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The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations

J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. -S. Shao, T. Stelzer, P. Torrielli, M. Zaro

arXiv:1405.0301v2hep-ph

TL;DR

The paper addresses the challenge of automating fixed-order cross sections, parton-shower matching, and multiplicity merging. It presents MadGraph5_aMC@NLO as a unified framework and concludes that automated NLO techniques are established, robust, fast, and less error-prone than process-by-process approaches.

  • Problem

    Merging samples with different hard-process multiplicities requires avoiding double counting, with NLO solutions inherently more complicated because they depend on the matching strategy.

  • Method

    The paper develops unified automated procedures for NLO QCD calculations, parton-shower matching, and FxFx merging within MadGraph5_aMC@NLO.

  • Results

    Automated NLO techniques are fully established, more robust, orders of magnitude faster, and less error-prone than traditional analytical process-by-process approaches.

  • Takeaways & Limitations

    Broad NLO process coverage could support wider PDF extraction and improved parton-shower tuning using NLO results.

  • Takeaways & Limitations

    The first public version restricts higher-order effects to QCD corrections for Standard Model processes, while automation for general renormalisable Lagrangians remains under implementation.

Abstract

from arXiv · show

We discuss the theoretical bases that underpin the automation of the computations of tree-level and next-to-leading order cross sections, of their matching to parton shower simulations, and of the merging of matched samples that differ by light-parton multiplicities. We present a computer program, MadGraph5_aMC@NLO, capable of handling all these computations -- parton-level fixed order, shower-matched, merged -- in a unified framework whose defining features are flexibility, high level of parallelisation, and human intervention limited to input physics quantities. We demonstrate the potential of the program by presenting selected phenomenological applications relevant to the LHC and to a 1-TeV $e^+e^-$ collider. While next-to-leading order results are restricted to QCD corrections to SM processes in the first public version, we show that from the user viewpoint no changes have to be expected in the case of corrections due to any given renormalisable Lagrangian, and that the implementation of these are well under way.

A. Technical prerequisites, setup, and structure · D. Third-party codes included in MadGraph5 · 1. Introduction

The paper motivates a unified, automated framework for tree-level and NLO calculations, with or without parton-shower matching, in response to the growing precision demands of LHC phenomenology. It realizes this goal in the public MadGraph5_aMC@NLO code, which combines existing capabilities and adds merging of samples with different light-parton multiplicities.

  • 1. Introduction: Beyond-Born QCD observables remain difficult because matrix-element complexity must be combined with infrared-singularity cancellation.The challenge involves both tree and loop matrix elements and the cancellation of their infrared singularities.
  • 1. Introduction: LHC precision measurements and the lack of clear beyond-Standard-Model signals increase reliance on accurate theoretical predictions.Large-pT tails can contain events with small probabilities while remaining phenomenologically relevant.
  • 1. Introduction: At NLO, a general solution exists for arbitrary process complexity, eliminating the need for ad-hoc strategies.The paper contrasts this with the absence of a general solution at arbitrarily high perturbative orders.
  • 1. Introduction: The package includes the ingredients needed for NLO calculations, optionally combined with showers through the MC@NLO formalism, while retaining broad theory flexibility.The underlying subtraction, one-loop, and shower-matching ideas require no or minimal changes for other renormalisable theories, and tree-level computations can handle user-defined Lagrangians.
  • 1. Introduction: The framework targets arbitrary observables and processes at tree level or NLO, with optional matching to parton showers, through full automation.The authors identify automation as the effective way to address the associated technological challenges.
  • 1. Introduction: MadGraph5_aMC@NLO unifies MadGraph5 and aMC@NLO, supersedes both codes, and adds merging of samples with different light-parton multiplicities.The code is presented as a fully automated and public implementation of the proposed framework.

2. Theoretical bases and recent progress

The section develops a process-independent framework for automated fixed-order and shower-matched calculations, extending MadGraph5_aMC@NLO’s flexibility to complex BSM models and physically consistent mixed-coupling predictions. It also describes computational techniques that improve NLO stability and enable broader virtual-correction automation.

