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REVIEW 2 major objections 4 minor 17 references

Meson spectrum in $QCD_2$ revisited

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that Lorentz-invariant VSR-type terms recently added to QCD2 leave the meson spectrum unchanged: the gluon term vanishes in light-cone gauge, the quark term only shifts the effective quark mass to $\widetilde M_a^2 =…

desk verdict The paper's central claim that the VSR gluon term vanishes in light-cone gauge is wrong—it is exactly the mg regulator—so the 'null result' only survives for the quark term; the finite-mg spectra are physical effects of the new term. read the letter →

arxiv 1908.07660 v3 pith:JFEZJJRS submitted 2019-08-21 hep-th

Jorge Alfaro , Alex Soto This is my paper · ORCID
classification hep-th
keywords QCDintwodimensionsmesonspectrumtHooftmodelVerySpecialRelativitylight-conegaugeinfraredregulatorBethe-SalpeterequationLorentzinvariance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether the Lorentz-invariant Very Special Relativity (VSR) terms that can be added in two spacetime dimensions change the bound-state spectrum of the standard large-N meson model of QCD2. It shows they do not: in light-cone gauge the VSR gluon term drops out entirely, and the VSR quark term survives only as a shift $M_a^2 \to M_a^2 + m^2$ in the effective quark mass. The paper also introduces a gluon mass as a regulator for the infrared divergence and proves that the standard integral equation for the meson spectrum is recovered when that regulator mass goes to zero. If correct, the new terms are spectroscopically invisible in two dimensions; their possible relevance must lie in higher dimensions.

What carries the argument

The load-bearing object is the null vector $n = (1,1)$, which transforms with a phase under two-dimensional Lorentz transformations and therefore permits VSR-style terms. In light-cone coordinates the VSR quark propagator develops a piece proportional to $\gamma^-$, and the coupling vertex is also $i g \gamma^-$; the paper argues that this shared gamma structure makes the new term drop out of every relevant diagram. The remaining machinery is the gluon-mass regulator: the self-energy becomes $\Sigma(p_-) = \frac{g^2}{\pi m_g} \arctan(p_-/m_g)$, and the generalized integral equation (21) contains an $\arctan$ term whose divergence cancels the principal-value regulator exactly, leaving the standard 't Hooft integral equation (26) as the $m_g \to 0$ limit.

What would settle it

Compute the loop contribution of a non-ladder diagram, such as a crossed-ladder or vertex correction, containing one $\gamma^-$ insertion from the VSR quark propagator with the same $\gamma^-$ vertices; a nonzero result would show the cancellation is incomplete and the spectrum shifts beyond the quark-mass redefinition.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the new Lorentz-invariant VSR terms do not modify the meson spectrum of QCD2 at all, except through a redefinition of the quark mass. The VSR gluon mass term vanishes in the light-cone gauge because $n \cdot A = 0$ sets $A_- = 0$, and the VSR quark term does not contribute to the ladder diagrams because its $\gamma^-$ factor is absorbed by the same $\gamma^-$ structure at the gauge vertex. What remains is a corrected denominator mass $\widetilde M_a^2 = M_a^2 + m^2$, so the generalized Bethe-Salpeter equation reduces exactly to the standard 't Hooft equation in the limit where the gluon regulator mass is removed.

Load-bearing premise

The load-bearing premise is that every $\gamma^-$ insertion from the VSR quark propagator cancels against the $\gamma^-$ vertex, so that no VSR piece survives in any diagram.

Editorial extensions

If this is right

  • Meson eigenvalues follow exactly the same integral equation as in the standard model, with quark masses replaced by $\widetilde M_a^2 = M_a^2 + m^2$; any VSR effect is absorbed into a mass shift.
  • A nonzero gluon regulator mass changes the eigenvalues, but as $m_g \to 0$ the arctangent divergence cancels the principal-value cutoff and the standard 't Hooft result is returned.
  • The VSR gluon mass term vanishes in light-cone gauge in two dimensions, so the gluon contribution can only become visible in higher dimensions.
  • With fixed nonzero $m_g$, the endpoint behavior remains $\phi(x) \sim x$ at $x = 0$ and $x = 1$, and the computed spectrum does not show a Regge trajectory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The cancellation argument is checked in the ladder approximation; a natural extension is to test a non-ladder diagram with a single $\gamma^-$ insertion, where a nonzero gamma trace would produce a genuine VSR correction beyond the mass shift.
  • Because the VSR quark term is indistinguishable from an ordinary mass shift, two-dimensional meson spectroscopy cannot constrain the VSR parameter $m$ by itself; only the higher-dimensional generalization of the gluon term could produce distinctive effects.
  • The gluon regulator mass works as a pure infrared cutoff, but the paper's concluding suggestion invites interpreting it as an effective mass whose physical origin a future model might explain.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This manuscript computes the meson spectrum of two-dimensional QCD in the large-N limit, incorporating Lorentz-invariant VSR-like nonlocal terms proposed in the authors' earlier work: a fermionic term proportional to m^2 /n·D and a gauge-field term proportional to m_g^2 (n^α F_{μα})(n·D)^{-2}(n^β F^{μβ}). Working in light-cone coordinates and light-cone gauge, the paper claims that the VSR gluon term vanishes in this gauge, that the VSR quark term only produces a mass shift M̃_a^2 = M_a^2 + m^2, and that the remaining calculation reduces to the standard 't Hooft equation once a gluon mass regulator is removed. The authors derive a generalized integral equation (21), solve it numerically in a sine basis, present eigenvalues for various μ_g and α, and show that in the μ_g→0 limit the 't Hooft equation (26) is recovered.

