---
title: Higgs–N_R Operators in Neutrino EFT
url: https://www.emergentmind.com/topics/higgs-n_r-operators
type: topic
---

# Higgs–N_R Operators in Neutrino EFT

The Higgs–$N_R$ operator sector encompasses all effective interactions involving the Standard Model Higgs doublet $H$ and SM-singlet right-handed neutrinos $N_R$. These operators provide a comprehensive EFT framework for analyzing neutrino mass generation mechanisms, collider signatures of heavy fermion states, and radiative phenomena beyond the renormalizable SM+singlet extension. The leading operators by mass dimension include a unique dimension-five Majorana term and a catalog of dimension-six gauge-invariant (lepton-number–conserving) structures, each with specific phenomenological import. Recent developments provide systematic FeynRules and UFO model files for collider simulations [2511.10733].

## 1. Operator Basis and Classification

The operator basis in the $\nu$SMEFT is naturally organized by lepton-number violation and mass dimension. For $n_s$ generations of $N_{jR}$ and Higgs $H$, the complete Higgs–$N_R$ set up to dimension six is [2511.10733]:

**Dimension-5 (LNV):**
\[
{\cal O}_{NNH}^{jk} = (\overline{N_{jR}^c}\; N_{kR}) (H^\dagger H), \qquad j,k=1,...,n_s.
\]

**Dimension-6 (LNC):**
\[
\begin{aligned}
{\cal O}_{HN}^{jk} & = (\overline{N_{jR}}\gamma^\mu N_{kR}) (H^\dagger\,i\overleftrightarrow{D}_\mu H), \\
{\cal O}_{HNe}^{ji}& = (\overline{N_{jR}}\gamma^\mu e_{iR}) (\tilde{H}^\dagger\,i D_\mu H), \\
{\cal O}_{LNH}^{ij}& = (\overline{L_{i}}\tilde{H} N_{jR})(H^\dagger H), \\
{\cal O}_{NB}^{ij} & = \overline{L_i}\sigma^{\mu\nu}N_{jR}\,\tilde{H}\,B_{\mu\nu}, \\
{\cal O}_{NW}^{ij} & = \overline{L_i}\sigma^{\mu\nu}\sigma^I N_{jR}\,\tilde{H}\,W^I_{\mu\nu}.
\end{aligned}
\]
Rotation to the photon/Z basis via the weak angle is standard for dipole operators.

The operators are parameterized in the effective Lagrangian,
\[
{\cal L} = {\cal L}_{\rm SM} + \overline{N_{jR}}i\slashed{\partial}N_{jR} - [y_\nu^{ij}\overline{L_i}\tilde{H}N_{jR}+ \tfrac{1}{2}m_{N_j}\overline{N_{jR}^c}N_{jR} +{\rm h.c.}] + \frac{1}{\Lambda}{\cal L}_5 + \frac{1}{\Lambda^2}{\cal L}_6,
\]
with $c_X$ denoting dimensionless Wilson coefficients [2511.10733]. 

## 2. Phenomenological Structure after Electroweak Symmetry Breaking

Upon EWSB, $H \to (0, (v+h)/\sqrt{2})^T$, operators generate both new mass contributions and Higgs/weak-boson mediated interaction vertices:

- **${\cal O}_{NNH}$:** Direct Majorana mass shift, $m_N \to m_N + (c_{NNH} v^2/\Lambda)$, and $hN_RN_R$ ($\Delta L=2$) coupling.  
- **${\cal O}_{LNH}$:** Generates $h \nu_L N_R$ vertex, shifts Dirac Yukawa $y_\nu^{ij} \to y_\nu^{ij} + c_{LNH}^{ij}v^2/(2\Lambda^2)$, alters light neutrino–Higgs coupling.
- **${\cal O}_{HN}$:** Induces $Z_\mu N_R N_R$ vertex, $h Z_\mu N_R N_R$, relevant for $Z$–pole and high-energy searches.
- **Dipoles ${\cal O}_{NB}, {\cal O}_{NW}$:** After rotation, yield $N \to \nu \gamma/Z$ transitions, enable $N$ production and decay via photons/Z bosons.
- **${\cal O}_{HNe}$:** Provides $W_\mu N_R e_R$ vertex and $h W_\mu N_R e_R$ term.

Feynman rules implement these structures for collider simulations and theoretical calculations, including proper Lorentz and flavor chaining as in the FeynRules and UFO conventions [2511.10733].

## 3. Collider Signatures and Calculational Formulas

### Rare Higgs Decays and Production

The singlet–seesaw model mediates the process $h \rightarrow N_R N_R$ via Higgs–singlet mixing:
\[
g_{h_1N_RN_R} = y_S \sin\theta, \qquad
BR(h_1\to N_RN_R) = \frac{\Gamma(h_1\to N_RN_R)}{\Gamma_{\rm SM} + \Gamma(h_1\to N_RN_R)}
\]
with
\[
\Gamma(h_1\to N_RN_R) = \frac{|g_{h_1N_RN_R}|^2}{16\pi m_{h_1}} \left(1-\frac{4m_N^2}{m_{h_1}^2}\right)^{3/2}
\]
[1904.12325].

At hadron colliders, the cross section is
\[
\sigma(pp\to h_1\to N_RN_R) = \sigma(gg\to h_1)\times BR(h_1\to N_RN_R),
\]
with numerical estimates, e.g., $\sigma_{14\,\rm TeV}\simeq 5.5$ fb for $BR=10^{-4}$ [1904.12325].

