From Higgs physics to lepton flavour violation: current bounds and future prospects for vector-like lepton models

This paper presents a comprehensive phenomenological study of six vector-like lepton models with seesaw-like mass contributions, systematically analyzing their constraints and correlations across Higgs physics, lepton flavor violation, and collider observables in light of current and upcoming experimental data.

The Gist

Imagine the Standard Model of particle physics as a massive, incredibly detailed instruction manual for how the universe is built. For decades, this manual has worked perfectly, explaining almost everything we see. But there are a few pages that are blurry, and some questions the manual simply doesn't answer: Why do particles have such different weights (masses)? Why is there more matter than antimatter? And why do some particles seem to change their "flavor" (like a muon turning into an electron) in ways the manual says shouldn't happen?

This paper is like a team of detectives (physicists from TU Dresden) proposing a new, secret chapter to add to that manual. They are investigating a specific theory involving "Vector-Like Leptons" (VLLs).

Here is the breakdown of their investigation using simple analogies:

1. The New Characters: The "Vector-Like Leptons"

In our current manual, particles like electrons and muons are like distinct actors with specific roles. The Standard Model says they can't just swap roles or change their weight easily.

The authors propose adding six new types of actors to the cast. These are the Vector-Like Leptons.

• The Analogy: Imagine the Standard Model particles are like regular employees in a company. They have a specific job, a specific salary (mass), and they can only interact in very specific ways.
• The VLLs are like "Super-Interns" or "Shadow Employees." They are heavy, they can interact with the regular employees in weird ways, and they have a special ability: they can act as a bridge between the "regular" world and a "hidden" world.

2. The "Seesaw" Mechanism: How Mass is Made

The paper focuses on how these new VLLs change how particles get their mass.

• The Analogy: Think of a playground seesaw. On one side, you have a very heavy adult (the new VLL). On the other side, you have a tiny child (the electron or muon).
• In the old theory, the child's weight was fixed. In this new theory, the heavy adult pushes down, which actually lifts the child up or changes how heavy they feel. This is called a "Seesaw Mechanism."
• The result? The new VLLs can explain why particles have the specific masses they do, and they do it in a way that creates a strong link between the particle's weight and how it interacts with the Higgs Boson (the particle that gives everything mass).

3. The Detective Work: Connecting the Dots

The authors didn't just invent these characters; they checked if they fit the clues we already have. They looked at two main types of evidence:

A. The "Higgs" Clues (High Energy)

The Large Hadron Collider (LHC) smashes particles together to create Higgs bosons.

• The Clue: The Higgs boson is like a "magnet" that pulls on particles. The strength of this pull determines the particle's mass.
• The Prediction: If these new VLLs exist, they should change how strongly the Higgs pulls on electrons and muons. The paper calculates exactly how much this pull should change for each of the six VLL models.
• The Twist: The paper notes that recent measurements of the Higgs interacting with muons are very precise. This acts like a strict bouncer at a club, letting only certain VLL models pass the door.

B. The "Flavor" Clues (Low Energy)

This is where it gets really interesting. In the Standard Model, a muon should never turn into an electron. It's like a rule that says "Apples can never turn into Oranges."

• The Clue: Experiments like MEG II, Mu2e, and Belle II are looking for "Apples turning into Oranges" (a muon decaying into an electron and a photon, for example).
• The Prediction: The VLLs are the "chefs" that might be secretly swapping the fruit. The paper predicts that if these VLLs exist, we should see these "flavor violations" happening at specific rates.

4. The "Chiral Enhancement": The Secret Sauce

One of the paper's most important findings is a concept called "Chiral Enhancement."

• The Analogy: Imagine you are trying to push a heavy car. Normally, it's hard. But if you have a special lever (the VLLs), you can push it with much less effort.
• In physics terms, the VLLs act as a lever that amplifies the effects of the new physics. This means that even if the new particles are very heavy (hard to find directly), their "shadow" (their effect on other particles) is huge and easy to detect.
• The Catch: This amplification creates a very specific pattern. It's not just random chaos; it's a specific "fingerprint."

5. The Verdict: Can We Tell the Models Apart?

The authors ran millions of computer simulations (scans) to see which of the six VLL models survive the current experimental rules.

