Gravitational transverse momentum distribution of proton

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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, tiny marble, but as a bustling, chaotic city made of tiny citizens called quarks and gluons. For decades, scientists have been trying to map this city. They've created maps showing where the citizens live (position), how fast they move forward (longitudinal momentum), and how they spin.

But there's a missing piece of the puzzle: How do these citizens push and pull on each other? What are the internal "forces" holding this city together?

This paper is like a new, high-tech survey that tries to map the internal pressure and stress inside the proton, but with a very specific, clever twist.

The Big Idea: Gravity as a Measuring Tape

You might be thinking, "Gravity? Isn't that just for planets and apples?"

Yes, but in the world of subatomic particles, gravity is a special tool. In physics, the way matter responds to gravity is described by something called the Energy-Momentum Tensor. Think of this as a "stress report" that tells you how much energy is moving where, and how hard the particles are pushing against their neighbors.

Usually, scientists look at the average pressure inside a proton (like the average air pressure in a tire). This paper, however, looks at the transverse momentum distributions.

The Analogy:
Imagine you are trying to understand the traffic in a busy city.

Standard maps tell you: "There are 100 cars on Main Street." (This is the average).
This paper's map tells you: "At 5:00 PM, the cars moving East at 60 mph are pushing harder against the cars moving West at 40 mph."

It's a map of the internal forces based on how fast the particles are moving sideways (transverse momentum) relative to the proton's direction.

The "Model" They Used: A Simplified City

Calculating this for a real proton is incredibly hard because the interactions are messy. So, the authors used a Light-Front Quark-Diquark Model.

The Metaphor:
Imagine the proton is a dance floor.

Usually, it's a chaotic mosh pit with hundreds of dancers (quarks and gluons) bumping into each other.
To make the math work, the authors simplified the scene. They imagined the proton as a dance duo: One active dancer (the "struck quark") and one partner (the "spectator diquark," which acts like a pair of dancers glued together).
They used a framework inspired by AdS/QCD (a theory that connects particle physics to the geometry of higher dimensions) to give the dancers a specific "dance style" (wave function) that mimics how real quarks behave.

What Did They Discover?

Using this simplified "dance duo" model, they calculated six new types of maps (called Gravitational TMDs) for the Up and Down quarks. Here is what they found in plain English:

1. The Pressure Map (Isotropic Pressure)
They calculated the transverse pressure—how much the quarks are pushing outward or inward in the sideways direction.

The Result: The pressure is negative.
The Meaning: In everyday language, negative pressure means compression. It's like a spring being squeezed tight. This confirms that the force holding the proton together is a powerful "squeezing" force (confinement). If you tried to pull the quarks apart, this internal pressure would snap them back together.
The Difference: The "Up" quarks (which are more common in a proton) create a stronger squeeze than the "Down" quarks.

2. The Momentum Map (Average Longitudinal Momentum)
They also looked at how much forward momentum the quarks carry.

The Result: The Up quarks carry significantly more forward momentum than the Down quarks.
The Meaning: This matches what we already know: A proton is made of two Up quarks and one Down quark. The "Up" citizens are doing most of the heavy lifting in the forward direction.

3. The "TMD" vs. "GFF" Distinction
The paper makes a crucial distinction between two ways of looking at the proton:

Gravitational Form Factors (GFFs): These are like a photograph of the proton's pressure in space. You see where the pressure is high or low in a specific location.
Gravitational TMDs (This paper): These are like a speedometer reading of the pressure. They tell you the pressure generated by particles moving at a specific speed.
Why it matters: It's the difference between knowing "there is a crowd in the room" (GFF) and knowing "the people running at 10 mph are pushing harder than the people walking at 2 mph" (TMD). This gives a much more dynamic, 3D understanding of the proton's internal mechanics.

Why Does This Matter?

This is the first study of its kind. Before this, scientists had the "static" maps (GFFs) but lacked the "dynamic" maps (TMDs) that show how momentum and pressure are linked.

