Neutrino masses, anomalous magnetic moments and dark matter with vector-like fermions and an inert scalar doublet

This paper proposes a beyond-the-Standard-Model framework incorporating vector-like fermions and an inert scalar doublet protected by a $Z_2$ symmetry to simultaneously explain neutrino masses, the anomalous magnetic moments of electrons and muons, and dark matter, while remaining consistent with experimental constraints and offering testable LHC signatures.

The Gist

Imagine the Standard Model of particle physics as a very successful, high-end smartphone. It handles almost everything we throw at it: making calls, taking photos, running apps. But it has three glaring bugs that the manufacturer (nature) hasn't fixed yet:

1. The "Ghost" Problem: There's invisible stuff in the universe (Dark Matter) that holds galaxies together, but the phone's software doesn't know how to detect it.
2. The "Magnetic" Glitch: Two specific particles (the electron and the muon) are spinning slightly faster or slower than the math predicts. It's like a gyroscope wobbling in a way the manual says is impossible.
3. The "Weight" Mystery: Neutrinos (tiny, ghost-like particles) are supposed to be weightless, but we know they have a tiny bit of mass. The phone's code says they should be zero, but reality says otherwise.

This paper proposes a "software patch" to fix all three bugs at once. The author, Vandana Sahdev, suggests adding two new types of digital components to the phone's operating system: Vector-like Fermions (think of them as "twin" particles that come in left and right-handed pairs) and an Inert Scalar Doublet (a "silent" partner particle that doesn't interact with light but hangs around in the background).

Here is how this patch works, using some everyday analogies:

1. The "Silent Partner" and the "Twin" System

The author introduces a strict rule called $Z_2$ symmetry. Imagine a club with a bouncer.

• Standard Particles (like electrons and quarks) are VIPs with a "Green Badge" (+1). They can mingle freely.
• New Particles (the twins and the silent partner) have "Red Badges" (-1).
• The Rule: A Red Badge can never turn into a Green Badge on its own. They can only interact with other Red Badges or appear in pairs.

Because of this rule, the lightest Red Badge particle can never decay into anything else. It is stuck in the universe forever. This makes it a perfect candidate for Dark Matter. It's the "ghost" that is stable, invisible, and everywhere.

2. Fixing the "Weight" Mystery (Neutrino Mass)

In the standard model, neutrinos are weightless. In this new patch, they get their weight through a "backdoor" process.

• Imagine neutrinos trying to walk through a wall. They can't.
• But, they can borrow a "twin" particle and a "silent partner" to build a temporary bridge (a loop) to get across.
• This bridge is only built at the quantum level (one loop). Because the bridge is so complex and involves heavy, expensive materials (the new heavy particles), the neutrinos only get a tiny, tiny bit of mass.
• The Result: The math finally explains why neutrinos have that small, non-zero weight we observe.

3. Fixing the "Magnetic" Glitch (Anomalous Magnetic Moments)

The electron and muon are wobbling because they are interacting with the new particles in the background.

• Think of the electron as a dancer. In the old model, the music (magnetic field) was predictable.
• In this new model, the dancer is constantly bumping into the new "twin" particles and "silent partners" in the crowd. These bumps change the dancer's spin slightly.
• The author shows that if the "twin" particles are heavy enough (around the size of a TeV, which is like a heavy weight in particle terms) and interact just right, these bumps perfectly explain the wobbling we see in experiments.

4. The "Unification" Bonus

The paper also checks if this patch helps the phone's battery (gauge couplings) last longer.

• In physics, there are three different "forces" (like different battery drain rates). In the standard model, they almost meet at a single point if you zoom out far enough, but they miss each other.
• By adding these new particles, the author shows that the three forces actually do meet at a single point at very high energies. This suggests the universe might be running on a single, unified "operating system" at its core, which is a huge bonus for the theory.

5. The "Safety" Check (Constraints)

Any new software patch needs to make sure it doesn't crash the phone.

• Flavor Violation: The author checks if these new particles cause particles to change identities in forbidden ways (like a muon turning into an electron and a photon). The math shows that by carefully tuning the "Red Badge" rules, these crashes are avoided.
• Direct Detection: If Dark Matter is everywhere, shouldn't we bump into it in labs? The author shows that because the new particles are "silent" and the mass differences are just right, they slip through our detectors without triggering an alarm, which matches what we see today.

