to large tree-level or resonant effects to 1% sensitivity to typical loop corrections to the Higgs self-coupling.Intermediate goals are 25–50% for heavy-scalar mixing and 5–10% for loop effects from light top squarks or other Higgs-coupled particles.
  • Status of the measurements at LHC: The LHC probes the Higgs self-coupling directly through Higgs-pair production and indirectly through radiative corrections to single-Higgs measurements, using complementary ATLAS and CMS searches.The experimental overview also includes direct searches for resonant states decaying to HH pairs and an initial combination of ATLAS and CMS results.

Detector objects, triggers and analysis techniques

ATLAS and CMS use complementary detector technologies and advanced object-reconstruction methods for di-Higgs analyses. Key developments include boosted-jet grooming and tagging, b-jet energy corrections, and improved τ identification with quantified performance gains and uncertainties.

  • Jet reconstruction: Grooming filters soft, wide-angle radiation and improves the reconstructed mass resolution of two-prong Higgs jets.The method targets QCD two-prong decays by reducing radiation that distorts the original invariant mass.
  • Jet reconstruction: Track-calorimeter-cluster jets combine inner-detector angular reconstruction with calorimeter energy measurements to significantly improve high-pT resolution.ATLAS validated this large-radius jet construction using combined tracking and calorimeter information.
  • b-tagging: At 70% b-tagging efficiency, the inclusive mis-tag rate falls from 4% to 1.2%-1.5%, more than a factor 2 below the BDT-based double-b tagger.The CMS approach exploits charged-particle-flow and secondary-vertex observables for boosted H →b ¯b candidates.
  • b-jet energy calibration: Corrected b-jet energies improve simulated Higgs mass resolution by about 15-25% and analysis sensitivity by 10–20%.The gains depend on the reconstructed Higgs pT and analysis strategy; regression power comes largely from neutrinos in semileptonic B-hadron decays.
  • b-jet energy calibration: CMS’s DNN regression improves b-jet resolution by about 13%, with roughly 20% improvement observed for di-jet invariant-mass resolution.Its loss function simultaneously trains an energy correction and a per-jet resolution estimator.
  • τ identification: τ identification uses BDTs and, since 2018, an ATLAS RNN, with efficiency uncertainties of approximately 5-6% and energy-scale precision of 1.2–2% for ATLAS and less than 1.2% for CMS.The algorithms combine tracker, calorimeter, and muon-detector information to suppress electron, muon, and jet misidentification.

J. Alison · Overview of HH searches at the LHC

The overview emphasizes that HH searches depend on efficient trigger-level identification of b-jets and τ leptons, while specialized reconstruction techniques improve mass resolution, sensitivity, and vertex assignment. ATLAS and CMS exploit diverse HH signatures, with trigger performance varying across channels and kinematic regimes.

  • J. Alison: Efficient trigger-level identification of b-jets and τ leptons is critical for the sensitive HH →b ¯bb ¯b and HH →b ¯bτ+τ− final states.The online identification challenges are addressed through dedicated trigger strategies.
  • 4.6.1 b-jet trigger: An increase of the jet threshold from 60 to 100 GeV reduces HH →b ¯bb ¯b sensitivity by a factor of two, especially in the low-mHH region sensitive to λ.This makes highly efficient trigger-level b-jet identification crucial despite the overwhelming QCD multi-jet rate.
  • 4.6.1 b-jet trigger: Because L1 lacks inner-detector tracks, b-jets are collected with inclusive high-rate seeds and identified at HLT when tracking information becomes available.At background rejection 100, online-minus-offline signal-efficiency differences are ∼2% for ATLAS and ∼6% for CMS.
  • 4.6.2 τ trigger: CMS developed a Run 2 L1 τ trigger using dynamic clustering, while ATLAS combines calorimeter-based L1 identification with HLT calorimeter clusters and BDT-based identification.The online τ working points target approximately 0.95 (0.70) efficiency after offline identification for one (three) associated tracks.
  • 4.7.1 Kinematic fit procedure: 20 to 40% resolution improvement in the CMS resonant HH →b ¯bb ¯b search yields 10–20% sensitivity improvement from the kinematic fit.The fit imposes Higgs-mass constraints and similar improvements are observed in HH →b ¯bτ+τ− and HH →b ¯bγγ searches.
  • 4.7.1 Kinematic fit procedure: About a factor of two improvement in mHH resolution is achieved for HH →b ¯bτ+τ− using a fit under the hypothesis of two 125 GeV Higgs bosons.The fit uses the four-momenta of τ and b-jets together with the pmiss_T vector and assumes collinear τ decay products.
  • 4.7.2 HH vertex reconstruction with H →γγ decay: Up to 99.9% correct primary-vertex identification is achieved in HH →b ¯bγγ because H →b ¯b hadronization provides additional information beyond the H →γγ algorithm.For comparison, the ggF H →γγ production vertex is correctly identified with an efficiency of about 80%.
  • Overview of HH searches at the LHC: ATLAS and CMS exploit a rich variety of Higgs decay signatures to search for HH pair production, with ATLAS trigger efficiency ranging from 65% at the 2mH threshold to ⪆99% above 600 GeV.The cited overview also reports CMS trigger efficiency ranging from 60% above the 2mH threshold to 60% above 800 GeV and 34% for the non-resonant hypothesis.

