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Gravitational D-form factors of the nucleon and pion with mixing between two-quark and four-quark states

This paper investigates the gravitational $D$ -form factors of the nucleon and pion within an extended linear sigma model incorporating two- and four-quark mixing, finding that while this mixing suppresses the nucleon $D$ -form factor to match lattice-QCD results (mimicking effects of chiral invariant mass), it uniquely modifies the pion's momentum-transfer dependence, thereby suggesting that a combined analysis of both form factors is necessary to distinguish between these distinct physical mechanisms.

Original authors: Mamiya Kawaguchi, Yao Ma, Yong-Liang Ma

Published 2026-10-02

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CC BY 4.0

Original authors: Mamiya Kawaguchi, Yao Ma, Yong-Liang Ma

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

Deep within the heart of every atom lies the proton and the neutron, the building blocks of the visible universe. For decades, physicists have known that these particles are made of even smaller pieces called quarks, held together by a powerful force. Yet, a fundamental mystery remains: where does the mass of these particles actually come from? While we know the quarks themselves are very light, the proton and neutron are heavy. The answer lies in the complex, invisible energy and forces swirling inside them. To map this hidden interior, scientists use a special tool called a gravitational form factor. Despite the name, this has nothing to do with gravity as we feel it on Earth; instead, it is a mathematical way to describe how a particle responds to pressure and stress, revealing its internal mechanical structure, much like an X-ray reveals the bones inside a body. One specific type of this map, known as the D-form factor, is particularly difficult to measure but holds the key to understanding the internal forces that keep matter from flying apart.

A team of researchers has taken a fresh look at this puzzle by building a new theoretical model to simulate how these particles behave. They focused on the nucleon, the collective name for protons and neutrons, and the pion, a lighter particle that helps carry the force between nucleons. In their model, they considered a specific possibility: that the particles inside these building blocks are not just simple pairs of quarks, but sometimes exist as more complex groups of four quarks. They investigated how these different internal configurations mix together and how that mixing changes the way the particles respond to stress. By running these simulations, they discovered that when these different internal states mix, it significantly reduces the strength of the internal pressure in the nucleon. This reduction brings their theoretical predictions into much closer alignment with data gathered from supercomputer simulations, known as lattice QCD, which are considered the gold standard for this type of calculation.

The researchers found that this mixing effect creates a suppression, or a dampening, of the nucleon's internal pressure map. In their calculations, this mixing reduced the strength of the effect by about thirty percent. This result is strikingly similar to a different theory that suggests the nucleon has a built-in mass component that exists independently of the forces holding it together. Because both theories produce nearly identical results for the nucleon alone, the scientists realized that looking at the proton or neutron in isolation is not enough to tell which of these two very different physical mechanisms is actually at work. It is like trying to identify a specific ingredient in a soup by tasting the final dish; if two different ingredients produce the same flavor, you cannot know which one was used without a different test.

To solve this ambiguity, the team turned their attention to the pion. Unlike the nucleon, the pion is sensitive to the mixing of quark states but is not directly affected by the other theory involving the built-in mass. When they applied their mixing model to the pion, they found that while the basic value at zero pressure remained the same, the way the pion's internal structure changed under increasing pressure was different. Specifically, the mixing made the pion's response to pressure stronger at higher levels of stress compared to models that did not include this mixing. This difference provides a crucial clue. By comparing the behavior of the nucleon and the pion together, scientists can now distinguish between a mass that comes from a fundamental, unchanging source and a mass that arises from the complex mixing of quark states.

The study suggests that the internal structure of matter is more nuanced than previously thought, with the possibility that four-quark states play a significant role in shaping the properties of the particles that make up our world. While the researchers acknowledge that their current model is a simplification that ignores some real-world details like the small mass of the quarks, the results offer a promising new path forward. By combining the data from both the heavy nucleons and the light pions, future experiments and simulations may finally be able to pinpoint the exact origin of the mass that holds the universe together. This work does not claim to have solved the entire mystery, but it provides a clear, testable way to separate two competing ideas, bringing us one step closer to understanding the fundamental architecture of matter.

Technical Summary: Gravitational $D$ -form factors of the nucleon and pion with mixing between two-quark and four-quark states

Problem Statement
Understanding the origin of hadron masses and their internal quark composition remains a fundamental challenge in low-energy Quantum Chromodynamics (QCD). Gravitational form factors (GFFs), derived from the matrix element of the energy-momentum tensor (EMT), serve as probes for these internal structures. Specifically, the $D$ -form factor encodes information regarding the internal mechanical forces and pressure distributions of hadrons. While previous studies within the Parity Doublet Model (PDM) suggested that a non-zero chiral-invariant mass component of the nucleon suppresses the nucleon $D$ -form factor, bringing it into agreement with lattice-QCD data, it remained unclear whether this suppression is unique to chiral-invariant mass or if it can also arise from other mechanisms, such as meson mixing involving multiquark components. Furthermore, the potential impact of scalar tetraquark degrees of freedom on the nucleon $D$ -form factor had not been fully explored.

