Decays of heavy scalars in the Grimus-Neufeld model

Imagine the universe as a giant, complex machine built from a standard set of Lego bricks. Physicists call this standard set the "Standard Model." For a long time, this model worked perfectly for explaining how most things in the universe behave. However, there are two big missing pieces in the puzzle: Dark Matter (the invisible glue holding galaxies together) and Neutrino Oscillations (tiny ghost-like particles that change their identity as they travel).

This paper introduces a new, slightly modified Lego set called the Grimus–Neufeld Model (GNM). The authors, Aurimas Vitkus, Simonas Draukšas, and Thomas Gajdosik, wanted to see if this new set could solve both problems at once.

Here is a simple breakdown of what they did and what they found:

1. The New Ingredients

To fix the Standard Model, the authors added two new pieces to their Lego set:

A Second Higgs Doublet: Think of the Higgs field as a "cosmic molasses" that gives particles mass. The Standard Model has one batch of this molasses. The GNM adds a second, secret batch.
A Sterile Neutrino: Imagine a neutrino that is so shy it doesn't even talk to the other particles in the Standard Model. This is the "sterile" one.

2. The Big Question: Is it Dark Matter?

In some versions of this new model (specifically when it looks like a model called the "Inert Doublet Model"), one of these new particles acts like a perfect Dark Matter candidate. It's heavy, invisible, and stable.

However, for something to be Dark Matter, it must be extremely stable. It needs to last longer than the entire age of the universe (about 13.8 billion years). If it decays (breaks apart) too quickly, it can't be the dark stuff holding galaxies together.

3. The Experiment: Calculating the "Break-Up" Rate

The authors acted like cosmic detectives. They asked: "If we create these heavy new particles, how fast will they break apart into lighter particles?"

They calculated every possible way these heavy particles could decay (break apart) at the most basic level of physics (called "tree-level"). They looked at:

Breaking into force-carrying particles (like W and Z bosons).
Breaking into other Higgs particles.
Breaking into charged particles (like electrons).
Breaking into neutrinos.

They used a mathematical "recipe" (the Lagrangian) to figure out the speed of these break-ups.

4. The Verdict: The Candidate is Too Short-Lived

Here is the punchline of their paper:

They focused on a specific particle in their model called the pseudoscalar (A). In a simplified version of their model (the "Inert Doublet" limit), this particle should be a Dark Matter candidate.

However, when they did the math, they found that this particle decays way too fast.

The Requirement: To be Dark Matter, it needs to live for billions of years.
The Reality: Their calculations showed that even under the most optimistic conditions, this particle would vanish in a fraction of a second (ranging from $10^{-20}$ seconds to just 13 seconds).

5. Why Did It Fail?

The reason for this failure is a bit like a security system.

In the "Inert Doublet Model" (the simpler version), there is a strict symmetry (a rule) that forbids the Dark Matter particle from breaking apart. It's like a vault that cannot be opened.
But in the Grimus–Neufeld Model, the authors needed to break that symmetry slightly to explain why neutrinos have mass. They had to add a tiny "crack" in the vault to let the neutrinos get their mass.
The Consequence: That tiny crack was enough to let the Dark Matter candidate escape and decay almost instantly. The very mechanism that gives mass to neutrinos also destroys the Dark Matter candidate.

Summary

The authors built a new theoretical model to explain Dark Matter and neutrino masses. They carefully calculated how long the new particles in this model would last. They concluded that while the model is mathematically interesting, the specific particle that could be Dark Matter is too unstable to actually be the Dark Matter we see in the universe. It breaks apart far too quickly to be the "cosmic glue" holding galaxies together.

In short: The Grimus–Neufeld Model is a clever idea, but the "Dark Matter" piece in this specific puzzle is too fragile to survive the age of the universe.

Technical Summary: Decays of Heavy Scalars in the Grimus–Neufeld Model

Problem Statement
The Standard Model (SM) fails to account for neutrino oscillations and dark matter (DM). While Inert Doublet Models (IDM) and sterile neutrino extensions offer potential solutions, the Grimus–Neufeld Model (GNM) proposes a specific extension combining a second Higgs doublet and a single sterile Majorana neutrino. In the GNM, the additional neutral scalars ( $H^0$ and $A$ ) and the sterile neutrino are Weakly Interacting Massive Particles (WIMPs). For these particles to serve as viable dark matter candidates, their lifetimes must exceed the age of the universe. However, the GNM requires specific symmetry-breaking parameters to generate neutrino masses (both tree-level seesaw and radiative masses), which may induce decays that render the scalars unstable. The central problem addressed is whether the pseudoscalar $A$ in the GNM can remain stable enough to function as a DM candidate, particularly in the limit where the model resembles the IDM.

