New physics in the $ZZh$ vertex: One-loop contributions from a radiative seesaw model

This paper investigates one-loop contributions from a radiative seesaw model to the $ZZh$ vertex, finding that while $CP$-conserving anomalous couplings could reach experimentally accessible levels of order $10^{-3}$ at future lepton colliders, $CP$-violating contributions remain too suppressed to be detected.

The Gist

Imagine the universe as a giant, intricate machine built from a set of blueprints called the Standard Model. For decades, physicists have been checking these blueprints to see if they hold up. Recently, they found a crucial piece of the machine: the Higgs boson (the "God particle" that gives other particles mass).

However, there's a glitch in the machine. We know for a fact that neutrinos (tiny, ghost-like particles that pass through everything) have mass, but the original blueprints say they shouldn't. This suggests there are hidden parts of the machine we haven't seen yet—"New Physics."

This paper is like a team of mechanics (the authors) trying to figure out how these invisible neutrinos might be messing with a specific connection in the machine: the ZZh vertex.

The Analogy: The Three-Way Junction

Think of the ZZh vertex as a three-way traffic intersection where two heavy trucks (Z bosons) and a delivery van (the Higgs boson) meet.

• The Standard Model says this intersection works perfectly according to a strict traffic law. The trucks and van interact in a very specific, predictable way.
• The New Physics suggests that invisible "ghost drivers" (Majorana neutrinos) might be sneaking into the intersection, changing the traffic flow just a tiny bit.

The authors are asking: If these ghost drivers exist, how much do they change the traffic rules at this intersection?

The Secret Mechanism: The "Radiative Seesaw"

The paper uses a specific theory called a Radiative Seesaw Model.

• The Seesaw: Imagine a playground seesaw. On one side, you have heavy weights (heavy neutrinos). On the other, you have light weights (light neutrinos). The theory says the heavy weights push the light ones down, making them almost weightless.
• Radiative: In this specific version, the light weights don't get their mass immediately. Instead, they gain it slowly over time through a "loop" process (like a child slowly pumping their legs to get the seesaw moving). This means the light neutrinos are essentially "zero mass" at the start and only get a tiny bit of mass later.

The authors calculated what happens when these heavy and light neutrinos run in a loop around the Z-Z-H intersection.

The Findings: The "Whisper" vs. The "Silence"

The team ran complex simulations (using supercomputers and math) to see how much the ghost drivers change the traffic. They looked at two types of changes:

1. The "Normal" Change (CP-Conserving)
This is like the ghost drivers slightly speeding up or slowing down the traffic, but keeping the direction the same.

• The Result: They found that the traffic does change, but only by a tiny amount (about 1 part in 1,000).
• The Good News: This is a "whisper" that future super-advanced traffic cameras (called Future Lepton Colliders, like the ILC or CLIC) might actually be able to hear. If we build these machines, we might finally catch the ghost drivers in the act!

2. The "Twisted" Change (CP-Violating)
This is like the ghost drivers making the trucks spin in the wrong direction or driving on the wrong side of the road. This is a much stranger, more exotic effect.

• The Result: The authors found this effect is incredibly, almost impossibly small (1 part in 100,000,000,000,000,000).
• The Bad News: Even our most powerful future cameras will never see this. It's so quiet it's effectively silent.

Why Does This Matter?

Think of this paper as a treasure map.

• It tells us that while we can't find the "twisted" treasure (the CP-violating effects), the "normal" treasure (the CP-conserving effects) is hidden just within the reach of our future tools.
• It gives specific numbers to look for. If the future colliders measure the Z-Z-H intersection and find a deviation of about 0.1%, it would be a smoking gun proving that these heavy, ghost-like neutrinos exist and that our "Radiative Seesaw" theory is correct.

In a Nutshell

The authors took a complex theory about invisible neutrinos and calculated how they would nudge a specific particle interaction. They found that while some effects are too small to ever see, others are just on the edge of what our next generation of particle accelerators can detect. It's a hopeful sign that if we build these machines, we might finally solve the mystery of why neutrinos have mass.

