Three-jet cross sections to next-to-leading order

S. Frixione; Z. Kunszt; A. Signer

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Three-jet cross sections to next-to-leading order

S. Frixione, Z. Kunszt, A. Signer

arXiv:hep-ph/9512328v1Dec 16, 1995hep-ph

TL;DR

The paper addresses the cumbersome subtraction treatment of next-to-leading-order three-jet production. It reformulates subtraction with angle and energy variables, supplies analytic soft and collinear results, and extends the formalism to n-jet production and other collision processes. The resulting expressions separate finite numerical terms from divergent contributions whose cancellation is explicit.

Problem
Next-to-leading-order three-jet predictions require analytically cancelling soft and collinear singularities, while the existing boost-invariant subtraction formulation is cumbersome for three or more jets.
Method
The paper uses angle and energy variables, exploits measurement functions to separate singular regions, and analytically computes soft and collinear subtraction contributions.
Results
The subtracted next-to-leading-order contribution is expressed as finite two-to-four and two-to-three kinematic terms, with remaining divergences cancelling in their sum.
Takeaways & Limitations
The formalism provides the analytical ingredients for efficient numerical three-jet calculations and extends to n-jet production in hadron, e+e−, and photon-hadron collisions.

Abstract

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One- and two-jet inclusive quantities in hadron collisions have already been calculated to next-to-leading order accuracy, using both the subtraction and the cone method. Since the one-loop corrections have recently been obtained for all five-parton amplitudes, three-jet inclusive quantities can also be predicted to next-to-leading order. The subtraction method presented in the literature is based on a systematic use of boost-invariant kinematical variables, and therefore its application to three-jet production is quite cumbersome. In this paper we re-analyze the subtraction method and point out the advantage of using angle and energy variables. This leads to simpler results and it has complete generality, extending its validity to $n$-jet production. The formalism is also applicable to $n$-jet production in $e^+e^-$ annihilation and in photon-hadron collisions. All the analytical results necessary to construct an efficient numerical program for next-to-leading order three-jet inclusive quantities in hadroproduction are given explicitly. As new analytical result, we also report the collinear limits of all the two-to-four processes.

S. Frixionea , Z. Kunszt

The paper lists affiliations with Theoretical Physics at ETH Zurich and SLAC, and acknowledges support from the National Swiss Foundation.

The authors are affiliated with Theoretical Physics at ETH, Zurich, Switzerland, and SLAC in Stanford, California.
The work was supported by the National Swiss Foundation.

1. Introduction

The paper motivates extending next-to-leading-order QCD predictions from one- and two-jet observables to three-jet production. It simplifies subtraction with angle and energy variables and supplies the analytical ingredients for numerical implementation.

Motivation: Three-jet data motivate next-to-leading-order analysis, including because three-to-two-jet rate ratios could provide a clean hadron-collider measurement of αS.
Motivation: Three-jet calculations require five-parton loop amplitudes, leading-order six-parton amplitudes, and analytical cancellation of soft and collinear singularities.
Subtraction method: The prior subtraction formulation becomes cumbersome for three or more jets because boost-invariant variables make soft integrals more complicated than necessary.
Subtraction method: Using angle and energy variables simplifies soft integrals and extends the subtraction formalism to inclusive production of any number of jets.
Subtraction method: The measurement function automatically separates the squared amplitude into terms with at most one or two contributing singular regions, avoiding impractical partial fractioning.
Paper organization and contributions: The paper constructs local subtraction terms, analytically evaluates soft and collinear contributions, and provides formulae for efficient numerical three-jet calculations.

2. The jet cross section

This section formulates the hadronic three-jet cross section, its measurement functions, jet algorithm, and collinear counterterms. It organizes the next-to-leading-order calculation so infrared singularities can cancel analytically.

