SU(2)L Doublet Scalar Overview

Updated 18 December 2025

SU(2)L doublet scalar is a field transforming as a fundamental doublet under the electroweak gauge group, crucial for symmetry breaking.
It underpins fermion mass generation and models like the Standard Model and left-right symmetric frameworks with radiative neutrino masses.
Lattice studies and phenomenological constraints emphasize its role in dark matter stability and maintaining precise electroweak observables.

An $SU(2)_L$ doublet scalar is a field transforming as a doublet (fundamental representation) under the $SU(2)_L$ electroweak gauge group. Its implementation and role are central to the Higgs mechanism, extensions of the Standard Model, and various beyond-Standard-Model (BSM) constructions. The doublet structure provides a template for electroweak symmetry breaking, fermion mass generation, dark matter candidates, and nontrivial vacuum/phase structures.

1. Quantum Numbers and Field Content

An $SU(2)_L$ doublet scalar field, denoted generically as $\Phi$ or $H$ , has the following transformation properties:

Under $SU(3)_c \times SU(2)_L \times U(1)_Y$ , it transforms as $[1,2,Y]$ for some hypercharge $Y$ .
The most common choice is $Y = +1/2$ , as in the Standard Model Higgs doublet.
The explicit component form is

$\Phi = \begin{pmatrix} \phi^+ \ \phi^0 \end{pmatrix}$

where the components carry electric charges $Q = T_3 + Y$ .

For instance, in the left-right symmetric model considered by Borboruah et al., the light-sector doublet is $H_L = (h_L^+, h_L^0)^T$ , a color singlet, $SU(2)_L$ doublet, $SU(2)_R$ singlet, with $B-L$ charge $+1$ . For hypercharge identification $Y = (B-L)/2$ so $Y(H_L) = +1/2$ (Borboruah et al., 11 Apr 2025).

In inert doublet constructions and minimal extensions, an extra scalar doublet $\Phi$ with identical quantum numbers but possibly different symmetry assignments (such as odd under a discrete $Z_2$ ) is introduced, with the prototypical form $\Phi = (\Phi^+, \Phi^0)^T \sim (2, +1/2)$ under $SU(2)_L \times U(1)_Y$ (Segre et al., 2011, Melara-Duron et al., 2023).

2. Scalar Potential and Electroweak Symmetry Breaking

The renormalizable scalar potential for one or more $SU(2)_L$ doublet scalars is highly constrained by gauge invariance:

For a single doublet $H$ , the Standard Model potential is

$V(H) = -\mu^2 H^\dagger H + \lambda (H^\dagger H)^2$

For two doublets $H_1, H_2$ , the general $SU(2)_L \times U(1)_Y$ invariant and renormalizable potential is

$\begin{aligned} V &= m_1^2 H_1^\dagger H_1 + m_2^2 H_2^\dagger H_2 \ &+ \frac{\lambda_1}{2} (H_1^\dagger H_1)^2 + \frac{\lambda_2}{2}(H_2^\dagger H_2)^2 \ &+ \lambda_3 (H_1^\dagger H_1)(H_2^\dagger H_2) + \lambda_4(H_1^\dagger H_2)(H_2^\dagger H_1) \ &+ \left[ \frac{\lambda_5}{2}(H_1^\dagger H_2)^2 + \text{h.c.} \right] \end{aligned}$

Specific symmetry assignments (such as a $Z_2$ in the inert doublet model) restrict terms and ensure, for example, an inert doublet never acquires a VEV (Melara-Duron et al., 2023, AbdusSalam et al., 2013).

For the left-right symmetric model (Borboruah et al., 11 Apr 2025):

$V(H_L, H_R) = -\mu_L^2 H_L^\dagger H_L - \mu_R^2 H_R^\dagger H_R + \lambda \left( (H_L^\dagger H_L)^2 + (H_R^\dagger H_R)^2 \right) + \beta (H_L^\dagger H_L)(H_R^\dagger H_R)$

Minimization yields VEVs $\langle H_L \rangle = (0, v_L/\sqrt{2})^T$ , $\langle H_R \rangle = (0, v_R/\sqrt{2})^T$ , with $v_L \approx 246$ GeV responsible for electroweak breaking.

3. Mass Spectrum and Mixing

Vacuum expectation values (VEVs) break electroweak symmetry, converting some scalar degrees of freedom into longitudinal components of $W$ and $Z$ , and leaving physical Higgs bosons. For a single doublet, one CP-even scalar remains (the 125 GeV Higgs). For models with multiple doublets:

Scalar mass matrices are determined by scalar potential parameters and VEVs.
In two-doublet models, CP-even, CP-odd, and charged scalars mix, yielding $h$ (SM-like), $H$ (heavier CP-even), $A^0$ (CP-odd), and $H^\pm$ .
In LR symmetric models, the CP-even neutral scalars $h_L^0, h_R^0$ mix via $(\beta v_L v_R)$ ; their mass matrix is

$M_H^2 = \begin{pmatrix} 2\lambda v_L^2 & \beta v_L v_R \ \beta v_L v_R & 2\lambda v_R^2 \end{pmatrix}$

with SM-like $h_1$ at $125$ GeV, $h_2$ heavier at $\sim 2.5$ TeV for $v_R \simeq 5$ TeV, given $\lambda \simeq 1/8$ and $\beta \ll 2\lambda$ (Borboruah et al., 11 Apr 2025).

