Leptoquark Variant of the Zee Model

• The paper introduces a leptoquark extension of the Zee model that generates radiative Majorana neutrino masses via two-loop diagrams using scalar leptoquarks and diquarks.
• The model details a specific scalar field content and Yukawa structure to correlate neutrino oscillation data with charged lepton flavor violation and distinct collider signatures.
• The framework connects low-energy observables with gauge coupling unification and predicts unique implications for neutrinoless double beta decay and flavor anomalies.

The leptoquark variant of the Zee model, often termed the colored Zee–Babu model (cZBM), generalizes the two-loop radiative neutrino mass construction of the original Zee–Babu scenario by introducing scalar leptoquarks and diquarks in place of singly and doubly charged scalar singlets. These colored scalars mediate new lepton-number-violating, baryon-number-conserving interactions that naturally yield small Majorana masses for the neutrinos, and predict distinctive correlations among charged lepton flavor violation (cLFV), collider observables, and rare processes such as neutrinoless double beta decay (0νββ). Additionally, variants of this framework accommodate connections to flavor anomalies observed in $B$ physics and to the muon anomalous magnetic moment.

1. Gauge Structure and Field Content

The minimal cZBM extends the Standard Model (SM) by two fundamental scalars:

• Scalar leptoquark (Δ): $\Delta \sim (3, 1, -1/3)$ under $SU(3)_c\times SU(2)_L\times U(1)_Y$
• Scalar diquark (S): $S \sim (6, 1, -2/3)$

In alternative conventions, the leptoquark may be denoted $S_1 \sim (\overline{\bf3}, {\bf1}, +1/3)$ and the diquark $\omega_1 \sim ({\bf6}, {\bf1}, -2/3)$, preserving the loop topology for neutrino mass generation (Chang et al., 2016, Chen et al., 2022).

Some versions introduce a $Z_2$ symmetry that controls the structure of Yukawa couplings, preventing tree-level contributions and enforcing radiative mass generation. For example, scalar leptoquarks $\phi\sim(3,1,-1/3)$ and vectorlike quark doublets $A_{L,R}\sim(3,2,-5/6)$ may be assigned specific $Z_2$ parities to forbid tree-level seesaw mechanisms (Popov et al., 2016).

2. Yukawa Sector and Scalar Potential

The renormalizable Yukawa and scalar terms governing the cZBM interactions comprise:

• Generic leptoquark and diquark Yukawa couplings:

$$\begin{aligned} \mathcal{L}_Y \supset &\; - [\overline{L_i^C}(Y_L)_{ij} i\sigma_2 Q_j + \overline{(\ell_R^i)^C}(Y_R)_{ij} u_{Rj}]\,\Delta^* \ &\; - \overline{(d_R^i)^C}(Y_S)_{ij} d_{Rj}\,S^* + y_\Delta^{ij} \overline{(u_R^i)^C} d_{Rj} \Delta + \mathrm{h.c.} \end{aligned}$$

Here $Y_S$ is symmetric in generation indices. For collider safety and minimal flavor violation, one can set $y_\Delta=0$ and $Y_R=0$ (Chang et al., 2016).

• The scalar potential includes a trilinear cubic interaction:

$V \supset \mu\,\Delta^*\Delta^* S + \mathrm{h.c.}$

with $\mu$ of order the TeV scale.

In the mass basis, flavor structure can be enforced via specific Yukawa textures to satisfy constraints from flavor-changing-neutral-current (FCNC) and 0νββ processes (Chen et al., 2022).

3. Two-Loop Neutrino Mass Generation

The cZBM realizes radiative Majorana neutrino mass via the two-loop diagram depicted as: $$\nu_{L_i} \xrightarrow{Y_L} \Delta \xrightarrow{d_j} \Delta \xrightarrow{Y_L^T} \nu_{L_{j'}} \xleftarrow{S; Y_S} d_{j'}^C$$ closed by the $\mu\,\Delta^*\Delta^* S$ cubic interaction.

