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Helac-Phegas: a generator for all parton level processes
Alessandro Cafarella, Costas G. Papadopoulos, Malgorzata Worek
TL;DR
Calculating many-parton matrix elements is challenging because standard Feynman-diagram methods grow rapidly. Helac-Phegas instead uses recursive amplitude computation and color configurations, with computational cost growing as ∼3^n rather than n!.
Problem
Standard Feynman-diagram calculations pose challenges because the number of diagrams grows rapidly for complex processes.
Method
Helac-Phegas computes color-summed squared amplitudes by summing color-connection configurations using a color matrix.
Results
∼3^n computational growth provides a substantial saving compared with the n! growth of Feynman-diagram methods.
Takeaways & Limitations
The recursive approach is efficient for processes involving many partons.
Takeaways & Limitations
The analogous color treatment becomes more complicated in the discussed formulation.
Abstract
from arXiv · showhide
The updated version of the Helac-Phegas event generator is presented. The matrix elements are calculated through Dyson-Schwinger recursive equations. Helac-Phegas generates parton-level events with all necessary information, in the most recent Les Houches Accord format, for the study of any process within the Standard Model in hadron and lepton colliders.
1 In tro du tion
Helac-Phegas addresses the computational challenges of multiparton matrix-element calculation and phase-space integration using recursive equations and automated multichannel mappings. The paper presents new functionality and user-interface developments for generating parton-level events and studying Standard Model processes.
- Motivation: Multiparton calculations become impractical with standard Feynman diagrams because their number grows factorially with the number of partons.The paper motivates recursive equations as the solution to this scaling problem.
- Method: Helac-Phegas combines Dyson-Schwinger recursive equations with automated multichannel phase-space mappings for arbitrary numbers of external particles.Helac calculates matrix elements, while Phegas generates phase space using information from Helac and scalarized Feynman graphs.
- Functionality: The updated program improves treatment of processes with many colored partons through explicit color configurations and introduces full QCD implementation in the color-connection representation.Consistent inclusion of all color configurations is identified as a central issue for QCD processes with colored partons.
- Functionality: New functionality includes automatic sub process summation, built-in CTEQ6l1 PDF support, parton-shower matching reweighting, Les Houches event files, and extended standard cuts.These developments target pp, p̄p, and e+e− colliders and support interfaces to LHAPDF and shower programs such as Pythia.
- User interface: The user interface adds script-managed single-file program control, numerical amplitudes at user-defined phase-space points, and parallel execution on computer clusters.These changes form the paper’s second development category, alongside functionality.
2 The Hela -Phegas Algorithm
Helac-Phegas combines a self-optimizing multichannel phase-space generator with a matrix-element evaluator based on Dyson–Schwinger recursion. Its recursive evaluation reduces the computational growth relative to the factorial growth of Feynman diagrams, while color degrees of freedom are treated through color representations and ordered amplitudes.
- Phegas phase-space generation: Phegas automatically constructs phase-space mappings corresponding one-to-one with contributing Feynman diagrams and optimizes them by channel variance.When many thousands of channels arise, optimization retains only a few tens that usually provide adequate phase-space efficiency.
- Phegas phase-space generation: For hadron collisions, Phegas also integrates over initial-parton momentum fractions weighted by parton distribution functions, using the optimized Parni algorithm.The passage states that Parni’s efficiency has been proven very good.
- Helac matrix-element evaluation: Helac evaluates amplitudes with Dyson–Schwinger recursive equations using a top-down skeleton construction followed by numerical subamplitude calculation.Phegas momenta are used to compute wave functions, and the result is obtained for each helicity and color-connection configuration.
- Helac matrix-element evaluation: The computational cost of Helac-Phegas grows as ∼3^n, compared with the n! growth of the Feynman-diagram approach.The recursive formulation therefore provides a substantial computational saving for large n.
