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Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS

A. Vogt

TL;DR

QCD-Pegasus provides a fast, flexible framework for evolving unpolarized and polarized parton distributions in perturbative QCD using Mellin-space solutions with a universal inverse-Mellin contour. The approach combines analytic N-space evolution with a fixed, efficient Mellin inversion, supporting LO, NLO, and NNLO (unpolarized) and polarized evolution in fixed- and variable-flavour schemes, including heavy-quark matching. It offers detailed handling of running coupling, flavour decomposition, and heavy-flavour thresholds, along with extensive, surface-level user and reference manuals that facilitate large-scale fits and cross-section studies. The package emphasizes speed and accuracy for repetitive computations and outlines pathways to integrate with coefficient functions for complete hadronic observables. Potential extensions include fragmentation-function evolution and photon-initiated processes, highlighting the method's versatility for perturbative QCD analyses.

Abstract

The Fortran package QCD-PEGASUS is presented. This program provides fast, flexible and accurate solutions of the evolution equations for unpolarized and polarized parton distributions of hadrons in perturbative QCD. The evolution is performed using the symbolic moment-space solutions on a one-fits-all Mellin inversion contour. User options include the order of the evolution including the next-to-next-to-leading order in the unpolarized case, the type of the evolution including an emulation of brute-force solutions, the evolution with a fixed number n_f of flavours or in the variable-n_f scheme, and the evolution with a renormalization scale unequal to the factorization scale. The initial distributions are needed in a form facilitating the computation of the complex Mellin moments.

Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS

TL;DR

QCD-Pegasus provides a fast, flexible framework for evolving unpolarized and polarized parton distributions in perturbative QCD using Mellin-space solutions with a universal inverse-Mellin contour. The approach combines analytic N-space evolution with a fixed, efficient Mellin inversion, supporting LO, NLO, and NNLO (unpolarized) and polarized evolution in fixed- and variable-flavour schemes, including heavy-quark matching. It offers detailed handling of running coupling, flavour decomposition, and heavy-flavour thresholds, along with extensive, surface-level user and reference manuals that facilitate large-scale fits and cross-section studies. The package emphasizes speed and accuracy for repetitive computations and outlines pathways to integrate with coefficient functions for complete hadronic observables. Potential extensions include fragmentation-function evolution and photon-initiated processes, highlighting the method's versatility for perturbative QCD analyses.

Abstract

The Fortran package QCD-PEGASUS is presented. This program provides fast, flexible and accurate solutions of the evolution equations for unpolarized and polarized parton distributions of hadrons in perturbative QCD. The evolution is performed using the symbolic moment-space solutions on a one-fits-all Mellin inversion contour. User options include the order of the evolution including the next-to-next-to-leading order in the unpolarized case, the type of the evolution including an emulation of brute-force solutions, the evolution with a fixed number n_f of flavours or in the variable-n_f scheme, and the evolution with a renormalization scale unequal to the factorization scale. The initial distributions are needed in a form facilitating the computation of the complex Mellin moments.

Paper Structure

This paper contains 72 sections, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The evolution equations and their solution
  3. The running coupling constant
  4. The general evolution equations
  5. The flavour decomposition
  6. The $N$-space solutions
  7. The evolution matrices $\,\hbox{\boldmath $U$}_{\!k}$
  8. Non-singlet cases and symmetry breaking
  9. The treatment of heavy flavours
  10. Complex moments and the Mellin inversion
  11. From moments to $x$-space
  12. Splitting functions and matching coefficients
  13. The initial distributions
  14. A brief Qcd-Pegasus user guide
  15. General initialization
  16. ...and 57 more sections

Figures (2)

  • Figure 1: Two integration contours for the inverse Mellin transformation in Eq. (\ref{['M-inv']}). The crosses schematically indicate the singularities of the moment-space function $a(N)$.
  • Figure 2: The up-valence and up-sea distributions at the initial scale (\ref{['std-as']}) obtained by the standard and fast Mellin inversions, divided by the exact $x$-space inputs in Eq. (\ref{['std-inp']}). The same four curves are shown in the left (right) part for small (large) values of $x$.