Sivers FunctionEdit
The Sivers function is a fundamental concept in the study of the spin structure of the nucleon within quantum chromodynamics (QCD). It lives in the broader framework of transverse momentum dependent distributions (TMDs) and captures a specific correlation: when the parent nucleon is transversely polarized, the quarks and gluons inside it can prefer a particular direction of intrinsic transverse momentum. This correlation manifests itself in observable single-spin asymmetries in high-energy scattering processes and provides a window into the orbital motion of partons inside the nucleon. The function is conventionally denoted f1T⊥(x, kT), where x is the longitudinal momentum fraction and kT is the parton's transverse momentum.
Historically, the Sivers idea was proposed to explain large transverse spin effects seen in hadronic experiments that were difficult to reconcile with simple, collinear pictures of parton dynamics. It is named after Dennis Sivers, who introduced the mechanism in the late 1980s and early 1990s. The Sivers effect is a quintessential example of how spin and momentum degrees of freedom intertwine in QCD, and it sits at the intersection of theory and experiment. In contemporary language it is one piece of the larger family of transverse momentum dependent distributions that enrich our understanding of nucleon spin structure and the three-dimensional momentum structure of hadrons.
Definition and formalism
Conceptual content: The Sivers function describes the correlation between the transverse spin vector S_T of the parent nucleon and the intrinsic transverse momentum kT of a parton. Intuitively, it tells you whether partons tend to carry more momentum to one side of the nucleon’s transverse plane when the nucleon is polarized in a particular transverse direction.
Notation and variables: f1T⊥(x, kT) depends on the light-cone momentum fraction x and the transverse momentum kT. When integrated over kT, the Sivers function does not survive as a conventional collinear PDF, reflecting its inherently transverse, angular nature.
Gauge invariance and Wilson lines: The presence of gauge links (Wilson lines) in the QCD definition of TMDs makes the Sivers function T-odd. These gauge links encode initial- and final-state interactions and are essential for a nonzero Sivers effect. See the general discussion of gauge invariance and the role of initial-/final-state interactions in generating T-odd observables.
Process dependence and sign structure: A striking consequence of the gauge-link structure is that the Sivers function is predicted to change sign between different processes, most famously between semi-inclusive deep inelastic scattering and Drell–Yan production. This sign change is a clean, testable prediction of QCD factorization with TMDs and has been a central target of experimental tests.
Phenomenology and connections: The Sivers function contributes to specific angular modulations in scattering cross sections, such as the sin(φh − φS) modulation in SIDIS, where φh is the hadron azimuth and φS is the spin azimuth. These modulations are how experiments access the underlying TMDs from data.
Theoretical framework and interpretation
TMD factorization: The Sivers function is part of the TMD factorization framework, which extends the conventional PDFs to include transverse momentum and spin correlations. Within this framework, cross sections for processes with transverse polarization can be written in terms of convolutions of TMD PDFs (like the Sivers function) and fragmentation functions.
Evolution and scale dependence: Like other TMDs, the Sivers function evolves with the hard scale Q of the process. The evolution is more intricate than in the collinear case and is described by a set of evolution equations (often discussed under the umbrella of Collins–Soper–Sterman, or CSS, evolution). Understanding this evolution is key to comparing extractions from experiments conducted at different energies.
Relation to orbital angular momentum: The Sivers effect is commonly interpreted as a manifestation of orbital motion of partons inside the nucleon. A nonzero f1T⊥ is compatible with nontrivial orbital angular momentum distributions, though the full picture of how orbital motion contributes to the nucleon spin remains an area of active research.
Lattice and modeling efforts: Computational approaches, including lattice QCD and phenomenological models, have sought to illuminate the Sivers function’s sign, magnitude, and flavor structure. While lattice studies face challenges due to the light-cone and nonlocal nature of TMDs, they provide independent cross-checks and constraints for global analyses.
Experimental status and phenomenology
SIDIS measurements: In experiments where a high-energy beam scatters off a transversely polarized nucleon target and a hadron is detected in the final state, sizeable single-spin asymmetries have been observed. Experiments such as HERMES and COMPASS (experiment) have reported nonzero Sivers asymmetries for pions and kaons, indicating a nonvanishing Sivers function for at least some quark flavors.
Flavor structure: Global analyses of SIDIS data suggest a pattern in the flavor decomposition of the Sivers function, typically with up-quark contributions and down-quark contributions of opposite sign. This flavor structure is consistent with expectations from simple valence-quark pictures and more sophisticated fits, and it interacts with fragmentation functions in the data interpretation.
Drell–Yan tests and sign change: A critical test of the TMD framework is the predicted sign change of the Sivers function between SIDIS and Drell–Yan processes. Ongoing and planned measurements in Drell–Yan production with transversely polarized targets or beams (for example at facilities such as COMPASS (experiment) and others) aim to confirm this fundamental aspect of QCD. Early results have started to probe the sign change, but high-precision confirmation requires continued data.
Cross-process consistency and evolution: A key goal of the field is a unified description of the Sivers function across different processes and scales, incorporating proper TMD evolution and consistent treatment of fragmentation functions. This consistency check is central to validating the universality properties predicted by QCD factorization, modulo the known process dependence.
Controversies and ongoing debates
Universality versus process dependence: The Sivers function is not universally the same across all processes in a naive sense because of the gauge-link structure. The established expectation is a sign change between SIDIS and Drell–Yan, but the full implications for other hadronic processes (including certain hadron-hadron collisions) are an active topic of theoretical work and experimental scrutiny. This remains a nuanced area where precise data and careful factorization considerations are essential.
Factorization validity in complex collisions: While TMD factorization is on solid footing for specific processes, there are ongoing discussions about its applicability in more complicated hadronic environments. Potential factorization-breaking effects or color entanglement phenomena could complicate the extraction of a universal Sivers function from some observables in hadron-hadron collisions.
Size and flavor sensitivity: The extracted magnitude of the Sivers function is sensitive to assumptions about fragmentation and other TMDs. Different global fits can yield somewhat different pictures of the relative sizes of f1T⊥ for u- and d-quarks, and uncertainties grow for sea-quark and gluon contributions. This motivates continued measurements and improved theory for TMD evolution and matching to collinear distributions.
Connections to orbital angular momentum: The interpretation of the Sivers function as a direct probe of partonic orbital angular momentum is appealing but not unambiguous. While nonzero Sivers signals imply orbital motion, disentangling this contribution from other dynamical effects requires careful, model-aware analyses and complementary observables.