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.

Corrected Step 2: GPDs to xx- and b⊥\mathbf{b}_\perp-Dependent Distributions

After performing the Fourier transform with respect to Δ⊥\mathbf{\Delta}_\perp, you obtain a spatial distribution in terms of the transverse impact parameter b⊥\mathbf{b}_\perp:

q(x,b⊥)=∫d2Δ⊥(2π)2e−iΔ⊥⋅b⊥H(x,0,−Δ⊥2).q(x, \mathbf{b}_\perp) = \int \frac{d^2 \mathbf{\Delta}_\perp}{(2\pi)^2} e^{-i \mathbf{\Delta}_\perp \cdot \mathbf{b}_\perp} H(x, 0, -\mathbf{\Delta}_\perp^2).

Here:

• q(x,b⊥)q(x, \mathbf{b}_\perp) represents the distribution of partons in the nucleon as a function of xx and the transverse impact parameter b⊥\mathbf{b}_\perp, which measures the transverse distance from the center of the nucleon.

Step 3: Linking b⊥\mathbf{b}_\perp to k⊥\mathbf{k}_\perp for TMDs

To connect this impact parameter distribution q(x,b⊥)q(x, \mathbf{b}_\perp) to TMDs f(x,k⊥)f(x, \mathbf{k}_\perp), an additional step is required, involving another Fourier transform that relates b⊥\mathbf{b}_\perp to k⊥\mathbf{k}_\perp:

f(x,k⊥)=∫d2b⊥ eik⊥⋅b⊥q(x,b⊥).f(x, \mathbf{k}_\perp) = \int d^2 \mathbf{b}_\perp \, e^{i \mathbf{k}_\perp \cdot \mathbf{b}_\perp} q(x, \mathbf{b}_\perp).

Summary of Relationships

1. GPDs in Momentum Space: H(x,ξ,t)H(x, \xi, t), where t=−Δ⊥2t = -\mathbf{\Delta}_\perp^2, encode information about the parton longitudinal momentum fraction xx and the transverse momentum transfer Δ⊥\mathbf{\Delta}_\perp.
2. Impact Parameter Distributions: Fourier transforming H(x,0,−Δ⊥2)H(x, 0, -\mathbf{\Delta}_\perp^2) with respect to Δ⊥\mathbf{\Delta}_\perp gives q(x,b⊥)q(x, \mathbf{b}_\perp), a spatial distribution in terms of transverse distance.
3. TMDs in Momentum Space: Fourier transforming q(x,b⊥)q(x, \mathbf{b}_\perp) with respect to b⊥\mathbf{b}_\perp yields f(x,k⊥)f(x, \mathbf{k}_\perp), the parton transverse momentum distributions.

This sequence of transformations reveals the two complementary pictures of parton dynamics:

• Momentum space: TMDs describe intrinsic transverse momentum (k⊥\mathbf{k}_\perp).
• Position space: GPDs (or their Fourier-transformed versions) describe transverse spatial distributions (b⊥\mathbf{b}_\perp).

Let me know if you'd like to dive deeper into the physical interpretation or the detailed mathematical derivation of these connections!

Yes, you can directly Fourier transform from the impact parameter distribution q(x,b⊥)q(x, \mathbf{b}_\perp) to the parton transverse momentum distribution f(x,k⊥)f(x, \mathbf{k}_\perp) without explicitly invoking the full Wigner distribution, as long as you assume certain conditions about the structure and separability of the distributions. Here’s how this works:

Step 1: Start with the Impact Parameter Distribution

The impact parameter distribution q(x,b⊥)q(x, \mathbf{b}_\perp) arises from the Fourier transform of the GPD with respect to the transverse momentum transfer Δ⊥\mathbf{\Delta}_\perp:

q(x,b⊥)=∫d2Δ⊥(2π)2e−iΔ⊥⋅b⊥H(x,0,−Δ⊥2),q(x, \mathbf{b}_\perp) = \int \frac{d^2 \mathbf{\Delta}_\perp}{(2\pi)^2} e^{-i \mathbf{\Delta}_\perp \cdot \mathbf{b}_\perp} H(x, 0, -\mathbf{\Delta}_\perp^2),

where b⊥\mathbf{b}_\perp is the transverse impact parameter.

