Source-linked AI summary
WHIZARD: Simulating Multi-Particle Processes at LHC and ILC
Wolfgang Kilian, Thorsten Ohl, Jurgen Reuter
TL;DR
Collider studies need automatic simulation of increasingly complex processes while controlling off-shell effects, backgrounds, and radiative contributions. WHIZARD addresses this with automatic tree-level matrix-element generation, adaptive integration, flexible event simulation, and broad model and collider support. The program handles high-multiplicity processes, including W+4 jets straightforwardly and W+5 jets with slightly more effort, while retaining complete matrix elements and correlations.
Problem
Collider analyses require off-shell multi-particle amplitudes, radiative effects, and background control beyond the capabilities of fixed libraries for simple processes.
Method
WHIZARD automatically constructs tree-level matrix elements, represents them efficiently with O’Mega, integrates phase space adaptively with VAMP, and generates events through modular interfaces.
Results
WHIZARD simulates high-multiplicity Standard Model processes, including W+4 jets straightforwardly and W+5 jets with slightly more effort.
Takeaways & Limitations
WHIZARD provides a universal framework for complete tree-level collider event simulation with model coverage, correlations, and interfaces for realistic event production.
Takeaways & Limitations
Adaptive sampling is statistically not guaranteed to reach an optimum, with fluctuations becoming overwhelming when the number of channels is too large.
Abstract
from arXiv · showhide
We describe the universal Monte-Carlo event generator WHIZARD. The program automatically computes complete tree-level matrix elements, integrates them over phase space, evaluates distributions of observables, and generates unweighted event samples that can be used directly in detector simulation. There is no principal limit on the process complexity; using current hardware, the program has successfully been applied to hard scattering processes with up to eight particles in the final state. Matrix elements are computed as helicity amplitudes, so spin and color correlations are retained. The Standard Model, the MSSM, and many alternative models such as Little Higgs, anomalous couplings, or effects of extra dimensions or noncommutative SM extensions have been implemented. Using standard interfaces to PDF, beamstrahlung, parton shower and hadronization programs, WHIZARD generates complete physical events and covers physics at hadron, lepton, and photon colliders.
1 The Need for Multi-Particle Event Generators
Modern collider studies require event generators that handle off-shell, multi-particle processes and radiative effects with greater precision than earlier tools. WHIZARD addresses these needs through automatic matrix-element construction, adaptive phase-space integration, and complete event-generation interfaces.
- Precision collider analyses require off-shell amplitudes, multi-particle processes, radiative corrections, and matrix-element/parton-shower matching.
- Earlier tools such as CompHEP and GRACE supported processes of limited complexity, while MadGraph introduced automatic multi-particle amplitude generation.
- WHIZARD combines the O’Mega matrix-element generator, VAMP adaptive multi-channel integration, and dedicated phase-space parameterizations.
- The program supports external shower and hadronization codes through the Les Houches interface or can internally call PYTHIA with matching.
2 Physics Simulation with WHIZARD
WHIZARD is a general-purpose tree-level collider simulator that automatically generates matrix elements, integrates processes, and produces analyzable events across supported physics models and collider environments. Its architecture combines flexible scripting, efficient helicity-amplitude generation, factorization, and interfaces to external physics software.
- WHIZARD computes tree-level cross sections and observable distributions and generates simulated event samples for hard scattering and decays.
- Supported physics includes the Standard Model, MSSM, Little Higgs models, anomalous couplings, and collider processes using PDFs and external QCD programs.
- For any tree-level process in the selected model, O’Mega generates helicity amplitudes as needed rather than relying on a fixed process library.
- WHIZARD 2 adds matrix-element factorization into production and decay subprocesses while retaining leading-order color and quantum-level spin correlations.
- SINDARIN unifies parameter specification, cuts, observables, reweighting, integration, simulation, and analysis workflows.
3 Checks and Applications
WHIZARD’s checks and applications demonstrate reliable complete tree-level calculations for multi-particle Standard Model processes and extend this capability to supersymmetric, Little Higgs, and exotic models. Its applications retain off-shell effects and non-resonant backgrounds in collider studies.
- Standard Model checks: WHIZARD reliably computes Standard Model partonic cross sections with four, six, or more final-state particles.Its results have been checked through internal tests and published physics studies.