Technical Summary

1. Problem Statement

The discovery of the Higgs boson at the LHC confirmed the Standard Model (SM) mechanism for electroweak symmetry breaking, yet the origin of neutrino masses remains unexplained. Neutrino oscillation experiments confirm non-zero neutrino masses, implying physics beyond the SM (BSM).
The authors investigate the $ZZh$ vertex (the interaction between two $Z$ bosons and the Higgs boson) as a sensitive probe for new physics. While the SM predicts a specific tree-level Lorentz structure for this vertex, BSM physics can induce anomalous couplings (deviations in strength and structure) and CP-violating effects.
Specifically, this paper focuses on a radiative variant of the Type-I seesaw mechanism (based on Ref. [33]) where:

• Light neutrino masses vanish at tree level and are generated radiatively.
• Heavy neutrinos form a quasi-degenerate spectrum.
• The goal is to calculate the one-loop contributions of both light and heavy Majorana neutrinos to the $ZZh$ vertex and determine if these effects are observable at future lepton colliders.

2. Methodology

Theoretical Framework

• Model: The authors utilize a specific seesaw variant where light neutrino masses are generated via two-point correlation functions, leading to a quasi-degenerate heavy neutrino spectrum ($m_{N1} \approx m_{N2} \approx m_{N3} \equiv m_N$).
• Effective Field Theory (EFT): The analysis employs the Strongly-Interacting Light Higgs (SILH) basis to parametrize the $ZZh$ vertex. The effective Lagrangian includes:
  • Dimension-6 operators: Generating CP-even couplings $\lambda_2, \lambda_3$.
  • Dimension-8 operators: Generating additional CP-even couplings $\lambda_4, \lambda_5$ (specific to this radiative model).
  • CP-odd operators: Generating the coupling $\tilde{\lambda}_1$.
• Vertex Parametrization: The effective vertex function $\Gamma_{\mu\nu}$ is decomposed into CP-even ($\Gamma^{even}$) and CP-odd ($\Gamma^{odd}$) parts, expressed in terms of the anomalous couplings $\lambda_j$ and $\tilde{\lambda}_1$.

Calculation Technique

• One-Loop Calculation: The authors compute the one-loop corrections to the $ZZh$ vertex arising from Majorana neutrino loops.
• Majorana Formalism: Special attention is paid to the Majorana nature of the neutrinos. Unlike Dirac fermions, Majorana fields allow for additional fermion contractions (Feynman diagrams) due to the charge-conjugation properties ($n^c = n$). The amplitude includes contributions from both Dirac-type and Majorana-type topologies.
• Regularization and Renormalization:
  • Dimensional Regularization: Used to handle ultraviolet (UV) divergences ($D = 4 - \epsilon$).
  • Passarino-Veltman Reduction: Implemented via FeynCalc and Package-X to reduce tensor integrals to scalar functions.
  • UV Finiteness: The anomalous couplings ($\lambda_{2,3,4,5}$ and $\tilde{\lambda}_1$) are found to be UV finite. Only the tree-level SM coupling ($\lambda_1$) requires renormalization.

Numerical Analysis

• Parameter Space:
  • Light neutrino mass: Fixed at the experimental upper bound $m_\nu = 0.45$ eV (degenerate).
  • Heavy neutrino mass ($m_N$): Ranged from 300 GeV to 3 TeV.
  • Mixing matrix ($C$): Parameterized via a complex matrix $\xi = \hat{\rho} e^{i\phi} \mathbb{1}_3$, with $\hat{\rho} = 0.65$ (consistent with CMS bounds).
• Kinematic Scenarios: Two production channels at future lepton colliders were analyzed:
  1. Higgsstrahlung ($e^-e^+ \to Zh$): One on-shell $Z$, one off-shell $Z^*$.
  2. Neutral-Current Vector Boson Fusion (VBF): Two off-shell $Z^*$ bosons fusing into a Higgs.