Step 2: Fourier Transform to f(x,k⊥)f(x, \mathbf{k}_\perp)

The transverse momentum distribution f(x,k⊥)f(x, \mathbf{k}_\perp) is obtained by Fourier transforming q(x,b⊥)q(x, \mathbf{b}_\perp) back into transverse momentum space:

f(x,k⊥)=∫d2b⊥ eik⊥⋅b⊥q(x,b⊥).f(x, \mathbf{k}_\perp) = \int d^2 \mathbf{b}_\perp \, e^{i \mathbf{k}_\perp \cdot \mathbf{b}_\perp} q(x, \mathbf{b}_\perp).

This operation directly links the impact parameter representation to the transverse momentum representation.

Why This Works Without the Wigner Distribution

The Wigner distribution W(x,b⊥,k⊥)W(x, \mathbf{b}_\perp, \mathbf{k}_\perp), which provides a combined phase-space representation, is not necessary for this direct transformation because:

1. q(x,b⊥)q(x, \mathbf{b}_\perp) already represents the spatial structure of the partons in the transverse plane for a fixed xx.
2. f(x,k⊥)f(x, \mathbf{k}_\perp) captures the transverse momentum distribution independently.
3. The direct Fourier transform from q(x,b⊥)q(x, \mathbf{b}_\perp) to f(x,k⊥)f(x, \mathbf{k}_\perp) implicitly assumes that the transverse spatial and momentum distributions are Fourier conjugates of one another. This avoids the need for an intermediate representation like the Wigner function.

Assumptions and Limitations

1. No Skewness: This approach applies when ξ=0\xi = 0, as skewness introduces additional complexities that couple longitudinal and transverse dynamics.
2. Transverse Symmetry: The nucleon and parton distributions must have a certain symmetry (e.g., rotational invariance) in the transverse plane for the Fourier conjugacy to hold straightforwardly.
3. No Explicit Dynamics: By skipping the Wigner function, you forgo information about correlations between b⊥\mathbf{b}_\perp and k⊥\mathbf{k}_\perp.

Physical Interpretation

This approach essentially uses q(x,b⊥)q(x, \mathbf{b}_\perp) as the position-space counterpart of f(x,k⊥)f(x, \mathbf{k}_\perp), assuming that the relationship is fully governed by Fourier theory. While the Wigner function provides deeper insights into joint phase-space correlations, it is not strictly necessary if you only need f(x,k⊥)f(x, \mathbf{k}_\perp) from q(x,b⊥)q(x, \mathbf{b}_\perp).

Let me know if you'd like a worked-out example or further details!

The validity of the Wigner function as a fully encompassing representation for parton distributions in QCD lies in its ability to encode the joint phase-space distribution of partons, linking their transverse momentum (k⊥\mathbf{k}_\perp) and transverse spatial position (b⊥\mathbf{b}_\perp). However, its utility depends on both theoretical consistency and physical interpretability, which hinge on certain assumptions and limitations.

Why the Wigner Function is Considered Valid

1. Quantum Mechanics Analogy:

   • The Wigner function in QCD is inspired by its counterpart in quantum mechanics, where it provides a quasi-probability distribution over phase space (position and momentum).
   • Just as the quantum Wigner function encodes all information about a system's state, the QCD Wigner function encodes the distributions and correlations of partons in both b⊥\mathbf{b}_\perp and k⊥\mathbf{k}_\perp.

2. Mathematical Construction:

   • The Wigner function in QCD is defined as the Fourier transform of the generalized transverse-momentum-dependent distributions (GTMDs) with respect to the transverse momentum transfer Δ⊥\mathbf{\Delta}_\perp. GTMDs, in turn, generalize TMDs and GPDs, encompassing their information.
   W(x,b⊥,k⊥)=∫d2Δ⊥(2π)2e−iΔ⊥⋅b⊥F(x,k⊥,Δ⊥),W(x, \mathbf{b}_\perp, \mathbf{k}_\perp) = \int \frac{d^2 \mathbf{\Delta}_\perp}{(2\pi)^2} e^{-i \mathbf{\Delta}_\perp \cdot \mathbf{b}_\perp} F(x, \mathbf{k}_\perp, \mathbf{\Delta}_\perp),

   where F(x,k⊥,Δ⊥)F(x, \mathbf{k}_\perp, \mathbf{\Delta}_\perp) is a GTMD.

