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Automation of next-to-leading order computations in QCD: the FKS subtraction

Rikkert Frederix, Stefano Frixione, Fabio Maltoni, Tim Stelzer

arXiv:0908.4272v2hep-ph

TL;DR

The paper addresses the difficulty of automating infrared-singularity cancellation for NLO QCD computations. It embeds the FKS subtraction formalism in MadGraph to automate cross-section calculations, leaving only the finite one-loop contribution external, and demonstrates a parallel-suitable implementation whose physical results are stable under subtraction-parameter choices.

  • Problem

    NLO multi-jet calculations require analytical cancellation and finite treatment of infrared singularities, with numerical integration becoming harder as external-leg multiplicity increases.

  • Method

    The authors embed the universal FKS subtraction formalism and MadGraph tree-level matrix-element generation into MadFKS, assuming ultraviolet-renormalized one-loop corrections are supplied.

  • Results

    MadFKS automates NLO QCD cross-section computation for processes expressible in MadGraph, with separately finite FKS regions suited to parallel computation and parameter-independent physical cross sections.

  • Takeaways & Limitations

    Dynamic phase-space partitioning limits each region to at most one soft and one collinear singularity, making subtraction numerically efficient and enabling independent parallel computations.

  • Takeaways & Limitations

    The implementation’s extension of FKS subtraction to hadronic collisions is deferred to future work.

Abstract

from arXiv · show

We present the complete automation of the universal subtraction formalism proposed by Frixione, Kunszt, and Signer for the computation of any cross section at the next-to-leading order in QCD. Given a process, the only ingredient to be provided externally is the infrared- and ultraviolet-finite contribution of virtual origin. Our implementation, currently restricted to the case of e+e- collisions, is built upon and works in the same way as MadGraph. It is particularly suited to parallel computation, and it can deal with any physical process resulting from a theory implemented in MadGraph, thus including the Standard Model as well as Beyond the Standard Model theories. We give results for some sample processes that document the performances of the implementation, and show in particular how the number of subtraction terms has an extremely mild growth with final-state multiplicity.

1. Introduction

The paper targets the need for reliable NLO QCD predictions for multi-jet processes, whose existing automated treatments were limited to LO accuracy. It fully automates FKS subtraction within the MadGraph framework, leaving only the finite one-loop contribution to be supplied externally.

  • Motivation: Multi-jet predictions were largely based on tree-level matrix elements, whose LO accuracy leaves large scale uncertainties for high-multiplicity cross sections.These uncertainties make absolute cross-section predictions unreliable and motivate extending calculations to NLO.
  • Contribution: The paper addresses the NLO implementation problem by assuming ultraviolet-renormalized one-loop corrections and automating the remaining computational steps.The automated steps include real-emission matrix elements and counterterms, finite combinations with loop terms, phase-space integration, and event-by-event output.
  • Implementation: Infrared-pole cancellation can be handled analytically because the pole structure is universal and its residues are proportional to automatically computable tree-level matrix elements.This enables the cancellation required when combining real and virtual contributions.
  • Implementation: The implementation combines MadGraph tree-level matrix elements with the universal Frixione, Kunszt, and Signer subtraction formalism.The resulting code is an NLO version of MadGraph/MadEvent up to the infrared- and ultraviolet-finite one-loop contribution.
  • Scope: The implementation is restricted to e+e- collisions but is designed for processes from MadGraph theories, including the Standard Model and user-defined Beyond the Standard Model theories.The formalism is written generally, and the authors state that extension to hadronic initial states requires only initial-state parton-emission implementation beyond parton-density factors.

2. Computations of multi-jet cross sections at the NLO

NLO multi-jet calculations require both loop corrections and finite, numerically integrable combinations of real-emission singularities. The paper uses FKS subtraction to automate these combinations while retaining a structure suited to parallel computation and parton-shower matching.

  • NLO challenges: NLO multi-jet calculations must cancel infrared singularities analytically and implement the resulting finite terms, including their phase-space integration.The first issue was already solved by universal cancellation formalisms, whereas the latter implementation challenge had received less attention for multi-jet cross sections.
  • FKS versus dipoles: FKS emphasizes collinear emissions and organizes local counterterms as sums over parton pairs, whereas dipole subtraction emphasizes soft emissions and sums over emitted partons and colour partners.The two methods are formally equivalent, but their local-counterterm building blocks are combined differently.
  • FKS structure: FKS partitions real-emission phase space into independent processes with at most one soft and one collinear singularity.Each resulting contribution can be computed separately, producing genuine parallelization rather than merely repeating one calculation with different random seeds.
  • Advantages: The paper advocates FKS for its collinear-emission structure, importance-sampling convenience, small and slowly growing subtraction-term count, parallel organization, and polarized-process support.These features are presented as reasons for using FKS in matrix-element predictions and for matching to parton showers.

