1. Introduction
As is known, the observations of neutrino oscillations indicate that neutrinos have non-zero masses and mixing effects [2]. One of the most natural ways of generating the small neutrino masses is via the type-I seesaw mechanism [3] which introduces some right-handed neutrinos ${N}_{I}^{}$ (for I = 1, 2, 3) into the Standard Model. For these newly introduced particles, one can arrive at a Dirac neutrino mass matrix ${M}_{{\rm{D}}}^{}$ [whose elements are denoted as ${\left({M}_{{\rm{D}}}^{}\right)}_{\alpha I}^{}]$ which connects them with the left-handed neutrinos ${\nu }_{\alpha }^{}$ (for α = e, μ, τ), and the Majorana mass matrix ${M}_{{\rm{R}}}^{}$ for themselves. In this paper, without loss of generality, we will work in the basis where ${M}_{{\rm{R}}}^{}$ is diagonal ${M}_{{\rm{R}}}^{}=\mathrm{diag}({M}_{1}^{},{M}_{2}^{},{M}_{3}^{})$ with ${M}_{I}^{}$ being the mass of ${N}_{I}^{}$. Then, the Majorana mass matrix for light neutrinos arises as ${M}_{\nu }^{}\simeq -{M}_{{\rm{D}}}^{}{M}_{{\rm{R}}}^{-1}{M}_{{\rm{D}}}^{{\rm{T}}}$ [whose elements are denoted as ${\left({M}_{\nu }^{}\right)}_{\alpha \beta }^{}$]. Subsequently, in the basis where the mass matrix of the charged leptons is diagonal, the unitary matrix for diagonalizing ${M}_{\nu }^{}$ is just the neutrino mixing matrix U:
$\begin{eqnarray}{U}^{\dagger }{M}_{\nu }^{}{U}^{* }={D}_{\nu }^{}=\mathrm{diag}({m}_{1}^{},{m}_{2}^{},{m}_{3}^{}),\end{eqnarray}$
with ${m}_{i}^{}$ being three light neutrino masses. U can be parameterized in terms of three mixing angles ${\theta }_{{ij}}^{}$ (for ij = 12, 13, 23), one Dirac CP phase δ and two Majorana CP phases ρ and σ as $\begin{eqnarray}\begin{array}{l}U=\left(\begin{array}{ccc}{c}_{12}^{}{c}_{13}^{} & {s}_{12}^{}{c}_{13}^{} & {s}_{13}^{}{{\rm{e}}}^{-{\rm{i}}\delta }\\ -{s}_{12}^{}{c}_{23}^{}-{c}_{12}^{}{s}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & {c}_{12}^{}{c}_{23}^{}-{s}_{12}^{}{s}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & {s}_{23}^{}{c}_{13}^{}\\ {s}_{12}^{}{s}_{23}^{}-{c}_{12}^{}{c}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & -{c}_{12}^{}{s}_{23}^{}-{s}_{12}^{}{c}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & {c}_{23}^{}{c}_{13}^{}\end{array}\right)\\ \quad \times \left(\begin{array}{ccc}{{\rm{e}}}^{{\rm{i}}\rho } & & \\ & {{\rm{e}}}^{{\rm{i}}\sigma } & \\ & & 1\end{array}\right),\end{array}\end{eqnarray}$
with ${c}_{{ij}}^{}=\cos {\theta }_{{ij}}^{}$ and ${s}_{{ij}}^{}=\sin {\theta }_{{ij}}^{}$.Thanks to the various neutrino oscillation experiments, the three neutrino mixing angles and neutrino mass squared differences ${\rm{\Delta }}{m}_{{ij}}^{2}\equiv {m}_{i}^{2}-{m}_{j}^{2}$ have been measured with a good degree of accuracy. See table 1 for the global-fit results of these parameters [4, 5]. And there is a preliminary result for δ, but it is subject to a large uncertainty. Note that the sign of ${\rm{\Delta }}{m}_{31}^{2}$ is still undetermined, thereby allowing for two possible neutrino mass orderings: the normal ordering (NO) ${m}_{1}^{}\lt {m}_{2}^{}\lt {m}_{3}^{}$ and inverted ordering (IO) ${m}_{3}^{}\lt {m}_{1}^{}\lt {m}_{2}^{}$. On the other hand, neutrino oscillations are completely insensitive to the neutrino mass themselves and the Majorana CP phases, whose values can only be inferred from some non-oscillatory processes (e.g. the neutrinoless double beta decay [6]). So far, there has not been any lower constraint on the lightest neutrino mass, nor any constraint on the Majorana CP phases.
