NLO Corrections to Double Higgs Production in the Higgs Singlet Model

I. INTRODUCTION

• For M_H ≳ 2m_h, a second scalar can resonantly enhance gluon-fusion double-Higgs production relative to the SM prediction.The singlet model is also studied in searches for heavy SM-like Higgs bosons and resonant di-Higgs production [8–14].Source: §I · p. 2
• The NLO QCD calculation uses the m_t → ∞ effective theory to predict the singlet-model cross section and hh invariant-mass distribution.The full NLO corrections would involve two-loop virtual diagrams with massive internal particles.Source: §I · p. 2
• Unknown m_t dependence of higher-order QCD corrections induces an O(±10%) uncertainty in SM predictions.The paper notes this uncertainty in the context of higher-order corrections.Source: §I · p. 3
• Resonance-region corrections are significant, making a constant K factor a poor approximation there.The paper examines the approximations behind the QCD corrections in the Higgs singlet model.Source: §I · p. 3
• Interference between heavy-scalar and SM-like contributions can be significant and should be included in searches for new heavy scalars.The paper investigates these interference effects alongside the QCD corrections.Source: §I · p. 3

II. MODEL

• A real gauge singlet S is added to the SM Higgs doublet, with a Z2 symmetry imposed under which S → −S.After symmetry breaking, the model has two physical scalar bosons, h and H.Source: A. Recap · p. 3
• Decays of H to SM particles have partial widths equal to the corresponding SM widths at M_H, rescaled by the scalar mixing.The model’s total widths include the H → hh decay channel.Source: A. Recap · pp. 4–5
• For small sin θ, BR(H → hh) is approximately 0.3–0.4 and relatively insensitive to tan β.The branching ratio is shown in Fig. 1.Source: A. Recap · p. 5, Figure 1 · p. 5
• The model takes m_h = 125 GeV, M_H, v = 246 GeV, tan β, and cos θ as its five free parameters.The lightest scalar h is assumed to be the SM-like Higgs particle.Source: A. Recap · pp. 4–5

B. Limits

• For M_H ≳ 400 GeV, W-mass measurements require |cos θ| > 0.96; h-coupling fits require |cos θ| > 0.93 at 95% confidence level.For M_H ≲ 300 GeV, heavy-Higgs search limits are stronger than the W-mass limits; interpreting them assumes no H → hh branching fraction.Source: B. Limits · p. 5
• Perturbativity of b4 imposes an upper limit on tan β that depends on M_H and θ; for sin θ = 0.1, tan β < 1.5 (0.5) at M_H = 200 (500) GeV.The paper generally presents results with cos θ = 0.96 and tan β = 0.5.Source: B. Limits · pp. 5–6

A. LO Results

• LO Higgs-pair production proceeds through the diagrams for gg → hh, with top-quark loops providing the dominant contribution in the singlet model.The amplitude is sensitive to the trilinear Higgs self-couplings and to new colored particles in the loops.Source: A. LO Results · pp. 6–7
• The gg → hh amplitude is expressed through spin-0 and spin-2 form factors depending on the partonic Mandelstam variables and loop-particle mass.The form factors F1 and F2 are known analytically [31, 32], and the partonic cross section is expressed in terms of them.Source: A. LO Results · p. 6

B. NLO Corrections

• The NLO rate is formed after real–virtual cancellation of infrared singularities and absorption of remaining collinear divergences into the NLO PDFs.The treatment covers gg, qg, and qq initial states with five light flavors.Source: B. NLO Corrections · pp. 7–8
• The finite-m_t approximation uses exact m_t-dependent quantities for virtual form factors and includes full mass dependence at LO.The virtual-correction coefficient is denoted C_mt.Source: B. NLO Corrections · p. 8
• The finite-m_t result is approximate because Eq. 25 adapts the large-m_t NLO calculation.Source: B. NLO Corrections · p. 9
• The paper defines an m_t-dependent differential K factor from Eqs. 22 and 25, with M_hh denoting the final-state double-Higgs invariant mass.Source: B. NLO Corrections · p. 9

IV. RESULTS

• The calculations use CT12NLO PDFs, μ_R = μ_F = μ = M_hh, m_t = 173.34 GeV, and m_b = 4.62 GeV.Source: §IV · p. 9
• The lightest scalar h is assumed to be the SM-like Higgs boson with m_h = 125 GeV.Source: §IV · p. 9
• Production includes only top-quark loops, the largest contribution, and the SM and singlet-model calculations are checked against HPAIR.The singlet-model check incorporates the resonance in HPAIR and compares the results.Source: §IV · p. 9

A. SM Results

• The heavy-top and approximate finite-m_t NLO calculations have different normalizations and shapes, but their K factors are almost identical.Real-gluon emission and virtual contributions are similar in size, while quark-initiated contributions are highly suppressed.Source: A. SM Results · p. 10
• NLO corrections significantly increase the SM rate; the heavy-top LO result overshoots the exact LO rate at high M_hh and misses its peak structure.Source: A. SM Results · p. 10
• In the SM, the differential K factor depends only slightly on M_hh and can be accurately approximated by a constant.Source: A. SM Results · p. 10
• The fractional scale dependence is reduced at NLO, and the differential K-factor scale uncertainty is about 11% at M_hh = 400 GeV and 15% at 800 GeV.Source: A. SM Results · p. 10, Figure 5 · p. 11