The effective Majorana mass matrix is (Chang et al., 2016): $$(M_\nu)_{ii'} = 24 \mu \sum_{j,j'} (Y_L)_{ij} m_{d_j} I_{jj'} (Y_S^\dagger)_{jj'} m_{d_{j'}} (Y_L^T)_{j'i'}$$ where $I_{jj'}$ denotes the two-loop integral over momenta, which can be approximated as $I_{jj'}\simeq I_\nu$ for $m_d \ll m_\Delta, m_S$: $$I_\nu \simeq \frac{1}{(4\pi)^4} \frac{\pi^2/3}{M^2} \tilde{I}\left(\frac{m_S^2}{m_\Delta^2}\right),\quad M \equiv \max(m_\Delta, m_S)$$ Defining $$\omega_{jj'}=24 \mu I_\nu m_j m_{j'} (Y_S^\dagger)_{jj'}$$, the mass matrix is compactly $M_\nu=Y_L\,\omega\,Y_L^T$.

A plausible implication is that the structure of $Y_L$ and the scalar sector directly correlates the observed neutrino oscillation data, charged-lepton flavor violation, and collider signals.

4. Phenomenological Correlations and Flavor Constraints

Charged Lepton Flavor Violation (cLFV)

One-loop diagrams induce branching ratios for $\ell\to\ell'\gamma$ and $Z\to\overline{\ell}\ell'$:

• For $Y_R=0$,

$$B(\ell\to\ell'\gamma) \simeq \frac{12\pi^2}{G_F^2 m_\ell^2} |d_R^{\ell\ell'}|^2$$

where

$$d_R^{\ell\ell'} = -\frac{N_c e}{16\pi^2 m_\Delta^2} \sum_{q=u,c,t} m_\ell (Y_L^*)_{\ell' q} (Y_L)_{\ell q} F_1(r_q)$$

with $F_1(0)=1/12$ and $r_q = m_q^2/m_\Delta^2$.

Lower bounds on cLFV branching ratios are robustly predicted, e.g. $B(\mu\to e \gamma)\gtrsim 3\times10^{-16}$ (normal hierarchy), while double-ratio observables such as $R_5=B(\tau\to\mu\gamma)/B(\tau\to e\gamma)$ can discriminate neutrino mass ordering (Chang et al., 2016).

Neutrinoless Double Beta Decay (0νββ)

Two-loop cZBM predicts both standard light-neutrino exchange ($\langle m_{ee}\rangle$) and short-range leptoquark-induced contributions to 0νββ. The latter arise via tree-level exchange of $S_1$ and $\omega_1$ (Chen et al., 2022): $$[T_{1/2}]^{-1}\propto\left[\epsilon_\nu\mathcal{M}_\nu + \epsilon_1^{RRL}\mathcal{M}_1 + \epsilon_2^{RRL}\mathcal{M}_2\right]^2$$ Matching coefficients depend on the cubic vertex and specific Yukawas, e.g.: $$\epsilon_1^{RRL}= \frac{1}{48} \frac{2m_p G_F^2 V_{ud}^2}{M_{S_1}^4 M_{\omega_1}^2} 4 (y_{1SL}^{\prime 11})^2 z_{1\omega}^{11} \mu_1$$ A nontrivial feature is that the 0νββ amplitude can be suppressed ("hidden 0νββ") if new-physics and light-neutrino contributions cancel for tuned values of $(y_{1SL}^{\prime 11})^2 z_{1\omega}^{11}$ and $\langle m_{ee}\rangle$.

Flavor Anomalies

cZBM also provides tree or loop-level contributions to flavor observables such as $R_{D^{(*)}}$, $(g-2)_\mu$, and $R_K$ via the exchange of scalar leptoquarks:

• The $(g-2)_\mu$ anomaly is addressed by chirally-enhanced Yukawa products
• Tree-level $b\to c\tau\bar{\nu}$ requires specific products of leptoquark couplings, constrained by $B$-physics data (Popov et al., 2016).

