1. Introduction
As we know, the observations of the phenomena of neutrino oscillations indicate that neutrinos are massive and three lepton flavors are mixed [1]. In the basis where the mass eigenstates of three charged leptons coincide with their flavor eigenstates, the lepton flavor mixing matrix U is identical with the unitary matrix for diagonalizing the neutrino mass matrix ${M}_{\nu }^{}$ 2(2 In this letter we consider the scenario of neutrinos being Majorana particles where ${M}_{\nu }^{}$ is symmetric.):
$\begin{eqnarray}{U}^{\dagger }{M}_{\nu }^{}{U}^{* }={D}_{\nu }^{}=\mathrm{diag}({m}_{1}^{},{m}_{2}^{},{m}_{3}^{}),\end{eqnarray}$
with ${m}_{i}^{}$ being three-light neutrino masses. In the standard form, U is parameterized in terms of three lepton flavor mixing angles ${\theta }_{{ij}}^{}$ (for ij = 12, 13, 23), the Dirac CP phase δ, and two Majorana CP phases ρ and σ: $\begin{eqnarray}\begin{array}{l}U=\left(\begin{array}{ccc}{c}_{12}^{}{c}_{13}^{} & {s}_{12}^{}{c}_{13}^{} & {s}_{13}^{}{{\rm{e}}}^{-{\rm{i}}\delta }\\ -{s}_{12}^{}{c}_{23}^{}-{c}_{12}^{}{s}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & {c}_{12}^{}{c}_{23}^{}-{s}_{12}^{}{s}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & {s}_{23}^{}{c}_{13}^{}\\ {s}_{12}^{}{s}_{23}^{}-{c}_{12}^{}{c}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & -{c}_{12}^{}{s}_{23}^{}-{s}_{12}^{}{c}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & {c}_{23}^{}{c}_{13}^{}\end{array}\right)\\ \,\times \,\left(\begin{array}{ccc}{{\rm{e}}}^{{\rm{i}}\rho } & & \\ & {{\rm{e}}}^{{\rm{i}}\sigma } & \\ & & 1\end{array}\right),\end{array}\end{eqnarray}$
where the symbols ${c}_{{ij}}^{}=\cos {\theta }_{{ij}}^{}$ and ${s}_{{ij}}^{}=\sin {\theta }_{{ij}}^{}$ have been employed.On the experimental side, the neutrino mass squared differences ${\rm{\Delta }}{m}_{{ij}}^{2}\equiv {m}_{i}^{2}-{m}_{j}^{2}$ and three lepton flavor mixing angles have been measured with good degrees of accuracy. And there has also been a preliminary result for δ. For the convenience of the reader, the global-fit results for these parameters [2, 3] are presented here in table 1. However, the sign of ${\rm{\Delta }}{m}_{31}^{2}$ remains unknown, leaving us with two possible neutrino mass orderings: the normal ordering (NO) ${m}_{1}^{}\lt {m}_{2}^{}\lt {m}_{3}^{}$ and inverted ordering (IO) ${m}_{3}^{}\lt {m}_{1}^{}\lt {m}_{2}^{}$. And there has not been any lower constraint on the smallest neutrino mass, nor any constraint on the Majorana CP phases, whose values can only be inferred from some non-oscillatory processes (e.g. the neutrinoless double beta decays [4]) or cosmological observations.