2.1. Introductory remarks: The hadronic cross section convolves parton densities with subtracted partonic cross sections whose initial-state collinear singularities are removed by counterterms.
2.1. Introductory remarks: In the MS scheme, the finite subtraction-scheme functions K_ad vanish, while the counterterms cancel initial-state collinear singularities against unsubtracted cross sections.
2.1. Introductory remarks: The measurement function S3 defines infrared-safe jet observables from unobservable parton momenta, while M(3,0) represents the leading-order two-to-three transition amplitude squared with flux normalization.
2.1. Introductory remarks: At next-to-leading order, virtual two-to-three and real two-to-four subprocesses are combined with modified four-parton measurement functions.
2.2. The jet-finding algorithm: The Ellis–Soper kT algorithm iteratively promotes protojets or merges them by minimum distance, using four-momentum addition and dropping jets below an event-dependent transverse-momentum threshold.
2.3. The measurement function: For four final-state partons yielding three jets, the measurement function includes one dropped parton or one merging occurring at one of three algorithmic stages.
2.4. Infrared singular regions: Analytical cancellation occurs only after real and virtual contributions are summed at the partonic level.

3. Virtual contribution

The virtual contribution has a simple divergent structure despite complicated loop amplitudes. Its dimensional-regularization conventions and finite terms are specified through color-linked Born amplitudes and flavor-dependent factors.

3. Virtual contribution: Although loop corrections to two-to-three subprocesses are complicated, their divergent terms have a simple structure.
3. Virtual contribution: The virtual contribution uses the renormalization scale μ and an arbitrary mass scale Q introduced to facilitate the expression of the loop result.
3. Virtual contribution: Color-linked Born squared amplitudes are symmetric in their indices and enter the organization of the virtual terms.
3. Virtual contribution: The finite non-divergent terms are collected in M(3,1)_NS, with explicit amplitudes reported for the relevant five-parton processes.
3. Virtual contribution: Converting dimensional-reduction results to conventional dimensional regularization requires evaluating Born quantities in 4−2ε dimensions and modifying the finite part.

4. Real contribution

The real contribution is decomposed into terms with controlled soft and collinear behavior, using angle- and energy-based variables to make each singular region analytically tractable. The resulting master equation separates singularities so the real, virtual, and counterterm contributions can ultimately be combined into a finite numerical calculation.

Kinematic variables: The partonic center-of-mass parametrization makes ξ_i → 0 the soft limit and y_i → ±1 the limits collinear to the incoming partons.The variables also include a transverse angular measure in 2 − 2ε dimensions.
Soft singularities: Soft singularities are regulated with ξ_i^2 and separated using an arbitrary cutoff ξ_cut, whose distributions isolate the soft contribution for analytic integration.Different ξ_cut values may be chosen for different partons.
Collinear singularities: Initial- and final-state collinear singularities are separated into terms containing only their respective singularities, with subtraction prescriptions rendering the remaining contribution finite and numerically integrable.The initial-state treatment uses Altarelli–Parisi kernels and a cutoff parameter δ_I.
Singularity decomposition: The real contribution is decomposed so every term receives contributions from at most two singular regions, enabling explicit singular integrations and eventual finiteness after combining real, virtual, and counterterm parts.The decomposition is treated as the master equation for the real contribution.
Cancellation of poles: The initial-state pole from the flavour-diagonal Altarelli–Parisi kernel cancels the corresponding soft-virtual term, leaving a finite remainder for numerical evaluation.The construction also shows cancellation of other singular terms against final- and initial-state contributions or the virtual contribution.
Final-state singularities: The final-state singular contribution factorizes into a two-to-three partonic cross section multiplied by a term that completely describes the Altarelli–Parisi collinear splitting.This factorization organizes the final-state singularity independently of the underlying reduced cross section.

5. Results

The subtracted next-to-leading-order contribution is organized into finite two-to-four and two-to-three kinematic components. Its initial-state collinear part includes an additional ξ folding, while the calculation is most conveniently implemented in the partonic center-of-mass frame with subtraction parameters providing numerical checks.