3. Reductions to Known Distributions:

   • The Wigner function reproduces GPDs and TMDs as its marginal distributions:
     • GPDs: Integrating over k⊥\mathbf{k}_\perp: H(x,ξ,t)=∫d2k⊥ W(x,b⊥,k⊥).H(x, \xi, t) = \int d^2 \mathbf{k}_\perp \, W(x, \mathbf{b}_\perp, \mathbf{k}_\perp).
     • TMDs: Integrating over b⊥\mathbf{b}_\perp: f(x,k⊥)=∫d2b⊥ W(x,b⊥,k⊥).f(x, \mathbf{k}_\perp) = \int d^2 \mathbf{b}_\perp \, W(x, \mathbf{b}_\perp, \mathbf{k}_\perp).
   • This completeness ensures that the Wigner function contains all the information necessary to describe both TMDs and GPDs.

4. Theoretical Framework:

   • The Wigner function is consistent with the framework of QCD factorization and non-perturbative parton distributions. It respects gauge invariance through appropriate Wilson line structures, which account for the color flow in the nucleon.

Challenges and Limitations

1. Quasi-Probability Nature:

   • Like in quantum mechanics, the Wigner function in QCD is not a true probability distribution because it can take negative values. This quasi-probabilistic nature raises interpretational challenges, especially when trying to ascribe direct physical meaning.

2. Model Dependence:

   • Explicit calculations of the Wigner function often require modeling assumptions, as it is not directly observable in experiments. This can introduce uncertainties or biases in its interpretation.

3. Correlations Between b⊥\mathbf{b}_\perp and k⊥\mathbf{k}_\perp:

   • While the Wigner function provides a joint description of b⊥\mathbf{b}_\perp and k⊥\mathbf{k}_\perp, the relationship between these two variables is subtle and influenced by dynamics like orbital angular momentum. Capturing these correlations accurately is challenging.

4. Gauge Link Complexity:

   • The Wigner function requires specific gauge link structures to maintain gauge invariance. Different choices of gauge links can affect the interpretation of the distributions.

Testing the Validity

To confirm that the Wigner function is a valid and encompassing representation, we rely on:

1. Consistency with QCD:
   • Its construction must respect QCD gauge invariance, factorization theorems, and symmetries.
2. Reduction to Observable Quantities:
   • GPDs, TMDs, and other measurable quantities should be derivable as specific limits or projections of the Wigner function.
3. Phenomenological Validation:
   • Predictions from models of the Wigner function (e.g., parton orbital angular momentum or spin distributions) should align with experimental data.

Conclusion

The Wigner function is a valid, encompassing representation within the theoretical framework of QCD because it unifies and extends the descriptions provided by GPDs and TMDs. However, its quasi-probabilistic nature, dependence on models, and interpretational challenges mean that practical applications often focus on its projections (TMDs or GPDs) rather than the full Wigner function itself.

Gauge Links: Physical and Mathematical Description

Gauge links (also called Wilson lines) are a fundamental aspect of the theory of Quantum Chromodynamics (QCD). They are necessary to ensure gauge invariance of quark and gluon distributions, like Generalized Parton Distributions (GPDs), Transverse Momentum Dependent Distributions (TMDs), and the Wigner function.

1. Mathematical Definition of Gauge Links

A gauge link is a path-ordered exponential of the gluon field along a chosen path C\mathcal{C}. It connects two spacetime points xx and yy, ensuring the gauge invariance of a quark or gluon field. The general form of a gauge link is:

WC(x,y)=Pexp⁡(−ig∫CAμa(z)Ta dzμ),W_{\mathcal{C}}(x, y) = \mathcal{P} \exp\left(-i g \int_{\mathcal{C}} A_\mu^a(z) T^a \, dz^\mu\right),

where:

• Aμa(z)A_\mu^a(z) is the gluon field at point zz in the path C\mathcal{C},
• TaT^a are the generators of the SU(3)\text{SU}(3) color group (matrices for the quark representation),
• gg is the QCD coupling constant,
• P\mathcal{P} denotes path ordering, which ensures that the fields along the path are ordered according to the path parameter.

