Physical Selection of Cartan Soft Breakings in a Broken SU (3) Yang–Mills Model

We revisit the broken-[Formula: see text] Yang–Mills model with a soft quadratic color deformation induced by the dimension-two operator [Formula: see text], previously studied for a constant source aligned along the color-3 direction. We extend the construction to a generic Cartan-valued source and determine the corresponding quadratic propagators and their associated [Formula: see text]-particle structure. Within the quadratic approximation and the lowest nontrivial one-loop analysis of the composite correlators, we show that the diagonal [Formula: see text] sector always reorganizes into a conjugate pair of complex modes and therefore supports a candidate composite channel with a Källén–Lehmann representation. By contrast, the replica-like off-diagonal channel built from the [Formula: see text] and [Formula: see text] sectors satisfies the same leading-order admissibility criterion only on a special locus in the Cartan plane. In this precise spectral sense, the original color-3 model is distinguished by the simultaneous survival of the off-diagonal and diagonal channels. We emphasize that this constitutes a leading-order selection criterion rather than a nonperturbative proof of the physical spectrum. Finally, we present a BRST-invariant spontaneous realization of the Cartan deformation and establish the algebraic renormalizability of this realization in the Landau gauge.