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The NNLO soft function for the pair invariant mass distribution of boosted top quarks

Andrea Ferroglia, Ben D. Pecjak, Li Lin Yang

TL;DR

The paper addresses the calculation of the NNLO corrections to the soft function governing the pair invariant mass distribution of boosted top quarks in hadron collisions, focusing on the m_t << M regime. The authors formulate the soft function as a four-Wilson-line vacuum expectation value in color space and compute the bare NNLO contributions by summing two-, three-, and four-line emission diagrams, leveraging non-abelian exponentiation to dramatically simplify the color structure. They perform a detailed renormalization using the Laplace-transformed soft function and verify consistency with known two-loop anomalous dimensions, yielding finite, renormalized NNLO results (with full expressions provided in the Appendix). The results enable a full virtual-plus-soft NNLO description of the invariant-mass distribution and suggest avenues for extending threshold resummation to related observables such as single-particle p_T distributions and dijet production.

Abstract

At high values of the pair invariant mass the differential cross section for top-quark pair production at hadron colliders factorizes into soft, hard, and fragmentation functions. In this paper we calculate the next-to-next-to-leading-order (NNLO) corrections to the soft function appearing in this factorization formula, thus providing the final piece needed to evaluate at NNLO the differential cross section in the virtual plus soft approximation in the large invariant-mass limit. Technically, this amounts to evaluating the vacuum expectation value of a soft Wilson loop operator built out of light-like Wilson lines for each of the four partons participating in the hard scattering process, with a certain constraint on the total energy of the soft radiation. Our result turns out to be surprisingly simple, because in the sum of all graphs the three and four parton contributions multiply color structures whose coefficients are governed by the non-abelian exponentiation theorem.

The NNLO soft function for the pair invariant mass distribution of boosted top quarks

TL;DR

The paper addresses the calculation of the NNLO corrections to the soft function governing the pair invariant mass distribution of boosted top quarks in hadron collisions, focusing on the m_t << M regime. The authors formulate the soft function as a four-Wilson-line vacuum expectation value in color space and compute the bare NNLO contributions by summing two-, three-, and four-line emission diagrams, leveraging non-abelian exponentiation to dramatically simplify the color structure. They perform a detailed renormalization using the Laplace-transformed soft function and verify consistency with known two-loop anomalous dimensions, yielding finite, renormalized NNLO results (with full expressions provided in the Appendix). The results enable a full virtual-plus-soft NNLO description of the invariant-mass distribution and suggest avenues for extending threshold resummation to related observables such as single-particle p_T distributions and dijet production.

Abstract

At high values of the pair invariant mass the differential cross section for top-quark pair production at hadron colliders factorizes into soft, hard, and fragmentation functions. In this paper we calculate the next-to-next-to-leading-order (NNLO) corrections to the soft function appearing in this factorization formula, thus providing the final piece needed to evaluate at NNLO the differential cross section in the virtual plus soft approximation in the large invariant-mass limit. Technically, this amounts to evaluating the vacuum expectation value of a soft Wilson loop operator built out of light-like Wilson lines for each of the four partons participating in the hard scattering process, with a certain constraint on the total energy of the soft radiation. Our result turns out to be surprisingly simple, because in the sum of all graphs the three and four parton contributions multiply color structures whose coefficients are governed by the non-abelian exponentiation theorem.

Paper Structure

This paper contains 13 sections, 64 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definitions and the NLO calculation
  3. The bare soft function at NNLO
  4. Two-Wilson-line integrals
  5. Three-Wilson-line integrals
  6. Four-Wilson-line integrals
  7. Renormalization
  8. Conclusions
  9. Calculation of $I_5$ as an example
  10. Constraints from non-abelian exponentiation
  11. Renormalized Soft Functions
  12. Quark Annihilation Channel
  13. Gluon Fusion Channel

Figures (6)

  • Figure 1: Diagram contributing to the NLO soft function.
  • Figure 2: Two-Wilson-line integrals required in the calculation of the NNLO soft matrix.
  • Figure 3: The abelian three-Wilson-line integrals required in the calculation of the NNLO soft matrix.
  • Figure 4: Example of a pair of mixed virtual-real one-particle cuts which adds up to a scaleless integral.
  • Figure 5: Examples of non-abelian three-Wilson-line integrals required in the calculation of the NNLO soft matrix.
  • ...and 1 more figures