Minimal Inverse Seesaw Model

• The minimal inverse seesaw model is an extension of the Standard Model that introduces extra singlet fermions and scalars to naturally generate a small, lepton number–violating μ term.
• It employs a two-loop radiative mechanism to produce a keV-scale μ parameter, ensuring sub-eV active neutrino masses without requiring fine-tuned Yukawa couplings.
• Its gauged B–L extension enforces anomaly cancellation, predicts a TeV-scale Z' boson, and provides a MeV-scale sterile neutrino as a promising dark matter candidate.

The minimal inverse seesaw model is an extension of the Standard Model (SM) in which the observed smallness of active neutrino masses is explained by introducing additional SM-singlet fermions—typically a right-handed neutrino and another sterile singlet—together with a small lepton number–violating parameter. In its "dynamical" realization, the crucial scale-breaking parameter is not set by hand but generated radiatively through the scalar sector, thereby addressing naturalness concerns. This construction links the neutrino mass scale to the presence of new scalar fields and, in its B–L-gauged extension, produces testable predictions at high-energy colliders and for dark matter.

1. Minimal Dynamical Inverse Seesaw: Field Content and Structure

In the minimal dynamical inverse seesaw model, the Standard Model is extended by:

• One right-handed neutrino ($\nu^c$)
• One extra sterile singlet fermion ($S$)
• A set of neutral scalar fields:
  • A real scalar $\phi$ (lepton number 0)
  • A complex scalar %%%%3%%%% (lepton number $-1$)
  • A complex, "inert" scalar $\Delta$ (lepton number $-2$)

The canonical (non-dynamical) inverse seesaw Lagrangian responsible for neutrino masses is: $$\mathcal{L} \supset y_\nu L H \nu^c + M\, \nu^c S + \tfrac{1}{2} \mu\, S S + \text{h.c.}$$ with $m_D = y_\nu v_W$ (where $v_W \approx 246$ GeV). In the dynamical version, the $\mu$ term is forbidden at tree level and radiatively generated via the new scalar sector, modifying the relevant Yukawa terms to: $$\mathcal{L} \supset y_\nu L H \nu^c + M \nu^c S + \tfrac{1}{2} y_S \Delta S S + \tfrac{1}{2} y_{\nu^c} \Delta^\dagger \nu^c \nu^c + \text{h.c.}$$ The neutrino mass matrix in the $(\nu_L,\,\nu^c,\,S)$ basis is then: $$M_\nu = \begin{pmatrix} 0 & m_D & 0 \ m_D & 0 & M \ 0 & M & \mu \end{pmatrix}$$ with the crucial lepton number–violating parameter $\mu$ induced at higher order.

2. Dynamical Origin of $\mu$ via Scalar Radiative Effects

The essential $\mu$ term is induced at two-loop order through neutral CP-even scalar mixing. The vacuum expectation values in the scalar sector are chosen as: $$\langle H \rangle = \frac{v_W}{\sqrt{2}}, \quad \langle \phi \rangle = v_\phi, \quad \langle \widetilde{\Delta} \rangle = \frac{v_{\widetilde{\Delta}}}{\sqrt{2}}, \quad \langle \Delta \rangle = 0.$$ The two-loop diagram (involving the inert $\Delta$, $\widetilde{\Delta}$, and $\phi$, with trilinear scalar couplings denoted symbolically by $A^3$) yields: $$\mu \sim y_S^2 y_{\nu^c} \left(\frac{1}{(16\pi^2)^2}\right) A^3\,\frac{v^2_{\widetilde{\Delta}}}{M^4}$$ where $M$ is the typical heavy mass scale ($\sim$TeV) and $A^3$ is a product of trilinear couplings with mass dimension 3 (of order $v_W^3$). For perturbative choices of Yukawa couplings ($y_S,\,y_{\nu^c}\lesssim 1$), $\mu$ naturally emerges in the keV range—precisely the scale required, via: $m_\nu \sim \mu\,\frac{m_D^2}{M^2}$ to guarantee sub-eV active neutrino masses. The dynamical suppression of $\mu$ through loop factors and scalar structure addresses the naturalness challenge without fine-tuning.

