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Initial State Parton Showers Beyond Leading Order

John C. Collins, Xiaomin Zu

TL;DR

The paper tackles the challenge of incorporating non-leading order corrections into Monte-Carlo event generators for initial-state collinear showering by introducing unintegrated parton correlation functions (pCf). It develops a comprehensive DIS framework within a φ^3 model, deriving a region-based hard-scattering factorization in terms of pCf, and establishes a recursive, subtraction-based factorization for both the cross section and the pCf itself. A practical Monte-Carlo algorithm is proposed where hard scattering is generated with exact kinematics and showering proceeds conditional on these momenta, with reweighting or vetoes ensuring consistency with the pCf-based factorization. The work demonstrates one-loop coefficient functions and pCf corrections, and outlines a clear path to extending the formalism to full QCD, including gauge-invariant definitions via Wilson lines and TMD-type factorization. The approach promises a framework for systematically including arbitrary non-leading corrections in MCEGs while preserving exact kinematics and momentum conservation in exclusive final states.

Abstract

We derive a new method for initial-state collinear showering in Monte-Carlo event generators which is based on the use of unintegrated parton correlation functions. Combined with a previously derived method for final-state showering, the method solves the problem of treating both the hard scattering and the evolution kernels to be used in arbitrarily non-leading order. Although we only treat collinear showering, so that further extensions are needed for QCD, we have discovered several new results: (1) It is better to generate exact parton kinematics in the hard scattering rather than with the subsequent parton showering, and similarly at each step of the showering. (2) Parton showering is then done conditionally on the exact energy-momentum of the initiating parton. (3) We obtain a factorization for structure functions in terms of parton correlation functions so that parton kinematics can be treated exactly from the beginning. (4) We obtain two factorization properties for parton correlation functions, one in terms of ordinary parton densities and one, suitable for event generation, in terms of parton correlation functions themselves.

Initial State Parton Showers Beyond Leading Order

TL;DR

The paper tackles the challenge of incorporating non-leading order corrections into Monte-Carlo event generators for initial-state collinear showering by introducing unintegrated parton correlation functions (pCf). It develops a comprehensive DIS framework within a φ^3 model, deriving a region-based hard-scattering factorization in terms of pCf, and establishes a recursive, subtraction-based factorization for both the cross section and the pCf itself. A practical Monte-Carlo algorithm is proposed where hard scattering is generated with exact kinematics and showering proceeds conditional on these momenta, with reweighting or vetoes ensuring consistency with the pCf-based factorization. The work demonstrates one-loop coefficient functions and pCf corrections, and outlines a clear path to extending the formalism to full QCD, including gauge-invariant definitions via Wilson lines and TMD-type factorization. The approach promises a framework for systematically including arbitrary non-leading corrections in MCEGs while preserving exact kinematics and momentum conservation in exclusive final states.

Abstract

We derive a new method for initial-state collinear showering in Monte-Carlo event generators which is based on the use of unintegrated parton correlation functions. Combined with a previously derived method for final-state showering, the method solves the problem of treating both the hard scattering and the evolution kernels to be used in arbitrarily non-leading order. Although we only treat collinear showering, so that further extensions are needed for QCD, we have discovered several new results: (1) It is better to generate exact parton kinematics in the hard scattering rather than with the subsequent parton showering, and similarly at each step of the showering. (2) Parton showering is then done conditionally on the exact energy-momentum of the initiating parton. (3) We obtain a factorization for structure functions in terms of parton correlation functions so that parton kinematics can be treated exactly from the beginning. (4) We obtain two factorization properties for parton correlation functions, one in terms of ordinary parton densities and one, suitable for event generation, in terms of parton correlation functions themselves.

Paper Structure

This paper contains 38 sections, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rationale for a new algorithm
  3. The model for DIS
  4. Hard-scattering factorization
  5. Regions
  6. Factorized approximation for one region
  7. Definition of projection onto massless momenta and its Jacobian
  8. Construction and proof of factorization
  9. Term for a given region
  10. Veto factor
  11. Factorization
  12. Choices of $\mu$ and $\mu_J$
  13. Collinear safety, renormalization group equations
  14. Factorization schemes
  15. Parton correlation functions (pCf's)
  16. ...and 23 more sections

Figures (2)

  • Figure 1: A graph like this with an initial-state parton self energy is not used.
  • Figure 3: Approximation for region $R_1$ of Fig. \ref{['fig:oneloop']}(a). The small square brackets indicate where the collinear approximation of Eq. (\ref{['eq:approx']}) is made.