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Effective field theory analysis of double Higgs production via gluon fusion

Aleksandr Azatov, Roberto Contino, Giuliano Panico, Minho Son

arXiv:1502.00539v2hep-ph

TL;DR

Double-Higgs production can probe Higgs interactions beyond the trilinear coupling, but its high-energy use requires assessing EFT validity. The paper simulates hh → b b̄ γγ at 14 and 100 TeV, finding about 30% trilinear-coupling precision at 100 TeV with 3 ab^-1, versus only an O(1) determination at the LHC.

  • Problem

    Double-Higgs production could reveal EFT interactions beyond the Higgs trilinear coupling, but exploiting high-energy distributions requires assessing where the EFT description remains valid.

    Source: §I · p. 2

  • Method

    The study simulates hh → b b̄ γγ signal and backgrounds, uses invariant-mass distributions to probe EFT coefficients, and assesses jet substructure and EFT validity.

    Source: §I · p. 2, §IV · p. 21

  • Results

    About 30% precision on c_3 is possible at a future 100 TeV collider with 3 ab^-1, while only an O(1) determination seems possible at the LHC with the same luminosity.

    Source: §VII · p. 43

  • Takeaways & Limitations

    The estimated trilinear-coupling precision is less optimistic than previously claimed, in part because the irreducible b b̄ γγ background has a sizable NLO k-factor of about 2.

    Source: §VII · p. 43

    Fake-photon backgrounds were not fully quantified, and a thorough quantitative analysis is needed to clarify whether they can be reduced to a subdominant level.

    Source: §VII · p. 44

Abstract

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We perform a detailed study of double Higgs production via gluon fusion in the Effective Field Theory (EFT) framework where effects from new physics are parametrized by local operators. Our analysis provides a perspective broader than the one followed in most of the previous analyses, where this process was merely considered as a way to extract the Higgs trilinear coupling. We focus on the $hh \to b\bar b\gamma\gamma$ channel and perform a thorough simulation of signal and background at the 14 TeV LHC and a future 100 TeV proton-proton collider. We make use of invariant mass distributions to enhance the sensitivity on the EFT coefficients and give a first assessment of the impact of jet substructure techniques on the results. The range of validity of the EFT description is estimated, as required to consistently exploit the high-energy range of distributions, pointing out the potential relevance of dimension-8 operators. Our analysis contains a few important improvements over previous studies and identifies some inaccuracies there appearing in connection with the estimate of signal and background rates. The estimated precision on the Higgs trilinear coupling that follows from our results is less optimistic than previously claimed in the literature. We find that a ~30% accuracy can be reached on the trilinear coupling at a future 100 TeV collider with 3 ab^-1. Only an O(1) determination seems instead possible at the LHC with the same amount of integrated luminosity.

I. INTRODUCTION

The paper uses an EFT perspective on double-Higgs production to probe electroweak symmetry-breaking dynamics beyond the Higgs trilinear coupling. It studies the b b̄ γγ channel at the LHC and a future 100 TeV collider, using kinematic distributions to assess EFT-operator sensitivity and validity.

  • The EFT framework parametrizes new-physics effects with local operators, whose coefficients can reveal properties of the underlying UV dynamics.The paper motivates EFT by the apparent gap between the electroweak scale and the new-physics scale, unless new physics couples very weakly to the Standard Model.Source: §I · p. 2
  • Double-Higgs production can probe Higgs trilinear and Higgs-pair couplings to gluons or top quarks that are inaccessible through single-Higgs processes.The paper describes it as the only potentially observable LHC process with access to these quartic couplings.Source: §I · p. 2
  • The study simulates signal and background for hh → b b̄ γγ at 14 TeV and 100 TeV, using kinematic distributions to improve sensitivity to EFT coefficients.It also assesses EFT validity and dimension-8 operators, compares with prior analyses, and examines the impact of jet substructure.Source: §I · p. 3

II. EFFECTIVE PARAMETRIZATION AND POWER COUNTING

The paper describes linear and nonlinear EFT parametrizations and uses SILH power counting to organize operator effects. The choice of parametrization and assumptions about the new-physics dynamics determine how the coefficients and Higgs couplings are interpreted.

