Non-vanishing U _e3 under S ₃ symmetry

Abstract

This work proposes two models of neutrino masses that predict non-zero θ ₁₃ under the non-Abelian discrete flavor symmetry $\mathbb{S}_{3}\otimes\mathbb{Z}_{2}$. We advocate that the size of θ ₁₃ is understood as a group theoretical consequence rather than a perturbed effect from the tri-bi-maximal mixing. So, the difference of two models is designed only in terms of the flavor symmetry, by changing the charge assignment of right-handed neutrinos. The PMNS matrix in the first model is obtained from both mass matrices, charged leptons giving rise to non-zero $\theta^{l}_{13}$ and neutrino masses giving rise to tri-bi-maximal mixing. The physical mixing angles are expressed by a simple relation between $\theta^{l}_{13}$ and tri-bi-maximal angles to fit the recent experimental results. The other model generates PMNS matrix with non-zero θ ₁₃, only from the neutrino mass transformation. The 5-dimensional effective theory of Majorana neutrinos obtained in this framework is tested with phenomenological bounds in the parametric spaces sinθ ₂₃,sinθ ₁₂ and m ₂/m ₃ vs. sinθ ₁₃.

Appendix: Higgs Potential

The contents of Higgs scalar particles and their representations under $\mathbb{S}_{3}\otimes\mathbb{Z}_{2}$ are

(39)

which commonly belong to (2,1/2) _G under SU(2)×U(1) gauge group. The full invariant Higgs potential can be organized into three parts as follows:

(40)

where V _e and V _o are the interactions of only ℤ₂-even particles and those of only ℤ₂-odd particles, respectively, while V _χ is the cross interactions of ℤ₂-even and ℤ₂-odd particles. Each contribution to the potential V is given as;

(41)

(42)

(43)

The subscript ‘1’ or ‘2’ in each term indicates that the product of two fields belongs to the representation 1 or 2 in $\mathbb{S}_{3}$. Each term with a subscript ‘r’ consists of three types of product, 1,1′ and 2 representations as in Eq. (9).

According to the product rules in Eqs. (1)–(3), $(\varPhi^{\dagger}\varPhi)_{1}=|\varphi_{1}|^{2}+|\varphi_{2}|^{2}, (\varPhi^{\dagger}\varPhi )_{1'}=\varphi_{1}^{*}\varphi_{2}-\varphi_{2}^{*}\varphi_{1}$, and $(\varPhi^{\dagger}\varPhi)_{2}=(|\varphi_{2}|^{2}-|\varphi_{1}|^{2} \varphi_{1}^{*}\varphi_{2}+\varphi_{2}^{*}\varphi_{1})^{T}$. The Higgs potential in Eq. (41) can be rephrased in terms of component fields $\{\varphi_{i}, \varphi_{i}^{\dagger}\}$ with i=1 and 2, and {H,H ^†}:

(44)

When the Higgs particles obtain their real vacuum expectation values such that 〈H〉=〈H ^†〉=v, 〈φ ₁〉=v ₁, and 〈φ ₂〉=v ₂, the potential can be expressed as follows:

(45)

where Λ _a =λ _a +λ _c , and Λ _b =λ+λ′+2λ″.

Following the same steps as in Eqs. (41)–(45), the potentials, V _o and V _χ , in terms of vevs, 〈h〉=u and (〈σ ₁〉,〈σ ₂〉)=(w ₁,w ₂), can be expressed as follows:

(46)

where $\varLambda_{c} \equiv\lambda_{s}+\lambda_{s}'+2\lambda_{s}''$.

(47)

where k ₁=χ+χ′+2χ″, $k_{2}=\lambda_{\chi}+\lambda_{\chi}'+2\lambda_{\chi}''$, $k_{3}=\eta_{\chi}+\eta_{\chi}'+2\eta_{\chi}''$, and k ₄=2(γ+γ′). The $k_{5}\ldots k_{5}'''$ are rather complicated polynomials of $\varGamma_{\chi},\varGamma'_{\chi}$, and Γ _χ in Eq. (43), such that $k_{5}=k_{5}(\varGamma_{\chi}, \varGamma'_{\chi})$, $k_{5}'=k_{5}'(\varGamma_{\chi}, \varGamma''_{\chi})$, $k_{5}''=k_{5}''(\varGamma_{\chi})$, and $k_{5}'''=k_{5}'''(\varGamma'_{\chi})$. Their details are not necessary for the following examination of the minimal condition. The first derivatives of the full potential given in Eq. (40) are

(48)

where each K denotes the part that corresponds to the coefficients of linear terms. The vevs, v ₁≠0 and v ₂=0, can make the potential minimum, when the following conditions are satisfied:

(49)

(50)

It is clear that any of w ₁ and w ₂ should not be zero to satisfy the above conditions. According to the symmetry of the potential under the interchange of σ ₁ and σ ₂, vevs can be taken as w ₁=w ₂=w. Thus, in summary, the following vevs of the fields in Eq. (39) can be adopted for the masses of leptons:

(51)

Then, the derivatives in Eqs. (48) reduce to the following conditions:

(52)

According to Eqs. (49) and (50), k ₄<0 and k ₅>0 are necessary. The mass matrices are examined upon the assumptions of 0.0001<u/w<5 and v/v ₁<1 with weak hierarchy. The assumptions do not show any conflicts with either the minimum conditions in Eq. (52) or the phenomenological constraints, $\mathcal{O}(m_{h}^{2}, m_{\varphi}^{2}, m_{s}^{2})>m_{H}^{2}$, since the constraints on masses and vevs contain a sufficient number of independent parameters.