3. Notation

The notation distinguishes Born and real-emission kinematics, particle classes, process labels, amplitudes, singular limits, and FKS pairs. FKS pairs identify the soft or collinear singularities that require subtraction after jet cuts.

  • Kinematics: Born contributions use 2 → n kinematics, while real-emission contributions use 2 → (n + 1) kinematics.The corresponding phase spaces and process sets are denoted separately for n-body and (n + 1)-body processes.
  • Particle content: The notation tracks non-strongly-interacting particles, massive strongly interacting particles, and light quarks or gluons separately across Born and real-emission processes.The real-emission class contains one additional light quark or gluon while preserving the corresponding non-strong and massive-particle content.
  • Process labels: A process is identified by the ordered list of particle identities, with equivalent final-state permutations represented only once.The notation also defines operations that remove an emitted parton or replace a sister particle according to the relevant QCD vertex.
  • FKS pairs: FKS pairs contain an FKS parton and sister whose soft or collinear limits generate singularities in the real-emission matrix element after jet cuts.Each pair maps to a separately finite subtraction contribution that can be integrated independently.
  • FKS pairs: For large multiplicities, many nominal FKS pairs are absent or identical, producing a much smaller set of independent subtraction terms.Absent pairs include combinations that cannot generate physical singularities, while matrix-element and phase-space symmetries make some remaining contributions identical.

4. Cross sections

The cross-section construction combines finite Born, soft, collinear, virtual, and subtracted real-emission contributions. FKS phase-space partitioning isolates singular limits into simple, independent terms and uses counterevents to obtain finite numerical integrands.

  • Cross-section decomposition: The factorization theorem separates collider-dependent incoming-particle functions from the short-distance n-body, (n + 1)-body, and degenerate contributions.For e+e- beams these functions may reduce to fixed-energy delta functions, while hadronic collisions use parton densities.
  • n-body contributions: The n-body contribution contains separately finite Born, collinear, soft, and one-loop-origin terms.The finite one-loop term is scheme-dependent and is defined in the Conventional Dimensional Regularization scheme used by the formula.
  • (n + 1)-body contributions: FKS partitions the real-emission phase space so each region contains at most one soft and one collinear singularity.The partition is implemented with positive-definite functions whose detailed choice does not affect the physical cross section.
  • Singular variables: The variables ξ_i and y_ij locate the soft and collinear limits through ξ_i = 0 and y_ij = 1, respectively.ξ_i is the rescaled FKS-parton energy, while y_ij is the cosine of its angle with the sister.
  • Degenerate contributions: Symmetry relations reduce repeated degenerate contributions and can lower computation time by a factor of p for p identical final-state particles.The resulting symmetry factors emerge naturally from the related counterevent configurations.

5. Implementation

The implementation automates the phase-space treatment and numerical evaluation of FKS-subtracted real-emission contributions. It constructs event and counterevent kinematics, evaluates finite singular-limit expressions, and uses parametrization choices that improve numerical stability.

  • Scope: The implementation described is applied to e+e− collisions, although the presented formulae can be used without modification for other incoming particles.Initial-state emissions require additional treatment, and the discussion of j = 1, 2 is deferred.
  • Kinematics: Integration of the subtracted real-emission matrix elements requires 3n −1 random numbers, including ξi, yij, and ϕi for the FKS parton.The remaining 3n −4 variables are denoted xα.
  • Kinematics: The generated configuration serves as the event from which soft, collinear, and soft-collinear counterevents are constructed.Counterevent momenta are related to an effectively 2 →n configuration through relabeling of the soft counterevent kinematics.
  • Kinematics: Three phase-space parametrizations were tested for convergence and numerical stability while preserving the required kinematic relations.The same ϕi and xα variables are used in event generation across parametrizations.
  • Kinematics: The required kinematic relations are optional for FKS implementation but improve differential-distribution stability and reduce computing time.They are motivated by the degeneracy of the counterevent configurations.
  • Singular limits: Smooth S functions are designed to remain well behaved across the full (n + 1)-body phase space, including soft and collinear limits.Their parameters aS and bS are arbitrary positive reals, while the physical cross section is independent of their choices.
  • Singular limits: Analytic cancellations of ξi and 1 −yij factors make the damped matrix elements numerically well behaved throughout phase space.The collinear limit can be used to construct the soft-collinear counterterm, and azimuthally nonzero terms must be retained in local collinear subtraction.

