What can break the Wandzura--Wilczek relation?

TL;DR

The paper analyzes the Wandzura-Wilczek relation for the $g_2$ structure function and its breaking in QCD, linking deviations to twist-3 quark-gluon-quark correlations and transverse momentum dependent distributions. By developing a covariant framework with parton correlation functions, Lorentz-invariance relations, and equations of motion relations, it identifies two distinct twist-3 contributions, $\tilde{g}_T$ and $\hat{g}_T$, as well as a quark-mass term that cause WW violation. An explicit quark-target calculation and phenomenological analysis of $g_2$ data show breaking at roughly 15–40% of $g_2$, underscoring that WW validity is not guaranteed and cannot alone reveal twist-3 sizes. The authors propose measuring $g_{1T}^{(1)}$ in SIDIS to separate the two twist-3 terms, enabling a deeper view into quark-gluon correlations and informing the evolution of twist-3 and TMDs. This work clarifies the interplay between twist-3 collinear functions and TMDs and guides future experiments.

Abstract

We analyze the breaking of the Wandzura-Wilczek relation for the g_2 structure function, emphasizing its connection with transverse momentum dependent parton distribution functions. We find that the relation is broken by two distinct twist-3 terms, and clarify how these can be separated in measurements of double-spin asymmetries in semi-inclusive deep inelastic scattering. Through a quantitative analysis of available g_2 data we also show that the breaking of the Wandzura-Wilczek relation can be as large as 15-30% of the size of g_2.

Figures (3)

• Figure 1: Top panels: Experimental proton and neutron $g_2$ structure functions compared to $g_2^{\rm WW}$. The crosses represent $g_2^{\rm WW}$ computed at the experimental kinematics, while the solid lines are $g_2^{\rm WW}$ computed at the average $Q^2$ of the E155x experiment. Data points for the proton target Abe:1998wqAnthony:2002hy have been slightly shifted in $x_B$ for clarity. For the neutron only the high-precision data from Anthony:2002hyZheng:2004ceKramer:2005qe are included. Bottom panels: The WW-breaking term $\Delta$ fitted to $\Delta_{\rm ex}$ computed using the LSS2006 $g_!^{\rm LT}$ (hashed region). The dashed line represents $g_2^{\rm WW}-(g_2^{\rm WW})'$, the spurious HT contribution to $\Delta$ that would be obtained using the total $g_1$ to compute $\Delta_{\rm ex}$.
• Figure 2: Diagrams in the quark-target calculation involving only real gluons. The Hermitean conjugate diagrams, which are not shown, are also taken into account in the calculation.
• Figure 3: As in Fig. \ref{['f:QTreal']} but for diagrams involving virtual gluons.