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 [4]. |
| 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 |
Among the particles in the Standard Model, neutrinos are the unique candidate for Majorana fermions. And the neutrino masses would be of the Majorana nature if they were really generated via the seesaw mechanism. Therefore, testing the (Majorana or Dirac) nature of neutrino masses will not only tell us if there exist Majorana fermions in nature but also can help us trace the origin of neutrino masses. The most promising way to tackle this issue is to search for the neutrinoless double beta (0νββ) decay [6]. Such a decay can be mediated by massive Majorana neutrinos. Its rate is controlled by the Majorana neutrino mass ${m}_{\beta \beta }^{}$. When a right-handed neutrino is much heavier than the typical scale of Fermi momentum of a nucleus ${{\rm{\Lambda }}}_{\beta }^{}\simeq 100\,\mathrm{MeV}$, its direct contribution to ${m}_{\beta \beta }^{}$ is negligibly small. But it can contribute to ${m}_{\beta \beta }^{}$ indirectly through its contribution to the light neutrino mass term ${\left({M}_{\nu }^{}\right)}_{{ee}}^{}$ via the seesaw mechanism. When a right-handed neutrino is lighter than ${{\rm{\Lambda }}}_{\beta }^{}$, as a result of the intrinsic property of the seesaw mechanism, its direct contribution to ${m}_{\beta \beta }^{}$ and its indirect contribution to ${m}_{\beta \beta }^{}$ through its contribution to ${\left({M}_{\nu }^{}\right)}_{{ee}}^{}$ exactly cancel out each other [7].
In [1], Asaka, Ishida and Tanaka put forward an interesting possibility that the 0νββ decay can be hidden in the minimal seesaw model (in which only two right-handed neutrinos are relevant for the generation of neutrino masses) with the two right-handed neutrinos having a hierarchical mass structure3(We note that the mass splitting between the two right-handed neutrinos can be realized by invoking the Froggatt–Nielsen mechanism [8, 9] or the extra dimensional theory [10].): the lighter one (to which we refer as ${N}_{1}^{}$, without loss of generality) is lighter enough than ${{\rm{\Lambda }}}_{\beta }^{}$ while the heavier one (to which we refer as ${N}_{2}^{}$) is sufficiently heavy to decouple from the 0νββ decay. Under the condition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$ 4(We note that a texture zero such as ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$ here can be realized by invoking some Abelian flavor symmetry [11].) , the 0νββ decay will be hidden: the direct and indirect contributions of ${N}_{1}^{}$ to ${m}_{\beta \beta }^{}$ exactly cancel out each other, while ${N}_{2}^{}$ has neither direct nor indirect contribution to ${m}_{\beta \beta }^{}$ [12]. This point can be understood more clearly as follows: ${m}_{\beta \beta }^{}$ receives two contributions as
$\begin{eqnarray}{m}_{\beta \beta }^{}={m}_{\beta \beta }^{\nu }+{m}_{\beta \beta }^{N},\end{eqnarray}$
with $\begin{eqnarray}{m}_{\beta \beta }^{\nu }={({M}_{\nu }^{})}_{{ee}}=-\displaystyle \frac{{\left({M}_{{\rm{D}}}^{}\right)}_{e1}^{2}}{{M}_{1}^{}}-\displaystyle \frac{{\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{2}}{{M}_{2}^{}},\end{eqnarray}$
and $\begin{eqnarray}{m}_{\beta \beta }^{N}=\displaystyle \frac{{{\rm{\Lambda }}}_{\beta }^{2}}{{{\rm{\Lambda }}}_{\beta }^{2}+{M}_{1}^{2}}\displaystyle \frac{{\left({M}_{{\rm{D}}}^{}\right)}_{e1}^{2}}{{M}_{1}^{}}+\displaystyle \frac{{{\rm{\Lambda }}}_{\beta }^{2}}{{{\rm{\Lambda }}}_{\beta }^{2}+{M}_{2}^{2}}\displaystyle \frac{{\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{2}}{{M}_{2}^{}}.\end{eqnarray}$
