⚛️ nuclear theory

Kernel moments and thermodynamic stability in nonlocal Nambu--Jona-Lasinio models

This paper investigates thermodynamic stability in nonlocal Nambu--Jona-Lasinio models by deriving an entropy-sign criterion from Euclidean kernel moments, demonstrating that while specific kernel choices can satisfy thermal positivity, they may still violate static equation-of-state constraints, thereby highlighting the limitations of mean-field approximations and the need for specific diagnostics over universal phenomenological kernels.

Original authors: Xiaozhu Yu, Xinyang Wang

Published 2026-10-14

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CC BY 4.0

Original authors: Xiaozhu Yu, Xinyang Wang

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

In the subatomic world, matter is not a solid, unchanging substance but a seething sea of particles and forces. Among the most fundamental of these are quarks, the tiny building blocks that bind together to form protons and neutrons. Physicists use mathematical models to describe how these quarks interact, particularly how they acquire mass and how they behave when heated to extreme temperatures, such as those found in the early universe or inside colliding stars. One of the most enduring tools for this description is a framework that treats these interactions as a collective dance of the entire system. However, a persistent challenge in this field is ensuring that these mathematical descriptions remain physically sensible. A valid model must not only predict how matter behaves but also obey the strict laws of thermodynamics, the rules governing heat, energy, and disorder. If a model predicts that a system becomes more ordered as it heats up, or that it can absorb heat while its temperature drops, it has broken the fundamental laws of physics, rendering its predictions useless for understanding reality.

Researchers Xiaozhu Yu and Xinyang Wang have taken a close look at a specific, sophisticated version of this model, known as a nonlocal Nambu–Jona-Lasinio model. They focused on a scenario where the interactions between quarks are not instantaneous but depend on the distance and momentum involved, a feature that makes the mathematics more complex but potentially more realistic. Their goal was to test whether this model remains stable and consistent when subjected to high temperatures and varying conditions. They approached this by treating the mass of the quarks as a variable source, essentially turning a dial to see how the system's entropy—a measure of its disorder—responds. They discovered a clear, mathematical rule: if the interaction pattern follows a specific, smooth shape known as a Gaussian mixture, the system inevitably breaks down at high source masses, developing negative entropy. In physical terms, this means the model predicts a state of impossible order, signaling that the description has failed.

The team then tested a different, more complex shape for these interactions, one that includes a quartic term, which is a mathematical way of describing a curve that bends more sharply than a simple exponential. When they fitted this model to known properties of pions, the particles that mediate the force between protons and neutrons, they found a surprising result. For this specific setup, the model successfully maintained positive entropy and a positive heat capacity across all temperatures, suggesting it was thermodynamically stable. This was a significant finding because it showed that a model could pass the basic tests of thermal stability. However, the researchers did not stop there. They applied a second, more stringent test: the speed of sound within the material. In any physical medium, the speed at which a pressure wave travels cannot exceed the speed of light. This speed is determined by how the pressure changes relative to the energy density.

In their stable, quartic model, the researchers found a narrow window of temperature, roughly between 93 and 96 MeV, where the ratio of pressure change to energy change exceeded one. This implies that a disturbance in this theoretical material would travel faster than light, violating the principle of causality. This finding is crucial because it demonstrates that a model can pass the standard tests for heat and disorder yet still fail the test of physical causality. The researchers showed that this violation is not a fluke but a direct consequence of the model's structure near a specific point where the heat capacity becomes very small. They further proved that this behavior is highly sensitive to the exact values used to calibrate the model. By slightly adjusting the input values for the pion mass and other properties, they could shift the model from being stable to being unstable, or vice versa. In fact, they constructed a specific, rigorously verified example where the model produced negative heat capacity, a clear sign of instability, even when using input values that are considered standard and acceptable in the field.

The study concludes that while these nonlocal models can be tuned to look stable, they do not offer a universal solution. The fact that a model can be stable in one sense but unstable in another means that simply fitting the model to vacuum data is not enough to guarantee it describes real-world matter correctly. The researchers also noted that their calculations focused only on the quarks themselves and did not include the thermal effects of the mesons, the particles that emerge from the quark interactions. They calculated that adding these missing particles would significantly alter the results, likely changing the speed of sound test at low temperatures. This suggests that the current mean-field approximation, which treats the quarks as moving in a smooth average field, is insufficient on its own. The work serves as a diagnostic tool, highlighting that thermodynamic stability and the speed of sound are separate, independent checks that a model must pass. It does not declare one specific mathematical shape for the interaction as the correct one, but rather warns that without including all the necessary physical components, such as the thermal mesons, any prediction about the behavior of hot, dense matter remains incomplete and potentially misleading.

Technical Summary: Kernel Moments and Thermodynamic Stability in Nonlocal Nambu–Jona-Lasinio Models

Problem Statement
The Nambu–Jona-Lasinio (NJL) model is a standard framework for describing chiral symmetry breaking, yet its thermal predictions are sensitive to regularization schemes. Previous studies have demonstrated that thermal regularization in local NJL models can lead to unphysical results, such as incorrect high-temperature heat capacity behavior and superluminal sound speeds. While nonlocal NJL models, which replace contact vertices with momentum-dependent form factors (kernels), offer ultraviolet convergence, they do not automatically guarantee thermodynamic stability. Specifically, complex-conjugate propagator poles can induce oscillating thermal contributions, potentially leading to negative entropy or heat capacity.

