Beyond a symmetry-restricted VEV ansatz: transverse stability of an $SU(5)$ special-subgroup vacuum

Abstract

Extended Higgs sectors can realize multistage breaking chains in grand unified theories (GUTs), and the resulting intermediate gauge phases can be relevant to cosmological defects. Restricting vacuum expectation values (VEVs) to configurations that preserve a chosen subgroup is an efficient way to identify candidate stationary points, while their stability against fluctuations outside this subspace is a separate question. We study this question in a renormalizable $SU(5)$ GUT Higgs sector with an adjoint $24_H$ and a complex symmetric $15_H$, motivated by a special-subgroup realization of Langacker-Pi monopole erasure. On the standard adjoint background, the $15_H$ VEVs that preserve the intermediate gauge group form a two-block family with independent color- and weak-block amplitudes. Portal interactions can make these amplitudes unequal without changing the unbroken gauge group. Two independent mixed quartic invariants become identical when evaluated on this two-block field space, although they contribute differently to color-weak fluctuations. We derive the complete spectrum of these modes and identify two bounded-from-below parameter choices that induce the same potential on the two-block field space but have different curvatures in transverse directions. The common stationary configuration is a local minimum in one case and a saddle with six negative physical modes in the other. Direct Hessian calculations in the full 54-dimensional real scalar field space reproduce the analytic spectrum and verify both the minimum-saddle distinction and locally stable benchmark vacua.