- Standard Model checks: Complete tree-level four-fermion and six-fermion cross sections agree with analytic calculations and with LUSIFER after accounting for unstable-particle treatments.The cross-checks cover W-pair production and an exhaustive set of e+e− six-fermion processes.
- Supersymmetry: Full matrix-element simulations enabled SUSY processes with six final-state particles, revealing off-shell effects, interference deviations from the narrow-width approximation, and ISR-jet combinatorial backgrounds.One studied process involved more than 30,000 Feynman diagrams, several thousand phase-space channels, and 22 color flows.
- Beyond the Standard Model: WHIZARD was the first arbitrary multi-leg generator to implement the NMSSM and the first multi-particle event generator to implement Little Higgs models.The Little Higgs model library includes Littlest Higgs and Simplest Little Higgs variants and supports studies at the LHC, ILC, and photon collider.
- Beyond the Standard Model: Complete tree-level matrix elements including vector-boson decays, angular correlations, and non-resonant backgrounds enabled less restricted ILC sensitivity estimates for anomalous couplings.WHIZARD also supports phenomenology of noncommutative Standard Model extensions, with care required for effective-vertex order counting.
4 O’Mega: Optimized Matrix Element Generator
O’Mega addresses the computational growth of complex tree-level amplitudes by constructing optimized symbolic representations that eliminate repeated subexpressions. Its helicity-amplitude and color-handling strategies make multi-particle calculations more efficient while preserving gauge cancellations and correlations.
- Requirements: Helicity amplitudes are preferred for complex processes because trace techniques become inefficient beyond 2 →4 processes, while numerical evaluation is efficient for massive particles.They also allow polarized initial and final states.
- Requirements: Gauge-cancellation failures can create large numerical uncertainties, so efficient grouping of terms is essential for realistic quantum field theories.The relevant cancellations occur among terms in perturbative scattering-amplitude expressions.
- Complexity: For 2 →6 processes, independent poles are two orders of magnitude fewer than Feynman diagrams, motivating complexity measures based on independent momenta.The φ3 comparison is summarized in Table 3.
- Complexity: The symbolic representation can approach the lower-bound complexity set by independent momenta, making amplitudes more compact than direct sums over Feynman diagrams.O’Mega constructs an optimally factored symbolic amplitude and later translates it into Fortran.
- Architecture: Compilation of symbolic Fortran code provides a two- to four-fold execution-speed increase, depending on the process and model.O’Mega also supports human-readable code editing for exotic models, radiative corrections, and form factors.
- Architecture: O’Mega exploits common subexpressions and represents amplitudes as directed acyclic graphs, with each subexpression appearing only once.Constructing a minimal DAG after forming the full diagram tree is impractical because expressions grow combinatorially.
- Architecture: O’Mega constructs DAGs from model topologies, derives flavorings from Feynman rules, and can generate amplitudes for all color flows together.The algorithm handles arbitrary vertex degree and enforces identical-particle symmetry properties.
5 The WHIZARD Architecture
WHIZARD combines matrix-element generation with phase-space integration, event sampling, physics-model support, beam descriptions, and external simulation interfaces. Its modular architecture links self-contained libraries and dynamically generated process code into an automated collider event generator.
- Core architecture: WHIZARD supplements O’Mega with phase-space parameterization, channel selection, integration with cuts, and exclusive unweighted-event generation.These components promote a matrix-element generator to a complete automated physics simulation program.
- External interfaces: WHIZARD interfaces with beam dynamics, showering, hadronization, event-file processing, detector simulation, and analysis tools.Its top-level routines combine these parts according to user-provided input, and a lightweight module plots observable distributions.
- Software structure: The core library is organized into modules, with additional libraries and process-code libraries typically generated and dynamically linked at runtime.O’Mega remains a standalone program that WHIZARD invokes to generate, compile, and organize process code.
- Core library: The architecture includes modules for physics models, helicity, color, flavor, correlated quantum numbers, particle data, beams, phase space, and event handling.Beam support includes beamstrahlung, initial-state radiation, EPA, and LHAPDF interfaces.
- Integration: VAMP is a self-contained multi-channel extension of VEGAS for integrating arbitrary real functions over a multidimensional hypercube.It has no dependence on scattering-specific physics objects such as four-vectors.
