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

• 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 $\widetilde{R}_2$ 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.

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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 $1/6$0 carries $1/6$1 (with $1/6$2 the color index). The renormalizable interaction Lagrangian is: $1/6$3 with $1/6$4 an arbitrary complex Yukawa matrix, $1/6$5 the right-handed down-type quark, $1/6$6 the lepton doublet, and $1/6$7 the $1/6$8 antisymmetric tensor. $1/6$9 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).

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2. Chirality Structure and Flavor Violation

Interactions with $\widetilde{R}_2$0 link right-handed down-type quarks to left-handed leptons. In loop-induced processes such as $\widetilde{R}_2$1, 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., $\widetilde{R}_2$2, $\widetilde{R}_2$3) are suppressed, scaling as $\widetilde{R}_2$4 (with $\widetilde{R}_2$5 the external lepton mass) (Arnold et al., 2013). This is in contrast to models such as $\widetilde{R}_2$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).

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3. Baryon Number Violation and Discrete Gauge Symmetries

While $\widetilde{R}_2$7 is safe from proton decay at the renormalizable level, nonrenormalizable dimension-five operators, such as

$\widetilde{R}_2$8

lead to baryon-number violating processes ($\widetilde{R}_2$9, $\widetilde{R}_2$0), potentially at rates exceeding experimental bounds for $\widetilde{R}_2$1 TeV unless couplings are finely tuned (Arnold et al., 2013). To control such operators, the imposition of a $\widetilde{R}_2$2 discrete gauge symmetry—defined as $\widetilde{R}_2$3—forbids them without affecting the allowed renormalizable couplings. This can be realized by gauging $\widetilde{R}_2$4 and breaking it appropriately to leave an unbroken $\widetilde{R}_2$5 subgroup.

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4. Phenomenological Roles in Charged and Neutral Current Anomalies

The $\widetilde{R}_2$6 doublet participates directly in B-physics anomalies and neutrino physics via its fundamental interactions:

4.1 B-Physics Anomalies

• $\widetilde{R}_2$7 and $\widetilde{R}_2$8: Tree-level exchange of $\widetilde{R}_2$9 modifies the neutral-current $\widetilde{R}_2$0 transitions via right-handed quark currents, affecting Wilson coefficients such as $\widetilde{R}_2$1 (Bečirević et al., 2016). Specifically, the operator

$\widetilde{R}_2$2

yields $\widetilde{R}_2$3 and distinctive $\widetilde{R}_2$4, both in agreement with certain experimental indications.

• $\widetilde{R}_2$5 and $\widetilde{R}_2$6: The coupling to right-handed neutrinos ($\widetilde{R}_2$7) allows $\widetilde{R}_2$8 to contribute scalar and tensor operators in $\widetilde{R}_2$9 transitions. Matching at high scale yields (Bečirević et al., 2024):

$SU(3)_C \times SU(2)_L \times U(1)_Y$0

but this necessarily implies large contributions to rare $SU(3)_C \times SU(2)_L \times U(1)_Y$1 decays, violating experimental bounds unless new cancellation mechanisms are invoked.

4.2 Radiative Neutrino Masses and Magnetic Moments

Mixing between the $SU(3)_C \times SU(2)_L \times U(1)_Y$2 singlet and $SU(3)_C \times SU(2)_L \times U(1)_Y$3 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

$SU(3)_C \times SU(2)_L \times U(1)_Y$4

where $SU(3)_C \times SU(2)_L \times U(1)_Y$5 parametrizes scalar mixing, and $SU(3)_C \times SU(2)_L \times U(1)_Y$6 are leptoquark Yukawa textures.

Similarly, the model predicts sizable neutrino transition magnetic moments

$SU(3)_C \times SU(2)_L \times U(1)_Y$7

with loop functions $SU(3)_C \times SU(2)_L \times U(1)_Y$8 and $SU(3)_C \times SU(2)_L \times U(1)_Y$9. Enhanced values up to $(\mathbf{3}, \mathbf{2}, 1/6)$0 are possible when the bottom quark dominates the loop (Sánchez-Vélez, 1 Aug 2025).

