Higgs–N_R Operators in Neutrino EFT

• Higgs–N_R Operators are effective field theory interactions between the Standard Model Higgs doublet and SM-singlet right-handed neutrinos, essential for neutrino mass models.
• They encompass both lepton-number–violating dimension-5 and lepton-number–conserving dimension-6 terms, each offering distinct phenomenological signatures at colliders.
• Simulation tools like FeynRules and UFO model files enable precise predictions of collider signals such as rare Higgs decays and heavy neutrino production.

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 (Titov, 13 Nov 2025).

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 (Titov, 13 Nov 2025):

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 (Titov, 13 Nov 2025).

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 (Titov, 13 Nov 2025).

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

$$\Gamma(h_1\to N_RN_R) = \frac{|g_{h_1N_RN_R}|^2}{16\pi m_{h_1}} \left(1-\frac{4m_N^2}{m_{h_1}^2}\right)^{3/2}$$

(Gao et al., 2019).

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 $BR=10^{-4}$ (Gao et al., 2019).

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 (Barducci et al., 2022):

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 (Barducci et al., 2022).

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 (Chala et al., 2020). 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 (Titov, 13 Nov 2025). 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 $S$ (Gao et al., 2019).
• ${\cal O}_{LNH}$, ${\cal O}_{HN}$, and dipole structures serve as probes of UV completions involving heavy scalars, vector-like leptons, or new gauge interactions at scale $\Lambda$ (Titov, 13 Nov 2025).
• 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 $v_S$ (Gao et al., 2019).

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 (Barducci et al., 2022).

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 (Titov, 13 Nov 2025). 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.