Completing the tensor gauge field (strong gravity) model of quark confinement: SU(3) color, dynamical quark sources, mode structure, horizons, and quantum consistency

In 1978 Salam and Strathdee proposed that hadrons are de Sitter "microuniverses": solutions of an Einstein-type equation for a strongly coupled tensor (vierbein) field $f_{\mu\nu}$ with a large "cosmological" constant, in which the spectrum of a test quark is discrete and essentially that of a harmonic oscillator, so that quarks are confined with no dissociation continuum. The paper itself acknowledges that this classical prototype is far from a realistic theory of confinement: the color-carrying solutions were constructed only for SU(2) with a fixed isospin direction, while the realistic SU(3) case (whose color charge points along two independent Cartan directions, $\lambda_3$ and $\lambda_8$) was explicitly not treated "due to its mathematical complexity"; quark sources appear only as point singularities or as a scalar-field model whose global regularity was merely conjectured; the perturbations on the confining backgrounds, needed to identify the massless graviton and the massive spin-2 modes, were never analyzed; event-horizon-induced partial confinement was noted but not quantified; and perturbative nonrenormalizability and the possible composite (even-gluon) interpretation of $f_{\mu\nu}$ were left open. This problem asks for the completion of that program. No published work is known to complete these five gaps within this framework: the program became dormant after quantum chromodynamics (QCD) established itself as the accepted theory of the strong interaction, with later related work confined to new classical two-tensor solutions, geometric reformulations of color, test-fermion spectra in fixed de Sitter-type bags, and Unruh-hadronization analogies (models of hadron formation built on the Unruh effect, the thermal radiation perceived by an accelerated observer).

Background

By the late 1970s, quark confinement — the empirical fact that colored objects are never observed in isolation — was a central puzzle of hadron physics. The MIT bag model of Chodos, Jaffe, Johnson, Thorn, and Weisskopf described hadrons as spherical cavities in which quarks move quasi-freely, but confinement there is imposed by an exterior boundary condition rather than derived from dynamics. Salam and Strathdee, building on the "f–g theory" of Isham, Salam, and Strathdee (1971) and on the "strong gravity" program of Sivaram and Sinha, proposed a gravitational mechanism instead. They postulated an Einstein-type equation, with a strong coupling $\kappa_f^2$ of order $1\,\mathrm{GeV}^{-2}$ (roughly $10^{38}$ times Newton's constant) and a large positive "cosmological" constant $\lambda_f$, for a tensor field $f_{\mu\nu}$ that couples to colored matter such as quarks, while the ordinary spacetime metric $g_{\mu\nu}$ couples to color-neutral fields. In this theory the de Sitter solution (a space of constant positive curvature, with symmetry group $O(3,2)$ in the source's static description) of the strong Einstein equation acts as a "microuniverse" of radius $R$, with $1/R^2 = \lambda_f \kappa_f^2/6$, embedded in physical spacetime through an exactly solvable two-tensor system that mixes $f_{\mu\nu}$ and $g_{\mu\nu}$. A test quark described by the Klein–Gordon (or Dirac) equation in this background has a purely discrete spectrum — essentially that of a harmonic oscillator — with no continuum and hence no dissociation: confinement emerges from geometry. A source quark is represented approximately by the Schwarzschild–de Sitter form (a point mass in a positive-curvature space) $f_{00} = 1 - 2\mu/r + r^2/R^2$, where $\mu$ is a mass-like "mechanical" parameter of the source, treated as an input. To incorporate color, the vierbein — the local frame field by which spin-$\tfrac{1}{2}$ fermions couple to gravity, in the extension of general covariance to local Lorentz symmetry introduced by Weyl for the treatment of fermions — is enlarged from a quartet to a color multiplet (an "n-bein"). The paper solves the resulting classical system explicitly for global SU(2) color, because the isovector components of the vierbein can be taken to point along one fixed direction in isospace, which lets the equations separate into Einstein-type systems; the parameters can then be arranged so that a source isodoublet (an isospin-doublet quark state) confines a test quark into color-singlet composites, while singlet composites exert no confining potential on one another. The paper closes, however, by acknowledging five linked gaps separating this prototype from a realistic theory of confinement: (a) the realistic color group is SU(3), whose diagonal generators (the Gell-Mann matrices $\lambda_3$ and $\lambda_8$) span two independent Cartan (maximally commuting) directions, so the single fixed color direction that made SU(2) solvable is unavailable — in the authors' words, "We have not been able to discuss the case of SU(3) color, due to its mathematical complexity"; (b) quark sources are point singularities or, at best, a scalar-field matter distribution whose global regularity is only conjectured — the fully coupled problem with fermionic quark fields is unsolved; (c) the model rests on classical equations at strong coupling, whose perturbative nonrenormalizability the authors acknowledge, and the alternative of interpreting $f_{\mu\nu}$ as a composite field (an effective even-gluon bound state) is left an open question; (d) for sufficiently large $\mu$ the Schwarzschild–de Sitter backgrounds possess event horizons (one-way causal surfaces that would radiate thermally, à la Hawking), which would make confinement partial rather than exact — a possibility the authors explicitly declined to quantify; and (e) the perturbations on the nontrivial confining backgrounds were never analyzed, so it is unknown which linear combinations of the $f$ and $g$ fields represent the massless long-range graviton and which represent short-range massive spin-2 mesons, nor how to define the gravitational mass of an $f$-microuniverse.

