⚛️ phenomenology

Electron-Ion Collision Environment: Distribution of Quark Spin and Orbital Angular Momentum

This theoretical study utilizes the Nambu--Jona-Lasinio model and light-front dressed quark model to calculate quark spin and orbital angular momentum distributions in proton and nuclear environments, introducing GTMD ratios between electron-proton and electron-ion collisions to quantify non-perturbative many-body nuclear density effects.

Original authors: Sujit Jana, Ashutosh Dwibedi, Vikash Kumar Ojha, Sabyasachi Ghosh

Published 2026-07-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Sujit Jana, Ashutosh Dwibedi, Vikash Kumar Ojha, Sabyasachi Ghosh

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

Imagine the proton not as a solid marble, but as a bustling, chaotic city made of tiny, energetic citizens called quarks. For a long time, scientists have been trying to figure out how these citizens contribute to the city's most important property: its spin (which way it's "spinning"). The big question is: Is the spin coming from the citizens just sitting still and spinning on their own axes, or is it coming from them zooming around the city in orbits?

This paper is a theoretical "flight simulator" for a future super-microscope called the Electron-Ion Collider (EIC). The authors, Sujit Jana and his team, are trying to predict what happens when we zoom in on a single proton (an electron-proton collision) versus when we zoom in on a proton trapped inside a heavy nucleus, like a crowded apartment building (an electron-ion collision).

The Map and the Mass
To run their simulation, the team needed to know how heavy the quarks feel in these two different environments. In the empty space of a lone proton, quarks have a certain "constituent mass" of about 313 MeV (a unit of energy). But when you pack them into a nucleus, the environment changes. The team used a mathematical tool called the Nambu-Jona-Lasinio (NJL) model to calculate how this mass shifts when the density of the neighborhood increases to the "nuclear saturation density" (a specific crowdedness of 0.16 fm⁻³).

Think of it like this: A quark in a lonely proton is like a runner in an empty park. A quark in a nucleus is like that same runner trying to jog through a packed mosh pit. The "drag" of the crowd changes how the runner moves and how heavy they feel.

The Three Clues
The authors focused on three specific "clues" hidden in the data, known as Generalized Transverse Momentum-Dependent Parton Distributions (GTMDs). These are like high-resolution maps showing where the quarks are, how fast they are moving sideways, and how they are spinning.

The Orbit Clue (OAM): This measures how much the quarks are circling around the center. The simulation suggests that in the crowded nucleus, the quarks' orbital motion actually gets stronger. The authors found that the magnitude of this orbital angular momentum increases by about 40% when moving from a lone proton to a nucleus. Interestingly, the simulation shows this orbital motion tends to spin in the opposite direction of the quark's own spin (anti-aligned), but the sheer amount of orbiting increases.
The Spin Clue: This measures the quark's own internal spin. The results suggest that in the crowded nuclear environment, the quark's spin contribution gets a boost, increasing by roughly 16% compared to the vacuum.
The Dance Clue (Spin-Orbit Correlation): This measures how well the quark's spin and its orbit are dancing together. The simulation indicates that this connection gets weaker in the nucleus, dropping by about 40%. It's as if the crowded environment makes it harder for the quarks to coordinate their spinning and orbiting perfectly.

The "Suppression" Ratio
To make these findings easier to spot in future experiments, the authors proposed a new way to look at the data, similar to a "nuclear suppression factor" used in heavy-ion collisions. They suggest comparing the measurements from the nucleus directly against the lone proton.

If you look at the ratio of these measurements:

The "Orbit" and "Spin-Orbit" signals in the nucleus are predicted to be about 10%–16% lower (suppressed) relative to the proton.
The "Spin" signal is predicted to be 0%–12% higher (enhanced).

What This Means (and What It Doesn't)
It is crucial to remember that this is a theoretical prediction based on a specific mathematical model (the light-front dressed quark model combined with the NJL model). The authors have not measured these numbers yet; they have calculated them to tell experimentalists what to look for when the Electron-Ion Collider comes online.

The paper explicitly rules out the idea that the internal structure of the proton stays exactly the same when it's inside a nucleus. Instead, the simulation suggests the "crowded" environment of the nucleus significantly reshapes how quarks move and spin. The authors are confident that these deviations from unity (the idea that nucleus = proton) are real effects of the non-perturbative quantum world, but they are waiting for the real-world data to confirm their simulation.

In short, this paper is a roadmap. It suggests that when we finally turn on the EIC, we won't just see the same old protons; we will see a dynamic, shifting landscape where the quarks' orbits and spins change their tune depending on how crowded their neighborhood is.

Technical Summary: Electron-Ion Collision Environment and Quark Spin/OAM Distributions

Problem Statement
The internal structure of the proton, specifically the origin of its spin, remains a central challenge in Quantum Chromodynamics (QCD). While the one-dimensional parton distribution functions (PDFs) describe longitudinal momentum, a complete understanding requires multidimensional information regarding quark spin, orbital angular momentum (OAM), and spin-orbit correlations. The future Electron-Ion Collider (EIC) aims to probe these structures by comparing electron-proton (eP) and electron-ion (eA) collisions. However, a theoretical framework is needed to map the differences between the vacuum environment (eP) and the nuclear medium environment (eA) to understand how many-body nuclear density effects modify quark distributions. Specifically, there is a need to quantify how non-perturbative QCD interactions at nuclear saturation density alter Generalized Transverse Momentum-Dependent Parton Distributions (GTMDs) associated with the proton's spin budget.

Methodology
The authors employ a hybrid theoretical approach combining the Nambu-Jona-Lasinio (NJL) model with the light-front dressed quark model to bridge the gap between vacuum and nuclear environments.