6. Optimization

The optimization reduces redundant FKS contributions and organizes n-body and (n + 1)-body terms for efficient integration. It also makes the resulting finite contributions suitable for separate computation and parallelization, while the presented derivation remains limited in part to e+e− implementations.

  • Non-redundant FKS pairs: The unreduced FKS contribution count scales approximately quadratically with the number of light final-state partons.The optimization targets this scaling by removing redundant contributions.
  • Symmetry reduction: Identical final-state particles produce identical FKS contributions, so the implementation retains one representative pair and multiplies by an integer symmetry factor.The reduction depends on the process and underlying theory.
  • Non-redundant FKS pairs: Non-redundancy conditions restrict FKS partons to be massless and remove the ordered-pair symmetry for non-gluon flavour combinations.For unordered pairs without soft singularities, only one ordered pair is retained.
  • Symmetry reduction: The optimized definitions of PFKS and the S-function weights are used throughout the MadFKS implementation.The modified formulae remain consistent with the earlier sections when the new definitions are applied.
  • n-body matrix elements: The n-body and (n + 1)-body contributions can be integrated together because the relevant n-body term has the same structure as the soft counterevent.This structure is also the default strategy in MC@NLO.
  • n-body matrix elements: The degenerate (n + 1)-body contribution can likewise be integrated with the other two contributions, but its derivation is deferred to future work on hadronic collisions.The paper states that this is the default strategy in MC@NLO.
  • n-body matrix elements: The implementation follows the underlying-Born-from-real-emission logic of MadGraph, while an inverse real-emission-from-Born organization is reserved for parton-shower matching.The inverse option is described in an appendix.
  • Multi-channel integration: FKS partitions real-emission integration into independent processes with at most one soft and one collinear singularity, enabling genuine parallel computation.The independent contributions are finite after subtraction, although integrating each can remain difficult at large multiplicity.

7. Results

MadFKS produces numerically stable NLO results for varied e+e− and µ+µ− processes, while its FKS-pair count grows only modestly with final-state multiplicity. The results validate the implementation and document its convergence, parameter independence, and parallelizable subtraction structure.

  • Scope: MadFKS results are intended to document convergence properties and implementation correctness rather than provide physical phenomenological predictions.The reported results use non-physical settings for this purpose.
  • Numerical checks: Three independent FKS pairs and five integration channels per pair contribute to e+e−→Z→u¯uggg.The calculation used 50,000 integration points and 10 iterations per channel, with about 15% of points passing the jet-finding conditions.
  • Numerical checks: The integration results are independent of the free parameters, and their errors scale approximately according to the Gaussian law expected for ordinary-function integrals.The authors use this behavior as evidence that Vegas estimates the integration errors correctly, despite the subtracted cross section being a distribution.
  • Numerical checks: The dσ(n+1)_0,β contribution is smaller than the statistical errors reported in table 1 and is omitted from those results.This occurs for the chosen pure-pole one-loop contribution in the CDR scheme.
  • Validation: Agreement with known e+e−→Z→2 jets and H→2 jets NLO cross sections validates all tested analytical formulae, apart from hadronic-only equations and separately checked non-massless eikonal integrals.The non-massless eikonal integrals were checked numerically with very high precision.
  • Scaling: The number of independent FKS pairs grows modestly with multiplicity: it remains constant when gluons are added in some processes and increases mainly through additional partonic subprocesses.For e+e−→(n+1)j processes with four, five, and six final-state particles, the numbers of contributing subprocesses are 7, 7, and 17; t¯tb¯bgg has one more FKS pair than t¯tb¯bg.
  • Distributions: The spectra use unsmoothed histograms with fine binning, exposing possible mis-binning when event and counterevent weights fall into different bins; most spectra are smooth, except polar-angle distributions with more evident t¯t fluctuations.The plots cover jet observables and, for t¯t processes, top-quark and hardest-jet energies and polar angles.

8. Conclusions

MadFKS automates NLO QCD cross-section calculations by embedding FKS subtraction into MadGraph, requiring only the finite one-loop contribution externally. Its design supports broad process coverage, parallel computation, parameter-based validation, and future extensions beyond the current scope.