### Operator-Induced Production Modes

Dimension-six operators enable processes such as $e^+e^-\to Zh\to ZN\nu$, $e^+e^-\to N\nu$ (via $\gamma^*, Z^*$), $e^+e^-\to NN$ through $Z$–mediated contact terms, and $e^+e^-\to N e$ via $W^*$ exchange [2201.11754]:

| Operator         | Main Vertex                                   | Production Channel                   | Leading Decays                       |
|------------------|-----------------------------------------------|--------------------------------------|--------------------------------------|
| ${\cal O}_{LNH}$ | $h \nu_L N_R$                                | $e^+e^- \to Zh \to Z N\nu$           | $N \to \nu f\bar{f}$                 |
| ${\cal O}_{LNB}/O_{LNW}$ | $A_\mu \nu N_R$ (dipole)              | $e^+e^- \to \gamma^*/Z^* \to N\nu$   | $N \to \nu\gamma,\ \nu Z^*,\ \ell W^*$|
| ${\cal O}_{HN}$  | $Z_\mu N_R N_R$                              | $Z \to N N$                          | $N \to 3f,\ \nu Z^*$                 |
| ${\cal O}_{HNe}$ | $W_\mu N_R e$                                | $e^+e^- \to W^* \to Ne$              | $N \to \ell W^* \to 3f$              |

Cutoff scales probed are $\sim$10–60 TeV depending on collider energy and channel [2201.11754].

## 4. Renormalization Group Evolution and Naturalness Constraints

One-loop RGE induces operator mixing in the dimension-six sector:
\[
\frac{d\alpha_i}{d\ln\mu} = \frac{1}{16\pi^2}\sum_j \gamma_{ij} \alpha_j
\]
with substantial mixing of dipole structures (${\cal O}_{NB}, {\cal O}_{NW}$) into Yukawa (${\cal O}_{LNH}$), leading to radiative neutrino masses,
\[
\delta m_\nu \sim \alpha_{LNH} v^3/\Lambda^2
\]
and invisible Higgs/Z decays ($\mathrm{Br}(h\to\text{inv}) \sim 10^{-14}–10^{-12}$) far below experimental sensitivities for $\Lambda\sim 1–100$ TeV [2006.14596]. Dipole moment requirements for XENON1T-scale anomalies produce radiative mass corrections $O(10^2-10^3)$ eV unless tuned.

## 5. Implementation in FeynRules/UFO and Simulation

Public model files systematically implement all Higgs–$N_R$ operators for collider-level event generation [2511.10733]. Key features:
- All vertex structures faithful to EFT expansion and flavor index structure.
- Parameter blocks for operator coefficients $c_X$, physical cutoff $\Lambda$.
- Vertices exported for MadGraph5 usage, allowing signal calculations for any specific parameter choices.
- Proper matching to low-energy seesaw relations; renormalization terms assure physical spectra without double counting.

Links for code and usage: https://github.com/arsenii-titov/vSMEFT.git

## 6. Connection to Neutrino Mass Models and UV Completions

The Higgs–$N_R$ sector is pivotal for both the minimal seesaw and extended EFT frameworks:
- ${\cal O}_{NNH}$ and singlet scalar/hybrid models generate the $N$ Majorana mass directly or through mixing with $S$ [1904.12325].
- ${\cal O}_{LNH}$, ${\cal O}_{HN}$, and dipole structures serve as probes of UV completions involving heavy scalars, vector-like leptons, or new gauge interactions at scale $\Lambda$ [2511.10733].
- Precision constraints and collider limits tightly bound the allowed parameters: e.g., $y_S\sin\theta \lesssim 10^{-4}$ at HL-LHC, $sin\theta \lesssim 10^{-6}$ at future 100 TeV hadron colliders for TeV-scale $v_S$ [1904.12325].

Discovery of rare Higgs decays such as $h\to N_R N_R$ or signals of displaced $N$ decays at future colliders would provide direct evidence for the scalar dynamics responsible for the seesaw-origin of neutrino mass, independent of active–sterile mixing angles. Current and future collider reach covers new physics scales $5–60$ TeV for various operator-induced channels [2201.11754].

## 7. Summary Table: Operator Landscape and Exclusion Reach

| Operator         | Production/Decay             | Collider Probes                  | $\Lambda$ Reach (TeV)            |
|------------------|-----------------------------|----------------------------------|----------------------------------|
| ${\cal O}_{NNH}$ | $h \to N_R N_R$             | LHC, future $pp$                 | $\sim$few $\to$ tens             |
| ${\cal O}_{LNH}$ | $h \nu_L N_R$               | FCC-ee, ILC, CLIC                | $10–30$                          |
| ${\cal O}_{LNB/LNW}$ | $N \to \nu \gamma$       | FCC-ee (Z-pole), high-energy      | $20–60$                          |
| ${\cal O}_{HN}$  | $Z \to N_R N_R$             | FCC-ee @ $Z$, ILC                | $\sim$5–6                        |
| ${\cal O}_{HNe}$ | $W N_R e$                   | CLIC, high-energy $e^+e^-/\mu\mu$| $10–30$                          |

All operator-induced phenomena are consistently simulated and tested in the published UFO/FeynRules models [2511.10733]. The Higgs–$N_R$ operator sector thus provides a structurally complete and phenomenologically rich avenue for probing both neutrino mass generation and new physics at colliders.

Source: https://www.emergentmind.com/topics/higgs-n_r-operators