On the validity of Lorentz invariance relations between parton distributions

TL;DR

This study questions the validity of Lorentz invariance (LI) relations that connect twist-3 distributions to moments of $k_{ot}$-dependent twist-2 distributions, showing that the necessary gauge-link structure introduces a light-like vector $n$ and additional amplitudes $B_i$ that spoil these relations. The authors present a model-independent decomposition of the quark-quark correlator with Wilson lines and demonstrate that LI-relations such as $g_T(x) = g_1(x) + (d/dx) g_{1T}^{(1)}(x)$ and $h_L(x) = h_1(x) - (d/dx) h_{1L}^{perp(1)}(x)$ fail when $n$-dependent terms are included. They corroborate this with explicit perturbative QCD calculations for a quark target and a simple diquark spectator model, finding that LI-relations hold at leading order but are violated by $ ext{O}(oldsymbol{\alpha_s})$ corrections for several relations, including $g_T$ vs $g_1$ and $h_L$ vs $h_1$; in the T-odd sector, the spectator model yields nonzero $f_{1T}^{ot}$ with $f_T$ vanishing by T-invariance, illustrating general LI-relations breakdown. The results highlight the essential role of the gauge-link structure and light-cone dynamics in parton distributions, with implications for phenomenology and the use of LI-relations to constrain twist-3 distributions.

Abstract

Lorentz invariance relations connecting twist-3 parton distributions with transverse momentum dependent twist-2 distributions have been proposed previously. These relations can be extracted from a covariant decomposition of the quark-quark correlator. It is argued, however, that the derivation of the Lorentz invariance relations fails if the path-ordered exponential is taken into account in the correlator. The model independent analysis is supplemented by an explicit calculation of the corresponding parton distributions in perturbative QCD with a quark target, and in a simple spectator model. We also clarify the status of a specific calculation of time-reversal even parton distributions in light-cone gauge.