Significance. If the central claim were correct, the paper would establish a compact and useful null result: the new VSR terms in QCD2 leave the meson spectrum unchanged apart from a renormalization of quark masses. The numerical solution of the generalized integral equation is transparent and the limiting check against the known 't Hooft result is valuable. The quark-sector cancellation is indeed valid in the ladder kernel, since γ^-γ^-=0 kills the VSR numerator between two γ^- vertices. However, the claimed vanishing of the VSR gluon term in light-cone gauge is incorrect on direct evaluation, so the paper's main conclusion is not supported. The calculation still has value as an exercise with a gluon-mass regulator, but the manuscript would need substantial reframing to be correct.

major comments (2)
  1. [Section III, paragraph after Eq. (4)] The claim that the m_g^2 term 'in these coordinates is zero' is not correct. In light-cone gauge with n^+ = √2 and n^- = 0, the condition n·A = 0 sets A_+ = 0 while A_- remains nonzero, and the only nonvanishing field strength is F_{+-} = ∂_+ A_-. Then n^α F_{μα} = √2 F_{-+} = -√2 ∂_+ A_- and n^β F^{μβ} = √2 F^{-+} = √2 ∂_+ A_-, so the quadratic part of the VSR gauge term in Eq. (4) is proportional to m_g^2 (∂_+ A_-)(∂_+^{-2})(∂_+ A_-), i.e., a nonzero A_- mass term, not zero. Consequently Eq. (10) omits a physical m_g-dependent contribution, and the gluon propagator i/((k_-)^2 + m_g^2) used in Eqs. (11) and (17) is precisely the propagator of this VSR gluon mass term. The conclusion in Section VI that the new terms 'do not modify anything, except a different mass for the quark' therefore fails for m_g ≠ 0; the m_g→0 limit in Eq. (26) simply sets the VSR gluon parameter to zero.
  2. [Sections IV and VI, Eqs. (11) and (21)] The manuscript uses the symbol m_g in two incompatible roles. In Eq. (4) it is introduced as the physical VSR gluon mass; after asserting that this term vanishes in light-cone gauge, Eq. (11) reintroduces 'a mass term for the gluon as a regulator' with the same symbol. If the Section III assertion were correct, the regulator would be external to the VSR theory and Table II would say nothing about VSR effects; if, as shown above, the assertion is false, then the regulator is the VSR gluon mass itself and the eigenvalue dependence on μ_g in Table II is a genuine modification of the theory. Either reading contradicts the abstract's statement that the new VSR terms do not affect the meson spectrum.
minor comments (4)
  1. [Section III, LCG paragraph] The sentence 'In the LCC it means A_- = A_+ = 0' is confusing: the gauge condition n·A = 0 fixes A_+ = 0, while A_- must be nonzero for F_{+-} = ∂_+ A_- and for the propagator i/(k_-)^2. Please correct this typo.
  2. [Section IV, Eq. (24)] Equation (24) is not an identity as printed: the right-hand side differs from the left by the term 2φ/λ. This appears to be a compressed pole subtraction; please rewrite the relation explicitly.
  3. [Section III, Feynman rules paragraph] The statement that the VSR numerator in the quark propagator decouples because the vertex has the same γ matrix is correct for the ladder kernel, since γ^-γ^- = 0, but a one-line gamma-algebra demonstration would make the argument easier to verify.
  4. [Abstract and Conclusions] The abstract and conclusions should clarify whether m_g is a technical regulator or the VSR gluon mass; the current text conflates the two, and the phrase 'these new terms does not affect' should be 'do not affect'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity is present: the central null result follows from a gamma-algebra cancellation and a coordinate-gauge assertion, benchmarked against 't Hooft's independent equation in the mg→0 limit.

