The solar^[1-7],atmospheric^[8],reactor^[9-12],and accelerator^[13-16]neutrino oscillation experiments have provided us with compelling evidences that neutrinos are massive and lepton flavors are mixed. What is of great interest to us is how to explain the observed lepton mass spectra and flavor mixing patterns. To account for the neutrino oscillation parameters (i.e.,Δm²_⊙,Δm_A²,θ_⊙,θ_A,θ₁₃), flavor symmetry has currently become a very popular approach to build models^[17-18]. Recently a number of authors have paid attention to the discrete flavor symmetries,among which the S₃ symmetry is the simplest one^[19-21]. A salient feature of the S₃ symmetry is that it requires the mass matrices of charged-leptons (M_l) and neutrinos (M_ν) to have a universal form M_x=ξ_xI+ζ_xD (for x=l,ν),where

$I=\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right),D=\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right),$

(1)

stand for the identity and “democracy”^[22-24] matrices,respectively.

A great variety of neutrino mass models have been used to explain the smallness of neutrino masses,in which the see-saw mechanism is the most natural one^[25-33]. Here,we propose a type-Ⅱ seesaw model by combining the S₃ flavor symmetry. In order to obtain the observed mass spectra and mixing patterns of lepton sector,we employ the explicit flavor symmetry breaking chain S₃→Z₂→ for both the charged-lepton and the neutrino matrices,in which Z₂ is a subgroup of S₃ and it has only a trivial subgroup^[21].

1 Lepton mass matrices and their construction

In type-Ⅱ seesaw mechanism,where we extend the standard electroweak model by introducing one Higgs triplet Δ,the gauge-invariant Lagrangian relevant to lepton masses reads as follows:

$\begin{align} & -{{L}_{lepton}}={{{\bar{L}}}_{L}}{{Y}_{l}}H{{E}_{R}}+\frac{1}{2}{{{\bar{L}}}_{L}}{{Y}_{\Delta }}\Delta i{{\sigma }_{2}}L_{L}^{c}- \\ & {{\lambda }_{\Delta }}{{M}_{\Delta }}{{H}^{T}}i{{\sigma }_{2}}\Delta H+h.c., \\ \end{align}$

(2)

where L_L is the left-handed doublet of leptons and E_R is the right-handed singlet of charged-leptons,H and Δ represent the Higgs doublet and triplet respectively. When the electroweak symmetry is broken spontaneously,all of the leptons become massive,and the corresponding mass terms for charged-leptons and neutrinos are

$-{{L}_{mass}}=\overline{{{E}_{L}}}{{M}_{l}}{{E}_{R}}+\frac{1}{2}\overline{{{\nu }_{L}}}{{M}_{\nu }}\nu _{L}^{c}+h.c.,$

(3)

where M_l=Y_l〈H〉 with 〈H〉≈174GeV and M_ν=Y_Δ〈Δ〉 with 〈Δ〉=2λ_Δ〈H〉²/M_Δ stand for the charged-lepton mass matrix and the Majorana neutrino mass matrix,respectively.

The simplest Non-Abelian discrete symmetry S₃,a permutation group of three objects,contains six elements,which can be categorized into three conjugacy classes C₀={S⁽¹²³⁾},C₁={S⁽²³¹⁾,S⁽³¹²⁾} and C₂={S⁽²¹³⁾,S⁽¹³²⁾,S⁽³²¹⁾}. Explicitly,the three-dimensional representations of the S₃ group elements are

$\begin{align} & {{S}^{(123)}}=\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right),{{S}^{(213)}}=\left( \begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right), \\ & {{S}^{(132)}}=\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix} \right),{{S}^{(321)}}=\left( \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{matrix} \right), \\ & {{S}^{(312)}}=\left( \begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{matrix} \right),{{S}^{(231)}}=\left( \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \end{matrix} \right). \\ \end{align}$

(4)

Taking advantage of the S₃ flavor symmetry and its explicit symmetry-breaking scheme,the lepton mass matrices can be decomposed into a symmetry-limit term M_x⁽⁰⁾ and a symmetry-breaking perturbation term ΔM_x (for x=l,ν):

${{M}_{l}}=M_{l}^{(0)}+\Delta {{M}_{l}},\text{ }{{M}_{\nu }}=M_{v}^{(0)}+\Delta {{M}_{\nu }}.$

(5)

In the S₃ symmetry-limit,we may easily find that the Lagrangian in Eq. (2) is invariant under the transformation L_L→S^(ijk)L_L and E_R→S^(ijk)E_R with S^(ijk) (for (ijk)=(123),(213),(132),(321),(312),(231)) being the group elements of S₃,if the following commutation conditions [_x,S^(ijk)]=0 (for x=l,Δ) are satisfied. It implies that the mass matrices M_x⁽⁰⁾ (for x=l,ν) have the same commutation conditions with S^(ijk) (i.e.,[M_x⁽⁰⁾,S^(ijk)]=0) and therefore take a general form