3. Key Contributions

1. Analytical Derivation: The paper provides the first complete analytical calculation of the one-loop $ZZh$ vertex corrections specifically within the radiative seesaw model where light neutrino masses are generated radiatively.
2. Dimension-8 Operators: The study identifies that this specific model induces effective interactions of mass dimension 8 (terms $\lambda_4$ and $\lambda_5$), which are distinct from the standard dimension-6 SMEFT contributions often studied.
3. CP-Violation Suppression: A rigorous derivation showing that CP-violating contributions in this specific model are heavily suppressed because they rely exclusively on interference between light and heavy neutrino sectors, whereas CP-conserving terms receive dominant contributions from heavy neutrino exchanges alone.
4. Phenomenological Mapping: The authors map the theoretical parameters to observable quantities ($\Delta_2, \Delta_3$ for Higgsstrahlung; $\lambda_{2,3,4,5}$ for VBF) and compare them against projected sensitivities of future colliders (ILC, CLIC, Muon Collider).

4. Results

CP-Conserving Effects

• Magnitude: The CP-conserving anomalous couplings can reach magnitudes of $\mathcal{O}(10^{-3})$.
  • In the Higgsstrahlung scenario, the linear combinations $|\Delta_2|$ and $|\Delta_3|$ reach $\sim 10^{-4}$ to $10^{-3}$.
  • In the VBF scenario, the individual couplings $|\lambda_2|$ and $|\lambda_3|$ reach $\sim 10^{-3}$.
• Dependence: These effects depend on the heavy neutrino mass ($m_N$) and the center-of-mass energy ($\sqrt{s}$). The contributions are largest for lower $m_N$ (near the 700 GeV lower bound) and higher $\sqrt{s}$.
• Observability: The predicted magnitude ($\sim 10^{-3}$) is potentially within the projected sensitivity of future lepton colliders like the ILC and CLIC, which aim for sensitivities in the range of $10^{-3}$ to $10^{-2}$.

CP-Violating Effects

• Magnitude: The CP-violating coupling $\tilde{\lambda}_1$ is found to be extremely small, of order $\mathcal{O}(10^{-15})$.
• Suppression Mechanism: This strong suppression arises because the CP-odd term requires interference between light and heavy neutrino sectors. Since light neutrino masses are tiny ($\sim$ eV) and the heavy masses are large ($\sim$ TeV), the interference term is negligible compared to the pure heavy-neutrino contributions that dominate the CP-even sector.
• Observability: These effects are completely unobservable with current or projected future experimental sensitivities (which are around $10^{-2}$ to $10^{-1}$).

5. Significance

• Testability of Radiative Seesaw: The paper demonstrates that while the radiative seesaw mechanism is a compelling solution to the neutrino mass problem, its specific signature in the $ZZh$ vertex is observable only in the CP-conserving sector at future high-energy lepton colliders.
• Differentiation from Other Models: The presence of dimension-8 operators ($\lambda_4, \lambda_5$) provides a unique fingerprint that could distinguish this radiative model from other BSM scenarios (like the standard Type-I seesaw or 2HDM) which typically only generate dimension-6 operators.
• Experimental Guidance: The results suggest that future lepton colliders should prioritize precision measurements of the CP-even $ZZh$ couplings in VBF and Higgsstrahlung channels. A deviation at the $10^{-3}$ level would be a strong hint of this specific radiative neutrino mass generation mechanism, whereas a null result for CP-violation is expected and consistent with the model.
• Theoretical Consistency: The work confirms that the anomalous couplings generated by this specific Majorana neutrino loop are UV finite, reinforcing the robustness of the effective field theory description for this process.

In conclusion, the paper establishes that the radiative seesaw model predicts detectable CP-conserving deviations in the $ZZh$ vertex at future colliders, while CP-violating effects remain negligible, offering a clear target for experimental verification of this specific neutrino mass generation mechanism.