⚛️ high-energy theory

Meson states in `t Hooft model: Hamiltonian approach

This paper demonstrates that the masses of highly excited meson states in the 't Hooft model can be accurately determined using a simple quantum-mechanical model with a linear potential, reproducing 't Hooft's mass spectrum in the ultrarelativistic limit while revealing exponentially suppressed finite widths for relativistic levels.

Original authors: A. V. Smilga

Published 2026-06-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: A. V. 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

Imagine the universe as a giant, invisible trampoline. In our real world, this trampoline is complex and bouncy in many directions. But in the paper you're asking about, the author, Andrei Smilga, zooms in on a very specific, simplified version of this trampoline: a universe with only two dimensions (one length, one width).

In this 2D world, the rules of physics are a bit different. There are no "glueballs" (particles made purely of force) and no "baryons" (like protons). The only things that exist are mesons. You can think of a meson as a pair of dancers: a quark and an antiquark, holding hands and spinning around each other.

Here is the simple breakdown of what Smilga discovered about these dancing pairs:

1. The "String" That Binds Them

In our 3D world, the force holding quarks together is like a rubber band that gets stronger the more you stretch it. In this 2D world, the force is even simpler: it's like a perfectly stiff string connecting the two dancers.

Smilga points out that we don't need to solve a massive, terrifyingly complex math problem (which the original physicist 't Hooft did 50 years ago) to figure out how heavy these dancers get as they spin faster. Instead, we can treat them like a simple quantum mechanical system where the "string" pulls them with a force that grows linearly with distance.

2. The Two Ways They Dance

Smilga looks at two different scenarios for how these quarks move:

Scenario A: The Slow Dancers (Non-Relativistic)
Imagine the quarks are heavy and moving slowly, like two people waltzing.

The Result: As they get more excited (spin faster), their energy levels don't go up in a straight line. Instead, they follow a specific curve. Smilga shows that if you use a simple "semi-classical" guess (like counting the steps in a dance), you get the exact same answer as the complex math. It's like predicting the height of a bouncing ball just by knowing how hard you threw it.

Scenario B: The Super-Fast Dancers (Ultra-Relativistic)
Now, imagine the quarks are very light and moving at nearly the speed of light.

The Result: Here, the math becomes incredibly clean. The mass of these excited states follows a simple, straight-line pattern: the square of the mass is directly proportional to the "excitement level" (the quantum number $n$ ).
The Connection: This simple pattern matches perfectly with the complex, 50-year-old solution 't Hooft found using a different method. It's as if Smilga found a shortcut through a maze that 't Hooft had to walk through the long way.

3. The "Leaky" Bucket (A New Discovery)

This is the most interesting part of the paper. In the original 2D theory (with an infinite number of colors, $N \to \infty$ ), these mesons are perfectly stable. They never fall apart. They are like a bucket with no holes.

However, Smilga's simple "string" model suggests something slightly different. Because the string can vibrate and the particles can move so fast, there is a tiny chance the dancers could "tunnel" through the string and escape to infinity.

The Analogy: Imagine a ball trapped in a deep valley. In the perfect theory, the walls are infinitely high, so the ball stays forever. In Smilga's model, the walls are slightly fuzzy. The ball can escape, but only if it tunnels through.
The Catch: For heavy quarks, this "leak" is so incredibly small (exponentially suppressed) that the particles are effectively stable. It's like a bucket with a microscopic pinhole; it will take billions of years to empty. But for very light quarks, the bucket leaks so fast that the particles don't really have a stable existence at all—they just dissolve into a continuous stream.

The Big Picture

Smilga's paper is essentially saying: "We don't need a supercomputer to understand the heavy, excited states of these particles. A simple model of a ball on a spring (or a string) gives us the right answer."

He confirms that the simple model reproduces the famous results of the complex theory for the masses of these particles. The only new twist is that his simple model reveals a tiny, theoretical instability (a "width") that the original theory smoothed over, though this instability is so small for heavy particles that it barely matters.

In summary: The paper takes a complex, high-level physics problem about particle masses and shows that a simple, intuitive picture of a string pulling two particles together explains the results just as well as the complicated original math, while adding a tiny new detail about how "leaky" these particles might be.