$\begin{align} & M_{l}^{(0)}=\frac{{{c}_{l}}}{3}\left[ \left( \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right)+{{r}_{l}}\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right) \right], \\ & M_{v}^{(0)}={{c}_{v}}\left[ \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right)+{{r}_{v}}\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right) \right], \\ \end{align}$

(6)

where c_l and c_ν measure the mass scales of charged-leptons and neutrinos respectively,r_x is real,and |r_x|$\ll $1 holds (for x=l and ν). Moreover,we can see that both of the mass matrices can be diagonalized by the same orthogonal matrix V₀ as following,

$\begin{align} & V_{0}^{T}M_{v}^{(0)}{{V}_{0}}=\frac{{{c}_{l}}}{3}\left( \begin{matrix} {{r}_{l}} & 0 & 0 \\ 0 & {{r}_{l}} & 0 \\ 0 & 0 & 3+{{r}_{l}} \\ \end{matrix} \right), \\ & V_{0}^{T}M_{v}^{(0)}{{V}_{0}}={{c}_{v}}\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1+3{{r}_{v}} \\ \end{matrix} \right), \\ \end{align}$

(7)

where

${{V}_{0}}=\frac{1}{\sqrt{6}}\left( \begin{matrix} \sqrt{3} & 1 & \sqrt{2} \\ -\sqrt{3} & 1 & \sqrt{2} \\ 0 & -2 & \sqrt{2} \\ \end{matrix} \right).$

(8)

It is in fair agreement with the observed mass hierarchies of charged-leptons (i.e.,m_τ$\gg $m_μ$\gg $m_e) and a nearly degenerate mass spectrum of neutrinos in the S₃ symmetry limit. However,in order to obtain more realistic mass spectra of leptons and their flavor mixing patterns,the symmetry-breaking perturbation terms ΔM_x are also necessary.

Since the S₃ symmetry has three conjugacy classes C₀={S⁽¹²³⁾},C₁={S⁽²³¹⁾,S⁽³¹²⁾}, and C₂={S⁽²¹³⁾,S⁽¹³²⁾,S⁽³²¹⁾}. Then we can easily show the three subgroups of S₃ group

$\begin{align} & Z_{2}^{(12)}=\left\{ {{S}^{(123)}},{{S}^{(213)}} \right\}, \\ & Z_{2}^{(23)}=\left\{ {{S}^{(123)}},{{S}^{(132)}} \right\}, \\ & Z_{2}^{(31)}=\left\{ {{S}^{(123)}},{{S}^{(321)}} \right\}. \\ \end{align}$

(9)

Depending on the line that the S₃ symmetry can be explicitly broken to its subgroups,we suggest to construct the perturbation terms as

$\begin{align} & \Delta {{M}_{l}}=\Delta M_{l}^{(1)}+\Delta M_{l}^{(2)}, \\ & \Delta {{M}_{v}}=\Delta M_{v}^{(1)}+\Delta M_{v}^{(2)}. \\ \end{align}$

(10)

We propose a simple perturbation ansatz to break the S₃ symmetry,in which the flavor symmetry is explicitly broken down via the chain

${{S}_{3}}\xrightarrow{\Delta M_{l,v}^{(1)}}{{Z}_{2}}\xrightarrow{\Delta M_{l,v}^{(2)}}\varnothing ,$

(11)

in both the charged-lepton sector and the neutrino sector.

We take Z₂⁽²³⁾ which is also called the μ-τ symmetry as an example. The first-order perturbation term ΔM_l,ν⁽¹⁾ in the lepton mass matrix can be obtained by solving the equations [ΔM_l,ν⁽¹⁾,S⁽¹²³⁾]=0 and [ΔM_l,ν⁽¹⁾,S⁽¹³²⁾]=0. With no loss of generality,we arrive at

$\begin{align} & \Delta M_{l}^{(1)}=\frac{{{c}_{l}}}{3}\left( \begin{matrix} {{\delta }_{l}} & 0 & 0 \\ 0 & 0 & {{\delta }_{l}} \\ 0 & {{\delta }_{l}} & 0 \\ \end{matrix} \right), \\ & \Delta M_{v}^{(1)}={{c}_{v}}\left( \begin{matrix} {{\delta }_{v}} & 0 & 0 \\ 0 & 0 & {{\delta }_{v}} \\ 0 & {{\delta }_{v}} & 0 \\ \end{matrix} \right). \\ \end{align}$

(12)

For the sake of simplicity,the second-order perturbation terms in the charged-lepton and the neutrino mass matrices can be assumed to take the diagonal forms