- Physics inputs: WHIZARD can access all installed LHAPDF structure functions and the corresponding αs function, but its leading-order framework limits the applicability of NLO structure functions.Matching structure functions and the hard process is therefore possible within the program’s leading-order scope.
6 Algorithms
WHIZARD combines automatic color treatment, multi-channel phase-space integration, and event construction to simulate complex collider processes while retaining complete matrix elements. Its algorithms are robust for signal-like applications, though adaptive sampling and some background-like cases have explicit limitations.
- Color treatment: WHIZARD represents color through a U(3) formulation with subtracted ghost diagrams, making color factors trivial while preserving the exact SU(3) squared matrix element.Color connections are obtained by dropping ghost diagrams.
- Phase-space integration: Multi-dimensional Monte Carlo integration is required for typical 2 →6 processes because their integration dimension exceeds 10 and can reach 20.The statistical error scales as c/N^0.5, so reducing the integration constant is central to performance.
- Phase-space integration: VAMP handles many phase-space channels by adapting channel weights to the squared-graph contributions and dropping unimportant channels after each iteration.This addresses the simultaneous presence of many, sometimes thousands, of Feynman graphs.
- Performance: For signal-like processes, WHIZARD includes all Feynman graphs in the matrix element while using dominant graphs only to guide integration.The resulting procedure is reported as stable after tuning and covers practical applications.
- Performance: In typical electroweak 2 →6 applications, 10^6 events can yield relative errors in the per-mil range, while reweighting efficiencies are usually per-mil to percent and can reach 10 % for favorable signal-like cases.The adaptation, integration, and event-generation costs average out at these efficiencies.
- Limitations: Adaptive sampling is not guaranteed to reach an optimum, especially when the number of channels becomes too large, although WHIZARD reports sufficient robustness for its target processes.The limitation is statistical and depends on the number of events and iterations.
7 User Interface
WHIZARD is controlled through SINDARIN, a domain-specific programming language for configuring models, processes, beams, parameters, cuts, integration, and simulation. Its interface supports generic syntax processing and model declarations, including a library of supported models and an extensible template.
- Installation: WHIZARD is distributed with auxiliary packages including O’Mega, VAMP, and CIRCE, and is built using Fortran and O’Caml compilers.Hadron-collider applications may also require LHAPDF, while HepMC or STDHEP support is optional when needed.
- SINDARIN: SINDARIN scripts specify the model, processes, beams, parameters, cuts, and Monte Carlo operations, and can run interactively or from files.The language includes commands such as integrate and simulate.
- SINDARIN: SINDARIN provides typed variables, particle aliases, subevents, expressions, conditional execution, file inclusion, visualization constructs, and parameter-scanning loops.These features support both Monte Carlo steering and event analysis.
- Language implementation: The generic lexer, top-down parser, and compiler transform SINDARIN input into evaluation trees and executable command objects.The same lexer and parser infrastructure also handles model-file and SLHA syntax.
- Physics models: WHIZARD supports the SM, MSSM, other beyond-Standard-Model theories, and a user-augmentable template for additional particles and interactions.The supported-model list includes SM relatives, simple subsets, the MSSM, and other BSM models.
8 Extensions and Extensibility
WHIZARD can incorporate external models, modify generated amplitudes, and include selected higher-order effects. Its extensibility reaches NLO lepton-collider calculations through external matrix elements, but automatic NLO support and QCD corrections beyond tree level remain unavailable.
- Model extensions: FeynRules models can be incorporated through WHIZARD interfaces, while manually adding a model requires coordinated edits to the WHIZARD model file and O’Mega driver.The model template provides the syntax structure for adding particles and interactions.
- Amplitude extensions: Generated O’Mega matrix-element code can be modified to replace propagators or couplings, add particle exchanges, or provide hooks for radiative corrections.Modified amplitudes must preserve the established vertex-and-propagator phase convention to maintain interference terms.
- Higher orders: WHIZARD accounts for lepton-collider radiation through an activatable ISR structure function, while complete one-loop calculations can be linked as external matrix elements.This approach has been used for chargino production with complete electroweak and supersymmetric corrections.
- Higher-order limitations: Automatic NLO implementation is not yet available, and the current version does not address QCD corrections beyond tree level for hadron colliders.The latter limitation is especially relevant because QCD radiation must be matched to hard processes and non-perturbative hadronization models.
