Baryon Bethe-Salpeter Equation in Minkowski-Space QCD$_2$

This paper formulates and numerically solves the three-quark Bethe-Salpeter equation for baryons in Minkowski-space QCD$_2$ using the light-cone gauge, demonstrating that the leading-order valence truncation reproduces the Bars-Durgut equation and yields a ground-state mass and Regge trajectory consistent with previous results and experimental trends while providing a framework for calculating various structure observables.

The Gist

Imagine the universe as a giant, complex machine made of tiny building blocks called quarks. These quarks stick together to form larger particles called baryons (like protons and neutrons, which make up the atoms in your body).

For a long time, physicists have struggled to write a single, perfect "instruction manual" (an equation) that describes exactly how these three quarks hold hands and dance together, especially when they are moving at near-light speeds. This is the Bethe–Salpeter Equation.

This paper is like a team of physicists trying to solve a very difficult puzzle by building a simplified, miniature version of the universe to test their tools. Here is what they did, explained simply:

1. The "Flatland" Laboratory

Real life has three dimensions of space and one of time (3+1). Calculating how quarks behave in this full space is incredibly hard, like trying to solve a Rubik's cube while blindfolded.

So, the authors decided to work in a 2D universe (1 dimension of space + 1 of time), which they call QCD2. Think of this as a "training wheels" version of reality. In this flat world, the rules of how quarks stick together (confinement) are much clearer and easier to write down mathematically. It's like practicing your golf swing on a putting green before trying to hit a ball on a full course.

2. The "Shadow" Trick (Light-Cone Projection)

The authors wanted to take their complex 2D equations and translate them into a format that looks like the way we usually think about particles: as a snapshot in time.

They used a mathematical technique called Light-Cone Projection. Imagine shining a bright light on a 3D object to cast a 2D shadow on the wall. The shadow is simpler than the object, but it still holds the essential shape.

• They took their complex "Minkowski-space" equations (the full 3D object) and projected them onto this "Light Front" (the shadow).
• The Result: They found that when they looked at the simplest version of the problem (only the three main quarks, ignoring extra "ghost" particles that pop in and out), their new equation looked exactly like an old, famous equation called the Bars–Durgut equation. This was a big "Aha!" moment, proving their method works.

3. The "Three-Quark Dance"

In this simplified world, they solved the equation for a baryon made of three quarks.

• The Ground State: They calculated the weight (mass) of the most stable baryon (the "ground state"). Their result matched previous calculations and real-world data very well. This suggests that for the basic building blocks of matter, you mostly just need to look at the three main quarks; you don't need to worry too much about the chaotic "sea" of extra particles for now.
• The Excited States: They also looked at "excited" baryons (particles that are wiggling or vibrating more). They found a pattern in their masses that looks like a Regge trajectory.
  • Analogy: Imagine plucking a guitar string. You get a low note (ground state) and then higher, harmonious notes (excited states). The authors found that their mathematical guitar string produces notes that line up surprisingly well with the actual notes (masses) of protons and neutrons we see in experiments.

4. Mapping the Interior

Once they had the solution, they didn't just stop at the weight. They used their equation to map out the internal structure of these particles:

• Parton Distribution Functions: They calculated how likely you are to find a quark moving at a certain speed inside the proton. When they compared this to real-world data from huge particle accelerators, it matched up very well.
• Double Distributions & Coordinate Space: They created "heat maps" showing where the quarks are likely to be found.
  • For the stable proton, the quarks like to huddle close together in the center.
  • For the excited states, the quarks spread out more, creating different patterns (like a doughnut shape or a star shape) depending on how much energy they have.

5. Why This Matters (According to the Paper)

The authors aren't claiming this solves the problem for our real, 3D universe yet. Instead, they are saying:

• It's a Test Bed: This 2D model is a perfect "training ground" to test new mathematical tools (Minkowski-space methods) that are supposed to handle confinement (the force that keeps quarks stuck together).
• Validation: Since their method worked perfectly in this 2D test and matched known results, it gives them confidence that these same tools can eventually be used to solve the much harder problem of baryons in our real, 3D universe.

In summary: The team built a simplified, 2D model of the universe to test a new way of calculating how three quarks stick together. They proved their math works by showing it predicts the correct weights and internal shapes of particles, matching both old theories and real experimental data. This gives them a solid foundation to try these same tools on the real, complex 3D world.

Technical Summary

Technical Summary: Baryon Bethe–Salpeter Equation in Minkowski-Space QCD2

Problem Statement
The paper addresses the challenge of formulating relativistic bound-state equations for baryons in Minkowski space, specifically within the context of Quantum Chromodynamics in (1+1) dimensions (QCD2). While the 't Hooft equation provides a well-understood framework for mesons in large-$N_c$ QCD2, and light-cone (LC) quantization has successfully solved the theory for finite $N_c$, extending covariant Bethe–Salpeter (BSE) methods to confining theories remains difficult. Existing BSE formulations often rely on non-confining kernels or require complex treatments of singularities. Furthermore, baryon problems are generally not closed within the valence sector, necessitating coupling to higher Fock components. The authors aim to bridge the gap between covariant Minkowski-space BSEs and light-front (LF) Hamiltonian approaches by deriving a baryon equation from first principles in QCD2 and comparing it to established results.

