Transverse Momentum Dependent PDFs (TMDs)

• Transverse Momentum Dependent Parton Distribution Functions (TMDs) extend conventional PDFs by incorporating intrinsic transverse momentum, offering a three-dimensional view of hadron structure.
• They are central to modeling semi-inclusive processes, spin–momentum correlations, and azimuthal asymmetries observed in high-energy scattering experiments.
• Their rigorous formulation, involving gauge-invariant operator definitions with Wilson lines, enables precise QCD evolution and robust comparisons with experimental data.

Transverse Momentum Dependent Parton Distribution Functions (TMDs) provide a multidimensional representation of the partonic structure of hadrons by encoding not only the longitudinal momentum fraction, $x$, carried by the parton but also its intrinsic transverse momentum, $k_\perp$. Extending the traditional collinear parton distribution functions, TMDs are central to the theoretical description of semi-inclusive processes, the understanding of spin–momentum correlations, and the interpretation of azimuthal and polarization asymmetries observed in high-energy scattering experiments. The modern theoretical framework for TMDs encompasses gauge-invariant operator definitions with Wilson lines, QCD evolution and resummation, model calculations, and extensive phenomenological and lattice studies.

1. Operator Definitions and Gauge Structure

TMDs are defined through nonlocal operator correlators that explicitly depend on the parton’s transverse momentum. For a generic unpolarized quark TMD PDF, the foundational field-theoretic definition in QCD is given by a light-cone correlator,

$$\tilde f_{i/p}^{0}(x,\mathbf{b},\tau,xP^+) = \int \frac{db^-}{2\pi} e^{-ib^- (xP^+)} \langle p(P) | [\bar{\psi}^0_i(b^\mu) W_\sqsubset(b^\mu,0) \frac{\gamma^+}{2} \psi^0_i(0)]_\tau | p(P) \rangle \,,$$

where $b^\mu$ has components $(0, b^-, \mathbf{b}_\perp)$, $W_\sqsubset$ is the staple-shaped gauge link required for color gauge invariance, and $\tau$ is a rapidity regulator. The corresponding TMD PDF is renormalized and soft-subtracted: $$\tilde f_{i/p}(x,\mathbf{b},\mu,\zeta) = \lim_{\tau\to 0} Z^i_{\rm uv}(\mu,\zeta) \frac{\tilde f_{i/p}^{0}(x,\mathbf{b},\tau,xP^+)}{\sqrt{\tilde S_{n_an_b}^{0}(b_T,\tau)}} \,,$$ with the denominator the Wilson-line soft function and $\zeta$ a Collins–Soper(CS) evolution scale introduced to regulate rapidity divergences. This operator construction is mirrored for gluon TMDs, and generalized to describe polarized distributions and fragmentation functions.

Wilson line structures impart process dependence to TMDs. For example, future-pointing gauge links for SIDIS and past-pointing links for Drell–Yan processes lead to the predicted sign change between the Sivers functions extracted from these processes. Gauge invariance and the explicit path of the Wilson lines, reflecting the hard process color flow, are crucial to the theoretical consistency and universality structure of TMDs (Boussarie et al., 2023).

2. Classification, Twist, and Spin–Momentum Structure

TMDs classify all possible two-parton correlators respecting QCD symmetries for given target spin and parton species. For a spin-$1/2$ hadron and quark parton, there are eight leading-twist TMDs, decomposed according to the Dirac structure: $$\begin{aligned} f_{i/p}^{[\gamma^+]}(x,k_T) &= f_1(x,k_T) - \frac{\epsilon_T^{\rho\sigma}\,k_{T\rho} S_{T\sigma}}{M} f_{1T}^{\perp}(x,k_T),\ f_{i/p}^{[\gamma^+\gamma_5]}(x,k_T) &= S_L g_1(x,k_T) - \frac{k_T \cdot S_T}{M} g_{1T}^{\perp}(x,k_T),\ f_{i/p}^{[i\sigma^{\alpha+} \gamma_5]}(x,k_T) &= S_T^\alpha h_1(x,k_T) + \frac{S_L\, k_T^\alpha}{M} h_{1L}^\perp(x,k_T) - \frac{k_T^\alpha k_T^\rho - \frac{1}{2} g_T^{\alpha\rho}k_T^2}{M^2} S_{T\rho} h_{1T}^\perp(x,k_T)\ &\qquad - \frac{\epsilon_T^{\alpha\rho} k_{T\rho}}{M} h_1^\perp(x,k_T). \end{aligned}$$ The Sivers ($f_{1T}^{\perp}$) and Boer–Mulders ($h_1^\perp$) functions are time-reversal odd (T-odd), arising due to the Wilson line structure. For gluons and higher-spin targets, analogous but richer decompositions exist, incorporating functions of definite rank in transverse momentum (Buffing et al., 2013, Buffing et al., 2013, Kumano et al., 2021).

