Minimal Left-Right Symmetric Model

• The minimal left-right symmetric model is a gauge theory extending the Standard Model to include SU(2)L × SU(2)R and a discrete parity symmetry, enabling parity restoration at high energies.
• It naturally explains light neutrino masses via the type-II seesaw mechanism by inducing a small vacuum expectation value in the left-handed triplet.
• The model connects inflationary dynamics, CMB constraints, and successful leptogenesis through a correlated set of scalar potential parameters and heavy right-handed neutrino decays.

The minimal left-right symmetric model (MLRSM) constitutes an extension of the Standard Model gauge structure to SU(2)$_L$ × SU(2)$_R$ × U(1)$_{B-L}$ × P, where P is a discrete left–right (parity) symmetry. This framework restores parity at high energies, providing a compelling solution to the origin of neutrino masses and offering correlated explanations for inflation, the cosmic microwave background (CMB) anisotropies, and the baryon asymmetry via leptogenesis. The model is distinguished by the inclusion of right-handed (RH) gauge interactions and a scalar sector consisting of a gauge singlet, bidoublet, and triplet representations. Its salient features are dictated by the mechanism of spontaneous D-parity breaking, the pattern of symmetry breaking, and the concomitant cosmological and low-energy implications.

1. Gauge Structure, Scalar Sector, and D-Parity Breaking

The MLRSM is built on the gauge group SU(2)$_L$ × SU(2)$_R$ × U(1)$_{B-L}$ × P. The scalar content comprises:

• a gauge-singlet scalar σ, responsible for spontaneous D-parity breaking,
• a right-handed triplet Δ$_R$ ∼ (1,3,2),
• a left-handed triplet Δ$_L$ ∼ (3,1,2),
• a bidoublet Φ ∼ (2,2,0).

The D-parity, implemented as a discrete left-right symmetry P, is spontaneously broken when the singlet σ acquires a vacuum expectation value (VEV):

$$\langle \sigma \rangle = \sigma_P = \mu/\sqrt{\lambda}$$

with a scalar potential

$$V_\sigma(\sigma) = -\frac{1}{2} \mu^2 \sigma^2 + \frac{1}{4} \lambda \sigma^4 + V_0$$

where λ is a quartic coupling and V$_0$ is set to ensure a vanishing vacuum energy at the minimum. This symmetry-breaking pattern decouples the left– and right–handed sectors, eventually enabling a hierarchy between their respective VEVs.

2. Inflationary Dynamics and CMB Constraints

The gauge-singlet scalar σ, while neutral under the extended gauge group, serves as the inflaton. The potential V$_\sigma$ supports a slow-roll inflationary phase characterized by the slow-roll parameters:

$$\epsilon = \frac{M_{Pl}^2}{16\pi} \left(\frac{V'}{V}\right)^2 , \qquad \eta = \frac{M_{Pl}^2}{8\pi} \frac{V''}{V}$$

where $M_{Pl}$ is the Planck scale and derivatives are with respect to σ. Inflation terminates when $\epsilon$ reaches unity, followed by oscillations and decay of σ, reheating the universe.

The amplitude of horizon-crossing density fluctuations is:

$$\delta_H = \sqrt{ \frac{V_H^3}{75\pi^2 M_{Pl}^6 (V'_H)^2} } \simeq 1.91 \times 10^{-5}$$

and the predicted scalar spectral index is

$$n_s \simeq 1 - \frac{\lambda M_{Pl}^2}{\pi \mu^2} - \frac{40\pi^2 \mu^4}{\lambda A_H^2 M_{Pl}^4}$$

where $A_H$ is fixed by the observed amplitude $\delta_H$ and $V_H, V'_H$ are the potential and its derivative at horizon exit. These expressions directly connect the inflaton sector and the neutrino mass scale since the VEV of σ parametrically enters the seesaw formula below.

3. Symmetry Breaking and Neutrino Masses

Symmetry breaking unfolds in two major stages:

1. Parity breaking by σ at a high scale.
2. SU(2)$_R$ × U(1)$_{B-L}$ → U(1)$_Y$ by a large VEV $v_R$ of Δ$_R$ at a lower scale (∼10$^{14}$ GeV), and subsequently SU(2)$_L$ × U(1)$_Y$ → U(1)$_{em}$ by the bidoublet Φ.

