Scale Invariant Standard Model

• Scale Invariant Standard Model is a framework where classical mass terms are omitted, and all mass scales emerge from quantum corrections or hidden sector dynamics.
• It employs mechanisms like the Coleman–Weinberg method and strong hidden sector dynamics to radiatively generate the electroweak scale and dark matter masses, ensuring high predictivity.
• Tight constraints on scalar couplings and portal interactions yield testable predictions for Higgs mixing, dark matter relics, and collider signatures consistent with current experiments.

The Scale Invariant Standard Model (SISM) and its variants comprise a class of extensions to the Standard Model (SM) in which all explicit mass parameters are forbidden at the classical level. Instead, all physical scales—including the electroweak scale, dark matter masses, and possible new sector scales—emerge via radiative or dynamical symmetry breaking consistent with classical or quantum scale invariance. These frameworks address the Higgs hierarchy problem and often provide viable dark matter candidates, an enhanced predictive structure, and tight phenomenological constraints.

1. Foundations of Scale Invariance in the Standard Model

A classically scale-invariant theory contains no dimensionful couplings; in the SM context, this requires the Higgs mass term $\mu_H^2\,H^\dagger H$ to be absent from the Lagrangian. The archetypical scale-invariant SM extension supplements the classical Higgs potential

$$V(H)_{\text{SM}} = -\mu_H^2\,(H^\dagger H) + \lambda_H\,(H^\dagger H)^2$$

with $\mu_H^2$ set to zero: $V(H) = \lambda_H\,(H^\dagger H)^2$ Here, all tree-level stationary points are at the origin, $H=0$, due to the absence of negative mass-squared driving spontaneous symmetry breaking. Thus, classical scale invariance forbids all dimensionful parameters in both bosonic and fermionic sectors (Ghorbani et al., 2015), and all allowed operators in the scalar sector are quartic.

Spontaneous breaking of scale invariance, and hence the emergence of mass scales, proceeds via quantum corrections. Two general mechanisms are considered:

1. Coleman–Weinberg Mechanism: Quantum corrections to the effective potential generate nontrivial minima along a "flat direction" of the tree-level potential, breaking scale invariance radiatively.
2. Strong Hidden Sector Dynamics: Hidden QCD-like sectors confine and/or undergo chiral symmetry breaking, inducing nonperturbative scales via dimensional transmutation, communicated to the visible sector through portal couplings.

2. Quantum Generation of Mass Scales

2.1 Coleman–Weinberg Radiative Breaking

In SISM scenarios extended by singlet scalars—real (scalon, dark matter) or complex—the tree-level scalar sector is designed to admit a flat direction in field space. The general renormalizable, scale-invariant potential for the SU(2) doublet $H$, scalon $s$, and $n$ real singlet fields $\varphi_i$ (dark matter candidates) takes the form (Ghorbani et al., 2015): $$\begin{aligned} V(H, s, \varphi_i) &= \frac{\lambda_H}{4} (H^\dagger H)^2 + \frac{\lambda}{2} s^2(H^\dagger H) + \frac{\lambda_s}{4} s^4 \ &\qquad + \sum_i \left[ \frac12 \lambda_i s^2\varphi_i^2 + \frac14 \lambda_{\varphi_i}\varphi_i^4 \right] + \mathrm{DM\,interactions} \end{aligned}$$ No dimensionful parameter appears. The model may impose one or two $\mathbb{Z}_2$ symmetries stabilizing $\varphi_i$, so that $\langle \varphi_i \rangle = 0$.