6. The "LHC" Test (Can we find them?)

The Large Hadron Collider (LHC) is like a high-speed crash test for these new particles.

• If we smash protons together hard enough, we might create these "twin" particles.
• Because of the "Red Badge" rule, they would decay into a cascade of lighter particles, eventually leaving behind the invisible Dark Matter.
• The signature would be a bunch of jets (debris) and leptons (electrons/muons) with a lot of "missing energy" (the invisible Dark Matter running away).
• The author suggests that if these particles exist at the predicted weights, the LHC might already have seen hints of them, or could find them soon by looking for these specific "missing energy" patterns.

Summary

The paper proposes a neat, single package deal:

1. Add two generations of "twin" particles and a silent scalar partner.
2. Use a symmetry rule to make the lightest one stable (Dark Matter).
3. Let the heavy twins interact with neutrinos to give them tiny mass.
4. Let them interact with electrons/muons to fix their magnetic wobble.
5. Show that this setup makes the universe's forces unify and doesn't break any existing safety rules.

It's a "one patch, three fixes" solution that keeps the universe's software running smoothly.

Technical Summary

Technical Summary: Neutrino masses, anomalous magnetic moments and dark matter with vector-like fermions and an inert scalar doublet

Problem Statement
The Standard Model (SM) fails to account for several critical observational phenomena: the existence of non-zero neutrino masses and mixings, the discrepancies in the anomalous magnetic moments ($g-2$) of the electron and muon, and the nature of Dark Matter (DM). While individual extensions of the SM can address these issues, constructing a unified framework that explains all three simultaneously without violating stringent constraints—such as those from charged lepton flavor violation (cLFV) processes (e.g., $\mu \to e\gamma$), direct DM detection, and cosmological bounds (BBN, CMB)—remains a significant challenge. Furthermore, the introduction of new fields often leads to large corrections to electroweak precision observables or requires fine-tuned Yukawa couplings.

Methodology and Model Construction
The authors propose a Beyond-the-Standard-Model (BSM) scenario extending the SM gauge group $SU(3)_C \otimes SU(2)_L \otimes U(1)_Y$ with:

1. Two generations of Vector-Like (VL) Fermions: These include VL leptons (doublets $L_{\alpha}$ and singlets $E^S_{\alpha}, N^S_{\alpha}$) and VL quarks (doublets $Q_{\alpha}$ and singlets $U^S_{\alpha}, D^S_{\alpha}$). The vector-like nature allows for gauge-invariant bare mass terms, avoiding the need for large Yukawa couplings that would otherwise disrupt electroweak precision tests.
2. An Inert Scalar Doublet ($\Phi$): A $Z_2$-odd scalar doublet that does not acquire a vacuum expectation value (VEV), ensuring the stability of the lightest $Z_2$-odd particle (LZ2OP).
3. $Z_2$ Symmetry: An unbroken discrete symmetry under which SM fields are even (+1) and all new fields are odd (-1). This forbids tree-level mixing between SM and new fermions, suppresses tree-level neutrino masses, and ensures the stability of the DM candidate.

The model is analyzed through several theoretical and phenomenological lenses:

• Gauge Coupling Unification: The renormalization group equations (RGEs) are solved up to two-loops to determine if the inclusion of these particles allows the SM gauge couplings to unify at a high scale ($M_G$). The authors estimate the required mass scales for the exotic quarks to achieve unification near $10^{16}$ GeV.
• Radiative Neutrino Mass Generation: Neutrino masses are generated at the one-loop level via the "scotogenic" mechanism, involving the inert scalar doublet and the heavy neutral VL fermions. The smallness of neutrino masses is attributed to the loop suppression and the specific mass hierarchy of the new particles, rather than tiny Yukawa couplings.
• Anomalous Magnetic Moments ($g-2$): The contributions to the electron and muon $g-2$ are calculated via one-loop diagrams involving the new fermions and scalars. The authors analyze the dependence on the scalar quartic coupling $\lambda_3$ and Yukawa textures to simultaneously explain the $g-2$ anomalies while suppressing cLFV decays.
• Dark Matter Phenomenology: The LZ2OP is identified as a DM candidate. The authors calculate the thermal relic abundance using the Boltzmann equation, accounting for co-annihilation processes with heavier $Z_2$-odd states. Constraints from direct detection (spin-independent cross-sections), indirect detection, Big Bang Nucleosynthesis (BBN), and the Cosmic Microwave Background (CMB) are applied.
• Collider Signatures: The production cross-sections for the exotic leptons at the LHC (13 TeV) are computed at Leading Order (LO) and Next-to-Leading Order (NLO). Decay cascades and final state topologies are mapped to existing LHC searches (specifically ATLAS and CMS SUSY searches) to derive mass limits.