It validates the theory: The authors checked their math against known rules (model-independent relations) and found that their simplified "dance duo" model actually holds up. This gives us confidence that our models of the proton are on the right track.
It prepares for the future: With the upcoming Electron-Ion Collider (EIC), scientists will be able to probe these internal structures with incredible precision. This paper provides a theoretical "cheat sheet" to help interpret those future experiments.

The Bottom Line

This paper is like building a simulator for the internal stress of a proton. By treating the proton as a simple pair of dancers and using a special "gravity-based" measuring tool, the authors created a new map showing that:

The inside of a proton is under immense compressive stress (it's being squeezed tight).
The Up quarks are the heavy lifters, carrying more momentum and creating more pressure than the Down quarks.
We can now look at the proton's internal forces not just by where the particles are, but by how fast they are moving.

It's a small step in a simplified model, but a giant leap toward understanding the invisible "glue" that holds the universe's building blocks together.

1. Problem Statement and Motivation

The internal structure of the proton is traditionally described by Parton Distribution Functions (PDFs), Generalized Parton Distributions (GPDs), and Transverse Momentum Dependent distributions (TMDs). While GPDs connect longitudinal momentum to transverse spatial distributions (providing a 3D tomographic view), and TMDs describe intrinsic transverse momentum, a unified framework connecting the Energy-Momentum Tensor (EMT) to transverse momentum is less explored.

Specifically, the paper addresses the lack of systematic investigation into Gravitational Transverse-Momentum-Dependent distributions (Gravitational TMDs). These distributions, introduced recently by Lorcé et al., encode the mechanical properties of hadrons (such as pressure and shear forces) in momentum space. While Gravitational Form Factors (GFFs) derived from GPDs provide spatial pressure distributions, Gravitational TMDs offer a complementary view in momentum space. The authors aim to:

Derive analytical expressions for unpolarized (T-even) Gravitational TMDs for up and down quarks.
Compute the corresponding Gravitational Parton Distribution Functions (Gravitational PDFs).
Investigate the connection between these distributions and the proton's mechanical properties (transverse pressure, shear forces) and average longitudinal momentum.

2. Methodology

The study employs the Light-Front Quark-Diquark Model (LFQDM) inspired by the Soft-Wall AdS/QCD framework.

Model Framework: The proton is modeled as a bound state of an active quark and a spectator diquark. The proton state is a superposition of isoscalar-scalar ( $|uS_0\rangle$ ), isoscalar-vector ( $|uA_0\rangle$ ), and isovector-vector ( $|dA_1\rangle$ ) diquark states.
Wave Functions: The Light-Front Wave Functions (LFWFs) are constructed using a modified soft-wall AdS/QCD ansatz. These functions include a Gaussian suppression factor in transverse momentum ( $k_\perp$ ) and specific power-law behaviors in the longitudinal momentum fraction ( $x$ ), constrained by Dirac and Pauli form factors.
Theoretical Formalism:
- The authors define the fully unintegrated light-front energy-momentum tensor (EMT) for a quark.
- They utilize a gauge-invariant canonical EMT operator (using the pure-gauge covariant derivative) to ensure a consistent interpretation of the quark four-momentum $k$ .
- The TMD-EMT is parametrized in terms of 32 independent scalar functions (Gravitational TMDs). The authors focus on the 10 unpolarized (T-even) functions.
- Analytical expressions are derived by calculating the forward matrix elements of the EMT operator using the overlap of the LFWFs.
Consistency Checks: The derived expressions are verified against model-independent relations established in Ref. [37], which link Gravitational TMDs to standard unpolarized TMDs (twist-2, twist-3, and twist-4).