Problem Statement

Complete the classical tensor gauge field (strong gravity) model of quark confinement of Salam and Strathdee (1978) into a consistent and realistic theory of hadron confinement. Within the framework of Einstein-type field equations with strong coupling $\kappa_f^2 \sim 1\,\mathrm{GeV}^{-2}$ and "cosmological" constant $\lambda_f$ for a color-carrying tensor (vierbein / n-bein) field $f_{\mu\nu}$, mixed with the weak gravitational tensor $g_{\mu\nu}$ as in the source paper, the following linked results are required. (i) SU(3) color: construct the explicit classical confining solutions for the full SU(3) color group, in which the color components of the n-bein point along the two independent Cartan directions ($\lambda_3$ and $\lambda_8$) rather than the single fixed isospin direction that made the SU(2) model solvable, and show that the parameters can be arranged so that colored objects are confined into color singlets while singlet composites exert no confining potential on one another. (ii) Dynamical sources: replace the point-source Schwarzschild singularity by dynamically treated quark fields — solve the coupled spinor–tensor system with back-reaction, establish or refute the existence of globally regular solutions extending to $r \to \infty$, and derive the source-strength parameter $\mu$ from the dynamics rather than treating it as an input. (iii) Mode structure: analyze the perturbations of the f–g system on the confining de Sitter-type backgrounds to determine which linear combinations describe the massless long-range graviton and which describe short-range massive spin-2 excitations, and define the gravitational mass of an f-microuniverse. (iv) Horizons and partial confinement: determine quantitatively for which source strengths the Schwarzschild–de Sitter microuniverses develop event horizons, and compute the resulting decay / partial-confinement effects for extended sources. (v) Consistency: assess the quantum and perturbative consistency of the model at the strong couplings involved — whether the Einstein-type structure can be controlled as a quantum effective theory, or whether $f_{\mu\nu}$ must emerge as a composite (effective even-gluon) field, and with what consequences for the confinement mechanism. These five gaps are linked: (i)–(ii) make the confinement model realistic, (iii)–(iv) fix its physical particle content and phenomenology, and (v) concerns its viability as a theory; the perturbation, horizon, and consistency analyses all presuppose the solutions delivered by (i)–(ii). Component (ii) is a decision problem: whether globally regular dynamical sources exist, and whether the source-strength parameter $\mu$ is dynamically derivable.

