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Automation of the Dipole Subtraction Method in MadGraph/MadEvent
Rikkert Frederix, Thomas Gehrmann, Nicolas Greiner
TL;DR
Dipole subtraction automation lacked a general tool for arbitrary processes that also handled massive partons. MadDipole generates Fortran code for dipole terms from a specified real-emission process in MadGraph/MadEvent. Checks show the subtraction term tracks the matrix element in the reported collinear limits, with a current MadGraph multiplicity constraint.
Problem
A general tool was lacking for generating dipole subtraction terms for arbitrary processes that also handles massive partons.
Source: §1 · p. 3Method
MadDipole automatically generates Fortran code for dipole subtraction terms from a specified (n+1)-particle process in the MadGraph/MadEvent framework.
Source: §1 · p. 3, §5 · p. 11Results
In the reported final-final and initial-final collinear limits, the subtraction term diverges with the matrix element and their ratio tends to one.
Source: §3 · p. 8Takeaways & Limitations
MadDipole automates real-radiation dipole subtraction terms required for NLO calculations, a component that could contribute to full automation of NLO calculations for collider processes.
Source: §5 · p. 12The process gg → 5g is not yet feasible in MadGraph because of the size of the color factors, and MadGraph cannot handle more than seven colored particles.
Source: §3 · p. 10
Abstract
from arXiv · showhide
We present the implementation of the dipole subtraction formalism for the real radiation contributions to any next-to-leading order QCD process in the MadGraph/MadEvent framework. Both massless and massive dipoles are considered. Starting from a specific (n+1)-particle process the package provides a Fortran code for all possible dipoles to all Born processes that constitute the subtraction term to the (n+1)-particle process. The output files are given in the usual "MadGraph StandAlone" style using helicity amplitudes.
1. Introduction
Multi-particle LHC searches require precise predictions for signals and indistinguishable Standard Model backgrounds, while NLO calculations are still performed process by process. MadDipole addresses the lack of a general tool for generating dipole subtraction terms for arbitrary processes, including massive partons.
1. Introduction
- Multi-particle LHC searches need accurate signal and background predictions, and high precision requires going beyond leading-order perturbation theory.Leading-order event generators were multipurpose, whereas NLO calculations were performed process by process.Source: §1 · p. 2
- NLO QCD corrections combine one-loop virtual contributions with real-emission contributions from (n+1)-parton scattering processes.The two contributions must be computed separately for numerical evaluation.Source: §1 · p. 3
- Real-emission soft and collinear singularities cancel against virtual one-loop singularities, yielding a finite NLO correction.Subtraction methods extract real-radiation singularities systematically.Source: §1 · p. 3
- No general tool then produced dipole terms for arbitrary processes while also handling massive partons.Existing automated approaches had included only massless dipoles or required users to supply process-dependent information.Source: §1 · p. 3
- MadDipole automatically generates Fortran subroutines for all dipole configurations associated with Born processes that can produce a specified (n+1)-particle process.The output follows the usual MadGraph style and returns squared amplitudes.Source: §1 · p. 3
2. Construction of dipole terms
Dipole terms combine splitting functions with reduced matrix elements, with color and helicity handled to fit MadGraph amplitudes. The construction also treats massive unresolved partons separately and allows user-controlled phase-space restrictions.
2. Construction of dipole terms
- Each dipole splitting function describes the collinear limit of an unresolved parton with its emitter and part of its soft limit between emitter and spectator.Multiplication by reduced matrix elements uses recombined pseudo-partons; summing dipole factors recovers the full soft behavior.Source: §2 · p. 4
2.1 Color and helicity management
- The implementation inserts the color operator by modifying the ket-side color labeling and squares the modified ket against the original complex conjugate.Off-diagonal helicity terms pair each stored helicity amplitude with the corresponding opposite-helicity amplitude; polarization vectors use existing HELAS routines.Source: §2.1 · pp. 4–5
2.2 Massive particles
- Dipoles with massive unresolved partons are calculated in a separate subroutine and omitted by default because their collinear limits are regulated by mass.Users can include these non-divergent dipoles at high center-of-momentum energy or small external masses to subtract associated large logarithms; the zero-mass limit agrees with generating massless code.Source: §2.2 · p. 5
2.3 Phase space restriction
- Four user-changeable α parameters restrict final-final, final-initial, initial-final, and initial-initial dipoles, defaulting to unity.The integrated dipole factors added to the virtual contribution also depend on α; calculating subtraction terms is only necessary near soft or collinear limits.Source: §2.3 · pp. 5–6
3. Checks
MadDipole’s checks show that subtraction terms reproduce the real-emission singular behavior in soft and collinear limits, and the implementation agrees with MCFM where expected. The tested examples include massless and massive dipoles, while code generation is not yet feasible for gg → 5g.
