Single Transverse Spin Asymmetry in QCD

• Single transverse spin asymmetry (STSA) quantifies left-right differences in scattering processes relative to a hadron’s transverse spin, highlighting key quark-gluon correlations.
• The twist-3 collinear factorization approach explains STSA via multi-parton correlations and ETQS functions, with pole contributions generating the necessary phase.
• The TMD framework, through the Sivers function, encodes spin-momentum correlations and validates the predicted sign reversal between SIDIS and Drell–Yan processes.

A single transverse spin asymmetry (STSA), typically denoted as $A_N$, quantifies the difference in cross section for a process involving scattering or production of final-state particles when a hadron’s spin is flipped from one transverse direction to the opposite. Specifically, it measures left-right asymmetries in the distribution of final-state particles relative to the polarization direction of an initial-state hadron, most commonly a proton. In perturbative Quantum Chromodynamics (QCD), STSAs probe dynamics beyond leading-twist collinear factorization, sensitively encoding quark-gluon correlations, spin–orbit couplings, color flow, and the gauge structure of QCD.

1. Definition and Significance of Single Transverse Spin Asymmetry

Single transverse spin asymmetry is formally defined for a generic observable as

$$A_N = \frac{d\sigma(S_T) - d\sigma(-S_T)}{d\sigma(S_T) + d\sigma(-S_T)}$$

where $d\sigma(S_T)$ represents the differential cross section with the initial hadron transversely polarized along $S_T$, and $d\sigma(-S_T)$ the cross section with the polarization reversed.

STSA is sensitive to spin–orbit correlations in partonic subprocesses and to QCD mechanisms generating the necessary interference phase—a requirement in hard processes with an odd number of spin flips under naive time reversal (T-odd observables). Historically, observations of large $A_N$ in hadron–hadron collisions, contrary to the expectations from leading-twist pQCD, demanded the development of advanced QCD factorization approaches and the incorporation of multi-parton correlations and color gauge-link effects.

2. Theoretical Frameworks for STSA: Twist-3 and TMD Approaches

Two major theoretical frameworks govern the modern calculation and interpretation of STSAs:

A. Twist-3 Collinear Factorization

In twist-3 collinear factorization, STSAs arise from multi-parton (quark–gluon–quark) correlations in the transversely polarized hadron, captured by so-called Efremov–Teryaev–Qiu–Sterman (ETQS) functions $T_F$ (chiral-even) and their chiral-odd counterparts $T_F^{(\sigma)}$. In this approach, the interference between scattering amplitudes with and without an extra gluon attachment produces the essential phase via pole contributions in internal partonic propagators. The crucial nonperturbative objects are given by

$$T_F(x,x_1) = \int \frac{dy^- dy_1^-}{4\pi} e^{-ixP^+y^- + i(x_1-x)P^+y_1^-} \langle PS | \bar{\psi}(y^-) \gamma^+ \epsilon_T^{\nu \mu} S_{T,\nu} g F^+_\mu(y_1^-) \psi(0) | PS \rangle$$

with an analogous definition for $T_F^{(\sigma)}$ via insertion of the $\sigma^{\mu+}$ Dirac structure.

B. Transverse Momentum-Dependent (TMD) Factorization and Sivers Effect

In the TMD framework, STSAs are generated by spin-dependent transverse-momentum–dependent PDFs and FFs (e.g., Sivers and Collins functions), reflecting initial or final state spin–momentum correlations. The Sivers function $f_{1T}^\perp(x, k_T^2)$ encodes the correlation between the nucleon’s transverse spin and the transverse momentum of its constituent partons, and is T-odd due to initial/final state interactions inducing a nontrivial gauge-link structure.

The major connection between frameworks is that the first $k_T^2$-moment of the Sivers function is related to the ETQS function: $$T_{q,F}(x,x) = -\int d^2k_T \frac{|k_T|^2}{M} f_{1T}^{\perp q}(x, k_T^2)$$ in the SIDIS convention.

3. Calculation and Extraction of STSA in Hard Processes

Drell-Yan Process

A rigorous twist-3 calculation of STSA in the angular distribution of Drell-Yan lepton pair production leads to the following expression (Zhou et al., 2010): $$A_N = -\frac{1}{2Q} \frac{\sin 2\theta \sin \phi_S}{1 + \cos^2 \theta} \frac{ \sum_q e_q^2 \int dx \left[ T_F^q(x,x)\ f_1^{\bar{q}}(x') + h_1^q(x)\ T_F^{(\sigma),\bar{q}}(x',x') \right] }{ \sum_q e_q^2 \int dx\ f_1^q(x)\ f_1^{\bar{q}}(x') }$$ with $Q$ the dilepton invariant mass, $f_1$ the unpolarized twist-2 PDF, $h_1$ the transversity distribution. Both the chiral-even (ETQS) and chiral-odd (transversity/three-parton) correlators contribute. This formula is precisely half the size of earlier results, a direct consequence of retaining $k_T$-dependence in both hadronic and leptonic tensors through the collinear expansion and careful accounting of phase space constraints.

