Doublet Left-Right Symmetric Model

• DLRSM is a gauge extension of the Standard Model that replaces conventional scalar triplets with a bidoublet and two scalar doublets, establishing left-right symmetry and realistic fermion masses.
• It features a complex scalar and gauge sector with a light SM-like Higgs and additional heavy states, offering clear predictions for collider experiments and precision measurements.
• The model provides a framework for neutrino mass generation, dark matter, and gravitational wave signatures from cosmological phase transitions, linking particle physics with early universe phenomena.

The Doublet Left-Right Symmetric Model (DLRSM) is a well-defined class of gauge extensions of the Standard Model (SM) that replace the conventional scalar triplets of the minimal Left-Right Symmetric Model (LRSM) with a bidoublet and two scalar doublets. This framework implements the gauge group $$SU(3)_\mathrm{c} \times SU(2)_\mathrm{L} \times SU(2)_\mathrm{R} \times U(1)_{B-L}$$, with discrete parity $\mathcal{P}$ often imposed to ensure left-right symmetry. The DLRSM achieves left-right symmetry breaking, charged-fermion and neutrino mass generation, and introduces a distinctive scalar and gauge sector linked to cosmological, collider, and flavor phenomena.

1. Gauge Structure, Field Content, and Symmetry Breaking

The gauge group of the DLRSM is $$SU(3)_\mathrm{c} \times SU(2)_\mathrm{L} \times SU(2)_\mathrm{R} \times U(1)_{B-L}$$, possibly supplemented by a discrete left-right parity $\mathcal{P}$ relating $L\leftrightarrow R$ sectors (Borah et al., 2020, Karmakar et al., 2023, Karmakar et al., 2022). The chiral fermion content is assigned as follows:

• Quark doublets: $Q_L = (u_L, d_L)^T \sim (3,2,1,+1/3)$, $Q_R = (u_R, d_R)^T \sim (3,1,2,+1/3)$.
• Lepton doublets: $L_L = (\nu_L, e_L)^T \sim (1,2,1,-1)$, $L_R = (\nu_R, e_R)^T \sim (1,1,2,-1)$.

The scalar sector comprises:

• A bidoublet: $\Phi \sim (1,2,2,0)$, which provides electroweak symmetry breaking by acquiring VEVs $$\langle\Phi\rangle=\mathrm{diag}(\kappa_1, \kappa_2)/\sqrt2$$.
• Left and right doublets: $\chi_L \sim (1,2,1,+1)$, $\chi_R \sim (1,1,2,+1)$, breaking $SU(2)_R\times U(1)_{B-L}$ at a high scale via $\langle\chi_R^0\rangle=v_R/\sqrt2$, and potentially contributing to electroweak symmetry via $\langle\chi_L^0\rangle=v_L/\sqrt2$.

The symmetry breaking chain is:

$$SU(2)_L\times SU(2)_R\times U(1)_{B-L} \ \xrightarrow{\langle\chi_R\rangle}\ SU(2)_L \times U(1)_Y \ \xrightarrow{\langle\Phi\rangle, \langle\chi_L\rangle}\ U(1)_\mathrm{em}$$

where $$\kappa_1^2 + \kappa_2^2 + v_L^2 \approx (246\,\mathrm{GeV})^2$$ (Karmakar et al., 2023, Karmakar et al., 2022).

2. Scalar Potential, Vacuum Structure, and Mass Spectrum

The most general renormalizable, CP-conserving and parity-symmetric potential involves quadratic, cubic, and quartic terms combining traces and products of $\Phi$, $\chi_L$, and $\chi_R$. Notable terms include (Frank et al., 2021, Karmakar et al., 2023, Karmakar et al., 2022):

• $V \supset \text{Tr}[\Phi^\dagger\Phi]$, $(\chi_L^\dagger\chi_L), (\chi_R^\dagger\chi_R)$,
• Mixed bilinears, e.g., $(\chi_L^\dagger \Phi\chi_R)$,
• Quartics mixing bidoublet and doublets.

After symmetry breaking, the scalar mass spectrum features:

• A light SM-like Higgs $h$ with $m_h\approx 125$ GeV.
• Three additional heavy neutral CP-even scalars $H_{1,2,3}$, with $m_{H_3}^2=2\rho_1 v_R^2$ (for $\rho_1$ the quartic coupling of $\chi_R$) and $m_{H_{1,2}}^2 \sim v_R^2$ with coefficients dictated by quartic couplings.
• Charged and CP-odd scalars, including Higgs doublet-like and singlet-like states.