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5. Collider Phenomenology and High-Energy Constraints

5.1 Vacuum Stability and Perturbativity

$(\mathbf{3}, \mathbf{2}, 1/6)$1 affects the running of the SM Higgs quartic coupling $(\mathbf{3}, \mathbf{2}, 1/6)$2 via its Yukawa and quartic couplings (Bandyopadhyay et al., 2021). With

$(\mathbf{3}, \mathbf{2}, 1/6)$3

$(\mathbf{3}, \mathbf{2}, 1/6)$4 stabilizes the EW vacuum for $(\mathbf{3}, \mathbf{2}, 1/6)$5 (three-generation case) and quartics $(\mathbf{3}, \mathbf{2}, 1/6)$6. 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, $(\mathbf{3}, \mathbf{2}, 1/6)$7 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 $(\mathbf{3}, \mathbf{2}, 1/6)$8), the charge $(\mathbf{3}, \mathbf{2}, 1/6)$9 components mix, leading to three leptoquark mass eigenstates:

• A pure $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$0 doublet state from $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$1,
• Two mixed $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$2 states,

with near-degenerate masses $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$3TeV in benchmarks. Pair and single productions at muon colliders, accompanied by two muons and at least four jets, can probe LQ masses up to $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$4 TeV for $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$5 Yukawa couplings, vastly surpassing HL-LHC reach (Saha et al., 4 Sep 2025).

5.3 Indirect Effects and Loop Suppression

The contribution of $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$6 to processes such as $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$7 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.

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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 $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$8 are usually ruled out or heavily constrained. Attempts to accommodate $$\widetilde{R}_2 = \begin{pmatrix} V_\alpha \ Y_\alpha \end{pmatrix},$$9 and $V_\alpha$0 anomalies unavoidably produce excessive contributions to $V_\alpha$1, violating experimental limits (Bečirević et al., 2024). Only mixed scenarios—where $V_\alpha$2 is accompanied by other leptoquarks ($V_\alpha$3, $V_\alpha$4)—and subject to judicious symmetry and texture choices, remain viable for simultaneous explanations of experimental observations.

The $V_\alpha$5 scenario contrasts with $V_\alpha$6, which benefits from top mass enhancement in loop-induced CLFV and g–2 observables, and with $V_\alpha$7, which is comparatively unconstrained and fully viable under current experimental data for $V_\alpha$8-physics anomalies, lepton flavor violation, and collider searches.

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7. Summary Table: $V_\alpha$9 Leptoquark—Key Properties

Feature	$+\frac{2}{3}$0 Prediction	Constraint/Comment
Renormalizable Couplings	$+\frac{2}{3}$1	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 $+\frac{2}{3}$2 symmetry	Needs discrete symmetry
$+\frac{2}{3}$3, $+\frac{2}{3}$4 Anomalies	Shift via right-handed current; $+\frac{2}{3}$5	Unique signature; fits possible
$+\frac{2}{3}$6, $+\frac{2}{3}$7 Anomalies	Coupling to $+\frac{2}{3}$8 allows scalar/tensor operators	Ruled out by $+\frac{2}{3}$9
Neutrino Mass Generation	One-loop with $1/6$00–$1/6$01 mixing	Textures control mass ordering and mixing
Collider Reach (Muon Coll.)	Single prod., $1/6$02 up to 6 TeV for $1/6$03 $1/6$04	Surpasses HL-LHC sensitivity
CP Violation in $1/6$05	Only in three-field models with mass splitting and complex phase	Loop and mass suppressed—tiny effect

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• Minimal scalar leptoquark phenomenology: (Arnold et al., 2013)
• LFV, Higgs decays, g–2 anomalies, leptoquark mixing: (Baek et al., 2015, Doršner et al., 2019, Zhang, 2021)
• $1/6$06-physics anomalies, right-handed neutrinos: (Bečirević et al., 2016, Chen et al., 2022, Bečirević et al., 2024)
• Collider, vacuum, and unitarity constraints: (Bandyopadhyay et al., 2021, Parashar et al., 2022, He, 27 Aug 2025, Saha et al., 4 Sep 2025)
• Neutrino magnetic moment and DSL models: (Sánchez-Vélez, 1 Aug 2025)