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

Xiaolin Li; Ying Li; Yu Jia; Zhewen Mo

arxiv: 2409.20555 · v2 · submitted 2024-09-30 · ✦ hep-ph · hep-th

Solving bound-state equations in QCD₂ with bosonic and fermionic quarks

Xiaolin Li , Yu Jia , Ying Li , Zhewen Mo This is my paper

Pith reviewed 2026-05-23 20:09 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords two-dimensional QCDbound-state equationstetraquarksbaryonsbosonic quarksfermionic quarkslight-front quantizationfinite momentum frame

0 comments

The pith

Bound-state equations for tetraquarks of bosonic quarks and baryons of mixed bosonic-fermionic quarks are derived and solved in two-dimensional QCD in both infinite and finite momentum frames.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the 't Hooft model of two-dimensional QCD to include bosonic quarks alongside fermionic ones. It derives the bound-state equations for a tetraquark composed of a bosonic quark and antiquark, and for a baryon made of a bosonic antiquark and fermionic quark, using the Hamiltonian approach in both light-front and equal-time quantizations. Numerical solutions yield mass spectra and wave functions for the lowest states in these exotic configurations. The work also demonstrates numerically that boosting these bound states causes their wave functions to approach the light-cone form.

Core claim

In the large Nc limit of QCD in two dimensions, the bound-state equations for exotic hadrons consisting of bosonic and fermionic quarks can be obtained in both the infinite momentum frame and the finite momentum frame. For the tetraquark of bosonic quark and antiquark, the equations are recovered in the finite momentum frame, while for the baryon of bosonic antiquark and fermionic quark, the finite momentum frame equations are presented for the first time. Numerical solutions of these equations provide the mass spectra and bound-state wave functions, with the property that continuous boosting leads the forward component to the light-cone wave function and the backward component to vanish.

What carries the argument

The bound-state equations (BSEs) derived via the Hamiltonian approach from the extended 't Hooft model, solved in both light-front and equal-time quantization.

If this is right

Mass spectra for tetraquarks and baryons are obtained by numerical solution of the equations.
Bound-state wave functions for the lowest-lying states are computed in both frames.
The forward-moving component of the wave function approaches the light-cone wave function upon continuous boosting while the backward component fades.
The baryon bound-state equation in the finite momentum frame is obtained for the first time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The demonstrated frame equivalence in this solvable model may guide similar checks in four-dimensional QCD calculations that mix statistics.
The numerical approach could be extended to study decay widths or scattering amplitudes of these exotic states within the same framework.
Patterns in the mass spectra might reveal how the bosonic versus fermionic nature affects binding energies across different quark combinations.

Load-bearing premise

The large-Nc limit is taken so that the Hamiltonian approach yields exact bound-state equations without frame-dependent corrections.

What would settle it

Numerical solutions of the boosted finite-momentum-frame wave functions that fail to match the independently computed light-cone wave functions would disprove the claimed continuity between frames.

read the original abstract

We investigate the bound-state equations (BSEs) in two-dimensional QCD in the $N_c\to \infty$ limit, viewed from both the infinite momentum frame (IMF) and the finite momentum frame (FMF). The BSE of a meson in the original 't Hooft model, viz., spinor $\text{QCD}_2$ containing only fermionc quarks, has been extensively studied in literature. In this work, we focus on the BSEs pertaining to two types of "exotic" hadrons, a "tetraquark" which is composed of a bosonic quark and bosonic antiquark, and a "baryon" which is composed of a bosonic antiquark and a fermionic quark. Utilizing the Hamiltonian approach, we derive the corresponding BSEs for both types of "exotic" hadrons, from the perspectives of the light-front and equal-time quantization, and confirm the known results. The recently available BSEs for "tetraquark" in FMF has also been recovered with the aid of the diagrammatic approach. For the first time we also present the BSEs of a "baryon" in FMF in the extended 't Hooft model. By solving various BSEs numerically, we obtain the mass spectra pertaining to "tetraquark" and "baryon" and the corresponding bound-state wave functions of the lowest-lying states. It is numerically demonstrated that, when a "tetraquark" or "baryon" is continuously boosted, the forward-moving component of the bound-state wave function approaches the corresponding light-cone wave function, while the backward-moving component fades away.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper claims a first-time baryon BSE in the finite-momentum frame for the mixed bosonic-fermionic 't Hooft model and shows numerical boosting to light-cone wave functions, but only the abstract is available so nothing can be checked.