• The Result: Most models are still in the game, but they are squeezed into a very specific "safe zone."
• The Future: The paper argues that we don't need to wait forever to solve this mystery. The next generation of experiments (like HL-LHC, Belle II, and Mu3e) will be sensitive enough to see these "Apples turning into Oranges."
• The Goal: By measuring exactly how often these flavor swaps happen, and comparing them to how the Higgs interacts with particles, we can distinguish between the six different VLL models. It's like having six different suspects; by looking at the specific pattern of footprints (the correlations between different experiments), we can identify exactly which one did it.

Summary

This paper is a roadmap for the next decade of particle physics. It says:

1. We have a theory (Vector-Like Leptons) that could fix the "blurry pages" of the Standard Model.
2. We have a prediction: These particles create a unique, amplified signal in how particles change flavor and how the Higgs behaves.
3. We have a plan: Upcoming experiments will be able to catch these particles in the act, not by seeing them directly, but by seeing the "footprints" they leave behind in the behavior of electrons, muons, and tau particles.

It's a story of how heavy, invisible ghosts might be the key to understanding why the universe has the weight and flavor it does today.

Technical Summary

1. Problem Statement

The Standard Model (SM) fails to explain the origin of fermion mass hierarchies, the smallness of CKM mixing angles, and the observed matter-antimatter asymmetry. While many Beyond-the-Standard-Model (BSM) scenarios extend the Higgs sector, this paper focuses on extending the fermion sector by introducing Vector-Like Leptons (VLLs).

Specifically, the authors investigate a class of six distinct VLL models where the mixing between SM leptons and VLLs generates seesaw-like contributions to lepton masses. These models introduce a rich Yukawa sector that significantly alters:

• Lepton mass generation mechanisms.
• Higgs-lepton couplings.
• Lepton dipole moments (magnetic and electric).
• Charged Lepton Flavour Violation (CLFV) processes.

The central challenge addressed is determining the viable parameter space of these six models in light of recent experimental updates, specifically:

• The re-evaluation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment ($a_\mu$), which has shifted the SM prediction to be consistent with the Fermilab measurement (reducing the tension).
• Current and projected limits on Higgs decays ($h \to \mu\mu, \tau\tau$) and CLFV processes (e.g., $\mu \to e\gamma$, $\mu \to 3e$).

2. Methodology

The authors employ a comprehensive phenomenological approach combining analytical approximations with numerical scans.

A. Model Definition and Mass Diagonalization

• Six Models: Defined by the $SU(2)_L \times U(1)_Y$ quantum numbers of the added VLL multiplets (singlets, doublets, and triplets). The models are: $L \oplus E$, $L \oplus N$, $L_{3/2} \oplus E$, $L \oplus E_a$, $L \oplus N_a$, and $L_{3/2} \oplus E_a$.
• Mass Matrices: After Electroweak Symmetry Breaking (EWSB), the authors construct mass matrices for neutral, singly charged, and doubly charged leptons.
• Perturbative Expansion: They derive effective couplings by expanding in powers of the mass suppression factors $\xi = \lambda v / M_{VLL}$, where $\lambda$ represents Yukawa couplings and $M_{VLL}$ the vector-like masses.

B. Effective Couplings and Correlations

• Chiral Enhancement: The models feature a "seesaw-like" mass term ($m_i^{HHH}$) that breaks chiral symmetry. This leads to chirally enhanced contributions to observables proportional to the new Yukawa coupling $\lambda$, rather than being suppressed by the heavy mass scale.
• Key Correlations: The authors derive analytical relations connecting:
  • Lepton masses ($m_i$).
  • Higgs couplings ($Y^h_{e,ii}$).
  • Dipole moments ($\Delta a_i, d_i$).
  • These are summarized by the correlation equation: $R_{ii} = (1 - \frac{\Delta a_i}{2\omega_i Q})^2 + (\frac{m_i d_i}{e \omega_i Q})^2$, where $Q$ is a model-dependent coefficient.

C. Constraints and Scans

• Flavor-Independent Constraints: Applied limits from Electroweak Precision Observables (EWPO) like $Z$-pole couplings and oblique parameters ($S, T, U$), as well as LHC direct search limits and vacuum stability requirements ($|\lambda| \lesssim 0.5$).
• Flavor-Violating Observables: Calculated branching ratios for radiative decays ($\ell_i \to \ell_j \gamma$), three-body decays ($\ell_i \to 3\ell$), $\mu-e$ conversion in nuclei, and bosonic decays ($Z, h \to \ell_i \ell_j$).
• Numerical Scans: Two random parameter scans were performed:
  • Scan I: CP-conserving (real couplings).
  • Scan II: CP-violating (complex phases).
  • Scans were filtered against current experimental bounds (LEP, LHC, MEG, Mu3e, etc.) to identify viable regions.