Beyond twist-2, twist-3 and twist-4 TMDs encode multi-parton and quark-gluon correlations, central to the description of power-suppressed but phenomenologically relevant azimuthal asymmetries. Their structure has been systematically studied in light-front, bag, and chiral models (Lorcé et al., 2014, Liu et al., 2021), and for spin-1 hadrons, an expanded hierarchy of TMDs appears, reflecting the tensor polarization of the target (Kumano et al., 2021).

3. TMD Evolution and Perturbative Matching

TMDs enter factorization theorems for semi-inclusive processes at scales where $k_\perp \ll Q$. The evolution of TMDs involves coupled renormalization group equations in the UV scale $\mu$ and the rapidity evolution scale $\zeta$: $$\mu^2 \frac{d}{d\mu^2} \tilde f(x,b_T,\mu,\zeta) = \frac{1}{2}\gamma^f(\mu,\zeta) \tilde f(x,b_T,\mu,\zeta), \qquad \zeta \frac{d}{d\zeta} \tilde f(x,b_T,\mu,\zeta) = -\mathcal{D}^f(\mu,b_T) \tilde f(x,b_T,\mu,\zeta),$$ with $\gamma^f$ and $\mathcal{D}^f$ known through NNLO (Echevarria et al., 2016). In the perturbative (small $b_T$) region, the TMD factorizes into a convolution of process-independent Wilson coefficients and collinear PDFs,

$$\tilde f_{1}(x,b_T,\mu,\zeta) = \sum_j \int_x^1 \frac{dy}{y}\, \tilde C_{ij}\Bigl(\frac{x}{y},b_T,\mu,\zeta\Bigr) f_j(y,\mu) + O(b_T^2\Lambda_{\mathrm{QCD}}^2),$$

with the matching coefficients calculated to NNLO and higher (Echevarria et al., 2016). The large logarithms of $Q^2 b_T^2$ (i.e., the hard scale vs. $1/b_T$) appearing in the perturbative regime are resummed via evolution kernels, e.g., using the Collins–Soper–Sterman (CSS) or modern rapidity renormalization group approaches, with model independence achieved by restricting to the perturbative region or using $b^*$-prescription with $b_{\max} \approx 1.5\,\mathrm{GeV}^{-1}$ as a phenomenological cutoff (Echevarria et al., 2012). For $Q_f \gg Q_i$, the nonperturbative contamination of the kernel is exponentially suppressed.

4. Universality, Process Dependence, and Gauge-Link Structures

The universality of TMDs is characterized by the structure of the gauge links in their definitions. While the collinear PDFs are universal, TMDs display an intrinsic process dependence embedded in their gauge links (future- or past-pointing staple-shaped Wilson lines, or more complex color flow configurations). This process dependence is captured quantitatively through gluonic pole factors and the theory of definite rank TMDs. For example, T-odd TMDs such as the Sivers and Boer–Mulders functions change sign between SIDIS and Drell–Yan due to their gauge-link structure (Buffing et al., 2012, Buffing et al., 2013, Buffing et al., 2013). Even some T-even functions (such as pretzelosity and its gluon analog) exhibit process dependence in general color environments (Buffing et al., 2012).

The decomposition into universal TMDs of definite rank, together with process-dependent gluonic pole factors, allows all observable TMDs to be written as

$$TMD^{[\mathrm{process}]} = \sum_{i} C^{[\mathrm{process}]}_i \cdot TMD^{(\mathrm{universal})}_i\,,$$

enabling robust cross-process predictions and factorization proofs.

5. Model Calculations and Nonperturbative Structure

Quark model calculations, such as those using the MIT bag model, light-front constituent quark models, spectator models, and chiral or instanton liquid models, provide key insights into the nonperturbative aspects of TMDs. In the bag model, all T-even leading and subleading twist TMDs are computed, and nontrivial linear and nonlinear relations among TMDs are found due to the limited number of independent wave function components (e.g., nine linear and two nonlinear relations among 14 T-even TMDs). Lorentz-invariance relations (LIRs) that must hold in quark-only models without explicit gluons are verified: $$g_T(x) = g_1(x) + \frac{d}{dx} g^{\perp(1)}_{1T}(x), \qquad h_T(x) = -\frac{d}{dx} h^{\perp(1)}_{1T}(x), \ldots$$ These relations are only model-exact; in QCD, TMDs are independent (Avakian et al., 2010, Avakian et al., 2011, Lorcé et al., 2014).