  • Process-independent framework: Embedding multiple simulation modes in one framework enables mutually consistent scenario comparisons while minimizing shared technical work.Consistency includes common physical parameters such as couplings and masses.
  • Process-independent framework: A process-independent cross-section structure lets MadGraph5_aMC@NLO encode common computational elements once while leaving masses and particle content as free parameters.This design reduces technical duplication across theories and perturbative orders.
  • BSM capabilities: MadGraph5_aMC@NLO extends model support for BSM quantities, including form factors and particles with spin larger than one.The supported spin set includes {0, 1/2, 1, 3/2, 2}.
  • Unstable particles and BSM calculations: The complex mass scheme is available for both LO and NLO simulations, allowing off-shell effects, spin correlations, and interference to be treated beyond the narrow-width approximation.Complete calculations are needed for broad resonances, strongly off-shell regions, and potentially large gauge cancellations.
  • NLO corrections: Mixed-coupling expansions require automatically including coupling structures beyond the user-selected terms so the resulting cross section remains physically meaningful.QCD and QED corrections generally cannot be interpreted separately when both contribute to a given coupling combination.
  • NLO integration and matching: Multi-channel integration parallelizes independent channels, while counterevent weights combine infrared and remainder contributions to reduce mis-binning and improve NLO numerical stability.The event organization is also necessary for MC@NLO matching with parton showers.

3. How to perform a computation

MadGraph5_aMC@NLO reduces cross-section computations to an automated three-step workflow—generation, output, and running—performed through online shell commands. Process generation determines LO- or NLO-type capabilities, while subsequent runtime choices support fixed-order, shower-matched, event-generation, and plotting tasks without regenerating the process.

  • Workflow: The computation proceeds through three shell-driven steps: generation constructs the process, output writes it to disk, and running integrates and executes it.The running stage can produce unweighted events and user-defined observable plots.
  • Generation: Users choose LO-type or NLO-type generation because the two modes expose different running options and involve different computational complexity.LO-type generation retains MadGraph5 short-distance cross-section capabilities while extending the overall scope.
  • Generation: Generation accepts a user-specified process and can switch from the default SM with massive b quarks to another imported model, such as the MSSM.Examples include pp → tt̄W+ and pp → tt̄n1n1 generation.
  • Output: A generated process is written with the output command into a user-named current-process directory, where subsequent operations are performed.The output name may be omitted, in which case the program chooses one, subject to reserved-name restrictions.
  • Running: After generation and output, runtime settings let users obtain fNLO, NLO+PS, LO+PS, or fLO results without regenerating the process.The program can also plot the same observable at NLO using all required weights and at LO using the Born contribution alone; LO options from either generation type yield the same physics, though default inputs can change the numbers.

4. Illustrative results

The illustrative studies demonstrate the breadth of first-time NLO and merged predictions, while showing substantial effects from non-resonant contributions, NLO corrections, spin correlations, shower choices, and matching systematics.

  • Novel calculations: Several processes receive NLO predictions or NLO QCD corrections here for the first time, including HV jj, HV V, and selected high-multiplicity reactions.All tabulated processes can yield corresponding NLO+PS event samples with MadGraph5_aMC@NLO.
  • Resonance effects: Non-3-resonant contributions induce dramatic shape modifications for M(6ℓ) < 500 GeV and 10 < pT(6ℓ) < 250 GeV.Agreement with resonant approximations requires simultaneously applying all three cuts so that the vector bosons remain near their mass shells.
  • NLO effects: NLO corrections are non-negligible in both rate and shape across the studied predictions.The comparison finds that replacing the full ME treatment with generation cuts and only one additional condition does not qualitatively alter the conclusions.
  • Decay and spin effects: Production spin correlations are sizable, with MadSpin- and Pythia8-decayed shapes sometimes barely within or slightly outside NLO theoretical-systematics bands.At LO, apparent compatibility is attributed solely to the larger LO scale dependence.
  • Parton-shower dependence: Third-jet observables in VBF+0j show poor agreement among Pythia8, Pythia6(Q2), and HERWIG6, whereas NLO-nature observables agree better and predictive capability is restored in VBF+1j.The discussion specifically identifies pT(j3) and σveto as problematic VBF+0j quantities.
  • Merging comparisons: 2.5% is the absolute cross-section difference between the compared FxFx and unmerged predictions, whose shapes remain close despite the FxFx result being slightly softer.The FxFx prediction does not appear sufficiently softer to agree with the CMS measurement.
  • Matching systematics: 3% is the largest deviation among the compared predictions, while differences between unmerged results show that matching systematics are not negligible.The passage states that all predictions are otherwise quite close to one another.