Mass Determination via NJL Model:
- The NJL model is utilized to determine the constituent quark mass ( $m^*$ ) under two distinct conditions: zero nuclear density ( $\rho_B = 0$ , representing the eP/vacuum environment) and nuclear saturation density ( $\rho_B = \rho_0 \approx 0.16 \text{ fm}^{-3}$ , representing the eA/nuclear environment).
- The model uses a two-flavor Lagrangian with four-fermion contact interactions. The constituent mass is derived from the gap equation, which accounts for spontaneous chiral symmetry breaking.
- Parameters ( $m=0.0027$ GeV, $G=1.95$ GeV $^{-2}$ , $\Lambda=0.95$ GeV) are fixed to reproduce vacuum pion properties ( $m_\pi = 135$ MeV, $f_\pi = 92.4$ MeV).
- For finite density, the quark chemical potential ( $\mu$ ) is introduced, allowing the calculation of $m^*$ as a function of baryon density.
GTMD Calculation via Light-Front Dressed Quark Model:
- Using the density-dependent constituent masses derived from the NJL model, the authors calculate GTMDs within the light-front dressed quark model.
- The target state is modeled as a dressed quark (a bound state of a bare quark and a gluon) expanded in Fock space up to the two-particle sector.
- The quark-quark correlator is computed using light-front wave functions (LFWFs) in a symmetric frame.
- The study focuses on three specific GTMDs at leading twist: $F_{1,4}$ $F_{1, 4}$ , $G_{1,1}$ $G_{1, 1}$ , and $G_{1,4}$ $G_{1, 4}$ . These are selected because their integrated values correspond to physical observables:
  - $F_{1,4}$ : Linked to quark Orbital Angular Momentum (OAM).
  - $G_{1,4}$ : Linked to quark spin contribution.
  - $G_{1,1}$ : Linked to quark spin-orbit (L-S) correlation.
Definition of Ratios:
- Inspired by the nuclear suppression factor ( $R_{AA}$ ) in heavy-ion collisions, the authors define a ratio of GTMDs between the nuclear environment ( $\rho_B = \rho_0$ ) and the vacuum environment ( $\rho_B = 0$ ). This ratio is intended to serve as an indirect measure of nuclear density effects.

Key Results
The study presents contour plots of the GTMDs as functions of the longitudinal momentum fraction ( $x$ ) and transverse momentum transfer ( $\Delta_\perp$ ), alongside integrated values for OAM, spin, and spin-orbit correlation.

GTMD Distributions:
- $F_{1,4}$ (OAM): The distribution is symmetric about $\Delta_\perp = 0$ and peaks in the positive $x$ region. While the magnitude of $F_{1,4}$ decreases in the nuclear medium, the integrated OAM ( $l_z^q$ ) increases due to the inverse dependence on the constituent mass ( $m^*$ ) in the integration formula.
- $G_{1,4}$ (Spin): This distribution is non-vanishing primarily in the positive $x$ region. Its magnitude is enhanced at finite baryonic density compared to the vacuum, indicating an increased quark spin contribution in the nuclear medium.
- $G_{1,1}$ (Spin-Orbit): Concentrated in the positive $x$ region, this distribution is suppressed at finite baryonic density relative to the vacuum, suggesting a weaker quark spin-orbit coupling in nuclear matter.
- Antiquark Contribution: The suppression in the negative $x$ region across all distributions suggests a reduced contribution from antiquarks in these correlations.
Integrated Quantities:
- As baryon density increases from 0 to $\rho_0$ , the magnitude of quark OAM, quark spin, and the magnitude of the spin-orbit correlation all increase.
- The authors estimate a relative enhancement of approximately 40% for quark OAM, 16% for quark spin, and 40% for the L-S contribution when moving from the proton (vacuum) to the nucleus environment.
- The negative sign of the OAM indicates that the quark OAM is anti-aligned with the system's spin.
GTMD Ratios:
- The ratio of GTMDs ( $\rho_B=\rho_0$ vs. $\rho_B=0$ ) shows a percentage deviation that decreases monotonically with increasing $x$ .
- At lower $x$ , the variation is larger, indicating higher sensitivity to transverse momentum transfer ( $\Delta_\perp$ ).
- The ratios for OAM and spin-orbit correlations show a suppression of approximately 10%–16% relative to vacuum values, while the ratio for quark spin shows an enhancement of 0%–12%.

Significance and Claims
The paper claims to provide the first theoretical prediction of GTMD ratios between eP and eA collisions within the dressed quark model framework. The primary significance lies in:

Bridging Models: Successfully integrating the NJL model (for density-dependent mass generation) with the light-front dressed quark model (for GTMD calculation) to simulate the transition from vacuum to nuclear environments.
Probing Nuclear Effects: Proposing that the deviation of GTMD ratios from unity serves as a direct signature of non-perturbative many-body nuclear density effects, analogous to how $R_{AA}$ probes hot/dense media in heavy-ion collisions.
Experimental Guidance: Offering specific theoretical predictions (GTMDs $F_{1,4}$ , $G_{1,1}$ , $G_{1,4}$ and their ratios) that can be tested at the future EIC (and EicC) by comparing eP and eA collision data. The authors suggest that these measurements could guide the methodology for integrating GTMDs to extract OAM and spin contributions in nuclear environments.

The authors remain modest regarding the integration limits, noting that while they used a fixed cutoff ( $Q=5$ GeV) for their calculations, future experimental data will be required to determine the appropriate density-dependent integration methodology.

Technical Summary: Electron-Ion Collision Environment and Quark Spin/OAM Distributions

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