The mathematical operation to go from a Generalized Parton Distribution (GPD) to a Transverse Momentum-Dependent parton distribution (TMD) is typically tied to Fourier transformations and kinematic projections, but the relationship is not straightforward due to their fundamentally different definitions. Here's an outline of the connections:
1. Reduction from GPDs to PDFs
- GPDs H(x,ξ,t)H(x, \xi, t), E(x,ξ,t)E(x, \xi, t), etc., depend on:
- The longitudinal momentum fraction xx,
- The skewness ξ\xi (a measure of the longitudinal momentum transfer),
- The momentum transfer squared t=Δ2t = \Delta^2.
- A standard parton distribution function (PDF) can be obtained from a GPD by setting ξ=0\xi = 0 (no skewness) and integrating out the transverse momentum transfer: f(x)=∫d2Δ⊥H(x,0,−Δ⊥2).f(x) = \int d^2 \mathbf{\Delta}_\perp H(x, 0, -\mathbf{\Delta}_\perp^2).
2. Fourier Transform to Connect Transverse Momentum
- TMDs f(x,k⊥)f(x, \mathbf{k}_\perp) explicitly depend on the transverse momentum k⊥\mathbf{k}_\perp of the parton. They are obtained by retaining the dependence on the transverse structure, often through a Fourier transform of the GPD in transverse momentum transfer Δ⊥\Delta_\perp: f(x,k⊥)=∫d2Δ⊥(2π)2e−iΔ⊥⋅b⊥H(x,ξ,−Δ⊥2).f(x, \mathbf{k}_\perp) = \int \frac{d^2 \mathbf{\Delta}_\perp}{(2\pi)^2} e^{-i \mathbf{\Delta}_\perp \cdot \mathbf{b}_\perp} H(x, \xi, -\mathbf{\Delta}_\perp^2). Here, b⊥\mathbf{b}_\perp is the transverse impact parameter.
3. Impact Parameter Space Connection
- GPDs in ξ=0\xi = 0 are sometimes interpreted in terms of the impact parameter distributions of partons. These can be linked to TMDs through Fourier transforms that bridge momentum and position space.
4. Key Differences
- GPDs describe correlations between incoming and outgoing partons in deeply virtual scattering processes, carrying information about spatial distributions.
- TMDs, on the other hand, describe the intrinsic transverse momentum of partons inside a nucleon.
5. Model-Dependent Relations
- Direct relationships between GPDs and TMDs are often derived within specific models, such as quark-diquark models, bag models, or spectator models. These models provide a framework to link the transverse spatial distributions in GPDs to the transverse momentum distributions in TMDs.
6. Practical Application
- In phenomenology, constraints on GPDs from lattice QCD, experimental data, or model calculations are used to generate TMD predictions under specific assumptions about the nucleon structure.
In summary, the main mathematical operation involves Fourier transforms and the appropriate choice of kinematics to bridge the transverse spatial and momentum representations, though the exact connection often depends on modeling and specific approximations.