Methodology
The authors employ a systematic reduction of the three-quark ladder Bethe–Salpeter equation in QCD2 using the light-cone gauge. The core methodology involves:

1. Quasi-Potential (QP) Expansion: The authors utilize the QP expansion to project the full Minkowski-space BSE onto the light front. This technique eliminates the relative light-front time of propagating particles, systematically reducing the full equation to an effective equation in the valence sector.
2. Leading Order (LO) Truncation: The study focuses on the leading order of the QP expansion, retaining only the valence three-quark intermediate state. This truncation neglects contributions from higher Fock sectors (e.g., quark-antiquark pairs) at this stage.
3. Derivation of the Effective Equation: By applying the QP expansion to the three-quark BSE with massless gluon exchange, the authors derive an effective mass-squared eigenvalue equation. They demonstrate that, at LO, this equation is mathematically equivalent to the Bars–Durgut equation (BDE), which was previously derived using string-inspired methods and planar Feynman diagrams.
4. Numerical Solution: The resulting integral equation is solved numerically for $N_c=3$ using a basis of non-orthogonal, totally symmetric Jacobi polynomials. The basis functions are constructed to satisfy the endpoint power-law behavior derived analytically.
5. Structure Observables: Using the solved wave functions, the authors compute various observables, including parton distribution functions (PDFs), double distribution amplitudes (DDAs), and coordinate-space densities in Ioffe-time coordinates. The PDFs are evolved to experimental scales ($\mu^2 = 10 \text{ GeV}^2$) using next-to-next-to-leading order (NNLO) DGLAP evolution.

Key Contributions and Results

• Equivalence to Bars–Durgut Equation: The primary theoretical contribution is the derivation showing that the light-front projection of the covariant three-quark BSE in QCD2, when truncated to the valence sector at leading order, yields an equation identical to the Bars–Durgut equation. This establishes a direct link between the covariant BSE framework and previous effective baryon models.
• Endpoint Behavior: The authors derive a transcendental equation for the power-law exponent ($s$) of the valence wave function near the endpoints ($x_i \to 0$). This equation, $m^2/g^2 \pi (2 - 1 + \pi s \cot(\pi s)) = 0$, is the baryon analogue of the 't Hooft equation for mesons, with a modified mass term coefficient.
• Mass Spectrum:
  • Ground State: The calculated ground-state baryon mass is in reasonable agreement with previous light-cone quantization results for QCD2. The authors note that their valence-only result is slightly higher (by roughly 10%), suggesting that higher Fock sectors provide a modest reduction in mass at the nucleon scale.
  • Excited States and Regge Trajectory: The excited-state spectrum yields a Regge trajectory that captures the overall trend of the experimental nucleon spectrum, though it exhibits a slight deviation from strict linearity, which the authors attribute to intrinsic properties of the LO effective Hamiltonian.
• Structure Observables:
  • PDFs: The evolved valence parton distribution functions show good agreement with global analyses (CT18, MSHT20, NNPDF40) at $\mu^2 = 10 \text{ GeV}^2$.
  • DDAs and Densities: The double distribution amplitudes and coordinate-space densities reveal distinct structural patterns for the ground state ($N(939)$) versus excited states ($N(1440)$, $N(1710)$). The excited states exhibit more complex nodal structures and spatial extents in Ioffe-time coordinates.

Significance and Claims
The paper positions QCD2 as a valuable "confining test bed" for Minkowski-space bound-state methods. The authors claim that their work demonstrates that Minkowski-space approaches, when combined with quasi-potential expansions, can successfully capture the dominant features of confined baryons, specifically reproducing the spectral and structural results of light-cone quantization.

The significance lies in:

1. Validation of Framework: It validates the use of the quasi-potential expansion to reduce covariant BSEs to effective light-front equations in a confining theory.
2. Foundation for 3+1 Dimensions: The authors suggest that this framework provides a necessary stepping stone for developing confining formulations in (3+1) dimensions beyond the valence truncation.
3. Baryon Structure: It offers a first-principles derivation of baryon structure observables (PDFs, DDAs) within a confining model, showing that the valence sector provides the dominant contribution to the ground state, while higher Fock components are required for precise mass corrections and more complex structural details.

The authors remain modest regarding the scope, acknowledging that the current work is restricted to the valence sector and that the full BSE solution would require including irreducible many-body terms arising from coupling to higher Fock components.