Key Contributions and Results

• Unified Explanation: The paper demonstrates that a single minimal extension with two generations of VL fermions and an inert scalar doublet can simultaneously generate neutrino masses, explain the electron and muon $g-2$ anomalies, and provide a viable DM candidate.
• Yukawa Textures and cLFV Suppression: A critical finding is the identification of specific Yukawa matrix textures that allow for large contributions to $g-2$ (via the $C_{LR}$ term in the effective operator) while suppressing cLFV branching ratios (e.g., $Br(\mu \to e\gamma)$). The authors show that for $\lambda_3 \sim \mathcal{O}(1)$, the dominant contributions to $g-2$ come from diagrams with charged fermions in the loop, which can be tuned to satisfy experimental bounds.
• Gauge Unification: The inclusion of the VL quarks is shown to be necessary for achieving gauge coupling unification at a scale $M_G \gtrsim 10^{16}$ GeV, consistent with proton decay limits. The authors provide specific mass hierarchies for the exotic quarks (e.g., singlet up-type quarks at very high scales, while doublet and down-type singlets are lighter) to achieve this.
• Dark Matter Viability: Two distinct mass hierarchies (MH1 and MH2) for the exotic particles are explored. In both scenarios, the DM candidate is a singlet-doublet admixture. The relic density is achieved either through significant self-annihilation (requiring large mixing) or, more efficiently, through co-annihilation with nearly degenerate heavier states. The model satisfies direct detection constraints (XENON/LZ limits) and cosmological bounds (BBN/CMB) by ensuring the next-to-lightest particle decays rapidly enough or is stable on cosmological timescales.
• Numerical Validation: Using a random scan of $10^4$ parameter points, the authors verify that the model can reproduce the observed neutrino oscillation data, $g-2$ anomalies, and relic density while respecting all cLFV and direct detection limits. Two benchmark points (MH1 and MH2) are presented with specific mass spectra and Yukawa couplings.
• Collider Constraints: The paper reinterprets ATLAS searches for electroweakinos and sleptons. For Scenario II (where doublet leptons are light), masses of the doublet charged and neutral leptons ($\tilde{E}_D, \tilde{N}_D$) above 160 GeV are allowed. For Scenario I, a qualitative bound excludes singlet charged lepton masses up to 100 GeV.

Significance and Claims
The authors claim that their work provides a "very simple, and natural" extension of the SM that resolves multiple outstanding problems without inordinate fine-tuning. The significance of the work lies in:

1. Naturalness: The vector-like nature of the fermions allows for TeV-scale masses without large Yukawa couplings, preserving the validity of perturbative calculations and evading electroweak precision constraints.
2. Consistency: The model successfully navigates the tension between explaining $g-2$ anomalies and suppressing cLFV, a common failure point in similar models.
3. Unification Potential: The inclusion of vector-like quarks, while not strictly required for the low-energy observables, is motivated by the possibility of Grand Unified Theory (GUT) embedding, offering a pathway to gauge coupling unification.
4. Testability: The paper outlines specific collider signatures (multi-lepton/jet + MET) and provides reinterpretations of existing LHC data, making the scenario testable in current and future runs.

The authors remain modest regarding the scalar DM candidate, noting that it does not satisfy relic density observations for masses up to 500 GeV in their analysis, and emphasize that the fermionic DM candidate is the primary viable solution within their framework. They also note that a full reinterpretation of LHC bounds for their specific signal topologies requires dedicated simulation (e.g., Delphes), which is left for future work.