Scalar Leptoquark Doublet ̃R₂ (3,2,1/6)

Updated 8 September 2025

Scalar leptoquark doublet ̃R₂ (3,2,1/6) is a minimal Standard Model extension that introduces a color triplet weak doublet with hypercharge 1/6 and renormalizable Yukawa couplings linking right-handed down quarks to left-handed leptons.
Its interactions yield suppressed charged lepton flavor violation, enable radiative neutrino mass generation, and offer distinctive contributions to B-physics anomalies through modified Wilson coefficients.
Implementation of a discrete Z₃ gauge symmetry and careful cancellation mechanisms control baryon-number violation, ensuring compatibility with experimental bounds and paving the way for testable collider signatures.

The scalar leptoquark doublet $\widetilde{R}_2({3},{2},1/6)$ is a minimal extension of the Standard Model (SM) by a single scalar multiplet that transforms as a color triplet, weak doublet, and has hypercharge $1/6$. Its interactions, flavor structure, phenomenological implications, and constraints have been systematically studied in multiple theoretical and experimental contexts. The %%%%2%%%% doublet’s defining property is its coupling of right-handed down-type quarks to left-handed lepton doublets via a renormalizable Yukawa interaction. This structure gives distinctive predictions for charged lepton flavor violation, baryon number violation, B-physics anomalies, neutrino mass generation, and collider signatures. Below, key aspects of the $\widetilde{R}_2$ doublet are synthesized and organized for advanced research readership.

1. Field Content and Renormalizable Interactions

The $\widetilde{R}_2$ doublet transforms under the SM gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$ as $(\mathbf{3}, \mathbf{2}, 1/6)$ , and is conventionally represented as: $\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$ where $V_\alpha$ carries electric charge $+\frac{2}{3}$ and $Y_\alpha$ carries $-\frac{1}{3}$ (with $\alpha$ the color index). The renormalizable interaction Lagrangian is: $\mathcal{L}_{\widetilde{R}_2} = - \lambda_d^{ij}\, \bar{d}_{R}^{\,i} (\widetilde{R}_2^T\, \epsilon)\, L_{L}^{\,j} + \mathrm{h.c.},$ with $\lambda_d^{ij}$ an arbitrary complex Yukawa matrix, $d_R^i$ the right-handed down-type quark, $L_L^j$ the lepton doublet, and $\epsilon$ the $SU(2)$ antisymmetric tensor. $\widetilde{R}_2$ does not couple to up-type quarks, nor is it capable of mediating proton decay at the renormalizable level due to gauge and Lorentz structure (Arnold et al., 2013).

2. Chirality Structure and Flavor Violation

Interactions with $\widetilde{R}_2$ link right-handed down-type quarks to left-handed leptons. In loop-induced processes such as $\mu\to e\gamma$ , there is no enhancement from the top quark mass; the chiral flip in the loop must occur on the external lepton line or, at best, the b-quark line for specific flavor assignments. As a result, predicted rates for charged lepton flavor-violating (CLFV) processes (e.g., $\mu\to e\gamma$ , $\tau\to\mu\gamma$ ) are suppressed, scaling as $m_{\ell_\text{ext}}^2$ (with $m_{\ell_\text{ext}}$ the external lepton mass) (Arnold et al., 2013). This is in contrast to models such as $R_2(3,2,7/6)$ that allow for top mass enhancement and thus much more stringent bounds. In multi-leptoquark scenarios, however, cancellation effects—enabled by mixing between doublets and triplets—can dramatically reduce CLFV amplitudes and allow for sizable Yukawa couplings while remaining under experimental constraints (Baek et al., 2015).

3. Baryon Number Violation and Discrete Gauge Symmetries

While $\widetilde{R}_2$ is safe from proton decay at the renormalizable level, nonrenormalizable dimension-five operators, such as

$\mathcal{O} = \frac{1}{\Lambda} g^{ab}\, d_R^a d_R^b (H^\dagger \widetilde{R}_2)\, \epsilon,$

lead to baryon-number violating processes ( $p\to K^+ \nu$ , $n\to e^- K^+$ ), potentially at rates exceeding experimental bounds for $m_{\widetilde{R}_2} \lesssim 10^4$ TeV unless couplings are finely tuned (Arnold et al., 2013). To control such operators, the imposition of a $Z_3$ discrete gauge symmetry—defined as $\exp[2\pi i(B-L)/3]$ —forbids them without affecting the allowed renormalizable couplings. This can be realized by gauging $B-L$ and breaking it appropriately to leave an unbroken $Z_3$ subgroup.

