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SusHi: A program for the calculation of Higgs production in gluon fusion and bottom-quark annihilation in the Standard Model and the MSSM
Robert V. Harlander, Stefan Liebler, Hendrik Mantler
TL;DR
SusHi addresses the need for precise Higgs-production predictions by calculating gluon-fusion and bottom-quark-annihilation cross sections in the SM and MSSM. It implements inclusive and differential higher-order QCD calculations with supersymmetric, electroweak, renormalization, and Higgs-mass interfaces. The program supports inclusive NNLO-level results, NLO differential predictions, and MSSM effects including squarks, gluinos, and tan β-enhanced bottom corrections.
Problem
Precise predictions for Higgs production are needed to identify the new LHC boson, including MSSM gluon-fusion and bottom-quark-annihilation processes.
Method
SusHi calculates gluon-fusion and bottom-quark-annihilation cross sections using higher-order QCD and electroweak contributions, selectable renormalization schemes, tan β-enhanced resummation, and optional FeynHiggs mass calculations.
Results
SusHi evaluates inclusive cross sections, differential distributions, and Higgs-momentum cuts in the SM and MSSM, including squark and gluino effects.
Takeaways & Limitations
The program provides a unified tool for dominant neutral-Higgs production mechanisms with full NLO QCD and partial NNLO and electroweak corrections.
Takeaways & Limitations
NNLO corrections are available only for total inclusive cross sections; only top-(s)quark effects are included fully, while top-squark effects are approximate.
Abstract
from arXiv · showhide
This article describes the code SusHi (for "Supersymmetric Higgs") which calculates the cross sections $pp/p\bar{p}\rightarrowφ+X$ in gluon fusion and bottom-quark annihilation in the SM and the MSSM, where $φ$ is any of the neutral Higgs bosons within these models. Apart from inclusive cross sections up to NNLO QCD, differential cross sections with respect to the Higgs' transverse momentum $p_T$ and (pseudo-)rapidity $y(η)$ can be calculated through NLO QCD. In case of gluon fusion, SusHi contains NLO QCD contributions from the third family of quarks and squarks, NNLO corrections due to top-quarks, approximate NNLO corrections due to top-squarks, and electro-weak effects. It supports various renormalization schemes for the sbottom sector and the bottom Yukawa coupling, as well as resummation effects of higher order $\tanβ$-enhanced sbottom contributions. SusHi provides a link to FeynHiggs for the calculation of the Higgs masses.
1. Introduction
SusHi is a program for calculating Higgs production through gluon fusion and bottom-quark annihilation in the SM and MSSM. It combines higher-order QCD and electroweak contributions with inclusive, differential, and cut-dependent predictions.
- Precise Higgs production predictions are important for identifying the new boson observed at the LHC.
- In the SM, gluon fusion is the main Higgs production mechanism and its inclusive cross section is known through NNLO QCD.Electroweak contributions reach up to 8% relative to the LO cross section.
- In the MSSM, gluon fusion receives contributions from quarks and squarks, with squark effects generally most relevant for small to moderate squark masses.Squark contributions to the CP-odd Higgs A are absent at LO.
- For large tan β, enhanced bottom-Higgs couplings increase the importance of bottom-sector contributions to gluon fusion and bottom-quark-associated production.Bottom-quark annihilation resums ln(m_b/m_φ) terms through b-parton distribution functions and is available through NNLO QCD in the SM.
- SusHi implements complete third-generation NLO QCD contributions for gluon fusion, including quark, squark, and genuine supersymmetric virtual corrections.
- The program provides inclusive and differential predictions, kinematic p_T and rapidity cuts, multiple sbottom and bottom-Yukawa renormalization schemes, and tan β-enhanced resummation.Its distinctive feature is full NLO QCD with partial NNLO and electroweak corrections for the dominant neutral-Higgs production mechanisms in the MSSM.
2. Physics background
SusHi’s physics background defines the SM and MSSM Higgs and squark sectors, their couplings and mass parameters, and the renormalization choices used for stop and sbottom contributions. The sbottom treatment determines on-shell masses and mixing through a dependent-parameter scheme, while allowing alternative renormalization schemes in the gluon-fusion amplitude.
- Standard Model: The SM Higgs sector is specified by the vacuum expectation value v ≈246 GeV and Higgs mass mH, with fermion Yukawa couplings Yf = 2mf/v.
