Consistency completion of the SL(6,C) gauge theory of the $2^+$ tensor-meson nonet: symmetry protection of unwanted degrees of freedom, ultraviolet behavior, vacuum stability, and canonical quantization

The 1973 Isham-Salam-Strathdee paper constructed an SL(6,C) gauge-invariant, nonpolynomial Lagrangian in which the $J^P=2^+$ tensor-meson nonet appears as massive gauge particles, with SL(6,C)-covariant constraints and spontaneous symmetry breaking arranged so that the bilinear part reduces to the Pauli-Fierz theory of nine massive spin-2 particles. The authors themselves listed the unresolved issues: the nonpropagation of the unwanted spin-1 and spin-0 components is enforced by parameter choice rather than guaranteed by gauge symmetry (with a conjecture that a higher group such as U(6,6) might guarantee it); the spin-2 theory is not renormalizable in the conventional sense and needs new, possibly nonpolynomial, ultraviolet methods; adding a Yang-Mills-type term may alter the particle spectrum and destabilize the vacuum; the Lagrangian is not unique; and no canonical (Hamiltonian) or quantum analysis was performed. Modern adjacent lines - ghost-free massive and multi-spin-2 gravity (Boulware-Deser; de Rham-Gabadadze-Tolley; Hinterbichler-Rosen) and SL(2N,C) gauge unification (Chkareuli), which cites the 1973 theory as its prototype - address generalized versions of the unwanted-degree-of-freedom problem within different theory families and do not complete the original construction.

Background

In the early 1970s, before quantum chromodynamics (QCD) became established as the theory of strong interactions, a major effort attempted to realize the approximate internal symmetries of hadrons together with spin as a local gauge symmetry. The nonrelativistic quark model classified hadrons by SU(6), the group acting jointly on the spin and flavor ('unitary spin') degrees of freedom of the light quarks; its relativistic extension was sought in the noncompact group SL(6,C) - the group of complex unimodular 6-by-6 matrices, containing the Lorentz group (realized as SL(2,C)) together with the internal flavor group SU(3). Isham, Salam and Strathdee developed this program through 'f-dominance of gravity' (Phys. Rev. D 3, 867 (1971)) and SL(6,C)-invariant Einstein-like Lagrangians (Lett. Nuovo Cim. 5, 969 (1972)), culminating in the 1973 paper.

The physical aim of the 1973 paper is to make the nine mesons of spin and parity $J^P=2^+$ - the tensor-meson nonet (the isoscalars now called $f$ and $f'$, the isovector $a_2$, and their strange partners; in modern language $q\bar q$ states with orbital angular momentum 1 and aligned quark spins) - the gauge particles of a local SL(6,C) symmetry, in the same spirit in which vector mesons had earlier been assigned a Yang-Mills role. A gauge theory 'of the second kind' is one whose symmetry parameters are arbitrary spacetime functions; the corresponding gauge potential is a tensor gauge field, denoted $B_{\mu[\alpha\beta]}^k$ and related fields, carrying a spacetime index $\mu$, a Lorentz-vector index pair $[\alpha\beta]$, and an SU(3) flavor index $k$. The construction uses three ingredients: (i) an SL(6,C)-invariant nonpolynomial Lagrangian (the paper's Eq. (2.9), containing a free parameter in addition to the coupling and mass); (ii) covariant constraints - nonlinear field constraints transforming covariantly under the gauge group - imposed to eliminate most of the a priori enormous multiplet of components; and (iii) spontaneous symmetry breaking, the mechanism whose spin-1 analog is the Higgs-Kibble mechanism (gauge bosons acquire mass without explicit mass terms), by which the vacuum fails to be invariant under the rigid SL(6,C) transformations so that the gauge particles emerge massive. A conserved set of currents closes on the SL(6,C) algebra - a current algebra without nonzero charges - reproducing the current-algebra phenomenology of the day.