In inert doublet models, mass splittings among $H^0$ , $A^0$ , $H^\pm$ are set by quartic couplings, and $H^0$ is the dark matter candidate if it is the lightest.

4. Yukawa Couplings and Fermion Masses

The mechanism of fermion mass generation and scalar-fermion Yukawa structure depends on the model:

In the Standard Model, $H$ couples directly to all SM fermions via Yukawa terms.
In left-right symmetric models without scalar bidoublet, charged fermion masses arise from a universal seesaw with vectorlike partners. The left and right sector doublets $H_L, H_R$ do not permit direct $H_L^{\dagger} H_R$ -type interactions; tree-level neutrino masses are forbidden (Borboruah et al., 11 Apr 2025).
For neutrino masses, left-handed neutrinos are coupled to gauge singlet Majorana fermions via Yukawa $Y_L \bar{\ell}_L \tilde{H}_L S$ , generating Dirac mass terms $M_{LS}=Y_L v_L/\sqrt2$ . The Majorana mass arises via a one-loop diagram with $H_L$ exchange and quartic $\lambda$ , producing

$m_\nu = \frac{v_L^2}{2} Y_L M_S^{-1} Y_L^T\, I_{\rm loop}$

where

$I_{\rm loop} = \frac{\lambda}{16\pi^2} \left( \ln \frac{M_S^2}{m_{h_1}^2} - 1 \right)$

and $M_S$ is the singlet Majorana mass matrix. Right-handed neutrino masses are mainly generated by a type-I seesaw with $v_R$ .

In many BSM scenarios (e.g. inert doublet), discrete symmetries forbid tree-level Yukawa couplings of the extra doublet to SM fermions, leading to dark matter stability (Melara-Duron et al., 2023, AbdusSalam et al., 2013).

5. Symmetries, Vacuum Structure, and Phenomenological Constraints

The vacuum and symmetry structure of $SU(2)_L$ doublet scalar models gives rise to rich phase diagrams and has profound consequences for phenomenology:

In left-right symmetric models, tree-level $\rho$ and $W$ - $Z$ observables remain SM-like due to the absence of triplet scalars; $W_L$ - $W_R$ mixing is suppressed by large $v_R$ (Borboruah et al., 11 Apr 2025).
Precision Higgs coupling measurements constrain doublet mixing angles; for left-right models, the $h_1$ - $h_2$ mixing is required to be $\lesssim 0.1$ , enforcing $\beta \ll 2\lambda$ .
The phase diagram in two-doublet lattice realizations reveals regions with spontaneous breaking of the global $SU(2)_1 \times SU(2)_2$ symmetry, separated by phase boundaries; e.g., symmetry breaking $SU(2)_1 \times SU(2)_2 \to SU(2)_{\mathrm{diag}}$ with three Goldstone bosons ( $R_{12}$ phase) (Lewis et al., 2010).
Scalar masses and mixings are constrained by electroweak precision and direct search limits; e.g., in the left-right model, $m_{h_2} \gtrsim 2.5$ TeV is above current LHC limits, and $v_R \gtrsim 5$ TeV is required to suppress $W_L$ - $W_R$ mixing (Borboruah et al., 11 Apr 2025).

6. Role in Beyond-Standard-Model Physics

$SU(2)_L$ doublet scalars are central to several classes of BSM phenomena:

Neutrino mass mechanisms: In the absence of bidoublets, neutrino masses can be radiatively induced at one loop via $SU(2)_L$ doublet coupling to singlet Majorana fermions and quartic scalar couplings (Borboruah et al., 11 Apr 2025).
Leptogenesis and dark matter: Appropriate choices of Yukawa couplings and heavy singlet Majorana masses allow resonant leptogenesis at the TeV scale; in the right-handed sector, the lightest right-handed neutrino may be a warm dark matter candidate ( $\sim$ keV) (Borboruah et al., 11 Apr 2025).
Collider signatures: Extra doublets with no VEV provide a minimal extension for new TeV-scale physics without altering electroweak symmetry breaking, and can provide WIMP dark matter, as in inert doublet models (Melara-Duron et al., 2023, AbdusSalam et al., 2013). Constraints from direct searches (e.g. LEP, LHC) and indirect precision observables (Higgs coupling fit, oblique parameters) filter the viable parameter space.

7. Lattice Studies and Nonperturbative Dynamics

Nonperturbative effects in $SU(2)_L$ doublet scalar models have been probed via lattice simulations:

Lattice models with one doublet and singlet interactions (via quartic and Yukawa-type couplings) show that operator-mixing populates the scalar spectrum in both ultra-light and ultra-heavy regions, but the mass spectrum is notably sparse in the 100–1000 GeV range.
The renormalized doublet propagator exhibits enhancement over its tree-level form, with robust nontrivial interactions persisting (no triviality) (Saad et al., 29 Jun 2025).
Classification of the field-expectation values under varying fundamental parameters reveals bifurcated branches for strong cubic couplings, but no clear thermodynamic phase transition is observed in the explored parameter regime (Saad et al., 29 Jun 2025).
In two-doublet lattice gauge theory, rigorous identification of continuous global symmetry breaking, Goldstone spectrum, and precise mapping from the lattice to continuum parameters has been accomplished (Lewis et al., 2010).

References