- Supersymmetry: The MSSM has two Higgs doublets, producing h, H, A, and H±, with the lowest-order spectrum determined by SM parameters, tan β, and mA.SusHi can optionally obtain two-loop MSSM Higgs-mass corrections through FeynHiggs, or accept Higgs masses as input.
- Supersymmetry: The angles α and β determine normalized Higgs–fermion couplings, while third-generation squarks enter through soft-breaking parameters, µ, quark masses, and electroweak charges.
- Squark sector: Squark mass eigenstates and mixing angles are obtained by diagonalizing the squark mass matrix, with the convention 0 ≤θq < π ordering m˜q1 < m˜q2.For the stop sector, SusHi expresses At in terms of on-shell top and stop masses and the stop mixing angle before renormalization.
- Sbottom renormalization: Sbottom renormalization is more subtle because replacing Ab before renormalization can produce potentially large corrections proportional to αsµ2 tan2 β/m˜g.The scheme instead uses Eq. (7) to eliminate mb, while sbottom masses are renormalized on shell and Ab may be defined through an A˜b1˜b2-vertex condition.
- Sbottom renormalization: After determining on-shell sbottom masses, SusHi permits several renormalization schemes for mb, Ab, and θ˜b, with either mb or Ab required to be dependent; scheme changes affect counterterms and parameter values.The bottom mass in the Higgs-bottom Yukawa coupling is renormalized independently of the sbottom-sector scheme.
3. Cross section for gluon fusion
SusHi computes gluon-fusion cross sections for neutral Higgs bosons in the SM and MSSM, incorporating quark, squark, QCD, and electroweak contributions across several perturbative orders. Its higher-order treatment combines exact or analytic ingredients with heavy-mass approximations and user-selectable electroweak prescriptions.
- NLO QCD contributions: SusHi formulates the hadronic gluon-fusion cross section for all neutral MSSM Higgs bosons at NLO QCD using LO normalization, virtual corrections, and real-radiation contributions.The real terms cover gg, gq, and q q̄ channels, while the strong coupling uses five active quark flavors.
- NLO QCD contributions: NLO gluon fusion includes third-generation quark and squark effects, with quark-induced terms implemented analytically and supersymmetric contributions combined from squark and mixed quark-squark-gluino diagrams.SusHi uses expansions in two heavy-mass limits for these amplitudes.
- NLO QCD contributions: The heavy-mass expansions apply to complementary regimes: mφ ≪ mq, m˜q1, m˜q2, m˜g or mφ, mq ≪ m˜q1, m˜q2, m˜g.The first is used for light Higgs production in the top-stop sector, while the second covers the bottom-sbottom sector and heavy-Higgs top-stop cases.
- NNLO corrections: NNLO corrections are available only for the total inclusive cross section, with top-quark effects included and top-squark effects treated approximately.The NNLO implementation uses effective-theory programs and approximations for top-squark and mixed top-stop-gluino contributions.
- Electroweak corrections: Electroweak corrections are incorporated through alternative combination prescriptions, while CP-odd Higgs electroweak corrections are not included because they are unknown.For a SM-like Higgs with mφ < 2mt, the available prescriptions give comparable NLO results.
4. Cross section for bottom-quark annihilation
SusHi evaluates bottom-quark annihilation in the MSSM using five-flavor-scheme results, which resum logarithms through bottom-quark PDFs and are reweighted by the appropriate supersymmetric bottom Yukawa coupling. It provides inclusive NNLO and differential NLO predictions.
- Motivation: For large tan β, enhanced bottom-Higgs couplings can make bottom-quark annihilation comparable to or more important than gluon fusion in the MSSM.The same coupling enhancement increases associated production with bottom quarks.
- Production schemes: The five-flavor scheme resums ln mb/mφ terms through bottom-quark PDFs and describes the leading process as b b̄ → φ.The four-flavor scheme instead starts from gg → b b̄ φ.
- Implementation: SusHi obtains the inclusive NNLO QCD prediction from bbh@nnlo and reweights it with the resummed supersymmetric bottom coupling.The implementation supports associated b b̄φ production in the five-flavor scheme.