5. Conclusions and outlook

MadGraph5_aMC@NLO establishes a unified, user-friendly framework for automated tree-level and NLO QCD computations, including parton-shower matching. Its modular design supports broader corrections, models, uncertainty studies, and future methodological improvements.

  • Framework and capabilities: MadGraph5_aMC@NLO extends MadGraph5 with NLO QCD corrections and optional parton-shower matching.It is presented as a successor rather than a plugin to MadGraph5.
  • Framework and capabilities: Tree-level and NLO QCD computations are treated on the same footing, with the perturbative order selected by an input switch.Users provide physics inputs such as masses, couplings, scales, observables, and the hard process.
  • Limitations and outlook: The current public implementation restricts higher-order effects to QCD corrections for Standard Model processes, while extensions to arbitrary NLO corrections from user-defined Lagrangians are planned.The stated limitation is connected to ultraviolet renormalisation and, typically, R2 counterterms in one-loop computations.
  • Impact and future development: The work argues that automated NLO techniques are fully established and provide perturbatively accurate, realistic predictions for a very large range of complex processes.Automation can free experts from increasingly involved perturbative computations and enable work on NNLO methods and improved parton showers.
  • Impact and future development: MadGraph5_aMC@NLO’s modularity enables improvements to recursive relations, matching and merging schemes, and integral-reduction tools, while systematic uncertainty assessment and extensions to new-physics models are identified as important applications.The uncertainties include scales, PDFs, and matching and merging methods.

A. Technical prerequisites, setup, and structure · Setup

MadGraph5_aMC@NLO runs on Linux and Mac OS-X with specified Perl, Python, and compiler prerequisites. After unpacking, users enter its shell and configure optional local tools through variables that can be set interactively or edited directly.

  • A. Technical prerequisites, setup, and structure: aMC@NLO is routinely run on Linux platforms and Mac OS-X systems.
  • A. Technical prerequisites, setup, and structure: The required scripting environments are Perl 5.8 or higher and Python 2.6 or higher but lower than 3.0.
  • A. Technical prerequisites, setup, and structure: The default compiler requirement is gfortran/gcc 4.6 or higher, while other modern Fortran/C++ compilers are acceptable with quadruple-precision support.
  • A. Technical prerequisites, setup, and structure: After the tarball is unpacked in the main directory, the code is ready to run without mandatory external-package installation.The tarball includes copies of the third-party codes listed in appendix D.
  • Setup: From a terminal shell in the main directory, users type the launch command and enter the MadGraph5_aMC@NLO shell, identified by the aMC> prompt.
  • Setup: A minimal setup phase defines configuration variables before the first process and needs to be performed at most once.The phase does not need to be repeated before generating later processes.
  • Setup: Local FastJet and LHAPDF installations are registered by setting their paths with the FASTJET and LHAPDF configuration commands.These commands associate the supplied values with variables in configuration.txt.
  • Setup: Users can inspect or edit configuration variables in configuration.txt, use shell auto-completion after set, or consult the advanced setup options in appendix B.1.

Structure · B. Advanced usage

The current-process directory organizes user inputs, NLO-specific analyses, shower controls, process codes, and run outputs. Advanced usage emphasizes editing the appropriate files before launch, supported output formats, automatic channel combination, and the distinction between current-process and template files.