  • The linear Lagrangian organizes new-physics deformations by operator dimension, while the nonlinear formulation treats the physical Higgs as a custodial singlet.The nonlinear coefficients resum corrections of order (v^2/f^2) and rely on the derivative expansion; the linear formulation also requires v/f < 1.Source: §II · pp. 3–5
  • SILH power counting assumes a new-physics mass scale m_* and coupling g_*; Higgs-doublet insertions scale as 1/f = g_*/m_*, while extra covariant derivatives are suppressed by m_*.The analysis also assumes baryon and lepton number conservation and focuses on CP-conserving operators.Source: §II · p. 4
  • In the nonlinear formulation, O_u modifies the top Yukawa coupling and also generates a quartic t̄t hh interaction.Source: §II · p. 5
  • The O_6 normalization differs from Ref., but the paper’s c_3 parametrization remains linear in c̄_6 and is useful when c̄_6 is large.The authors anticipate constraints on c̄_6 values larger than 1 at the LHC.Source: §II · p. 5

A. Modified power counting for the Higgs trilinear coupling

The section considers power-counting scenarios in which new physics can modify the Higgs trilinear coupling while leaving other Higgs couplings near their Standard Model values. The possible enhancement depends on the scenario and can carry a tuning or perturbativity cost.

A. Modified power counting for the Higgs trilinear coupling

  • A generic composite Higgs that is not a Nambu-Goldstone boson can have an enhanced c̄_6, but keeping the Higgs mass light requires tuning.For (v/f)^2 = 0.05, the paper gives c̄_6 ∼ 3.5 (g_*/3)^2 and c_3 ≃ 0.93 + 3.5 (g_*/3)^2.Source: A. Modified power counting for the… · p. 6
  • A Higgs-mass-portal scenario can enhance the trilinear coupling without tuning the Higgs mass, but c̄_6 is bounded at O(1) without fine-tuning.At saturation, all orders in λ become equally important and perturbation theory is lost.Source: A. Modified power counting for the… · pp. 6–7
  • Different power counting can allow large trilinear-coupling modifications while other Higgs couplings remain close to their Standard Model values.In a generic composite-Higgs scenario, c̄_6 can be of order 1 or larger without invalidating the perturbative expansion in Higgs-field powers.Source: A. Modified power counting for the… · pp. 5–7

B. The relevance of higher-order operators

The section assesses when the EFT description of double-Higgs production remains valid and when higher-dimensional operators matter. In this process, high-energy effects can make dimension-8 contributions relevant below the EFT cutoff, constraining analyses that retain only dimension-6 operators.

B. The relevance of higher-order operators

  • At high energies below the cutoff, energy-growing new-physics interactions can exceed the top Yukawa contribution, unlike dimension-6 corrections to on-shell Higgs rates, which remain at most O(v^2/f^2).The high-energy region is difficult to exploit in gg → hh → b b̄ γγ because of falling gluon PDFs at large x and the small final-state branching ratio.Source: B. The relevance of higher-order… · pp. 8–9
  • For λ ≲ √(g_* y_t), dimension-8 effects can become the largest contributions above the crossover energy, even while the EFT remains valid up to E ∼ m_*.The operators mediate gluon-to-Higgs-pair scatterings with angular momentum 0 and 2; the dimension-8 operator O_gD2 contributes to the J_z = ±2 amplitude.Source: B. The relevance of higher-order… · pp. 8–9
  • Dimension-6-only analyses are sensible only in a bounded region of (g_*, λ) parameter space; the light-blue region requires dimension-8 operators, and the gray region has no sensible EFT description.Reducing g_min by improving precision at fixed energy expands the viable region; the λ = y_t and λ = √g_* y_t cases require g_min < y_t.Source: B. The relevance of higher-order… · p. 9, Figure 1 · p. 10
  • Figure 2 shows diagrams contributing to double-Higgs production via gluon fusion, including an additional crossed-box contribution, a ¯tthh coupling in the last first-line diagram, and Higgs–gluon contact interactions in the second-line diagrams.Source: Figure 2 · p. 10

C. Cross section of double Higgs production

EFT operators leave distinct signatures in the invariant-mass spectrum, while angular information provides little practical leverage on the dimension-8 contribution. The analysis therefore uses mhh as its main kinematic variable and assesses the cross section through a polynomial fit in EFT parameters.

