⚛️ phenomenology

Energy-momentum tensor at small $x$ from gravitational transverse-momentum distributions

This paper investigates the momentum-space mechanical properties of small- $x$ gluons and sea quarks by deriving gravitational transverse-momentum distributions from the light-front gauge-invariant canonical energy-momentum tensor and numerically evaluating their momentum densities and fluxes using the McLerran-Venugopalan model and Balitsky-Kovchegov evolution, revealing significant departures from free-particle relations.

Original authors: Tomoya Uji

Published 2026-10-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tomoya Uji

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

Inside every atom lies a proton, a tiny, dense core that holds the universe together. For decades, physicists have known that protons are not solid balls but swirling clouds of even smaller particles called quarks and gluons. These particles zip around at nearly the speed of light, bound together by the strong force, the most powerful interaction in nature. While scientists have mapped out where these particles are and how they spin, a deeper question remains: how do they carry energy and momentum? Imagine a crowded room where people are running in all directions; knowing who is running is one thing, but understanding how their collective motion creates pressure, stress, and flow is another. This is the realm of the energy-momentum tensor, a mathematical description of how energy and force are distributed inside matter. Understanding this internal mechanical structure is crucial for a complete picture of the proton, and it is a key goal for the upcoming Electron-Ion Collider, a massive machine designed to peer into the heart of matter with unprecedented clarity.

A researcher at the University of Tokyo has taken a significant step toward this goal by investigating the mechanical properties of gluons and sea quarks when they are moving at extremely high speeds. In the language of particle physics, this high-speed regime is known as "small-x," a state where the particles carry only a tiny fraction of the proton's total forward momentum. In this chaotic environment, the researcher focused on a specific type of measurement called gravitational transverse-momentum distributions. While the name sounds exotic, the concept is straightforward: it is a way to map how the momentum of these fast-moving particles flows sideways, perpendicular to their main direction of travel. By analyzing these distributions, the researcher could determine how much "inertia" these particles carry sideways and how they push against one another, effectively creating a mechanical map of the proton's interior.

To do this, the researcher used a sophisticated theoretical framework based on the light-front gauge-invariant canonical energy-momentum tensor. In simpler terms, they developed a new set of rules to calculate how gluons and sea quarks move and interact, ensuring their calculations respected the fundamental laws of quantum mechanics and symmetry. They applied these rules to a model of the proton that simulates the dense, high-energy conditions found at small-x. The model started with a specific initial state known as the McLerran–Venugopalan model, which describes a proton as a dense collection of color charges, and then evolved it using a mathematical process called Balitsky–Kovchegov evolution to see how the particles behave as the energy increases. This allowed the researcher to calculate the momentum densities and the flow of inertia for both gluons and the sea quarks that pop in and out of existence within the proton.

The results revealed a striking difference between how gluons and sea quarks behave in this high-speed regime. For the gluons, the researcher found that while they possess a clear momentum density—meaning they are definitely moving sideways—this motion does not generate a corresponding flow of inertia. In the language of the study, the transverse flux of inertia vanishes. This means that at this level of approximation, the sideways movement of gluons does not behave like a free particle flowing through space; instead, it remains locked in a specific configuration that does not transport inertia in the expected way. The study notes that the finding that the ratio of flux to momentum is zero is "stronger than a mere violation" of the free-particle condition, indicating that the nonzero canonical transverse momentum density has no accompanying transverse flux of inertia at this order.

In contrast, the sea quarks told a different story. When the researcher looked at the high-momentum tail of the sea quark distribution, they found that these particles do carry a nonzero transverse flux of inertia. However, this flow is not equal to their momentum density. Instead, the study calculated that the inertia flux is exactly one-quarter of the canonical transverse momentum density. This specific ratio emerged from the mathematical structure of how gluons split into quark-antiquark pairs in this high-energy environment. It is a precise finding derived from the specific approximations used in the simulation, indicating that while sea quarks do transport inertia, they do so in a way that is distinct from both free particles and the gluons surrounding them.