Methodology
The authors perform a tree-level calculation of all possible two-body decays for the heavy scalars ( $H^0$ and $A$ ) within the GNM framework.

Lagrangian Formulation: The study utilizes the GNM Lagrangian, which includes the SM gauge sector, two Higgs doublets ( $H_1, H_2$ ), and a sterile neutrino $N$ . The potential is parametrized using Higgs doublet bilinears with complex couplings ( $\lambda_i$ ).
Mass Basis Rotation: The authors transition from the flavor basis to the mass basis. This involves diagonalizing the gauge boson sector (defining $W^\pm, Z, \gamma$ ) and the Higgs sector. The neutral Higgs fields ( $h_1, h_2, h_3$ ) are rotated into mass eigenstates ( $h^0, H^0, A$ ) via an orthogonal matrix $R$ dependent on mixing angles. The neutrino sector is diagonalized using a unitary matrix $U$ derived from Takagi factorization, incorporating the seesaw mechanism and radiative mass generation.
Decay Rate Calculation: The authors derive analytical expressions for the decay widths ( $\Gamma$ $Γ$ ) of the neutral scalars into:
- Gauge bosons ($VV'$).
- Lighter Higgs bosons ( $S'S''$ ).
- Charged Higgs and $W$ bosons.
- Charged fermions (leptons).
- Two Majorana neutrinos ( $n_j n_k$ ).
  Phase space integrals are computed for two-body final states, assuming massless final-state fermions where appropriate.
IDM Limit Analysis: To test the DM hypothesis, the authors evaluate the decay rate of the pseudoscalar $A$ in the "IDM limit." This limit is defined by specific parameter choices: CP conservation ( $s_{13}=0$ ), no mixing between the second doublet and the SM-like Higgs ( $s_{12}=1$ ), and vanishing $\lambda_6, \lambda_7$ . Crucially, they retain the non-zero Yukawa couplings required for the GNM's neutrino mass generation, which break the protective symmetries of the IDM.

Key Contributions

General Decay Amplitudes: The paper provides the first explicit tree-level expressions for the decay rates of GNM scalars into all two-body final states. Notably, the general amplitude for Higgs decays into other Higgs bosons (Eq. 44-45) is presented as a new result, though it reduces to known literature values in specific limits.
Lepton Flavor Violation (LFV): The analysis of decays into charged fermions (Eq. 58) demonstrates that LFV decays are directly proportional to the off-diagonal entries of the second Higgs doublet's Yukawa coupling matrix, with no possibility of cancellation between terms.
Neutrino Decay Channels: Due to the Majorana nature of the neutrinos, the decay amplitudes into neutrino pairs are symmetrized. The authors derive specific rates for decays into the five kinematically allowed neutrino pairs ( $n_2n_3, n_2n_4, n_3n_3, n_3n_4, n_4n_4$ ), highlighting that the lightest massless neutrino state does not couple to the Higgs bosons in this framework.
Lifetime Bounds: The paper derives a lower bound on the decay rate of the pseudoscalar $A$ in the IDM limit (Eq. 80). This bound is proportional to the light neutrino masses and the symmetry-breaking parameter $\Lambda$ (related to the mass splitting between $H^0$ and $A$ ).

Results
The calculation of the decay rates leads to a definitive conclusion regarding the dark matter viability of the GNM scalars:

Instability of the Pseudoscalar: In the limit where the GNM mimics the IDM, the requirement to generate neutrino masses (via the seesaw mechanism and radiative corrections) necessitates non-zero Yukawa couplings and a non-zero mass splitting between the scalar and pseudoscalar. These factors break the symmetries that would otherwise stabilize the pseudoscalar $A$ .
Quantitative Exclusion: Using extreme parameter values ( $m_4 = 10$ TeV, $m_A = 60$ GeV, $m_{H^0} = 800$ GeV), the authors calculate an upper limit on the lifetime of the pseudoscalar $A$ to be approximately 13 seconds.
Comparison to DM Requirements: A lifetime of 13 seconds is orders of magnitude shorter than the age of the universe ( $\sim 10^{17}$ seconds). Even with more conservative mass assumptions (e.g., $m_{H^0} = 300$ GeV), the lifetime remains far too short. Consequently, the pseudoscalar $A$ in the GNM cannot serve as a dark matter candidate.

Significance
The paper concludes that while the GNM shares a similar coupling structure with the IDM and the scoto-seesaw model, the specific symmetry breaking required to generate neutrino masses in the GNM is sufficient to destroy the stability of the scalar dark matter candidate. The "slight symmetry breaking" needed for the GNM's neutrino sector has a "large phenomenological impact," effectively excluding the GNM scalars as viable dark matter. The work establishes that the lower bounds on decay rates derived from neutrino mass constraints are incompatible with the stability requirements for dark matter in this specific model extension.