3. Extension to Gauged $B$–$L$ and Anomaly Cancellation

Promoting lepton number to a local U(1)$_{B-L}$ symmetry requires anomaly cancellation. This necessitates two right-handed neutrinos ($\nu^c_1$, $\nu^c_2$) for each $S$, leading to the generalization: $$\mathcal{L} \supset y_{\nu_1} L H \nu^c_1 + y_{\nu_2} L H \nu^c_2 + M_1 \nu^c_1 S + M_2 \nu^c_2 S +$$

$$+\tfrac{1}{2} y_S \Delta S S + \tfrac{1}{2} y_{\nu^c}^a \Delta^\dagger \nu^c_1 \nu^c_1 + \tfrac{1}{2} y_{\nu^c}^a \Delta^\dagger \nu^c_2 \nu^c_2 + y_{\nu^c}^b \Delta^\dagger \nu^c_1 \nu^c_2 + \text{h.c.}$$

A discrete permutation symmetry selects one linear combination, e.g. $\hat\nu^c_1 = (\nu^c_1+\nu^c_2)/\sqrt{2}$, to participate in the standard inverse seesaw. The orthogonal state, $\hat\nu^c_2$, decouples from SM interactions, receiving a two-loop radiative Majorana mass of order $0.1$–$1$ MeV (protected by a $Z_2$ symmetry). This sterile state is stable and a viable dark matter candidate. The gauging of $B-L$ also introduces a $Z'$ boson with model-dependent mass and interactions.

4. Phenomenological Signatures and Testability

Distinctive phenomenological signals arise:

• $Z'$ production at colliders: For TeV-scale $Z'$ masses, $pp \to Z'$ has possible decays into SM particles or new scalars, with characteristic signatures accessible at the LHC.
• Extended scalar sector: Mixing of the neutral scalars ($\phi$, $\widetilde{\Delta}$, $\Delta$) with each other and the Higgs may alter Higgs couplings and produce additional scalar resonances.
• Sterile neutrino dark matter: The MeV-scale $\hat\nu^c_2$ is stable, with weak couplings ensuring compatibility with cosmological and direct detection constraints.
• Collider observables: Production/decay channels of $Z'$ and new scalars, as well as possible deviations in Higgs and lepton flavor observables.

These elements provide experimentally accessible tests of the model's core predictions.

5. Mathematical Formulation and Scaling Relations

The key mass formulae are:

• Standard ISS light neutrino mass (for $m_D \ll M$, and $\mu \ll M$): $m_\nu \simeq \mu\, \frac{m_D^2}{M^2}$
• Dynamical $\mu$ scale from two-loop radiative correction: $$\mu \sim y_S^2 y_{\nu^c} \frac{A^3\,v^2_{\widetilde{\Delta}}}{(16 \pi^2)^2 M^4}$$
• Generalized mass matrix (in the B–L extension, after permutation symmetry): $$\mathcal{L} \supset \sqrt{2}y_\nu L H \hat{\nu}^c_1 + \sqrt{2} M \hat{\nu}^c_1 S +$$

$$+\frac{1}{2}(y_{\nu^c}^a+y_{\nu^c}^b)\Delta^\dagger \hat{\nu}^c_1 \hat{\nu}^c_1 + \frac{1}{2}(y_{\nu^c}^a-y_{\nu^c}^b)\Delta^\dagger \hat{\nu}^c_2 \hat{\nu}^c_2 + \text{h.c.}$$

with $\hat\nu^c_1$ involved in ISS mass generation and $\hat\nu^c_2$ a DM candidate.

6. Interplay of Neutrino Mass, Dark Matter, and Collider Physics

The minimal dynamical inverse seesaw connects sub-eV neutrino masses, new scalar and vector bosons, and keV–MeV sterile states:

• The loop-induced $\mu$ parameter (naturally in the $\sim$keV range) enables $M \sim$ TeV neutrino mass scales to account for observed mass splittings without excessively small Yukawa couplings or new mass scales.
• Gauged $B$–$L$ resolves anomaly issues and dynamically selects sterile neutrino DM, linking cosmic dark matter to the structure of the neutrino sector.
• The enlarged particle content—extra scalars, $Z'$, and heavy singlets—ensures rich collider phenomenology.

This framework thus unifies the origin of neutrino masses, the presence of dark matter, and possible LHC signals within a minimal and technically natural structure.