Table 1. The best-fit values, 1σ errors and 3σ ranges of six neutrino oscillation parameters extracted from a global analysis of the existing neutrino oscillation data [2]. |
| Normal ordering | Inverted ordering | |||
|---|---|---|---|---|
| bf ± 1σ | 3σ range | bf ± 1σ | 3σ range | |
| ${\sin }^{2}{\theta }_{12}^{}$ | ${0.318}_{-0.016}^{+0.016}$ | 0.271 → 0.370 | ${0.318}_{-0.016}^{+0.016}$ | 0.271 → 0.370 |
| ${\sin }^{2}{\theta }_{23}^{}$ | ${0.566}_{-0.022}^{+0.016}$ | 0.441 → 0.609 | ${0.566}_{-0.023}^{+0.018}$ | 0.446 → 0.609 |
| ${\sin }^{2}{\theta }_{13}^{}$ | ${0.02225}_{-0.00078}^{+0.00055}$ | 0.02015 → 0.024 17 | ${0.02250}_{-0.00076}^{+0.00056}$ | 0.02039 → 0.024 41 |
| δ/π | ${1.20}_{-0.14}^{+0.23}$ | 0.80 → 2.00 | ${1.54}_{-0.13}^{+0.13}$ | 1.14 → 1.90 |
| ${\rm{\Delta }}{m}_{21}^{2}/({10}^{-5}\,{\mathrm{eV}}^{2})$ | ${7.50}_{-0.20}^{+0.22}$ | 6.94 → 8.14 | ${7.50}_{-0.20}^{+0.22}$ | 6.94 → 8.14 |
| $| {\rm{\Delta }}{m}_{31}^{2}| /({10}^{-3}\,{\mathrm{eV}}^{2})$ | ${2.56}_{-0.04}^{+0.03}$ | 2.46 → 2.65 | ${2.46}_{-0.03}^{+0.03}$ | 2.37 → 2.55 |
From table 1 we see that ${\theta }_{12}^{}$ and ${\theta }_{23}^{}$ are close to some special values: ${\sin }^{2}{\theta }_{12}^{}\sim 1/3$ and ${\sin }^{2}{\theta }_{23}^{}\sim 1/2$ (i.e. ${\theta }_{23}^{}\sim \pi /4$), and ${\theta }_{13}^{}$ is not far from zero. As is well known, the μ-τ symmetry (under which ${M}_{\nu }^{}$ keeps invariant with respect to the ${\nu }_{\mu }^{}\leftrightarrow {\nu }_{\tau }^{}$ transformation [5, 6]) can naturally accommodate ${\theta }_{23}^{}=\pi /4$ and ${\theta }_{13}^{}=0$. After the observation of a relatively large ${\theta }_{13}^{}$ (compared to zero) and a preliminary hint for δ ∼ − π/2 [7], its variant—the μ-τ reflection symmetry [6, 8] has attracted particular attention. This symmetry is defined in a way that ${M}_{\nu }^{}$ keeps invariant with respect to the following transformations of three neutrino fields
$\begin{eqnarray}\begin{array}{l}{\nu }_{e}^{}\leftrightarrow {\nu }_{e}^{c},\quad {\nu }_{\mu }^{}\leftrightarrow {\nu }_{\tau }^{c},\\ {\nu }_{\tau }^{}\leftrightarrow {\nu }_{\mu }^{c},\end{array}\end{eqnarray}$
where the superscript c denotes the charge conjugation of the relevant neutrino fields. To be explicit, ${M}_{\nu }^{}$ takes a form as $\begin{eqnarray}{M}_{\nu }^{}=\left(\begin{array}{ccc}A & B & {B}^{* }\\ B & C & D\\ {B}^{* } & D & {C}^{* }\end{array}\right),\end{eqnarray}$
with A and D being real. Such an ${M}_{\nu }^{}$ leads to the following interesting predictions for the lepton flavor mixing parameters $\begin{eqnarray}\begin{array}{l}{\theta }_{23}^{}=\displaystyle \frac{\pi }{4},\quad \delta =\pm \displaystyle \frac{\pi }{2},\\ \quad \rho ,\sigma =0\ \mathrm{or}\ \displaystyle \frac{\pi }{2}.\end{array}\end{eqnarray}$
Throughout this letter, we will take δ = − π/2 (which is more favored by the experimental results) out of ± π/2.Considering that the μ-τ reflection symmetry has no predictive power for the neutrino masses, in this letter we make an attempt to embed it in the interesting Friedberg-Lee (FL) neutrino model which employs a translational flavor symmetry and keeps one neutrino mass vanishing (${m}_{1}^{}=0$ or ${m}_{3}^{}=0$ in the NO or IO case) and study the consequences of such a combination. In this scenario, both the lepton flavor mixing parameters and neutrino masses will be constrained by the flavor symmetries.
