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Next-to-Leading Order Evolution of Polarized and Unpolarized Fragmentation Functions

M. Stratmann, W. Vogelsang

TL;DR

The paper derives the next-to-leading order (NLO) time-like evolution kernels for both polarized and unpolarized fragmentation functions by analytically continuing the space-like counterparts. It shows that while the leading-order (LO) Gribov-Lipatov relation (GLR) and analytic continuation relation (ACR) hold, they break beyond LO due to phase-space and dimensional regularization effects, and it demonstrates how scheme transformations can restore a consistent framework. The authors provide explicit NLO expressions for unpolarized and polarized time-like splitting functions, along with endpoint contributions and Mellin moments, and they verify a supersymmetric consistency relation in dimensional reduction as a cross-check. This work enables a consistent NLO QCD evolution of fragmentation functions in e+e− annihilation and related processes, with implications for extracting polarized fragmentation functions such as ΔD_f^Λ.

Abstract

We determine the two-loop 'time-like' Altarelli-Parisi splitting functions, appearing in the next-to-leading order Q^2-evolution equations for fragmentation functions, via analytic continuation of the corresponding 'space-like' splitting functions for the evolution of parton distributions. We do this for the case of unpolarized fragmentation functions and - for the first time - also for the functions describing the fragmentation of a longitudinally polarized parton into a longitudinally polarized spin-1/2 hadron such as a Lambda baryon. Our calculation is based on the method proposed and employed by Curci, Furmanski and Petronzio in the unpolarized case in which we confirm their results.

Next-to-Leading Order Evolution of Polarized and Unpolarized Fragmentation Functions

TL;DR

The paper derives the next-to-leading order (NLO) time-like evolution kernels for both polarized and unpolarized fragmentation functions by analytically continuing the space-like counterparts. It shows that while the leading-order (LO) Gribov-Lipatov relation (GLR) and analytic continuation relation (ACR) hold, they break beyond LO due to phase-space and dimensional regularization effects, and it demonstrates how scheme transformations can restore a consistent framework. The authors provide explicit NLO expressions for unpolarized and polarized time-like splitting functions, along with endpoint contributions and Mellin moments, and they verify a supersymmetric consistency relation in dimensional reduction as a cross-check. This work enables a consistent NLO QCD evolution of fragmentation functions in e+e− annihilation and related processes, with implications for extracting polarized fragmentation functions such as ΔD_f^Λ.

Abstract

We determine the two-loop 'time-like' Altarelli-Parisi splitting functions, appearing in the next-to-leading order Q^2-evolution equations for fragmentation functions, via analytic continuation of the corresponding 'space-like' splitting functions for the evolution of parton distributions. We do this for the case of unpolarized fragmentation functions and - for the first time - also for the functions describing the fragmentation of a longitudinally polarized parton into a longitudinally polarized spin-1/2 hadron such as a Lambda baryon. Our calculation is based on the method proposed and employed by Curci, Furmanski and Petronzio in the unpolarized case in which we confirm their results.

Paper Structure

This paper contains 7 sections, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General framework and LO results
  3. NLO results for the unpolarized case
  4. The breaking of the ACR revisited
  5. NLO results for the polarized case
  6. A supersymmetric property of the NLO time-like splitting functions
  7. Summary

Figures (3)

  • Figure 1: Basic topologies of the diagrams that contribute to $\Gamma_{ij}^{\epsilon}$ (as defined in Eq. (\ref{['ACRNLO1']})) in NLO.
  • Figure 2: Graphs of topology (i) contributing to $P_{qq,PS}^{(U),(1)}$ in the space-like (left) and time-like (right) situations.
  • Figure 3: Comparison of the spin-dependent NLO ($\overline{\rm{MS}}$) time-like singlet splitting functions $\Delta P_{ij}^{(T),(1)}[n]$ as functions of Mellin-$n$ according to Eqs. (A.1)-(A.6) with the corresponding unpolarized ones, as taken from [\ref{['grvfr']}], for $f=3$ flavors.