C. Cross section of double Higgs production

  • Dimension-6 effects reshape the mhh spectrum differently: trilinear-coupling terms matter near threshold, while modified top-Yukawa and contact interactions tend to enhance its high-mass tail.Dimension-8 operators add a contribution growing as ŝ^2, also distorting the tail and helping a shape analysis break total-rate degeneracies.Source: C. Cross section of double Higgs… · p. 11
  • The gluon-fusion cross section is fitted as a polynomial in effective-Lagrangian parameters, with LO coefficients reported for 14 and 100 TeV colliders.The study starts with the non-linear Lagrangian and neglects higher-derivative terms, then assesses their impact on angular distributions and finds it small given limited high-mhh sensitivity.Source: C. Cross section of double Higgs… · pp. 10–11, Table I · p. 12
  • The gg → hh amplitude separates into Jz = 0 and Jz = ±2 components; the SM Jz = ±2 contribution is negligible near mhh ≃ 400 GeV and below about 20% near 700 GeV.The leading angular dependences are approximately constant for M0 and sin^2 θ for M2.Source: C. Cross section of double Higgs… · p. 12
  • At the benchmark, the OgD2 contribution reaches the SM rate near mhh ≃ 1.3 TeV; angular cuts lower this crossover by at most about 10%.The analysis finds little practical angular leverage, particularly given the limited mhh range realistically probed in the b¯bγγ channel.Source: C. Cross section of double Higgs… · p. 13
  • Because angular cuts only modestly increase the dimension-8 contribution and the b¯bγγ channel probes a limited mhh range, the analysis uses mhh rather than angular distributions.The dimension-8 operator OgD2 contributes to the Jz = 2 amplitude.Source: C. Cross section of double Higgs… · p. 13

III. DOUBLE HIGGS PHENOMENOLOGY AT 14 TEV AND 100 TEV

The study examines double-Higgs phenomenology at 14 and 100 TeV, focusing on the clean but low-rate b¯bγγ channel. The higher-energy collider increases rates and access to boosted, high-mass events, motivating an initial assessment of jet substructure.

III. DOUBLE HIGGS PHENOMENOLOGY AT 14 TEV AND 100 TEV

  • The analysis focuses on hh → γγb¯b, whose SM branching ratio is about 0.264%, because it is considered the cleanest channel despite its small rate.The higher-rate b¯bb¯b and b¯bWW* channels face large backgrounds; the b¯bτ+τ− channel depends on tau-pair reconstruction.Source: §III · pp. 14–15
  • The study concentrates on interpreting the b¯bγγ search in the EFT framework and leaves exploration of other available channels for future work.The authors describe the b¯bγγ analysis strategy as relatively simple, allowing them to focus on its EFT interpretation.Source: §III · p. 15
  • At 100 TeV, the SM cross section is about 40 times its 14 TeV value, with the mhh distribution changing mainly through enhancement of its tail above roughly 700 GeV.The peak remains around mhh ≃ 400 GeV, largely determined by the falling gluon parton distributions.Source: §III · p. 15
  • The 100 TeV analysis is not fully optimized; the study offers only a first assessment of how jet substructure could improve results for more highly boosted events.The higher cross section and enhanced tail make higher mhh values accessible, potentially increasing sensitivity to energy-growing operators.Source: §III · p. 16
  • For Higgs transverse momenta above roughly 500–600 GeV, decay partons are often unresolved as separate standard jets, making jet-substructure techniques likely helpful.Their angular separation scales as ΔR ∼ 2mh/pT(h), compared with reconstruction cones of 0.4 for ATLAS and 0.5 for CMS.Source: §III · p. 17
  • At 100 TeV, SM signal reach extends to mhh ∼ 1.5 TeV in γγb¯b and ∼ 4 TeV in b¯bb¯b, where many Higgs bosons are boosted.At 14 TeV, boosted phase space is almost negligible for γγb¯b and very limited for the other listed channels.Source: §III · p. 18