Checks of singular limits
- The matrix-element-to-dipole ratio approaches unity near tested soft and collinear limits, and MadDipole found no inconsistencies across massless and massive checks.The included check program follows random phase-space points toward a singular limit and reports the matrix element, summed dipoles, and their ratio.Source: §3 · p. 6
Performance and scope
- Choosing smaller α values, such as 0.1, improves computation time and subtraction convergence, while code-generation time ranges from seconds to minutes.Generation of gg → 5g is not yet feasible in MadGraph because of the size of its color factors.Source: §3 · pp. 7, 10
Final-final dipoles
- In the e+e− → Z → qq̄g scans, the ratio D/|MR|2 tends to one and the average difference tends to zero in collinear and soft limits.The maximal pointwise difference rises near the limits but does not produce a divergent phase-space integral.Source: Figure 1 · p. 7, §3 · p. 8
Initial-final and massive dipoles
- For e−q → e−qg, both the matrix element and subtraction term diverge as 1/s25 while their ratio tends to one; the massive t t̄g scan shows similar behavior to the massless case.Restricting dipoles with α < 1 includes only those needed to cancel the divergence and yields a smaller constant contribution to the subtraction term.Source: §3 · pp. 8–9, Figure 2 · p. 9
Comparison with MCFM
- Random phase-space comparisons with MCFM agree close to singularities where expected; differing terms arise when MadDipole sums spectator dipoles for purely collinear limits.MCFM uses a single spectator momentum for these cases, whereas MadDipole sums over all spectators.Source: §3 · p. 9, Table 1 · p. 10
4. How to use the package
Users specify the desired (n+1)-particle process in a process card, and MadDipole generates its matrix element and dipole code. The generated routines accept (n+1) momenta, perform momentum remapping internally, and apply configurable α restrictions.
Installation
- Installation follows the StandAlone MadGraph pattern: build MadGraphII, copy the Template directory, and specify the process in the new directory.The package is distributed as MG_ME_DIP_V4.4.3.tar.gz.Source: §4 · p. 10
Specify and generate a process
- Users specify the requested (n+1)-particle process in ./Cards/proc_card.dat, then run the process-generation step to create code for its matrix element and dipole terms.The generated files are placed under ./Subprocesses/P0_yourprocess.Source: §4 · pp. 10–11
Use the generated routines
- Each dipole routine is called with (n+1) momenta, like the matrix element, and remaps them internally to the n-parton phase space.dipolsum.f provides DIPOLSUM and DIPOLSUMFINITE; the latter contains dipoles that contribute potentially large logarithms but no real singularity.Source: §4 · p. 11
Configure dipole restrictions
- The four dipole-type α parameters restrict which dipoles enter the sum, and are set to unity by default.Their values can be changed in the DIPOLSUM and DIPOLSUMFINITE subroutines.Source: §4 · p. 11
5. Conclusions
MadDipole automates massless and massive dipole subtraction in MadGraph/MadEvent, generating Fortran code for all dipoles and compatible Born processes from a specified real-emission process. Tests confirm the expected singular-limit behavior and point-wise agreement with MCFM wherever anticipated.
Conclusions
- MadDipole automates dipole subtraction for massless and massive partons, generating Fortran code for dipoles and possible Born processes from a specified (n+1)-particle process.The implementation reuses MadGraph color routines and combines helicity amplitudes to calculate off-diagonal contributions.Source: §5 · p. 11
Validation
- Across numerous massive and massless processes, the matrix-element-to-dipole ratio approaches unity in soft or collinear limits, and comparisons with MCFM agree point-wise wherever anticipated.Differences with MCFM are attributed to implementation details.Source: §5 · pp. 11–12
Implications
- Together with developing automated one-loop calculations, MadDipole could contribute to full automation of NLO collider calculations.The authors describe such automation as important for precision phenomenology at the LHC.Source: §5 · p. 12