  • MadFKS automates NLO QCD cross sections for any MadGraph-supported physical process, except for the finite infrared- and ultraviolet-finite one-loop contribution.This includes Standard Model and user-defined Beyond the Standard Model theories.
  • Dynamic phase-space partitioning limits each partonic process to at most one soft and one collinear singularity, simplifying subtraction.The construction also reduces the chance that oppositely signed, large weights land in different histogram bins.
  • The physical cross section is independent of FKS free parameters, although separate (n + 1)-body and n-body contributions depend on them.This parameter independence provides a correctness check and permits choices that do not compromise numerical stability.
  • Numerical integrations converge quickly and can be parallelized by assigning integration channels to separate jobs.The authors identify adaptive allocation of statistics to dominant or high-error channels as a straightforward improvement.
  • MadFKS can be made independent of Feynman-diagram information, while recursive amplitudes could extend multiplicity limits beyond current MadGraph constraints.Only the present multi-channel sampling step strictly depends on Feynman graphs.

A. Eikonal integrals

This appendix defines the eikonal-factor integrals, separating divergent and finite terms and extending the analytic treatment to massive configurations. The normalization and kinematic conventions are specified for consistency with earlier FKS results.

  • The eikonal integrals separate divergent terms in ˆEkl from finite terms in Ekl.The divergent part includes an ε-dependent prefactor, while the normalization follows the convention of earlier FKS work up to a stated factor.
  • The normalization of the eikonal-integral definition is conventional and differs from ref. [12] by a factor 8π2.The difference is compensated by a different normalization of the colour-linked Born contribution.
  • The parton energy and angular measure are defined in the colliding-parton centre-of-mass frame in 3 − 2ε dimensions.These conventions specify the kinematic frame and dimensional measure used in the integrals.
  • The appendix supplies the previously missing analytic result for the case of nonzero mass in the eikonal-integral calculation.Earlier work covered massless configurations and selected mixed-mass cases, but had not published this remaining result.
  • For equal emitter indices, the generalized equations coincide with the corresponding earlier equations.This provides a consistency check on the generalized expressions.

B. Finite one-loop contribution

The finite one-loop contribution is obtained after universal infrared poles are determined and ultraviolet divergences are removed, with scheme conventions fixing its unambiguous definition. The appendix relates CDR and DR treatments and specifies the required input conventions.

  • The divergent part of the ultraviolet-renormalized one-loop contribution is determined from real-emission divergences using KLN cancellation.The resulting poles have infrared, rather than ultraviolet, origin because ultraviolet divergences are removed by renormalization.
  • Equation (B.2) generalizes the infrared-pole structure to arbitrary particle masses and is consistent with prior results.Ultraviolet renormalization is assumed to have already eliminated ultraviolet divergences.
  • The Ellis-Sexton scale Q rewrites logarithm arguments in terms of sij/Q2 rather than ratios involving another invariant scale.Introducing Q is optional in the original virtual-amplitude computation, but the appendix explains how to insert its dependence afterward.
  • After interference with the Born amplitude, the finite one-loop contribution is solved for only after specifying a regularization scheme.The paper discusses Conventional Dimensional Regularization and Dimensional Reduction, whose quantities are evaluated in 4−2ε and 4 dimensions, respectively.
  • The finite contribution must use the CDR convention for the initial-state finite term F IN in eq. (4.14).This is a scheme-specific input condition for the subtraction formula.

C. How to set µF̸ = µR

The appendix extends the calculation to independent factorization, renormalization, and Ellis-Sexton scales by imposing scale invariance of the physical cross section. It provides conventions for reconstructing the finite one-loop contribution from a single-scale result.

  • The default convention assumes µ = µF = µR before the appendix generalizes to unequal scales.This establishes the baseline convention for the short-distance cross sections.
  • When µF ≠ µR, short-distance cross sections use µF while the strong coupling uses µR.The PDFs are likewise evaluated at µF.
  • The cross-section formulae account for the Born-level power b of αS, including b = 0 for purely electroweak Born processes.This power enters the scale-dependence conditions used in the appendix.
  • The scale-dependent one-loop terms are determined by imposing invariance of the physical cross section under renormalization and factorization scales.The derivation uses the explicit short-distance cross sections and PDF evolution equations.
  • Ellis-Sexton substitutions rewrite log^p(tk/tl) as [log(tk/Q2) − log(tl/Q2)]^p for p = 1, 2.This inserts the dependence on Q into one-loop expressions originally written with invariant ratios.
  • The method reconstructs a convention-compliant finite one-loop contribution from a result computed with a single scale M.The resulting calculation can use independent µF, µR, and Q assignments.

D. Azimuthal terms in collinear limits

The collinear-limit treatment retains an azimuthal interference term needed for local counterterms and handles convention matching between Mangano–Parke and HELAS polarizations. For off-shell-quark branchings, the azimuthal contribution vanishes because helicity is conserved along a light-quark line.

