Solving bound-state equations in $\text{QCD}_2$ with bosonic and fermionic quarks

This paper investigates bound-state equations for exotic hadrons (tetraquarks and baryons) composed of bosonic and fermionic quarks in two-dimensional QCD using both light-front and equal-time quantization, deriving new equations in the finite momentum frame and numerically demonstrating how their wave functions converge to light-cone forms under continuous boosting.

The Gist

Imagine the universe is a giant, complex Lego set. Physicists call the rules that govern how these Lego bricks snap together Quantum Chromodynamics (QCD). It's the rulebook for how quarks (the tiny bricks) and gluons (the glue) stick together to form protons, neutrons, and other particles we call hadrons.

The problem? The real-world rulebook (QCD in our 3D space + 1D time) is incredibly messy and hard to solve. It's like trying to solve a Rubik's Cube while it's spinning, on fire, and the instructions are written in a language you don't speak.

To get around this, physicists use a "training wheels" version of the universe. This paper is about solving a specific, simplified version of the Lego rules in a 2D universe (just a line and time). This simplified world is called the 't Hooft model.

Here is what the authors did, explained simply:

1. The New Characters: Bosonic Quarks

In our real world, quarks are "fermions" (like electrons). They follow strict rules: no two can sit in the same seat at the same time.
But in this paper, the authors introduced a new type of character: Bosonic quarks.

• Analogy: Imagine fermions are like introverts who hate sharing a room. Bosons are like extroverts who love sharing a room and can all pile into the same space.
• Why do this? In the real world, we think two quarks can sometimes act like a single "boson" (called a diquark). By studying these "bosonic quarks" directly, the authors can learn how these diquark pairs behave, which helps us understand how protons and neutrons are built.

2. The Two Exotic Creatures

The authors looked at two strange new "animals" made from these Lego bricks:

• The "Tetraquark" (The Double-Date): A creature made of two bosonic quarks (a pair of extroverts). In the real world, these are rare, exotic particles that might be made of two quarks and two antiquarks.
• The "Baryon" (The Hybrid): A creature made of one fermionic quark (the introvert) and one bosonic antiquark (the extrovert). This is the authors' way of modeling a real-world proton or neutron, treating the proton as a mix of a single quark and a "diquark" pair.

3. The Two Perspectives: The Train vs. The Station

To understand these creatures, you have to look at them from different angles. The authors did this from two viewpoints:

• The Infinite Momentum Frame (IMF): Imagine the creature is on a train moving at the speed of light. From this view, time slows down, and the creature looks like a flat, 2D pancake. This is the "easy" view, already well-studied by others.
• The Finite Momentum Frame (FMF): Imagine the creature is sitting still on a platform, or moving at a normal speed. This is the "hard" view. The math gets messy because the creature is wobbling and changing shape as it moves.

The Big Breakthrough:
The authors successfully wrote down the "blueprints" (equations) for these creatures in the hard view (FMF).

• The Analogy: Think of the "pancake view" (IMF) as a shadow puppet on a wall. It's easy to see the shape. The "platform view" (FMF) is the actual 3D puppet. The authors figured out how to translate the 3D puppet's movements back into the 2D shadow, proving that the shadow is just a special case of the real thing.

4. The "Boost" Experiment

One of the coolest things they did was simulate boosting the creatures.

• What they did: They started with the creature sitting still (FMF) and gradually accelerated it to near the speed of light (IMF).
• What happened: As the creature sped up, the "backward-moving" parts of its wave function (the wobbly, messy parts) faded away. The "forward-moving" part (the clean, simple part) became identical to the "pancake" shadow we already knew.
• Why it matters: This confirms a modern theory called LaMET (Large Momentum Effective Theory). It proves that if you study a particle moving fast enough, you can learn everything about its internal structure without needing the messy, slow-motion math.

5. The Results: The Mass Spectrum

Finally, they solved the equations to find the mass (weight) of these creatures.

• They found that as you add more energy (like stacking more Lego bricks), the mass increases in a very predictable, straight-line pattern.
• The Analogy: It's like a guitar string. When you pluck it, it makes a specific note. If you tighten it, the note goes up. The authors showed that these exotic particles have a "musical scale" of masses, just like the real particles we see in nature.

Summary

In short, this paper is a mathematical proof-of-concept.

1. They built a simplified 2D universe with new types of particles.
2. They figured out how to describe these particles when they are moving slowly and when they are moving at light speed.
3. They proved that the "slow" view smoothly turns into the "fast" view, validating our modern methods for studying the universe.
4. They provided the first-ever detailed map of how a "hybrid" particle (one fermion + one boson) behaves, which helps us understand the deep structure of protons and neutrons in our real 4D world.

It's like solving a puzzle in a small, quiet room to understand how a massive, noisy machine works in the real world.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding hadron structure within Quantum Chromodynamics (QCD) by studying a solvable toy model: two-dimensional QCD ($QCD_2$) in the large-$N_c$ limit (the 't Hooft limit). While the original 't Hooft model (containing only fermionic quarks) is well-understood in the Infinite Momentum Frame (IMF), the behavior of "exotic" hadrons composed of mixed bosonic and fermionic constituents remains less explored, particularly in the Finite Momentum Frame (FMF).

Specifically, the authors aim to:

• Derive Bound-State Equations (BSEs) for two types of "exotic" hadrons in the extended 't Hooft model (which includes both scalar/bosonic and spinor/fermionic quarks):
  1. "Tetraquark": A bound state of a bosonic quark and a bosonic antiquark.
  2. "Baryon": A bound state of a fermionic quark and a bosonic antiquark (mimicking a diquark-quark system).
• Formulate these equations in both the Infinite Momentum Frame (IMF) (Light-Front quantization) and the Finite Momentum Frame (FMF) (Equal-time quantization).
• Investigate the relationship between the wave functions in FMF and the light-cone wave functions in IMF, specifically testing the convergence predicted by Large Momentum Effective Theory (LaMET).
• Address subtleties regarding quark mass renormalization in different quantization schemes.