  • Structure: The current-process directory contains all operations for the generated process, divided into user input-type operations before launch and program-managed output-type operations afterward.Input cards are located in MYPROC/Cards, while output operations are handled by MadGraph5_aMC@NLO.
  • Structure: Input cards are plain-text, commented files that steer MadGraph5_aMC@NLO and define theory parameters, with values editable before launch or through the interactive shell.When both methods are used, the values stored in the cards at the end of the talk-to phase determine the run.
  • Structure: NLO generations require pre-launch Fortran-aware edits in FixedOrderAnalysis, MCatNLO, and SubProcesses for analyses, shower control, cross-section code, and parton-level cuts.MCatNLO is ignored for externally showered LHE files but exposes shower drivers and analysis code when MadGraph5_aMC@NLO steers showering.
  • Structure: Run-specific subdirectories store input summaries, integration results, Les Houches events, shower outputs, and user histograms according to the run type.Outputs include StdHEP/HepMC or topdrawer files for internally steered showers, and Root or topdrawer histograms for fixed-order runs.
  • Structure: Root and topdrawer outputs are supported for automatic summation across multi-channel analyses, whereas other formats require manual combination code and scripts.Each channel is non-physical individually, so its analysis output must be summed; n-tuples may also require weight rescaling.
  • Structure: Template-directory edits affect subsequent process generations, while compilation and input cards for the current run come from the current-process directory.Changing Template files therefore has no effect on an already generated current run.
  • B. Advanced usage: The Advanced usage section explains selected MadGraph5_aMC@NLO features to help users exploit the code’s physics potential, rather than serving as a complete usage manual.It briefly expands on subjects only touched upon in the main text.

B.1 Models, basic commands, and extended options · Setup

MadGraph5_aMC@NLO requires an imported model for process generation, supports switching among related models and user-defined restrictions, and limits first-public-version NLO corrections to QCD corrections for SM processes. Setup options are inspectable and configurable, while widths and, for some UFO models, mass-matrix diagonalisation are handled as separate setup operations.

  • B.1 Models, basic commands, and extended options: A model must be loaded before MadGraph5_aMC@NLO can generate a process.The default shell assumes the SM, but users can import another model with import model ModelName.
  • B.1 Models, basic commands, and extended options: Available models can be listed with <TAB>, and users may add their own models under the models directory.Related models are grouped in a ModelClass directory, such as SM variants with massless or massive charm quarks.
  • B.1 Models, basic commands, and extended options: Users can implement non-extensive SM modifications by creating a restriction file and importing it as a modified SM model.The passage gives models/sm/restrict-XXX.dat and import model sm-XXX as the mechanism.
  • B.1 Models, basic commands, and extended options: First-public-version NLO corrections are restricted to QCD corrections to SM processes, and NLO generation automatically switches to the corresponding NLO model when available.If the corresponding model is unavailable, the code issues a warning and stops.
  • Setup: Setup options affecting physics schemes can be displayed with display options and changed using set Option Value.The complex mass scheme is disabled by default and can be enabled by setting scheme True.
  • Setup: Computing unstable-particle widths is a mandatory pre-running setup operation performed independently of generation through the widths command.MadWidth operates at tree level and in the narrow-width approximation, so NLO simulations may require manually setting widths.
  • Setup: Mass-matrix diagonalisation is another setup operation, but it is available only for a restricted class of UFO models containing the AsperGe module.Its inputs are accessible during the interactive talk-to phase.

Generation · Output · Running

The interface organizes process generation, code output, and execution through commands that specify processes, coupling orders, NLO contributions, and output or running modes. It also supports retrieving saved processes and repeatedly showering selected event files, while warning that several diagram-selection syntaxes can produce non-physical results.