2. FL neutrino model with μ-τ reflection symmetry
Let us first briefly recapitulate the salient features of the FL neutrino model [9-11]. Under the FL symmetry, the lagrangian relevant for the neutrino masses is given by7 )3(3 We thank the anonymous referee for pointing out this point.):
$\begin{eqnarray}\begin{array}{l}{{ \mathcal L }}_{\mathrm{mass}}^{}=a({\eta }_{\tau }^{* }{\bar{\nu }}_{\mu }^{}-{\eta }_{\mu }^{* }{\bar{\nu }}_{\tau }^{})({\eta }_{\tau }^{* }{\nu }_{\mu }^{c}-{\eta }_{\mu }^{* }{\nu }_{\tau }^{c})\\ \,+\,b({\eta }_{\mu }^{* }{\bar{\nu }}_{e}^{}-{\eta }_{e}^{* }{\bar{\nu }}_{\mu }^{})({\eta }_{\mu }^{* }{\nu }_{e}^{c}-{\eta }_{e}^{* }{\nu }_{\mu }^{c})\\ \,+\,c({\eta }_{\tau }^{* }{\bar{\nu }}_{e}^{}-{\eta }_{e}^{* }{\bar{\nu }}_{\tau }^{})({\eta }_{\tau }^{* }{\nu }_{e}^{c}-{\eta }_{e}^{* }{\nu }_{\tau }^{c})+{\rm{h}}.{\rm{c}}.,\end{array}\end{eqnarray}$
where a, b and c are mass parameters and ${\eta }_{e,\mu ,\tau }^{}$ are coefficients. One can easily see that ${{ \mathcal L }}_{\mathrm{mass}}^{}$ keeps invariant with respect to the following translational transformations of three neutrino fields $\begin{eqnarray}\begin{array}{l}{\nu }_{e}^{}\to {\nu }_{e}^{}+{\eta }_{e}^{}\xi ,\quad {\nu }_{\mu }^{}\to {\nu }_{\mu }^{}+{\eta }_{\mu }^{}\xi ,\\ \quad {\nu }_{\tau }^{}\to {\nu }_{\tau }^{}+{\eta }_{\tau }^{}\xi ,\end{array}\end{eqnarray}$
where ξ is a space-time independent element of the Grassmann algebra and anti-commutates with the neutrino fields. It is worth noting that the following neutrino mass terms also keep invariant with respect to the translational transformations of three neutrino fields in equation ( $\begin{eqnarray}({\eta }_{\tau }^{* }{\bar{\nu }}_{\mu }^{}-{\eta }_{\mu }^{* }{\bar{\nu }}_{\tau }^{})({\eta }_{\mu }^{* }{\nu }_{e}^{c}-{\eta }_{e}^{* }{\nu }_{\mu }^{c}).\end{eqnarray}$
However, considering that these terms are not present in the original FL model and otherwise they would introduce additional parameters, we will not include them in our analysis.As a result of the FL symmetry, ${M}_{\nu }^{}$ takes a form as
$\begin{eqnarray}{M}_{\nu }^{}=\left(\begin{array}{ccc}b{\eta }_{\mu }^{* 2}+c{\eta }_{\tau }^{* 2} & -b{\eta }_{e}^{* }{\eta }_{\mu }^{* } & -c{\eta }_{e}^{* }{\eta }_{\tau }^{* }\\ -b{\eta }_{e}^{* }{\eta }_{\mu }^{* } & a{\eta }_{\tau }^{* 2}+b{\eta }_{e}^{* 2} & -a{\eta }_{\mu }^{* }{\eta }_{\tau }^{* }\\ -c{\eta }_{e}^{* }{\eta }_{\tau }^{* } & -a{\eta }_{\mu }^{* }{\eta }_{\tau }^{* } & a{\eta }_{\mu }^{* 2}+c{\eta }_{e}^{* 2}\end{array}\right).\end{eqnarray}$
It is direct to verify that the relation ${M}_{\nu }^{}{({\eta }_{e}^{* },{\eta }_{\mu }^{* },{\eta }_{\tau }^{* })}^{T}\,=\,0$ holds. This means that one eigenvalue of ${M}_{\nu }^{}$ (i.e. one neutrino mass) is vanishing and the eigenvector corresponding to it is proportional to ${({\eta }_{e}^{* },{\eta }_{\mu }^{* },{\eta }_{\tau }^{* })}^{T}$. Therefore, in the NO (or IO) case with ${m}_{1}^{}=0$ (or ${m}_{3}^{}=0$), the first (or third) column of U will be proportional to ${({\eta }_{e}^{* },{\eta }_{\mu }^{* },{\eta }_{\tau }^{* })}^{{\rm{T}}}$.Now, we are ready to study the embedding of the μ-τ reflection symmetry in the FL neutrino model. It is found that in order for ${M}_{\nu }^{}$ in equation (9 ) to conform to the pattern described in equation (4 ) the following relations must hold10 ):
