Flavourful Vector-Like Lepton Model

• The Flavourful Vector-Like Lepton Model is a SU(5)-based extension that introduces new lepton doublets and down-type quarks to break the SU(5) mass degeneracy between charged leptons and down quarks.
• It modifies right-handed Z-boson couplings and imposes strict lepton flavor violation bounds, such as |Y^e₁₂| < 1.6×10⁻⁷, thereby impacting precision electroweak observables.
• Restricting mass rotations to three free parameters, the model offers robust predictions for fermion hierarchies, proton decay rates, and preserves gauge coupling unification via distinct scalar or fermion extensions.

A flavourful vector-like lepton model enhances the Standard Model (SM) gauge and flavour structure by introducing new lepton doublets (and often accompanying down-type vector-like quarks) that interact with the SM chiral states. Such extensions, especially when embedded in a unified gauge framework like minimal SU(5), can address long-standing issues in fermion mass relations, impact precision flavour and electroweak observables, and lead to distinct predictions for rare processes such as proton decay, all while preserving gauge coupling unification (Dorsner et al., 2014).

1. Embedding of Vector-Like Lepton Doublets in SU(5)

In the minimal Georgi–Glashow SU(5) scenario, all SM fermions are embedded within three copies of $\mathbf{10}$ and $\overline{\mathbf{5}}$ representations. However, this setup predicts degenerate GUT-scale masses for down-type quarks and charged leptons in each family ($m_d = m_e$), which is excluded empirically. By extending the matter sector with a single pair $(\mathbf{5}_4, \overline{\mathbf{5}}_4)$, the model introduces:

• A vector-like isosinglet down-type quark,
• A vector-like isodoublet lepton.

These extra states reside within the same SU(5) multiplet—crucial for maintaining gauge unification—and allow additional mixings in the charged lepton and down-type quark sectors. The low-energy representation under the SM gauge group yields an isodoublet lepton ($L_4$) among the five fields. The model structure is engineered such that mixing with this state breaks the undesired SU(5) GUT mass degeneracy by enabling distinct unitary rotations in flavour space for quarks and leptons.

2. Effects on Flavor Physics: Couplings and Constraints

The mixing of the vector-like lepton doublet with SM leptons modifies experimental observables primarily via induced changes in the couplings to electroweak gauge bosons:

• Z-boson Couplings: The effective Lagrangian for charged-lepton Z interactions receives new contributions through a right-handed matrix $Y^e$:

$$\mathcal{L}_Z \supset -\frac{g}{c_W} \left[ -Y^e_{ij} \bar{e}^i \gamma^\mu P_R e^j - \frac{1}{2}\bar{e}^i \gamma^\mu P_L e^i + s_W^2 \bar{e}^i \gamma^\mu e^i \right] Z_\mu$$

In contrast to the quark sector, where left-handed FCNCs dominate, the leading new physics alterations for leptons appear in the right-handed sector.

• Lepton Flavor Violation (LFV) Bounds: Off-diagonal entries of $Y^e$ (e.g., $|Y^e_{12}|$) are constrained at the level of $1.6 \times 10^{-7}$ due to nonobservation of processes like $\mu \to e$ conversion in nuclei. LFV three-body decays such as $\tau \to 3e$ and $\tau \to 3\mu$ also limit flavor-mixing elements. These constraints require the mixing angles introduced by the vector-like lepton to be especially small, sharply restricting the corresponding unitary rotations and directly impacting the possible structure of the mass matrices.

3. Mass Matrices, Flavor Rotations, and Predictivity

Charged lepton and down-type quark mass generation involves two classes of Yukawa operators:

• The usual $$\mathbf{10} \cdot \overline{\mathbf{5}} \cdot \mathbf{5}_H$$ term,
• New mixings due to the vector-like $\mathbf{5}_4, \overline{\mathbf{5}}_4$ multiplets.

The resulting $4 \times 4$ mass matrices for the charged lepton $(\mathcal{M}_E)$ and down quark $(\mathcal{M}_D)$ sectors take the form: $$\mathcal{M}_E = \begin{pmatrix} \text{diag}(y_i v') & (M^e_1, M^e_2, M^e_3)^\intercal \ 0 & |M^e_4| \end{pmatrix}, \qquad \mathcal{M}_D = \begin{pmatrix} \text{diag}(y_i v') & (M^D_1, M^D_2, M^D_3)^\intercal \ 0 & |M^D_4| \end{pmatrix}$$ where the $M^e_i, M^D_i$ encode further mixing from Higgs and adjoint VEVs.

Crucially, the mass diagonalization and associated flavour rotations between the gauge and mass eigenbases are parameterized by only three free parameters $(m_1, m_2, m_3)$, which enter a triplet of algebraic relations relating GUT-scale and low-energy masses: $$(m^{E}_1)^2 + (m^{E}_2)^2 + (m^{E}_3)^2 = \frac{m_1^2(1+|x_2|^2+|x_3|^2) + m_2^2(1+|x_3|^2+|x_1|^2) + m_3^2(1+|x_1|^2+|x_2|^2)}{1+|x|^2}$$ along with corresponding formulas for the products and determinants involving the mixing parameters $x_i$. This structure allows precise "detuning" of the naive SU(5) mass predictions, but only within a tightly constrained domain. The high degree of predictivity arises from the dependence of all relevant rotations on this triplet.

