⚛️ phenomenology

Kaon gravitational form factors and mechanical structure in a three-flavor NJL model

Using a three-flavor Nambu–Jona-Lasinio model with proper-time regularization, this study investigates the kaon's gravitational form factors and mechanical structure, revealing that explicit flavor-symmetry breaking leads to a dominant strange-quark contribution, a less negative $D$ -term, and more compact mechanical distributions with enhanced central pressure compared to the pion.

Original authors: Zhibo Liu, Hiroaki Abuki

Published 2026-09-01

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CC0 1.0

Original authors: Zhibo Liu, Hiroaki Abuki

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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, protons and neutrons hold together not just by glue, but by a complex internal architecture of forces. To understand how these particles are built, physicists look at how they carry energy, momentum, and spin. While scientists have long studied how particles interact with electricity and magnetism, a newer line of inquiry asks how they respond to gravity. This might sound strange, as gravity is incredibly weak at the scale of a single particle, but the mathematical tools used to describe gravity reveal the hidden mechanical structure of matter. These tools, called gravitational form factors, act like a map showing where energy is stored and how internal pressures push and pull from the inside. Just as a pressure gauge tells a mechanic if an engine is running smoothly, these maps tell physicists if a particle is stable and how its internal parts share the load.

The focus of this research is the kaon, a short-lived particle made of two different types of quarks: a light one and a strange one. This makes the kaon a unique laboratory for studying what happens when the building blocks of matter are not equal. In the simplest particle, the pion, both parts are light and identical, creating a perfectly balanced system. The kaon, however, breaks this symmetry. Because the strange quark is significantly heavier than its partner, the researchers wanted to see how this imbalance reshapes the particle's internal landscape. Does the heavier part carry more of the weight? Does it push harder against the lighter part? Answering these questions helps scientists understand the fundamental rules that govern how matter holds itself together.

In a study published in the journal of high-energy physics, researchers Zhibo Liu and Hiroaki Abuki used a theoretical model to simulate the kaon and calculate its internal mechanical properties. They treated the kaon as a bound state of a light quark and a strange quark, using a framework known as the Nambu–Jona-Lasinio model. This model is a powerful tool for describing how quarks interact, but calculating the mechanical forces inside a particle requires extreme precision. The researchers had to ensure that their calculations respected the fundamental laws of physics, specifically the conservation of energy and momentum. To do this, they included every possible way the particles could interact, including a specific type of contact interaction that often gets overlooked. By carefully accounting for these details, they ensured their results were physically consistent and free from mathematical artifacts that could distort the picture.

The simulation revealed a clear and striking difference between the kaon and the pion. The heavier strange quark in the kaon does not just sit passively; it actively dominates the particle's structure. The calculations showed that the strange quark carries about 56.6 percent of the kaon's total momentum, while the lighter quark carries the remaining 43.4 percent. This imbalance extends to the mechanical forces as well. The strange sector contributes more to the internal pressure and the shear forces that hold the particle together. Consequently, the kaon is not just a heavier version of the pion; it is a fundamentally different shape. The internal stress distribution in the kaon is more compact, meaning the forces are concentrated in a smaller region near the center.

When the researchers mapped out the pressure inside the kaon, they found a central pressure of 6.9 GeV/fm³, which is higher than the 5.6 value found in the pion. The region where the pressure drops to zero is also smaller, shrinking from 0.47 femtometers in the pion to 0.36 femtometers in the kaon. Similarly, the maximum shear force, which represents the strongest internal tension, is higher in the kaon at 2.5 compared to 2.0 in the pion. These numbers indicate that the kaon is a tighter, more tightly wound system. The researchers also calculated the size of the particle using two different methods. One method, which looks at the particle as a static 3D object, suggested the kaon is much smaller than the pion. However, another method, which looks at the particle from a moving perspective, showed that both particles have very similar sizes. This discrepancy highlights that the way we measure the size of these particles depends heavily on how we look at them.

The study confirms that the difference between the kaon and the pion is not just a matter of scale but a fundamental reshaping of the internal stress response caused by the unequal masses of their quarks. The heavier strange quark pulls the structure inward, creating a more compact and pressurized core. This finding aligns with other recent theoretical approaches, suggesting that the compact nature of the kaon is a robust consequence of the symmetry breaking in the universe. While the model used in this study has limitations, such as not including all the complex interactions of the strong force, the results provide a clear, symmetry-preserving picture of how mass asymmetry dictates the mechanical structure of matter. The work demonstrates that even in the subatomic world, the balance of forces is delicate, and a change in the weight of a single component can dramatically alter the shape and stability of the whole.

Technical Summary: Kaon Gravitational Form Factors and Mechanical Structure in a Three-Flavor NJL Model

Problem Statement
The internal mechanical structure of hadrons, described by the matrix elements of the QCD energy-momentum tensor (EMT), is encoded in gravitational form factors (GFFs). While the pion has served as a primary benchmark for understanding these structures due to chiral symmetry constraints, the kaon introduces a qualitatively new element: explicit $SU(3)_F$ flavor symmetry breaking. The unequal masses of the light ( $u, d$ ) and strange ( $s$ ) quarks in the kaon provide a unique system to resolve how flavor asymmetry redistributes momentum and internal forces. However, calculating these properties in effective field theories like the Nambu–Jona-Lasinio (NJL) model presents specific challenges. Evaluating the EMT within a contact interaction requires satisfying the gravitational Ward–Takahashi identity (GWTI) and accounting for interaction-induced contact diagrams. Omitting these contributions leads to a non-conserved total EMT and regulator-dependent artifacts, particularly in the $D$ -term channel, which governs pressure and shear-force distributions.