With the parameters fixed by hand, the bilinear (free-field) part of the Lagrangian reduces to the Pauli-Fierz form - the standard, unique free field theory of a massive spin-2 particle, propagating exactly five physical polarizations - for a nonet of positive-energy $2^+$ particles. The trouble, stated candidly by the authors, is that nothing in the gauge symmetry itself guarantees this: an unconstrained tensor gauge multiplet contains components that would describe rest-frame spin-1 and spin-0 fields, and nonlinear interactions could excite them. The same issue was identified by Boulware and Deser (1972) for ordinary massive gravity: generic nonlinear completions of the Pauli-Fierz theory propagate an extra ghostlike scalar mode, now called the Boulware-Deser ghost. A further structural obstacle to merging spacetime and internal symmetries is the Coleman-Mandula theorem (1967), which forbids such nontrivial mixing in the S-matrix symmetry group of an interacting relativistic theory with a mass gap - a theorem that in fact originated from these very SU(6)-relativization attempts.

Within this setting, the 1973 paper closes with a precisely stated elaboration program.

Problem Statement

The problem is the consistency completion of the SL(6,C) gauge theory of the $2^+$ tensor-meson nonet, exactly as posed by Isham, Salam and Strathdee at the close of their 1973 construction ('the theory presented in the present paper will need further elaboration'). The common objective is to determine whether this gauge theory can be completed into a fully consistent theory of a nonet of interacting massive spin-2 particles - or to prove that it cannot. The following linked issues, jointly, must be settled:

1. Degree-of-freedom protection. Establish - rather than arrange by the choice of special parameter values and constraints - that the unwanted components of the tensor gauge multiplet, namely the rest-frame spin-1 fields and the spin-0 fields, never propagate once interactions are included. In particular, settle the authors' conjecture: construct (or rule out) an extension of the gauge symmetry to a higher group, 'possibly U(6,6)', whose gauge invariance alone guarantees the nonpropagation of all unwanted degrees of freedom.

2. Ultraviolet behavior. The theory, like all interacting spin-2 theories, is not renormalizable in the conventional sense, and the spontaneous-breaking mechanism does not change this. Determine whether the ultraviolet infinities can be controlled - or prove that no controlled quantum treatment of this Lagrangian exists.

3. Vacuum stability and spectrum under derivative interactions. Adding the Yang-Mills-type term (4.1), which introduces derivatives of $B_{\mu[\alpha\beta]}^k$ into the interaction terms, converts fields that were algebraic composites into independent dynamical variables, so that the particle spectrum read off from the bilinear terms is probably altered and the stability of the vacuum 'can no longer be taken for granted'. Characterize when such derivative terms are admissible and what spectrum and vacuum they yield, or show that none is admissible.

4. Selection of the Lagrangian. The SL(6,C)-invariant Lagrangian is not unique (the paper's Eq. (2.9) with its free parameter and the alternative family of its Appendix B are examples). Determine whether any principle selects one member of the family, or whether the physically meaningful conclusions are family-independent.

5. Canonical and quantum analysis. Perform the complete canonical (Dirac constraint/Hamiltonian) analysis of the constrained gauge system - explicitly deferred by the authors - and on that basis the quantization, including the fate of the current algebra of charges.

The five linked issues above are jointly posed, as one problem rather than as five separate ones. The question's content is not narrowed by the historical supersession of its hadronic motivation by QCD: it stands today as a question in the theory of constrained tensor (higher-spin) gauge systems and interacting massive spin-2 fields, with exactly the five linked components above.

References

1. C. J. Isham, Abdus Salam, and J. Strathdee, "2+ Nonet as Gauge Particles for SL(6,C) Symmetry", Physical Review D 8, 2600-2609 (1973). DOI: 10.1103/PhysRevD.8.2600, https://doi.org/10.1103/PhysRevD.8.2600
2. C. J. Isham, Abdus Salam, and J. Strathdee, "f-Dominance of Gravity", Physical Review D 3, 867-873 (1971). DOI: 10.1103/PhysRevD.3.867, https://doi.org/10.1103/PhysRevD.3.867
3. C. J. Isham, Abdus Salam, and J. Strathdee, "SL(6,C) gauge invariance of Einstein-like Lagrangians", Lettere al Nuovo Cimento 5, 969-972 (1972). DOI: 10.1007/BF02798862, https://doi.org/10.1007/BF02798862
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