Transverse Momentum Dependent DistributionsEdit
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Transverse Momentum Dependent Distributions are a family of functions in quantum chromodynamics (QCD) that describe the probability to find a parton (quark or gluon) inside a hadron with a given longitudinal momentum fraction x and a specific intrinsic transverse momentum kT, taking into account the spin states of the parton and the parent hadron. They extend the conventional parton distribution functions (PDFs), which are integrated over transverse momentum, to provide a three-dimensional picture of nucleon structure in momentum space. In practice, TMDs appear in factorization theorems for processes where the transverse momentum of the observed final-state system is much smaller than the hard scale, such as semi-inclusive deep inelastic scattering and Drell–Yan production. They connect to observable azimuthal and spin asymmetries and thus offer a window into the inner dynamics of QCD.
Transverse Momentum Dependent Distributions are defined in a gauge-invariant way via quark and gluon correlators that include Wilson lines (gauge links) extending along light-like directions. The presence and orientation of these gauge links depend on the color flow of the hard scattering process, which leads to a crucial feature: certain TMDs exhibit process dependence. A well-known example is the Sivers function, which changes sign between semi-inclusive deep inelastic scattering (SIDIS) and the Drell–Yan process due to the different gauge-link structures. This subtlety touches on the broader issue of universality in TMD physics and has motivated careful theoretical and experimental studies of how TMDs are extracted from data across different processes.
TMDs are related to other ways of encoding parton structure. They reduce to ordinary PDFs when integrated over transverse momentum, and, in a mixed sense, they connect to generalized parton distributions (GPDs) and Wigner distributions, providing a pathway to 3D imaging of nucleons in momentum and coordinate space. In particular, Wigner distributions offer a quantum phase-space description whose projections yield both TMDs and GPDs, linking momentum structure to spatial information. The overall framework sits within the broader context of factorization in QCD, which separates short-distance, perturbative physics from long-distance, nonperturbative physics encoded in TMDs.
Theoretical foundations
Operator definitions and gauge invariance
TMDs arise from gauge-invariant operator definitions that take the form of correlators of quark or gluon fields connected by Wilson lines. These Wilson lines are essential to preserve gauge invariance in QCD and reflect the color flow in the specific process under consideration. The resulting TMDs depend on the renormalization scale μ and on the energy scale of the process, requiring a dedicated evolution formalism to describe how they change with the hard scale.
Gauge links, universality, and process dependence
Because the gauge links depend on the scattering process, some TMDs are not universally the same across all processes. This does not negate the usefulness of TMDs but does require careful attention when comparing extractions from SIDIS to Drell–Yan or to other reactions. The Sivers function, for instance, is predicted to reverse sign between SIDIS and Drell–Yan, a prediction that has been a focal point for experimental tests of TMD factorization and universality.
Relation to Wigner distributions and 3D imaging
TMDs provide a practical handle on the momentum-space structure of hadrons, while GPDs provide coordinate-space information. Through their connection to Wigner distributions, TMDs are part of a broader program to map the full quantum phase-space structure of nucleons, enabling a more complete three-dimensional picture of parton dynamics.
Evolution and factorization
Factorization theorems
TMD factorization theorems specify regimes in which a cross section can be written as a convolution of TMDs, a hard-scattering kernel, and possibly a soft factor. This factorization is most robust in processes with a single small transverse momentum scale relative to a large hard scale, such as SIDIS and Drell–Yan at low qT. The precise form of the factorization depends on the process, the polarization states involved, and the treatment of soft radiation.
Evolution equations and soft factors
The scale dependence of TMDs is governed by evolution equations akin to the Collins–Soper–Sterman (CSS) framework. This evolution describes how TMDs change with the energy scale and with a parameter controlling the separation between perturbative and nonperturbative physics. In many formulations, a soft factor is introduced to absorb soft gluon radiation and to separate perturbative contributions from genuinely nonperturbative input. Different schemes exist for implementing CSS evolution and for matching TMDs onto collinear PDFs in the region where the transverse momentum is large.
Non-perturbative inputs and matching
At small kT, non-perturbative physics dominates, and TMDs are typically parameterized and constrained by data. At larger kT, perturbative calculations can be applied and matched to the non-perturbative input in a way that preserves factorization. This matching is crucial for connecting TMD phenomenology with the broader framework of collinear factorization and for ensuring a consistent description across kT ranges.
Experimental access and phenomenology
Key processes
- SIDIS (Semi-Inclusive Deep Inelastic Scattering): a lepton scatters off a hadron, and a hadron is detected in the final state. SIDIS is a primary laboratory for accessing various TMDs through azimuthal and spin asymmetries.
- Drell–Yan: a quark–antiquark annihilation into a lepton pair, providing complementary access to TMDs and offering strong tests of process-dependent sign changes predicted by gauge-link structure.
- e+e− annihilation into hadron pairs: used to constrain fragmentation functions that appear in TMD factorization when one or both hadrons are detected in the final state.
Observables and asymmetries
TMDs imprint characteristic azimuthal and spin-dependent modulations in cross sections. Examples include: - Sivers asymmetry, associated with the correlation between the nucleon spin and the parton transverse momentum, observable as a sin(phi_h - phi_S) modulation in SIDIS. - Boer–Mulders effect, leading to cos(2 phi) modulations in unpolarized SIDIS due to a correlation between transverse momentum and quark transverse polarization. - Transversity-related asymmetries and related TMDs that appear in polarized SIDIS and related processes.
Global analyses and data sources
Global analyses combine data from SIDIS, Drell–Yan, and other processes to extract parameterizations of TMDs, subject to evolution and gauge-link constraints. Experimental programs at facilities such as fixed-target experiments, electron–ion colliders, and high-energy colliders contribute to a growing map of TMDs across x, kT, and polarization.
Controversies and debates
- Universality vs process dependence: While some TMDs are largely universal, the gauge-link structure implies process-dependent effects for others. This interplay is a central topic in theoretical development and in interpreting cross-process comparisons.
- Factorization validity in complex processes: Factorization theorems are well-established for SIDIS and Drell–Yan, but there is ongoing discussion about factorization in multi-hactor hadron-hadron processes with multiple observed final-state particles, where color entanglement can threaten clean factorization.
- Evolution schemes and non-perturbative input: Different groups adopt various schemes for TMD evolution and soft factors. Disagreements over non-perturbative parameterizations and the matching to collinear PDFs affect the extraction of TMDs and their interpretation.
- Lattice QCD and model connections: Lattice calculations and phenomenological models provide complementary insights, but connecting lattice results to the full, gauge-link–dependent TMD definitions used in phenomenology remains an active area of research.
- Experimental tensions and future tests: Ongoing and planned experiments aim to sharpen tests of sign changes (e.g., Sivers function) and to map TMDs over a wider range of x and kT, helping to discriminate between competing theoretical approaches.