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arxiv: 2605.07095 · v1 · submitted 2026-05-08 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Baryon Bethe-Salpeter Equation in Minkowski-Space QCD₂

Satvir Kaur , Sreeraj Nair , Chandan Mondal , Jiangshan Lan , Xingbo Zhao , J. P. B. C. de Melo , Tobias Frederico
Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:01 UTC · model grok-4.3

classification ✦ hep-ph
keywords Bethe-Salpeter equationQCD in two dimensionsbaryon bound stateslight-cone gaugevalence truncationRegge trajectoryparton distribution functions
0
0 comments X

The pith

The three-quark Bethe-Salpeter equation in two-dimensional QCD reduces to the Bars-Durgut equation at leading order in the valence truncation.

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

The paper develops a Minkowski-space formulation of the three-quark ladder Bethe-Salpeter equation for baryons in QCD in two dimensions, formulated in the light-cone gauge. Application of the quasi-potential expansion followed by light-front projection produces a mass-squared eigenvalue equation that matches the earlier Bars-Durgut result exactly when only the leading valence term is kept. Numerical solution of this reduced equation for three colors yields a ground-state mass consistent with prior light-cone quantization work and an excited-state spectrum that follows a Regge trajectory resembling the nucleon spectrum. The framework also supplies endpoint power-law behavior for the valence wave function and selected observables such as parton distributions.

Core claim

Using the quasi-potential expansion of the ladder Bethe-Salpeter equation in light-cone gauge, the mass-squared eigenvalue problem for the three-quark system in QCD2 is shown to be equivalent to the Bars-Durgut equation when truncated to the leading valence sector. Numerical solution for Nc=3 gives a ground-state mass in reasonable agreement with light-cone quantization results, and the excited states follow a Regge trajectory that tracks the experimental nucleon spectrum trend.

What carries the argument

The quasi-potential expansion of the three-quark ladder Bethe-Salpeter equation, truncated at leading order in the valence sector after light-front projection, which converts the Minkowski-space bound-state equation into a mass-squared eigenvalue problem.

If this is right

  • The ground-state baryon mass agrees with previous light-cone quantization results, indicating the valence sector dominates the ground state.
  • The excited-state spectrum produces a Regge trajectory that follows the overall trend of the nucleon spectrum.
  • The valence wave function obeys an endpoint power-law behavior fixed by the quark mass and the coupling, paralleling the 't Hooft meson analysis.
  • Structure observables including parton distribution functions, double distribution amplitudes, and coordinate-space densities follow directly from the numerical solutions.

Where Pith is reading between the lines

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

  • The demonstrated equivalence offers a controlled setting to test whether higher-order terms in the quasi-potential expansion remain small for ground states in related confining models.
  • The same reduction procedure could be applied to other lower-dimensional gauge theories to isolate the contribution of valence versus higher Fock components.
  • Numerical techniques developed here for solving the reduced eigenvalue problem may transfer to Minkowski-space formulations of baryons once systematic truncations are available in 3+1 dimensions.

Load-bearing premise

The quasi-potential expansion truncated at leading order in the valence sector captures the dominant physics of the ground-state baryon.

What would settle it

A direct numerical solution of the full three-quark Bethe-Salpeter equation in QCD2 without the valence truncation that produces a ground-state mass differing substantially from the truncated result.

Figures

Figures reproduced from arXiv: 2605.07095 by Chandan Mondal, Jiangshan Lan, J. P. B. C. de Melo, Satvir Kaur, Sreeraj Nair, Tobias Frederico, Xingbo Zhao.

Figure 1
Figure 1. Figure 1: Diagrammatic representation of the three-quark BSE in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic iteration of the LO valence equation. Left: an [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Endpoint exponent s as a function of the ratio m/g, ob￾tained from the transcendental Eq. (9), where m and g are the mass of quark and the strength of confinement, respectively. 4. Numerical method The LO equation for the valence baryon LF wave func￾tion, Eq. (6), is analogous in form to the ’t Hooft equation for mesons. A key difference is that, unlike the meson case in the large-Nc limit, this equation a… view at source ↗
Figure 4
Figure 4. Figure 4: Ground-state baryon mass in SU(3) QCD2. The dashed curve is an interpolation of the light-cone quantization results re￾ported in Ref. [2], while the solid curve shows our results obtained by varying both m and gSU(3). 5. Baryon phenomenology We solve the eigenvalue equation for the three-quark Hamiltonian, Eq. (13), using the basis expansions defined in Eqs. (10) and (12). The calculation is performed with… view at source ↗
Figure 5
Figure 5. Figure 5: Regge trajectory of the squares of the nucleon mass spec [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Top: Distribution amplitudes of the nucleon states: N(939) (left), N(1440) (middle) and N(1710) (right). Bottom: Co-ordinate [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
read the original abstract

