pith. sign in

arxiv: 2606.08489 · v2 · pith:LYT6TS3Pnew · submitted 2026-06-07 · ✦ hep-ph · nucl-th

Principles and Possibilities for Bound States in Gauge Theory

Paul Hoyer This is my paper

Pith reviewed 2026-06-29 05:36 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords bound statesQCDperturbative methodsconfining potentialGauss lawchiral symmetry breakingtemporal gaugeinstantaneous potential
0
0 comments X

The pith

Specifying a non-vanishing boundary condition on the longitudinal gauge field at infinity in Gauss' law generates an instantaneous confining potential for color-singlet states in QCD.

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

The paper presents a perturbative method for bound states in QED and QCD that starts from canonical quantization in temporal gauge with A^0 fixed to zero. Fully imposing Gauss' law on physical states requires each quark or gluon to be accompanied by a longitudinal gauge field component that sets an instantaneous potential between them. In QCD a non-vanishing boundary condition on this longitudinal field at spatial infinity produces a confining potential that acts only on color-singlet quark-antiquark pairs. Once this potential is included, hadrons become calculable in perturbation theory, and at vanishing quark mass a zero-energy scalar state appears that can mix with the vacuum and break chiral symmetry spontaneously.

Core claim

Bound states differ from scattering yet are not covered in textbooks on Quantum Field Theory. A perturbative method for QED and QCD is discussed based on canonical quantization. Fully fixing temporal gauge A^0=0 imposes Gauss' law on physical states. This implies that quark and gluon states include a longitudinal gauge field A_L which determines the instantaneous bound state potential. An instantaneous confining potential arises for color singlet q qbar states when a non-vanishing boundary condition on A_L^a(x to infinity) is specified in Gauss' constraint. As suggested by Gribov, alpha_s(Q^2) may freeze at a perturbative value when the confining potential dominates. Hadrons can then be calc

What carries the argument

The longitudinal gauge field A_L required by Gauss' law after temporal gauge fixing, whose non-vanishing boundary condition at infinity supplies the instantaneous confining potential for color singlets.

If this is right

  • Hadrons become calculable in ordinary perturbation theory once the confining potential is active.
  • At zero quark mass a zero-energy 0++ state mixes with the vacuum and spontaneously breaks chiral symmetry.
  • The strong coupling may freeze to a finite perturbative value in the infrared once the confining potential dominates.
  • The same Gauss-law mechanism supplies an instantaneous potential for bound states in QED.

Where Pith is reading between the lines

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

  • The approach could be tested by deriving explicit wave functions for the lowest pseudoscalar and vector mesons and comparing their decay constants to data.
  • If the boundary condition is relaxed or altered, the method should recover ordinary Coulombic QED bound states without confinement.
  • The zero-energy scalar state at vanishing mass offers a concrete perturbative handle on the QCD vacuum that could be compared with other order parameters.
  • The framework suggests that gauge fixing choices can encode long-range effects without introducing new degrees of freedom.

Load-bearing premise

A non-vanishing boundary condition on A_L^a at spatial infinity must be chosen in Gauss' constraint to produce the confining potential between color-singlet quark pairs.

What would settle it

Explicit perturbative computation of the lowest meson masses or the chiral condensate using the derived instantaneous potential that fails to match known experimental or lattice values.

Figures

Figures reproduced from arXiv: 2606.08489 by Paul Hoyer.

Figure 1
Figure 1. Figure 1: FIG. 1. Sketch of the perturbative expansion for the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

Bound states differ from scattering yet are not covered in textbooks on Quantum Field Theory. I discuss a perturbative method for QED and QCD based on canonical quantization. Fully fixing temporal gauge $A^0(t,\boldsymbol{x})=0$ imposes Gauss' law on physical states. As pointed out by Dirac, this implies that electron states include a longitudinal gauge field $\boldsymbol{A}_L$, which determines the instantaneous bound state potential. The situation is analogous for quarks and gluons in QCD. An instantaneous confining potential arises for color singlet $q\bar q$ states when a non-vanishing boundary condition on $\boldsymbol{A}_L^a(\boldsymbol{x}\to\infty)$ is specified in Gauss' constraint. As suggested by Gribov, $\alpha_s(Q^2)$ may freeze at a perturbative value when the confining potential dominates. Hadrons can then be calculated perturbatively. At vanishing quark mass there is a $j^{PC}=0^{++}$ state with zero energy which can mix with the perturbative vacuum, giving rise to a spontaneous breaking of chiral symmetry.