$\begin{eqnarray}\begin{array}{l}{\phi }_{\mu }^{}+{\phi }_{\tau }^{}={\phi }_{a}^{},\quad {\phi }_{b}^{}+{\phi }_{c}^{}=2{\phi }_{e}^{}+{\phi }_{a}^{},\\ | b| | {\eta }_{\mu }^{}| =| c| | {\eta }_{\tau }^{}| ,\\ | {\eta }_{\mu }^{}| \sin ({\phi }_{b}^{}-2{\phi }_{\mu }^{})+| {\eta }_{\tau }^{}| \\ \times \,\sin ({\phi }_{c}^{}-2{\phi }_{\tau }^{})=0,\\ | a| (| {\eta }_{\mu }^{}{| }^{2}-| {\eta }_{\tau }^{}{| }^{2}){{\rm{e}}}^{{\rm{i}}({\phi }_{\mu }^{}-{\phi }_{\tau }^{})}=| {\eta }_{e}^{}{| }^{2}\\ \times \,\left[| b| {{\rm{e}}}^{{\rm{i}}({\phi }_{a}^{}-{\phi }_{c}^{})}-| c| {{\rm{e}}}^{{\rm{i}}({\phi }_{b}^{}-{\phi }_{a}^{})}\right],\end{array}\end{eqnarray}$
with ${\phi }_{\alpha }^{}\equiv \arg ({\eta }_{\alpha }^{})$ and ${\phi }_{a}^{}=\arg (a)$ (and similarly for ${\phi }_{b}^{}$ and ${\phi }_{c}^{}$). In the present paper, we will study the following instructive solution of equation ( $\begin{eqnarray}\begin{array}{l}b={c}^{* },\quad {\eta }_{\mu }^{}={\eta }_{\tau }^{* },\\ {\phi }_{e}^{}={\phi }_{a}^{}=0.\end{array}\end{eqnarray}$
For the sake of simplicity, without loss of generality, hereafter we will take ${\eta }_{e}^{}=1$. Therefore, in the FL neutrino model with the μ-τ reflection symmetry, ${M}_{\nu }^{}$ can be expressed as $\begin{eqnarray}{M}_{\nu }^{}=\left(\begin{array}{ccc}2\mathrm{Re}(b{\eta }_{\mu }^{* 2}) & -b{\eta }_{\mu }^{* } & -{b}^{* }{\eta }_{\mu }^{}\\ -b{\eta }_{\mu }^{* } & a{\eta }_{\mu }^{2}+b & -a| {\eta }_{\mu }^{}{| }^{2}\\ -{b}^{* }{\eta }_{\mu }^{} & -a| {\eta }_{\mu }^{}{| }^{2} & a{\eta }_{\mu }^{* 2}+{b}^{* }\end{array}\right),\end{eqnarray}$
with a being real.By diagonalizing the above ${M}_{\nu }^{}$ according to equation (1 ), one can derive the expressions of the resulting lepton flavor mixing angles and neutrino masses in terms of the model parameters a, b and ${\eta }_{\mu }^{}$. Before doing that, we first transform ${M}_{\nu }^{}$ to the following form
$\begin{eqnarray}\begin{array}{l}{M}_{\nu }^{{\prime} }=\left(\begin{array}{ccc}1 & & \\ & 1 & \\ & & -1\end{array}\right){M}_{\nu }^{}\left(\begin{array}{ccc}1 & & \\ & 1 & \\ & & -1\end{array}\right)\\ \,=\,\left(\begin{array}{ccc}2\mathrm{Re}(b{\eta }_{\mu }^{* 2}) & -b{\eta }_{\mu }^{* } & {b}^{* }{\eta }_{\mu }^{}\\ -b{\eta }_{\mu }^{* } & a{\eta }_{\mu }^{2}+b & a| {\eta }_{\mu }^{}{| }^{2}\\ {b}^{* }{\eta }_{\mu }^{} & a| {\eta }_{\mu }^{}{| }^{2} & a{\eta }_{\mu }^{* 2}+{b}^{* }\end{array}\right),\end{array}\end{eqnarray}$
via a redefinition of the phases of three neutrino fields (which are unphysical). Then, diagonalization of ${M}_{\nu }^{{\prime} }$ leads us to $\begin{eqnarray}\begin{array}{l}\tan {\theta }_{13}^{}=-\displaystyle \frac{\mathrm{Im}(b)+a\mathrm{Im}({\eta }_{\mu }^{2})}{\sqrt{2}\mathrm{Re}(b{\eta }_{\mu }^{* })},\\ \tan 2{\theta }_{13}^{}=-\displaystyle \frac{2\sqrt{2}\mathrm{Im}(b{\eta }_{\mu }^{* })}{\mathrm{Re}(b)+a\mathrm{Re}({\eta }_{\mu }^{2})+a| {\eta }_{\mu }^{}{| }^{2}+2\mathrm{Re}(b{\eta }_{\mu }^{* 2})},\\ \tan 2{\theta }_{12}^{}=\displaystyle \frac{2B}{C-A},\\ {m}_{1}^{}=\left|{c}_{12}^{2}A+{s}_{12}^{2}C-2{c}_{12}^{}{s}_{12}^{}B\right|,\\ {m}_{2}^{}=\left|{s}_{12}^{2}A+{c}_{12}^{2}C+2{c}_{12}^{}{s}_{12}^{}B\right|,\\ {m}_{3}^{}=\left|{c}_{13}^{2}[\mathrm{Re}(b)+a\mathrm{Re}({\eta }_{\mu }^{2})+a| {\eta }_{\mu }^{}{| }^{2}]\right.\\ \,\left.