Background modelling

Background modelling relies on data-driven constructions and parametric fits, but is limited by statistical constraints, extrapolation assumptions, and event-level correlations. These limitations can materially affect sensitivity, while alternative derivations and improved reconstruction offer ways to control uncertainties.

  • Data-driven background models: Hemisphere mixing builds artificial events by combining similar hemispheres from different events after dividing events along the transverse-thrust axis.The method is designed to model background while removing event-level correlations associated with the HH signal process.
  • Systematic uncertainties: Background-model shape and normalisation uncertainties affect the final result by about 9% and 30%, respectively.Minor t̄tH, ZH, and b̄bH contamination has negligible impact because it does not produce a signal-like BDT distribution.
  • Model validation: Statistically limited control regions make background-model assumptions difficult to validate, especially when extrapolation changes distribution shapes beyond normalisation.An ATLAS procedure derived the model twice using orthogonal kinematic selections and used the signal-region variation to estimate systematic uncertainties.
  • Model extrapolation: The ATLAS background model can misrepresent correlations because its correction factors iteratively reweight multiple one-dimensional distributions across b-jet multiplicity.This avoids the statistical limitations of high-dimensional histograms but may not correctly account for correlations between reweighted variables.
  • Statistical limitations: 33% reduction in sensitivity to SM HH production occurs when the background and data have the same statistical uncertainty.Hemisphere mixing also cannot reuse each source event, limiting the background model’s statistical precision to that of the true background.
  • Parametric background fits: CMS’s 2D fit improves search sensitivity by ≈10% by exploiting the full m_jj distribution, whereas ATLAS avoids dependence on modelling m_γγ–m_jj correlations.The two approaches therefore trade correlation-modelling independence against stronger constraints on the non-resonant background.

Results presentation

Di-Higgs search results are difficult to reinterpret because profile-likelihood fits depend on signal-shape assumptions, motivating publication of additional statistical information and reusable analysis implementations. These practices can extend the impact of searches beyond their initial publications and enable timely testing of new models.

  • Challenges: Profile-likelihood fits make di-Higgs searches difficult to reinterpret because they use full m_HH or MVA-score distributions and require assumptions about signal shapes.Without the signal-distribution shape, an upper cross-section limit cannot readily be provided.
  • Alternative presentations: “Cut and count” methods reduce dependence on signal shape but are disfavoured for di-Higgs results because distribution shapes provide important background-discrimination information.Parameterized efficiencies and analysis selections can yield predicted signal counts for comparison with model-independent cross-section limits.
  • Reproducible likelihoods: Publishing statistical objects and analysis code, including human-readable JSON workspaces through PYHF, can improve the re-interpretability of published searches.PYHF describes the underlying HISTFACTORY probability distributions without requiring ROOT.
  • Reproducible likelihoods: RECAST can archive a search’s selection and statistical analysis so supplied LHE signal files are automatically simulated and converted into a final CLs value.This provides a container-based implementation of the full analysis and statistical interpretation for a particular BSM signal.
  • Validation: Truth-level and fully simulated inputs agreed within 10% in an ATLAS SUSY stop one-lepton search when processed through the same Boosted Decision Tree.This demonstrates that, in at least some cases, truth-level reinterpretation can achieve accuracy comparable to other reinterpretation approaches.
  • Outlook: Reinterpretation is expected to become critical as publication timescales increase, while collaboration practices can extend search impact beyond initial publications and improve timely model testing.The reviewed options differ in workload and accuracy, with likelihood reproduction described as less preferred than the first option because it can require more analysis-team work and yield lower reinterpretation accuracy.

LHC results … J. Tian

The paper reviews LHC di-Higgs searches and self-coupling constraints, then projects HL-LHC and future e+e− collider sensitivity, emphasizing complementary channels and the impact of additional couplings and EFT effects.