IV. ANALYSIS OF pp →hh →γγb¯b

The analysis simulates the hh → b¯bγγ signal and backgrounds at 14 and 100 TeV, using selection cuts and reconstructed hh-mass categories to assess EFT sensitivity. It finds different leading backgrounds and jet-veto choices at the two collider energies.

Background simulation

  • Allowing one extra matched jet increases the b¯bγγ cross section by a factor of ∼2, accounting for real emissions that form the bulk of its NLO correction.This background is dominant after cuts, and the extra-jet effect had been omitted in most previous studies.Source: §IV · p. 19

Signal-rate estimation

  • The signal-rate estimate for non-SM parameter points rescales parton-level samples using the SM hadron-to-parton ratio, applied separately in each mhh bin.The approach assumes the ratio is approximately the same for SM and BSM points; for boosted analysis, a different pre-cut cross-section ratio procedure was checked to be accurate at a few-percent level in high-mhh categories.Source: §IV · pp. 19–20

14 TeV strategy

  • At 14 TeV, angular cuts on the b-jet and photon pairs efficiently reduce backgrounds, especially b¯bγγ, while retaining most of the signal.The cuts require ∆R(b,b) < 2, ∆R(γ,γ) < 2, and ∆R(b,γ) > 1.5.Source: §IV · p. 21, Figure 7 · p. 21
  • At 14 TeV, a jet-multiplicity veto removes roughly 80% of t¯th while keeping ∼70% of signal; a hadronic-W veto removes ∼50% while retaining over 90% of signal.Neither veto was used at 14 TeV because the significance gain was modest and the analysis prioritized retaining signal events.Source: §IV · p. 23

Invariant-mass categories

  • Events are divided into six reconstructed hh-mass categories to use their differing kinematic dependence when constraining effective-operator coefficients.The category yields are reported separately for the 14 and 100 TeV analyses.Source: §IV · pp. 23–24

100 TeV strategy

  • At 100 TeV, t¯th becomes the dominant background, and vetoing hadronic W bosons is the most efficient tested way to maximize signal significance.The background changes from b¯bγγ at 14 TeV because t¯th grows more steeply with collider energy; the W veto is applied in the final selection.Source: §IV · p. 24

A. Recovering the boosted topologies

At high energy, boosted Higgs decays can make the two b-jets or photons too close for a traditional resolved-object analysis. The study compares jet substructure with traditional reconstruction and finds its clearest benefit at 100 TeV in the high-mass tail.

A. Recovering the boosted topologies

  • At high mhh, boosted Higgs decays can fail a dijet-style analysis when the b-jets are insufficiently separated, and the diphoton system can face the same issue.The partonic ∆R distributions show that many events with mhh > 1000 GeV do not resolve into two well-separated partons.Source: Figure 10 · p. 26, A. Recovering the boosted topologies · p. 27
  • The substructure strategy clusters R = 1.5 Cambridge/Aachen fat-jets and applies BDRS declustering to identify three subjets in a Higgs-mass window.Candidates require pT(j) > 150 GeV and two subjet b-tags; the study also tests reduced or mutual photon isolation.Source: A. Recovering the boosted topologies · p. 27
  • Jet substructure is crucial mainly at 100 TeV, while its impact is minor at the 14 TeV LHC; at lower reconstructed hh masses, traditional reconstruction is more efficient.The study reports that some events with mreco_hh ≲ 1000 GeV are not correctly reconstructed by substructure because of the finite R = 1.5 fat-jet size.Source: A. Recovering the boosted topologies · p. 28
  • At 14 TeV, about 0.5 signal events above mreco_hh = 850 GeV correspond to the same yield at 100 TeV above 1540 GeV with traditional jets or 2560 GeV with substructure and Riso = 0.2.The higher threshold with substructure approaches the partonic-level reach more closely than the traditional analysis.Source: A. Recovering the boosted topologies · pp. 28–29
  • The boosted analysis includes only the dominant b¯bγγ and t¯th backgrounds; dedicated boosted top- and W-tagging to further reduce t¯th is left beyond the study’s scope.The authors also describe a hybrid strategy combining traditional and boosted reconstruction as future work.Source: A. Recovering the boosted topologies · pp. 25, 26, 29