Technical Summary: Meson states in 't Hooft model: Hamiltonian approach

Problem Statement
The paper addresses the determination of the mass spectrum for highly excited bound quark-antiquark states in two-dimensional Quantum Chromodynamics (QCD $_2$ ) with an $SU(N)$ gauge group in the limit $N \to \infty$ . In this limit, the theory contains only mesons (bound states of a quark and an antiquark), as gluon self-interactions and baryon production are absent or suppressed. While 't Hooft previously derived the exact spectrum by solving the Bethe–Salpeter equation, the author seeks to demonstrate that the asymptotic behavior of these masses can be recovered using a simpler, semiclassical quantum-mechanical model involving a linear potential $V(x) = \sigma|x|$ .

Methodology
The author employs a Hamiltonian approach, contrasting the exact integral equation method with a relativistic quantum mechanical model. The methodology proceeds as follows:

Model Definition: The author defines the system as two particles (quark and antiquark) of mass $m$ bound by a string with tension $\sigma$ . The analysis is conducted in the center-of-mass frame.
Nonrelativistic Limit: For energies $\epsilon_n \ll m$ , the system is treated using the nonrelativistic Schrödinger equation with the Hamiltonian $H_{cm} = p^2/m + \sigma|x|$ . The energy spectrum is estimated using semiclassical (WKB) quantization conditions.
Relativistic Limit: For the ultrarelativistic case ( $E \gg m$ ), the author utilizes the relativistic Hamiltonian $H_{cm}^{rel} = 2\sqrt{p^2 + m^2} + \sigma|x|$ . The spectrum is again evaluated via semiclassical approximation, calculating the action integral $S(E)$ .
Width Estimation: Unlike the exact 't Hooft analysis where states are stable in the $N \to \infty$ limit, the relativistic Hamiltonian approach allows for tunneling. The author calculates the tunneling amplitude through the classically forbidden region to estimate the finite widths ( $\Gamma_n$ ) of these states.
Parameter Identification: The string tension parameter $\sigma$ in the mechanical model is identified with the QCD $_2$ coupling by comparing the Abelian and non-Abelian Coulomb potentials, establishing the relation $\sigma_{QCD2} = g^2N/4$ .

Key Results

Ultrarelativistic Spectrum: In the limit where the bound state mass $\mu_n$ is much larger than the quark mass $m$ and the scale $g^2N$ , the semiclassical relativistic model yields the mass squared spectrum:
$\mu_n^2 = \frac{g^2N}{2\pi} n$
This result coincides exactly with the asymptotic law derived by 't Hooft [1] from the Bethe–Salpeter equation.
Nonrelativistic Spectrum: For the nonrelativistic case ( $\epsilon_n = \mu_n - 2m \ll m$ ), the energy levels follow the asymptotic behavior:
$\epsilon_n \propto n^{2/3}$
Specifically, $\epsilon_n \approx \left( \frac{3\pi\sigma n}{4\sqrt{m}} \right)^{2/3}$ . This matches the WKB asymptotics of the 't Hooft equation found in recent literature [6].
Finite Widths: A distinct finding of this Hamiltonian approach is that the relativistic energy levels possess finite widths due to tunneling, whereas the exact $N \to \infty$ QCD $_2$ states are absolutely stable. The width is estimated as:
$\Gamma_n \sim \frac{\sigma}{E_n} \exp\left\{ -\frac{2\pi m^2}{\sigma} \right\}$
For heavy quarks ( $m^2 \gg \sigma$ ), these widths are exponentially suppressed. However, for light quarks, the widths become comparable to the level spacings, suggesting a transition to a continuous spectrum in the Hamiltonian model, unlike the discrete spectrum of the exact theory.

Significance and Claims
The paper claims that the linear mass-squared trajectory (Regge-like behavior) of the 't Hooft model can be derived "in a very simple way" as a solution to a relativistic quantum problem with a linear potential in the semiclassical approximation.

The author notes two specific observations regarding this derivation:

The states obtained via this semiclassical Hamiltonian method are quasi-stationary (having finite widths) rather than strictly stationary, a feature not previously highlighted in similar string-framework derivations [2, 3].
The paper provides a direct comparison confirming that the semiclassical results align with the exact 't Hooft equation results for the mass spectrum, despite the difference in stability properties (widths).

The work does not propose new applications or experimental tests but serves as a pedagogical and conceptual bridge, showing that complex field-theoretic results in the large- $N$ limit can be captured by simple quantum-mechanical analogs.

1. The "String" That Binds Them

2. The Two Ways They Dance

3. The "Leaky" Bucket (A New Discovery)

The Big Picture

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