Majorana Sterile Fermions Overview

• Majorana sterile fermions are neutral, gauge-singlet fermions with self-conjugate mass terms that offer a framework for neutrino mass generation and dark sector interactions.
• They underpin seesaw mechanisms and complex neutrino mixing, leading to observable lepton-number-violating signals in oscillation experiments and collider searches.
• Their effects are probed via EDM measurements, neutrinoless double beta decay, and other rare processes, linking CP violation and dark matter phenomenology.

A Majorana sterile fermion is a gauge-singlet (sterile) fermionic field, typically denoted by $N$ or $N_R$, that possesses a Majorana mass term (hence is its own antiparticle under charge conjugation), and interacts with the Standard Model (SM) primarily through small mixings with active fields or via higher-dimensional operators. Majorana sterile fermions are motivated by extensions of the SM that accommodate neutrino masses and provide portals to dark sectors. They play pivotal roles in seesaw mechanisms, lepton-number-violating (LNV) processes, baryogenesis, and dark matter models.

1. Theoretical Foundations and Mass Structures

Majorana sterile fermions are introduced as gauge singlet fields, typically denoted as $N_i$ (with $i$ labeling generations), that admit Majorana mass terms: $$\mathcal{L}_{\rm new} = -Y_{\alpha i}\,\overline{L_\alpha}\,\widetilde H\,N_i - \tfrac12\,M_{ij}\,\overline{N_i^c} N_j + {\rm h.c.}$$ where $L_\alpha$ are SM lepton doublets, $\widetilde H = i\,\sigma_2 H^*$ is the conjugate Higgs doublet, $Y_{\alpha i}$ are neutrino Yukawa couplings, and $M_{ij}$ is a symmetric Majorana mass matrix for the sterile states (Abada et al., 2015).

Upon electroweak symmetry breaking, one forms a Dirac mass matrix $m_D$, and the full $(3+N)\times(3+N)$ neutrino mass matrix is

$$\mathcal{M}_\nu = \begin{pmatrix} 0 & m_D \ m_D^T & M \end{pmatrix}\,,$$

which is diagonalized by a unitary matrix $U$. The physical eigenstates are admixtures of active and sterile fields, leading to a mass spectrum of up to $3+N$ Majorana neutrinos.

Related frameworks, such as the Dirac–Majorana Standard Model (DM ν SM), further allow both Dirac and Majorana terms for active and sterile fields, leading to rich spectra including pseudo-Dirac, fully Majorana, and "Diracian" limits with mass degeneracies (Nieuwenhuizen, 2018, Kimura et al., 2021).

2. Neutrino Mixing, Phases, and Oscillation Phenomenology

When sterile fermions possess Majorana masses and couple to the SM via Dirac mass terms, the resulting neutrino mixing matrix is extended to a larger unitary transformation ($U$), including PMNS-type angles, Dirac and Majorana phases. In minimal $3+N$ models, the parametrization includes: $$U = R_{45}\cdots R_{12} \times \mathrm{diag}(1, e^{i\varphi_2},...,e^{i\varphi_{3+N}})$$ with each $R_{ij}$ a complex rotation (angle $\theta_{ij}$, phase $\delta_{ij}$), and $\varphi_k$ are Majorana phases (Abada et al., 2015).

Majorana phases are unobservable in standard oscillation experiments but directly impact lepton-number-violating observables (see §5). In the "Diracian" limit all mass eigenstates are pairwise degenerate and the Majorana phases transmute into Dirac-type phases affecting phenomenology (Nieuwenhuizen, 2018).

Oscillation probabilities in scenarios with both Dirac and Majorana mass terms show that large active–sterile mixing and oscillations can arise if mass terms are comparable, with the presence of unsuppressed transitions serving as a direct probe for Majorana mass terms (Kimura et al., 2021).

3. Impact on Electric Dipole Moments and CP Violation

Majorana sterile fermions induce charged-lepton electric dipole moments (EDMs) via CP-violating two-loop diagrams. The minimal scenario requires at least two non-degenerate Majorana sterile states to yield observable EDMs; the EDM operator is

$$d_\alpha = -\frac{g_2^4 e m_\alpha}{4 (4\pi)^4 m_W^2} \sum_{\beta,i,j} \left[ J_{ij\alpha\beta}^M\,I_M + J_{ij\alpha\beta}^D\,I_D \right]$$

with the Majorana-type CP-odd phase factor

$$J_{ij\alpha\beta}^{M} =\mathrm{Im}\left[ U_{\alpha j}U_{\beta j} U_{\beta i}^* U_{\alpha i}^* \right]$$

(Abada et al., 2015). The Majorana nature is essential: diagrams involving lepton-number violation (Majorana mass insertions) vanish for Dirac neutrinos, and only the antisymmetric combinations enabled by Majorana phases survive the two-loop structure.

Electron EDMs can approach future experimental sensitivities ($d_e/e \sim 10^{-30}$ cm) for sterile masses $m_{4,5} \gtrsim 100$ GeV and sizable mixing, subject to strong constraints from flavor violation, EW precision observables, and unitarity.