full rationale

The derivation chain is self-contained. The VSR phase property of the null vector n=(1,1) is re-derived in Section II (Λn=e^θ n), so the citation to the authors' prior work [8] is not load-bearing. The quark VSR numerator is removed by the identity γ^-γ^-=0 at the vertices, leaving the denominator mass shift M_ea^2=M_a^2+m^2; this is a derived cancellation, not a fitted input. The generalized integral equation (21) is then solved numerically, and the mg→0 limit is shown to reduce Eq. (21) to 't Hooft's Eq. (26), an independent external benchmark; no parameter is tuned to force that agreement. The only unsupported assertion is 'we notice the term with m_g^2 in these coordinates is zero' (Section III, after Eq. (4)). This premise is not derived and, if false, would undermine the null result, but the conclusion 'these new terms do not modify anything' is not used to justify the premise. It is therefore a correctness risk rather than a circular reduction. The paper's self-citations ([8], [12], [15], [16]) define the VSR framework but do not carry the load of the calculation. No circular step is exhibited.

Assumptions & free parameters 2 free parameters · 4 assumptions · 2 invented entities

The central claim rests on the VSR framework inherited from the authors' prior papers (the phase transformation of n=(1,1)), on a gauge choice that eliminates the VSR gluon term, on the gamma-matrix cancellation that removes the VSR quark term from ladder diagrams, and on a principal-value regularization to recover the 't Hooft limit. No parameters are fitted to data; m and m_g are theoretical inputs, and the only numerical tuning is basis truncation N=60 with a one-point convergence check.

free parameters (2)
  • VSR quark mass parameter m = not fitted
    Introduced in Lagrangian (4); enters only through M~^2 = M^2 + m^2 in the quark propagator, so its effect is a redefinition of the quark mass. No data are used to set it.
  • Gluon regulator mass m_g (dimensionless μg) = 0 in the physical limit; illustrative values 0.25, 0.5, 0.75, 1
    Added in Eq. (11) as an infrared regulator. The central claim is the m_g→0 limit recovers 't Hooft; finite values are used only to study the generalized equation numerically.
assumptions (4)
  • domain assumption In two dimensions, the null vector n=(1,1) transforms with a phase under proper Lorentz transformations, and VSR-like ratios built from n·(...) are Lorentz invariant.
    Section II, Eqs. (2)-(3), relying on the authors' prior work [8]. If this property fails, the new terms in Lagrangian (4) are not Lorentz invariant and the whole setup collapses.
  • standard math In light-cone gauge, the nonlocal gluon VSR mass term (n^α F_{μα})(n·D)^{-2}(n^β F^{μβ}) vanishes identically.
    Section III: 'the term with m_g^2 in these coordinates is zero.' This removes the gluon VSR term from all subsequent diagrams.
  • domain assumption The gamma^- piece of the VSR quark propagator does not contribute to the diagrams because the gauge vertex is also gamma^-.
    Section III: 'Since the vertex has the same gamma matrix, this new term will not contribute.' This is the key structural input for the null result; no explicit gamma-algebra proof is given.
  • standard math The singular kernel in the mg→0 limit is regularized by a principal-value split with λ=2mg/π that cancels the divergent arctan term.
    Section IV, Eqs. (23)-(26). This is the standard 't Hooft principal-value prescription, though written with some notational looseness.
invented entities (2)
  • VSR nonlocal quark mass term (i m^2/2) nslash/(n·D)
    purpose: Introduces a Lorentz-invariant mass-like correction to the quark propagator.
    No falsifiable prediction is attached to m in this paper; the term's only effect is the effective mass shift M~^2=M^2+m^2, which is absorbed into the quark mass.
  • VSR nonlocal gluon mass term (m_g^2/2)(n^α F_{μα})(n·D)^{-2}(n^β F^{μβ})
    purpose: Introduces a gauge-invariant gluon mass term.
    The term vanishes in light-cone gauge for this computation, and the paper provides no independent observable for it.

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Pith. "Pith review of Meson spectrum in $QCD_2$ revisited." pith.science (2026). https://pith.science/paper/JFEZJJRS

@misc{pith2026190807660,
  author       = {Pith},
  title        = {Pith review of: Meson spectrum in $QCD_2$ revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JFEZJJRS}},
  note         = {Machine review of arXiv:1908.07660}
}
abstract

Recently it has been shown that in two dimensions is possible to add new Lorentz invariant terms built with fractions containing the null vector $n= (1, 1)$. In this work, we have computed the meson spectrum following the 't Hooft model in $QCD_2$ incorporating these new kinds of terms. We found these new terms does not affect the meson spectrum. We have computed the 't Hooft model with a new regulator. We have introduced a gluon mass and we have recovered the 't Hooft result when this parameter is set to zero.

Figures

Figures reproduced from arXiv: 1908.07660 by the authors.

Figure 1
Figure 1. FIG. 1: Bethe-Salpeter equation in diagram. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Plot for the different values of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Plot of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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