The study also examined the internal stresses within the proton, looking at how these particles push and pull on each other. In the eikonal approximation used for the gluons, the transverse stresses were found to vanish, reinforcing the picture of a system where the usual mechanical forces of pressure and shear are suppressed. For the sea quarks, the analysis showed that their mechanical properties are governed by a combination of factors that result in a finite but reduced flow of inertia. These findings provide a concrete, numerical picture of the proton's interior mechanics, moving beyond abstract theory to show exactly how energy and momentum are distributed and transported in the most extreme conditions.

By combining these theoretical calculations with numerical simulations, the researcher has provided a detailed snapshot of the proton's mechanical life at small-x. They have shown that the proton is not a uniform soup of particles but a complex system where different types of particles obey different mechanical rules. The gluons, despite their abundance and energy, do not flow with inertia in the way one might expect, while the sea quarks exhibit a specific, quantifiable relationship between their momentum and their inertia. These results lay the groundwork for future experiments at the Electron-Ion Collider, offering a clear set of predictions for how the internal forces of the proton will manifest when probed with high-energy beams. The work does not claim to have solved the entire puzzle of the proton's structure, but it has successfully mapped a critical piece of the terrain, revealing the unique mechanical signatures of the particles that make up our world.

Technical Summary: Energy-momentum tensor at small x from gravitational transverse-momentum distributions

Problem Statement
The internal mechanical structure of hadrons—specifically how energy, momentum, and internal forces are distributed among quarks and gluons—is a central goal of Quantum Chromodynamics (QCD). While the spatial organization of these properties is studied via Generalized Parton Distributions (GPDs) and Gravitational Form Factors (GFFs), a complementary question concerns the distribution of mechanical properties over the parton's longitudinal momentum fraction ( $x$ ) and intrinsic transverse momentum ( $k_T$ ). This momentum-space resolution is distinct from spatial imaging. Previous studies have addressed helicity and orbital angular momentum in the small- $x$ regime, and the gluon GFF $A(t)$ has been investigated, but a comprehensive description of the Energy-Momentum Tensor (EMT) resolved in both $x$ and $k_T$ for small- $x$ gluons and sea quarks remains to be established. This paper addresses the formulation and evaluation of these mechanical properties using Gravitational Transverse-Momentum-Dependent distributions (gravitational TMDs).

Methodology
The author formulates gluon gravitational TMDs using the light-front gauge-invariant canonical (gic) energy-momentum tensor, $T^{\mu\nu}_{g, \text{gic}}$ , rather than the kinetic (Belinfante–Rosenfeld) tensor. This choice allows for a direct interpretation in terms of canonical partonic momentum densities.

Formalism:
- The gic EMT is constructed by separating the gauge field into pure-gauge and physical components ( $A^\mu = A^\mu_{\text{pure}} + A^\mu_{\text{phys}}$ ) and imposing $F^{\mu\nu}_{\text{pure}} = 0$ to define a gauge-covariant canonical momentum operator.
- A gauge-invariant bilocal extension of the EMT is introduced, incorporating Weizsäcker–Williams (WW) gauge links.
- The EMT is parametrized in terms of gravitational TMDs ( $a_g^i, a_g^{\perp i}$ ), which are related to ordinary gluon TMD correlators ( $f_1^g, f_{1T}^{\perp g}$ , etc.) through Lorentz decomposition.
- Key mechanical quantities are identified: longitudinal and transverse momentum densities ( $T^{++}, T^{+i}$ ), transverse flux of inertia ( $T^{i+}$ ), and transverse stresses ( $T^{ij}$ ).
Small- $x$ Approximation:
- The analysis is applied to the small- $x$ regime using the Color Glass Condensate (CGC) framework in the eikonal approximation.
- Fast partons are treated as color sources generating a classical gluon field. The field strength components are restricted, leading to the survival of only the unpolarized WW distribution ( $f_1^g$ ) in the canonical EMT components relevant to the mechanical discussion.
- The Weizsäcker–Williams distribution is expressed in terms of Wilson-line correlators.
Numerical Evaluation:
- A common initial condition based on the McLerran–Venugopalan (MV) model is used for both gluons and sea quarks.
- The system is evolved using fixed-coupling leading-order (LO) Balitsky–Kovchegov (BK) evolution.
- The WW gluon distribution is reconstructed using a nonlinear Gaussian approximation for Wilson-line correlations.
- The sea-quark distribution is derived from the same dipole correlator via small- $x$ matching formulas.
- The sea-quark transverse flux of inertia is evaluated separately in the high- $k_T$ region ( $k_T \gg Q_s$ ) by calculating the perturbative tail of the twist-three quark TMD ( $f_{\perp q}$ ) arising from $g \to q\bar{q}$ splitting.

