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REVIEW 3 major objections 5 minor 1 cited by

't Hooft model in the temporal gauge

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read In QCD with one spatial dimension and many colors, the equal-time meson wave function, Fourier transformed in the infinite-momentum frame, satisfies the 't Hooft bound-state equation with the quark momentum fraction integrated over the…

desk verdict A serious but conditional challenge to the 't Hooft model; the negative-energy components may be an artifact of the order of limits in the IMF Fourier transform. read the letter →

arxiv 2501.18352 v2 pith:LKNUZRCE submitted 2025-01-30 hep-ph nucl-th

Paul Hoyer This is my paper · ORCID
classification hep-phnucl-th
keywords tHooftmodelQCDin1+1dimensionstemporalgaugelargeNclimitboundstatewavefunctionnegativekineticenergyinfinitemomentumframestringbreaking
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 tries to establish that the 't Hooft model—QCD in one spatial dimension with a large number of colors—has meson wave functions with quark and antiquark components of negative kinetic energy, and that these components survive in the infinite-momentum frame. Starting from equal-time color-singlet $q\bar q$ states in temporal gauge and avoiding quark and gluon propagators, the author derives an analytic wave function whose Fourier transform satisfies the 't Hooft equation, but with the quark momentum fraction integrated from $-\infty$ to $\infty$ rather than from 0 to 1. If this is right, the standard solutions of the 't Hooft model are incomplete: the same potential that binds the quark and antiquark can create pairs, so mesons overlap with multi-meson states and acquire widths of order $g^2N_c$ even in the $N_c\to\infty$ limit. The larger point is that confinement physics in the simplest soluble gauge theory can be studied directly through bound states, without propagators, and may carry over to a confinement mechanism in 3+1 dimensions.

What carries the argument

The machinery is the equal-time, color-singlet $q\bar q$ wave function $\Phi^{(P)}(x)$ built with a gauge link, whose linear potential $V(x)=V'|x|$ makes the bound-state equation (2.13) solvable by a confluent hypergeometric function: $\phi_1(\tau_P)\propto V'\tau_P e^{-i\tau_P/4}{}_1F_1(1-im^2/2V',2,i\tau_P/2)$, with $\tau_P=(E_P-V)^2-P^2$. A local boost parameter $\zeta_P(x)$ packs all frame dependence into a spin rotation, and the Fourier transform in the infinite-momentum frame uses an integral representation of ${}_1F_1$ to produce a closed form for $\phi^{(\infty)}(x_B)$. Energy projection onto quark and antiquark spinors of both kinetic-energy signs then turns the potential into the convolution in (5.15), where the $\Theta$ functions mix positive- and negative-energy components.

What would settle it

Evaluate the momentum-space wave function at $x_B=1.2$ using a convergence factor $e^{-\epsilon u}$ in the Fourier integral (5.7) and take $\epsilon\to0^+$; if the value is zero or depends on the regularization, the negative-energy components are an artifact.

Watch

Extended reading notes

Core claim

The central claim is that equation (5.15) is the infinite-momentum-frame bound-state equation of QCD$_2$ in the $N_c\to\infty$ limit, with the same singular kernel as 't Hooft's equation (5.16) but with the integration over the quark momentum fraction $y_B$ running from $-\infty$ to $\infty$ instead of 0 to 1. The negative-kinetic-energy components that produce the extended integration are present in the rest frame as well, and they do not vanish under boosts. Consequently the usual boundary condition $\phi(x_B)\to0$ at $x_B\to0,1$ is not a property of the wave function; rather, the wave function takes nonzero values at momentum fractions outside the physical region, which gives nonvanishing overlaps between different bound states and suggests string breaking and meson widths of order $g^2N_c$.

Load-bearing premise

The derivation treats as a discrete physical bound state a wave function that is not globally normalizable—it oscillates with undamped amplitude at large separation—and defines the momentum-space wave function by analytically continuing the Fourier transform; if that continuation is not legitimate, the full-line equation and its negative-energy components are artifacts.