- Implementation: Differential bottom-annihilation cross sections include NLO virtual corrections for b b̄ → φ and LO real-radiation processes b b̄ → gφ and bg → bφ combined with dipole subtraction.The resulting cross sections are multiplied by resummed supersymmetric couplings for MSSM predictions.
5. Differential cross sections
SusHi supports differential Higgs-production predictions and kinematic cuts in transverse momentum, rapidity, and pseudo-rapidity. These non-inclusive results are available through NLO, subject to perturbative and numerical-precision constraints.
- Kinematic cuts: SusHi applies upper and lower cuts on the Higgs transverse momentum pT, rapidity y, and pseudo-rapidity η for gluon fusion and bottom-quark annihilation.The variables are defined in the hadronic reference frame.
- Differential observables: For gluon fusion, SusHi provides dσ/dpT, dσ/dy, and d2σ/(dpTdy), or corresponding distributions using η instead of y.Rapidity and pseudo-rapidity distributions are treated symmetrically about zero.
- Perturbative accuracy: Non-inclusive predictions are calculated through NLO, at O(αs^3) for gluon fusion and O(αs) for bottom-quark annihilation.For bottom-quark annihilation, the Higgs transverse momentum is always pT = 0.
- Validity and numerical precision: Reliable fixed-order differential results require pT/mφ ≳ 0.1 because smaller pT values can introduce large logarithms and spoil perturbative convergence.Differential calculations also require higher numerical integration precision than inclusive cross sections.
6. The program SusHi
SusHi is a Fortran program that calculates Higgs-production cross sections in the SM and MSSM through configurable perturbative orders and kinematic selections. Its workflow accepts a single input file, processes model and renormalization choices, computes the requested cross sections, and writes screen and file outputs.
- SusHi calculates inclusive Higgs-production cross sections in gluon fusion and bottom-quark annihilation for the SM and MSSM.
- SusHi processes a single SLHA-inspired input file, initializes parameters, applies renormalization and tan β-enhanced bottom-Yukawa resummation, then computes requested perturbative orders.
- External components include FeynHiggs for MSSM Higgs masses and LHAPDF for selectable PDF sets.FeynHiggs is optional for Higgs-mass calculation, whereas LHAPDF is required for PDF selection.
- Compilation uses the provided Makefile and configure script, with library paths for LHAPDF and optionally FeynHiggs specified before building.
- The input specifies the model, Higgs type, collider energy, perturbative orders for gluon fusion and bottom-quark annihilation, and electroweak options.
- The program supports total cross sections, gluon-fusion distributions, and kinematic cuts involving Higgs transverse momentum and rapidity.Differential cross sections are available only for gluon fusion, while cuts can be applied to both production mechanisms.
7. Conclusion
The conclusion presents SusHi as a tool for Higgs-production calculations in the SM and MSSM. It combines inclusive and differential predictions with higher-order corrections, supersymmetric contributions, renormalization choices, and FeynHiggs integration.
- SusHi calculates Higgs-production cross sections in gluon fusion and bottom-quark annihilation at hadron colliders in both the SM and MSSM.
- The program evaluates inclusive cross sections, distributions, and kinematical cuts on the Higgs 4-momentum.
- SusHi includes higher-order QCD and electroweak corrections together with squark and gluino effects.
- Users can choose renormalization schemes for the sbottom sector and bottom Yukawa coupling, including resummation of tan β-enhanced effects.
- SusHi can be linked to FeynHiggs to calculate MSSM Higgs masses.
Appendix A. Formulas: Higgs-squark couplings
The appendix specifies the quark and squark couplings of the three neutral MSSM Higgs bosons implemented in SusHi. It organizes the squark couplings into components and relates the heavy-Higgs expressions to the light-Higgs formulas.
- The appendix presents the couplings of the three neutral MSSM Higgs bosons to quarks and squarks implemented in SusHi.
- The Higgs-quark couplings are expressed relative to the corresponding SM Higgs-boson coupling.
- The squark couplings to the light and heavy Higgs bosons are split into components indexed by squark states.
- For the light Higgs h, the appendix gives the implemented coupling formulas using abbreviations sx = sin x and cx = cos x.
- The heavy-Higgs H couplings follow from the preceding formulas through the replacement α → α − π/2.
- The CP-odd Higgs A has separately specified couplings, with mb partly interpreted as the bottom mass within Higgs-sbottom couplings.