Determine the ordering of the protected odd spectral sectors

Determine whether the lowest gauge-invariant odd pole in the isolated broken G(2) gauge–Higgs theory belongs to the 1^{+-} two-vector/glue sector or to the 0^{--} three-vector sector, by resolving the possible deeply bound or near-threshold scalar state relative to the vector sector.

Background

The paper constructs two distinct gauge-invariant dark-G-parity-odd sectors. The 1{+-} sector contains a two-vector FMS operator and competing glue-dominated operators, while the 0{--} sector contains a three-vector FMS operator and derivative-dressed gluonic operators. Heavy-particle inequalities, Y-junction arguments, and the decoupled pure-SU(3) glue spectrum favor the 1{+-} channel in controlled regimes, but these arguments do not determine the exact strongly confining spectrum.

The unresolved issue is intrinsically nonperturbative because the scalar channel can be affected by the 1{+-}+0{++} scattering threshold and by channel-specific self-energy effects. The authors identify separate scattering-aware generalized-eigenvalue problems, including explicit multiparticle operators, as the calculation required to decide the ordering.

References

The scalar alternative is therefore genuine; what remains uncertain is its mass relative to the $1{+-}$ sector.

— Dark matter glueball candidate from a $G(2)$--$E_6$--$E_7$ exceptional grand unified theory: $G$-parity, spin, mass and stability  (2609.04296 - Masi, 3 Sep 2026) in Section 2, subsection “The physical spectroscopy problem: separate 1^{+-} and 0^{--} GEVPs”; Conclusions, paragraph “Odd-state spectroscopy”

A first-principles determination of their ordering requires Monte-Carlo correlators for the broken $G(2)$ gauge--Higgs theory, which are not presently available.

— Dark matter glueball candidate from a $G(2)$--$E_6$--$E_7$ exceptional grand unified theory: $G$-parity, spin, mass and stability  (2609.04296 - Masi, 3 Sep 2026) in Section 2, subsection “The physical spectroscopy problem: separate 1^{+-} and 0^{--} GEVPs”; Conclusions, paragraph “Open calculations”