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HAWK 2.0: A Monte Carlo program for Higgs production in vector-boson fusion and Higgs strahlung at hadron colliders
Ansgar Denner, Stefan Dittmaier, Stefan Kallweit, Alexander Mück
TL;DR
Precision predictions for Higgs production in vector-boson fusion and Higgs strahlung require accurate treatment of QCD, electroweak, and off-shell effects. HAWK addresses this with Monte Carlo calculations of fully differential cross sections and reweighting factors, while supporting only a limited Higgs-decay treatment in its current version.
Problem
Precision predictions are needed for Higgs production channels used to measure Higgs properties and test Standard Model expectations.
Method
HAWK uses Monte Carlo integration to calculate complete NLO QCD and electroweak predictions, differential distributions, and electroweak reweighting factors.
Results
HAWK provides hadron-collider predictions for VBFH and VH production, including full NLO strong and electroweak corrections, integrated cross sections, and binned distributions.
Takeaways & Limitations
HAWK enables fully differential electroweak corrections to be combined with advanced QCD predictions through differential reweighting.
Takeaways & Limitations
The current version supports only Higgs decay into a pair of massless singlets without corrections to that decay.
Abstract
from arXiv · showhide
The Monte Carlo integrator HAWK provides precision predictions for Higgs production at hadron colliders in vector-boson fusion and Higgs strahlung, i.e. in production processes where the Higgs boson is Attached to WeaK bosons. The fully differential predictions include the full QCD and electroweak next-to-leading-order corrections. Results are computed as integrated cross sections and as binned distributions for important hadron-collider observables.
PROGRAM SUMMARY
HAWK 2.0 is a Fortran-based Monte Carlo program for precision Higgs-production cross sections and differential distributions at hadron colliders. It uses multi-channel integration of matrix elements with higher-order QCD and electroweak corrections, while imposing channel-specific decay restrictions.
- HAWK is written in Fortran 77 and Fortran 90 and runs on Linux or Mac OS with less than 1 GB of RAM.
- HAWK computes Higgs-production cross sections and differential distributions for vector-boson fusion and Higgs strahlung at the LHC.
- The program uses multi-channel Monte Carlo integration of Feynman-diagrammatic matrix elements with higher-order QCD and electroweak corrections.
- For vector-boson fusion, only Higgs decay into a pair of massless singlets is supported; for Higgs strahlung, decay products are unsupported, although the Higgs may be off shell.
- Meaningful results can typically be obtained within a few hours on a single core, with statistical uncertainty improving as the square root of runtime.
1. Introduction
The introduction motivates precision predictions for VBFH and Higgs strahlung and presents HAWK as a tool including complete channel diagrams and NLO QCD and electroweak corrections. Its predictions support differential studies and can be combined with advanced QCD calculations.
- Precision Higgs-production predictions are important for testing Higgs properties and searching for deviations from Standard Model expectations.
- HAWK includes the full s-, t-, and u-channel diagrams, all interferences, and corresponding NLO QCD and electroweak corrections for VBFH.
- Leptonic weak-boson decays make Higgs strahlung experimentally promising by providing trigger objects and enabling background suppression.
- HAWK provides fully differential electroweak corrections that can be combined by differential reweighting with advanced QCD predictions.
- The program calculates electroweak corrections with complex masses and combines real and virtual contributions using dipole subtraction.
2. EW corrections and off-shell effects in HAWK
HAWK carefully treats electroweak and off-shell effects, including infrared-safe photon handling and differential electroweak reweighting. Its current Higgs-decay implementation remains restricted to massless singlet decays without decay corrections.
- HAWK supports careful treatment of electroweak corrections and off-shell effects in its predictions.
- 2.1. αGF scheme: The αGF scheme derives the electromagnetic coupling from the Fermi constant, incorporating some higher-order effects at tree level.
- 2.3. Final-state photons: Infrared safety in VBFH is maintained by treating unresolved photons as partons in the jet algorithm and merging photon–jet pairs into jets.
- 2.3. Final-state photons: For charged-lepton final states, HAWK isolates photons from muons but recombines sufficiently collinear photons and electrons or positrons.
- 2.3. Final-state photons: HAWK supplies differential electroweak correction factors for reweighting QCD-based predictions beyond NLO, using approximate QCD–electroweak factorization.
- 2.5. Higgs-boson decays: The current version supports only Higgs decay into massless singlets without decay corrections, with the branching ratio supplied externally.
3.1. HAWK installation
HAWK is a stand-alone Fortran 77/90 program that can be compiled with standard Fortran compilers. LHAPDF is optional for up-to-date PDF sets, while two PDF sets are included for operation without it.