V. SENSITIVITY ON THE EFT COEFFICIENTS

This section uses Bayesian fits to assess sensitivities to Higgs effective operators across three collider benchmarks, examining the role of differential $m_{hh}$ information and correlated couplings while treating the anomalous $tthh$ coupling as negligible in the SM case.

V. SENSITIVITY ON THE EFT COEFFICIENTS

  • The fits omit theoretical signal-cross-section uncertainty and experimental background-extraction systematics, and assume the Standard Model signal is injected.The likelihood uses signal and background counts in six m_hh categories; the omitted uncertainty is an explicit limitation of the sensitivity estimates.Source: §V · p. 30
  • In the nonlinear parametrization, c2t and c3 are strongly anti-correlated, c2t is more precisely determined than c3, and jet substructure improves c2g sensitivity dramatically but scarcely changes c2t or c3.The authors attribute the c2g improvement to boosted, large-m_hh events; c2g is determined more accurately than c2t and c3.Source: §V · pp. 31–32, Figure 13 · p. 32
  • At 100 TeV, an exclusive m_hh analysis breaks the c2t–c3 degeneracy that persists inclusively; high-m_hh categories constrain c2t and c2g, while threshold events chiefly constrain c3.An inclusive analysis is much less powerful, especially at 100 TeV.Source: §V · pp. 32–33, Figure 14 · p. 33
  • δ(c2g) is 0.15, 0.1, 0.05, and 0.016 for LHC14, HL-LHC, FCC100, and FCC100 with boosted jets, respectively.The corresponding mean event energies are about 1.0 TeV for the first three scenarios and 1.8 TeV for boosted FCC100; g_min is 1.6, 1.3, 0.9, and 0.9, respectively. The authors infer that dimension-6-only LHC analyses cannot probe pseudo-Nambu–Goldstone Higgs scenarios with fully or partially composite t_R, though they can probe a generic composite Higgs.Source: §V · pp. 33, 35
  • In the Higgs-doublet parametrization, double-Higgs production breaks the c_g degeneracy left by single-Higgs measurements including tth and improves constraints on c_u and c_g, especially at 100 TeV.tth is crucial for reducing overall uncertainty and mainly constrains c_u; the bbar-gamma-gamma channel alone is less powerful but competitive on c_u, and other double-Higgs final states could strengthen the constraints.Source: §V · pp. 35–37, Figure 17 · p. 36
  • At 14 TeV, the constraint is on c̄6 > 1; even at HL-LHC, the likelihood is far from Gaussian, with a second maximum near c̄6 ≃ 4.5, while a fully exclusive analysis removes the degeneracy of the inclusive analysis without m_hh categories.Source: §V · pp. 37–38
  • At 100 TeV, marginalizing over other couplings materially affects c6 precision: its uncertainty grows as σ(c_u) increases at small values, then plateaus for σ(c_u) ≳ 0.2.In that limit, double-Higgs production alone determines c_u more precisely than the single-Higgs input, independently of it; poor c_u determination can therefore weaken c6 measurement.Source: §V · pp. 38–39, Figure 20 · p. 39
  • The top-Yukawa uncertainty is sufficiently small to have negligible impact on double-Higgs production when the SM case is considered without the anomalous tthh coupling.The top Yukawa coupling can be more precisely determined from the top-quark mass.Source: §V · p. 40

VI. COMPARISON WITH PREVIOUS WORKS

The study reports less optimistic trilinear-coupling precision than earlier work, attributing differences to background estimates, analysis choices, and how the coupling uncertainty is derived. Comparisons also identify disagreements in signal and background rates across prior studies.

