---
title: Scalar Leptoquark Doublet ̃R₂ (3,2,1/6)
url: https://www.emergentmind.com/topics/scalar-leptoquark-doublet-widetilde-r-_-2-3-2-1-6
type: topic
---

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

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 $Y_\alpha$ 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 [1304.6119].

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## 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) [1304.6119]. 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 [1509.07410].

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## 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 [1304.6119]. 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.

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## 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 $C'_9 = -C'_{10}$ [1608.08501]. 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 [2404.16772]:
  \[
  C^{(1)}_{Nldq}(m_{\widetilde{R}_2}) = 4\, C^{(3)}_{Nldq}(m_{\widetilde{R}_2}) = -\frac{1}{2}\widetilde{y}_R^{sN}\widetilde{y}_L^{b\tau*},
  \]
  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 [2105.08670, 2508.00226]. 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 [2508.00226].

---

## 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 [2111.03872]. 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 [2209.05890, 2509.04579]. 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 [2509.04579].

### 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 [2508.19642]. 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 $\widetilde{R}_2$ 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 [2404.16772]. 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.

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## 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           |

---

## References

- Minimal scalar leptoquark phenomenology: [1304.6119]
- LFV, Higgs decays, g–2 anomalies, leptoquark mixing: [1509.07410], [1910.03877], [2105.08670]
- $B$-physics anomalies, right-handed neutrinos: [1608.08501], [2205.15794], [2404.16772]
- Collider, vacuum, and unitarity constraints: [2111.03872], [2209.05890], [2508.19642], [2509.04579]
- Neutrino magnetic moment and DSL models: [2508.00226]

Source: https://www.emergentmind.com/topics/scalar-leptoquark-doublet-widetilde-r-_-2-3-2-1-6