By parameterizing the flat direction as $h = \phi\cos\alpha$, $s = \phi\sin\alpha$, and $\varphi_i = 0$, the tree-level potential vanishes if: $$\lambda^2 = \lambda_H \lambda_s,\qquad \cos^2\alpha = \frac{\lambda_H}{\lambda_H - \lambda}$$ The one-loop RG-improved potential along $\phi$ is then of the Coleman–Weinberg form,

$$V_{\mathrm{eff}}(\phi) = A\phi^4 + B\phi^4 \ln\left(\frac{\phi^2}{\Lambda^2}\right)$$

where $A$ and $B$ are renormalization group functions of the field content: $$\begin{aligned} A &= \frac{1}{64\pi^2 v_\phi^2} \left[m_h^4\left(-\frac23 + \ln\frac{m_h^2}{v_\phi^2}\right) + \dots\right] \ B &= \frac{1}{64\pi^2 v_\phi^4} \left[m_h^4 + \dots \right] \end{aligned}$$ with all scalar, gauge boson, and top-quark contributions included (Ghorbani et al., 2015).

Extremizing the potential ($\partial V_{\mathrm{eff}}/\partial \phi=0$) dynamically sets the renormalization scale and produces vacuum expectation values (VEVs) for $H$ and $s$. The Higgs mass arises radiatively: $$m_H^2 = 2(\lambda_H - \lambda)\,v_H^2, \qquad v_H = v_\phi \cos\alpha$$ demonstrating that the electroweak scale is generated via quantum transmutation.

2.2 Dynamical Scale Genesis via Hidden Sectors

An alternative route realizes all scales via strong coupling in a hidden gauge theory. In this scenario, the SM is extended with new QCD-like gauge dynamics and possibly new scalars or fermions. Dimensional transmutation in the hidden sector generates a confinement or chiral symmetry breaking scale $\Lambda_H$. Scalar condensates (e.g., $\langle \bar{Q}Q \rangle$ in hidden QCD) induce effective tadpoles and mass terms for scalar portals, which then trigger electroweak symmetry breaking in the visible sector (Hur et al., 2011, Kubo et al., 2016, Haba et al., 2017).

The mass scales of the visible and hidden sectors are then closely tied, and the resulting mass spectrum and couplings are determined by anomaly matching, naive dimensional analysis, or holographic matching to the hidden sector (Hatanaka et al., 2016).

3. Scalar Mass Spectrum and Coupling Constraints

The SISM contains, after spontaneous breaking, a massless (at tree level) scalon $s$—the pseudo-Goldstone boson of broken scale invariance—which acquires a radiatively induced mass: $$\delta m_s^2 = \frac{d^2 V_{\mathrm{eff}}}{d\phi^2}\Big|_{\phi = v_\phi} = 2B v_\phi^2$$ Explicitly, incorporating heavy scalar and gauge boson contributions, one finds (Ghorbani et al., 2015): $$\delta m_s^2 = -\frac{\lambda}{32\pi^2 m_H^2} \left[m_H^4 + \sum_i m_{\varphi_i}^4 + 6m_W^4 + 3m_Z^4 - 12m_t^4\right]$$ Positivity of $\delta m_s^2$ imposes a lower bound on the sum of dark matter scalar masses. The dark matter singlets $\varphi_i$ have masses

$$m_{\varphi_i}^2 = -\frac{\lambda_H \lambda_i}{\lambda} v_H^2$$

Positivity (requiring $m^2 > 0$) mandates $\lambda_i>0$ for $\lambda<0$.

Portal couplings between the visible and hidden sectors, and/or direct interactions among dark sector states, are stringently constrained by vacuum stability, perturbativity, and phenomenology. The couplings must satisfy

$$-0.128 < \lambda < 0,\qquad \lambda_H = \lambda + 0.128,\qquad \lambda_i > -1.65\frac{\lambda}{\lambda_H}$$

In models with two or more scalar singlets, the general mass matrix is block diagonal with the scalon and Higgs mixing set by the portal coupling, while the physical dark matter states remain unmixed due to the preserved $\mathbb{Z}_2$ symmetries (Ghorbani et al., 2015, Ishiwata, 2011).