  • Collinear-limit structure: The collinear limit contains a second term that is equally important for constructing local counterterms, although it is often omitted in customary expressions.The reduced matrix element is computed following the procedure described in Appendix B of ref. [12].
  • Collinear-limit structure: The relevant interference involves the plus and minus helicity states of parton j ⊕i, while the remaining helicities are summed.The first curly-bracket term is a pure phase and remains numerically well defined in the soft limit Ei →0.
  • Universal kernels: For final-state and initial-state branchings, the Q kernels are treated separately, with the ⋆ marking the off-shell parton.The notation distinguishes timelike from spacelike branching forms already at leading order.
  • Azimuthal contribution: For branchings involving an off-shell quark, the azimuthal term vanishes identically because helicity is conserved along a light-quark line.This is the stated reason the corresponding interference contribution disappears.
  • Implementation conventions: The implementation must use identical spinor conventions for the amplitudes and pure-phase factor; MadGraph amplitudes use HELAS conventions.Explicit comparison with the Mangano–Parke representations yields the required conversion relations.

E. Possible variants in the implementation

The implementation admits alternative constructions of Born and real-emission processes, with generalized bookkeeping that can reduce repeated n-body computations and supports matching to parton showers. One proposed integration alternative may help processes with many massive strongly interacting particles, but it was not explored here.

  • Possible variants in the implementation: Implementation parameters define different computer realizations of the subtraction procedure, and two non-parametric variants are proposed for possible future use.The variants aim at further reducing computational effort.
  • Computational implications: For many massive strongly-interacting particles, an alternative integration form may reduce the number of independent integrations, but the paper does not explore it.The authors defer this study to future work.
  • Process construction: Real-emission processes can be generated from Born processes through formal a →bc branchings, even when the resulting pair has no collinear singularity.The example t →tg produces an extra-gluon final state despite the absence of a collinear singularity for the tg pair.
  • Process construction: For a real-emission process, particle pairs yielding an underlying Born structure are identified as FKS pairs through generalized process bookkeeping.The generalized Kronecker symbol organizes this identification while respecting process equivalence under final-state permutations.
  • Cross-section organization: The generalized construction assigns an (n + 1)-body cross section an n-body process as an argument and extends analogously to degenerate contributions.The same procedure applies because these contributions have the structure of initial-state collinear limits.
  • Cross-section organization: The Kronecker symbol makes the sum over (n + 1)-body processes nearly trivial by reducing it to counting branchings allowed by particle identities.The approach uses the FKS-pair sets that must already be constructed for the subtraction procedure.
  • Computational implications: Using the alternative equations computes n-body contributions fewer times, potentially reducing CPU time when the one-loop finite term V(n,1) is expensive.The same equations are required for POWHEG matching and can also be used for MC@NLO S-event generation.

F. Integration of matrix elements of given helicities

The FKS formalism can be extended to matrix elements with fixed helicities using polarized splitting kernels and modified collinear terms. The implementation requires careful helicity bookkeeping and, for the available polarized kernels, is restricted to MS PDFs.

  • Motivation and setup: Monte Carlo helicity summation is suited to large-multiplicity final states when matrix elements are available for every fixed-helicity configuration.MadGraph provides these fixed-helicity matrix elements, requiring only trivial modifications to the FKS formalism.
  • Motivation and setup: For fixed-helicity partonic processes, the general equations remain applicable after removing the implicit helicity sum from the matrix elements.The n-body and (n + 1)-body contributions retain the subtracted organization of the unpolarized formalism.
  • Polarized subtraction terms: The soft-limit counterterm is essentially unchanged for a fixed helicity configuration, whereas the collinear limit must resolve the helicity of the branching parton.At fixed hi and hj, the reduced branching parton j ⊕i can have either helicity.
  • Polarized subtraction terms: In the polarized collinear term, the interference contribution sums neither over the branching helicity nor over the two interfering helicity states, while the W kernels are helicity-independent.The quantity f M(n,0) otherwise follows the unpolarized definition without the helicity sum.
  • Polarized kernels: The T and W kernels are the fully polarized four-dimensional counterparts of the Altarelli–Parisi and Q kernels.Fixed-helicity Altarelli–Parisi kernels follow from the T kernels, and the stated identities recover the unpolarized relations.
  • Polarized kernels: The kernel relations used for these constructions hold in general in 4 −2ǫ dimensions.The four-dimensional fixed-helicity relations connect the T kernels to the relevant Altarelli–Parisi expressions.
  • Scope and scheme: The polarized extension is obtained from earlier work on FKS subtraction for polarized collisions, but the paper limits fixed-helicity use to MS PDFs because polarized K kernels are unavailable in other schemes.A finite scheme transformation affects the O(ǫ) term of the qq kernel.
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