[Paper Review] On the validity of Lorentz invariance relations between parton distributions

Marc Schlegel, Andreas Metz|ArXiv.org|Jun 25, 2004

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper challenges the validity of Lorentz invariance (LI) relations connecting twist-3 parton distributions to moments of transverse-momentum-dependent (TMD) twist-2 distributions. It demonstrates that these relations fail due to an overlooked dependence on a light-cone vector introduced by the gauge link in the quark-quark correlator, confirmed via perturbative QCD and a spectator model, with explicit violations found in both T-even and T-odd cases.

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.

Motivation & Objective

• To assess the validity of Lorentz invariance relations (LI-relations) linking twist-3 and TMD twist-2 parton distributions.
• To identify the origin of the failure in previous derivations of LI-relations, particularly the neglect of path-ordered exponential (Wilson line) dependence.
• To provide a model-independent analysis of the quark-quark correlator with proper gauge invariance via covariant decomposition.
• To verify the findings with explicit calculations in perturbative QCD and a simple spectator model.
• To clarify the status of time-reversal even parton distributions in light-cone gauge, resolving prior gauge-invariance issues.

Proposed method

• Conducts a covariant decomposition of the gauge-invariant quark-quark correlator, including the path-ordered exponential (Wilson line), to identify all Lorentz structures.
• Derives parton distributions via projections of the correlator, using momentum-space decomposition with $k^+$ fixed to $xP^+$.
• Analyzes the role of the light-cone vector $n$ in the Wilson line, showing it introduces additional amplitudes $B_i$ absent in naive derivations.
• Performs one-loop perturbative QCD calculations in $4-2ar{ u}$ dimensions with a quark target, computing UV-divergent $1/ar{ u}$ terms.
• Applies a diquark spectator model to evaluate T-odd distributions, using T-invariance arguments to test LI-relations.
• Compares results with prior work (e.g., Ref. [10]) to confirm consistency and validate the absence of light-cone vector dependence in specific cases.

Experimental results

Research questions

• RQ1 Do Lorentz invariance relations between twist-3 and TMD twist-2 parton distributions remain valid when the full gauge link structure is included?
• RQ2 What is the role of the light-cone vector $n$ in the Wilson line in breaking the LI-relations?
• RQ3 Are the LI-relations violated in perturbative QCD at $α_s$ order, particularly in the UV-divergent $1/ε$ terms?
• RQ4 Can model calculations in a spectator model confirm the violation of LI-relations for T-odd and T-even distributions?
• RQ5 Why do previous calculations in light-cone gauge yield inconsistent results, and how does gauge invariance affect the derivation of these relations?

Key findings

• The LI-relations are invalidated because the derivation neglects the dependence on the light-cone vector $n$ introduced by the path-ordered exponential in the gauge link.
• In perturbative QCD, the LI-relation $g_T(x) = g_1(x) + d/dx hinspace g_{1T}^{(1)}(x)$ is violated at $α_s$ order due to $1/ε$ divergences in the $g_{1T}^{(1)}(x)$ and $g_T(x)$ contributions.
• The $g_T(x)$ and $g_{1T}^{(1)}(x)$ distributions receive $α_s$ corrections proportional to $\frac{1+2x-x^2}{1-x}\frac{1}{\varepsilon}$ and $-x(1-x)\frac{1}{\varepsilon}$, respectively, breaking the LI-relation.
• For T-odd distributions, the LI-relation $f_T(x) = -d/dx hinspace f_{1T}^{\perp(1)}(x)$ is violated because $f_{1T}^{\perp(1)}(x)$ is non-zero in the spectator model while $f_T(x)$ vanishes due to T-invariance.
• Similarly, $h(x) = -d/dx hinspace h_{1}^{\perp(1)}(x)$ is violated because $h_{1}^{\perp}(x,\vec{k}_\perp)$ is non-zero in the model while $h(x)$ is zero by T-invariance.
• The presence of the $n$-dependent term in the Wilson line is the root cause of the violation, and removing it would restore the LI-relations, but only in the absence of $n$-dependence.