2. Methodology

The authors employ a Hamiltonian approach utilizing both Light-Front (LF) and Equal-Time (ET) quantization. They also utilize diagrammatic techniques (Dyson-Schwinger/Bethe-Salpeter equations) to cross-verify results.

• Model Setup: The Lagrangian includes $SU(N_c)$ gauge fields, a Dirac fermion field ($\psi$), and a complex scalar field ($\phi$). The limit $N_c \to \infty$ with the 't Hooft coupling $\lambda = g_s^2 N_c / 4\pi$ fixed is applied.
• Light-Front Quantization (IMF):
  • The light-cone gauge $A^+ = 0$ is imposed.
  • The Hamiltonian is constructed, and the non-dynamical gluon field is integrated out, leading to an instantaneous interaction potential.
  • Bosonization/Fermionization: The authors introduce color-singlet compound operators (e.g., $W$ for tetraquarks, $K$ for baryons) to diagonalize the Hamiltonian.
  • The BSEs are derived by enforcing the diagonalization of the Hamiltonian in the basis of physical bound states, leading to integral equations for the light-cone wave functions.
• Equal-Time Quantization (FMF):
  • The axial gauge $A_z = 0$ is imposed.
  • The Hamiltonian is derived via Legendre transformation.
  • Mass-Gap Equation: A variational approach is used to derive the mass-gap equation, determining the dispersion relation of dressed quarks.
  • Bogoliubov Transformation: To handle the mixing of forward and backward moving components in FMF, the authors introduce a Bogoliubov transformation. This connects the bare quark operators to physical "tetraquark" and "baryon" creation/annihilation operators.
  • This leads to a system of coupled integral equations for the forward-moving ($\chi^+, \Phi^+$) and backward-moving ($\chi^-, \Phi^-$) wave functions.
• Diagrammatic Approach: For the "tetraquark" in FMF, the authors also provide a derivation using Dyson-Schwinger/Bethe-Salpeter equations. A key technical challenge addressed here is the presence of "seagull" vertices in scalar QCD, which are handled by decomposing them into quark-gluon vertices and modifying the quark propagator structure.
• Renormalization: The paper carefully treats ultraviolet (UV) divergences. It distinguishes between the renormalized quark mass in LF quantization ($m_r$) and ET quantization ($m_{\tilde{r}}$), establishing the relationship between them.

3. Key Contributions

1. First Derivation of "Baryon" BSE in FMF: The paper presents, for the first time, the coupled bound-state equations for a "baryon" (fermion-boson bound state) in the Finite Momentum Frame within the extended 't Hooft model. These are the counterparts to the Bars-Green equations for mesons in the original 't Hooft model.
2. Diagrammatic Derivation in FMF: The authors successfully reproduce the "tetraquark" BSE in FMF using a diagrammatic approach, overcoming the difficulty of non-vanishing seagull vertices in the axial gauge.
3. Renormalization Scheme Connection: The work explicitly clarifies the relationship between the renormalized bosonic quark mass in Light-Front quantization and Equal-Time quantization, showing they differ by a finite term dependent on the renormalization scheme.
4. Numerical Verification of LaMET: The paper provides numerical evidence supporting the Large Momentum Effective Theory (LaMET) conjecture: as the hadron momentum increases, the backward-moving components of the FMF wave functions vanish, and the forward-moving components converge to the Light-Cone wave functions.

4. Results

• Analytical Results:
  • Derived the Shei-Tsao equation (for bosonic mesons) and the Aoki equation (for fermion-boson "baryons") in the IMF.
  • Derived the coupled Bars-Green-like equations for both "tetraquarks" and "baryons" in the FMF.
  • Proved analytically that in the $P \to \infty$ limit, the FMF equations reduce to their IMF counterparts.
• Numerical Results:
  • Mass Spectra: Calculated the mass spectra for "tetraquarks" and "baryons" with various quark masses. The results exhibit linear Regge trajectories ($M^2 \propto n$), consistent with the original 't Hooft model and phenomenological expectations.
  • Wave Functions:
    • Plotted the wave functions for ground and first excited states.
    • Demonstrated that for low momenta, the wave functions have significant backward-moving components ($\chi^-, \Phi^-$).
    • Showed that as the momentum $P$ increases, the backward components fade away rapidly, and the forward components ($\chi^+, \Phi^+$) converge to the Light-Front wave functions ($\chi, \Phi$).
  • Regge Trajectories: Confirmed that the squared mass of the excited states grows linearly with the principal quantum number $n$.

5. Significance

• Theoretical Laboratory: The extended 't Hooft model serves as a rigorous theoretical laboratory for studying non-perturbative QCD phenomena, specifically the structure of exotic hadrons (tetraquarks) and the diquark picture of baryons.
• Validation of LaMET: The numerical demonstration of the convergence of FMF wave functions to Light-Cone wave functions provides strong support for the validity of LaMET, which is crucial for extracting parton distribution functions from Euclidean lattice QCD calculations.
• Methodological Advancement: The successful application of the Hamiltonian approach to mixed boson-fermion systems in both IMF and FMF, including the handling of seagull vertices and mass renormalization, offers a robust framework for future studies of more complex hadronic systems.
• Insight into Exotic Hadrons: By modeling tetraquarks as boson-antiboson bound states and baryons as fermion-boson bound states, the paper offers insights into the mass spectra and wave function structures of these exotic states, which are relevant to the ongoing experimental search for tetraquarks and pentaquarks in 4D QCD.