Research of the Behavior of the Effective Potential in Systems with Phase Transitions through the Prism of A--D--E Type Singularities

Abstract

Detecting a scalar singlet interacting through the Higgs portal demands a pivot from conventional particle detection strategies to a comprehensive examination of the effective potential's landscape. The presence, intensity, and first-order nature of the electroweak phase transition are dictated by the critical manifold, with its universal traits encapsulated in the Milnor number $\mu$ -- the dimensionality of the local Jacobian algebra. Throughout the parameter space consistent with experimental observations, the portal potential exhibits a non-simple singularity with $\mu =9$, maintaining topological stability amid substantial fluctuations in mixing angle, singlet mass, and cubic interactions. High-precision assessments of the Higgs trilinear self-coupling (${\kappa }_{\lambda }$), the uniform rescaling of Higgs couplings ($c_H$), and the stochastic gravitational-wave spectrum (${\Omega }_{\mathrm{GW}}$) collectively delineate the catastrophe, extending beyond mere mass matrix analysis. Projections for 2027--2040 collider and LISA capabilities indicate that no viable region supporting a strong first-order transition will evade scrutiny; thus, the singlet will either be identified or conclusively dismissed via direct interrogation of the electroweak vacuum's critical structure.

16 pages, 10 figures, presented at the KINR Annual Workshop 2026 on "High Energy Physics. Theoretical and Experimental Challenges" (HEP-TEC-2026)