Fermion and Neutrino Portals

A neutral fermion can connect a hidden sector to Standard Model leptons through the gauge-invariant Yukawa interaction $\bar L\widetilde HN_R$. After electroweak breaking this interaction produces active–sterile mass mixing; whether it also generates Majorana masses, lepton-number violation, nonunitary light mixing, or long lifetimes depends on the singlet mass terms and symmetries. Exact diagonalization comes before the seesaw or small-mixing expansion.

Required background. Consistency Checklist for Standard Model Extensions supplies the consistency gates. Neutrino Mass Mechanisms supplies Majorana/Takagi conventions and the Weinberg-operator matching.

Helpful background. Integrating Out Heavy Fields supplies tree and loop matching beyond the leading formulas below.

The renormalizable neutrino portal

For gauge-singlet right-handed fermions $N_{RI}$,

$$\mathcal L\supset -\overline{L_\alpha}\,\widetilde H\,Y_{\alpha I}N_{RI} -\frac12\overline{N_{RI}^{,c}}\,M_{IJ}N_{RJ} +\text{h.c.}, \qquad \widetilde H=i\sigma_2H^*.$$

Assigning lepton number $L(N_R)=1$ makes the Yukawa interaction conserving; the Majorana matrix $M$ then breaks lepton number by two units. If an exact symmetry forbids $M$, the neutral states can instead be Dirac. Approximate lepton number can protect small symmetry-breaking masses, but the symmetry, charge assignment, and all allowed operators must be explicit.

After $\langle H\rangle=(0,v/\sqrt2)^{\mathsf T}$, $m_D=vY/\sqrt2$ and the symmetric neutral mass matrix is

$$\mathcal M_\nu= \begin{pmatrix} 0&m_D\\ m_D^{\mathsf T}&M \end{pmatrix}.$$

A unitary Takagi factor $\mathcal U$ satisfies $\mathcal U^{\mathsf T}\mathcal M_\nu\mathcal U=\operatorname{diag}(m_i)$ with $m_i\ge0$. An ordinary real-symmetric eigenvalue may be negative; the corresponding Majorana field is rephased so that the physical Takagi mass is positive.

Exact one-generation solution and seesaw residual

For real positive $m_D$ and $M$,

$$\mathcal M=\begin{pmatrix}0&m_D\\m_D&M\end{pmatrix}, \qquad \lambda_\pm=\frac{M\pm\sqrt{M^2+4m_D^2}}{2}.$$

Here $\lambda_-<0$. Rephasing its eigenstate by $i$ gives

$$m_{\rm light}=\frac{\sqrt{M^2+4m_D^2}-M}{2}, \qquad m_{\rm heavy}=\frac{\sqrt{M^2+4m_D^2}+M}{2}.$$

The real rotation obeys $\tan2\theta=2m_D/M$ before the Majorana phase is attached. For $m_D/M\ll1$,

$$m_{\rm light}=\frac{m_D^2}{M}-\frac{m_D^4}{M^3} +O\!\left(\frac{m_D^6}{M^5}\right),$$ $$m_{\rm heavy}=M+\frac{m_D^2}{M}-\frac{m_D^4}{M^3} +O\!\left(\frac{m_D^6}{M^5}\right), \qquad \theta=\frac{m_D}{M}+O\!\left(\frac{m_D^3}{M^3}\right).$$

At $(m_D,M)=(1,10)$ the exact masses are $\sqrt{26}\mp5$. The displayed truncations are $99/1000$ and $10099/1000$, and each absolute residual is

$$\sqrt{26}-\frac{5099}{1000} =1.95135928\ldots\times10^{-5}.$$

This checks both branches, the Takagi sign, and the expected next power, and it can be reproduced with exact symbolic arithmetic.

For several generations, define $\Theta=m_DM^{-1}$ when $M$ is nonsingular and $\|\Theta\|\ll1$. Then

$$m_\nu=-m_DM^{-1}m_D^{\mathsf T}+O(\Theta^4M),$$

while the light charged-current matrix is approximately

$$N_{\rm light}=\left(I-\frac12\Theta\Theta^\dagger\right)U_\nu +O(\Theta^4).$$

The complete matrix $\mathcal U$ is unitary; “nonunitarity” refers to the truncated light block after inaccessible heavy states are omitted. Integrating out $N_R$ produces the Weinberg operator at dimension five and a dimension-six kinetic/current correction proportional to $YM^{-1}M^{-1\dagger}Y^\dagger$. Their correlated origin is derived in Broncano, Gavela, and Jenkins 2003, §§2–3.

Hidden-sector completion and conserved charges

The neutrino portal can mediate communication with further singlets. For example, a hidden fermion $\chi$ and scalar $\phi$ may couple through

$$\mathcal L_{\rm hidden}\supset-y_\chi\phi\,\overline\chi_LN_R+\text{h.c.},$$

provided their exact gauge and global charges make the operator invariant. This addition changes widths and branching fractions and can stabilize a hidden particle. It does not alter the requirement that every new charge assignment pass anomalies, that the scalar potential be stable, and that accidental stable charged states be identified. A “neutrino portal” names an interaction channel, not a complete dark-sector model.

Width and lifetime regimes

Mixing inserts the heavy mass eigenstate into weak currents. Below electroweak-boson thresholds, representative three-body widths scale as

$$\Gamma_N\sim G_F^2M_N^5 \sum_\alpha|\Theta_{\alpha N}|^2,$$

with channel-dependent phase space, flavor factors, hadronic matrix elements, and Majorana/Dirac multiplicities. Well above those thresholds, two-body widths scale instead as

$$\Gamma_N\sim G_FM_N^3 \sum_\alpha|\Theta_{\alpha N}|^2.$$

Detailed exclusive formulas and their domains are organized in Atre et al. 2009, §§3–4. The proper lifetime is $\tau=1/\Gamma_N$ and a laboratory decay scale is $\beta\gamma/\Gamma_N$. “Prompt,” “displaced,” and “detector-stable” are therefore analysis-relative comparisons with declared spatial and timing scales, not universal intervals in $(M_N,\Theta)$.

If $\Gamma_N/M_N$ is not small, production and decay do not factorize into a narrow on-shell rate. If multiple neutral fermions overlap, their coherent propagator matrix can carry oscillation and CP phases. A lifetime classification never substitutes for an amplitude calculation or detector response.

Checks and failure modes

• Verify gauge invariance and lepton number before inserting $v$; the Majorana and Dirac limits are different symmetry statements.
• Compare exact singular values with the seesaw series and report a residual before using $m_D/M$ as a mixing angle.
• Check that $Y\to0$ restores zero active–sterile mixing and that $M\to\infty$ reproduces the matched operators.
• Preserve the full-unitarity identity before interpreting the nonunitary light subblock.
• Compute every open width with consistent mixing, phase space, and Majorana/Dirac counting; do not infer an experimental category from $c\tau$ alone.

Detailed light-neutrino mass and PMNS physics remains in Neutrino and Lepton Physics. Leptogenesis and thermal production belong to the thermal volume; current heavy-neutral-lepton limits belong to Effective Field Theory and Tests of the Standard Model.

References

• Atre, Anupama, Tao Han, Silvia Pascoli, and Bin Zhang. “The Search for Heavy Majorana Neutrinos.” Journal of High Energy Physics 2009, no. 5 (2009): 030. DOI.
• Broncano, A., M. B. Gavela, and E. Jenkins. “The Effective Lagrangian for the Seesaw Model of Neutrino Mass and Leptogenesis.” Physics Letters B 552 (2003): 177–184. DOI.