The Minimal Scale Invariant Extension of the Standard Model

TL;DR

The paper develops the Minimal Scale Invariant extension of the Standard Model (MSISM), adding a complex singlet S to realize a scale-invariant framework at tree level whose quantum corrections trigger electroweak symmetry breaking. It systematically classifies flat directions (Type I, II, III), constructs the one-loop effective potential via the Gildener–Weinberg approach, and derives scalar spectra under various symmetry realizations, with emphasis on perturbativity and electroweak precision constraints. The study demonstrates the viability of MSISM scenarios with explicit or spontaneous CP violation, and shows how right-handed neutrinos can enable a seesaw mechanism and how parity symmetries can yield a dark matter candidate in certain Type-II maximal SCPV realizations, with some models remaining perturbative up to the Planck scale. Overall, MSISM offers a minimal, predictive framework addressing the hierarchy problem without intermediate scales, while yielding rich phenomenology testable at colliders and in cosmology.

Abstract

We perform a systematic analysis of an extension of the Standard Model that includes a complex singlet scalar field and is scale invariant at the tree level. We call such a model the Minimal Scale Invariant extension of the Standard Model (MSISM). The tree-level scale invariance of the model is explicitly broken by quantum corrections, which can trigger electroweak symmetry breaking and potentially provide a mechanism for solving the gauge hierarchy problem. Even though the scale invariant Standard Model is not a realistic scenario, the addition of a complex singlet scalar field may result in a perturbative and phenomenologically viable theory. We present a complete classification of the flat directions which may occur in the classical scalar potential of the MSISM. After calculating the one-loop effective potential of the MSISM, we investigate a number of representative scenarios and determine their scalar boson mass spectra, as well as their perturbatively allowed parameter space compatible with electroweak precision data. We discuss the phenomenological implications of these scenarios, in particular, whether they realize explicit or spontaneous CP violation, neutrino masses or provide dark matter candidates. In particular, we find a new minimal scale-invariant model of maximal spontaneous CP violation which can stay perturbative up to Planck-mass energy scales, without introducing an unnaturally large hierarchy in the scalar-potential couplings.

Figures (16)

• Figure 1: Numerical estimates of $m_{h}$ (upper plot) and $m_{\sigma, J}$ (lower plot) as functions of $\lambda_{3}(\Lambda)$ in the U(1)-symmetric Type-I MSISM. The solid/black $\beta_{\lambda_{3}} <1$ line shows the perturbative values of $\lambda_{3}(\Lambda) \le 5.84$, whilst the dashed/gray $\beta_{\lambda_{3}} > 1$ line shows the non-perturbative values of $\lambda_{3}(\Lambda) > 5.84$. The area between the horizontal blue LEP line and the horizontal red $\delta S$ line is allowed by experimental considerations of the LEP2 mass limit on the SM-like $h$ boson and the $\delta S$ parameter respectively. The area above the horizontal red $\delta T$ line is excluded by the $\delta T$ parameter constraint.
• Figure 2: The RG scale $\Lambda$ as a function of $\lambda_{3}(\Lambda)$ in the U(1)-symmetric Type-I MSISM. The solid/black $\beta_{\lambda_{3}} <1$ line shows the perturbative values of $\lambda_{3}(\Lambda) \le 5.84$, whilst the dashed/gray $\beta_{\lambda_{3}} > 1$ line shows the non-perturbative values. The areas lying to the right of the red $\delta S$ and $\delta T$ lines are excluded, and similarly to the left of the blue LEP line is also excluded by the LEP2 Higgs mass limit.
• Figure 3: Numerical estimates of $m_{h}$ (upper panel), $m_{H_{1}}$ (middle panel) and $m_{H_{2}}$ (lower panel) versus $\lambda_{3}(\Lambda)$ in the general Type-I MSISM. The white area between the black lines show the regions which correspond to perturbative values of $\lambda_{3}(\Lambda)$ and $|\lambda_{4}(\Lambda)|$ and positive scalar masses (\ref{['eqn:TypeIUnot1positivemasses']}), whilst the gray-shaded areas show their non-perturbative regions. The areas lying to the right of the red lines for $\delta S$ and $\delta T$ are excluded. Likewise, the area left of the blue $LEP$ line is ruled out by the LEP2 Higgs-mass limit.
• Figure 4: The RG scale $\Lambda$ as a function of $\lambda_{3}(\Lambda)$ in the general Type-I MSISM. The white area between the black lines shows the region that corresponds to perturbative values of $\lambda_{3}(\Lambda)$ and $|\lambda_{4}(\Lambda)|$, whilst the gray-shaded area shows the non-perturbative region. The area between the red $\delta S$ and blue $LEP$ lines is permitted by the oblique parameters and the LEP2 Higgs-mass limit. The area to the right of the red $\delta T$ line is excluded by the $\delta T$ limit.
• Figure 5: Theoretical and experimental exclusion contours in the $\lambda_{1}(\Lambda)$-$\lambda_{3}(\Lambda)$ parameter space in the U(1)-invariant Type-II MSISM. The upper panel shows the full perturbative parameter space, whilst the lower panel focuses on the region with small $\lambda_{3}(\Lambda)$. The theoretically allowed areas are enclosed by the black lines which correspond to keeping $\beta_{\lambda_{1,2}} \le 1$, $\beta > 0$ and $\lambda_{3}(\Lambda) \le 0$. The LEP2 limit is given by the blue (grey) LEP line and above (below) is excluded for the upper (lower) panel. The blue and grey shaded areas are allowed by the theoretical constraints, the LEP2 Higgs-mass limit and the oblique parameters. The region of parameter space which remains perturbative to GUT (Planck) scale is enclosed by the solid (dashed) green Pert lines.