Comparison with previous works

  • 200% at the HL-LHC and 30% at FCC100 are the study’s trilinear-coupling uncertainties, versus Yao’s 50% and 8%, respectively.Using post-cut rate slopes revises Yao’s estimates to 90% and 16%; the remaining difference reflects the HL-LHC’s nonlinear fit and FCC100 marginalization over additional coefficients.Source: §VI · pp. 40–42
  • Ref. estimates lower non-resonant backgrounds partly through a narrower photon-mass window, while its LO rescaling underestimates the b̄bγγ background enhanced by an NLO k-factor of about 2.The study also contrasts Ref. ’s b-tagging and light-jet mis-tag assumptions with its own.Source: §VI · pp. 40–41
  • Ref. reports 47 HL-LHC signal events after cuts versus the study’s approximately 13, and its limited Monte Carlo statistics make the b̄bγγ continuum appear negligible.The study could not fully identify the signal-rate difference; it attributes part of its lower rate to including photon efficiency.Source: §VI · p. 41
  • The study finds Yao’s signal and background rates consistent with its own, while the ATLAS analysis also reports consistent rates but finds fake-photon and fake-b backgrounds sizable.The study cautions that the Yao-based precision estimate is more optimistic because it uses a pre-cut signal-rate dependence; ATLAS’s reducible backgrounds may be mitigated by future strategies.Source: §VI · pp. 41–42

VII. SUMMARY AND OUTLOOK

The study uses an EFT framework to analyze double-Higgs production beyond extracting the trilinear coupling, while assessing operator sensitivity and EFT validity. It finds that c3 is measurable to about 30% at a future 100 TeV collider with 3 ab^-1, but only to O(1) at the LHC with the same luminosity.

VII. SUMMARY AND OUTLOOK

  • The EFT analysis encodes heavy-New-Physics effects in local-operator deformations and assesses sensitivity beyond the Higgs trilinear coupling.It examines SILH power counting and alternative scenarios with large trilinear-coupling shifts but small changes to other Higgs couplings.Source: §VII · pp. 42–43
  • The EFT validity range requires assessment because dimension-8 operators can matter when dimension-6 contributions are suppressed, especially in high-energy tails.The paper notes this possibility for double-Higgs production when Og is suppressed by Higgs shift symmetry in a pseudo Nambu-Goldstone boson scenario.Source: §VII · p. 42
  • About 30% precision on c3 is possible at 100 TeV with 3 ab^-1, while the LHC permits only an O(1) determination at the same luminosity.The authors describe the 100 TeV precision as less optimistic than claims usually made in the literature.Source: §VII · p. 43
  • A sizable NLO enhancement of the irreducible b¯bγγ background and fits allowing other dimension-6 corrections contribute to the less optimistic c3 precision.The top-Yukawa operator also generates a quartic t¯thh interaction; an approximately 10% uncertainty on ¯cu can nearly double the c3 error at 100 TeV.Source: §VII · p. 43
  • The t¯thh and gghh quartic couplings, parametrized by c2t and c2g, can be determined more accurately than c3.The paper treats these as independent parameters in a non-linear effective Lagrangian.Source: §VII · p. 43
  • The exclusive mhh distribution helps disentangle operator contributions, whereas the Higgs-pair angular separation has a marginal role in the analysis.The angular distribution is nearly flat in the SM and nearly unchanged by New Physics; dimension-8 effects are a possible exception but are unlikely to be accessible at the LHC in the rare b¯bγγ mode.Source: §VII · pp. 43–44
  • Jet substructure is relevant for mhh ≳1 TeV and may significantly increase sensitivity to Og and possibly dimension-8 operators at future higher-energy colliders.Its impact is marginal at the LHC.Source: §VII · p. 44
  • The analysis leaves mass-window optimization, fake-photon backgrounds, advanced multivariate methods, and combined double-Higgs channels for further study.The authors note that non-resonant backgrounds scale roughly linearly with mass-window size and that a quantitative fake-photon study is needed; combining channels may improve sensitivity.Source: §VII · pp. 44–45