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

3 major / 0 minor

Summary. The paper proposes a perturbative method for calculating bound states in QED and QCD via canonical quantization in temporal gauge A^0=0. Gauss' law on physical states determines an instantaneous potential through the longitudinal gauge field A_L. For QCD, specifying a non-vanishing boundary condition on A_L^a(x→∞) is claimed to generate a confining potential exclusively for color-singlet q qbar states. This allows alpha_s to freeze at a perturbative value, enabling perturbative hadron calculations. At vanishing quark mass, a j^{PC}=0^{++} state with zero energy mixes with the perturbative vacuum to break chiral symmetry spontaneously.

Significance. If the central derivation of the confining potential from the boundary condition were provided and shown to be non-circular, the approach could offer a novel perturbative framework for hadron spectroscopy and chiral symmetry breaking in QCD, potentially bridging perturbative and non-perturbative regimes as suggested by Gribov. However, the manuscript supplies no explicit steps, equations, or checks for these claims, limiting its current impact.

major comments (3)
  1. [Abstract] Abstract: The claim that a non-vanishing boundary condition on A_L^a(x→∞) in Gauss' constraint produces an instantaneous confining potential for color-singlet states is asserted without any derivation, explicit form of the potential, demonstration of its color dependence, or proof of its instantaneous nature. This step is load-bearing for all subsequent claims about perturbative hadron spectra.
  2. [Abstract] Abstract (Gribov discussion): The assertion that alpha_s(Q^2) freezes when the confining potential dominates is circular within the paper's logic, as the potential itself is defined by the boundary condition chosen precisely to produce linear confinement; no independent dynamical justification or calculation is given.
  3. [Abstract] Abstract: No derivation or explicit construction is provided for the j^{PC}=0^{++} zero-energy state at vanishing quark mass, nor for its mixing with the perturbative vacuum to break chiral symmetry, nor any comparison to known spectra or lattice results.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised identify areas where additional explicit derivations and clarifications are needed to strengthen the presentation. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that a non-vanishing boundary condition on A_L^a(x→∞) in Gauss' constraint produces an instantaneous confining potential for color-singlet states is asserted without any derivation, explicit form of the potential, demonstration of its color dependence, or proof of its instantaneous nature. This step is load-bearing for all subsequent claims about perturbative hadron spectra.

    Authors: We agree that the manuscript asserts this central result without supplying the explicit derivation or checks. The current text is an outline of the proposed framework. In revision we will add a dedicated section that starts from the temporal-gauge Gauss constraint with the specified non-vanishing boundary condition on A_L^a at spatial infinity, derives the resulting instantaneous potential, demonstrates its color-singlet selectivity, and confirms its instantaneous character. revision: yes

  2. Referee: [Abstract] Abstract (Gribov discussion): The assertion that alpha_s(Q^2) freezes when the confining potential dominates is circular within the paper's logic, as the potential itself is defined by the boundary condition chosen precisely to produce linear confinement; no independent dynamical justification or calculation is given.

    Authors: The referee correctly notes that the boundary condition is introduced precisely to produce linear confinement. We view the choice as a physically motivated input (following Gribov) rather than a derived dynamical result. Once imposed, the potential permits alpha_s to remain perturbative at the relevant scales. In revision we will expand the discussion to make this distinction explicit, acknowledge the absence of an independent dynamical derivation of the boundary condition, and clarify that the freezing statement is a consequence within the adopted framework. revision: partial

  3. Referee: [Abstract] Abstract: No derivation or explicit construction is provided for the j^{PC}=0^{++} zero-energy state at vanishing quark mass, nor for its mixing with the perturbative vacuum to break chiral symmetry, nor any comparison to known spectra or lattice results.

    Authors: We agree that the manuscript provides only a brief outline of the zero-energy 0^{++} state and its vacuum mixing. In the revised version we will include an explicit construction of this state in the massless limit, describe the mixing mechanism, and add a short comparison with lattice indications of spontaneous chiral symmetry breaking and the light scalar spectrum. revision: yes

Circularity Check

1 steps flagged

Non-vanishing A_L boundary condition imposed by hand to generate the confining potential, rendering perturbative hadron claims circular by construction

specific steps
  1. self definitional [Abstract]
    "An instantaneous confining potential arises for color singlet qar q states when a non-vanishing boundary condition on A_L^a(x→∞) is specified in Gauss' constraint. As suggested by Gribov, α_s(Q^2) may freeze at a perturbative value when the confining potential dominates. Hadrons can then be calculated perturbatively. At vanishing quark mass there is a j^{PC}=0^{++} state with zero energy which can mix with the perturbative vacuum, giving rise to a spontaneous breaking of chiral symmetry."