Editorial extensions

If this is right

  • The usual 't Hooft equation restricted to $0<x_B<1$ does not give the complete wave function; negative-kinetic-energy components at $x_B<0$ and $x_B>1$ are required by the equal-time solution and survive boosts.
  • Because the binding potential is of order $g^2N_c$, the pair production that generates overlaps between different bound states is not suppressed at large $N_c$, so mesons acquire finite widths of order $g^2N_c$.
  • The wave function is not globally normalizable and oscillates with undamped amplitude at large separation, so the bound-state spectrum has features analogous to relativistic electron states in a linear potential, including negative-energy (positron-like) components.
  • Including a homogeneous solution of Gauss' law in the temporal gauge adds a linear confining potential with a scale $\Lambda$ and an isotropic vacuum energy density, which in 3+1 dimensions suggests a confining instantaneous potential and a bag constant.
  • The Fourier-transformed wave function does not vanish at $x_B\to0$ and $x_B\to1$, in contrast to the boundary behavior assumed in the original 't Hooft solution.

Reading between the lines

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

  • Beyond the paper: if (5.15) is correct, the 't Hooft meson spectrum should be reinterpreted as resonances rather than stable poles; their widths could be computed from the overlap integrals and compared with lattice spectra of QCD$_2$.
  • Beyond the paper: computing the full-line wave function for excited states and different quark masses would test how the negative-energy support grows; the paper only illustrates one ground state with a $\sim 1/x_B^2$ tail.
  • Beyond the paper: the homogeneous Gauss-law solution suggests computing the static $q\bar q$ potential in 3+1 dimensions in the same formalism and comparing its slope with lattice determinations.
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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

3 major / 5 minor

Summary. The paper derives an equal-time q\bar q bound-state wave function for QCD_2 in temporal gauge in the N_c→∞ limit. The bound-state equation (2.13) is solved analytically by a confluent hypergeometric function whose frame dependence is carried by a quadratic variable τ_P(x) (Eqs. (3.9)–(3.14)). The central new claim is that the infinite-momentum-frame Fourier transform of this wave function satisfies an equation of the 't Hooft form, but with the momentum-fraction integral extended over the whole real line, producing negative kinetic-energy components for x_B<0 and x_B>1. The paper further argues that these components survive boosts, that they induce overlaps between bound states suggestive of string breaking, and that they give the bound states widths of O(g^2N_c). The manuscript also proposes a homogeneous solution of Gauss' law that adds a bag-constant-like energy density and suggests a mechanism for confinement in D=3+1.

Significance. If the negative-energy extension of the 't Hooft equation were established, the result would be significant: it would imply that the standard 't Hooft solution is incomplete, that the equal-time wave function contains negative kinetic-energy components even in the infinite-momentum frame, and that leading-1/N_c mesons have nonvanishing widths. The paper's strengths are its explicit analytic wave function, the equal-time derivation that avoids singular quark propagators, and the numerical check in Table I showing that the regularized φ^(∞) satisfies Eq. (5.15) at selected x_B values. However, the physical validity of the central claim hinges on a mathematically delicate issue: the coordinate wave function is not globally normalizable (Sec. III.C), and the momentum-space wave function is defined through a specific iε analytic continuation of a divergent Fourier integral. The paper does not establish that this regularization is unique or that it corresponds to the physical large-P limit; consequently the negative-energy components, the width estimate, and the D=3+1 confinement proposal rest on assumptions that need to be explicitly justified.