The crucial induced VEV for Δ$_L$:

$v_L \simeq -\frac{\beta v^2 v_R}{2 M \sigma_P}$

with $v = \sqrt{k_1^2 + k_2^2} \approx 174$ GeV, β an O(1) coupling, and M a mass scale. This small v$_L$ enables naturally light neutrino masses via the type-II seesaw mechanism. The relevant Yukawa couplings in the Lagrangian are

$$-\mathscr{L}_{\mathrm{Yuk}} \supset h_{ij} \bar{\psi}_{iL} \Phi \psi_{jR} + \tilde{h}_{ij} \bar{\psi}_{iL} \tilde{\Phi} \psi_{jR} + f_{ij}[\psi_{iR}^T C i\tau_2 \Delta_R \psi_{jR} + (L \leftrightarrow R)] + h.c.$$

The light neutrino mass matrix is:

$$m_{\nu} \simeq m_{\nu}^{(II)} = -\frac{\beta v^2 v_R}{2 M \sigma_P} f$$

The observed structure of PMNS mixing and neutrino mass-squared differences can be matched by appropriate choices of the Majorana coupling matrix f and the parameters $v_R$, M, σ$_P$, and β.

The mass-squared differences relevant for oscillations are:

$$\Delta m^2_\odot \propto \left[\frac{\beta v^2 v_R}{2 M \sigma_P}\right]^2 \Delta f_{12}^2$$

$$\Delta m^2_{\mathrm{atm}} \propto \left[\frac{\beta v^2 v_R}{2 M \sigma_P}\right]^2 |\Delta f_{23}^2|$$

Precision cosmology via CMB measurements of n$_s$, together with oscillation data, places correlated constraints on the inflationary (μ, λ), seesaw, and symmetry-breaking parameters.

4. Leptogenesis and Baryon Asymmetry Generation

After the breakdown to the Standard Model gauge group, the out-of-equilibrium decay of the lightest right-handed neutrino N$_1$ generates a net lepton asymmetry:

$$N_1 \rightarrow e^-_{iL} + \phi_1^+, ~~~ N_1 \rightarrow e^+_{iL} + \phi_1^-$$

The CP asymmetry per decay is:

$$\delta_{\mathrm{CP}} \simeq -\frac{1}{8\pi} \frac{f_1}{f_2} \frac{\mathrm{Im}[(h^\dagger h)_{12}^2]}{(h^\dagger h)_{11}}$$

where $f_1$, $f_2$ are Majorana coupling eigenvalues and $h$ is the Dirac-type Yukawa matrix. The resulting lepton asymmetry is partially converted via electroweak sphalerons, which conserve B−L, to produce the baryon asymmetry compatible with observed values. Successful leptogenesis in this scenario requires $M_1 \gtrsim 4.8 \times 10^8$ GeV.

5. Unified Connections and Predictive Interplay

The structure of the MLRSM with D-parity breaking gives rise to multiple predictive correlations:

• The inflaton dynamics and scalar singlet VEV σ$_P$ set the scale for inflation and are directly involved in the type-II seesaw expression for neutrino masses.
• Parameters (μ, λ, β, f, h) are simultaneously testable via cosmological, neutrino, and baryogenesis observables.
• The small induced $v_L$ needed for sub-eV neutrino masses is exactly the scale at which CMB anisotropies are imprinted through the inflationary dynamics.

These connections yield a quantitative bridge between early-universe cosmology and low-energy neutrino physics.

6. Summary Table of Key Relations

Physical Quantity	Formula/Parameterization	Impact/Constraint
Inflaton VEV	$$\langle \sigma \rangle = \sigma_P = \mu/\sqrt{\lambda}$$	Sets D-parity breaking, inflation scale
Slow-roll parameters	$\epsilon,\eta$ as defined above	Determine end of inflation, $n_s$
Light neutrino mass	$m_\nu^{(II)} = -(\beta v^2 v_R)/(2M\sigma_P) f$	Tied to inflation, fit by oscillation data
CMB amplitude	$$\delta_H = \sqrt{V_H^3/(75\pi^2 M_{Pl}^6 (V'_H)^2)}$$	Observable density fluctuations
Spectral index	$$n_s \simeq 1 - (\lambda M_{Pl}^2)/(\pi \mu^2) - (40\pi^2 \mu^4)/(\lambda A_H^2 M_{Pl}^4)$$	Links inflation/ν parameters
CP asymmetry (leptogenesis)	$$\delta_{CP} \simeq -\frac{1}{8\pi} \frac{f_1}{f_2} \frac{\operatorname{Im}[(h^\dagger h)_{12}^2]}{(h^\dagger h)_{11}}$$	Baryon asymmetry generation
RH neutrino mass bound	$M_1 \gtrsim 4.8 × 10^8\ \mathrm{GeV}$	For successful baryogenesis

The predictive framework of the MLRSM with spontaneous D-parity breaking, therefore, links cosmic inflation, neutrino mass and mixing, the origin of CMB perturbations, and the matter-antimatter asymmetry through a minimal set of parameters, all embedded in a renormalizable gauge theory context.