3. Key Contributions

1. Updated Phenomenology: The paper updates previous studies by incorporating the 2025 Theory Initiative White Paper results for $a_\mu$ and the latest Higgs signal strength measurements. This shifts the preferred parameter space significantly, disfavoring large opposite-sign contributions to $a_\mu$ that were previously popular.
2. Chiral Enhancement vs. Dipole Dominance: The authors rigorously distinguish between chiral enhancement (large $\lambda$) and dipole dominance (where CLFV is driven solely by dipole operators). They demonstrate that while chiral enhancement is common in these models, dipole dominance is often excluded because tree-level $Z$-exchange contributions (non-chirally enhanced) compete with loop-induced dipole terms.
3. Model Discrimination Patterns: The study identifies unique "fingerprints" for each of the six models. These include specific correlations between:
   • Higgs signal strengths ($R_{ii}$) and dipole moments ($\Delta a_i$).
   • CLFV ratios (e.g., $BR(\mu \to 3e) / BR(\mu \to e\gamma)$).
   • $Z$-coupling corrections and CLFV rates.

4. Key Results

A. Impact of Updated $a_\mu$ and Higgs Data

• The new $a_\mu$ measurement (consistent with SM) combined with $h \to \mu\mu$ constraints forces the VLL contribution to $\Delta a_\mu$ to be small ($\approx 0$).
• This restricts the BSM Yukawa coupling $\lambda$ to moderate values ($|\lambda| < 0.5$), avoiding the large coupling regime required for the "opposite-sign" solution.
• Consequently, the viable parameter space is confined to regions where the new physics contribution to the muon mass is moderate.

B. CLFV Predictions and Future Prospects

• Observability: In the viable parameter space, future experiments like MEG II, Mu3e, Mu2e/COMET, and Belle II have a high probability of detecting signals.
• Dipole Dominance Ratios: The ratio $D_{3e} = BR(\mu \to 3e)/BR(\mu \to e\gamma)$ is a powerful discriminator.
  • If $D_{3e} \approx 1$, it implies dipole dominance (requiring large $\lambda$ or specific field dominance).
  • If $D_{3e}$ aligns with "single-field limits" (specific values listed in Table 7), it suggests hierarchical couplings.
  • Intermediate values indicate contributions from both VLL fields.
• Electron vs. Muon EDM: The electron EDM ($d_e$) provides a stringent constraint on CP-violating phases, often excluding large parameter regions. The muon EDM ($d_\mu$) is less constrained but correlated with $h \to \mu\mu$.

C. Model Discrimination

• $L \oplus E$ and $L \oplus N_a$: Predict identical correlations (same $Q$ coefficient).
• $L_{3/2} \oplus E$ and $L_{3/2} \oplus E_a$: Share a unique correlation pattern.
• $L \oplus E_a$ and $L \oplus N$: Each predicts a unique correlation.
• Tau Sector: CLFV decays involving $\tau$ leptons (e.g., $\tau \to 3\mu$, $Z \to \tau\mu$) offer distinct separation power, particularly when combined with measurements of $Z$-coupling corrections.

5. Significance

This work provides a roadmap for the next generation of experiments to test Vector-Like Lepton models.

• Complementarity: It highlights that no single observable is sufficient; a combination of Higgs physics, precision electroweak measurements, and low-energy CLFV searches is required to distinguish between the six models.
• Theoretical Insight: It clarifies the interplay between chiral symmetry breaking and loop effects, showing that the "dipole dominance" assumption common in other BSM scenarios does not automatically apply here.
• Experimental Guidance: The paper identifies specific observables (e.g., the ratio of $Z \to \tau\mu$ to $\tau \to 3\mu$, or the correlation between $h \to \mu\mu$ and $a_\mu$) that can definitively rule out or confirm specific VLL representations in the coming decade at the HL-LHC, FCC-ee, and dedicated flavor factories.

In summary, the paper demonstrates that while the six VLL models share a common mechanism for mass generation, they yield distinct, testable phenomenological patterns that will be accessible to upcoming high-precision experiments, offering a robust path to discovering physics beyond the Standard Model.