Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Variable-Flavour Number Scheme for NNLO

R. S. Thorne

TL;DR

The paper addresses the need for a robust NNLO Variable-Flavour Number Scheme to correctly treat heavy quarks across flavour thresholds, where fixed-flavour and zero-mass schemes fail or are incomplete. It develops a VFNS by enforcing FFNS–VFNS equivalence via matching matrices A_{jk}, combines ACOT(chi) with Thorne–Roberts matching to ensure continuity of physical observables at Q^2 = m_H^2, and provides explicit NNLO constructions for F_2 and F_L with controlled modeling of missing fixed-flavour coefficients. The approach handles both heavy-quark and light-flavour sectors, as well as charged-current processes, and yields improved agreement with data at low x and low Q^2, enabling precise NNLO global parton analyses (MRST06). The work demonstrates that while parton distributions may be discontinuous at flavour transitions, the resulting structure functions remain continuous, validating the practical viability of NNLO VFNS implementations.

Abstract

At NNLO it is particularly important to have a Variable-Flavour Number Scheme (VFNS) to deal with heavy quarks because there are major problems with both the zero mass variable-flavour number scheme and the fixed-flavour number scheme. I illustrate these problems and present a general formulation of a Variable-Flavour Number Scheme (VFNS)for heavy quarks that is explicitly implemented up to NNLO in the strong coupling constant alpha_S, and may be used in NNLO global fits for parton distributions. The procedure combines elements of the ACOT(chi) scheme and the Thorne-Roberts scheme. Despite the fact that at NNLO the parton distributions are discontinuous as one changes the number of active quark flavours, all physical quantities are continuous at flavour transitions and the comparison with data is successful.

A Variable-Flavour Number Scheme for NNLO

TL;DR

The paper addresses the need for a robust NNLO Variable-Flavour Number Scheme to correctly treat heavy quarks across flavour thresholds, where fixed-flavour and zero-mass schemes fail or are incomplete. It develops a VFNS by enforcing FFNS–VFNS equivalence via matching matrices A_{jk}, combines ACOT(chi) with Thorne–Roberts matching to ensure continuity of physical observables at Q^2 = m_H^2, and provides explicit NNLO constructions for F_2 and F_L with controlled modeling of missing fixed-flavour coefficients. The approach handles both heavy-quark and light-flavour sectors, as well as charged-current processes, and yields improved agreement with data at low x and low Q^2, enabling precise NNLO global parton analyses (MRST06). The work demonstrates that while parton distributions may be discontinuous at flavour transitions, the resulting structure functions remain continuous, validating the practical viability of NNLO VFNS implementations.

Abstract

At NNLO it is particularly important to have a Variable-Flavour Number Scheme (VFNS) to deal with heavy quarks because there are major problems with both the zero mass variable-flavour number scheme and the fixed-flavour number scheme. I illustrate these problems and present a general formulation of a Variable-Flavour Number Scheme (VFNS)for heavy quarks that is explicitly implemented up to NNLO in the strong coupling constant alpha_S, and may be used in NNLO global fits for parton distributions. The procedure combines elements of the ACOT(chi) scheme and the Thorne-Roberts scheme. Despite the fact that at NNLO the parton distributions are discontinuous as one changes the number of active quark flavours, all physical quantities are continuous at flavour transitions and the comparison with data is successful.

Paper Structure

This paper contains 5 sections, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Variable-Flavour Number Scheme
  3. Light-Flavour Sector
  4. Charged-Current Structure Functions
  5. Conclusions

Figures (4)

  • Figure 1: The discontinuity in $F_2^c(x,Q^2)$ (left) and $F_2(x,Q^2)$ (right) using the zero-mass variable-flavour number scheme at NNLO.
  • Figure 2: Comparison between the ACOT choice of ordering and the Thorne-Roberts choice at NLO.
  • Figure 3: Comparison of the NLO and NNLO predictions for $F_2^c(x,Q^2)$ compared with data.
  • Figure 4: Comparison of the NLO and NNLO predictions for $F_b^c(x,Q^2)$ compared with H1 data.