-\,2{s}_{13}^{2}\mathrm{Re}(b{\eta }_{\mu }^{* 2})-2\sqrt{2}{c}_{13}^{}{s}_{13}^{}\mathrm{Im}(b{\eta }_{\mu }^{* })\right|,\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{l}A=2{c}_{13}^{2}\mathrm{Re}(b{\eta }_{\mu }^{* 2})-{s}_{13}^{2}\left[\mathrm{Re}(b)+a\mathrm{Re}({\eta }_{\mu }^{2})\right.\\ \,\left.+\,a| {\eta }_{\mu }^{}{| }^{2}\right]-2\sqrt{2}{c}_{13}^{}{s}_{13}^{}\mathrm{Im}(b{\eta }_{\mu }^{* }),\\ B=-\sqrt{2}{c}_{13}^{}\mathrm{Re}(b{\eta }_{\mu }^{* })\,+\,{s}_{13}^{}\left[\mathrm{Im}(b)+a\mathrm{Im}({\eta }_{\mu }^{2})\right],\\ C=\mathrm{Re}(b)+a\mathrm{Re}({\eta }_{\mu }^{2})-a| {\eta }_{\mu }^{}{| }^{2}.\end{array}\end{eqnarray}$
Conversely, one can derive the expressions of the model parameters in terms of the lepton flavor mixing angles and neutrino masses by comparing ${M}_{\nu }^{{\prime} }$ in equation (13 ) and ${M}_{\nu }^{}$ obtained via the reconstruction relation ${M}_{\nu }^{}={{UD}}_{\nu }^{}{U}^{{\rm{T}}}$:
$\begin{eqnarray}\begin{array}{l}{\left({M}_{\nu }^{}\right)}_{{ee}}^{}={m}_{1}^{}{\eta }_{\rho }^{}{c}_{12}^{2}{c}_{13}^{2}+{m}_{2}^{}{\eta }_{\sigma }^{}{s}_{12}^{2}{c}_{13}^{2}-{m}_{3}^{}{s}_{13}^{2},\\ {\left({M}_{\nu }^{}\right)}_{e\mu }^{}=-\displaystyle \frac{1}{\sqrt{2}}{m}_{1}^{}{\eta }_{\rho }^{}{c}_{12}^{}{c}_{13}^{}\left({s}_{12}^{}-{\rm{i}}{c}_{12}^{}{s}_{13}^{}\right)\\ \,+\,\displaystyle \frac{1}{\sqrt{2}}{m}_{2}^{}{\eta }_{\sigma }^{}{s}_{12}^{}{c}_{13}^{}\left({c}_{12}^{}+{\rm{i}}{s}_{12}^{}{s}_{13}^{}\right)+\displaystyle \frac{1}{\sqrt{2}}{\rm{i}}{m}_{3}^{}{c}_{13}^{}{s}_{13}^{},\\ {\left({M}_{\nu }^{}\right)}_{\mu \mu }^{}=\displaystyle \frac{1}{2}{m}_{1}^{}{\eta }_{\rho }^{}{\left({s}_{12}^{}-{\rm{i}}{c}_{12}^{}{s}_{13}^{}\right)}^{2}\\ \,+\,\displaystyle \frac{1}{2}{m}_{2}^{}{\eta }_{\sigma }^{}{\left({c}_{12}^{}+{\rm{i}}{s}_{12}^{}{s}_{13}^{}\right)}^{2}+\displaystyle \frac{1}{2}{m}_{3}^{}{c}_{13}^{2},\\ {\left({M}_{\nu }^{}\right)}_{\mu \tau }^{}=-\displaystyle \frac{1}{2}{m}_{1}^{}{\eta }_{\rho }^{}\left({s}_{12}^{2}+{c}_{12}^{2}{s}_{13}^{2}\right)\\ \,-\displaystyle \frac{1}{2}{m}_{2}^{}{\eta }_{\sigma }^{}\left({c}_{12}^{2}+{s}_{12}^{2}{s}_{13}^{2}\right)+\displaystyle \frac{1}{2}{m}_{3}^{}{c}_{13}^{2},\end{array}\end{eqnarray}$
with ${\eta }_{\rho }^{}=1$ or −1 for ρ = 0 or π/2 (and similarly for ${\eta }_{\sigma }^{}$), ${\left({M}_{\nu }^{}\right)}_{e\tau }^{}=-{\left({M}_{\nu }^{}\right)}_{e\mu }^{* }$, and ${\left({M}_{\nu }^{}\right)}_{\tau \tau }^{}={\left({M}_{\nu }^{}\right)}_{\mu \mu }^{* }$. With the help of these expressions, by inputting the experimental results of the lepton flavor mixing angles and neutrino mass squared differences, we can calculate the values of ${({M}_{\nu }^{})}_{\alpha \beta }$ and subsequently the values of the model parameters.Let us first consider the NO case (i.e. ${m}_{1}^{}=0$). For σ = 0, allowing the lepton flavor mixing angles and neutrino mass squared differences to vary in their 3σ ranges, the ranges of ${({M}_{\nu }^{})}_{\alpha \beta }$ are found to be16 ) by taking ${m}_{1}^{}=0$. Subsequently, the ranges of the model parameters are obtained as13 ), (17 ). It is interesting to note that $| {\eta }_{\mu }^{}| $ can take the special value of 1/2 that corresponds to the so-called TM1 mixing [12]: after a proper redefinition of the phases of three neutrino fields, the first column of U will be proportional to ${\left(\mathrm{2,1,1}\right)}^{{\rm{T}}}$ as predicted by the TM1 mixing.