  • Generation: The generate command requires a Process specifying initial- and final-state particles, while AmpOrders bounds coupling powers in the scattering amplitudes.In NLO generation, these coupling bounds apply to Born amplitudes, with one-loop and real-emission couplings determined automatically.
  • Generation: Syntaxes s.3–s.5 select diagrams by excluding particle-containing diagrams, requiring a particle type, or excluding s-channel occurrences.These selections generally risk violating gauge invariance and require particular caution, especially for NLO computations.
  • Generation: The Mode option controls NLO contributions: all includes one-loop and FKS-subtracted real-emission terms, real includes only real emission, and virt includes only one-loop terms.The all setting is the only one recommended for non-experts because it leads to physical results; virt supports standalone extraction of virtual pole residues and finite parts.
  • Generation: The Couplings option specifies which NLO corrections are computed, but the current version accepts only QCD.The general syntax permits multiple coupling names, while the currently valid setting is Couplings ≡ QCD.
  • Output: The output command creates a target directory for the generated process, choosing a name automatically when MYPROC is omitted.Reserved names are interpreted as OutputForm keywords for creating executables or standalone libraries rather than the usual integration and event-unweighting code.
  • Running: The launch command runs a previously generated process directory through specified run modes and options, with further syntax documented through help and tutorials.ProcDir must identify one of the current-process directories generated earlier.
  • Running: The launch -i command reopens a saved process in running mode, which can be re-entered indefinitely and supports showering event files from selected Events subdirectories.The shower command may be repeated with the same argument to change seeds or parameters in the parton-shower Monte Carlo.

B.2 Setting the hard scales at the NLO

MadGraph5_aMC@NLO requires users to configure renormalisation, factorisation, and Ellis-Sexton hard scales, choosing fixed or event-dependent definitions before running. The framework supports flexible scale functions, with specific NLO constraints and separate prescriptions for FxFx-merged simulations.

  • Scale definitions: The three hard scales are the renormalisation (µR), factorisation (µF), and Ellis-Sexton (QES) scales, with QES variations leaving cross sections unchanged.QES is intended for developer validation studies and should normally be set equal to the factorisation scale.
  • Fixed and dynamical scales: Users must set hard scales before compiling and running, choosing fixed values independent of event kinematics or dynamical values that vary with final-state four-momenta.Whether a scale is fixed or dynamical is controlled by an input parameter.
  • Fixed and dynamical scales: Dynamical reference-scale definitions can be customized in setscales.f, but edits must be made before the launch command to affect the current run.The file contains example dynamical scales and allows process-specific functions defined by the user.
  • NLO constraints: At NLO, the two incoming-hadron factorisation scales must be equal; otherwise, the code stops.At LO they can differ because the scales enter only the relevant PDFs, whereas NLO logarithmic terms create ambiguities.
  • FxFx merging: FxFx-merged simulations use different scale-setting prescriptions inherent to the method and require separate documentation.The section directs users to the FxFx documentation for further details.

B.3 Scale and PDF uncertainties: the NLO case

The NLO uncertainty framework factorizes scale- and PDF-dependent cross sections into reusable weights and basis members, enabling essentially instantaneous evaluations for alternative choices. MadGraph5_aMC@NLO stores these variations event-by-event in LHE files and propagates them to fixed-order and shower-matched analyses.

  • Reusable scale and PDF weights: Scale and PDF dependence is represented exactly through weights independent of scales and PDFs and basis members containing their dependence.The weights may be expensive to compute, whereas basis members are straightforward to recompute for each desired choice.
  • Reusable scale and PDF weights: MadGraph5_aMC@NLO computes a central result while optionally storing weights for later theoretical-uncertainty evaluations.Alternative cross sections are reconstructed by combining the stored weights with basis members according to the exact decomposition.
  • LHE reweighting and envelopes: For (N)LO+PS, event-by-event LHE weights cover all selected scale and PDF combinations alongside the central cross section.Scale uncertainty is typically obtained from bin-by-bin extrema, while PDF uncertainty uses an envelope defined according to the PDF-author prescription.
  • LHE reweighting and envelopes: PDF dependence from parton-shower Sudakov factors is not included in the decomposition, but is expected to be particularly small for PDF uncertainties.This limitation is explicitly identified in the discussion of the exact weight-based representation.
  • Fixed-order event weights: 1 + (1 + Nµ) + NPDF reweight weights are associated with each kinematic configuration in f(N)LO runs.Because these runs cannot use unweighted events, each configuration carries an array rather than a single event weight.

B.4 Scale and PDF uncertainties: the LO case

At LO, scale and PDF uncertainty handling is simplified because the cross section contains one PDF-and-coupling combination, making the weight w1 as straightforward to use as σ. SysCalc automates these studies through configurable scale, PDF, and merging-scale variations.