Methodology
The authors investigate the gravitational $D$ -form factors of the nucleon and pion using an Extended Linear Sigma Model (eLSM). This model extends the conventional Linear Sigma Model (LSM) by incorporating both two-quark ( $q\bar{q}$ ) and four-quark ( $qq\bar{q}\bar{q}$ ) states, allowing for mixing between these sectors in both scalar and pseudoscalar channels. The study operates within the "scalar-meson-dominance" picture, where the nucleon matrix element of the EMT is assumed to be saturated by scalar-meson exchanges.

Key methodological steps include:

Model Construction: The eLSM is formulated with chiral nonet fields for two-quark ( $\Phi'$ ) and four-quark ( $\hat{\Phi}'$ ) states. Spontaneous chiral symmetry breaking is realized through vacuum expectation values ( $\alpha$ and $\beta$ ), inducing mixing between the isosinglet scalar components ( $\sigma'$ and $\hat{\sigma}'$ ) and pseudoscalar components ( $\pi'$ and $\hat{\pi}'$ ).
Coupling Derivation: The authors derive the Yukawa couplings of the scalar mesons to the nucleon and the couplings of these mesons to the EMT. These couplings depend on the mixing angles ( $\theta_S$ for scalars and $\theta_P$ for pseudoscalars) and the vacuum parameters.
Calculation of Form Factors:
- Nucleon: The nucleon $D$ -form factor is calculated by summing the contributions from the lightest scalar meson ( $\sigma$ , identified with $f_0(500)$ ) and the heavier scalar meson ( $\hat{\sigma}$ , identified with $f_0(1500)$ ).
- Pion: The pion $D$ -form factor is calculated in the chiral limit, considering the modified couplings of the scalar mesons to pions due to mixing.
Comparison: The results are compared against the conventional LSM (which lacks four-quark states and chiral-invariant mass), the PDM (which includes chiral-invariant mass but no four-quark mixing), and existing lattice-QCD data.

Key Contributions and Results

Suppression of the Nucleon $D$ -form Factor: The study finds that meson mixing between two-quark and four-quark states significantly suppresses the magnitude of the nucleon $D$ -form factor. Using a specific parameter set (el-g30g) from neutron-star physics applications, the suppression factor is calculated to be $C(\theta) \approx 0.685$ . This reduction brings the theoretical prediction into quantitative agreement with lattice-QCD data, similar to the agreement achieved in the PDM with a chiral-invariant mass of $m_0 \approx 600$ MeV.
Indistinguishability via Nucleon Data Alone: The suppression factor derived from meson mixing ( $C(\theta) \approx 0.685$ ) is nearly identical to that derived from the chiral-invariant mass in the PDM ( $C(m_0) \approx 0.690$ for $m_0=600$ MeV). The authors conclude that the nucleon $D$ -form factor alone is insufficient to distinguish whether the observed suppression originates from meson mixing or a chiral-invariant mass component.
Pion $D$ -form Factor as a Discriminator: To resolve this ambiguity, the authors analyze the pion $D$ $D$ -form factor.
- In the forward limit ( $t=0$ ), the value remains fixed at $D_\pi(0) = -1$ in both the LSM/PDM and the eLSM, consistent with Chiral Perturbation Theory (ChPT).
- However, the momentum-transfer dependence differs. In the eLSM, meson mixing modifies the slope at the forward limit, reducing it by a factor of $\sin^2(\theta_S - \theta_P) \approx 0.835$ . This results in an enhancement of the magnitude of the pion $D$ -form factor at finite momentum transfer compared to the LSM result.
- Crucially, the chiral-invariant mass in the PDM does not directly affect the pion $D$ -form factor because the mesonic sector in the PDM follows the conventional LSM.
Role of Heavy Scalar Mesons: While the contribution of the heavier scalar meson ( $\hat{\sigma}$ ) to the nucleon $D$ -form factor is suppressed by its large mass, it is non-negligible for the pion $D$ -form factor. The $1/M^2_{\hat{\sigma}}$ suppression from the propagator is canceled by the $M^2_{\hat{\sigma}}$ dependence of the coupling, meaning both scalar exchanges are required to preserve the forward limit value of $-1$.

Significance and Claims
The paper claims that a combined analysis of the nucleon and pion $D$ -form factors offers a pathway to distinguish between mechanisms affecting hadron structure. While the nucleon $D$ -form factor is sensitive to both the chiral structure of the nucleon mass and meson mixing, the pion $D$ -form factor provides more direct information on the mesonic sector, as it is unaffected by the chiral-invariant mass but sensitive to meson mixing.

The authors modestly note that their analysis is performed in the chiral limit, whereas lattice-QCD data is obtained at finite pion mass. Therefore, a direct quantitative comparison requires future work that includes explicit chiral-symmetry-breaking effects. Additionally, the study is limited to the scalar-meson-dominance picture, which is valid at low momentum transfer; contributions from heavier scalar and tensor mesons at larger momentum transfers are acknowledged as a necessary extension for future research.

Ultimately, the work suggests that the interplay between baryon and meson sectors, specifically the mixing of two-quark and four-quark states, plays a critical role in the mechanical properties of hadrons, and that $D$ -form factors serve as a valuable tool for investigating the quark composition of mesons and the decomposition of nucleon mass.

Technical Summary: Gravitational DDD-form factors of the nucleon and pion with mixing between two-quark and four-quark states

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Technical Summary: Gravitational $D$ -form factors of the nucleon and pion with mixing between two-quark and four-quark states