We study the three-quark ladder Bethe--Salpeter equation in Minkowski-space QCD$_2$ in the light-cone gauge. Using the quasi-potential expansion, we project the baryon equation onto the light front and show that, at leading order in the valence truncation, the resulting mass-squared eigenvalue equation is equivalent to the Bars--Durgut equation. We also derive the endpoint power-law behavior of the valence wave function in terms of the quark mass and coupling, closely paralleling the original 't Hooft analysis for mesons. The resulting three-quark equation is solved numerically for $N_c=3$, and the ground-state baryon mass is found to be in reasonable agreement with previous light-cone quantization results in QCD$_2$, suggesting that the valence sector provides the dominant contribution to the ground state. The excited-state spectrum further yields a Regge trajectory that captures the overall trend of the experimental nucleon spectrum, and we compute selected structure observables, including parton distribution functions, double distribution amplitudes, and coordinate-space densities. This framework provides a useful confining test bed for Minkowski-space bound-state methods and for future developments toward confining formulations in 3+1 dimensions beyond the valence truncation.

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.

Referee Report

0 major / 3 minor

Summary. The paper studies the three-quark ladder Bethe-Salpeter equation in Minkowski-space QCD_2 in the light-cone gauge. Using the quasi-potential expansion, it projects the baryon equation onto the light front and demonstrates that at leading order in the valence truncation, the resulting mass-squared eigenvalue equation is equivalent to the Bars-Durgut equation. It derives the endpoint power-law behavior of the valence wave function, solves the equation numerically for N_c=3 obtaining ground-state mass in reasonable agreement with prior light-cone quantization results, and computes the excited-state spectrum showing a Regge trajectory, along with structure observables such as parton distribution functions, double distribution amplitudes, and coordinate-space densities.

Significance. If the results hold, this provides a useful confining test bed for Minkowski-space bound-state methods. The explicit equivalence to the Bars-Durgut equation and the parallel derivation of the endpoint power-law behavior with the meson case are strengths. The numerical agreement with independent prior results supports the approach, and the computation of structure observables adds value for applications in bound-state physics.

minor comments (3)
  1. [Abstract] The phrase 'reasonable agreement' for the ground-state mass should be quantified with the actual numerical values and the difference from previous results to allow readers to assess the level of agreement.
  2. [Numerical results] The Regge trajectory is said to capture the 'overall trend' of the experimental nucleon spectrum; this comparison should be qualified as qualitative due to the model being in 1+1 dimensions versus real-world 3+1 dimensions.
  3. Check for consistency in notation between the quasi-potential expansion and the valence truncation throughout the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript on the baryon Bethe-Salpeter equation in Minkowski-space QCD_2. We are pleased that the equivalence to the Bars-Durgut equation under valence truncation, the derivation of the endpoint power-law behavior, the numerical agreement with prior light-cone quantization results, and the computation of structure observables are recognized as strengths. The recommendation for minor revision is noted, and we will incorporate improvements to clarity and presentation in the revised version.

Circularity Check

0 steps flagged

No significant circularity; equivalence derived by explicit projection and truncation

full rationale

The paper's central derivation projects the three-quark ladder BSE onto the light front via quasi-potential expansion and shows equivalence to the Bars-Durgut equation at leading valence order through explicit truncation steps. The resulting eigenvalue equation is solved numerically for masses and wave functions, with results compared to independent prior light-cone quantization calculations rather than fitted to them. Endpoint power-law behavior is derived in parallel with 't Hooft's meson analysis using the same coupling and mass parameters. No load-bearing step reduces to a self-citation chain, a fitted parameter renamed as prediction, or an ansatz smuggled via prior work by the same authors. The framework is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the light-cone gauge choice, the quasi-potential expansion, and the valence truncation; these are standard domain assumptions in light-front QCD rather than new postulates.

axioms (2)
  • domain assumption Light-cone gauge for the gluon propagator in QCD2
    Simplifies the interaction kernel and is invoked when projecting the Bethe-Salpeter equation.
  • ad hoc to paper Leading-order valence truncation in the quasi-potential expansion
    Restricts the Fock space to the three-quark component and is the key step that produces equivalence to the Bars-Durgut equation.

pith-pipeline@v0.9.0 · 5541 in / 1477 out tokens · 63434 ms · 2026-05-11T01:01:32.000737+00:00 · methodology

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Reference graph

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