major comments (3)
  1. [V.A] The definition of the IMF momentum-space wave function φ^(∞)(x_B) is obtained by an iε prescription applied to the Fourier integral of a coordinate-space function that is not globally normalizable and oscillates as exp(iV′x²/4) at large |x| (Eq. (3.15)). In Eq. (5.1) the P→∞ limit is taken before the Fourier transform, and this order of limits discards stationary-phase contributions from the oscillatory tail that are of order one at finite large P. A different regularization, for example damping the tail first and then taking P→∞, could produce a φ^(∞) supported only on 0<x_B<1, which would eliminate the negative-energy components and invalidate Eq. (5.15) as a statement about the physical wave function. The paper must justify that the iε prescription is the correct large-P limit of the finite-P state, or show that the result is independent of the regularization; otherwise Eq. (5.15) describes only an auxiliary regularized object.
  2. [III.C/VII] Because the wave function (3.14)–(3.15) has constant-amplitude oscillations at large |x|, the states (1.6) are not elements of the physical Hilbert space. The discrete mass spectrum is inferred from continuity at x=0 and regularity of φ_2,3 at τ_P=0 (Sec. III.B), but no condition at infinity can select discrete eigenvalues for an oscillatory solution of the second-order equation (3.10). Consequently the overlap ⟨B C|A⟩ invoked in Sec. VII is ill-defined, and the claimed O(g²N_c) widths are not a consequence of the presented calculation. A limiting definition of the state space (for example, through wave packets or a box normalization) is needed before these physical conclusions can be drawn.
  3. [III.B/Table I] The numerical example fixes M=2.51954√V′ (Fig. 1) rather than determining it from Eq. (5.15); Table I verifies that the regularized φ^(∞) solves Eq. (5.15) at selected x_B values. This is a useful consistency check, but it does not establish the discrete mass spectrum within the temporal-gauge formalism, since M is imported from earlier work [13,15] and the regularity conditions are imposed on a non-normalizable solution. The paper should either derive the spectrum within the present framework or identify M as an input from previous results.
minor comments (5)
  1. [§IV.A, text before Eq. (4.6)] The sentence 'according to the sign of the of the quark' contains a duplicated article and should be corrected.
  2. [§VII, paragraph on D=3+1] 'vith' should read 'with'.
  3. [§VI.B, Eqs. (6.13)–(6.17)] The homogeneous solution introduces a free parameter Λ with no constraint from the equations of motion; the D=3+1 confinement mechanism should be presented as an ansatz rather than as a derivation, since Eq. (6.16) fixes κ only by fiat.
  4. [§II.C, Eq. (2.11)] The statement that the quark-loop contribution from the −gψ†α¹A_aT^aψ term is suppressed as g→0 in the N_c→∞ limit is the only place where large-N counting enters; a few lines of explanation with the standard g²N_c scaling would make the argument self-contained.
  5. [Table I and Fig. 1] The numerical integration scheme, the treatment of the principal-value singularity in Eq. (5.15), and the error estimates are not described; a brief description would facilitate reproduction of the check.

Circularity Check

1 steps flagged · score 4.0 of 10

Core QCD2 result is self-consistent, but the D=3+1 confinement mechanism reduces to a hand-added homogeneous solution, with key wave-function input cited from the author's own earlier work.

  1. ansatz smuggled in via citation [Section VI B, Eqs. (6.13)-(6.17), and Section VII.]
    "The normalization κ may depend on the state |M, P⟩ that Ea(x) acts on. Linearity in x ensures that the homogeneous (sourceless) part of Ea(x) is independent of x. ... Defining the universal constant Λ by κ ≡ Λ2/(gCF) 1/|xq − x¯q| EΛ = Λ4/(2g2CF) ... gives ... HV |q(xq)¯q(x¯q)⟩ = (Λ2 + 1/2 g2CF)|xq − x¯q| |q(xq)¯q(x¯q)⟩. Inclusion of the homogeneous solution to Ea(x) (6.13) added Λ2 to the slope V′ of the linear potential."

    Equation (6.17) is not derived from QCD: it follows solely from choosing the homogeneous solution (6.13), whose free normalization κ is renamed as the new constant Λ. The paper then presents this same linear term as a confinement mechanism in Sec. VII ('This allows a perturbative expansion in g2, which at lowest order gives all the previous results with V′ = Λ2') and cites the author's own [13] for the D=3+1 analogue. The conclusion 'confinement from Λ' is exactly the input assumption 'add a sourceless linear-potential term with strength Λ'. A chosen homogeneous solution cannot be both the premise and the derived mechanism; this is an ansatz presented as a derivation.

full rationale

The main QCD2 derivation is not circular: the bound-state equation (2.13) is obtained from the temporal-gauge Hamiltonian, the mass M is fixed by continuity and regularity conditions rather than by fitting Eq. (5.15), and Table I numerically verifies that the Fourier-transformed wave function satisfies (5.15). The negative-energy components outside 0<x_B<1 follow from the analytic form of the wave function and the i-epsilon-regulated Fourier transform, not from a fitted parameter. That part of the paper has independent content. However, two inputs weaken the overall independence. First, the analytic wave function (3.14) and its large-separation behavior (3.15) are imported from the author's earlier works [13,15]; this is standard special-function material that can be checked, so it reduces but does not eliminate independence. Second, and more importantly, the D=3+1 confinement 'mechanism' in Sec. VI B and VII is constructed by explicitly adding a homogeneous solution with arbitrary normalization κ, redefining it as Λ, and then presenting the resulting linear potential as a prediction, with [13] cited for the same construction. That step is circular-by-construction. Because the core two-dimensional result stands independently, the overall score is moderate rather than high.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The main 1+1 calculation uses only the model inputs m and V' plus boundary conditions to determine the mass M. The most significant additional input is the new constant Lambda introduced via a homogeneous solution of Gauss' law, which is an ad hoc addition with no independent support. The axioms reflect standard large-Nc and gauge-quantization assumptions plus the explicit analytic continuation of the Fourier transform.