The stability of the vacuum and positivity of the scalar mass spectrum are governed by bounded-from-below (BFB) and copositivity criteria for quartic parameters, enforced in all phenomenologically viable scenarios (Frank et al., 2021).

3. Fermion Mass Generation and Neutrino Sector

In the minimal DLRSM, all charged fermion and Dirac neutrino masses arise from renormalizable Yukawa couplings to the bidoublet:

$$\mathcal{L}_Y = h_{ij}\,\bar{L}_{L,i} \Phi L_{R,j} + \tilde h_{ij}\, \bar{L}_{L,i} \tilde\Phi L_{R,j} + {\rm q\text{--}sector\ terms} + h.c.$$

where $\tilde\Phi = \sigma_2\Phi^*\sigma_2$. After symmetry breaking, this yields

$$M_\nu \approx \frac{h\kappa_1}{\sqrt{2}}, \quad M_\ell \approx \frac{\tilde h\kappa_1}{\sqrt{2}}$$

with sub-eV Dirac neutrino masses requiring $h \lesssim 10^{-12}$ (Borah et al., 2020).

Extensions may incorporate sterile singlet fermions, modular flavor symmetries ($A_4$), or invoke double/inverse/double-seesaw structures via new gauge singlets and their couplings to $\chi_R$. Such variants can yield:

• keV-scale sterile neutrino warm dark matter (Kakoti et al., 26 Feb 2025).
• Radiative Majorana masses for right-handed neutrinos via two-loop diagrams, if additional charged singlet scalars are present (Babu et al., 2020).
• Universal seesaw mechanisms for charged fermion masses, in models lacking the bidoublet (Patra, 2012, Borboruah et al., 11 Apr 2025).

4. Electroweak Precision Tests and Collider Phenomenology

The DLRSM preserves custodial symmetry and protects the electroweak ρ parameter from large corrections, in contrast to triplet LR models. After diagonalizing the gauge boson mass matrices (in the basis of $W^\pm_L, W^\pm_R$ and neutral $W^3_{L,R}, B$), the physical charged and neutral states $W_1 \simeq W_L$, $W_2 \simeq W_R$, $Z_1 \simeq Z$, $Z_2 \simeq Z_R$ emerge, with:

$$M_{W_1}^2 \approx \frac{g^2}{4}(\kappa_1^2 + \kappa_2^2 + v_L^2), \quad M_{W_2}^2 \approx \frac{g^2}{4}v_R^2$$

and similar formulas for $Z_{1,2}$ (Karmakar et al., 2022, Bernard et al., 2020).

The model is highly constrained by:

• Direct collider searches for $W_R$ and $Z_R$ in resonance production, $M_{W_R/Z_R}\gtrsim 4$–$6$ TeV (LHC), translating to $v_R\gtrsim 6$–$8$ TeV.
• Higgs coupling measurements restrict the ratio of vacuum expectation values $r\equiv \kappa_2/\kappa_1$ and $w\equiv v_L/\kappa_1$ to $0.1\lesssim r \lesssim 0.8$, $0.1\lesssim w \lesssim 3.5$ with an absolute bound $w\lesssim6.7$ from perturbativity and the $m_h$ fit. Large $w\sim \mathcal{O}(1)$ is allowed and sometimes preferred (Karmakar et al., 2022).
• Precision observables from $Z$-pole data and low-energy weak-interaction measurements set further constraints, requiring alignment of heavy spectra and decoupled Higgs/gauge mixing (Bernard et al., 2020).

New scalars can be probed at future hadron colliders via vector-boson fusion and decays such as $H_3\to$ SM, with sensitivity up to $m_{H_3}\sim$ few TeV at $100$ TeV colliders (Karmakar et al., 2023).

