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SusHi Bento: Beyond NNLO and the heavy-top limit

Robert V. Harlander, Stefan Liebler, Hendrik Mantler

arXiv:1605.03190v2hep-ph

TL;DR

Precise Higgs-production cross sections are important for measuring the 125 GeV Higgs boson and searching for additional Higgs bosons. SusHi 1.6.0 adds higher-order and mass-suppressed gluon-fusion predictions, heavy-quark annihilation, scale-uncertainty evaluation, and dimension-5 operator studies across supported models. The release provides soft-limit and high-energy information through N3LO, top-mass effects through NNLO, and broader studies of Higgs production and scalar interactions.

  • Problem

    High-precision production cross sections are needed to measure the 125 GeV Higgs boson's properties and search for additional Higgs bosons.

  • Method

    SusHi 1.6.0 combines soft expansions, N3LO gluon-fusion terms, inverse-top-mass expansions through NNLO, heavy-quark annihilation, scale-dependence evaluation, and dimension-5 operator calculations.

  • Results

    SusHi 1.6.0 provides improved fixed-order gluon-fusion predictions and uncertainty estimates, including soft-limit and high-energy matching through N3LO and top-quark mass effects through NNLO.

  • Takeaways & Limitations

    The code supports Higgs cross-section predictions in the SM, 2HDM, MSSM, and NMSSM, including studies of arbitrary dimension-5 operators and heavy-quark annihilation.

  • Takeaways & Limitations

    SusHi does not include effects beyond fixed order such as soft-gluon resummation, and its N3LO scale dependence inherits approximations made at NNLO.

Abstract

from arXiv · show

Version 1.6.0 of the code SusHi is presented. Concerning inclusive CP-even Higgs production in gluon fusion, the following new features with respect to previous versions have been implemented: expansion of the partonic cross section in the soft limit, i.e. around $x=M_H^2/\hat{s}\to 1$; N$^3$LO QCD corrections in terms of the soft expansion; top-quark mass suppressed terms through NNLO; matching to the cross section at $x\to 0$ through N$^3$LO. For CP-even and -odd scalars, an efficient evaluation of the renormalization-scale dependence is included, and effects of dimension-5 operators can be studied, which we demonstrate for the SM Higgs boson and for a CP-even scalar with a mass of $750$ GeV. In addition, as a generalization of the previously available $b\bar{b}\to H$ cross section, SusHi 1.6.0 provides the cross section for charged and neutral Higgs production in the annihilation of arbitrary heavy quarks. At fixed order in perturbation theory, SusHi thus allows to obtain Higgs cross-section predictions in different models to the highest precision known today. For the SM Higgs boson of $M_H=125$ GeV, SusHi yields $48.28$ pb for the gluon-fusion cross section at the LHC at $13$ TeV. Simultaneously, SusHi provides the renormalization-scale uncertainty of $\pm 1.97$ pb.

PROGRAM SUMMARY

SusHi 1.6.0 supersedes earlier versions and expands the program’s model coverage, gluon-fusion predictions, uncertainty estimation, and heavy-quark annihilation capabilities.

  • SusHi 1.6.0 supports the 2HDM and NMSSM while retaining features from previous versions.
  • The release adds top-quark mass terms through NNLO, soft and N3LO heavy-top corrections, top-squark corrections through NNLO, and dimension-5 operators.
  • SusHi 1.6.0 analytically determines renormalization-scale dependence and calculates Higgs production through heavy-quark annihilation.

1. Introduction

SusHi 1.6.0 addresses the need for precise Higgs production cross sections by extending fixed-order predictions and uncertainty tools across several models and production mechanisms.

  • High-precision Higgs production cross sections are important for measuring the 125 GeV Higgs boson and searching for additional Higgs bosons.
  • SusHi calculates Higgs production through gluon fusion and bottom-quark annihilation in the SM, 2HDM, MSSM, and NMSSM.
  • SusHi includes N3LO gluon-fusion terms for CP-even Higgs bosons in the heavy-top limit.
  • The soft expansion is available around x = 1 through N3LO, while exact x-dependence remains the default at NLO and NNLO.
  • Top-quark mass effects are included through NNLO using an inverse-mass expansion, with exact mass dependence factored at leading order.
  • Soft-expansion results can be matched to the high-energy limit x → 0 through N3LO.
  • Dimension-5 operators affect inclusive gluon fusion through N3LO and Higgs transverse-momentum and rapidity distributions at LO and NLO.
  • Heavy-quark annihilation is implemented at NNLO for inclusive cross sections and up to NLO for more exclusive cross sections.