B. Singlet Model Results

• For M_hh below M_H, the H-resonance and box diagrams interfere constructively, while above M_H their interference is destructive.Source: B. Singlet Model Results · p. 12
• For M_H < 2m_h, both resonance diagrams interfere destructively with the box near M_hh ≈ 2m_h, producing the strongest dip at M_H = 200 GeV.Source: B. Singlet Model Results · p. 12
• Changing the sign of sin θ barely changes distribution shapes or interference, but for M_H > 2m_h the negative-sign total cross section is about 70–80% of the positive-sign rate.Source: B. Singlet Model Results · p. 12
• The sign dependence is suppressed because sin θ enters with tan β and is suppressed relative to cos θ terms in the triple couplings.Source: B. Singlet Model Results · p. 12
• For M_H < 2m_h, the total cross sections for the two signs of sin θ are nearly the same.Source: B. Singlet Model Results · p. 12
• Large rate enhancements are possible near the resonances, making the singlet model clearly distinguishable from the SM.Source: B. Singlet Model Results · p. 13

C. Interference effects

• For Mhh < MH, the H propagator interferes constructively with the box diagram, but for Mhh > MH the interference is destructive.The two scalar propagators instead interfere destructively below MH.Source: C. Interference effects · pp. 13–14
• For Mhh < MH, the H-resonance and SM-like contributions interfere constructively, and for Mhh > MH they interfere destructively.The H-resonance contribution is suppressed as MH increases, so its fractional contribution falls while the fractional interference contribution rises.Source: C. Interference effects · pp. 13, 15, 17
• The infinite-top-mass limit fails to reproduce the interference structure near and just above the resonance peak and overshoots the rate at high Mhh.It also shifts the interference dip slightly to larger Mhh.Source: C. Interference effects · p. 15
• For Mhh ≪ MH, the interference contribution is independent of MH at fixed θ and tan β.The absolute interference contribution is nearly independent of MH for MH ≳ 500 GeV.Source: C. Interference effects · pp. 16–17
• Below 2mh, the H-resonance and interference contributions are larger for cos θ = 0.9 than for cos θ = 0.96.For MH > 2mh in the narrow-width approximation, the H-resonance amplitude remains larger at cos θ = 0.9.Source: C. Interference effects · p. 17
• Once MH approaches 2mh, the H-resonance contribution dominates; its fractional contribution then decreases as MH rises above roughly 2mt.The propagator suppresses the resonance contribution, while the top-loop production enhancement and propagator suppression largely offset one another for 2mh ≲ MH ≲ 2mt.Source: C. Interference effects · p. 17

D. NLO Effects

• For tan β = 1 and cos θ = 0.9, enhancements of the SM rate up to a factor of ∼22 are possible.For tan β = 5 and cos θ = 0.96, the maximum enhancement is O(8) for MH ≲ 500 GeV and falls rapidly to O(1) at larger MH.Source: D. NLO Effects · p. 17
• The narrow-width approximation for the H-resonance contribution is accurate for MH ≲ 400 GeV.The two NLO methods increasingly agree as MH increases.Source: D. NLO Effects · pp. 17, 21
• The full mt-dependent NLO rate is not known, so the reported NLO results use an approximate finite-mt dependence.The leading corrections come from gg and virtual contributions, while q contributions are subleading.Source: D. NLO Effects · pp. 17–18
• The approximate mt dependence of the NLO rate is mostly captured by weighting the exact LO rate with Kmt→∞, so the rates overlap almost exactly.Most contributions to the NLO rate are proportional to the LO rate.Source: D. NLO Effects · p. 19
• The exact-LO-weighted Kmt→∞ approximation does not reproduce the NLO result at the strongest interference dip, especially for MH = 200 GeV.Agreement between the two methods improves as MH increases because the nonproportional virtual contribution decreases.Source: D. NLO Effects · pp. 20–21
• The scale dependence is fairly flat across the invariant-mass distribution, including the resonance and strong-destructive-interference regions.The distributions are evaluated over Mhh/2 < µ < 2Mhh.Source: D. NLO Effects · p. 21
• The differential K-factor in the mt → ∞ limit is 2–2.2 except for spikes in regions of strongest destructive interference.The K-factor from Eq. 29 agrees with Kmt→∞ except in those regions.Source: D. NLO Effects · p. 21

E. Results at 100 TeV

• Because the K-factor ratio at 100 TeV is similar to that at 13 TeV, the comparison of the exact-LO-weighted Kmt→∞ rate with the Eqs. 25,26 rate carries over between the energies.Source: E. Results at 100 TeV · p. 23
• At 13 TeV and 100 TeV, the paper presents normalized invariant-mass distributions.Source: E. Results at 100 TeV · p. 23
• The distribution tails are enhanced at 100 TeV relative to 13 TeV because the increased gluon parton luminosity lengthens the tail.At fixed invariant mass, the PDFs are evaluated at smaller x at 100 TeV.Source: E. Results at 100 TeV · p. 24

• Interference alters invariant-mass distributions for all MH, contributes ∼10−20% to the total rate for MH ≳450 GeV, and can suppress the total cross section by up to ∼30% for a viable parameter point.The authors state that these effects should be included in searches for heavy scalars.Source: §V · p. 24
• The two NLO approximations give similar results except in regions with large destructive interference.One weights the exact mt-dependent LO cross section by a K factor computed in the mt →∞ limit; the other inserts exact mt-dependent form factors into NLO contributions.Source: §V · pp. 24–25
• For tan β = 0.5 and cos θ = 0.96, the total cross section is increased by factors between 5 −10 above the SM rate; for tan β = 1 and cos θ = 0.9, enhancements are 10 −20 for MH < 500 GeV.Source: §V · p. 25
• At large MH, the resonant approximation to the total cross section underestimates the enhancement by about a factor of 2.Source: §V · p. 25
• The singlet model significantly alters outgoing Higgs-pair kinematics from the SM, and higher-order QCD corrections differ from the SM near the resonance peak.Source: §V · p. 25