Methodology
The authors investigate the kaon GFFs using a proper-time-regularized three-flavor NJL model, excluding the Kobayashi–Maskawa–'t Hooft determinant interaction. The calculation employs the following key methodological steps:

Framework: The model utilizes isospin-symmetric light quarks ( $m_\ell = m_u = m_d$ ) and a heavier strange quark ( $m_s$ ), with constituent masses generated dynamically. Loop integrals are regularized using a proper-time cutoff ( $\Lambda_{UV}$ ) to handle the non-renormalizable nature of the model, while an infrared cutoff ( $\Lambda_{IR}$ ) removes unphysical thresholds.
Vertex Construction: A dressed quark–graviton vertex is constructed to satisfy the GWTI. This involves including the driving term generated by the four-fermion interaction, ensuring the vertex is algebraic in the local theory.
Diagrammatic Inclusion: The calculation explicitly includes three topologies for the kaon EMT matrix element:
- Light-quark triangle diagrams.
- Strange-quark triangle diagrams.
- Interaction-induced "bicycle" (contact) diagrams arising from the direct coupling of the EMT to the local four-fermion interaction.
Symmetry Preservation: A critical procedural step involves performing the GWTI algebraic reduction on the loop integrand before applying the proper-time cutoff. This specific ordering is necessary to eliminate spurious local surface terms in the $C$ -term channel that would otherwise violate the soft-pion low-energy theorem ( $C_\pi(0) = -1$ ).
Decomposition: The total form factors are separated into light-quark and strange-quark contributions to isolate the effects of explicit flavor symmetry breaking.

Key Results

Conservation and Normalization: By retaining the contact contribution alongside the triangle diagrams and enforcing the kaon pole condition, the non-conserved form factor $\bar{c}_K(t)$ is shown to vanish identically, ensuring total EMT conservation. The momentum sum rule $A_K(0) = 1$ is satisfied.
Flavor Asymmetry: The strange quark carries a dominant share of the kaon's momentum ( $A_K^s(0) \approx 0.566$ ) compared to the light quark ( $A_K^\ell(0) \approx 0.434$ ). Similarly, the strange sector contributes more significantly to the magnitude of the $D$ -term.
$D$ -term and Mechanical Structure: The kaon $D$ -term at zero momentum transfer is calculated as $C_K(0) = -0.688$ , which is substantially less negative than the pion value ( $C_\pi(0) = -0.965$ ).
Pressure and Shear Distributions: In the Breit frame, the kaon exhibits a more compact mechanical structure than the pion.
- The central pressure is higher ( $p(0) \approx 6.9$ GeV/fm $^3$ for kaon vs. $5.6$ GeV/fm $^3$ for pion).
- The peak shear force is larger ( $s_{max} \approx 2.5$ GeV/fm $^3$ vs. $2.0$ GeV/fm $^3$ ).
- The spatial distributions are more localized, with the first zero of the pressure distribution shifting from $0.47$ fm (pion) to $0.36$ fm (kaon).
Mass Radii: The study highlights a significant discrepancy between frame-dependent radii. The Breit-frame mass radius for the pion is anomalously large ($1.24$ fm) due to sensitivity to the small pion mass in the $C_\pi(0)$ term. In contrast, the boost-invariant transverse light-front radius is similar for both mesons ( $\approx 0.25-0.27$ fm), suggesting a comparable intrinsic transverse size.

Significance and Claims
The paper claims that the primary distinction between the kaon and pion in this framework is not a uniform spatial rescaling but a fundamental reshaping of the internal stress response driven by unequal constituent masses. The explicit $SU(3)_F$ symmetry breaking leads to a more compact kaon stress distribution and an enhanced central pressure.

The authors emphasize that their methodology rigorously preserves the soft-pion low-energy theorem ( $C_\pi(0) = -1$ ) without ad hoc subtractions, provided the GWTI reduction is performed prior to regularization. This validates the necessity of including contact diagrams and proper ordering of regularization in effective models.

The results are presented as consistent with recent continuum, dispersive, and light-front studies, suggesting that the enhanced compactness and modified mechanical response of the kaon are robust consequences of flavor symmetry breaking. However, the authors modestly note the intrinsic limitations of the approach: as a leading-order quark-model evaluation, it lacks explicit gluon degrees of freedom, omits the 't Hooft determinant, and neglects meson-loop corrections. Consequently, the flavor-separated form factors are interpreted at the intrinsic model scale rather than a perturbative QCD scale, and coordinate-space profiles depend on the chosen ultraviolet extrapolation scheme.

Technical Summary: Kaon Gravitational Form Factors and Mechanical Structure in a Three-Flavor NJL Model

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