3. Key Contributions

First Model Calculation: This is the first study to calculate unpolarized quark Gravitational TMDs within a phenomenological model (LFQDM).
Analytical Derivations: The authors provide explicit analytical formulas for six unpolarized Gravitational TMDs ( $a_1, a_3, a_5, a_6, a_7, a_8$ ) and the corresponding Gravitational PDFs ( $A_1, A_3, A_5$ ).
Mechanical Interpretation in Momentum Space: They explicitly connect the Gravitational TMD $a_3(x, k_\perp^2)$ to the transverse isotropic pressure and shear-force distributions in momentum space, distinguishing this from the spatial pressure derived via GFFs.
Momentum Density: They relate the Gravitational TMD $a_1(x, k_\perp^2)$ to the average longitudinal momentum carried by quarks.

4. Key Results

A. Gravitational TMDs and PDFs

Behavior of $a_1, a_3, a_6, a_7$ : These distributions exhibit positive peaks for both up and down quarks. They vanish at the kinematic endpoints ( $x \to 0$ and $x \to 1$ ) due to the Gaussian suppression in the wave function.
Behavior of $a_5, a_8$ : These distributions are predominantly negative. They show a divergent-like behavior as $x \to 0$ (due to $1/x$ and $1/x^2$ factors) but become negative at larger $x$ as the second term in their analytical expressions dominates.
Flavor Asymmetry: The up-quark contributions are significantly larger than the down-quark contributions across the entire $x$ range, reflecting the proton's valence structure.
Gravitational PDFs: Upon integrating over $k_\perp$ , only three PDFs ( $A_1, A_3, A_5$ ) are non-zero. $A_1$ and $A_3$ show valence-like structures (peaking at $x \sim 0.4-0.5$ ), while $A_5$ changes sign, becoming negative at large $x$ .

B. Mechanical Properties (Pressure and Shear)

Transverse Pressure: The isotropic transverse pressure $\sigma(x, k_\perp)$ $σ (x, k_{⊥})$ is derived from $a_3$ $a_{3}$ .
- The pressure distribution is circularly symmetric in the $k_\perp$ plane.
- It is negative throughout, indicating a compressive internal force, consistent with confinement dynamics.
- The magnitude of the pressure is maximum at intermediate transverse momentum and vanishes at $k_\perp = 0$ and large $k_\perp$ .
- The up-quark pressure is more negative (stronger compressive force) than the down-quark pressure.
Shear Force: The shear-force distribution is found to be twice the magnitude of the pressure distribution ( $\Pi = 2\sigma$ ), exhibiting the same qualitative features.

C. Average Longitudinal Momentum

By integrating $a_1$ $a_{1}$ , the authors calculated the average longitudinal momentum carried by quarks at a center-of-mass energy of $\sqrt{s} = 100$ $s = 100$ GeV (relevant for the future Electron-Ion Collider).
- Up quark: $\langle k^+ \rangle_u \approx 368.44$ GeV.
- Down quark: $\langle k^+ \rangle_d \approx 169.91$ GeV.
This confirms that up quarks carry a larger fraction of the proton's longitudinal momentum, consistent with the valence quark model.

5. Significance and Conclusion

This work establishes a crucial link between the mechanical properties of the proton and its transverse momentum structure.

Conceptual Distinction: It highlights the difference between pressure distributions in position space (derived from GFFs via Fourier transform of momentum transfer) and momentum space (derived from Gravitational TMDs). The latter provides information about partons with specific momentum fractions without access to an average spatial coordinate.
Validation: The results satisfy model-independent relations between Gravitational TMDs and standard TMDs, validating the LFQDM approach for these complex observables.
Future Outlook: The study provides a baseline for future experimental investigations, particularly at the Electron-Ion Collider (EIC), where such distributions could potentially be accessed. The authors plan to extend this work to polarization-dependent (T-odd) Gravitational TMDs.

In summary, the paper successfully maps the mechanical stress and momentum distribution of the proton's constituent quarks in momentum space, offering a new perspective on the confinement dynamics and internal forces within the nucleon.