free parameters (1)
  • Lambda = not determined
    Introduced in Eq. (6.16) as a universal constant fixing the homogeneous solution of Gauss' law; it sets the additional linear potential slope in the D=3+1 proposal and has no independent determination.
assumptions (5)
  • domain assumption Gauss' law constraint Ga(t,x)|phys>=0 is the correct condition for physical states in temporal gauge.
    Sec. I, Eq. (1.5); needed to select physical states and to derive the linear potential from E^2 commutators.
  • domain assumption Large Nc limit with Nc to infinity at fixed g^2 Nc, with neglect of quark-loop and non-planar contributions.
    Sec. I and Eq. (2.11); used to drop the quark self-energy term in the Hamiltonian commutator, following 't Hooft and Witten.
  • domain assumption Free-field spinor basis can be used to define positive and negative kinetic energy projections of the interacting wave function.
    Sec. IV A, Eqs. (4.4)-(4.6); the bound state is expanded in free spinors, an assumption about the validity of the Fock basis for a confining state.
  • ad hoc to paper The Fourier transform of the non-normalizable wave function is defined by analytic continuation xB to xB +/- i epsilon.
    Sec. V, Eq. (5.7)-(5.8); this regularization of a formally divergent integral is needed to obtain the momentum-space BSE.
  • ad hoc to paper A homogeneous solution of Gauss' law with constant Lambda may be added for color singlet states, preserving Poincare covariance.
    Sec. VI B, Eqs. (6.13)-(6.17); this is the basis for the D=3+1 confinement suggestion and is not derived from QCD dynamics.
invented entities (1)
  • Isotropic vacuum gluon field energy density E_Lambda (bag constant) from homogeneous solution for Ea
    purpose: Generate an additional linear confining potential with slope Lambda^2 in QCD and in D=3+1 dimensions.
    Introduced in Sec. VI B, Eq. (6.16); no observable outside the model is predicted and Lambda is a free parameter chosen by hand.

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Cite this review

Pith. "Pith review of 't Hooft model in the temporal gauge." pith.science (2026). https://pith.science/paper/LKNUZRCE

@misc{pith2026250118352,
  author       = {Pith},
  title        = {Pith review of: 't Hooft model in the temporal gauge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LKNUZRCE}},
  note         = {Machine review of arXiv:2501.18352}
}
abstract

I consider QCD$_2$ in the $N_c \to \infty$ limit at fixed $g^2N_c$. The derivation starts from equal-time $q\bar q$ bound states in coordinate space and temporal ($A^0=0$) gauge, avoiding the use of quark and gluon propagators. The wave function is given analytically by a $_1F_1$ function with an explicit frame dependence. In the infinite momentum frame the Fourier transformed wave function satisfies the 't~Hooft equation, however with contributions also from quarks with negative kinetic energy. Such contributions are present also in the rest frame, and do not vanish under boosts.

Figures

Figures reproduced from arXiv: 2501.18352 by the authors.

Figure 1
Figure 1. (right) shows the IMF (P → ∞) wave function ϕ (∞) (xB) = ϕ (∞) (1 − xB) (5.8) as a function of the quark momentum fraction xB = q/P. The quark momentum ranges over −∞ < q < ∞ at any P (4.3), and thus also −∞ < xB < ∞. The deep inelastic scattering (DIS) cross section is described by |ϕ (∞) (0 < xB < 1)| 2 , with xB identified as the Bjorken variable [13, 15]. In analogy to Dirac wave functions the strong potential g… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Functional renormalization of QCD in $1 + 1$ dimensions: four-fermion interactions from quark-gluon dynamics

    hep-ph 2024-12 conditional novelty 7.0 of 10

    First functional renormalization group study of two-dimensional QCD with a mass-like regulator, producing flow equations for the gauge coupling, quark mass, and a Fierz-complete set of four-fermion interactions.

Reference graph

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