read the letter

The headline result is the new bound-state equation for a baryon built from one bosonic antiquark and one fermionic quark, written in the finite-momentum frame. The authors also recover the tetraquark BSE in that frame and solve the equations numerically for the lowest masses and wave functions in both frames. They then boost the states continuously and show the forward component approaching the light-cone wave function while the backward piece disappears. That last numerical check is a concrete consistency test on the formalism they are using. Recovering the known meson and tetraquark results from both Hamiltonian and diagrammatic routes is also useful as a cross-check inside the large-Nc limit. The work sits squarely inside the long line of exactly solvable 2D QCD models and supplies new benchmark spectra for the mixed-statistics case. The main limitation is that only the abstract exists. Without the actual equations, the discretization or basis choices used in the numerics, or any convergence plots, it is impossible to judge whether the FMF derivation introduces extra terms or whether the boosting demonstration is robust. The assumption that the Hamiltonian approach carries over without frame-dependent corrections at finite Nc is stated but untestable here. If the full paper contains clean derivations and reproducible numerics, the new baryon equation would be a modest but usable addition for people who build toy models of exotics. This is narrow-gauge work aimed at specialists already working on light-front or equal-time quantization in two dimensions. A reader who needs reference spectra or wave-function shapes in the extended 't Hooft model could extract something concrete from it. The specific claim of a new BSE plus the boosting test is concrete enough that a serious referee should see the full text rather than desk-reject on the abstract alone.

Referee Report

0 major / 1 minor

Summary. The paper investigates bound-state equations (BSEs) in two-dimensional QCD in the large-Nc limit from both the infinite-momentum frame (IMF) and finite-momentum frame (FMF). It derives BSEs for a tetraquark (bosonic quark-antiquark pair) and a baryon (bosonic antiquark plus fermionic quark) in the extended 't Hooft model using the Hamiltonian approach in light-front and equal-time quantization, recovers known results, presents the first BSEs for the baryon in FMF via the diagrammatic approach, solves the equations numerically for mass spectra and lowest-lying wave functions, and shows that continuous boosting causes the forward component to approach the light-cone wave function while the backward component vanishes.

Significance. If the derivations and numerical solutions hold, the work extends the 't Hooft model to mixed bosonic-fermionic constituents and supplies the first FMF baryon BSEs together with explicit spectra and boosting demonstrations. This could strengthen the case for frame-independent descriptions of exotic states in 2D QCD and provide concrete wave-function data for further analytic or lattice comparisons.

minor comments (1)

The abstract places quotation marks around 'tetraquark', 'baryon', and 'exotic' without defining the precise quantum numbers or color representations; a short clarification in the introduction would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their detailed summary of our work and for recognizing its potential significance in extending the 't Hooft model. The report lists no specific major comments, so we have no points to address point-by-point. We note the 'uncertain' recommendation but would welcome any additional feedback to clarify concerns.

Circularity Check

0 steps flagged

No significant circularity; derivations from Hamiltonian/diagrammatic methods with no reduction shown

full rationale

Only the abstract is available, which states that BSEs are derived via the Hamiltonian approach (light-front and equal-time quantization) and diagrammatic approach, confirming known results and presenting new BSEs for baryons in FMF. No equations, parameters, or self-citations are quoted that reduce any claimed result to a fitted input, self-definition, or author-prior ansatz by construction. The central claims (new BSEs, numerical spectra, boosting behavior) are presented as outputs of standard methods applied to the extended 't Hooft model, with no evidence of the derivation chain collapsing to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard large-Nc limit of 2D QCD and the applicability of the Hamiltonian formalism in both frames; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Large Nc limit of QCD2
Explicitly invoked as Nc to infinity limit throughout the abstract.
domain assumption Validity of Hamiltonian approach for bound-state equations in both IMF and FMF
Used to derive the BSEs for tetraquarks and baryons.

pith-pipeline@v0.9.0 · 5816 in / 1498 out tokens · 36716 ms · 2026-05-23T20:09:19.973683+00:00 · methodology