9 Conclusions and Outlook
WHIZARD addresses the need for flexible, accurate simulation of complex collider processes by combining multi-particle event generation with redesigned QCD capabilities and a streamlined architecture. Its demonstrated LHC applications are tractable on current workstations, while the new framework supports further improvements in shower matching and higher-order automation.
- WHIZARD targets the flexibility and theoretical accuracy needed to investigate TeV-scale physics at the LHC and ILC.
- Version 2 builds QCD in the color-flow representation into WHIZARD from the ground up, extending the program beyond its ILC-oriented predecessor.
- Version 2 simplifies installation and usage through a streamlined architecture, a central executable, and a single physics-oriented analysis input file.
- W+4 jet processes are straightforward to simulate, while W+5 jets are possible with slightly more effort; computing time and memory rise roughly tenfold per added jet.
- Current-workstation calculations remain tractable with a few GB of memory, several days of integration, and additional CPU time for unweighted samples corresponding to 1 fb^-1 of LHC luminosity.
- The new framework is intended to support improved matrix-element/parton-shower matching and automated incorporation of higher perturbative orders, although automated NLO event generation remains less mature than the LO case.
A Conventions
WHIZARD's helicity-amplitude conventions are implemented through omegalib but can be replaced for specialized applications.
- WHIZARD uses conventions for helicity amplitudes and polarized off-shell wave functions that are implemented in omegalib.
- The conventions can be replaced by another set for specialized applications.
A.1.1 Dirac and Majorana fermions
The appendix specifies spinor, polarization, and gravitino-wavefunction conventions used to construct and constrain the relevant helicity states. It emphasizes consistency of quantization axes and distinguishes the conventions used for different fermion implementations.
- Dirac and Majorana fermions: Two-component Weyl spinors are used to construct four-component Dirac or Majorana spinors.
- Dirac and Majorana fermions: Conjugated spinors for purely Dirac fermions are constructed directly, whereas mixed Dirac/Majorana implementations use the charge-conjugation matrix.
- Polarization vectors: The spin-1 conventions are similar to HELAS but do not fit the spinor-wavefunction definitions because they correspond to a different angular-momentum quantization axis.
- Polarization vectorspinors: Correct spin-3/2 states require the same quantization axis for spin-1/2 and spin-1 components.
- Polarization vectorspinors: The resulting gravitino wavefunctions obey the Dirac equation, are transverse, and satisfy the stated irreducibility condition.
- Polarization vectorspinors: The polarization vectors used for constructing these spin-3/2 states are specified explicitly in the convention formulas.
A.1.4 Polarization tensors
The appendix defines spin-2 polarization tensors through their symmetry and trace properties and identifies the polarization-vector conventions used to construct them.
- Spin-2 polarization tensors are symmetric, transversal, and traceless.
- The allowed spin projections are m = −2, −1, 0, 1, 2.
- The construction uses the polarization vectors defined in appendix section A.1.2.
A.2 Propagators
The section specifies propagator and vertex conventions used in WHIZARD, including gauge choices for spin-1 fields and momentum-flow conventions for fermions.
- Fermionic lines use negative momentum because all momenta are treated as outgoing and charge flow is opposite to momentum.
- Spin-1 propagators are provided for massive fields in unitarity and Rξ gauges and for massless fields in Feynman gauge.
- Table 5 lists Fortran functions for fermionic vector and axial currents, plus scalar and pseudoscalar currents.
- Fermionic vertices use the chiral representation employed in HELAS.
B Sample Matrix Element Code
The sample Fortran code generated by O’Mega computes the e+e−→µ+µ− Standard Model amplitude through helicity loops, wavefunctions, fusions, and bracket calculations.
- The example implements the e+e−→µ+µ− process in the Standard Model as process-specific Fortran code compiled and linked with WHIZARD.
- The generated code is equivalent to the sum of Feynman diagrams because this process has no common subexpressions.
- The decomposition into numbered subroutines speeds compilation, which matters for large process codes.
- The calculate_amplitudes subroutine accepts amplitudes, momenta, and a mask, then initializes helicity-state and spin-state arrays.
- For each helicity configuration, the code constructs external fermion wavefunctions and invokes fusion and bracket subroutines.