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Next-to-leading Order Evolution of Transversity Distributions and Soffer's Inequality

W. Vogelsang

TL;DR

This paper computes the next-to-leading order (NLO) splitting functions for the QCD evolution of twist-2 transversity distributions using a Bjorken-x space, light-cone gauge approach, and projector techniques. It provides analytic MS-bar expressions for the NLO kernels $\Delta_T P_{qq,\pm}^{(1)}(x)$, establishes $\Delta_T P_{qq,PS}^{(1)}(x)=0$, and derives the corresponding Drell–Yan coefficient $\Delta_T C_q^{\rm DY}(x)$, enabling consistent NLO evolution of transversity. By analyzing these results within a controlled factorization scheme, the authors test Soffer's inequality beyond LO and find that, for valence densities, the inequality is preserved at NLO if it holds at the input scale. The work also notes the small-$x$ behavior leading to a mild divergence in a derived quantity, emphasizes the need for future numerical studies of the singlet sector, and reports agreement with independent two-loop anomalous-dimension results obtained via OPE methods.

Abstract

We present a calculation of the two-loop splitting functions for the evolution of the twist-2 `transversity' parton densities of transversely polarized nucleons. We study the implications of our results for Soffer's inequality for the case of valence quark densities.

Next-to-leading Order Evolution of Transversity Distributions and Soffer's Inequality

TL;DR

This paper computes the next-to-leading order (NLO) splitting functions for the QCD evolution of twist-2 transversity distributions using a Bjorken-x space, light-cone gauge approach, and projector techniques. It provides analytic MS-bar expressions for the NLO kernels , establishes , and derives the corresponding Drell–Yan coefficient , enabling consistent NLO evolution of transversity. By analyzing these results within a controlled factorization scheme, the authors test Soffer's inequality beyond LO and find that, for valence densities, the inequality is preserved at NLO if it holds at the input scale. The work also notes the small- behavior leading to a mild divergence in a derived quantity, emphasizes the need for future numerical studies of the singlet sector, and reports agreement with independent two-loop anomalous-dimension results obtained via OPE methods.

Abstract

We present a calculation of the two-loop splitting functions for the evolution of the twist-2 `transversity' parton densities of transversely polarized nucleons. We study the implications of our results for Soffer's inequality for the case of valence quark densities.

Paper Structure

This paper contains 5 sections, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. Calculation of the NLO transversity splitting functions
  4. Results
  5. Conclusions

Figures (2)

  • Figure 1: The functions $\tilde{D}(x)$, $D (x)$ (see Eqs. (\ref{['master1']}),(\ref{['masterxms']})) for $f=3$ active flavours.
  • Figure 2: The Mellin-$n$ moments of the NLO transversity splitting functions $\Delta_T P_{qq,-}^{(1)}$ (solid line) and $\Delta_T P_{qq,+}^{(1)}$ (dashed line) versus real $n\geq 1$. For comparison the dash-dotted curve shows the LO result $\Delta_T P_{qq}^{(0),n}$.