⚛️ phenomenology

Twist-3 T-Even TMD Distributions in Quark Target Model

This paper calculates T-even twist-3 quark transverse momentum dependent parton distributions using the quark target model to address the lack of direct phenomenological fits needed for understanding beam single spin asymmetries and the 3-D structure of the proton.

Original authors: Siddhesh Padval, Anuradha Misra, Abhirup Karmakar, Deepesh Bhamre

Published 2026-09-24

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CC BY 4.0

Original authors: Siddhesh Padval, Anuradha Misra, Abhirup Karmakar, Deepesh Bhamre

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand the matter that makes up our world, physicists often look at the proton, the tiny, positively charged particle found in the nucleus of every atom. For decades, scientists have mapped the proton's internal structure by treating it as a bag of smaller particles called quarks and gluons. They have successfully charted how these particles move in straight lines, but a more complex picture has emerged: the proton is a spinning, three-dimensional object where particles also move sideways. To see this full picture, researchers study how quarks are distributed not just by their energy, but by their sideways motion and their spin. This detailed map is known as a transverse momentum dependent distribution. While scientists have already mapped the simplest version of this structure, a deeper, more intricate layer remains largely unexplored. This deeper layer is crucial for explaining a specific phenomenon observed in high-energy experiments: why beams of spinning particles behave differently when they smash into targets. Without a clear map of this deeper layer, the full story of the proton's shape remains incomplete.

The challenge lies in the fact that no one has yet directly measured these complex distributions from experimental data. Unlike the simpler maps that are well-established, these deeper patterns have not been extracted from real-world collisions. Consequently, scientists must rely on simplified models to predict what these distributions look like. For some time, the most popular tool for this job has been a model that treats the proton as a quark orbiting a pair of other particles. While useful, this approach has a significant blind spot: it cannot account for the gluons, the particles that carry the force holding the quarks together, because the model does not include them explicitly. To overcome this limitation, a team of researchers led by Siddhesh Padval turned to a different approach called the Quark Target Model. In this framework, the target particle is treated as a single, "dressed" quark—a quark that is already surrounded by a cloud of gluons and other interactions. This setup naturally includes the gluons and their complex interactions, offering a more complete theoretical laboratory to calculate the missing pieces of the proton's puzzle.

In their work, the researchers used this model to calculate the specific mathematical expressions for these elusive, deeper distributions. They focused on a set of patterns that do not change when time is reversed, known as T-even distributions, which are essential for understanding the spin-related asymmetries seen in experiments. By analyzing the interactions between the dressed quark and the gluons within this model, they derived precise formulas for eight different types of these distributions. These formulas describe how the probability of finding a quark with a certain sideways motion and spin changes depending on the energy it carries. The team found that several of these distinct distributions actually share the exact same mathematical form, suggesting a deep underlying symmetry in how these particles behave within the model. They also identified how the mass of the quark and the strength of the force between particles shape these patterns.

The results of this study provide a new set of theoretical tools. While these specific calculations apply to a simplified model of a single quark rather than a real proton, the researchers explain that these results can be combined with other models to build a realistic picture of the proton itself. By connecting these new calculations with existing theories of how quarks behave inside hadrons, scientists can now generate more accurate predictions for future experiments. This work does not claim to have solved the mystery of the proton's shape, nor does it present a final answer derived from direct measurement. Instead, it offers a necessary step forward: a clear, calculated set of predictions that can be tested against future data. With these new expressions in hand, the scientific community is better equipped to interpret the results of high-energy collisions and, eventually, to fully reveal the three-dimensional structure of the matter that makes up our universe.

Technical Summary: Twist-3 T-Even TMD Distributions in Quark Target Model

Problem Statement
Transverse momentum dependent (TMD) parton distributions are essential for understanding the three-dimensional structure of the proton. While twist-2 distributions (such as unpolarized PDFs and the Sivers distribution) have been successfully extracted from experimental data, twist-3 distributions remain unextracted due to a lack of direct phenomenological fits. These twist-3 distributions are theoretically crucial for explaining phenomena like Beam Single Spin Asymmetries (BSSAs) observed in semi-inclusive deep-inelastic scattering (SIDIS), such as recent measurements by the CLAS collaboration. Existing theoretical approaches, primarily the quark-diquark spectator model, lack explicit gluon degrees of freedom, preventing the calculation of gluon distributions and limiting the framework's completeness. Consequently, there is a need for alternative models that incorporate gluonic interactions to provide theoretical predictions for twist-3 BSSAs.

Methodology
This work employs the Quark Target Model (QTM) to calculate twist-3 T-even quark distributions. In the QTM, the target hadron is modeled as a single dressed quark, and the relevant Lagrangian reduces to the QCD Lagrangian for a single quark flavor, albeit with fields dressed by a low-energy effective theory of QCD. This framework naturally incorporates gluonic degrees of freedom, including the three-gluon vertex, which is absent in simpler models like the scalar diquark model.

The calculation proceeds perturbatively in QCD. The target state is defined as a dressed quark of mass $M$ and spin $S$ , composed of a bare quark (momentum $k$ ) and a gluon (momentum $P-k$ ). The authors compute the quark-quark correlator up to twist-3 using handbag diagrams. Specifically, the calculation includes:

Fig. 1a: The lowest-order perturbative diagram.
Fig. 1b: A diagram contributing at the same order but containing a higher-order Wilson line contribution.
Fig. 1c: A higher-order diagram noted as relevant for T-odd distributions but excluded from the current T-even analysis.

The authors utilize the methodology established in Ref. [7] to evaluate these correlators. The required TMD distributions are extracted by taking specific Dirac $\gamma$ -projections (traces) of the correlators $\Phi^{(1a)}$ and $\Phi^{(1b)}$ .

Key Contributions and Results
The primary contribution of this work is the derivation of analytical expressions for the twist-3 T-even quark TMD distributions within the QTM framework. The authors provide explicit formulas for the following distributions, expressed in terms of the strong coupling constant $g_s$ , the color factor $C_F$ , the dressed quark mass $m$ , the longitudinal momentum fraction $x$ , and the transverse momentum squared $k_\perp^2$ :

Scalar and Vector Distributions: $e(x, k_\perp^2)$ and $f^\perp(x, k_\perp^2)$ .
Axial-Vector Distributions: $g_T(x, k_\perp^2)$ and $g_L^\perp(x, k_\perp^2)$ .
Tensor Distributions: $h_L(x, k_\perp^2)$ , $h_T(x, k_\perp^2)$ , and $h_T^\perp(x, k_\perp^2)$ .

A notable result highlighted in the paper is that certain distributions, specifically $e$ , $g_T$ , and $h_L$ , are found to be given by the identical mathematical expression within this model. The expressions generally involve denominators of the form $[m^2(1-x)^2 + k_\perp^2]^2$ or similar structures, reflecting the dynamics of the dressed quark-gluon system.

Significance and Claims
The paper claims that while the distributions calculated in the QTM cannot be directly applied to SIDIS phenomenology (as the target is a quark, not a hadron), they serve as a necessary intermediate step. The authors propose that realistic hadronic distributions can be obtained by convoluting these QTM results with quark-in-hadron distributions derived from low-energy effective models, such as the Nambu-Jona-Lasinio (NJL) model.

The significance of this work lies in providing the "quark-in-dressed quark" component required for this convolution procedure. By establishing these expressions, the authors enable future studies to calculate actual quark-in-hadron distributions and subsequently perform phenomenological studies of BSSAs. The paper concludes modestly, stating that the results presented are the calculated expressions for the T-even distributions, with the application to phenomenology and the final convolution results reserved for future work to be presented elsewhere.

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