5. Cosmology: Gravitational Waves, Domain Walls, and Early Universe Constraints

DLRSM realisations naturally admit a cosmological history with distinctive signatures:

• First-order phase transitions associated with $SU(2)_R\times U(1)_{B-L}\to U(1)_Y$ breaking can yield a stochastic gravitational wave background. For $v_R=20$–50 TeV, nucleation temperatures $T_n\sim 2$–16 TeV, and favorable quartic couplings ($\rho_1 \lesssim 0.1$), signal strengths are potentially detectable at mHz–Hz frequencies by BBO, DECIGO, and similar (Karmakar et al., 2023). The correlation between first-order transitions and a light CP-even scalar $H_3$ (sub–10 TeV) is explicit.
• Formation and annihilation of domain wall networks sourced by spontaneous parity breaking generate a gravitational wave background with a peak in the nHz regime. PTA data from NANOGrav and EPTA constrain the parity-breaking scale to $v_0\sim 10^5$ GeV and the bias energy to $V_\text{bias}\sim(100\ \mathrm{MeV})^4$. The corresponding right-handed sector is inaccessible at colliders but testable by PTAs and planned space interferometers (Ringe, 19 Jul 2024).
• Cosmological thermalization of right-handed neutrinos places a lower bound $M_{W_R}\gtrsim 4$ TeV (from $\Delta N_\mathrm{eff}$ at $2 \sigma$ C.L.), rising to $\gtrsim 25$ TeV for future Stage IV CMB experiments (Borah et al., 2020). These bounds are competitive with direct collider searches.

6. Neutrino Masses, Lepton Flavor Violation, and Dark Matter

DLRSM variants support a rich phenomenology in flavor and dark matter:

• The minimal scenario yields sub-eV Dirac neutrino masses, with extremely small Yukawa couplings.
• Modular-symmetry–based extensions allow double/inverse seesaw mechanisms, enabling keV-scale sterile neutrino dark matter. The allowed parameter space is $m_\mathrm{DM}\sim10$–12 keV, $M_{W_R}\gtrsim 12.6$–$13.3$ TeV, with relic abundance and X-ray constraints satisfied (Kakoti et al., 26 Feb 2025).
• Neutrinoless double beta decay, if right-handed neutrinos have Majorana masses induced, can exhibit contributions from both light and heavy sectors, with future experiments probing the predicted rates.
• Radiative and universal seesaw models generate neutrino and charged-fermion masses at loop-level or via mixing with vector-like states, leading to testable effects in $0\nu\beta\beta$ and rare decays (Babu et al., 2020, Borboruah et al., 11 Apr 2025, Patra, 2012).
• Lepton flavor-violating Higgs decays $h\to \mu\tau$ mediated by heavy neutrino loops are extremely suppressed, with branching ratios $<10^{-11}$ for $v_R\gtrsim 10$ TeV, far below current LHC bounds (Zeleny-Mora et al., 2 Aug 2025).

The DLRSM can also accommodate stable dark matter candidates in the form of real $SU(2)_R$ quintuplets with TeV-scale masses, compatible with relic-density and cosmic-microwave-background constraints (Borah et al., 2020).

7. Theoretical Extensions and Model Variants

The DLRSM admits several structurally distinct variants, with implications for phenomenology:

• Radiative scenarios with forbidden tree-level Yukawas and effective, loop-induced, low-energy couplings via hidden sectors and messenger fields solve the strong CP and hypercharge problems (Gabrielli et al., 2016).
• Dirac neutrino models with multiple bidoublets allow vanishing Majorana mass at all orders and introduce inert scalar doublets stabilized by parity, providing WIMP dark matter candidates. Hierarchical bidoublet VEVs can remove fine-tuning of Yukawa couplings (Chavez et al., 2019).
• Embedding in grand-unified frameworks such as $SO(10)$ naturally accommodates the full field content, including triplet and singlet fermions necessary for realistic neutrino mass generation (Gu, 2011).
• Models with vectorlike quark doublets enable TeV-scale Higgs boson masses consistent with suppressed flavor-changing neutral currents, opening search channels at the LHC (Mohapatra et al., 2013).
• Scenarios omitting the bidoublet rely on universal seesaw mechanisms for SM fermion masses and predict resonant leptogenesis, warm dark matter, and observable $0\nu\beta\beta$ rates (Borboruah et al., 11 Apr 2025, Patra, 2012).

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The DLRSM constitutes a central and highly flexible platform for model-building at the intersection of BSM flavor, collider, and cosmological phenomenology. Its predictions for Higgs, gauge, and flavor sectors, as well as cosmological signatures, are actively probed by ongoing and future experiments (Borah et al., 2020, Karmakar et al., 2023, Karmakar et al., 2022, Kakoti et al., 26 Feb 2025, Ringe, 19 Jul 2024, Frank et al., 2021, Gabrielli et al., 2016).