$\begin{align} & \Delta M_{l}^{(2)}=\frac{{{c}_{l}}}{3}\left( \begin{matrix} -i{{\in }_{l}} & 0 & 0 \\ 0 & +i{{\in }_{l}} & 0 \\ 0 & 0 & +{{\varepsilon }_{l}} \\ \end{matrix} \right), \\ & \Delta M_{v}^{(2)}={{c}_{v}}\left( \begin{matrix} -i{{\in }_{v}} & 0 & 0 \\ 0 & +i{{\in }_{v}} & 0 \\ 0 & 0 & +{{\varepsilon }_{v}} \\ \end{matrix} \right), \\ \end{align}$

(13)

where c_x>0,δ_x,|∈_x|$\ll $|ε_x|<1 (for x=l,ν). Note that the second-order symmetry-breaking perturbation term ΔM_l⁽²⁾ is complex. We expect the CP violation in the charged-lepton sector.

At this moment,we have constructed the lepton mass matrices with a symmetry-limit term M_l,ν⁽⁰⁾ and a symmetry-breaking perturbation term ΔM_l,ν.

2 Lepton mass spectra and flavor mixing matrix

Now the question is how to diagonalize the lepton mass matrices M_l,ν, in order to obtain the lepton mass spectra and the lepton flavor mixing matrix which arises from the mismatch between the diagonalization of M_l and M_ν.

Let us first consider the charged-lepton sector. As can be seen above,the mass matrix of charged-lepton can be written as

$\begin{align} & {{M}_{l}}=M_{l}^{(0)}+\Delta {{M}_{l}}=\frac{{{c}_{l}}}{3}\left[ \left( \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right)+{{r}_{l}}\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right) \right]+ \\ & \frac{{{c}_{l}}}{3}\left( \begin{matrix} {{\delta }_{l}} & 0 & 0 \\ 0 & 0 & {{\delta }_{l}} \\ 0 & {{\delta }_{l}} & 0 \\ \end{matrix} \right)+\frac{{{c}_{l}}}{3}\left( \begin{matrix} -i{{\in }_{l}} & 0 & 0 \\ 0 & +i{{\in }_{l}} & 0 \\ 0 & 0 & +{{\varepsilon }_{l}} \\ \end{matrix} \right), \\ \end{align}$

(14)

where we set c_l>0 and the dimensionless parameters obey the hierarchy r_l,δ_l,|∈_l|$\ll $|ε_l|<1.

The charged-lepton mass matrix M_l is a complex symmetric matrix,and therefore can be diagonalized by a unitary matrix V_l via V_l^†M_lV_l^*=Diag{m_e,m_μ,m_τ} generally,where m_α (for α=e,μ,τ) stand for charged-lepton masses. To diagonalize M_l, we transform it into the so-called hierarchy basis first,

$\begin{align} & M_{l}^{'}\equiv V_{0}^{T}{{M}_{l}}{{V}_{0}}= \\ & {{c}_{l}}\left( \begin{matrix} \frac{{{\delta }_{l}}}{6}+\frac{{{r}_{l}}}{3} & \frac{{{\delta }_{l}}}{2\sqrt{3}}-\frac{i{{\in }_{l}}}{3\sqrt{3}} & -\frac{1}{3}\sqrt{\frac{2}{3}}{{\in }_{l}} \\ \frac{{{\delta }_{l}}}{2\sqrt{3}}-\frac{i{{\in }_{l}}}{3\sqrt{3}} & -\frac{{{\delta }_{l}}}{6}+\frac{2{{\varepsilon }_{l}}}{9}+\frac{{{r}_{l}}}{3} & -\frac{\sqrt{2}{{\varepsilon }_{l}}}{9} \\ -\frac{1}{3}i\sqrt{\frac{2}{3}}{{\in }_{l}} & -\frac{\sqrt{2}{{\varepsilon }_{l}}}{9} & 1+\frac{{{\delta }_{l}}}{3}+\frac{{{\varepsilon }_{l}}}{9}+\frac{{{r}_{l}}}{3} \\ \end{matrix} \right), \\ \end{align}$

(15)

where V₀ has been given in (8).

The 3-dimensional almost-diagonal matrix M_l^′ can be diagonalized exactly. However,we will use perturbative method to diagonalize it instead,since we are more interested in how the perturbation terms in mass matrix behave in the diagonalization. The fact that M_l′ contains several small parameters,which are hierarchical,implies that we need to do higher order perturbation than commonly-used first order approximation to reach the diagonalization consistently. In this paper,we use Brillouin-Wigner (BW) perturbation theory to diagonalize the mass matrix. The tedious analytic perturbation formulae are listed in Appendix A.