5. Collider Signatures and Experimental Searches

The decay branching ratios of the scalar leptoquark $\Delta$ are sharply predicted when $Y_R=0$:

• $$\Gamma(\Delta\to \ell_i u_j) = \Gamma(\Delta\to \nu_i d_j)$$, leading to
• The branching fraction to charged lepton + quark is $50\%$ (Chang et al., 2016).

Nuanced neutrino hierarchy-dependent patterns arise:

• Inverted hierarchy: either $B_{\Delta e}\approx 1$ or $$(B_{\Delta\mu}\approx 0.55, B_{\Delta\tau}\approx 0.45)$$
• Normal hierarchy: $B_{\Delta e}\lesssim 0.3$ with $B_{\Delta \mu}+B_{\Delta \tau}\approx 0.7 – 1.0$
• Pure muon or tau exclusive decays are disallowed.

Collider limits (e.g., from LHC searches) are directly correlated, with $m_\phi\gtrsim625$ GeV for $b$-jet decays and $m_\phi\gtrsim850$ GeV for $\mu$-jet final states (Popov et al., 2016).

6. Gauge Coupling Unification and Vacuum Stability

The presence of colored leptoquarks and vector-like quarks leads to significant shifts in the gauge β-functions:

• For $\phi\sim(3,1,-1/3)$: $\Delta b_1=+1/30$, $\Delta b_2=0$, $\Delta b_3=+1/3$
• For each $A_{L,R}\sim (3,2,-5/6)$: $\Delta b_1=+5/6$, $\Delta b_2=+2$, $\Delta b_3=+4/3$

With $A_{L,R}$ and $\phi$ at the TeV scale, the three SM gauge couplings unify at $\sim 10^{15.5}$ GeV with unification quality $|\delta U| \simeq 0.015$ (Popov et al., 2016).

Vacuum stability is improved: the $\phi$-Higgs portal coupling $g_{h\phi}$ provides a positive one-loop correction to the Higgs quartic $\lambda_H$, sufficient for $g_{h\phi}\gtrsim0.4-0.6$ to preserve $\lambda_H>0$ up to the GUT scale. Larger Yukawas and two-loop terms can threaten stability but remain safe for perturbative couplings.

7. Experimental Constraints and Prospects

Tree-level four-fermion processes, neutral-meson mixing, and cLFV searches place stringent limits on the relevant Yukawa couplings:

• For $M_{S_1}\sim1.5$ TeV, $|y_{1SL}^{\prime 11}|<0.12$ and $|y_{1SL}^{\prime 31}|<0.16$ from rare decays (Chen et al., 2022).
• $(g-2)_\mu$ enhancement requires $$Re[y_{1SR}^{32} y_{1SL}^{\prime * 32}]\sim8\times10^{-2}$$ for $M_{S_1}\approx 1.5$ TeV.

Next-generation experiments with $0\nu\beta\beta$ sensitivities reaching $T_{1/2}\sim10^{27}–10^{28}$ yr can probe the tuning between new physics and light-neutrino exchange, especially using multiple isotopes to address the possibility of "hidden" 0νββ (Chen et al., 2022). Collider searches are also refined by the predicted $50\%$ lepton + jet branching fraction.

8. Synthesis and Significance

The leptoquark variant of the Zee model establishes an integrated framework for addressing radiative neutrino masses, lepton flavor violation, and TeV-scale collider phenomenology. The correlated predictions for low-energy flavor observables, distinctive collider signatures, and gauge unification are tightly tied to the underlying scalar and Yukawa structure. The possibility of tuning short-range contributions to neutrinoless double beta decay against the light-neutrino amplitude underscores the relevance of multi-isotope searches.

A plausible implication is that signal nulls in one isotope for $0\nu\beta\beta$ do not rule out Majorana neutrino mass in this framework, and combined data are necessary for robust exclusion or confirmation.

The model accommodates connections to observed flavor anomalies and $(g-2)_\mu$, further stimulating experimental programs in cLFV, colliders, and rare process detection (Chang et al., 2016, Popov et al., 2016, Chen et al., 2022).