[Paper Review] Polarized twist-three distributions $g_T$ and $h_L$ and the role of intrinsic transverse momentum

R. D. Tangerman, P. J. Mulders|arXiv (Cornell University)|Aug 16, 1994

Particle physics theoretical and experimental studies1 references16 citations

TL;DR

This paper reanalyzes twist-three polarized quark distributions $g_T$ and $h_L$ using nonlocal matrix elements to explicitly include quark intrinsic transverse momentum, revealing that assumptions of zero transverse momentum lead to unphysical results. It derives new sum rules $\int g_2(x)dx = 0$ and $\int h_2(x)dx = 0$, and shows that neglecting transverse momentum significantly distorts the Drell-Yan double-spin asymmetry $A_{LT}$, with bag model estimates showing substantial deviations from standard approximations.

ABSTRACT

In a nonstandard way we split up the polarized quark distributions $g_T$ and $h_L$ into their twist-two, quark-mass, and interaction-dependent parts, emphasizing the sensitivity to quark intrinsic transverse momentum. We show how to derive the Burkhardt-Cottingham sum rule in this approach and derive a similar sum rule for the chiral-odd distribution $h_2$. The effect of intrinsic transverse momentum in experimental observables is illustrated in the calculation of the ${\cal O}(1/Q)$ double-spin asymmetry $A_{LT}$ in Drell-Yan scattering.

Motivation & Objective

• To re-express twist-three polarized distributions $g_T$ and $h_L$ in terms of twist-two, quark-mass, and interaction-dependent parts using nonlocal matrix elements.
• To investigate the role of intrinsic transverse momentum in twist-three distributions and their impact on physical observables.
• To derive sum rules for $g_2(x)$ and $h_2(x)$, analogous to the Burkhardt-Cottingham sum rule.
• To analyze the $\mathcal{O}(1/Q)$ double-spin asymmetry $A_{LT}$ in Drell-Yan scattering beyond the zero-transverse-momentum approximation.
• To quantify the discrepancy between results assuming zero transverse momentum and those including nonzero intrinsic transverse momentum using the bag model.

Proposed method

• Uses nonlocal quark-quark and quark-gluon-quark matrix elements to decompose $g_T(x)$ and $h_L(x)$ into twist-two, quark-mass, and interaction-dependent components.
• Applies Lorentz symmetry, discrete symmetries, and QCD equations of motion to derive the decomposition and sum rules.
• Calculates the hadronic tensor for polarized Drell-Yan scattering, including transverse momentum dependence via nonlocal matrix elements.
• Derives the double-spin asymmetry $A_{LT}$ in the form $A_{LT} = \lambda_A \frac{\sin 2\theta \cos\phi}{1 + \cos^2\theta} \frac{\overline{U}^{LT}_{2,1}}{\overline{W}_T}$, with $\overline{U}^{LT}_{2,1}$ containing $g_T$, $h_L$, and their correlation terms $\tilde{g}_T$, $\tilde{h}_L$.
• Compares results with and without the assumption of $\delta(\mathbf{k}_T^2)$ transverse momentum distributions, using bag-model estimates for quark distributions.
• Performs numerical comparisons via plots (Figs. 11 and 12) showing the magnitude of differences in $A_{LT}$ when transverse momentum is included.

Experimental results

Research questions

• RQ1 How does intrinsic transverse momentum affect the structure of twist-three distributions $g_T(x)$ and $h_L(x)$?
• RQ2 What sum rules can be derived for $g_2(x)$ and $h_2(x)$, and how do they relate to the Burkhardt-Cottingham sum rule?
• RQ3 How does the $\mathcal{O}(1/Q)$ double-spin asymmetry $A_{LT}$ in Drell-Yan scattering change when transverse momentum is explicitly included?
• RQ4 To what extent do standard approximations assuming zero transverse momentum fail in describing $A_{LT}$?
• RQ5 How do bag-model estimates of quark distributions reveal the sensitivity of $A_{LT}$ to transverse momentum effects?

Key findings

• The paper derives a new sum rule $\int_0^1 dx\, h_2(x) = 0$, analogous to the Burkhardt-Cottingham sum rule for $g_2(x)$.
• Assuming zero transverse momentum leads to unphysical results, such as $g_2(x) = 0$, which are corrected by including intrinsic transverse momentum.
• The double-spin asymmetry $A_{LT}$ in Drell-Yan scattering depends on both $g_T(x)$ and $h_L(x)$, as well as their correlation terms $\tilde{g}_T(x)$ and $\tilde{h}_L(x)$, which vanish under the zero-transverse-momentum assumption.
• Bag model calculations show that the $A_{LT}$ asymmetry is significantly shifted and altered when transverse momentum is included, indicating a large quantitative difference from standard approximations.
• The discrepancy between the zero-transverse-momentum and full transverse-momentum treatments is substantial, as confirmed by plots of the asymmetry in the $(x_A, x_B)$ plane and along the diagonal $x_A = x_B$.
• The analysis confirms that intrinsic transverse momentum is essential for a correct description of twist-three distributions and their physical observables, especially at $\mathcal{O}(1/Q)$.