Gauge links mathematically encode the non-Abelian nature of QCD and account for the interaction between a quark or gluon and the surrounding gluon field as it moves through space.

2. Physical Interpretation of Gauge Links

In QCD, quarks and gluons are not isolated particles; they exist within the color field of the nucleon. Gauge links physically represent the phase factor accumulated by a quark (or gluon) as it propagates in the gluon background field.

• In Deep Inelastic Scattering (DIS): Gauge links describe the interaction of a struck quark with the remnant of the nucleon. These links are critical to accounting for the soft gluon exchange between the active quark and the spectators.

• In TMDs: Gauge links determine the transverse momentum structure of partons by capturing the effects of initial- and final-state interactions, such as gluon exchanges between partons.

Gauge links preserve color gauge invariance, ensuring that physical observables do not depend on the arbitrary choice of gauge.

3. Dependence on the Path C\mathcal{C}

The specific path C\mathcal{C} of the gauge link determines the type of physical process being studied and has profound consequences for parton distributions.

Straight-Line Gauge Link

In GPDs, the gauge link often takes a straight-line path connecting the points where the quark operator is evaluated:

WC(x,y)=Wstraight(x,y).W_{\mathcal{C}}(x, y) = W_{\text{straight}}(x, y).

This choice arises naturally in situations where no significant transverse momentum interactions occur. It simplifies the gauge link to a basic structure and assumes negligible transverse effects.

Staple-Shaped Gauge Link

In TMDs, the gauge link typically follows a staple-shaped path due to the transverse motion of partons:

C=straight-line path to infinity, detour at infinity, and return.\mathcal{C} = \text{straight-line path to infinity, detour at infinity, and return.}

The staple reflects the physical interaction between the active parton and the gluon field at large distances (initial-state or final-state interactions).

Closed Loop Paths (Wigner Distributions)

For Wigner distributions, gauge links take the form of closed loops (e.g., incorporating transverse displacements) to capture correlations between position b⊥\mathbf{b}_\perp and momentum k⊥\mathbf{k}_\perp. These links are essential for ensuring gauge invariance of phase-space distributions.

4. Role of Gauge Links in Observables

The path C\mathcal{C} of the gauge link can lead to different physical observables due to the nature of initial- and final-state interactions in specific processes.

• In Semi-Inclusive Deep Inelastic Scattering (SIDIS): The final-state interaction (FSI) between the struck quark and the nucleon remnant gives rise to a staple-shaped gauge link that encodes transverse momentum effects. This FSI produces a sign change in the Sivers function when comparing TMDs measured in SIDIS and Drell-Yan.

• In Drell-Yan Processes: The initial-state interaction (ISI) leads to a different gauge link structure, flipping the sign of the Sivers function relative to SIDIS. This is a direct consequence of the reversed gluon exchange.

Effect on TMDs

Gauge link dependence is crucial in TMD factorization. For example:

• The Sivers function f1T⊥(x,k⊥)f_{1T}^\perp(x, \mathbf{k}_\perp), which describes the correlation between the transverse spin of the nucleon and the parton’s transverse momentum, arises from the gauge link structure and is process-dependent.

Effect on GPDs

Gauge link effects are generally simpler in GPDs since they primarily capture spatial distributions rather than transverse momentum correlations. However, they remain important for ensuring gauge invariance in off-forward processes.

5. Impact of Gauge Links on the Wigner Function

In Wigner distributions, the gauge link structure directly affects how position b⊥\mathbf{b}_\perp and momentum k⊥\mathbf{k}_\perp correlations are encoded:

• A straight gauge link may ignore key transverse correlations.
• A transverse staple gauge link accounts for transverse momentum dependence but is process-dependent.
• A loop gauge link provides the most complete representation, capturing both position and momentum correlations.