- Installation: HAWK is distributed as a stand-alone Fortran 77/90 code and has been tested on Linux and Mac OS.It has been successfully tested with several GNU, PGI, Intel, and OpenMPI compiler configurations.
- Installation: Running make in the HAWK-2.0 directory compiles the source and generates the hawk-2.0 executable.The source, executable, object, module, and sample-run files occupy separate distribution directories.
- PDF support: LHAPDF provides the interface for using up-to-date parton distribution functions but is not included in the HAWK distribution.The Makefile must enable includeLHAPDF and link to the selected LHAPDF installation.
- PDF support: Without LHAPDF, HAWK includes code and tables for the MRST2004QED and CTEQ6 PDF sets.The alternative configuration uses only the bundled implementations and tables.
3.2. HAWK execution
HAWK is run from an input file supplied through standard input, and it writes results to standard output or to a specified output file. Sample input files provide defaults and relevant parameters.
- Execution: HAWK requires an input file supplied through standard input before execution can begin.Parallel execution via MPI is discussed separately in Section 3.4.
- Execution: Results are written to standard output unless an output file is specified in the input file.The specified file stores the central results.
- Input files: The general input format is illustrated by the default inputfile inputdefault and the sample-run inputs.Only values differing from defaults need to be specified.
- PDF files: When LHAPDF is used, the relevant PDF files must be installed in LHAPDF’s PDFsets directory.For bundled PDF sets, the corresponding table files must instead be available in HAWK’s working directory.
3.3. HAWK input
HAWK input controls the collider process, event generation, physical parameters, jet and lepton treatment, phase-space cuts, and selected contributions. Defaults depend on the chosen process and cut scheme.
- Global parameters: Collider energy, collider type, Standard Model parameters, and output-file naming are specified through dedicated input values.The default centre-of-mass energy is 14000 GeV, and the listed masses and widths define the model inputs.
- Global parameters: The selprocess flag selects VBFH, W+H, W−H, or ZH production, while changing it resets options and cuts to process defaults.Hadronic Higgs strahlung requires selecting selprocess=0 and then changing the kinematic cuts.
- Global parameters: The default event count is 10000000 weighted events, while at least 10^7 events are recommended for reliable integration results and error estimates.Published distributions used 10^9 events.
- Physical parameters: Weak-boson input masses and widths are internally converted to pole values used in propagators, the complex weak mixing angle, and derived couplings.The output reports pole masses rather than the input on-shell masses.
- Recombination: HAWK constructs IR-safe VBFH observables with the kT jet algorithm, including photons as partons when electroweak corrections are included.For WH/ZH, no jet algorithm is applied because NLO final states contain at most one jet or photon.
- Recombination: Lepton–photon isolation or recombination is controlled by sbarelep and the D parameter, with photons recombined when Rγl < D.The separation Rγl is defined using rapidity and azimuthal-angle differences.
- Cuts: The scuts setting selects predefined cuts or no cuts, with VBFH-specific requirements including a minimum rapidity separation of 4.0 between the leading jets.VBFH jets may be ordered by transverse momentum or energy, depending on the selected scheme.
3.3.5. Off-shell Higgs boson, Higgs decays and corresponding cuts
HAWK offers configurable off-shell Higgs treatments, singlet decays, PDF choices, scale schemes, and switches for LO and NLO contributions. Its phase-space-dependent scale choices are not supported.
- Off-shell Higgs treatment: The shtr flag selects on-shell Higgs production, off-shell production, or off-shell production with an isotropic decay into two massless singlets.The singlet option is available only for VBFH.
- Off-shell Higgs treatment: For off-shell Higgs production, the width must be supplied or calculated internally, and electroweak corrections use an on-shell momentum projection required by gauge invariance.The shbw flag determines the off-shell propagator treatment.
- Higgs width and decay: The sgh flag selects the input width, an interpolated width, or a complex-pole-scheme width, with the complex-pole scheme as default.Hbr rescales the complete cross section as the branching ratio into singlets.
- Higgs width and decay: HAWK supports Higgs-decay-product cuts and an optional requirement that the products lie between the tagging jets in rapidity.The higgsbetweenjets flag controls this selection criterion.
- Parton distributions: LHAPDF can evaluate multiple PDF members for cross-section uncertainty calculations, but distributions are produced only for the central member.Using only LHpdf implementations sequentially avoids repeated-initialization runtime growth.
- Parton distributions: Photon-initiated channels require a photon PDF; otherwise their predicted contribution is zero.Photon PDFs are available for MRST2004QED or suitable LHAPDF sets such as the default NNPDF2.3QED.