After diagonalizing the above matrix,we can get the three charged-lepton masses in terms of parameters in oringial M_l

$\begin{align} & {{m}_{\tau }}\approx {{c}_{l}}\left( 1+\frac{{{\varepsilon }_{l}}}{9}+\frac{{{r}_{l}}}{3}+\frac{{{\delta }_{l}}}{3} \right), \\ & {{m}_{\mu }}\approx {{c}_{l}}\left( \frac{2{{\varepsilon }_{l}}}{9}+\frac{{{r}_{l}}}{3}-\frac{{{\delta }_{l}}}{6} \right), \\ & {{m}_{e}}\approx {{c}_{l}}\left| \frac{{{r}_{l}}}{3}+\frac{{{\delta }_{l}}}{6}+\frac{\in _{l}^{2}}{6{{\varepsilon }_{l}}}-\frac{3\delta _{l}^{2}}{8{{\varepsilon }_{l}}} \right|. \\ \end{align}$

(16)

Defining m₀=c_l(2r_l+δ_l)/6, we can re-express the small parameters ε_l,δ_l, and ∈_l by

$\begin{align} & {{\varepsilon }_{l}}\approx \frac{9}{2}\frac{{{m}_{\mu }}-{{m}_{0}}}{{{m}_{\tau }}-{{m}_{0}}}, \\ & \frac{\in _{l}^{2}}{\varepsilon _{l}^{2}}-\frac{9\delta _{l}^{2}}{4\varepsilon _{l}^{2}}\approx \frac{4}{3}\frac{\left| {{m}_{e}}-\left| {{m}_{0}} \right| \right|}{{{m}_{\mu }}-{{m}_{0}}}. \\ \end{align}$

(17)

Through a little perturbation calculation,the unitary matrix V_l which is used to diagonalize the charged-lepton mass matrix M_l can be obtained as

$\begin{align} & {{V}_{l}}\approx {{V}_{0}}-\frac{i}{\sqrt{6}}{{x}_{l}}\left( \begin{matrix} 1 & \sqrt{3} & 0 \\ 1 & -\sqrt{3} & 0 \\ -2 & 0 & 0 \\ \end{matrix} \right)- \\ & \frac{3{{\delta }_{l}}}{2{{\in }_{l}}}\frac{{{x}_{l}}}{\sqrt{6}}\left( \begin{matrix} 1 & \sqrt{3} & 0 \\ 1 & -\sqrt{3} & 0 \\ -2 & 0 & 0 \\ \end{matrix} \right)+\frac{{{y}_{l}}}{2\sqrt{3}}\left( \begin{matrix} 0 & \sqrt{2} & -1 \\ 0 & \sqrt{2} & -1 \\ 0 & \sqrt{2} & 2 \\ \end{matrix} \right), \\ \end{align}$

(18)

where we define x_l≡$\sqrt{\left| {{m}_{e}}-\left| {{m}_{0}} \right| \right|}/\sqrt{\left( {{m}_{\mu }}-{{m}_{0}} \right)}$ and y_l≡(m_μ-m₀)/(m_τ-m₀).

In the case of the neutrino sector,the mass matrix

$\begin{align} & {{M}_{v}}=M_{v}^{\left( 0 \right)}+\Delta {{M}_{v}}={{c}_{v}}\left[ \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right)+{{r}_{v}}\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right) \right]+ \\ & {{c}_{v}}\left( \begin{matrix} {{\delta }_{v}} & 0 & 0 \\ 0 & 0 & {{\delta }_{v}} \\ 0 & {{\delta }_{v}} & 0 \\ \end{matrix} \right)+{{c}_{v}}\left( \begin{matrix} -{{\in }_{v}} & 0 & 0 \\ 0 & +{{\in }_{v}} & 0 \\ 0 & 0 & +{{\varepsilon }_{v}} \\ \end{matrix} \right), \\ \end{align}$

(19)

in which c_ν>0,r_ν,δ_ν,|∈_ν|$\ll $|ε_ν|<1, can be diagonalized by the unitary matrix V_ν via V_ν^†M_νV_ν^*=Diag{m₁,m₂,m₃}. Approximately,V_ν^{[20-21, 34]} can be found to be

$\frac{1}{{{\varepsilon }_{\nu }}}\left( \begin{matrix} {{\varepsilon }_{\nu }}{{c}_{\theta }} & {{\varepsilon }_{\nu }}{{s}_{\theta }} & {{r}_{\nu }} \\ -{{\varepsilon }_{\nu }}{{s}_{\theta }} & {{\varepsilon }_{\nu }}{{c}_{\theta }} & {{r}_{\nu }}+{{\delta }_{\nu }} \\ ({{r}_{\nu }}+{{\delta }_{\nu }}){{s}_{\theta }}-{{r}_{\nu }}{{c}_{\theta }} & -({{r}_{\nu }}+{{\delta }_{\nu }}){{c}_{\theta }}-{{r}_{\nu }}{{s}_{\theta }} & {{\varepsilon }_{\nu }} \\ \end{matrix} \right),$

(20)

where c_θ≡cosθ and s_θ≡sinθ with tan2θ≡2r_ν/(2∈_ν-δ_ν).