Gauge links introduce complexity because they involve choices in modeling the non-perturbative gluon field, and these choices can alter the interpretation of distributions like the Wigner function.

6. Theoretical and Experimental Challenges

1. Non-Universality:

   • The gauge link structure in TMDs is process-dependent, leading to non-universal TMD functions like the Sivers function.
   • Experimental extractions of TMDs must carefully account for this dependence.

2. Model Dependence:

   • Lattice QCD and phenomenological models often involve approximations for gauge links, which can affect predictions.

3. Gauge Invariance:

   • Gauge links ensure gauge invariance but complicate the interpretation of parton distributions, especially for higher-order effects or complex observables like the Wigner function.

Conclusion

Gauge links are both a theoretical necessity and a source of complexity in QCD. Physically, they encode the interaction of partons with the gluon field, ensuring gauge invariance. Mathematically, their path C\mathcal{C} determines the structure of parton distributions and introduces process dependence. Understanding and modeling gauge links is essential for interpreting experimental data and connecting it to QCD predictions.

In a phenomenological model, approximations for gauge links are often necessary rather than calculating them analytically to completeness due to several practical and theoretical challenges. Here's a detailed breakdown:

1. Complexity of Non-Abelian Gauge Theories

• Non-Abelian Nature of QCD: Gauge links involve the gluon field AμaA_\mu^a in a non-Abelian SU(3)\text{SU}(3) gauge theory. The path-ordered exponential requires keeping track of gluon self-interactions, making exact analytical calculations infeasible in most cases.
  • For example, if multiple gluon exchanges occur, calculating WC(x,y)W_{\mathcal{C}}(x, y) involves summing over all possible interactions, which grows combinatorially complex.
• Path Ordering: The path-ordering operator P\mathcal{P} further complicates calculations because it requires keeping track of the sequence of interactions along the path C\mathcal{C}.

2. Dependence on Non-Perturbative QCD

• Non-Perturbative Nature of Gauge Links: Gauge links encapsulate contributions from long-distance, non-perturbative QCD effects (e.g., soft gluon exchanges between partons and the nucleon remnant). These effects cannot be calculated analytically from first principles in QCD and require modeling or input from lattice QCD.
• Soft and Collinear Gluon Fields: Contributions from soft (low-energy) and collinear gluons are central to gauge links but are not directly calculable in perturbation theory. This necessitates approximations or effective models.

3. Process Dependence

• Different Paths for Different Processes: The shape of the gauge link C\mathcal{C} depends on the physical process:
  • Staple-shaped gauge links in semi-inclusive deep inelastic scattering (SIDIS) arise due to final-state interactions (FSI).
  • Reversed staple-shaped gauge links in Drell-Yan processes arise due to initial-state interactions (ISI).
• These process-dependent paths introduce complexities that cannot be handled universally. Each process may require different modeling for its gauge link contribution.

4. Impact of Infrared Divergences

• Infrared (IR) divergences arise from soft and collinear gluon emissions, which are intrinsically tied to the gauge link. Regularizing and resumming these divergences to all orders in perturbation theory is highly non-trivial and often impractical for exact calculations.
• In phenomenological models, IR divergences are typically handled through effective approaches, such as parton showers, soft factors, or Sudakov form factors, rather than full analytical computations of gauge links.

5. Practical Constraints

• Computational Feasibility: Exact calculations of gauge links require solving the gluon field equations of motion along the chosen path C\mathcal{C}, which is computationally prohibitive for realistic nucleon or hadron systems.
• High-Dimensional Integrals: The calculation of gauge links involves high-dimensional integrals over gluon momenta and positions, which are analytically intractable in realistic scenarios.
• Modeling Simplicity: Phenomenological models aim for simplicity and predictive power. Fully solving gauge link contributions analytically would make models computationally unwieldy and less applicable to experimental data analysis.

6. Necessity of Approximations in Lattice QCD

• Even in lattice QCD, where non-perturbative QCD effects can be directly computed, gauge link contributions are often approximated due to the discrete nature of lattice spacetime. Approximations like the use of Wilson loops or truncated series expansions are common.