2. The program SusHi

SusHi combines higher-order QCD results for Higgs production in gluon fusion and heavy-quark annihilation, with model rescaling and optional electroweak corrections.

  • SusHi consistently combines literature results through N3LO for gluon fusion and bottom-quark annihilation in the MSSM.
  • N3LO gluon-fusion contributions have recently become available in the heavy-top effective theory, alongside known NLO general-mass and NNLO heavy-top results.
  • Soft-gluon resummation addresses effects beyond fixed order but is not included in SusHi.
  • SM results can be used in the 2HDM, MSSM, and NMSSM through appropriate Yukawa-coupling rescaling, with additional squark contributions in supersymmetric models.
  • Bottom-quark annihilation is especially relevant when bottom Yukawa couplings are enhanced, such as at large tan β, and its inclusive cross section is implemented at NNLO.

3. Higgs production through gluon fusion

SusHi 1.6.0 combines exact low-order mass dependence with heavy-top higher-order approximations, soft expansions, top-mass corrections, high-energy matching, and faster renormalization-scale evaluation for gluon-fusion Higgs production.

  • Cross-section construction: SusHi convolves partonic cross sections with parton densities to construct the hadronic gluon-fusion cross section.The calculation includes quark-initiated channels and uses order-specific PDF evaluations.
  • Cross-section construction: The exact NLO result retains top-, bottom-, and charm-mass dependence, while higher-order terms use the large-top-mass approximation with electroweak corrections.SusHi offers an alternative electroweak treatment in beyond-Standard-Model scenarios because full factorization may not be justified.
  • Soft expansion: The soft expansion describes CP-even gluon-fusion cross sections through N3LO and is available through order (1−x)16, while exact x-dependence remains available through NNLO.The heavy-top approximation formally loses validity at sufficiently high partonic energies, motivating the threshold expansion.
  • N3LO terms: N3LO QCD corrections are implemented through the soft expansion, including factorization- and renormalization-scale-dependent terms.The implementation uses order-(1−x)16 terms; higher terms were found not to change the result within the associated uncertainty.
  • Top-quark mass effects: Subleading inverse-top-mass terms through NNLO are included for CP-even Higgs production to study gluon-fusion theoretical uncertainties.These settings are controlled by the GGHMT block, while top-mass-suppressed terms are unavailable at N3LO.
  • High-energy matching: High-energy matching merges the soft expansion with the x→0 limit, changing the final Standard Model Higgs cross section by about 0.5%.The matched curve approaches the high-energy behavior smoothly, although unknown coefficients introduce theoretical uncertainty.
  • Renormalization-scale dependence: SusHi provides a faster renormalization-scale evaluation, but its N3LO scale dependence inherits approximations made at NNLO and requires on-shell non-strong-coupling parameters.The NNLO renormalization-scale dependence is exact under the described procedure.

4. Heavy-quark annihilation

SusHi 1.6.0 adds inclusive Higgs-production cross sections through arbitrary heavy-quark annihilation, supporting neutral and charged Higgs states with configurable perturbative orders and scales.

  • The QQH block activates inclusive NNLO cross sections for heavy-quark annihilation, Q′Q̄ → φ.Its presence selects the heavy-quark-annihilation calculation instead of gluon fusion.
  • Initial- and final-state quark flavors are specified explicitly, while the Q′Q̄φ coupling is given in the MS scheme at a chosen scale.The coupling is normalized so the Standard Model q̄qH value equals m_q(μ)/GeV.
  • The implementation supports rapidity and transverse-momentum cuts, including pT distributions, through the DISTRIB settings.These distributions are available up to O(α^3_s).
  • Because all quarks are treated as massless, scalar and pseudoscalar Higgs results coincide and the SUSHI(2) setting is irrelevant.The same assumption makes the underlying theory chirally symmetric.

5. Numerical results

The numerical results examine soft-expansion convergence, top-mass effects, scale dependence, and dimension-5 operators for Higgs production at 13 TeV. The SM Higgs prediction reaches 45.80 pb with high-energy matching and a renormalization-scale uncertainty of ±1.87 pb.