The central problem addressed is the thermodynamic consistency and stability of a covariant two-flavor nonlocal NJL model at zero chemical potential within the homogeneous quark mean-field approximation. The authors investigate whether a common thermodynamic potential can yield acceptable thermal responses (positive entropy and heat capacity) and whether these responses ensure the standard equation of state (EOS) slope bound ( $dP/d\epsilon \leq 1$ ).

Methodology
The study employs a two-pronged approach: a controlled analytical expansion and numerical refitting of light-source models.

Common Potential Framework: The authors define a thermodynamic potential density $\Omega$ derived from a single source-dependent potential, ensuring consistency between condensates and entropy. This potential includes the quark determinant and a homogeneous scalar mean field $\sigma$ , with vacuum pion matching performed without adding a thermal pion gas.
Large-Source Expansion: At fixed couplings and kernel parameters, the authors perform a controlled expansion in the limit of a large current quark mass source ( $m$ ). This expansion expresses the equilibrium field $\sigma^*$ and the entropy $s$ in terms of Euclidean kernel moments ( $A, A_1, H_g$ ). This method allows for the derivation of a sufficient criterion for the sign of the total entropy directly from kernel properties, without needing to locate propagator poles.
Kernel Analysis: The study examines specific kernel classes, including positive Gaussian mixtures and a covariant quartic-exponent kernel ( $g_\alpha(Q^2) = e^{-y-\alpha y^2}$ ).
Numerical Refitting and Interval Analysis: The model is refitted to light-source observables (pion mass $m_\pi$ , decay constant $f_\pi$ , and subtracted condensate $C$ ). The authors use rigorous interval arithmetic and certified numerical enclosures to track thermal zeros (where entropy or heat capacity vanishes) and verify positivity bounds across temperature ranges.

Key Contributions and Results

Entropy Sign Criterion: The authors derive a sufficient condition for negative total entropy based on the temperature derivative of the first kernel moment, $A'(T)$ . For any admissible positive Gaussian mixture kernel, $A'(T) < 0$ holds, implying that the total entropy becomes negative at sufficiently large source mass for any fixed positive temperature. This establishes that positive Gaussian mixtures are thermodynamically unstable in the heavy-source limit.
Decoupling of Thermal Positivity and EOS Slope: Using a covariant quartic-exponent kernel ( $\alpha=1$ ) with specific vacuum inputs ( $m_\pi=139$ MeV, $f_\pi=92.4$ MeV, $c=225$ MeV), the authors construct a case where the entropy $s$ and heat capacity $C_V$ are strictly positive for all $T > 0$ . However, the static EOS slope $dP/d\epsilon = s/C_V$ exceeds unity on a finite temperature interval ( $93.5 \leq T \leq 93.5625$ MeV). This demonstrates that full thermal integration and positive thermal responses do not guarantee the standard causality bound on the static derivative.
Sensitivity to Vacuum Inputs: The stability of the model is highly sensitive to the matching inputs. Numerical refits show that the enhancement of the EOS slope weakens as the condensate input increases. Furthermore, a "certified counterexample" is constructed using exact scale covariance: by rescaling a model with negative heat capacity, the authors produce a fit with exactly the same condensate input ( $c=225$ MeV) but negative heat capacity at $T \approx 93.75$ MeV. This proves that uniform thermal positivity cannot be guaranteed over the entire domain of adopted matching inputs.
Role of Mean-Field Relaxation: The study highlights that the positive heat capacity in the nominal fit is maintained only after including the mean-field relaxation term (the response of the equilibrium field $\sigma^*$ to temperature). The fixed-field contribution alone can be negative, and the relaxation term reverses the sign, underscoring the necessity of consistent minimization.

Significance and Claims
The paper positions its results as diagnostics for the mean-field functional rather than a selection of a preferred phenomenological kernel. The authors claim:

Separation of Diagnostics: The work distinguishes four separate stability criteria: common-potential consistency, stability in the retained scalar direction, thermal positivity (sign of $s$ and $C_V$ ), and the static EOS slope bound. A model can satisfy the first three while failing the fourth.
Limitations of Static Derivatives: The authors emphasize that the static derivative $dP/d\epsilon$ does not determine a real-time front velocity or microscopic causality in temporally nonlocal actions.
Omitted Physics: The study explicitly notes that vacuum pion matching does not include thermal mesons. Estimates suggest that adding an ideal pion gas could significantly alter the low-temperature slope test, potentially resolving the superluminal slope issue, though this requires a consistent fluctuation potential beyond the current mean-field approximation.
No Universal Positivity: The results indicate that no single kernel guarantees uniform thermal positivity across all physically motivated vacuum matching domains. The stability of the thermodynamic description is contingent on specific parameter choices and input constraints.

In conclusion, the paper provides a rigorous framework for testing thermodynamic stability in nonlocal NJL models using kernel moments and interval analysis, revealing that positive thermal responses and static EOS bounds are independent conditions that are not automatically satisfied by standard nonlocal formulations.

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