4. Phenomenological Roles in Charged and Neutral Current Anomalies

The $\widetilde{R}_2$ doublet participates directly in B-physics anomalies and neutrino physics via its fundamental interactions:

4.1 B-Physics Anomalies

$R_K$ and $R_{K^*}$ : Tree-level exchange of $\widetilde{R}_2$ modifies the neutral-current $b\to s\mu^+\mu^-$ transitions via right-handed quark currents, affecting Wilson coefficients such as $C'_9 = -C'_{10}$ (Bečirević et al., 2016). Specifically, the operator

$\mathcal{O}_V^{RL} = (\bar{s}_R \gamma^\mu b_R)(\bar\ell_L\gamma_\mu \ell_L)$

yields $R_K < 1$ and distinctive $R_{K^*} > 1$ , both in agreement with certain experimental indications.

$R_D$ and $R_{D^*}$ : The coupling to right-handed neutrinos ( $N_R$ ) allows $\widetilde{R}_2$ to contribute scalar and tensor operators in $b\to c\tau N$ transitions. Matching at high scale yields (Bečirević et al., 25 Apr 2024):

$C^{(1)}_{Nldq}(m_{\widetilde{R}_2}) = 4\, C^{(3)}_{Nldq}(m_{\widetilde{R}_2}) = -\frac{1}{2}\widetilde{y}_R^{sN}\widetilde{y}_L^{b\tau*},$

but this necessarily implies large contributions to rare $B\to K^{(*)}\nu\nu$ decays, violating experimental bounds unless new cancellation mechanisms are invoked.

4.2 Radiative Neutrino Masses and Magnetic Moments

Mixing between the $S_1$ singlet and $\widetilde{R}_2$ doublet gives rise to radiative neutrino masses via one-loop diagrams with down-type quarks (Zhang, 2021, Sánchez-Vélez, 1 Aug 2025). The induced neutrino mass matrix is

$(M_\nu)_{\alpha\beta} \propto \sin2\theta\, \ln\left(\frac{M_2^2}{M_1^2}\right) [ (\lambda^{\prime\dagger} m_d \lambda^{\prime*})_{\alpha \beta} + (\lambda^{\prime\dagger} m_d \lambda^{\prime*})_{\beta \alpha} ],$

where $\theta$ parametrizes scalar mixing, and $\lambda'$ are leptoquark Yukawa textures.

Similarly, the model predicts sizable neutrino transition magnetic moments

$\mu^{1/3}_\text{int} \propto -\frac{N_c m_e \mu_B}{16\pi^2 m_\text{LQ}^2} \sum_{i=1}^3 \{(m_\alpha + m_\beta)[...] \mathcal{F}(m_{d_i}^2/m_\text{LQ}^2) - 2 m_{d_i}\sin2\theta_\text{LQ} ...\mathcal{G}(m_{d_i}^2/m_\text{LQ}^2)\},$

with loop functions $\mathcal{F}$ and $\mathcal{G}$ . Enhanced values up to $O(10^{-12})\mu_B$ are possible when the bottom quark dominates the loop (Sánchez-Vélez, 1 Aug 2025).

5. Collider Phenomenology and High-Energy Constraints

5.1 Vacuum Stability and Perturbativity

$\widetilde{R}_2$ affects the running of the SM Higgs quartic coupling $\lambda_h$ via its Yukawa and quartic couplings (Bandyopadhyay et al., 2021). With

$\Delta\beta(\lambda_h) \sim -\frac{3}{8\pi^2}\mathrm{Tr}(Y_2 Y_2^\dagger) + \frac{3}{8\pi^2}\left(\lambda_2^2 + \lambda_2 \tilde{\lambda}_2 + \frac{1}{2}\tilde{\lambda}_2^2\right),$

$\widetilde{R}_2$ stabilizes the EW vacuum for $Y_2 \lesssim 1.36$ (three-generation case) and quartics $\lambda_2,\tilde{\lambda}_2 \lesssim 0.2$ . The RG evolution avoids Landau poles below the Planck scale, even when contrasted with triplet leptoquarks.