- Scales and schemes: QCD and QED factorization schemes, scale rescaling factors, and default process-dependent scales are configurable.The default scales are MW for VBFH, MW+MH for WH, and MZ+MH for ZH.
- Scales and schemes: The current HAWK version does not support phase-space-dependent, or dynamical, scale choices.This is an explicit scope limitation of the scale configuration.
3.3.9. Anomalous HVV couplings
HAWK supports anomalous HVV couplings through parameterized inputs for W and Z bosons, with form factors and restricted QCD-correction settings for these extensions.
- Coupling inputs: HAWK enables anomalous HVV couplings for V = W, Z through the shvv switch, while retaining an optional Standard Model rescaling rsm.The default settings disable anomalous couplings and set rsm to one.
- Coupling inputs: The implementation relates anomalous coupling coefficients to d, db, dt, and dtb, or accepts direct a1hww, a2hww, and related inputs.The selected shvv mode determines which parameterization is supplied.
- Regulation and corrections: A virtuality-dependent form factor suppresses anomalous neutral-boson couplings at small momentum transfer to avoid infrared singularities.The form factor uses the two intermediate-boson virtualities and m0 = 1 GeV.
- Regulation and corrections: When anomalous couplings are enabled, selected QCD corrections are set to zero because they are unsupported for anomalous Higgs couplings.The text notes that these corrections are tiny in the Standard Model.
3.4. Parallel execution using the MPI standard
HAWK supports parallel Monte Carlo execution through MPI, with compiler configuration and a file-based input requirement for the parallel version.
- MPI configuration: MPI parallel execution is enabled by setting FC=$(MPIFC) in the Makefile, where MPIFC is the MPI Fortran compiler.The parallel version was tested with Intel Fortran 11.1 and SUN MPI 8.2.
- MPI configuration: The MPI version requires input in a file literally named inputfile beside the hawk-2.0 executable rather than standard input.This differs from the usual ./hawk-2.0 < inputfile invocation.
4. HAWK output and sample runs
HAWK reports integrated cross sections, corrections, and binned distributions, while providing configurable histograms and sample runs for VBFH and Higgs-strahlung processes.
- Output: HAWK writes selected-process information, Standard Model parameters, anomalous couplings, off-shell treatment, and Monte Carlo configuration details to its output.The configuration includes event counts, collider, PDFs, scales, contributions, corrections, and jet-algorithm information.
- Output: The central total-cross-section result is accompanied by Born, complete LO + NLO EW + NLO QCD, QCD, gluon, and electroweak contribution summaries.Cross sections are reported in femtobarns, with relative corrections normalized to the LO cross section.
- Differential distributions: With shisto=1, HAWK generates binned histograms for Higgs transverse momentum, rapidity, and pseudo-rapidity.Histogram ranges are listed at the end of the output and the distributions are written to files.
- Differential distributions: For VBFH, available distributions include off-shell Higgs invariant mass and leading-jet transverse momentum, rapidity, separation, azimuthal difference, and invariant mass.Leading and subleading jets are ordered by transverse momentum.
- Differential distributions: For Higgs strahlung, HAWK provides vector-boson transverse momentum and charged-lepton, neutrino, and missing-momentum distributions.The available observables include transverse momentum, rapidity, pseudo-rapidity, and azimuthal-angle differences, with analogous W−H distributions.
- Reweighting and customization: Differential electroweak corrections are obtained bin by bin as the ratio of virtual-plus-real EW to Born Monte Carlo averages, with incoming-photon contributions reported separately.Histograms can be adapted or extended by modifying the relevant settings subroutines.
- Sample runs: Sample runs cover on-shell and off-shell VBFH, W±H, ZH, and anomalous-HVV configurations using default inputs and cuts.The examples include lepton-decay and bare-lepton cases for Higgs-strahlung processes.
5. Conclusions
HAWK predicts Higgs production through weak vector-boson fusion and Higgs strahlung with complete NLO QCD and electroweak corrections, while supporting selected off-shell and anomalous-coupling extensions.
- Scope and accuracy: HAWK provides hadron-collider predictions for weak vector-boson fusion and Higgs strahlung, including the full NLO strong and electroweak corrections.In Higgs strahlung, leptonic W and Z decays and corresponding off-shell effects are included at this accuracy.
- Scope and accuracy: Off-shell Higgs effects can be simulated, but VBFH supports only Higgs decay into a pair of massless singlets.This restriction applies to the current version.
- Model scope: Predictions use the Standard Model and can optionally include anomalous Higgs–gauge-boson interactions.Generalizations to extended Higgs sectors are planned for future HAWK updates.