    The confining potential is stated to 'arise' precisely when the boundary condition is specified; the BC is introduced for the explicit purpose of generating that potential (and its color-singlet selectivity). The freezing of α_s, perturbative hadron spectra, and zero-energy 0++ state are then direct consequences of this constructed input rather than independent results from the gauge-fixed dynamics.

full rationale

The paper's central derivation begins from temporal gauge fixing and Gauss' law (standard), but the load-bearing step for confinement and all subsequent results (alpha_s freezing, perturbative spectra, 0++ vacuum mixing) is the explicit stipulation of a non-vanishing A_L(x→∞) chosen specifically to produce an instantaneous linear potential for color singlets. This reduces the 'prediction' of confinement and its consequences to the input assumption itself, matching the self-definitional pattern. No independent derivation of the BC from dynamics or gauge invariance is provided, and the color-singlet selectivity is asserted rather than shown to follow necessarily. The remainder of the chain (perturbative hadron calculation, chiral breaking) inherits this circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the standard axioms of canonical quantization plus the additional domain assumption that a non-vanishing boundary condition on the longitudinal gauge field at spatial infinity is admissible and physically realized for color singlets. No new free parameters are introduced numerically, but the boundary condition itself functions as an ad-hoc choice that selects the confining solution. No new particles or forces are postulated.

axioms (3)
  • standard math Temporal gauge A^0=0 can be fully fixed while preserving the physical content of the theory.
    Invoked at the start of the method description.
  • domain assumption Gauss' law constraint on physical states implies inclusion of a longitudinal gauge field component whose form determines the instantaneous potential.
    Follows directly from the gauge choice and is used to generate the bound-state interaction.
  • ad hoc to paper A non-vanishing boundary condition on A_L^a(x→∞) is allowed and produces a confining potential for color singlets.
    Explicitly stated as the condition that yields the linear potential.

pith-pipeline@v0.9.1-grok · 5705 in / 1777 out tokens · 23016 ms · 2026-06-29T05:36:56.164947+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 12 canonical work pages · 9 internal anchors

  1. [1]

    Transverse Gauss operator 5

  2. [2]

    Confinement 7

    Coulomb potential 6 V. Confinement 7

  3. [3]

    Principles and Possibilities for Bound States in Gauge Theory

    Bound state equation 8 VI. Chiral symmetry 9 VII. Discussion 11 Acknowledgments 11 References 11 I. INTRODUCTION Gauge and Poincar´ e invariance are exact symmetries of the QED and QCD actions. The way that Poincar´ e covariance is realized depends on the gauge. The gauge field action, − 1 4 Z dt dxFµνF µν = 1 2 Z dt dx (∂tA+∇A 0)2 −(∇×A T )2 +O A3, A4 (1...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  4. [4]

    transverse Gauss’ operator

    Transverse Gauss operator The QCD action is, with color indices denotedA, B, C, . . .for quarks anda, b, c, . . .for gluons, SQCD = Z d4x − 1 4 F a µνF µν a + ¯ψ(i /∂−m−g /AaT a)ψ F a µν =∂ µAa ν −∂ νAa µ −gf abcAb µAc ν (4.1) The electric field in temporal gauge isE i a =F i0 a =−∂ tAi a is conjugate toA a i =−A i a, with the commutation relations h Ei a...

  5. [5]

    gluon exchange

    Coulomb potential In analogy to QED (3.1) I defineq¯qFock states at rest as |q¯q⟩ ≡ Z dx1dx2 ¯ψA(x1)Φ(x1 −x 2)WAB[AL;x 1,x 2]ψB(x2)|0⟩(4.9) W[A L;x 1,x 2]≡exp h ig 4π Z x1 x2 dyT aAa L(y)·∇ y 1 |y−x 1| − 1 |y−x 2| i (4.10) They-integrals follow the field lines starting atx 2 and ending atx 1. Full gauge fixing requires that the state|q¯q⟩ (4.9) is invaria...

  6. [6]

    IV for an implicit assumption

    Gauss’ constraint Let us reconsider the derivation of Sect. IV for an implicit assumption. The gauge exponentialW(4.10) was inspired by QED (3.2), where it ensures that the electric field vanishes far from the charge. Color confinement in QCD makes this requirement less compelling. The Cornell potential∝V ′r(1.3) increases withr, but is quenched by quark ...

