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Threshold Hadronic Event Shapes with Effective Field Theory

Randall Kelley, Matthew D. Schwartz

TL;DR

This paper develops a Soft-Collinear Effective Theory (SCET) framework to study hadronic dijet event shapes, introducing threshold thrust as a global, parameter-free observable. It constructs a factorization theorem with color-space hard and soft functions, inclusive jet functions, and PDFs, diagonalizes color evolution to enable NNLL resummation, and explicitly computes the NLO soft function to validate RG consistency. The work highlights dynamical threshold enhancement, showing how threshold-resummed effects can influence phenomenology away from strict threshold and proposing asymmetric thrust as a globally well-behaved observable to exploit these resummations. Together, these results provide a blueprint for NNLL resummation of hadronic event shapes and guide future explorations of non-global logs and broader observables at hadron colliders.

Abstract

Hadronic event shapes, that is, event shapes at hadron colliders, could provide a great way to test both standard and non-standard theoretical models. However, they are significantly more complicated than event shapes at e+e- colliders, involving multiple hard directions, multiple channels and multiple color structures. In this paper, hadronic event shapes are examined with Soft-Collinear Effective Theory (SCET) by expanding around the dijet limit. A simple event shape, threshold thrust, is defined. This observable is global and has no free parameters, making it ideal for clarifying how resummation of hadronic event shapes can be done in SCET. Threshold thrust is calculated at next-to-leading fixed order (NLO) in SCET and resummed to next-to-next-to-leading logarithmic accuracy (NNLL). The scale-dependent parts of the soft function are shown to agree with what is expected from general observations, and the factorization formula is explicitly shown to be renormalization group invariant to 1-loop. Although threshold thrust is not itself expected to be phenomenologically interesting, it can be modified into a related observable which allows the jet pT distribution to be calculated and resummed to NNLL+NLO accuracy. As in other processes, one expects resummation to be important even for moderate jet momenta due to dynamical threshold enhancement. A general discussion of threshold enhancement and non-global logs in hadronic event shapes is also included.

Threshold Hadronic Event Shapes with Effective Field Theory

TL;DR

This paper develops a Soft-Collinear Effective Theory (SCET) framework to study hadronic dijet event shapes, introducing threshold thrust as a global, parameter-free observable. It constructs a factorization theorem with color-space hard and soft functions, inclusive jet functions, and PDFs, diagonalizes color evolution to enable NNLL resummation, and explicitly computes the NLO soft function to validate RG consistency. The work highlights dynamical threshold enhancement, showing how threshold-resummed effects can influence phenomenology away from strict threshold and proposing asymmetric thrust as a globally well-behaved observable to exploit these resummations. Together, these results provide a blueprint for NNLL resummation of hadronic event shapes and guide future explorations of non-global logs and broader observables at hadron colliders.

Abstract

Hadronic event shapes, that is, event shapes at hadron colliders, could provide a great way to test both standard and non-standard theoretical models. However, they are significantly more complicated than event shapes at e+e- colliders, involving multiple hard directions, multiple channels and multiple color structures. In this paper, hadronic event shapes are examined with Soft-Collinear Effective Theory (SCET) by expanding around the dijet limit. A simple event shape, threshold thrust, is defined. This observable is global and has no free parameters, making it ideal for clarifying how resummation of hadronic event shapes can be done in SCET. Threshold thrust is calculated at next-to-leading fixed order (NLO) in SCET and resummed to next-to-next-to-leading logarithmic accuracy (NNLL). The scale-dependent parts of the soft function are shown to agree with what is expected from general observations, and the factorization formula is explicitly shown to be renormalization group invariant to 1-loop. Although threshold thrust is not itself expected to be phenomenologically interesting, it can be modified into a related observable which allows the jet pT distribution to be calculated and resummed to NNLL+NLO accuracy. As in other processes, one expects resummation to be important even for moderate jet momenta due to dynamical threshold enhancement. A general discussion of threshold enhancement and non-global logs in hadronic event shapes is also included.

Paper Structure

This paper contains 21 sections, 112 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Hadronic event shapes in SCET
  3. Wilson coefficients and the hard function
  4. Jet functions
  5. Parton distribution functions
  6. Soft function: general observations
  7. Threshold Thrust
  8. Kinematics
  9. Threshold limit
  10. Soft function
  11. NLO soft function
  12. Soft RGE
  13. Checking RGE invariance at NLO
  14. Threshold Thrust distribution
  15. Dynamical Threshold Enhancement
  16. ...and 6 more sections

Figures (2)

  • Figure 1: The hemispheres are defined so that the remnants of proton ${{\color{darkred} 1}}$ go left and the remnants of proton ${{\color{darkred} 2}}$ to right. For $u>t$, the ${{\color{darkred} 3}}$-jet is in the right hemisphere and the ${{\color{darkred} 4}}$-jet in the left hemisphere, for $t<u$ the jets switch sides.
  • Figure 2: Diagrams contributing to the soft function at NLO.