Taking advantage of the condition that m₁<m₂<m₃ (the so-called normal ordering),we can easily obtain the three neutrino mass eigenvalues

$\begin{align} & {{m}_{3}}\approx {{c}_{\nu }}(1+{{r}_{\nu }}+{{\varepsilon }_{\nu }}), \\ & {{m}_{2}}\approx {{c}_{\nu }}\left( 1+{{r}_{\nu }}+{{\delta }_{\nu }}/2+\sqrt{{{\left( {{\in }_{\nu }}-{{\delta }_{\nu }}/2 \right)}^{2}}+r_{v}^{2}} \right), \\ & {{m}_{1}}\approx {{c}_{\nu }}\left( 1+{{r}_{\nu }}+{{\delta }_{\nu }}/2-\sqrt{{{\left( {{\in }_{\nu }}-{{\delta }_{\nu }}/2 \right)}^{2}}+r_{v}^{2}} \right). \\ \end{align}$

(21)

As indicated by (21),we can simply obtain that the neutrino masses are nearly degenerate,and the neutrino mass-square differences are

$\begin{align} & \Delta m_{31}^{2}\approx 2c_{\nu }^{2}{{\varepsilon }_{\nu }}, \\ & \Delta m_{21}^{2}\approx 2c_{\nu }^{2}\sqrt{{{\left( 2{{\in }_{v}}-{{\delta }_{v}} \right)}^{2}}+4r_{v}^{2}}. \\ \end{align}$

(22)

From (18) and (20),we can derive the flavor mixing matrix V_MNS≡V_l^†V_ν^[35]

$\begin{align} & {{V}_{MNS}}\approx \frac{1}{\sqrt{6}}\left( \begin{matrix} \sqrt{3}\left( {{c}_{\theta }}+{{s}_{\theta }} \right) & -\sqrt{3}\left( {{c}_{\theta }}-{{s}_{\theta }} \right) & 0 \\ \left( {{c}_{\theta }}-{{s}_{\theta }} \right) & \left( {{c}_{\theta }}+{{s}_{\theta }} \right) & -2 \\ \sqrt{2}\left( {{c}_{\theta }}-{{s}_{\theta }} \right) & \sqrt{2}\left( {{c}_{\theta }}+{{s}_{\theta }} \right) & \sqrt{2} \\ \end{matrix} \right)+\frac{1}{2\sqrt{3}}{{y}_{l}}\left( \begin{matrix} 0 & 0 & 0 \\ \sqrt{2}\left( {{c}_{\theta }}-{{s}_{\theta }} \right) & \sqrt{2}\left( {{c}_{\theta }}+{{s}_{\theta }} \right) & \sqrt{2} \\ -\left( {{c}_{\theta }}-{{s}_{\theta }} \right) & -\left( {{c}_{\theta }}+{{s}_{\theta }} \right) & 2 \\ \end{matrix} \right)+ \\ & \frac{i}{\sqrt{6}}{{x}_{l}}\left( \begin{matrix} \left( {{c}_{\theta }}-{{s}_{\theta }} \right) & \left( {{c}_{\theta }}+{{s}_{\theta }} \right) & -2 \\ \sqrt{3}\left( {{c}_{\theta }}+{{s}_{\theta }} \right) & -\sqrt{3}\left( {{c}_{\theta }}-{{s}_{\theta }} \right) & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right)-\frac{3{{\delta }_{l}}}{2{{\in }_{l}}}\frac{1}{\sqrt{6}}{{x}_{l}}\left( \begin{matrix} \left( {{c}_{\theta }}-{{s}_{\theta }} \right) & \left( {{c}_{\theta }}+{{s}_{\theta }} \right) & -2 \\ -\sqrt{3}\left( {{c}_{\theta }}+{{s}_{\theta }} \right) & \left( {{c}_{\theta }}-{{s}_{\theta }} \right) & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right)+ \\ & \frac{1}{\sqrt{6}}\frac{{{r}_{v}}}{{{\varepsilon }_{v}}}\left( \begin{matrix} 0 & 0 & 0 \\ 2\left( {{c}_{\theta }}-{{s}_{\theta }} \right) & 2\left( {{c}_{\theta }}+{{s}_{\theta }} \right) & 2 \\ -\sqrt{2}\left( {{c}_{\theta }}-{{s}_{\theta }} \right) & -\sqrt{2}\left( {{c}_{\theta }}+{{s}_{\theta }} \right) & 2\sqrt{2} \\ \end{matrix} \right)+\frac{1}{\sqrt{6}}\frac{{{\delta }_{v}}}{{{\varepsilon }_{v}}}\left( \begin{matrix} 0 & 0 & -\sqrt{3} \\ -2{{s}_{\theta }} & 2{{c}_{\theta }} & 1 \\ \sqrt{2}{{s}_{\theta }} & -\sqrt{2}{{c}_{\theta }} & \sqrt{2} \\ \end{matrix} \right), \\ \end{align}$

