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Numerical evaluation of virtual corrections to multi-jet production in massless QCD

Simon Badger, Benedikt Biedermann, Peter Uwer, Valery Yundin

arXiv:1209.0100v1hep-ph

TL;DR

NLO predictions for multi-jet production require one-loop virtual corrections, whose computation for many final-state particles remains challenging. NJET uses generalized unitarity and off-shell recursion to construct multi-fermion primitive amplitudes and full-colour squared-amplitude formulae. The library provides formulae for channels contributing to up to five-jet production and performs favourably against existing public generators for multi-gluon states.

  • Problem

    One-loop virtual corrections for multi-particle processes are theoretically challenging, while automation still needs improvements in efficiency and flexibility.

    Source: §1 · p. 2

  • Method

    NJET combines generalized unitarity and off-shell recursion for multi-fermion primitive amplitudes, then matches primitives to partial amplitudes to construct full-colour formulae.

    Source: §5 · pp. 21–22

  • Results

    NJET shows favourable performance compared with existing public one-loop amplitude generators for multi-gluon states, with full-colour formulae covering channels contributing to up to five-jet production.

    Source: §5 · p. 22

  • Takeaways & Limitations

    The full-colour formulae provide the colour information needed to use multi-fermion primitive amplitudes in NLO QCD computations for the covered channels.

    Source: §5 · p. 22

    Heavy-quark corrections are left for future development.

    Source: §5 · p. 22

Abstract

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We present a C++ library for the numerical evaluation of one-loop virtual corrections to multi-jet production in massless QCD. The pure gluon primitive amplitudes are evaluated using NGluon. A generalized unitarity reduction algorithm is used to construct arbitrary multiplicity fermion-gluon primitive amplitudes. From these basic building blocks the one-loop contribution to the squared matrix element, summed over colour and helicities, is calculated. No approximation in colour is performed. While the primitive amplitudes are given for arbitrary multiplicities we provide the squared matrix elements only for up to 7 external partons allowing the evaluation of the five jet cross section at next-to-leading order accuracy. The library has been recently successfully applied to four jet production at next-to-leading order in QCD.

1 Introduction

NLO corrections are essential for quantitative predictions of large QCD backgrounds, but one-loop virtual corrections with four or more final-state particles remain challenging. NJET addresses the continuing need for more efficient and flexible automation by providing a generalized-unitarity library for multi-jet amplitudes.

1 Introduction

  • NLO corrections to multi-particle Standard Model processes at hadron colliders are essential for quantitative predictions of large QCD backgrounds to new-physics scenarios.Source: §1 · p. 2
  • One-loop virtual corrections with four or more final-state particles remain a theoretical challenge, although unitarity, generalized unitarity, integrand reduction, and refined traditional approaches have advanced.Source: §1 · p. 2
  • Automated approaches and on-shell methods have enabled more phenomenological studies, but efficiency and flexibility still lag the automation of tree-level predictions.Source: §1 · p. 3
  • NJET is a C++ library for one-loop multi-jet amplitudes in massless QCD, using generalized unitarity and covering primitive amplitudes with arbitrary numbers of fermion pairs.The paper also describes colour- and helicity-summed squared amplitudes, package usage, validation, and performance tests.Source: §1 · p. 3

2 Methods

NJET extends generalized unitarity to construct fermion–gluon primitive amplitudes and combines primitives into full-colour, helicity-summed squared amplitudes. Its methods use recursive tree amplitudes, diagram matching, and caching, with scheme conversion handled through universal pole structure.

2 Methods

  • Generalized unitarity maps one-loop amplitudes to linear systems built from on-shell amplitudes, avoiding the complexity growth of individual Feynman diagrams.NGLUON supplies the pure-gluon primitive amplitudes; the extension adds primitives with external and internal quarks.Source: §2 · p. 3, §2.1 · pp. 3–4

2.1 Multiple Fermion Primitive Amplitudes

  • The fermion extension constructs primitive amplitudes with arbitrary external multiplicity and internal quarks.Parent diagrams define the ordered propagator structure; cut factorization and cached Berends–Giele recursion scale as O(n^4), while four-dimensional massive cuts recover D-dimensional rational terms in FDH.Source: §2.1 · pp. 3–4

2.2 Partial Amplitudes and Colour Summation

  • The partial amplitudes are combined with colour matrices to form colour-summed squared amplitudes, without restricting the colour treatment to leading colour.The leading-colour approximation reduces the number of primitives, but its validity remains for future study.Source: §2.2 · pp. 5, 9
  • Matching coefficients are +1, −1, or 0 when an ordered primitive diagram matches a kinematic topology (even-ordered, odd-ordered, or no match); RREF guarantees a minimal primitive basis.Exploiting Furry-type symmetries yields fewer primitive amplitudes in multi-fermion channels than the approach of Ita and Ozeren.Source: §2.2 · pp. 6–8

2.3 Cache system for tree amplitudes

  • Partial trees are cached for reuse when external momenta, flavours, and helicities match and the cut topology has the same momentum-sum set Kτ.The implementation stores the four helicity combinations of the two loop-connecting partons together.Source: §2.3 · pp. 9–10

2.4 Universal Pole Structure and Scheme Dependence

  • Universal infrared and ultraviolet pole structure supports conversion from FDH to CDR and renormalization, both proportional to the Born amplitude.The pole relations are given for colour-summed squared amplitudes in massless QCD.Source: §2.4 · pp. 10–11

3 Installation and Usage

Section 3 documents installation and two usage paths: BLHA for process-level evaluation and NParton for primitive amplitudes.