4. Dark Matter Phenomenology

4.1 Single and Multi-component Scalar Dark Matter

Each real singlet $\varphi_i$ is stabilized by its own $\mathbb{Z}_2$ symmetry. The dominant interaction with SM particles is through $s^2\varphi_i^2$ (Higgs portal) and mixing between $s$ and the Higgs. The dark matter relic density is determined by thermal freeze-out, and parameter space is constrained by Planck/WMAP observations and direct detection experiments such as LUX and XENON100: $$\Omega_{\mathrm{DM}}h^2 = 0.1172\,\text{--}\,0.1226$$

Direct detection proceeds via spin-independent elastic scattering, mediated by $h$ and $s$ exchange. The cross section on nucleons is: $$\sigma_{\mathrm{SI}}^{N} = \frac{\alpha_N^2\mu_N^2}{\pi\,m_{\mathrm{DM}}^2}$$ with the coupling

$$\alpha_q = m_q\,\frac{2\lambda_H\lambda_i}{\lambda_H-\lambda} \left(\frac{1}{m_s^2} + \frac{1}{m_H^2}\right)$$

Direct detection limits require $m_{\mathrm{DM}}\gtrsim 2$ TeV for the single-scalar case and $m_{\mathrm{DM}}\gtrsim300$ GeV for two-component scenarios (Ghorbani et al., 2015).

Benchmark points exemplifying all constraints include $\lambda\simeq -0.08$, $\lambda_1\simeq 0.12$, $m_s\simeq 1$ GeV, $m_{\mathrm{DM}}\simeq 2.5$ TeV.

4.2 Generalizations

SISM extensions with two real singlets, one acquiring a VEV and triggering EWSB ($S_1$), and another stabilized as $Z_2$-odd dark matter candidate ($S_2$), permit viable TeV-scale WIMP dark matter. The relic density and direct detection cross section can be computed analogously and match experimental limits for appropriate portal couplings (e.g., $\lambda_{H2}\sim0.1$–1, $m_{S_2}\sim1$ TeV) (Ishiwata, 2011).

Variants with both scalar and fermion dark matter—realized by supplementing the SISM with additional singlet fermions—yield two-component dark sectors exhibiting reduced parameter spaces and characteristic direct and indirect signatures (Ayazi et al., 2018).

5. Cosmological and Theoretical Implications

The SISM approach provides an intrinsic solution to the gauge hierarchy problem by prohibiting tree-level Higgs mass terms. The electroweak scale, Higgs mass, and dark matter masses are all set dynamically by quantum effects, protected by the underlying scale symmetry (Ghorbani et al., 2015, Foot et al., 2010). The presence of radiatively generated, naturally light scalon fields offers novel collider signatures, though their couplings are suppressed unless the dark sector is heavy.

Dark matter scenarios are highly predictive due to the severe parameter reduction imposed by classical scale invariance; typically only two or three independent couplings remain once the Higgs and DM masses are fixed.

These models remain consistent with current collider constraints (Higgs and dark matter sector searches), cosmological relic density bounds, and direct detection limits, but generally require relatively heavy dark matter (multi-hundred GeV to several TeV) (Ghorbani et al., 2015).

6. Extensions, Related Models, and Future Prospects

Scale invariant scenarios generalize efficiently to frameworks including:

• Strongly interacting hidden sectors (hidden QCD/dark QCD), in which chiral or confinement phenomena generate scales transmitted through scalar portals (Hur et al., 2011, Hatanaka et al., 2016).
• Minimal scale invariant axion solutions, realizing spontaneous breaking near the Planck scale and providing solutions to the strong CP problem, often with suppressed dark matter axion abundance (Tokareva, 2017).
• Flatland and criticality models predicting relations between the top quark mass, criticality of the Higgs quartic, and small cosmological constants (Foot et al., 2014).
• Models predicting strong first-order phase transitions and observable gravitational wave backgrounds from EW or scale genesis, tightly connected to the EWSB origin (Kubo et al., 2016).
• Scale-invariant models that remain viable for inflationary dynamics and the transmission of Planck or seesaw scales, often dynamically predicting mass hierarchies via decoupled hidden sectors (Foot et al., 2010).

The SISM paradigm tightly constrains the properties and parameter space for additional scalar and dark matter fields. Ongoing and future advances in direct detection, LHC searches for additional (possibly sub-TeV) scalars, indirect detection, and precision cosmology will continue to test and refine these predictive frameworks (Ghorbani et al., 2015).