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

TL;DR

The paper analyzes how ADE-type singularities organize the vacuum structure of two-field Higgs-portal potentials and introduces the Milnor number $μ$ as a physically meaningful invariant of the local deformation space. Using Gröbner-basis methods, it shows that for the minimal real-singlet Higgs portal the effective potential exhibits a non-simple singularity with $μ = 9$, robust across variations in mixing, singlet mass, and cubic couplings. It connects this topology to observable signatures in the Higgs sector via $κ_λ$ and $c_H$, and to stochastic gravitational waves via $Ω_{GW}$, outlining a no-lose theorem for 2027–2040 collider and LISA capabilities. The work provides a model-independent framework to catalog critical points in multi-scalar theories and demonstrates that the electroweak vacuum in this minimal setup cannot realize an ADE catastrophe, making precision Higgs measurements and GW observations decisive in confirming or ruling out EWBG via a real-singlet portal.

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 $μ$ -- the dimensionality of the local Jacobian algebra. Throughout the parameter space consistent with experimental observations, the portal potential exhibits a non-simple singularity with $μ= 9$, maintaining topological stability amid substantial fluctuations in mixing angle, singlet mass, and cubic interactions. High-precision assessments of the Higgs trilinear self-coupling ($κ_λ$), the uniform rescaling of Higgs couplings ($c_H$), and the stochastic gravitational-wave spectrum ($Ω_{\mathrm{GW}}$) 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.

Figures (10)

• Figure 1: Parameter space for $m_h = 125\,\mathrm{GeV}$2.. The abscissa is the mixing parameter $X_t \equiv A_t - \mu \cot\beta$, the ordinate is the average stop soft mass $\tilde{m} \equiv \sqrt{m_{Q_3} m_{U_3}}$. The grey band marks the region where the two-loop NNLL calculation of FeynHiggs 2.19.03. reproduces $m_h = 125\,\mathrm{GeV}$ within $2\sigma$. The parabolic shape follows $\Delta m_h^2 \simeq \frac{3 y_t^2}{8\pi^2} \tilde{m}^2 \left(1 - \frac{X_t^2}{6\tilde{m}^2}\right)^2 \ln(\tilde{m}^2/m_t^2)$.
• Figure 2: The SM transition as a smooth crossover and FOPT that involves the coexistence of two phases.
• Figure 3: Parameter space of the singlet scalar as a function of the universal shift to all Higgs couplings ($c_H$) and Higgs self-coupling ($\kappa_\lambda$).
• Figure 4: $\kappa_\lambda$ versus $c_H$ for assorted singlet-model parameters.
• Figure 5: Parameter space for the direct (solid regions) and indirect (dashed lines) constraints on the scalar field.