- Color treatment: Color-ordered amplitudes have polynomial computational growth, but calculating the full color-summed matrix element squared requires all (n−1)! terms.A U(N_c) color-connection representation also provides a unified description of amplitudes involving quarks and gluons.
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The practical implementation assigns color labels to external particles, constructs colored ordered A-functions for permutations, and obtains the color-summed squared amplitude through color-connection configurations. For unweighted-event generation, color-connection information is selected for parton-shower matching using A-functions without a neutral gluon and a defined probability distribution.
- Implementation: External particles receive color labels according to their flavor, and Feynman rules build the higher-level subamplitudes.The labels have the form (i, σ_i).
- Implementation: Each permutation produces a new colored ordered A-function corresponding to its color factor F_I.
- Color summation: The total color-summed squared amplitude is obtained by summing over all n_l! color-connection configurations using the color matrix C.
- Unweighted events: Unweighted-event generation requires color-connection information for proper matching with parton-shower algorithms.
- Unweighted events: Color structures are selected from A-functions corresponding to color connections without a neutral gluon, according to a defined probability distribution.
J |AJ|2, PI > 0,
Helac-Phegas automatically generates Les Houches Accord files containing the information needed to interface with Pythia and HERWIG. The section also introduces matching procedures to prevent jet double counting when combining matrix elements with parton showers.
- Les Houches interface: Helac-Phegas automatically generates Les Houches Accord files with the information needed to interface with Pythia and HERWIG.The files contain the necessary event information for interfacing with parton-shower and hadronization programs.
- Matrix-element and shower matching: Matching or merging algorithms remove double counting between jets from hard shower emissions and higher-order matrix elements.They provide a smooth transition between phase-space regions covered by parton showers and matrix elements.
- MLM matching: The implementation briefly presents the incorporated matching framework, while the MLM approach extends beyond Helac-Phegas capabilities and relies on interfacing with a parton-shower algorithm.The MLM framework is described as a broader approach rather than a capability contained entirely within Helac-Phegas.
- Matching procedure: The matching procedure begins by generating all parton-level configurations for final-state parton multiplicities up to a chosen N.Parton-level kinematic cuts are required because of collinear and soft singularities.
T > pmin
Helac-Phegas applies CKKW-based scale reweighting during event generation and produces an LHA file for independent showering and hadronization. In the matching procedure, the jet transverse-energy threshold controls the balance between shower and matrix-element contributions.
- T > pmin: Generation parameters are common to all multiplicities n = 1, . . . , N.
- T > pmin: At this level, Helac-Phegas creates an LHA file that can be processed independently for showering and hadronization.
- T > pmin: A higher jet transverse-energy matching threshold rejects fewer events through the extra-jet veto, yielding weaker Sudakov suppression and a more dominant shower approximation.
- T > pmin: A lower matching threshold enhances the matrix-element contribution and corresponds to stronger Sudakov suppression, reducing the parton shower’s role in inclusive-jet production.
3 Running the o de
Helac-Phegas runs in two phases: integer Dyson–Schwinger recursion constructs a reusable skeleton, then Phegas generates phase-space points and evaluates matrix elements for each color configuration. The program supports complete Standard Model particle and coupling content, consistent unstable-particle treatments, and exact spin/color correlations.
- Two-phase operation: The first phase uses completely integer arithmetic to construct a skeleton, providing a solution of the recursive equations for subsequent calculations.The program retains the necessary information in memory, avoiding extra subdirectories and contributing to a compact source directory.
- Two-phase operation: In the second phase, Phegas automatically generates phase-space points and calculates the corresponding matrix elements separately for each color connection configuration.This makes both total weights and individual color-connection weights available, so color and phase-space un-weighting is straightforward.
- Physics coverage: Helac-Phegas incorporates all Standard Model particles and couplings in unitary and Feynman gauges, with unstable particles treated using fixed widths or the complex-mass scheme.For final states, spin and color correlations are included automatically without approximation.