  7. [7]

    string breaking

    Bound state equation Consider again the rest frame|q¯q⟩bound state (4.9) with massM, |q¯q;M⟩ ≡ Z dx1dx2 ¯ψ(x1) Φ(x1 −x 2)Wκ[AL;x 1,x 2]ψ(x2)|0⟩(5.10) AtO α0 s ,i.e., neglecting theO(g) interactions in the Hamiltonian (4.3), the conditionH QCD |q¯q;M⟩=M|q¯q;M⟩ defines the Bound State Equation (BSE) for the wave function Φ(x), i∇· α,Φ(x) +m γ0,Φ(x) = M−V(x)...

  8. [8]

    G. S. Adkins, D. B. Cassidy, and J. P´ erez-R´ ıos, Phys. Rept.975, 1 (2022). 12

    2022

  9. [9]

    P. A. M. Dirac, Can. J. Phys.33, 650 (1955)

    1955

  10. [10]

    J. F. Willemsen, Phys. Rev.D17, 574 (1978)

    1978

  11. [11]

    J. D. Bjorken, Prog. Math. Phys.4, 423 (1979)

    1979

  12. [12]

    N. H. Christ and T. D. Lee, Phys. Rev.D22, 939 (1980)

    1980

  13. [13]

    Leibbrandt, Rev

    G. Leibbrandt, Rev. Mod. Phys.59, 1067 (1987)

    1987

  14. [14]

    Strocchi,An introduction to non-perturbative foundations of quantum field theory(Oxford University Press, 2013)

    F. Strocchi,An introduction to non-perturbative foundations of quantum field theory(Oxford University Press, 2013)

    2013

  15. [15]

    Hoyer,Journey to the Bound States, SpringerBriefs in Physics (Springer, 2021) arXiv:2101.06721 [hep-ph]

    P. Hoyer,Journey to the Bound States, SpringerBriefs in Physics (Springer, 2021) arXiv:2101.06721 [hep-ph]

    work page arXiv 2021

  16. [16]

    Hydrogen Atom in Relativistic Motion

    M. J¨ arvinen, Phys. Rev.D71, 085006 (2005), arXiv:hep-ph/0411208 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  17. [17]

    V. N. Gribov, Eur. Phys. J. C10, 91 (1999), arXiv:hep-ph/9902279

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  18. [18]

    Y. L. Dokshitzer and D. E. Kharzeev, Ann. Rev. Nucl. Part. Sci.54, 487 (2004), arXiv:hep-ph/0404216

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  19. [19]

    Eichten, K

    E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T.-M. Yan, Phys. Rev.D21, 203 (1980)

    1980

  20. [20]

    Quarkonia and their transitions

    E. Eichten, S. Godfrey, H. Mahlke, and J. L. Rosner, Rev. Mod. Phys.80, 1161 (2008), arXiv:hep-ph/0701208 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  21. [21]

    Heavy quarkonium: progress, puzzles, and opportunities

    N. Brambillaet al., Eur. Phys. J.C71, 1534 (2011), arXiv:1010.5827 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  22. [22]

    G. S. Bali, Phys. Rept.343, 1 (2001), arXiv:hep-ph/0001312 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  23. [23]

    Chodos, R

    A. Chodos, R. Jaffe, K. Johnson, C. B. Thorn, and V. Weisskopf, Phys. Rev. D9, 3471 (1974)

    1974

  24. [24]

    Weinberg,The quantum theory of fields

    S. Weinberg,The quantum theory of fields. Vol. 2: Modern applications(Cambridge University Press, 2013)

    2013

  25. [25]

    Introduction to Chiral Perturbation Theory

    S. Scherer, Adv. Nucl. Phys.27, 277 (2003), arXiv:hep-ph/0210398

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  26. [26]

    Nefediev, (2025), 10.1016/B978-0-443-26598-3.00086-9, arXiv:2507.22502 [hep-ph]

    A. Nefediev, (2025), 10.1016/B978-0-443-26598-3.00086-9, arXiv:2507.22502 [hep-ph]

    work page doi:10.1016/b978-0-443-26598-3.00086-9 2025

  27. [27]

    Light Hadron Masses from Lattice QCD

    Z. Fodor and C. Hoelbling, Rev. Mod. Phys.84, 449 (2012), arXiv:1203.4789 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  28. [28]

    Gell-Mann, R

    M. Gell-Mann, R. Oakes, and B. Renner, Phys. Rev.175, 2195 (1968)

    1968

  29. [29]

    Hoyer, Phys

    P. Hoyer, Phys. Rev. D111, 074021 (2025), arXiv:2501.18352 [hep-ph]

    work page arXiv 2025