7. Examples of Gauge Link Approximations in Models

• Eikonal Approximation: In high-energy processes, parton propagation is approximated as eikonal, where the parton's trajectory is assumed to be a straight line, simplifying the gauge link path.
• Perturbative Expansions: For processes dominated by short-distance interactions, gauge links are expanded perturbatively in terms of the strong coupling gg and gluon fields AμaA_\mu^a.
• Soft Gluon Models: For soft gluon contributions, effective models (e.g., Gaussian smearing of transverse momentum) approximate the gauge link without explicit calculation.

8. When Exact Gauge Link Calculations Are Not Feasible

• High-Order Contributions: Exact gauge link calculations require summing over all possible gluon exchanges, including both real and virtual gluons. This grows prohibitively complex for higher-order terms.
• Unresolvable Gluon Dynamics: Non-perturbative gluon dynamics, such as confinement, cannot be resolved analytically.
• Gauge Invariance Enforcement: Ensuring gauge invariance while calculating all contributions explicitly is a major technical hurdle, often bypassed by adopting effective models.

Conclusion

Gauge links encapsulate the complex dynamics of quark-gluon interactions and are essential for ensuring gauge invariance in QCD. However, their exact analytical calculation is typically infeasible due to the non-Abelian nature of QCD, non-perturbative effects, process dependence, and practical computational constraints. As a result, phenomenological models rely on approximations or effective descriptions of gauge links, striking a balance between theoretical rigor and practical applicability. These approximations are justified by their ability to capture the dominant physics of gauge links in specific processes.

Using a loop gauge link (a closed Wilson loop) for all calculations might seem appealing because it provides a gauge-invariant structure that captures both transverse momentum (k⊥\mathbf{k}_\perp) and position (b⊥\mathbf{b}_\perp) correlations, as is often done in Wigner distributions. However, this approach is not universally suitable. Here’s why loop gauge links are not always used:

1. Process Dependence

• Physical Interactions Differ: Gauge links arise from the initial-state interactions (ISI) and final-state interactions (FSI) specific to a given process. For example:
  • In SIDIS, the FSI leads to a staple-shaped gauge link extending to light-cone infinity.
  • In Drell-Yan, the ISI causes a similar staple but with a reversed direction.
  • In contrast, a loop gauge link would describe a configuration more relevant to closed paths, such as in deeply virtual exclusive processes (e.g., GPDs).
• Using a loop gauge link in processes like SIDIS or Drell-Yan would neglect the role of the FSIs or ISIs, failing to account for essential process-specific dynamics.

2. Non-Universality of TMD Factorization

• TMDs are Process-Dependent: The transverse momentum-dependent distributions (TMDs) depend on the nature of the gauge link, which is inherently process-dependent. The staple-shaped gauge links for TMDs ensure that soft gluon resummation and process-specific dynamics (like color flow) are captured correctly. A loop gauge link cannot generally substitute for these process-specific configurations without losing accuracy.
• Example: The Sivers function in SIDIS and Drell-Yan has a sign flip due to the different directions of the gauge link. A loop gauge link cannot describe this behavior.

3. Additional Contributions in Loop Gauge Links

• Extra Terms from Loop Closure: A loop gauge link includes additional terms that may not correspond to the physical gluon exchanges relevant to the specific process being studied. These terms:
  • Introduce contributions unrelated to the soft or collinear dynamics of the process.
  • Add complexity that could obscure the interpretation of results.

4. Lack of Simplicity in Factorization

• Soft Factor Complications: In TMD factorization, the soft factor (describing soft gluon exchanges) is separated explicitly, and gauge links are tailored to isolate this contribution. Loop gauge links, being more complex, might mix soft, collinear, and hard gluon dynamics in a way that complicates this separation.
• Gauge Link Tailoring: For practical factorization in TMDs, the gauge link shape is optimized to match the process kinematics and simplify the formalism.