$\begin{eqnarray}\begin{array}{l}1.0\leqslant {\left({M}_{\nu }^{}\right)}_{{ee}}^{}/\mathrm{meV}\leqslant 2.2,\\ 21\leqslant {\left({M}_{\nu }^{}\right)}_{\mu \tau }^{}/\mathrm{meV}\leqslant 23,\\ 5.8\leqslant | {\left({M}_{\nu }^{}\right)}_{e\mu }^{}| /\mathrm{meV}\leqslant 6.7,\\ 0.33\leqslant \arg [{\left({M}_{\nu }^{}\right)}_{e\mu }^{}]/\pi \leqslant 0.37,\\ 27\leqslant | {\left({M}_{\nu }^{}\right)}_{\mu \mu }^{}| /\mathrm{meV}\leqslant 28,\\ 0.006\leqslant \arg [{\left({M}_{\nu }^{}\right)}_{\mu \mu }^{}]/\pi \leqslant 0.008.\end{array}\end{eqnarray}$
We see that there exist large hierarchies among ${({M}_{\nu }^{})}_{\alpha \beta }$: $| {\left({M}_{\nu }^{}\right)}_{\mu \mu }^{}| \sim | {\left({M}_{\nu }^{}\right)}_{\mu \tau }^{}| \gg | {\left({M}_{\nu }^{}\right)}_{e\mu }^{}| \gg | {\left({M}_{\nu }^{}\right)}_{{ee}}^{}| $. And ${\left({M}_{\nu }^{}\right)}_{\mu \mu }^{}$ is nearly purely real. These results can be understood from equation ( $\begin{eqnarray}\begin{array}{l}69\leqslant a/\mathrm{meV}\leqslant 110,\\ 11\leqslant | b| /\mathrm{meV}\leqslant 14,\\ 0.27\leqslant {\phi }_{b}^{}/\pi \leqslant 0.29,\\ 0.45\leqslant | {\eta }_{\mu }^{}| \leqslant 0.56,\\ 0.92\leqslant {\phi }_{\mu }^{}/\pi \leqslant 0.94.\end{array}\end{eqnarray}$
One can see that the relations ∣a∣ ≫ ∣b∣ and ${\phi }_{b}^{}\sim \arg [{\left({M}_{\nu }^{}\right)}_{e\mu }^{}]$ hold. And ${\eta }_{\mu }^{}$ is nearly purely real. These results can be understood with the help of equations (For σ = π/2, the ranges of ${({M}_{\nu }^{})}_{\alpha \beta }$ turn out to be18 ). This is because the change of σ from 0 to π/2 does not affect the first column of U which is in turn proportional to ${\left(1,{\eta }_{\mu }^{* },{\eta }_{\mu }^{}\right)}^{{\rm{T}}}$ in the case under consideration.
$\begin{eqnarray}\begin{array}{l}-4.5\leqslant {\left({M}_{\nu }^{}\right)}_{{ee}}^{}/\mathrm{meV}\leqslant -3.2,\\ 27\leqslant {\left({M}_{\nu }^{}\right)}_{\mu \tau }^{}/\mathrm{meV}\leqslant 28,\\ 5.4\leqslant | {\left({M}_{\nu }^{}\right)}_{e\mu }^{}| /\mathrm{meV}\leqslant 6.0,\\ 0.64\leqslant \arg [{\left({M}_{\nu }^{}\right)}_{e\mu }^{}]/\pi \leqslant 0.69,\\ 21\leqslant | {\left({M}_{\nu }^{}\right)}_{\mu \mu }^{}| /\mathrm{meV}\leqslant 23,\\ -0.010\leqslant \arg [{\left({M}_{\nu }^{}\right)}_{\mu \mu }^{}]/\pi \leqslant -0.008.\end{array}\end{eqnarray}$
In this case, $| {\left({M}_{\nu }^{}\right)}_{{ee}}^{}| $ becomes comparable to $| {\left({M}_{\nu }^{}\right)}_{e\mu }^{}| $. Subsequently, the ranges of a and b are obtained as $\begin{eqnarray}\begin{array}{l}87\leqslant a/\mathrm{meV}\leqslant 142,\\ 9.8\leqslant | b| /\mathrm{meV}\leqslant 13,\\ 0.57\leqslant {\phi }_{b}^{}/\pi \leqslant 0.62.\end{array}\end{eqnarray}$
On the other hand, the results of ${\eta }_{\mu }^{}$ are completely the same as in equation (We then consider the IO case (i.e. ${m}_{3}^{}=0$). In this case, only the difference between ρ and σ is of physical meaning. For σ − ρ = 0, the ranges of ${({M}_{\nu }^{})}_{\alpha \beta }$ are found to be16 ) by taking ${m}_{3}^{}=0$. Subsequently, the ranges of the model parameters are obtained as13 ), (21 ).