(23)

where x_l≡|$\sqrt{\left| {{m}_{e}}-\left| {{m}_{0}} \right| \right|}/\sqrt{\left( {{m}_{\mu }}-{{m}_{0}} \right)}$ and y_l≡(m_μ-m₀)/(m_τ-m₀) have been defined,and we have discarded the high-order corrections to the elements of V_MNS. Comparing this result with the standard parametrization of neutrino mixing matrix U^[36] which is

$U=\left( \begin{matrix} {{c}_{12}}{{c}_{13}} & {{s}_{12}}{{c}_{13}} & {{s}_{13}}{{e}^{-i\delta }} \\ -{{s}_{12}}{{c}_{23}}-{{c}_{12}}{{s}_{23}}{{s}_{13}}{{e}^{i\delta }} & {{c}_{12}}{{c}_{23}}-{{s}_{12}}{{s}_{23}}{{s}_{13}}{{e}^{i\delta }} & {{s}_{23}}{{c}_{13}} \\ {{s}_{12}}{{s}_{23}}-{{c}_{12}}{{c}_{23}}{{s}_{13}}{{e}^{i\delta }} & {{c}_{12}}{{s}_{23}}-{{s}_{12}}{{c}_{23}}{{s}_{13}}{{e}^{i\delta }} & {{c}_{23}}{{c}_{13}} \\ \end{matrix} \right)\times diag(1,{{e}^{i\frac{\alpha 21}{2}}},{{e}^{i\frac{\alpha 31}{2}}}),$

(24)

(c_ij=cosθ_ij,s_ij=sinθ_ij, the angles θ_ij=[0,π/2],δ=[0,2π] is the Dirac CP violating phase and α₂₁,α₃₁ are two Majorana CP violating phases),we can immediately get the three neutrino mixing angles

$sin{{\theta }_{12}}\approx \left| {{V}_{e2}} \right|\approx \approx \frac{1}{\sqrt{2}}\left| cos\theta -sin\theta \right|,$

(25)

$\begin{align} & sin{{\theta }_{23}}\approx \left| {{V}_{\mu 3}} \right| \\ & \approx \frac{2}{\sqrt{6}}\left( 1-\frac{1}{2}{{y}_{l}}-\frac{{{r}_{v}}}{{{\varepsilon }_{v}}}-\frac{1}{2}\frac{{{\delta }_{v}}}{{{\varepsilon }_{v}}} \right), \\ \end{align}$

(26)

$\begin{align} & sin{{\theta }_{13}}\approx \left| {{V}_{e3}} \right| \\ & \approx \left| -i\frac{2}{\sqrt{6}}{{x}_{l}}+\frac{3}{\sqrt{6}}{{x}_{l}}\frac{{{\delta }_{l}}}{{{\in }_{l}}}-\frac{1}{\sqrt{2}}\frac{{{\delta }_{v}}}{{{\varepsilon }_{v}}} \right|, \\ \end{align}$

(27)

and the Jarlskog invariant of CP violation^[37-38]

$\begin{align} & J=\frac{1}{8}cos{{\theta }_{13}}sin2{{\theta }_{12}}sin2{{\theta }_{23}}sin2{{\theta }_{13}}sin\delta \\ & =Im[{{V}_{\mu 2}}{{V}_{\tau 3}}V_{\mu 3}^{*}V_{\tau 2}^{*}]. \\ \end{align}$

(28)

It is worth emphasizing that there are ten real parameters in our model,i.e.,five real parameters in the charged-lepton sector (i.e.,c_l,r_l,δ_l,∈_l, and ε_l), and other five real parameters in the neutrino sector (i.e.,c_ν,r_ν,δ_ν,∈_ν, and ε_ν). All of them can be determined by the ten relevant observable quantities,in which there are three charged-lepton masses (i.e.,m_e,m_μ,m_τ), three neutrino masses (i.e.,m₁,m₂,m₃), three neutrino mixing angles (i.e.,θ₁₂,θ₂₃,θ₁₃) and one Dirac CP-violating phase δ. Some detailed discussions are given below.

