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The chirally-odd twist-3 distribution e(x)

A. V. Efremov, P. Schweitzer

TL;DR

The paper surveys the chirally odd twist-3 distribution e^q(x), emphasizing a delta-function contribution at x=0 predicted by QCD equations of motion and its formal connection to the pion–nucleon sigma-term. It analyzes the operator structure, scale evolution, and sum rules, showing that the first moment is carried by the δ(x) piece and that the second moment is governed by the current-quark mass, with mass effects and Regge-inspired small-x behavior complicating naive interpretations. Through a tour of models (NR, bag, spectator, Gross–Neveu, χQSM) it demonstrates a generally sizable valence-like e^q(x) at low scales, with χQSM uniquely producing a δ-term tied to vacuum condensates and large-N_c guiding patterns. Experimentally, e^q(x) can be accessed via Collins-type asymmetries in SIDIS, with CLAS data hinting at a nonzero signal, but direct extraction remains challenging due to the twist-3 suppression and the δ(x) singularity, underscoring the need for targeted measurements and refined theoretical frameworks.

Abstract

Properties of the nucleon twist-3 distribution function e(x) are reviewed. It is emphasized that the QCD equations of motion imply the existence of a delta-function at x=0 in e(x), which gives rise to the pion-nucleon sigma-term. According to the resulting ``practical'' DIS sum rules the first and the second moment of e(x) vanish, a situation analogue to that of the pure twist-3 distribution function $\bar{g}_2(x)$.

The chirally-odd twist-3 distribution e(x)

TL;DR

The paper surveys the chirally odd twist-3 distribution e^q(x), emphasizing a delta-function contribution at x=0 predicted by QCD equations of motion and its formal connection to the pion–nucleon sigma-term. It analyzes the operator structure, scale evolution, and sum rules, showing that the first moment is carried by the δ(x) piece and that the second moment is governed by the current-quark mass, with mass effects and Regge-inspired small-x behavior complicating naive interpretations. Through a tour of models (NR, bag, spectator, Gross–Neveu, χQSM) it demonstrates a generally sizable valence-like e^q(x) at low scales, with χQSM uniquely producing a δ-term tied to vacuum condensates and large-N_c guiding patterns. Experimentally, e^q(x) can be accessed via Collins-type asymmetries in SIDIS, with CLAS data hinting at a nonzero signal, but direct extraction remains challenging due to the twist-3 suppression and the δ(x) singularity, underscoring the need for targeted measurements and refined theoretical frameworks.

Abstract

Properties of the nucleon twist-3 distribution function e(x) are reviewed. It is emphasized that the QCD equations of motion imply the existence of a delta-function at x=0 in e(x), which gives rise to the pion-nucleon sigma-term. According to the resulting ``practical'' DIS sum rules the first and the second moment of e(x) vanish, a situation analogue to that of the pure twist-3 distribution function .

Paper Structure

This paper contains 25 sections, 45 equations.

Table of Contents

  1. Introduction
  2. $e^q(x)$ in theory
  3. Definition.
  4. Evolution.
  5. Sum rules for the $1^{\rm st}$ and $2^{\rm nd}$ moment.
  6. Twist-3 inequality.
  7. The large $N_c$ limit.
  8. Decomposing $e^a(x)$ by means of the QCD equations of motion.
  9. Small-$x$ behaviour of $e^q(x)$.
  10. Conclusions from the use of equations of motion.
  11. Finite current quark mass effects.
  12. The $\delta(x)$ singularity in $e^q(x)$.
  13. $e^q(x)$ in models
  14. Non-relativistic quark model.
  15. Bag model.
  16. ...and 10 more sections