$\begin{eqnarray}\begin{array}{l}48\leqslant ({M}_{\nu }^{}{)}_{{ee}}^{}/\mathrm{meV}\leqslant 50,\\ -26\leqslant ({M}_{\nu }^{}{)}_{\mu \tau }^{}/\mathrm{meV}\leqslant -25,\\ 4.9\leqslant | ({M}_{\nu }^{}{)}_{e\mu }^{}| /\mathrm{meV}\leqslant 5.5,\\ 0.48\leqslant \arg [({M}_{\nu }^{}{)}_{e\mu }^{}]/\pi \leqslant 0.49,\\ 24\leqslant | ({M}_{\nu }^{}{)}_{\mu \mu }^{}| /\mathrm{meV}\leqslant 25,\\ 0.0006\leqslant \arg [({M}_{\nu }^{}{)}_{\mu \mu }^{}]/\pi \leqslant 0.0008.\end{array}\end{eqnarray}$
We see that the relation $| ({M}_{\nu }^{}{)}_{{ee}}^{}| \sim 2| ({M}_{\nu }^{}{)}_{\mu \mu }^{}| \sim 2| ({M}_{\nu }^{}{)}_{\mu \tau }^{}| \,\gg | ({M}_{\nu }^{}{)}_{e\mu }^{}| $ holds. And $({M}_{\nu }^{}{)}_{e\mu }^{}$ is nearly purely imaginary, while $({M}_{\nu }^{}{)}_{\mu \mu }^{}$ is nearly purely real. These results can be understood from equation ( $\begin{eqnarray}\begin{array}{l}-1.3\leqslant a/\mathrm{meV}\leqslant -1.0,\\ \quad 1.0\leqslant | b| /\mathrm{meV}\leqslant 1.2,\\ \quad 0.98\leqslant {\phi }_{b}^{}/\pi \leqslant 0.99,\\ 4.5\leqslant | {\eta }_{\mu }^{}| \leqslant 4.9,\\ \quad {\phi }_{\mu }^{}/\pi =-\displaystyle \frac{1}{2}.\end{array}\end{eqnarray}$
It is found that the relation ∣a∣ ∼ ∣b∣ holds. And b is nearly purely real. In particular, ${\eta }_{\mu }^{}$ is purely imaginary. These results can be understood with the help of equations (For σ − ρ = π/2, the ranges of ${({M}_{\nu }^{})}_{\alpha \beta }$ become20 ), the results of ${\eta }_{\mu }^{}$ are completely the same as in equation (22 ).