A particularly notable result is the operator-level relation between the pretzelosity distribution and quark orbital angular momentum (OAM),

$L_q^3 = -\int dx\, h_{1T}^{\perp(1)q}(x),$

suggesting that the measurement of azimuthal asymmetries associated with pretzelosity may probe the OAM contribution in the nucleon (Avakian et al., 2010, Avakian et al., 2011). Bag model studies also show that, in the valence-$x$ region ($0.2 \lesssim x \lesssim 0.5$), the transverse momentum dependence of TMDs is effectively Gaussian with a nearly constant width $p_T(x) \sim 0.25$ GeV, supporting the widespread Gaussian Ansatz used in phenomenology.

Other approaches, including the instanton liquid model (for pions, kaons, and rho mesons), chiral effective theory, and the statistical model (including energy sum rules and Melosh–Wigner rotations), address the modeling of TMDs at low resolution, the connection to underlying QCD dynamics, and relevant sum rules. Consistently, such models highlight the rapidity- and scale-dependent evolution of TMDs and their connection to partonic and chiral substructure (Bourrely et al., 2013, Copeland et al., 23 May 2024, Liu et al., 13 Feb 2025, Liu et al., 15 Mar 2025).

6. Experimental Access and Phenomenological Analysis

TMDs are experimentally accessed in processes where the transverse momentum of a detected hadron or lepton, or angular modulations, are measured—most notably, in semi-inclusive deep inelastic scattering (SIDIS), low-$q_T$ Drell–Yan, and $e^+e^-$ collisions. The measured cross sections are differential in azimuthal angles and transverse momenta, and are factorized as convolutions of TMD PDFs and fragmentation functions (FFs), often in $b_T$-space: $$\frac{d\sigma}{d^2 q_T} \sim \int d^2 b_T\, e^{i q_T\cdot b_T} e^{-S(b_T, Q)}f_1(x_1, b_T; \mu, \zeta) f_2(x_2, b_T; \mu, \zeta).$$ These formulas underpin the extraction of fundamental distributions such as the Sivers, Collins, and Boer–Mulders functions (Angeles-Martinez et al., 2015, Boussarie et al., 2023, Hautmann et al., 2014). Experimentally accessible observables include multi-dimensional single spin asymmetries (SSAs), which reveal spin-orbit correlations, and the sign change of T-odd TMDs is a direct test of QCD color dynamics.

Global fits and computational tools such as TMDlib provide repository interfaces, parameterizations, and evolution modules for both phenomenological and Monte Carlo applications, facilitating direct comparison with data and systematic uncertainty analysis (Hautmann et al., 2014, Angeles-Martinez et al., 2015).

7. Recent Developments and Theoretical Directions

New theoretical developments include the computation of matching coefficients to NNLO (and beyond) for TMD PDFs and FFs, improved understanding of the evolution kernel structure, and efforts to define and compute TMDs on the lattice via quasi-TMD or Lorentz-invariant schemes (Echevarria et al., 2016, Boussarie et al., 2023). For spin-1 hadrons, expanded TMD bases with up to 40 functions (including twist-3 and 4) have been classified, introducing new sum rules for time-reversal-odd functions and opening the paper of color entanglement effects (Kumano et al., 2021). Chiral effective theory frameworks and nonperturbative QCD vacuum models (e.g., instanton liquid) have provided explicit constructions of TMDs for light mesons, baryons, and vector mesons (Liu et al., 13 Feb 2025, Liu et al., 15 Mar 2025, Copeland et al., 23 May 2024).

A consistent picture has emerged: TMDs provide a three-dimensional, process-conscious mapping of hadron structure, sensitive to nonperturbative QCD dynamics, spin–orbit correlations, and color flow. Their extraction and application underpin precision measurements in collider physics, tests of QCD factorization, and the interpretation of spin phenomena in hadronic reactions.

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Key Relations and Formulas

Relation	Formula	Context
Pretzelosity–OAM	$L_q^3 = -\int dx\, h_{1T}^{\perp(1)q}(x)$	Bag model, OAM access (Avakian et al., 2010)
Effective Gaussian Width	$p_T^2(x)_{\rm Gauss} = \pi f_1^q(x,0)/f_1^q(x)$	Valence-$x$ region (Avakian et al., 2010)
Spin-Universal Evolution Kernel	Identical for all leading-twist TMDs, kernel is process- and spin-independent	NNLL evolution (Echevarria et al., 2012)
Generalized Universality	$$TMD^{[\text{process}]} = \sum_i C_i^{[\text{process}]} TMD_i^{(\text{universal})}$$	Gauge-link structure (Buffing et al., 2013, Buffing et al., 2013)

TMDs thus represent a well-defined, gauge-invariant, and phenomenologically rich sector of QCD, essential for multidimensional nucleon and hadron imaging, and for the interpretation of precision collider and fixed-target data.