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Transverse momentum dependent quark distributions and polarized Drell-Yan processes

Jian Zhou, Feng Yuan, Zuo-Tang Liang

TL;DR

This work develops a unified description of spin-dependent quark distributions at large transverse momentum by linking collinear twist-three quark–gluon correlations to transverse-momentum dependent distributions. It computes the large-$k_\perp$ behavior of the naive-time-reversal-even but $k_\perp$-odd distributions $g_{1T}$ and $h_{1L}$, expressing them through $G_D$, $\tilde{G}_D$, $H_D$, $E_D$, and the novel $\tilde{g}(x)$ and $\tilde{h}(x)$, and shows that these results match corresponding TMD predictions in the intermediate region $\Lambda_{QCD}\ll q_\perp\ll Q$. The paper then applies this framework to the angular distribution of Drell–Yan lepton pairs with polarized beams, deriving the single-spin and double-spin asymmetries $A_{UT}$ and $A_{LT}$ within twist-three formalism and confirming consistency with TMD-factorization expectations. Overall, the findings reinforce the compatibility of collinear and TMD factorization in the overlapping regime and provide a path to broader applications in spin-dependent hadronic processes. key mathematical relations involve the twist-three correlators $G_D$, $\tilde{G}_D$, $H_D$, $E_D$, $T_F$, $\tilde{T}_F$, and the functions $\tilde{g}(x)$ and $\tilde{h}(x)$, connecting to $g_{1T}$ and $h_{1L}$ through perturbative kernels and soft-logarithmic terms.

Abstract

We study the spin-dependent quark distributions at large transverse momentum. We derive their transverse momentum behaviors in the collinear factorization approach in this region. We further calculate the angular distribution of the Drell-Yan lepton pair production with polarized beams and present the results in terms of the collinear twist-three quark-gluon correlation functions. In the intermediate transverse momentum region, we find that the two pproaches: the collinear factorization and the transverse momentum dependent factorization approaches are consistent in the description of the lepton pair angular distributions.

Transverse momentum dependent quark distributions and polarized Drell-Yan processes

TL;DR

This work develops a unified description of spin-dependent quark distributions at large transverse momentum by linking collinear twist-three quark–gluon correlations to transverse-momentum dependent distributions. It computes the large- behavior of the naive-time-reversal-even but -odd distributions and , expressing them through , , , , and the novel and , and shows that these results match corresponding TMD predictions in the intermediate region . The paper then applies this framework to the angular distribution of Drell–Yan lepton pairs with polarized beams, deriving the single-spin and double-spin asymmetries and within twist-three formalism and confirming consistency with TMD-factorization expectations. Overall, the findings reinforce the compatibility of collinear and TMD factorization in the overlapping regime and provide a path to broader applications in spin-dependent hadronic processes. key mathematical relations involve the twist-three correlators , , , , , , and the functions and , connecting to and through perturbative kernels and soft-logarithmic terms.

Abstract

We study the spin-dependent quark distributions at large transverse momentum. We derive their transverse momentum behaviors in the collinear factorization approach in this region. We further calculate the angular distribution of the Drell-Yan lepton pair production with polarized beams and present the results in terms of the collinear twist-three quark-gluon correlation functions. In the intermediate transverse momentum region, we find that the two pproaches: the collinear factorization and the transverse momentum dependent factorization approaches are consistent in the description of the lepton pair angular distributions.

Paper Structure

This paper contains 9 sections, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Twist-3 correlation matrix elements and TMD distributions
  3. quark distributions at large transverse momentum
  4. Transverse momentum dependent quark distributions $g_{1T}$ and $h_{1L}$
  5. Contributions from $\tilde{T}_{\partial_\perp}$ and $T_{\partial_\perp}$
  6. Contributions from $\tilde{T}_{A_\perp}$ ($\tilde{T}_{\partial^+A_\perp}$) and $T_{A_\perp}$ ($T_{\partial^+A_\perp}$)
  7. Contributions from $\tilde{T}_{\partial_\perp A^+}$ and $T_{\partial_\perp A^+}$
  8. single spin $A_{UT}$ and double spin $A_{LT}$ asymmetries in the Drell-Yan lepton pair production process
  9. summary

Figures (2)

  • Figure 1: Generic diagram interpretations for the twist expansions in the high energy scattering amplitudes up to twist-three level: (a) corresponds to a leading twist matrix element $\langle \bar{\psi} \psi\rangle$; (b)-(d) for twist-three contributions, (b) for $\langle \bar{\psi} \partial_\perp\psi\rangle$, (c) for $\langle \bar{\psi} A_\perp \psi\rangle$, and (d) for $\langle \bar{\psi} \partial_\perp A^+\psi\rangle$. Additional $A^+$ gluon connection between hard partonic part and the non-perturbative nucleon structure part can be added to these diagrams. This is because they do not change the power counting in these diagrams. The contributions from these diagrams (b-d) are not gauge invariant individually. However, they will combine into the gauge invariant results in terms of the correlation functions introduced in Sec.II.
  • Figure 2: Feynman diagrams for the TMD quark distributions at large transverse momentum calculated from the twist-three quark-gluon correlation functions. The mirror diagrams of (b1)-(e4) are not shown, but included in the final results. (a1)-(a4) correspond to the contributions from the matrix elements of $\langle \bar{\psi} \partial_\perp\psi\rangle$; (b1)-(b4), (e1)-(e4), (c1) and (c3) correspond to the diagrams contributions from $\langle \bar{\psi} A_\perp \psi\rangle$; and (b1)-(e4) for $\langle \bar{\psi} \partial_\perp A^+\psi\rangle$.