Key Contributions and Results

Formulation of Gluon Gravitational TMDs: The paper establishes the relations between the light-front gic gluon EMT and ordinary gluon TMDs. It identifies that the EMT components depend on specific combinations of twist-two, twist-three, and twist-four TMDs. Notably, the gluon EMT includes an additional scalar contribution (metric term) absent in the quark case, which affects the transverse stress but not the traceless shear part.
Eikonal Approximation Results:
- In the eikonal approximation, the longitudinal and transverse canonical momentum densities ( $T^{++}$ and $T^{+i}$ ) are determined solely by the unpolarized WW gluon distribution.
- Crucially, the transverse flux of inertia ( $T^{i+}$ ) and transverse stresses ( $T^{ij}$ ) vanish in this approximation.
- This leads to a ratio $R_v^g = x f_{\perp}^g / f_1^g = 0$ . Consequently, the nonzero canonical transverse momentum density is not accompanied by a transverse flux of inertia, indicating a departure from the free-particle relation where momentum density and inertia flux coincide.
Numerical Momentum Densities:
- Numerical results for the longitudinal momentum-fraction density ( $T^{++}$ ) and transverse momentum density ( $T^{+i}$ ) for gluons and sea quarks are presented.
- Both distributions decrease at large $k_T$ , but the high- $k_T$ regions are enhanced as $x$ decreases (due to the increase in saturation momentum $Q_s$ ).
- The transverse momentum density grows approximately as $1/x$ relative to the longitudinal density, reflecting the different momentum weights ( $k_T$ vs. $xP^+$ ) in the EMT components.
Sea-Quark Transverse Flux of Inertia:
- Unlike the gluon case in the eikonal limit, the sea-quark transverse flux of inertia is nonzero when evaluated through the high- $k_T$ tail of the twist-three TMD ( $f_{\perp q}$ ).
- Under a specific longitudinal approximation (neglecting the dependence of the weighted gluon input on the parent momentum fraction within the convolution), the ratio of the transverse flux of inertia to the canonical transverse momentum density is found to be $R_{v, \text{tail}}^q = 1/4$ .
- This result ( $1/4$ ) differs from both the vanishing gluon flux ($0$) and the free-particle reference value ($1$), illustrating that the effective transverse velocity governing inertia flow does not necessarily coincide with the canonical transverse momentum.

Significance
The paper claims to provide a phase-space description of the mechanical properties of small- $x$ gluons and sea quarks by resolving the EMT in longitudinal momentum fraction and transverse momentum. The primary significance lies in demonstrating that the mechanical properties of small- $x$ systems deviate significantly from free-particle expectations. Specifically:

For gluons in the eikonal approximation, the canonical transverse momentum exists without a corresponding transverse flux of inertia.
For sea quarks, even when a flux exists, it does not equal the momentum density (ratio of 1/4 in the perturbative tail), challenging the simple free-particle interpretation of inertia flow.

The work serves as a first step toward a phase-space description of hadron mechanics on the light front. The author notes that the current implementation uses a simple small- $x$ dynamics model and suggests that future extensions could include running coupling, direct JIMWLK evolution, subeikonal corrections, and spin-dependent TMDs to further explore these mechanical properties.

Technical Summary: Energy-momentum tensor at small x from gravitational transverse-momentum distributions

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