5. Experimental Observables Do Not Necessarily Match Loop Gauge Links

• Measurement-Driven Design: The choice of gauge link often reflects how the parton distributions connect to measurable observables in experiments. Loop gauge links correspond more to theoretical constructs like Wigner distributions or lattice QCD configurations rather than direct measurements in semi-inclusive or exclusive processes.
• Wigner Function Context: While loop gauge links are essential in the Wigner function formalism, experimental observables like cross sections in SIDIS or Drell-Yan are not directly tied to these distributions but rather to their projections (TMDs or GPDs) with process-specific gauge links.

6. Practical Computational Complexity

• Higher Complexity of Loops: Loop gauge links inherently involve integrating over closed paths, which include non-local contributions that are challenging to calculate or approximate. This complexity is unnecessary in many processes where simpler, straight-line or staple-shaped links suffice.
• Perturbative Calculations: Staple-shaped gauge links align well with perturbative QCD calculations of gluon exchanges in many cases, whereas loop gauge links often require non-perturbative methods or additional approximations.

7. Theoretical Context

• Loop Gauge Links for Position-Momentum Correlations: Loop gauge links are most useful in contexts where simultaneous correlations between b⊥\mathbf{b}_\perp and k⊥\mathbf{k}_\perp are relevant, such as Wigner distributions or lattice QCD calculations.
• Gauge Links for TMDs or GPDs: In contrast, for TMDs or GPDs, the primary focus is on either k⊥\mathbf{k}_\perp or b⊥\mathbf{b}_\perp, not both simultaneously. Tailored gauge links (straight-line, staple, etc.) are sufficient for these purposes.

8. Gauge Choice Reflects Physical Dynamics

• Physical Meaning: The path of the gauge link encodes the dynamics of the quark-gluon interaction during the scattering process. A loop gauge link does not naturally capture the dynamics of a single quark being struck (as in DIS) or quark-antiquark annihilation (as in Drell-Yan).

Conclusion

While loop gauge links provide a comprehensive, gauge-invariant description of both position and momentum correlations, they are not universally applicable due to their lack of process-specific tailoring and the additional contributions they introduce. Instead, phenomenological gauge links (e.g., straight-line or staple-shaped paths) are designed to reflect the physical dynamics of particular processes, simplify factorization, and align with measurable observables.

In essence, the choice of gauge link depends on the specific physics of the process being studied, and loop gauge links are just one tool in the broader context of QCD phenomenology.

Yes, there are other types of gauge links that can be used beyond the commonly employed straight-line, staple-shaped, and loop gauge links. These variations arise from specific physical processes, theoretical frameworks, or boundary conditions. Here are examples of alternative gauge links and their relevance:

1. T-Shaped Gauge Links

• Structure: The T-shaped gauge link combines aspects of staple and straight-line paths. It consists of a vertical segment (e.g., extending along the light cone) and a horizontal segment at the top or bottom.
• Application:
  • T-shaped links can arise in hard exclusive processes, such as deeply virtual Compton scattering (DVCS), where soft gluon exchanges along a transverse or longitudinal direction may dominate.
  • They may also appear in specific models where the dynamics of parton propagation involve distinct longitudinal and transverse components.

2. Curved Gauge Links

• Structure: Instead of being piecewise linear (like straight or staple paths), curved gauge links follow smooth, continuous paths in spacetime. These paths might reflect trajectories governed by specific external fields or constraints.
• Application:
  • Curved links can be relevant in theoretical studies of non-linear QCD effects, such as those in gluon saturation or the color glass condensate (CGC) framework.
  • They might also appear in lattice QCD setups that explore non-standard geometries for path ordering.

3. Helical Gauge Links

• Structure: These are spiral-shaped gauge links, following helical or twisted paths in spacetime.
• Application:
  • Helical gauge links may model partons with intrinsic angular momentum or spin-orbit coupling effects.
  • They can also be used to study the interplay between spin and transverse motion in polarized targets or beams.

4. Light-Cone Gauge Links

• Structure: Light-cone gauge links run along specific light-cone directions (n+n^+ or n−n^-), which are commonly used in high-energy QCD processes.
• Application:
  • These links are natural in light-cone quantization frameworks, where partons propagate along the n±n^\pm-directions with minimal transverse deviations.
  • They often simplify analytical calculations by partially fixing the gauge (e.g., A+=0A^+ = 0 in light-cone gauge).