Appendix A: Angular decomposition

The appendix decomposes the scattering amplitude into angular-momentum components and identifies the higher-derivative contributions associated with dimension-8 operators.

Appendix A: Angular decomposition

  • Angular-momentum conservation decomposes the scattering amplitude into M0 and M2 contributions mediating Jz = 0 and Jz = ±2 transitions.The corresponding projectors are orthogonal.Source: Appendix A · p. 46
  • The appendix gives expressions for M0 and M2 in terms of form factors and the transverse momentum of the Higgses.The form factors F2, F∆, and G2 are referenced from Ref..Source: Appendix A · p. 46
  • The higher-derivative coefficients cgD0 and cgD2 provide next-to-leading corrections to the non-linear Lagrangian and relate to dimension-8 operator coefficients.The stated relations are cgD0 = (4π/α2)¯cgD0 and cgD2 = (4π/α2)¯cgD2.Source: Appendix A · p. 46

1. Cross sections of signal and backgrounds

This appendix reports generation-level cross sections for signal and backgrounds at 14 and 100 TeV and summarizes the associated sample setup and cuts.

1. Cross sections of signal and backgrounds

  • One extra matrix-element jet is simulated for backgrounds except t¯th and γγjj, and no generation-level cuts are applied to resonant t¯th and Zh samples.The b¯bh samples require pT(b) > 20 GeV at 14 TeV, increased to 30 GeV at 100 TeV, within |η(b)| < 3 and ∆R(b,¯b) > 0.3.Source: §1 · pp. 46–47
  • Table X summarizes signal and background cross sections in fb after generation-level cuts at 14 and 100 TeV.The supplied passage gives the table scope but not its cell values.Source: Table X · p. 47
  • The γγjj sample has pT(j) thresholds of 15 GeV at 14 TeV and 25 GeV at 100 TeV, while photons require pT(γ) thresholds of 20 and 30 GeV, respectively.Jets and photons are separated by ∆R > 0.3; the listed Z and Higgs branching fractions are folded into Table X cross sections.Source: §1 · p. 47

2. NLO estimate of γγb¯b background and k-factor

The appendix compares tree-level matched and full NLO simulations of the b¯bγγ background. The matched sample reproduces the approximate factor-two rate enhancement and overall differential shapes, supporting its use in the analysis.

2. NLO estimate of γγb¯b background and k-factor

  • The b¯bγγ background cross section is enhanced by roughly a factor of 2 when diagrams with one extra matrix-element parton are included.This is the most important background included in the analysis.Source: §2 · p. 47
  • The three samples comprise a full NLO simulation with real and virtual corrections, a tree-level matched sample, and a LO sample without extra matrix-element partons.The full NLO sample is showered with PYTHIA8 and the unmatched LO sample with PYTHIA6; generation-cut differences make direct comparison subtle.Source: §2 · p. 47
  • The k-factor is approximately 2 at each cut-flow step, and the matched-to-LO rate ratio is similar, indicating that real emissions supply most of the NLO correction.The first-row cross sections in Table XI should not be directly compared because their samples use different cuts and procedures.Source: §2 · pp. 48–49, Table XI · p. 48
  • The matched and full NLO differential shapes agree overall despite fluctuations in higher-value bins, justifying use of the less CPU-intensive matched sample.The authors expect the same results to apply at 100 TeV; Figure 21 compares several variables at 14 TeV within the generation-level mass window.Source: §2 · p. 49
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