  • LO uncertainty structure: At LO, the cross section contains one PDF-and-coupling combination, so handling w1 is as simple as handling σ.This single basis member explains the simplicity of the LO uncertainty treatment relative to the NLO case.
  • SysCalc setup: SysCalc is installed from the MadGraph5_aMC@NLO shell and enabled by saving weights in the intermediate LHE file.The run card option syst ! Enable systematics studies in run card.dat triggers automatic invocation at the end of the run.
  • Scale variations: Scale variations use multiplicative factors such as 0.5, 1, and 2, with sys_scalecorrelation selecting all combinations or only µR = µF.The -1 setting includes all µR and µF combinations, whereas -2 restricts the calculation to correlated variations.
  • PDF and merging uncertainties: PDF systematics can use all members of a specified error set or selected individual members, while SysCalc also supports αS-argument and Qmatch variations.The sys_pdf and sys_matchscale entries control these PDF and tree-level merging studies.

B.5 Other LO reweighting applications · B.6 Output formats and standalone libraries · C. Features of one-loop computations

MadGraph5_aMC@NLO supports LO matrix-element reweighting and MadWeight likelihood computations, while also providing standalone outputs for matrix elements and other calculation ingredients. Its reweighting is constrained to a single model and a non-expanding kinematical region, whereas output formats can be extended for specialized uses.

  • B.5 Other LO reweighting applications: LO matrix-element reweighting is enabled through the MadGraph5_aMC@NLO shell with reweight=ON and parameter modifications specified via reweight_card.dat.Multiple modifications and parameter changes can be entered through the set command.
  • B.5 Other LO reweighting applications: Reweighting must remain within one model, and the new hypothesis cannot access a larger kinematical region than the benchmark calculation.The accessible region under the new hypothesis must be equal to or smaller than the original one.
  • B.5 Other LO reweighting applications: MadWeight handles matrix-element-method computations, with increased speed from subprocess combination, Monte-Carlo jet-parton assignment sums, and simultaneous execution.The MadWeight executable is produced with the output madweight keyword and then launched through the shell interface.
  • B.6 Output formats and standalone libraries: MadGraph5_aMC@NLO provides a self-contained framework for cross sections and event generation while also exposing individual calculation ingredients for external use.This inherited MadGraph capability includes supplying objects such as matrix elements for calculations performed elsewhere.
  • B.6 Output formats and standalone libraries: The standalone output supplies a self-contained Fortran77 library for tree-level or one-loop matrix elements and includes a pointwise test program.The one-loop option follows an NLO-type generation with virt=coupling1.
  • B.6 Output formats and standalone libraries: The cpp output provides the standalone functionality in C++ but works only for tree-level matrix elements.The pythia8 output instead supplies tree-level matrix elements in a format directly usable by the Pythia8 PSMC, with a driver for sample runs.
  • B.6 Output formats and standalone libraries: The madweight output keyword sets up a computation with MadWeight, while the listed output formats are not exhaustive and specialized formats may be developed on request.Dedicated outputs for MatchBox and EventDeconstruction were being developed.

C.1 TIR and IREGI

This section describes TIR and IREGI for reducing loop tensor integrals to scalar integrals through Lorentz-covariant decomposition and recursive algebraic relations. IREGI supports both Passarino–Veltman and Davydychev reduction, with a fallback when the default procedure is unstable.

  • Tensor decomposition: IREGI reduces the original tensor integral to scalar integrals using a Lorentz-covariant decomposition into symmetric tensors of metrics and external momenta.All non-equivalent Lorentz-index permutations contribute with weight one.
  • Recursive reduction: Contracting tensor integrals with metrics and external four-momenta generates relations to lower-rank tensor or lower-point scalar integrals.Scalar products involving the loop momentum are rewritten to cancel denominators or simplify numerator dependence, yielding an algebraic system in scalar integrals.
  • Recursive reduction: IREGI implements recursive reduction through either Passarino–Veltman or Davydychev methods, with additional scalar-integral relations obtainable using integration by parts.The Davydychev approach uses generalized loop-tensor and basic-scalar integrals and can relate scalar integrals in different dimensions.
  • Implementation and stability: The calling code selects IREGI’s reduction method, while the default MadGraph5_aMC@NLO configuration uses Passarino–Veltman reduction controlled by an input-card parameter.If the Passarino–Veltman procedure is flagged unstable, IREGI switches to Davydychev’s method; this does not replace MadLoop5’s stability control.