$\begin{eqnarray}\begin{array}{l}12\leqslant ({M}_{\nu }^{}{)}_{{ee}}^{}/\mathrm{meV}\leqslant 22,\\ \quad 6.4\leqslant ({M}_{\nu }^{}{)}_{\mu \tau }^{}/\mathrm{meV}\leqslant 12,\\ 31\leqslant | ({M}_{\nu }^{}{)}_{e\mu }^{}| /\mathrm{meV}\leqslant 34,\\ \quad 0.97\leqslant \arg [({M}_{\nu }^{}{)}_{e\mu }^{}]/\pi \leqslant 0.99,\\ 9.6\leqslant | ({M}_{\nu }^{}{)}_{\mu \mu }^{}| /\mathrm{meV}\leqslant 14,\\ \quad -0.84\leqslant \arg [({M}_{\nu }^{}{)}_{\mu \mu }^{}]/\pi \leqslant -0.73.\end{array}\end{eqnarray}$
In this case, the relation $| ({M}_{\nu }^{}{)}_{e\mu }^{}| \gg | ({M}_{\nu }^{}{)}_{{ee}}^{}| \sim | ({M}_{\nu }^{}{)}_{\mu \mu }^{}| \,\sim | ({M}_{\nu }^{}{)}_{\mu \tau }^{}| $ holds. And $({M}_{\nu }^{}{)}_{e\mu }^{}$ is nearly purely real. Subsequently, the ranges of a and b are obtained as $\begin{eqnarray}\begin{array}{l}0.3\leqslant a/\mathrm{meV}\leqslant 0.6,\\ \quad 6.3\leqslant | b| /\mathrm{meV}\leqslant 7.7,\\ \quad -0.53\leqslant {\phi }_{b}^{}/\pi \leqslant -0.51.\end{array}\end{eqnarray}$
It turns out that ∣b∣ is much larger than ∣a∣. And b is nearly purely imaginary. For the same reason as mentioned below equation (3. Summary
To summarize, in this letter we have made an attempt to embed the μ-τ reflection symmetry which predicts maximal atmospherical mixing angle and Dirac CP phase in the FL neutrino model which employs a translational flavor symmetry and keeps one neutrino mass vanishing (${m}_{1}^{}=0$ or ${m}_{3}^{}=0$ in the NO or IO case). We have first formulated such a combination (see ${M}_{\nu }^{}$ in equation (12 )) and then studied its consequences. Such a scenario is highly restrictive and predictive: ${M}_{\nu }^{}$ only contains three effective parameters (complex parameters b and ${\eta }_{\mu }^{}$, real parameter a); both the lepton flavor mixing parameters (see equation (5 )) and neutrino masses are constrained by the flavor symmetries.
We have derived the expressions of the resulting lepton flavor mixing angles and neutrino masses in terms of the model parameters by diagonalizing ${M}_{\nu }^{{\prime} }$ in equation (13 ), while the expressions of the latter in terms of the former can be derived by comparing ${M}_{\nu }^{{\prime} }$ in equation (13 ) and ${M}_{\nu }^{}$ obtained via the reconstruction relation ${M}_{\nu }^{}={{UD}}_{\nu }^{}{U}^{{\rm{T}}}$ (see equation (16 )). By varying the lepton flavor mixing angles and neutrino mass squared differences in their 3σ ranges, we have calculated the ranges of ${({M}_{\nu }^{})}_{\alpha \beta }$ and subsequently the ranges of the model parameters. The results are summarized as follows.
| • | In the NO case with σ = 0, we have $| ({M}_{\nu }^{}{)}_{\mu \mu }^{}| \sim | ({M}_{\nu }^{}{)}_{\mu \tau }^{}| \,\gg | ({M}_{\nu }^{}{)}_{e\mu }^{}| \gg | ({M}_{\nu }^{}{)}_{{ee}}^{}| $, ∣a∣ ≫ ∣b∣, ${\phi }_{b}^{}\sim \arg [({M}_{\nu }^{}{)}_{e\mu }^{}]$, and $({M}_{\nu }^{}{)}_{\mu \mu }^{}$ being nearly purely real. And $| {\eta }_{\mu }^{}| $ can take the special value of 1/2 that corresponds to the so-called TM1 mixing. |
| • | In the NO case with σ = π/2, $| ({M}_{\nu }^{}{)}_{{ee}}^{}| $ becomes comparable to $| ({M}_{\nu }^{}{)}_{e\mu }^{}| $. Note that the results of ${\eta }_{\mu }^{}$ are completely same as in the σ = 0 case. |
| • | In the IO case with ρ − σ = 0, one arrives at $| ({M}_{\nu }^{}{)}_{{ee}}^{}| \,\sim 2| ({M}_{\nu }^{}{)}_{\mu \mu }^{}| \sim 2| ({M}_{\nu }^{}{)}_{\mu \tau }^{}| \gg | ({M}_{\nu }^{}{)}_{e\mu }^{}| $, ∣a∣ ∼ ∣b∣, $({M}_{\nu }^{}{)}_{e\mu }^{}$ being nearly purely imaginary, and $({M}_{\nu }^{}{)}_{\mu \mu }^{}$ and b being nearly purely real. In particular, ${\eta }_{\mu }^{}$ is purely imaginary. |
| • | In the IO case with ρ − σ = π/2, we have $| ({M}_{\nu }^{}{)}_{e\mu }^{}| \gg | ({M}_{\nu }^{}{)}_{{ee}}^{}| \sim | ({M}_{\nu }^{}{)}_{\mu \mu }^{}| \sim | ({M}_{\nu }^{}{)}_{\mu \tau }^{}| $, ∣b∣ ≫ ∣a∣, $({M}_{\nu }^{}{)}_{e\mu }^{}$ being nearly purely real, and b being nearly purely imaginary. And the results of ${\eta }_{\mu }^{}$ are completely the same as in the ρ − σ = 0 case. |