⚛️ high-energy theory

QCD_2: 't Hooft equation vs. relativistic Schroedinger equation

This paper compares the meson mass spectrum of 2D QCD in the 't Hooft limit derived from the Bethe-Salpeter equation against that obtained from a relativistic Schrödinger equation, finding that while both methods agree in the leading semiclassical approximation, they yield different preasymptotic WKB corrections that are significant for the ground state but converge rapidly for higher excited states.

Original authors: Andrei Smilga

Published 2026-10-07

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrei Smilga

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 held together by forces that are far too strong to be seen directly. One of these forces, known as the strong interaction, binds quarks together to form protons and neutrons, the building blocks of all visible matter. To understand how this force works, physicists often turn to simplified models. Imagine a universe with only two dimensions instead of our familiar three. In this flat, two-dimensional world, the rules of the strong force become much easier to study because the chaotic interactions that usually make the math impossible simply do not happen. Here, particles called mesons, which are pairs of a quark and its antimatter partner, exist as stable, long-lived objects. For decades, scientists have used a specific mathematical tool, developed fifty years ago, to predict the exact masses of these mesons. This tool, known as the 't Hooft equation, has been the gold standard for understanding how these particles behave in this simplified universe.

Recently, a physicist named Andrei Smilga revisited this classic problem to test a different, more intuitive approach. While the original method involves solving a complex integral equation that describes how quarks interact, Smilga asked whether a simpler method, based on a relativistic version of the Schrödinger equation, could yield the same results. This alternative method treats the meson as a single object moving under the influence of a string-like force that pulls the quarks together, a concept familiar from the idea of a rubber band snapping back. The question was not just about which method is easier to calculate, but whether the simpler picture captures the true physics of the system, especially when the quarks are moving at speeds close to the speed of light.

Smilga's investigation revealed that the two methods are not identical, though they are remarkably close. In the simplest scenarios, where the quarks are heavy and moving slowly, or where the mesons are in very high energy states, both approaches predict the exact same masses. This agreement confirms that the intuitive picture of a string pulling quarks together is a valid way to think about the system in these limits. However, the story changes when looking at the details of the middle ground. When the quarks are lighter and moving faster, or when the meson is in a low-energy state, the two methods begin to diverge. The simpler relativistic equation predicts masses that are slightly different from those calculated by the original, more rigorous 't Hooft equation.

To settle the question of which prediction was correct, the author performed precise numerical simulations. By breaking the problem down into thousands of small steps and using a computer to solve the equations, the study compared the results of both methods against each other. The simulations showed that the original 't Hooft equation provides the more accurate description of reality. The simpler relativistic approach, while useful, misses subtle corrections that become important for the lowest energy states. For the ground state, the difference is significant enough to matter. However, as the energy of the meson increases, the two methods converge again. By the time the meson is in its third excited state, the difference between the two predictions drops to less than two and a half percent, and it continues to shrink as the energy gets higher.

The study concludes that while the intuitive string model is a powerful tool for understanding the general behavior of these particles, it is not a perfect substitute for the original, more complex equation. The discrepancy is not a failure of the simple model, but rather a reminder that nature often hides small, pre-asymptotic corrections that only appear when looking closely at specific states. For the vast majority of excited states, the simple model works beautifully, but for the most fundamental states, the full complexity of the original theory is required to get the answer right. This work refines our understanding of how to model the strong force, showing us exactly where our shortcuts work and where we must rely on the full mathematical machinery to see the truth.

Technical Summary: 't Hooft Equation vs. Relativistic Schrödinger Equation in 2D QCD

Problem Statement
This paper investigates the mass spectrum of meson states in two-dimensional Quantum Chromodynamics (QCD) with a fundamental quark of mass $m$ , specifically within the 't Hooft limit ( $N \to \infty$ , $g^2 \to 0$ , $g^2N = \text{const}$ ). In this limit, the theory contains no glueballs or baryons, only stable mesons (quark-antiquark bound states). The standard method for determining this spectrum is solving the integral Bethe-Salpeter equation derived by 't Hooft. The paper addresses the validity of an alternative, simpler approach: solving the Schrödinger equation for a relativistic Hamiltonian with a linear confining potential (string tension $\sigma = g^2N/4$ ). The core question is whether the spectra derived from the exact 't Hooft integral equation and the approximate relativistic Schrödinger (Salpeter) equation coincide beyond the leading semiclassical approximation.