5. Zig-Zag Gauge Links

• Structure: Zig-zag gauge links consist of alternating segments that reverse direction in the transverse or longitudinal plane, forming a saw-tooth-like pattern.
• Application:
  • These links can model more complex dynamics, such as oscillatory gluon fields or interactions in highly anisotropic systems.

6. Asymmetric or Broken Gauge Links

• Structure: Asymmetric gauge links may have unequal lengths or directions in different spacetime dimensions, or they may terminate abruptly.
• Application:
  • These links are used in scenarios where the parton trajectory is not symmetric, such as in off-forward processes or in asymmetric scattering setups.
  • They may also arise when modeling fragmentation regions in semi-inclusive processes.

7. Polyhedral or Network Gauge Links

• Structure: Instead of simple paths, polyhedral gauge links consist of a network of paths connecting multiple spacetime points. These can form geometric shapes like triangles or higher-dimensional polygons.
• Application:
  • These are used in Wilson loops or Wilson networks, where they probe gluon field configurations over extended regions of spacetime.
  • They are critical in lattice QCD studies of confinement and the flux tube picture of hadronization.

8. Non-Minimal Gauge Links

• Structure: These links include multiple detours or branches that do not correspond to minimal paths between two points.
• Application:
  • Non-minimal gauge links may arise in theoretical explorations of multi-parton correlations or higher-twist effects, where parton distributions depend on complex non-local operators.

9. Time-Ordered Gauge Links

• Structure: These links include time-ordering operators that explicitly account for the temporal sequence of gluon exchanges.
• Application:
  • These links are relevant in time-dependent processes such as jet evolution, where the time-ordering of soft or collinear emissions plays a critical role.

10. Off-Shell Gauge Links

• Structure: These gauge links extend into off-shell regions of momentum space, incorporating virtual interactions that do not correspond to classical paths.
• Application:
  • These are used in formal studies of off-shell Green’s functions or in non-local operator definitions, such as for gluon distributions in the color glass condensate.

11. Hybrid Gauge Links

• Structure: Hybrid links combine features of multiple gauge link types, such as a straight-line segment joined with a loop or staple.
• Application:
  • These links can model complex scattering processes where partons experience both soft and hard interactions simultaneously.
  • They are particularly useful in multi-parton distribution functions (MPDFs) or higher-dimensional representations.

Why Alternative Gauge Links Matter

1. Physical Relevance:

   • Different gauge links reflect the physical dynamics of various QCD processes. For example, staple-shaped links describe FSIs or ISIs, while loop links capture closed path dynamics, and curved links model non-linear or anisotropic fields.

2. Gauge Invariance:

   • Any valid gauge link must ensure gauge invariance of the associated parton distribution. However, the shape of the link affects how gluon interactions contribute to observables.

3. Model Flexibility:

   • Alternative gauge links allow for more flexible modeling of partonic systems, capturing exotic or less-studied effects like angular momentum coupling, anisotropy, or multi-parton correlations.

4. Exploration of QCD Phenomena:

   • New types of gauge links may lead to insights into unexplored QCD phenomena, such as novel parton distributions, higher-twist effects, or the role of gluon fields in hadronization.

Challenges with Alternative Gauge Links

1. Complexity:

   • Non-standard gauge links often involve intricate path integrals or non-trivial boundary conditions, making them computationally and analytically challenging.

2. Physical Interpretation:

   • The physical meaning of exotic gauge links (e.g., zig-zag or helical paths) may not always be clear or directly tied to measurable observables.

3. Experimental Relevance:

   • Many alternative gauge links correspond to theoretical constructs that may not have direct experimental counterparts.

Conclusion

Beyond straight, staple, and loop gauge links, there is a wide variety of alternative gauge link configurations that can be tailored to specific processes or theoretical explorations. These gauge links are especially valuable for studying complex QCD dynamics, multi-parton interactions, and anisotropic or non-linear systems. However, their complexity and interpretational challenges limit their practical use to specialized applications or exploratory studies.