5.2 LHC, FCC, and Muon Collider Signatures

At hadron and muon colliders, $\widetilde{R}_2$ can be accessed via direct production, virtual effects, and characteristic decay modes (Parashar et al., 2022, Saha et al., 4 Sep 2025). In models with mixing (e.g., with $S_1$ ), the charge $-1/3$ components mix, leading to three leptoquark mass eigenstates:

A pure $+2/3$ doublet state from $\widetilde{R}_2$ ,
Two mixed $-1/3$ states,

with near-degenerate masses $\sim1.5\,$ TeV in benchmarks. Pair and single productions at muon colliders, accompanied by two muons and at least four jets, can probe LQ masses up to $6$ TeV for $\mathcal{O}(1)$ Yukawa couplings, vastly surpassing HL-LHC reach (Saha et al., 4 Sep 2025).

5.3 Indirect Effects and Loop Suppression

The contribution of $\widetilde{R}_2$ to processes such as $gg\to Zh$ is characterized by new tensor structures, arising only in multi-field scenarios where off-diagonal scalar mixing and CP-violating phases exist (He, 27 Aug 2025). These effects are loop-suppressed and strongly diminished for TeV-scale leptoquark masses, rendering them challenging to disentangle at colliders.

6. Comparison to Alternative Leptoquark Scenarios and Constraints

Within frameworks seeking unified explanations for B-physics anomalies and neutrino masses, single leptoquark solutions utilizing only $\widetilde{R}_2$ are usually ruled out or heavily constrained. Attempts to accommodate $R_{D^{(*)}}$ and $R_K$ anomalies unavoidably produce excessive contributions to $B\to K^{(*)}\nu\nu$ , violating experimental limits (Bečirević et al., 25 Apr 2024). Only mixed scenarios—where $\widetilde{R}_2$ is accompanied by other leptoquarks ( $S_1$ , $S_3$ )—and subject to judicious symmetry and texture choices, remain viable for simultaneous explanations of experimental observations.

The $\widetilde{R}_2$ scenario contrasts with $R_2(3,2,7/6)$ , which benefits from top mass enhancement in loop-induced CLFV and g–2 observables, and with $S_1(\bar{3},1,1/3)$ , which is comparatively unconstrained and fully viable under current experimental data for $B$ -physics anomalies, lepton flavor violation, and collider searches.

7. Summary Table: $\widetilde{R}_2(3,2,1/6)$ Leptoquark—Key Properties

Feature	$\widetilde{R}_2$ Prediction	Constraint/Comment
Renormalizable Couplings	$\bar{d}_R\, \widetilde{R}_2^T\, \epsilon\, L_L$	No proton decay at renormalizable level
Chirality Flip Enhancement	Absent (no top-quark coupling)	CLFV rates suppressed
Baryon Number Violation	Dimension-5 operators allowed; must be forbidden by $Z_3$ symmetry	Needs discrete symmetry
$R_K$ , $R_{K^*}$ Anomalies	Shift via right-handed current; $R_{K^*}>1$	Unique signature; fits possible
$R_D$ , $R_{D^*}$ Anomalies	Coupling to $N_R$ allows scalar/tensor operators	Ruled out by $B\to K^{(*)}\nu\nu$
Neutrino Mass Generation	One-loop with $S_1$ – $\widetilde{R}_2$ mixing	Textures control mass ordering and mixing
Collider Reach (Muon Coll.)	Single prod., $m_{\widetilde{R}_2}$ up to 6 TeV for $\mathcal{O}(1)$ $Y$	Surpasses HL-LHC sensitivity
CP Violation in $gg\to Zh$	Only in three-field models with mass splitting and complex phase	Loop and mass suppressed—tiny effect