4. Lepton Number Violation and Neutrinoless Double Beta Decay

Majorana sterile fermions mediate LNV processes, most notably neutrinoless double beta ($0\nu\beta\beta$) decay. In $3+1$ or $3+2$ frameworks, the effective mass parameter is

$$m_{ee} = \left| \sum_{i=1}^{3+N} U_{ei}^2\,m_i \right|$$

where contributions from light sterile Majorana states enter with their own mixing and phase structure (Li et al., 2011, Hernandez-Galeana, 2014).

Tuning of masses, mixings, and Majorana phases can lead to cancellations in $m_{ee}$, modifying the interpretation of null $0\nu\beta\beta$ results relative to three-neutrino scenarios. In models such as the Dirac–Majorana SM, leading-order contributions from mixed-chirality terms cancel for certain mass structures ("Diracian" limit), suppressing $0\nu\beta\beta$ decay (Nieuwenhuizen, 2018). In models with radiatively-generated sterile masses, sterile phases and masses can dominate the effective mass within the reach of next-generation searches (Hernandez-Galeana, 2014, Li et al., 2011).

5. Role in Dark Matter and Portal Interactions

Sterile Majorana fermions provide natural portals to dark matter sectors, mediated by renormalizable or effective operators. In the presence of a Majorana dark fermion $\chi$, dimension-6 portal operators can be classified as lepton-number-conserving ($\mathcal{O}_1$) or lepton-number-violating ($\mathcal{O}_{2,3}$): $$\begin{aligned} \mathcal{O}_1 &= (\overline{N_R} \chi_L)(\overline{\chi_L} N_R)\ \mathcal{O}_2 &= (\overline{N_R} \chi_L)(\overline{N_R} \chi_L)\ \mathcal{O}_3 &= (\overline{N_R^c} N_R)(\overline{\chi_L^c} \chi_L) \end{aligned}$$ Dirac versus Majorana nature of both sterile and active neutrinos crucially determines which operators are present, the symmetry structure, and dark matter annihilation patterns (Coito et al., 2022). Thermal freeze-out through $\chi\chi \to NN$ can match cosmological relic abundance in broad parameter ranges, with lepton-number violation encoded in the effective field theory and UV completions. Tree-level and radiative mechanisms can generate the sterile Majorana masses, intimately connecting dark matter phenomenology and LNV processes.

6. Experimental Signatures and Discrimination of Majorana Nature

Majorana sterile fermions can manifest in multiple experimental channels, distinguished from Dirac cases by unique LNV signals. At colliders, purely leptonic decays such as $W^\pm \to e^\pm e^\pm \mu^\mp \nu$ display rate differences between "no-OSSF" tri-lepton channels if the sterile neutrino is Majorana, owing to additional LNV diagrams. Comparing both channels at the LHC with $\sim3000\,{\rm fb}^{-1}$, at least a 3$\sigma$ Dirac exclusion can be achieved for moderate disparities in mixings, e.g., $|U_{Ne}|^2 < 0.7 |U_{N\mu}|^2$ (Dib et al., 2016).

Low-energy experiments search for EDMs (e.g., ACME for electrons), LFV decays (Mu3e/Mu2e), and $0\nu\beta\beta$; high-energy searches include lepton-number-violating signals (e.g., same-sign dileptons at LHC), displaced vertices, and direct production (ILC/FCC-ee). Cosmological and astrophysical bounds also provide crucial constraints, particularly for dark matter models employing sterile Majorana portals.

7. Ultraviolet Completions, Symmetry Realization, and Model Variants

Majorana sterile fermions naturally arise in a wide class of theoretical frameworks:

• Flavor gauge models spontaneously break chiral family symmetries (e.g., SU(3)$_{\rm F}$), leading to dynamical majorana mass generation, seesaw structures, and a spectrum of majorons—some of which may be dark matter candidates (Smetana, 2011).
• SU(3) family symmetry models yield radiative corrections to tree-level Dirac seesaw structures, generating eV-scale sterile Majorana states simultaneously compatible with oscillation data and measurable 0νββ rates (Hernandez-Galeana, 2014).
• Dirac–Majorana models interpolate between purely Dirac and Majorana limits, allowing "pseudo-Dirac" and "Diracian" regimes and providing phenomenological flexibility in interpreting neutrino data and ultra-high energy neutrino anomalies (Nieuwenhuizen, 2018).
• Quantum Field Theory (QFT) perspectives reveal that standard "active/sterile" Majorana mass fields are actually not strictly self-conjugate under full charge conjugation but carry pseudoscalar-type lepton number, and only genuine Majorana fields (equal admixtures) are truly neutral (Ziino, 2014).

The choice of mass basis, presence of lepton-number symmetry breaking, and parameter tuning critically affect both phenomenology and the nature of the sterile sector.

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Majorana sterile fermions thus constitute a theoretically robust and phenomenologically rich class of SM extensions. Their implications range from neutrino oscillations and rare processes to cosmology and dark matter. Experimental searches and theoretical modeling jointly determine the landscape of viable scenarios within this sector.