C.2 Quantitative profile of MadLoop performances

MadLoop5 performance is characterized quantitatively through table 15, complementing earlier benchmark results. The profile emphasizes topology-based optimization and documents computational costs, resource usage, accuracy, and user-accessible diagnostics.

  • Performance overview: Table 15 summarizes quantitative characteristics of MadLoop5 for the scattering processes studied in section 4.3.These characteristics complement the earlier results, which serve as benchmarks for validating other codes.
  • Topology optimization: The number of independent loop reductions is much smaller than the number of Feynman diagrams, highlighting the optimization induced by equation (2.76).Topologies provide an upper bound on the relevant reduction sum for one kinematic configuration.
  • Resource and accuracy metrics: MadLoop5 profiles generation time, running time, output-code size, peak RAM usage, and relative accuracy from internal stability tests.Running time is measured for one helicity configuration and scales linearly with the number of helicity combinations.
  • User diagnostics: Users can obtain these performance data through the check profile command in the MadGraph5_aMC@NLO shell, including for virtual t¯tZ corrections.The documented example is check profile g g > t t~ z [virt=QCD].

C.3 Computation of the integrand polynomial coefficients · D. Third-party codes included in MadGraph5 aMC@NLO

MadLoop5 computes loop-integrand polynomial coefficients numerically by promoting loop currents to polynomial objects, iteratively multiplying and symmetrising them, and closing the Lorentz trace. MadGraph5_aMC@NLO is self-contained through included third-party codes, while FastJet is provided only in a stripped core version.

  • C.3 Computation of the integrand polynomial coefficients: In renormalisable theories and Feynman gauge, a 2 → 6 loop numerator requires at most 495 coefficients.The maximal loop-momentum rank is set by the number of loop propagators, giving Ncoeff(rmax = 8) = 495.
  • C.3 Computation of the integrand polynomial coefficients: MadLoop5 computes integrand polynomial coefficients entirely numerically, following MadGraph’s procedure for evaluating Feynman diagrams.Loop currents are promoted from ordinary currents to objects embedding polynomials in the loop momentum.
  • C.3 Computation of the integrand polynomial coefficients: MadLoop5 does not enforce the renormalisable-theory bound that vertex-polynomial rank is at most one, enabling one-loop computations in effective theories such as HEFT.ALOHA automatically generates the numerical routines for vertex-polynomial evaluation from the UFO model specification.
  • C.3 Computation of the integrand polynomial coefficients: Each subsequent loop current is formed by multiplying the previous current with an intervening vertex polynomial and summing matching expanded-product terms.MadLoop5 symmetrises coefficients after every loop vertex to limit coefficient proliferation and computing time.
  • C.3 Computation of the integrand polynomial coefficients: The iterative construction continues to the second L-cut leg, after which loop-diagram coefficients are obtained by closing the Lorentz trace.Shared loop currents are computed only once, with the L-cut location chosen to maximise their reuse.
  • C.3 Computation of the integrand polynomial coefficients: All integrand-representation optimisations can be disabled through the loop option, producing slower output useful for self-consistency checks and debugging.The non-optimised mode generates a completely different code structure.
  • D. Third-party codes included in MadGraph5 aMC@NLO: MadGraph5_aMC@NLO is self-contained and ready-to-run because it includes third-party codes such as ALOHA, CutTools, FastJet core, HELAS, HERWIG6, MINT, OneLoop, Pythia6, QCDloop, RAMBO, StdHEP, and Vegas.The listed components support the framework’s physics calculations, integration, showering, jet reconstruction, and event handling.
  • D. Third-party codes included in MadGraph5 aMC@NLO: The included FastJet is a stripped version of the code, so users needing extended jet-reconstruction capabilities may install the full FastJet package with its plugins.The paper directs users to FastJet’s website for further details.
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