References

1. A. Salam and J. Strathdee, "Confinement through tensor gauge fields", Physical Review D 18 (12), 4596–4609 (1978). DOI: 10.1103/PhysRevD.18.4596, https://doi.org/10.1103/PhysRevD.18.4596
2. A. Salam and J. Strathdee, "Classical confining solutions of a tensor gauge theory incorporating color", Physical Review D 18 (4), 1323–1326 (1978). DOI: 10.1103/PhysRevD.18.1323, https://doi.org/10.1103/PhysRevD.18.1323
3. A. Salam and J. Strathdee, "Remarks on high-energy stability and renormalizability of gravity theory", Physical Review D 18 (12), 4480–4485 (1978). DOI: 10.1103/PhysRevD.18.4480, https://doi.org/10.1103/PhysRevD.18.4480
4. C. J. Isham, Abdus Salam, and J. Strathdee, "f-Dominance of Gravity", Physical Review D 3 (4), 867–873 (1971). DOI: 10.1103/PhysRevD.3.867, https://doi.org/10.1103/PhysRevD.3.867
5. A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, "New extended model of hadrons", Physical Review D 9 (12), 3471–3495 (1974). DOI: 10.1103/PhysRevD.9.3471, https://doi.org/10.1103/PhysRevD.9.3471
6. C. Sivaram and K. P. Sinha, "Strong gravity, black holes, and hadrons", Physical Review D 16 (6), 1975–1978 (1977). DOI: 10.1103/PhysRevD.16.1975, https://doi.org/10.1103/PhysRevD.16.1975
7. C. Sivaram and K. P. Sinha, "Strong spin-two interaction and general relativity", Physics Reports 51 (3), 111–187 (1979). DOI: 10.1016/0370-1573(79)90037-1, https://doi.org/10.1016/0370-1573(79)90037-1
8. M. Gürses, "New class of f−g fields relevant to quark confinement", Physical Review D 20 (4), 1019–1021 (1979). DOI: 10.1103/PhysRevD.20.1019, https://doi.org/10.1103/PhysRevD.20.1019
9. M. Gürses, "Cosmological f-g fields relevant to quark confinement", Journal of Physics A: Mathematical and General 13 (6), L223–L225 (1980). DOI: 10.1088/0305-4470/13/6/013, https://doi.org/10.1088/0305-4470/13/6/013
10. Eckehard W. Mielke, "The eightfold way to color geometrodynamics", International Journal of Theoretical Physics 19 (3), 189–209 (1980). DOI: 10.1007/BF00670675, https://doi.org/10.1007/BF00670675
11. E. van Beveren, C. Dullemond, and T. A. Rijken, "The Dirac equation and its solutions for quarks confined by an SO(3,2)-invariant geometry", Physical Review D 30 (5), 1103–1106 (1984). DOI: 10.1103/PhysRevD.30.1103, https://doi.org/10.1103/PhysRevD.30.1103
12. C. Dullemond, "Induced geometry and confinement", General Relativity and Gravitation 20 (2), 139–170 (1988). DOI: 10.1007/BF00759324, https://doi.org/10.1007/BF00759324
13. P. Caldirola, M. Pavšič, and E. Recami, "Classical quark confinement from general relativity", Physics Letters A 66 (1), 9–12 (1978). DOI: 10.1016/0375-9601(78)90087-7, https://doi.org/10.1016/0375-9601(78)90087-7
14. E. Recami, F. Raciti, W. A. Rodrigues Jr., and V. T. Zanchin, "Micro-universes and 'strong black holes': a purely geometric approach to elementary particles", arXiv:gr-qc/9509005 (1995). https://arxiv.org/abs/gr-qc/9509005
15. Eckehard W. Mielke, "Geometric Model of Quark Confinement?", in Geometrodynamics of Gauge Fields, Mathematical Physics Studies (Springer, Cham), pp. 347–358 (2017). DOI: 10.1007/978-3-319-29734-7_16, https://doi.org/10.1007/978-3-319-29734-7_16
16. Antonia Micol Frassino, Marcus Bleicher, and Robert B. Mann, "Unruh thermal hadronization and the cosmological constant", Physics Letters B 780, 93–99 (2018). DOI: 10.1016/j.physletb.2018.02.051, https://doi.org/10.1016/j.physletb.2018.02.051
17. Paolo Castorina, Alfredo Iorio, and Helmut Satz, "Hunting Quantum Gravity with Analogs: The Case of High-Energy Particle Physics", Universe 8 (9), 482 (2022). DOI: 10.3390/universe8090482, https://doi.org/10.3390/universe8090482
18. Jong-Ping Hsu, "A model of unified quantum chromodynamics and Yang-Mills gravity", Chinese Physics C 36 (5), 403–409 (2012). DOI: 10.1088/1674-1137/36/5/005, https://doi.org/10.1088/1674-1137/36/5/005