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First Forcer results on deep-inelastic scattering and related quantities

B. Ruijl, T. Ueda, J. A. M. Vermaseren, J. Davies, A. Vogt

TL;DR

Problem: determine N$^3$LO (αs^4) QCD corrections to DIS splitting and coefficient functions. Approach: use Forcer to compute fixed-Mellin-N moments of fourth-order quantities, decompose into non-singlet and singlet sectors, and employ LLL-based Diophantine fitting to recover all-$N$ behavior and rough x-space forms. Key results include explicit low-$N$ results such as the $N=4$ $ extgamma_{gg}^{(3)}$ and the complete nf$^2$ contributions to the four-loop cusp anomalous dimension, with evidence for quartic group invariants. Significance: validates Forcer, enhances DIS precision, informs α_s determinations, and provides data to support resummations and approximate N$^3$LO kernels.

Abstract

We present results on the fourth-order splitting functions and coefficient functions obtained using Forcer, a four-loop generalization of the Mincer program for the parametric reduction of self-energy integrals. We have computed the respective lowest three even-N and odd-N moments for the non-singlet splitting functions and the non-singlet coefficient functions in electromagnetic and nu+nu(bar) charged-current deep-inelastic scattering, and the N=2 and N=4 results for the corresponding flavour-singlet quantities. Enough moments have been obtained for an LLL-based determination of the analytic N-dependence of the nf^3 and nf^2 parts, respectively, of the singlet and non-singlet splitting functions. The large-N limit of the latter provides the complete nf^2 contributions to the four-loop cusp anomalous dimension. Our results also provide additional evidence of a non-vanishing contribution of quartic group invariants to the cusp anomalous dimension.

First Forcer results on deep-inelastic scattering and related quantities

TL;DR

Problem: determine NLO (αs^4) QCD corrections to DIS splitting and coefficient functions. Approach: use Forcer to compute fixed-Mellin-N moments of fourth-order quantities, decompose into non-singlet and singlet sectors, and employ LLL-based Diophantine fitting to recover all- behavior and rough x-space forms. Key results include explicit low- results such as the and the complete nf contributions to the four-loop cusp anomalous dimension, with evidence for quartic group invariants. Significance: validates Forcer, enhances DIS precision, informs α_s determinations, and provides data to support resummations and approximate NLO kernels.

Abstract

We present results on the fourth-order splitting functions and coefficient functions obtained using Forcer, a four-loop generalization of the Mincer program for the parametric reduction of self-energy integrals. We have computed the respective lowest three even-N and odd-N moments for the non-singlet splitting functions and the non-singlet coefficient functions in electromagnetic and nu+nu(bar) charged-current deep-inelastic scattering, and the N=2 and N=4 results for the corresponding flavour-singlet quantities. Enough moments have been obtained for an LLL-based determination of the analytic N-dependence of the nf^3 and nf^2 parts, respectively, of the singlet and non-singlet splitting functions. The large-N limit of the latter provides the complete nf^2 contributions to the four-loop cusp anomalous dimension. Our results also provide additional evidence of a non-vanishing contribution of quartic group invariants to the cusp anomalous dimension.

Paper Structure

This paper contains 4 sections, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Low-$N$ results for splitting functions and coefficient functions
  3. Large-${n^{}_{\! f}}$ all-$N$ parts of N$^3$LO splitting functions & the cusp anomalous dimension
  4. Summary and outlook

Figures (4)

  • Figure 1: nameref-Fig1 fith LAB: Fig1 The successive large-$N$ approximations in Eq. (\ref{['largeN']}) compared to the full NLO and NNLO results.$\!$
  • Figure 2: nameref-Fig2 fith LAB: Fig2 The lowest three even-$N$ and odd-$N$ values, respectively, of the anomalous dimensions $\gamma_{\,\rm ns}^{\,(3)+}$ and $\gamma_{\,\rm ns}^{\,(3)-}$ in Eqs. (\ref{['Pns']}) and (\ref{['largeN']}), compared to Padé estimates derived from the NNLO results of Ref. Pnnlo.
  • Figure 3: nameref-Fig3 fith LAB: Fig3 The moments calculated so far of the fourth-order coefficient functions $c_{2,\rm ns\,}^{\,(4)}$, $c_3^{\,(4)}$ and $c_{L,\rm ns}^{\,(4)}$ for $\nu\!+\!\bar{\nu}$ charged-current DIS at ${n^{}_{\! f}}=4$. Also show are the contributions provided by large-$N$ resummations.
  • Figure 4: nameref-Fig4 fith LAB: Fig4 Fermionic contributions to the N$^3$LO anomalous dimensions $\gamma^{\,(3)\pm}(N)$, compared to their calculated (left two panels) and conjectured (right panel) large-$N$ limits given by the respective parts of $\gamma^{\,(3)}_{\,\rm cusp}$.