Finite NLO organization: The subtracted next-to-leading-order contribution is written as a sum of a two-to-four partonic term and a two-to-three partonic term.The two-to-four term is finite term by term, while divergences in the two-to-three components cancel in their sum.
Finite NLO organization: Every term in the two-to-four representation is finite and can be numerically integrated, whereas the divergent two-to-three ingredients combine to give a finite partonic cross section.This provides the structure needed for numerical evaluation after analytic cancellation.
Numerical implementation: The finite initial-state collinear contribution is more complicated than ordinary two-to-three kinematics because it contains an additional folding in ξ, motivating a separate numerical treatment.The authors split the two-to-three contribution into folded and non-folded terms.
Scope and implementation: Although the derivation uses the partonic center-of-mass frame, the associated constraint on independent Bjorken x variables can be relaxed without affecting correctness.The authors nevertheless find the partonic frame advantageous for numerical computations.
Numerical implementation: Independence of the physical result from ξ_cut, δ_I, and δ_O supplies a numerical correctness check, while suitable parameter choices can reduce computing time.These parameters define the soft and collinear subtractions.

6. Conclusions

The paper simplifies NLO three-jet calculations through angle and energy variables and measurement-function-based singularity separation, while organizing results for numerical implementation and broader jet-production applications.

6. Conclusions: Angle and energy variables simplify eikonal-factor integration, while the measurement function separates soft and collinear singularities without partial fractioning.The resulting cross section is split into single-singular contributions.
6. Conclusions: The analytical results are organized for numerical computations, with real-plus-virtual contributions separated by final-state partonic kinematics.Terms may be redefined by finite pieces without changing the physically meaningful sum.
6. Conclusions: The formalism extends beyond explicit three-jet hadroproduction to n-jet production in hadron, e+e−, and photon-hadron collisions.

APPENDIX A: Soft and collinear integrals

Appendix A collects the soft and collinear integral results used in the subtraction formalism, including angular-variable decompositions that isolate poles from finite contributions.

APPENDIX A: Soft and collinear integrals: The appendix supplies integrated eikonal factors and Altarelli-Parisi kernels for insertion into the subtraction formulae.The collinear integrals use the prescriptions defined in eqs. (4.84) and (4.85).
APPENDIX A: Soft and collinear integrals: The angular measure is defined in 3 −2ǫ dimensions, and rotational invariance permits a convenient redefinition of the emitted momentum’s angular variables.The appendix parameterizes the momentum using polar angles.
APPENDIX A: Soft and collinear integrals: Soft-integral decompositions isolate a collinear pole in one term while subtractions regulate another term, leaving a finite contribution.
APPENDIX A: Soft and collinear integrals: The final integrated expressions follow from algebraic manipulation of the preceding equations.

APPENDIX B: Collinear limits

Appendix B derives collinear limits for final- and initial-state emissions, relating n-particle amplitudes to lower-multiplicity amplitudes through splitting functions and normalization factors.

APPENDIX B: Collinear limits: The collinear-limit results are general because the unspecified remaining system X does not affect the derivation.Their validity is therefore not restricted to jet physics.
APPENDIX B: Collinear limits: Final-state collinear emissions are treated through splitting processes, beginning with the i and j collinear limit and considering three cases.The splitting functions encode the relevant parton flavours and momentum fraction z.
APPENDIX B: Collinear limits: Table 1 enumerates splitting processes for possible parton flavours and helicities, with splitting functions obtained by dividing entries by <ab>[ab].
APPENDIX B: Collinear limits: The appendix connects the general n-particle amplitude formula to the three-jet NLO quantities M(4) and M(3,0).
APPENDIX B: Collinear limits: For q →qg splitting, no R term appears because helicity conservation along the quark line forbids both helicities for the exchanged virtual particle.
APPENDIX B: Collinear limits: The collinear amplitude structure combines a splitting kernel with a lower-multiplicity amplitude, including an off-shell flavour label for the eventual splitter.The ∆ term is identified with the corresponding helicity-dependent lower-multiplicity contribution.
APPENDIX B: Collinear limits: Initial-state collinear emission is obtained by exploiting crossing symmetry, with z and colour-spin normalization factors restoring the correct amplitude normalization.