  • Soft expansion up to N3LO: At NLO, a = 0 gives deviations below 2.5% for N ≥9 and 1.3% at N = 16, while a = 1 remains over 7% off.These deviations refer to the correction term's exact x-dependence.
  • Soft expansion up to N3LO: At NNLO, a = 0 (a = 1) approximates the correction term within 5% (2%), corresponding to about 0.9% (0.3%) for the total cross section.Terms beyond (1−x)^6 have no significant impact for either parameter choice.
  • Soft expansion up to N3LO: Above N = 11 at N3LO, the spread among a = 0, 1, 2, 3 is about 3% of δσ/σ, or roughly 0.1% of the total cross section.The comparison is also shown for the dominant gg channel alone.
  • Top-quark mass effects through NNLO and matching to the high-energy limit: At NLO, the soft expansion converges to the exact result for a = 0, while at NNLO available terms support permille-level accuracy when a = 0 is reliable.For N3LO with high-energy matching, convergence is slightly worse than without matching but behaves similarly across orders.
  • Top-quark mass effects through NNLO and matching to the high-energy limit: Matching to the high-energy limit affects NLO and NNLO results more than genuine 1/M_t terms, and top-mass effects are bounded at about 1% of the heavy-top limit.Including higher-order mass terms can move the NLO approximation away from the exact result after matching.
  • Cross section prediction for the SM Higgs boson and scale dependence: 45.80 pb ± 1.87 pb(µR) is obtained with matching at x →0 through NLO, NNLO, and N3LO for the SM Higgs setup.The uncertainty includes only renormalization-scale dependence, defined over µR/MH ∈ [1/4, 1] relative to µR = MH/2.
  • Cross section prediction for the SM Higgs boson and scale dependence: Renormalization-scale dependence decreases successively from NLO to N3LO, while the flat behavior near µR = MH/2 motivates symmetrizing the uncertainty band.The RGE procedure provides scale dependence at N^nLO with the precision of the N^{n−1}LO calculation.
  • Dimension 5 operators: Higher pT cuts increasingly break the inclusive-cross-section degeneracy between κt and c5,H, while point-like interactions distort pT shapes relative to massive-quark loops.The distortion is stated for the dimension-5 interaction relative to top- and bottom-quark induced contributions.

6. Conclusions

SusHi 1.6.0 extends precision predictions for Higgs production through improved gluon-fusion treatments, scale-uncertainty evaluation, dimension-5 operators, and arbitrary heavy-quark annihilation. The paper demonstrates these capabilities for 125 GeV and 750 GeV scalars.

  • SusHi 1.6.0 provides soft-threshold expansions and high-energy matching for CP-even gluon-fusion production through N3LO QCD.Top-quark mass effects beyond the infinite-mass limit are included at NLO and NNLO.
  • The code simultaneously calculates renormalization-scale uncertainty for CP-even and CP-odd Higgs gluon-fusion cross sections.
  • Dimension-5 operator effects can be studied in every model supported by SusHi, including new CP-even and CP-odd scalars.For a 125 GeV Higgs, large transverse momentum can break the degeneracy between top-mass and point-like dimension-5 contributions.
  • The perturbative series converges for inclusive gluon-fusion production of a 750 GeV scalar at the 13 TeV LHC.

Appendix A. Example input

The appendix documents SusHi input blocks for model selection, perturbative orders, soft expansions, scale variations, dimension-5 operators, top-mass effects, and heavy-quark annihilation.

  • Model and calculation selection: Block SUSHI selects the physics model, Higgs state, collider energy, perturbative orders, and electroweak contributions.The documented model choices include the SM, MSSM, 2HDM, and NMSSM.
  • Gluon-fusion improvements: Block GGHMT controls top-quark mass expansions at LO, NLO, and NNLO across individual partonic channels.The example retains the first four terms in the 1/Mt expansion beyond LO and can match to the high-energy limit.
  • Gluon-fusion improvements: Block GGHSOFT activates the threshold soft expansion at NLO, NNLO, or N3LO through order (1 − x)^16 with a = 0.
  • Scale dependence: Block SCALES sets central renormalization and factorization scales and defines the intervals used for renormalization-scale uncertainty.In the example, the renormalization scale varies within µR ∈ [0.5, 2] · 0.5 · MH.
  • Dimension-5 operators: Block DIM5 controls perturbative running and the coefficients of dimension-5 operators coupling a CP-even Higgs boson to gluons.DIM5(0)=1 evolves the operator perturbatively, while DIM5(11)=1 fixes its lowest-order contribution at Mφ.
  • Heavy-quark annihilation: Block QQH specifies incoming heavy-quark partons, their Yukawa coupling, and the input scale for heavy-quark annihilation.The example computes c cbar → H in the SM.
  • Extended models: The appendix also documents 2HDM and NMSSM settings, including Higgs-state labels, masses, mixing parameters, and model-specific inputs.The 2HDMC link supports λ-basis, physical-basis, and H2-basis input.
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