[Paper Review] Automated Calculation of ${\pmb N}$-jet Soft Functions

Guido Bell, Bahman Dehnadi+2 more|arXiv (Cornell University)|Aug 22, 2018

Computability, Logic, AI Algorithms4 references4 citations

TL;DR

This paper presents a systematic, automated framework for computing N-jet soft functions in Soft-Collinear Effective Theory (SCET), extending prior dijet methods to general N ≥ 2 jets. The approach leverages non-Abelian exponentiation and dimensional regularization to compute next-to-next-to-leading order (NNLO) corrections for the N-jettiness soft function in hadron collisions, yielding new numerical results for 2-jettiness and full agreement with existing NNLO predictions for 1-jettiness.

ABSTRACT

We present a systematic framework for the calculation of soft functions that are defined in terms of $N\geq2$ light-like Wilson lines. The formalism represents an extension of a method that we developed earlier for the calculation of dijet soft functions to the general $N$-jet case. We discuss the technical aspects of this generalisation, focussing on SCET-1 soft functions that obey the non-Abelian exponentiation theorem in this contribution. As a first application of our method, we consider the $N$-jettiness observable and present numerical results for the $1$-jettiness and $2$-jettiness hadron-collider soft functions to next-to-next-to-leading order in the perturbative expansion.

Motivation & Objective

• To extend a prior method for dijet soft functions to the general N-jet case in SCET.
• To develop a systematic, automated framework for computing soft functions with N ≥ 2 light-like Wilson lines.
• To compute the N-jettiness soft function at next-to-next-to-leading order (NNLO) for hadron collider processes.
• To validate results against renormalization group equations and existing NNLO calculations for 1-jettiness.
• To provide a general-purpose tool applicable to other event shapes and boosted observables in high-energy QCD.

Proposed method

• The method uses a formalism based on non-Abelian exponentiation (NAE) for SCET-1 soft functions, which are free from rapidity divergences.
• It computes soft functions in Laplace space using dimensional regularization with ε = (4−d)/2, expressing the perturbative expansion in powers of αs.
• At NLO, the soft function is computed via dipole contributions from real emissions between different Wilson lines, with matrix elements |A_ij(k)|² = 16π²n_ij/(k_i k_j).
• At NNLO, the method computes virtual corrections and triple-collinear (tripole) contributions, using color conservation to express the final result in terms of structure constants f_ABC and color generators T_i.
• The calculation employs numerical integration via pySecDec, with results cross-checked against renormalization group equations and known fit functions.
• The framework is general and can be applied to other N-jet observables, including boosted top quark or event shape measurements.

Experimental results

Research questions

• RQ1 How can the automated computation of N-jet soft functions be systematically extended beyond the dijet case in SCET?
• RQ2 What are the NNLO corrections to the 1-jettiness and 2-jettiness soft functions for hadron collider processes?
• RQ3 How do the divergent and finite parts of the soft function at NNLO compare with predictions from the renormalization group equation?
• RQ4 To what extent can the new framework reproduce known results for 1-jettiness, and what new results does it yield for 2-jettiness?
• RQ5 Can the method be generalized to other N-jet event shapes and observables in high-energy QCD?

Key findings

• The NLO soft function coefficients are computed via dipole contributions S_ij^(1)(ε) = (n_ij/2)^ε S_ij^(1)(ε), with the dipole integrals evaluated using boost-invariant variables k_T and y.
• The NNLO soft function for 1-jettiness is in perfect agreement with the fit functions from [20], validating the numerical implementation.
• The NNLO soft function for 2-jettiness is computed for the first time, providing the final missing ingredient for applying N-jettiness subtraction to dijet processes.
• The divergent parts of the bare soft function at NNLO match the predictions from the renormalization group equation, confirming consistency of the framework.
• The finite, non-logarithmic term at NNLO (proportional to δ(𝒯₁)) is computed numerically and matches the fit functions from [20] with high precision.
• The tripole contributions to the 2-jettiness soft function are computed and found to be in perfect agreement with the RG equation predictions, confirming the correctness of the color structure and integration.