3 Oscillation parameters: a detailed numerical analysis

For convenience,all the best-fit values (at the level of ～1.0σ for normal ordering) for three neutrino oscillation parameters^[39]① and the values of charged-lepton masses at the electroweak scale^[40] used in our numerical analysis are listed in Table 1.

①It is important to stress that the values we use here are the latest global fit to three neutrino oscillation parameters,and we just make some calculations that are not very strict in order to test the correctness of our model. The recent results from T2K (Salzgeber M R,[T2K Collaboration],Antineutrino oscillations with T2K,talk given at European Physical Society HEP conference,Vienna,22-29 July 2015) and NOνA (Patterson R,[NOνA Collaboration],First oscillation results from NOVA,talk given at the Joint Experimental-Theoretical Physics Seminar,Fermilab,6 August 2015) experiments which indicate at the 2σ level that θ₂₃>45° and δ≈270° of a normal neutrino mass ordering imply that the value of θ₂₃ favors the second octant and the CP violation effects in neutrino oscillations are relatively large.

while

$\begin{align} & {{m}_{e}}\approx 0.4866MeV,{{m}_{\mu }}\approx 102.718MeV, \\ & {{m}_{\tau }}\approx 1746.17MeV \\ \end{align}$

Now,let us illustrate how to calculate all the parameters in our model.

First of all,from (25),we can easily obtain θ₁₂≈θ-$\frac{\pi }{4}$ (then θ≈78.48°) when inputting the best-fit value θ₁₂≈33.48°^[39]. One may estimate the magnitude of 2r_ν/(2∈_ν-δ_ν) with the help of (25) and the condition tan2θ=2r_ν/(2∈_ν-δ_ν) defined above. The result is

$\frac{2{{r}_{\nu }}}{2{{\in }_{v}}-{{\delta }_{\nu }}}=-cot2{{\theta }_{12}}\approx -0.425_{-0.031}^{+0.032},$

(29)

where the best-fit value θ₁₂≈33.48° has been input. With the help of (22),from which r_ν/ε_ν≈c_θs_θΔm₂₁²¹/Δm₃₁³¹ can be obtained,the ratio r_ν/ε_ν can also be determined

$\frac{{{r}_{\nu }}}{{{\varepsilon }_{\nu }}}\approx (5.97_{-0.41}^{+0.43})\times {{10}^{-3}},$

(30)

where Δm₂₁²¹≈7.50×10^-5eV² and Δm₃₁³¹≈2.457×10^-3eV^2[39].

Because of the strong mass hierarchy of charged-leptons m_τ$\gg $m_μ$\gg $m_e and the magnitude of m₀ is about the same as that of m_e, we can safely neglect the m₀-induced correction in (26). By inputting m_μ≈102.718 MeV and m_τ≈1746.17 MeV at the electroweak scale^[40] and the best-fit value of θ₂₃≈42.3°^[39],we obtain

$\frac{{{\delta }_{v}}}{{{\varepsilon }_{v}}}\approx 0.281_{-0.051}^{+0.095}.$

(31)

Taking advantage of the recent results from T2K and NOνA experiments which indicate a maximal CP-violating phase δ≈270° and combining (27) with (28),we can immediately obtain

${{x}_{l}}=\frac{\sqrt{\left| {{m}_{e}}-\left| {{m}_{0}} \right| \right|}}{\sqrt{\left( {{m}_{\mu }}-{{m}_{0}} \right)}}\approx 0.182_{-0.0055}^{+0.0057},$

(32)

$\frac{{{\delta }_{l}}}{{{\varepsilon }_{l}}}\approx 0.448_{-0.08}^{+0.15}.$

(33)

Using (32) and adopting the condition m₀=c_l(2r_l+δ_l)/6, we can write

$\left| 2{{r}_{l}}+{{\delta }_{l}} \right|\approx 0.0134_{-0.00071}^{+0.00073}$

(34)

where c_l～m_τ≈1746.17MeV.

According to (17),the sizes of ε_l and ∈_l can be obtained ε_l≈0.265,(35)

${{\varepsilon }_{l}}\approx 0.265$

(35)

${{\in }_{l}}\approx 0.0752_{-0.011}^{+0.021}$

(36)

Making use of (33) and (34),we finally find

${{\delta }_{l}}\approx 0.034_{-0.008}^{+0.015},$

(37)

${{r}_{l}}\approx -0.0103_{-0.0040}^{+0.0075}(or~{{r}_{l}}\approx -0.0237_{-0.0040}^{+0.0075}).$

(38)

Finally,from (21),we find that the neutrino masses are nearly degenerate,which implies that the effective neutrino-mass in tritium beta decays 〈m〉_e and that in neutrinoless double-beta decays 〈m〉_ee are on the same order of the mass-scale parameter c_ν. With the current cosmological observations^[36],we can get the strong constraint: 〈m〉_e≈〈m〉_ee≈c_ν～O(0.1eV). Then,the magnitude of ε_ν can be determined by (22)