4. Parameter Space, Predictions for Proton Decay, and Numerical Results

• Allowed Parameter Space: The $(m_1, m_2, m_3)$ parameters are numerically limited so that the resulting low-energy fermion spectrum matches experimental data. Vector-like masses are benchmarked at $m_4^E = 400$ GeV (lepton) and $m_4^D = 800$ GeV (quark).
• Proton Decay:
  • The same flavour rotations required for mass diagonalization also control the rates for $d=6$ proton decay modes mediated by superheavy gauge bosons.
  • The partial widths for $p \to \pi^0 e^+$, $p \to \pi^0\mu^+$, $p \to\pi^+\bar\nu$, $p \to K^0\mu^+$ are explicitly computed and shown to depend sensitively on $m_1$ and on the symmetry structure of the up-type sector.
  • Notably, $\Gamma(p \to \pi^+ \bar{\nu})$ remains almost constant due to symmetries, whereas $\Gamma(p \to \pi^0 e^+)$ and $\Gamma(p \to \pi^0 \mu^+)$ vary with the required flavour rotations.
  • Conservative lower bounds on $m_{GUT}$ compatible with current experimental limits are given; e.g., with $m_4^E = 400$ GeV, $m_{GUT} \in [4.2, 5.7]\times 10^{15}$ GeV for $\alpha_{GUT} \simeq 0.033$.

5. Realistic SU(5) Extensions and Gauge Coupling Unification

Gauge coupling unification can be preserved in two fully realistic extensions:

• Scenario I: Introduction of a 50-dimensional scalar. The $50_H$ brings threshold corrections that realign the running of the couplings, enabling unification for light vector-like leptons and quarks. An explicit solution achieves $m_{GUT}\simeq 10^{16.25}$ GeV, $\alpha_{GUT}^{-1} \simeq 27.2$.
• Scenario II: Addition of two 24-dimensional fermion representations. Besides affecting running, these provide heavy fermions suitable for seesaw mechanisms (neutrino mass). In this setup, unification is possible for light vector-like quarks, but not for light leptons. For these parameters, unification yields $m_{GUT}\lesssim 5.3\times 10^{15}$ GeV, $\alpha_{GUT}^{-1} \simeq 35.0$.

Both models are consistent with SM mass and mixing data. They differ in the spectrum of allowed vector-like fermions and the corresponding $m_{GUT}$ and proton decay signature bands.

6. Phenomenological and Theoretical Implications

• Fermion Mass Relations: The introduction of a single vector-like pair can address the failure of simple SU(5) mass relations without invoking high-dimensional operators or complicated Higgs sectors. The scenario is highly predictive due to the restriction to three free parameters for the relevant flavour rotations; any further improvements in fermion spectroscopy or proton lifetime bounds will strongly constrain the model.
• Proton Decay: Branching ratios for $p \to \pi^0 e^+$ and $p \to \pi^0 \mu^+$ are sharply predicted, and the current experimental limits prefer relatively high GUT scales for models with light vector-like leptons.
• Electroweak and Flavor Physics: Modifications in Z couplings, especially in the right-handed charged lepton sector, provide powerful probes. LFV processes like $\mu \to e$ conversion restrict off-diagonal mixings at the $10^{-7}$ level.
• Gauge Coupling Unification: The two viable SU(5) completions with light vector-like fermions showcase alternative routes to reconciling TeV-scale new physics with precise unification, either via threshold corrections from new scalars or via additional adjoint fermions (enabling neutrino mass generation).
• Experimental Signatures: Tighter experimental bounds on proton decay, improved charged-lepton flavor-violating searches, and precision measurements of mass and mixing will further probe or constrain this framework.

7. Summary Table: Model Ingredients and Their Phenomenological Roles

Extension/Parameter	Phenomenological Impact	Comments
$(\mathbf{5}_4, \overline{\mathbf{5}}_4)$ pair	Breaks $m_d = m_e$ GUT degeneracy	Maintains gauge unification
3 parameters $(m_1, m_2, m_3)$	Controls all new mass diagonalization rotations	Highly predictive/severely constrained
Z right-handed coupling $Y^e_{ij}$	Modifies LFV and precision EW observables	$|Y^e_{12}| < 1.6 \times 10^{-7}$ (LFV limit)
Scalar $\mathbf{50}_H$	Realigns RGEs to enable unification, supports light VLLs	Scenario I
Two $\mathbf{24}_F$ fermions	Unification and neutrino seesaw mechanism	Scenario II, light VLLs not allowed
$m_4^E = 400$ GeV, $m_4^D = 800$ GeV	Benchmark vector-like lepton/quark masses	Used for numerical predictions
$p$ decay partial widths	Sensitive to $m_1$ parameter and VLL/VLQ scenario	Strong test for model and GUT scale

The minimal SU(5) "Flavourful Vector-Like Lepton Model" thus provides a predictive and tightly constrained solution to charged fermion mass relations, while offering correlated signatures in low-energy LFV observables, proton decay channels, and unification-scale physics, with all key flavour rotations determined by a small set of parameters (Dorsner et al., 2014).