Methodology
The author employs a multi-faceted approach combining analytical derivations, semiclassical (WKB) expansions, and numerical simulations:

Classical Equivalence Proof: The paper first establishes a theoretical link between the two approaches. It proves that the classical counterparts of the 't Hooft Hamiltonian (derived from the integral equation) and the relativistic Hamiltonian $\hat{H} = 2\sqrt{\hat{p}^2 + m^2} + \sigma|x|$ are related by a canonical transformation. This explains why their leading-order WKB asymptotics are identical.
Analytical Expansion: The author performs a quasi-nonrelativistic expansion of the mass spectrum in powers of a small parameter $\gamma = (\sigma/m^2)^{2/3}$ $γ = (σ / m^{2})^{2/3}$ . This involves:
- Solving the 't Hooft equation using known results (Ziyatdinov) for preasymptotic WKB corrections.
- Deriving the corresponding corrections for the relativistic Hamiltonian by expanding the kinetic energy term ( $\sqrt{p^2+m^2}$ ) and applying perturbation theory to the non-relativistic Airy function solutions.
Numerical Verification: To test the analytical predictions, the author numerically solves the discretized 't Hooft equation. The integral equation is reformulated into a symmetric form and discretized using a grid method (trading the Hadamard finite part for a principal value integral). The resulting large matrices are diagonalized for various grid sizes ( $K$ ) and extrapolated to $K \to \infty$ .

Key Contributions and Results

Leading Order Agreement: The paper confirms that in the leading semiclassical approximation (large excitation number $n$ ) and in the nonrelativistic limit ( $g^2 \ll m^2$ ), the mass spectra from both methods coincide.
Divergence in Preasymptotic Corrections: The primary finding is that the spectra diverge when considering preasymptotic WKB corrections (terms of order $1/n$ $1/ n$ or $\gamma^2$ $γ^{2}$ ).
- For odd parity states, the relativistic Hamiltonian prediction matches the 't Hooft equation result up to the second order in $\gamma$ .
- For even parity states, a discrepancy arises at the second order. The 't Hooft equation yields a term proportional to $1/z'_k$ (where $z'_k$ relates to the Airy function zeros) that is absent in the relativistic Hamiltonian prediction.
Numerical Confirmation: Numerical calculations strongly favor the 't Hooft equation results (Eqs. 4.2 and 4.3) over the relativistic Hamiltonian predictions (Eq. 4.10). The numerical data for ground and excited states align with the 't Hooft expansion, showing that the meson masses derived from the integral equation are systematically larger than those from the Salpeter Hamiltonian.
Magnitude of Discrepancy:
- For the ground state ( $n=0$ ), the difference is significant.
- For the first excited state ( $n=1$ ), the ratio of squared masses $R = M^2_{\text{Salpeter}}/M^2_{\text{Hooft}}$ remains close to 1 but deviates.
- By $n=3$ , the relative difference in masses is less than 2.5%.
- As $n$ increases, the ratio approaches unity, consistent with the asymptotic behavior where the difference is suppressed as $1/n^2$ .
Ultrarelativistic Limit: In the ultrarelativistic limit ( $m \to 0$ ), the paper notes that the 't Hooft equation yields $M^2_n \propto (n + 3/4)$ , whereas the relativistic Schrödinger equation (Klein-Fock-Gordon) yields $M^2_n \propto (n + 1/2)$ . This explains the persistent offset in the ratio $R$ even at large $\gamma$ .

Significance and Claims
The paper claims that while the relativistic Schrödinger equation with a linear potential is a valid approximation for the 't Hooft spectrum in the leading semiclassical and nonrelativistic limits, it is not exact for finite $N$ or intermediate excitation numbers. The discrepancy arises specifically in the preasymptotic corrections.

The author emphasizes that the 't Hooft integral equation provides the correct spectrum for 2D QCD in the large $N$ limit. The relativistic Hamiltonian approach, while intuitive and useful for leading-order estimates, fails to capture the precise subleading corrections required for an exact description of the meson spectrum, particularly for low-lying states. The work serves as a rigorous check on the validity of using simple relativistic potential models to approximate complex gauge-theory bound states, highlighting that even in solvable 2D models, the "standard" relativistic Schrödinger equation is an approximation with quantifiable errors.

Technical Summary: 't Hooft Equation vs. Relativistic Schrödinger Equation in 2D QCD

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