  • LHC results: 6.9 (10) and 22.2 (13) times the SM are the observed (expected) ATLAS and CMS upper limits on non-resonant HH production from 2015–2016 data.The combinations use multiple final states and correlate theoretical and experimental systematic uncertainties.
  • LHC results: 30% is the improvement in each experiment’s combined sensitivity relative to its best single channel, with b ¯bτ+τ− best for ATLAS and b ¯bγγ best for CMS.The comparable sensitivities of b ¯bγγ, b ¯bτ+τ−, and b ¯bb ¯b drive the gain, while systematic uncertainties remain non-negligible.
  • LHC results: No evidence for resonant HH production is observed, and 95% CL upper limits are set for spin-0 resonances across the analysed decay channels.At higher resonance masses, b ¯bb ¯b dominates sensitivity, while boosted b ¯bτ+τ− contributes above 1 TeV.
  • Higgs boson potential at future colliders: 3.5 times the SM prediction at the end of Run 2 and 2.4 times the SM at the end of Run 3 are the projected combined non-resonant cross-section sensitivities if systematic uncertainties remain sub-dominant.If systematic uncertainties are not sub-dominant, the end-of-Run-3 sensitivity would be about 5 times the SM expectation.
  • Higgs self-coupling at HL-LHC: 105 HH pairs per experiment are expected at the HL-LHC, with b ¯bγγ and b ¯bτ+τ− providing the best channel sensitivity, followed by b ¯bb ¯b.The b ¯bW W ∗ and b ¯bZ Z ∗ analyses add sensitivity despite their smaller branching fractions.
  • Higgs self-coupling at HL-LHC: 0.1 ≤κλ ≤2.3 is the projected 95% CL HL-LHC interval, while 0.5 ≤κλ ≤1.5 is the projected 68% CL interval; κλ =0 would be excluded at 95% CL.The combined ATLAS–CMS projection reaches 4σ including all systematic uncertainties and 4.5σ when those uncertainties are neglected.
  • J. Tian: 5% is the systematic error on extracting c6 from non-c6 SMEFT coefficients, enabling e+e−→Z HH to be interpreted as a model-independent c6 measurement within the SMEFT.Additional dimension-6 effects can be controlled by precision single-Higgs, electroweak, and other measurements, although including extra new-physics parameters can weaken self-coupling sensitivity.

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

Future colliders can determine the Higgs self-coupling through single-Higgs global fits and direct HH production, while higher-energy machines also probe quartic interactions and achieve substantially improved precision. These projections depend on controlling other new-physics effects and experimental systematics, but indicate robust, complementary constraints across collider programs.

  • C. Grojean: Single-Higgs measurements at e+e− colliders can extract c6 with 40–60% uncertainty, provided data at two different centre-of-mass energies constrain other SMEFT parameters.The determination is expected to be statistics-limited and largely model-independent within the SMEFT.
  • 9.9 The quartic Higgs self-coupling: Future e+e− colliders can set first coarse bounds on c8 and λ4, although Standard Model triple-Higgs production is not measurable at the considered energies.Combining double- and triple-Higgs production improves constraints on the cubic and quartic couplings.
  • C. Grojean: 44% precision is projected for FCC-ee single-Higgs self-coupling determination, while ILC at 500 GeV reaches 27% from ZHH and 24% when combined with single-Higgs data.ILC at 1 TeV or CLIC at 1.5 and 3 TeV is projected to reach 10% using ZHH and ν¯νHH.
  • Higgs self-coupling at future colliders: Summary: The FCC-ee(4IP), ILC, and CLIC programs can provide > 4σ evidence for or against a factor-2 increase in the Higgs self-coupling, with e+e− and pp measurements supplying complementary information.All four proposed Higgs factories are expected to measure the Hgg coupling to 1% precision, constraining cΦG relevant to gg → HH.
  • Higgs self-coupling at future hadron colliders: Double- and triple-Higgs production provide complementary constraints on (κ3,κ4), whose combination improves bounds on the Higgs self-interaction plane.At 100 TeV, the tt̄HH cross section increases by a factor of ∼75 from 14 TeV to 100 TeV.
  • Higgs self-coupling at future colliders: Summary: At the end of Run 3, SM di-Higgs production is not expected to be observed, whereas the HL-LHC can reach 5σ observation with only about 50% self-coupling precision.Run 3 projections allow an upper limit of 1–3 times the SM cross section at 95% CL; comparisons of future single-Higgs projections must account for the HL-LHC’s projected 50% error and differing SMEFT assumptions.
  • Higgs self-coupling at future hadron colliders: 5% precision on the Higgs self-coupling is projected at FCC-hh and 15% at HE-LHC from HH production cross-section measurements.At 100 TeV, several channels and techniques support δκλ = 5%, including boosted bbτ+τ− and bbγγ analyses.
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