- Process selection and summation: In summation mode, Helac-Phegas finds all partonic subprocesses producing the selected final state and sums over them; for t¯t, the example yields 9 subprocesses.The default excludes the b quark as an initial-state parton, while qnum controls the number of quark flavors considered.
- Process selection and summation: The number of subprocesses increases rapidly with many final-state jets, although independent subprocess generation makes the program trivially parallelizable.LSF scripts are provided as an example implementation for parallel execution.
General options
The general options configure Helac-Phegas process generation, phase-space and helicity treatments, physics schemes, event weighting, Monte Carlo integration, PDF interfaces, output, and collider cuts. Defaults and mode-specific restrictions are specified for these controls.
- General and collider-specific controls: General controls define executable and output names, phase selection, process flavor and gluon restrictions, LHA event-weight parameters, user constants, and collider-specific kinematic cuts.Cuts include transverse-momentum, rapidity, angular, ΔR, and invariant-mass requirements, with pp/pbarp and e+e− options read under their respective collider settings.
- Physics and model options: Physics options select fixed or running αs, Feynman or unitary gauge, Higgs contributions, fixed or complex-mass width schemes, and electroweak, combined, or QCD interactions.The defaults are running-coupling selection off, unitary gauge, Higgs contributions off, fixed-width scheme, and both electroweak and QCD interactions.
- PDF and event-interface options: The generator supports standalone CTEQ6L1 operation or an LHAPDF-selected PDF set, with default PDF identifier 10042 when the interface is enabled.The PDF identifier must match the desired LHAPDF set; n = 0 runs without the selected interface.
Ph ysi al onstan ts
Helac-Phegas lets users configure many physical constants through keywords, with default electroweak couplings and optional custom definitions. Advanced users can edit a constants header to specify the CKM matrix and particle masses and widths.
- Users can set many physical constants using the reported keywords.
- The electroweak couplings are provided by default, while users may optionally supply αem.
- Advanced users are recommended to edit constants_std.h or provide their own constants.A prototype is included in constants_std.h.
- The constants file can also define the CKM matrix, masses, and widths for all particles.
- The value −1 for sin2thetaw and alphaem indicates that these quantities are defined as described in the text.
4 Outlo ok
The outlook describes version 1.2.0 as intended for public use and outlines upcoming releases adding Higgs couplings, many-colored-particle processes, and MSSM particles and couplings.
- Release plans: Version 1.2.0 is intended for public use.The passage also invites bug reports or simple questions.
- Release plans: Version 1.3.0 will include all Higgs-gluon and Higgs-photon couplings in the large-m_top limit.It has already been tested and is expected to become available on the Helac-Phegas website soon.
- Release plans: Version 2 will incorporate processes with more than 9 equivalent gluons, including gg → 8g and higher multiplicities.The new version has already been developed and tested.
- Further extensions: The developers are also working to include MSSM particles and couplings.
A kno wledgmen ts
The authors acknowledge individual contributions, institutional hospitality, and support from several research programmes and funding bodies.
- The authors thank Andre van Hameren for providing Parni routines and Michelangelo Mangano for useful discussions on matching algorithms.
- The work received support from the Transfer of Knowledge programme ALGOTOOLS (MTKD-CT-2004-014319) and the RTN HEPTools (MRTN-2006-CT-035505).
- M.W. was supported in part by BMBF grant 05 HT6VKC and received hospitality from the Galileo Galilei Institute for Theoretical Physics, with partial INFN support.The INFN support covered completion of the work.
Referen es
This section lists 54 numbered references supporting the paper’s background, computational methods, phenomenology, and event-generation context. The bibliography includes journal articles, proceedings, and preprints spanning 1986–2008.
- References: References [14]–[18] include works by Duhr, Höche, Maltoni, Papadopoulos, Kleiss, Pittau, and Mangano and collaborators.The cited publications include JHEP, Computer Physics Communications, and Nuclear Physics B entries, with associated hep-ph identifiers.