3 Installation and Usage

  • The library is distributed as a tar archive or public Git source, with autoconf configuration, optional demos, and basic installation tests.Source: §3 · pp. 11–12

3.1 Interface via the Binoth Les Houches Accord

  • The BLHA interface uses an order file to generate a contract, then evaluates a selected subprocess at a supplied phase-space point.Source: §3.1 · pp. 12–13

3.1.1 Additional BLHA options

  • BLHA options select the returned matrix-element type, request accuracy estimates, set the precision-switch threshold, and choose the light-flavour count.Source: §3.1.1 · p. 13

3.2 The NParton library for multi-parton primitives

  • NParton evaluates arbitrary-multiplicity multi-parton primitive amplitudes at tree level and one loop.Source: §3.2 · p. 14

3.2.1 Initialization and Namespaces

  • NParton supports double and quadruple precision in parallel, with static initialization objects required before use.Source: §3.2.1 · p. 14

3.2.2 Constructor

  • The NParton constructor takes the number of external legs and is instantiated in the namespace for the desired precision.Source: §3.2.2 · p. 14

3.2.3 Member functions

  • After setAmp supplies momenta and flavours, eval or evalTree computes helicity amplitudes; getters expose finite, pole, tree, rational, and cut-constructible components.Source: §3.2.3 · pp. 14–15
  • IRcheck compares the 1/ε² and 1/ε poles with analytic universal pole formulae after eval.Source: §3.2.3 · p. 15

4 Results

NJET is checked against pole structures, analytic amplitudes, and existing codes, while accuracy tests characterize precision requirements and speed tests report scaling and evaluation costs.

4.1 Validation

  • Pole-structure tests and independent analytic and code comparisons support NJET’s validation, with four-jet results matching BLACKHAT in all reliable digits and every digit in quadruple precision.The checks also cover primitive amplitudes and colour-summed results.Source: §4.1 · pp. 16–17
  • Five-jet virtual amplitudes had not previously appeared in the literature, so the paper reports numerical values at a single phase-space point.Source: §4.1 · p. 18

4.2 Accuracy

  • The scaling test estimates valid digits on sequentially generated, collinear-preferring phase-space points; more quark pairs improve accuracy, while Gram determinants are only one instability source.About 1% of points in the most complex channels require quadruple-precision reevaluation.Source: §4.2 · p. 18
  • 99.99% of five-jet points achieve the required four-digit accuracy in quadruple precision, though octuple precision might be needed for phase-space integration.Source: §4.2 · p. 18

4.3 Speed Tests

  • The cached algorithm scales polynomially in the tested range up to 13 external legs; its asymptotic powers are n^8 for mixed loops and n^7 for closed quark loops, reached above 20 legs.The reported scaling analysis averages helicity configurations and uses fixed parton ordering.Source: §4.3 · pp. 18–19
  • Quadruple-precision evaluation takes 7–8 times longer per evaluation, but reevaluation only doubles total runtime even for the seven-gluon amplitude.De-symmetrized sums exploit final-state gluon Bose symmetry while retaining the same full colour information after phase-space integration or explicit summation.Source: §4.3 · p. 21

5 Conclusions

NJET combines generalized unitarity and off-shell recursion to construct multi-fermion amplitudes, then assembles full-colour results for multi-jet NLO calculations. Its interface supports integration with Monte Carlo programs, and it has been applied to three- and four-jet cross sections.

5 Conclusions

  • Generalized unitarity with off-shell recursion constructs multi-fermion primitive amplitudes at computational cost growing polynomially with time.Source: §5 · p. 21
  • Momentum-rescaling accuracy estimates identify when higher precision is needed, while caching makes helicity and permutation sums more efficient.Source: §5 · p. 22
  • Full-colour formulae cover all channels contributing to up to five-jet production, with symmetry techniques reducing colour sums and favourable performance for multi-gluon states.The authors report favourable performance in comparison with existing public one-loop amplitude generators.Source: §5 · p. 22
  • The Binoth Les Houches Accord interface allows NJET to link with parton-level Monte Carlo programs; the library has been used for three- and four-jet NLO cross sections.Benchmark points are also provided for the one-loop corrections needed for five-jet NLO calculations.Source: §5 · p. 22
  • Heavy-quark corrections are identified as a possible future development.Source: §5 · p. 22

A Numerical cross checks

The numerical checks compare NJET with HELAC-1LOOP across available dijet, three-jet, and four-jet channels, and report good numerical agreement. The appendices also list checks for additional multi-parton channels.

A Numerical cross checks

  • For the available dijet, three-jet, and four-jet production channels, NJet-demo--Helac1l provides a numerical cross-check against HELAC-1LOOP [90], with good numerical agreement; the exact phase-space points are displayed to double precision when running the program.Source: A Numerical cross checks · p. 23
  • For 0 → g g g g, the finite-term relative difference between NJET and HELAC-1LOOP is 1.6e-16.Source: A Numerical cross checks · p. 23
  • The five-parton checks include gluon and quark channels, with NJET and HELAC-1LOOP values and relative differences listed for each.Source: A Numerical cross checks · pp. 23–24
  • The six-parton checks list NJET and HELAC-1LOOP results for quark-gluon and multiple quark-pair channels.Source: A Numerical cross checks · pp. 24–25
  • For 0 → u~ u u~ u u~ u, the finite-term relative difference between NJET and HELAC-1LOOP is 5.8e-13.Source: A Numerical cross checks · p. 25

B Numerical evaluation for 7-parton virtual corrections

The appendix evaluates the independent channels contributing to virtual corrections for pp → 5 jets at a single phase-space point. It specifies the input parameters and presents the numerical amplitudes in Table 6.

B Numerical evaluation for 7-parton virtual corrections

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