Key Aspects of the Derivation

• The expansion retains terms linear in transverse momentum $k_T$ in both the hard scattering and the lepton tensor, since only these terms can generate the required spin asymmetry at twist-3 accuracy.
• Imaginary parts necessary for the nonzero asymmetry are produced by the pole (e.g., $1/(x_g + i\epsilon)$, providing $\sim \pi\delta(x_g)$ on taking the imaginary component).
• The overall sign, structure of angular modulation $(\sin 2\theta \sin \phi_S)/(1+\cos^2\theta)$, and the factor $1/2$ (relative to earlier literature) are all fixed by systematic collinear expansion and consistent kinematic treatment.

4. Physical Interpretation and Implications

The sign and magnitude of STSA in Drell-Yan processes encode fundamental features of QCD dynamics, particularly the role of initial-state interactions (process-dependent color flow), and the predicted sign reversal of the Sivers asymmetry between SIDIS and Drell-Yan channels. The chiral (even/odd) decomposition of the contributing correlation functions allows direct sensitivity to both longitudinal and transverse partonic spin structure.

The measurement of the sign of $A_N$ in Drell-Yan processes serves as a stringent test of QCD gauge invariance and non-universality predictions for spin–momentum correlations. The observed angular structure, linked to tensor contractions such as $$\epsilon_{\mu\nu\rho\sigma} P^\mu \bar{P}^\nu S^\rho R^\sigma$$ (with $R$ the lepton-momentum difference), further disentangles the dynamical origin of the asymmetry from typical Sivers-induced structures involving $\epsilon^{\mu\nu\rho\sigma} q_\mu S_\nu$.

5. Gauge Invariance and Factorization Consistency

A crucial aspect of any twist-3 calculation of $A_N$ is the explicit verification of electromagnetic gauge invariance. The calculation in (Zhou et al., 2010) has been performed in both covariant and light-cone gauges, verifying the conservation of the QED current ($q_\mu W^{\mu\nu} = 0$) and thus the absence of spurious gauge artifacts in the result. Such a check is nontrivial when collinear expansion and transverse-momentum corrections are retained in both hadronic and lepton tensors.

This robust treatment not only reinforces the validity of the twist-3 framework, but also clarifies the resolution of discrepancies and ambiguities present in previous calculations.

6. Comparison to Previous Literature and Experimental Relevance

Earlier theoretical treatments (e.g., Boer et al.) had predicted an asymmetry larger by a factor of two. The present analysis demonstrates—by systematic inclusion of all $k_T$–dependent terms and proper treatment of delta-function constraints—that the correct coefficient is $1/(2Q)$. The resolution of this normalization discrepancy is critical for matching theoretical predictions to experimental measurements and for extracting the underlying nonperturbative correlators from data.

The emphasis on the overall sign is as important as the magnitude: its measurement in Drell-Yan experiments directly tests predictions for the sign flip of the Sivers function, probing fundamental QCD color-flow effects and serving as a benchmark for the twist-3 paradigm.

7. Summary Table: Key Components of the Drell-Yan STSA Calculation

Component	Mathematical Expression	Physical Role
Asymmetry definition	$$A_N = [d\sigma(S_T)-d\sigma(-S_T)] / [d\sigma(S_T)+d\sigma(-S_T)]$$	Measurement of STSA
Chiral-even ETQS contribution	$T_F(x,x)$	Initial-state quark–gluon correlation
Chiral-odd twist-3 (transversity)	$h_1^q(x) T_F^{(\sigma),\bar{q}}(x',x')$	Chiral-odd quark-gluon correlator
Angular modulation	$$-(1/2Q)\cdot (\sin2\theta\sin\phi_S)/(1+\cos^2\theta)$$	Characteristic of Drell-Yan STSA
Gauge invariance check	QED current conservation in multiple gauges	Consistency of twist-3 factorization

This comprehensive treatment under twist-3 collinear factorization—incorporating systematic $k_T$-expansion, rigorous gauge invariance, and careful phase extraction—provides a unified and reliable framework for predicting and interpreting single transverse spin asymmetries in hard scattering processes such as Drell-Yan lepton pair production.