${{\varepsilon }_{\nu }}\approx \frac{\Delta m_{31}^{2}}{2c_{v}^{2}}\approx 0.123_{-0.002}^{+0.0024}$

(39)

The model parameters r_ν and δ_ν can in turn be determined in terms of (30) and (31)

${{r}_{\nu }}\approx (7.34_{-0.52}^{+0.55})\times {{10}^{-4}},$

(40)

${{\delta }_{\nu }}\approx 0.0337_{-0.006}^{+0.012}.$

(41)

From (29),one can find that

${{\in }_{\nu }}\approx 0.0152_{-0.003}^{+0.006}.$

(42)

It is easy to find that all the model parameters are in good agreement with the condition that c_x>0,r_x,δ_x,|∈_x|$\ll $|ε_x|<1 (for x=l,ν).

However,the latest results from T2K and NOνA experiments indicate that θ₂₃ favors the second octant. At the 2σ level,the T2K data imply that θ₂₃≈45.8°±3.2° in the case of normal neutrino mass ordering. Replacing the best-fit value of θ₂₃ with its latest experimental result and repeating the above calculation,we obtain

$\begin{align} & {{\varepsilon }_{l}}\approx 0.265 \\ & {{\in }_{l}}\approx 0.0613_{-0.007}^{+0.007} \\ & {{\delta }_{l}}\approx 0.017_{-0.009}^{+0.009}, \\ & {{r}_{l}}\approx -0.0103_{-0.0040}^{+0.0045} \\ & (or~{{r}_{l}}\approx -0.0153_{-0.0045}^{+0.0045}), \\ & {{\varepsilon }_{\nu }}0.123_{-0.0024}^{+0.0024} \\ & {{r}_{\nu }}\approx (7.34_{-0.52}^{+0.55})\times {{10}^{-4}}, \\ & {{\delta }_{\nu }}\approx 0.0208_{-0.012}^{+0.012}. \\ & {{\in }_{\nu }}\approx 0.0087_{-0.006}^{+0.006}. \\ \end{align}$

(43)

which are also in agreement with c_x>0,r_x,δ_x,|∈_x|$\ll $|ε_x|<1 (for x=l,ν). Since the parameters δ_l,∈_l,δ_ν, and ∈_ν exist in the perturbation terms,we prefer to choose their more smaller values,that is to say,our model is more suitable for θ₂₃>45°.

A few remarks that we would like to say in the consistency of our model are listed as follows.

1) In (28),we find that x_l is essential for obtaining maximal CP-violating phase (i.e. δ≈$\frac{\pi }{2}$). It is also essential for obtaining large sinθ₁₃ in (27).

2) Just considering the leading-order in (26),we have θ₂₃≈54.7°. Apparently,the value of θ₂₃ we obtain in this case is much larger than the current experimental data. In order to get correct value of θ₂₃, we need slightly larger δ_ν/ε_ν. That implies that large δ_l/∈_l, which is some what unexpected to cancel the contribution of δ_ν/ε_ν to sinθ₁₃ is also necessary,that is to say,the S₃→Z₂→ symmetry breaking chain in our ansatz is necessary.

3) In consideration of the relation between x_l and δ_l, the parameter r_l in our model is important too.

4 Summary

In this paper,we combine the permutation symmetry S₃ with the type-Ⅱ seesaw mechanism to lepton mass matrices,and propose a simple perturbation ansatz to break the S₃ symmetry,in which the flavor symmetry is explicitly broken down via S₃→Z₂→ in both the charged-lepton sector and the neutrino sector. Based on this hypothesis we can see that the mass matrices of charged-leptons are strongly hierarchical while the spectrum of three neutrinos is nearly degenerate and the lepton flavor mixing matrix contains two large mixing angles. It is shown that the ansatz can naturally lead to the three neutrino mixing angles (i.e.,θ₁₂,θ₂₃,θ₁₃) to be in good agreement with the current experiment data. One may also incorporate other mechanisms of neutrino mass generation,such as the type-Ⅰ seesaw mechanism and the type-(Ⅰ+Ⅱ) seesaw mechanism,with S₃ flavor symmetry to explain the lepton mass spectra and flavor mixing patterns.

Finally,we have to point out that our results are all dependent on the patterns of the perturbation matrices ΔM_l and ΔM_ν. Of course,one may also carry out some other different forms of ΔM_x (for x=l,ν), which might be off-diagonal or partially off-diagonal.

We are grateful to Prof. Xing Zhizhong and Prof. Zhou Shun for helpful discussions.