It is easy to see that, in the case of ${M}_{1}^{}\ll {{\rm{\Lambda }}}_{\beta }^{}\ll {M}_{2}^{}$, one will have ${m}_{\beta \beta }^{}\to 0$ under the condition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$. For convenience, in the following we refer to such a model as the Asaka-Ishida-Tanaka (AIT) model.A simple number counting can tell us that the condition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$ itself gives no constraint on the neutrino parameters (i.e. the light neutrino masses and mixing parameters) encoded in ${M}_{\nu }^{}$: in the minimal seesaw model, ${M}_{{\rm{D}}}^{}$ contains 6 complex entries, while ${M}_{\nu }^{}$ only contains 5 independent complex entries5(Although ${M}_{\nu }^{}$ has 6 independent complex entries in the general seesaw model, it only has 5 independent complex entries, as a result of $\det ({M}_{\nu }^{})=0$, in the minimal seesaw model.). Therefore, even after the imposition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$, ${M}_{{\rm{D}}}^{}$ still has enough degrees of freedom to match any experimental results for the neutrino parameters encoded in ${M}_{\nu }^{}$ (provided that the minimal seesaw model has not been ruled out). Just for this reason, it will be an interesting attempt, on top of the AIT mode, to further impose some meaningful conditions on ${M}_{{\rm{D}}}^{}$ so that some constraints on the neutrino parameters encoded in ${M}_{\nu }^{}$ and the properties of the right-handed neutrinos can be obtained. Along this direction, in this paper we study the interesting scenario that ${M}_{{\rm{D}}}^{}$ of the AIT model further obeys the TM1 symmetry or μ–τ reflection symmetry which are well motivated by the experimental results for the neutrino mixing parameters.
2. Scenario of ${M}_{{\rm{D}}}^{}$ obeying the TM1 symmetry
In this section, we study the scenario that ${M}_{{\rm{D}}}^{}$ of the AIT model further obeys the TM1 symmetry which is responsible for the TM1 mixing.
The neutrino oscillation experimental results reveal that ${\theta }_{12}^{}$ and ${\theta }_{23}^{}$ are around some special values: ${\sin }^{2}{\theta }_{12}^{}\sim 1/3$ and ${\sin }^{2}{\theta }_{23}^{}\sim 1/2$. Furthermore, before the experimental determination of the value of ${\theta }_{13}^{}$, it was believed by many people to be vanishingly small. Therefore, at that time, the tribimaximal (TBM) mixing [13]
$\begin{eqnarray}{U}_{\mathrm{TBM}}^{}=\displaystyle \frac{1}{\sqrt{6}}\left(\begin{array}{ccc}2 & \sqrt{2} & 0\\ 1 & -\sqrt{2} & -\sqrt{3}\\ 1 & -\sqrt{2} & \sqrt{3}\end{array}\right),\end{eqnarray}$
that can accommodate ${\sin }^{2}{\theta }_{12}^{}=1/3$, ${\sin }^{2}{\theta }_{23}^{}=1/2$ (i.e. ${\theta }_{23}^{}=\pi /4$) and ${\theta }_{13}^{}=0$ was very popular. However, the observation of a relatively large ${\theta }_{13}^{}$ compels us to modify this mixing pattern. A possible way out is to keep its first or second column unchanged while modifying the other two columns under the unitary constraints, giving the first or second trimaximal (TM1 or TM2) mixing [14] $\begin{eqnarray}\begin{array}{rcl}{U}_{\mathrm{TM}1}^{} & = & \displaystyle \frac{1}{\sqrt{6}}\left(\begin{array}{ccc}2 & \cdot & \cdot \\ 1 & \cdot & \cdot \\ 1 & \cdot & \cdot \end{array}\right),\\ {U}_{\mathrm{TM}2}^{} & = & \displaystyle \frac{1}{\sqrt{3}}\left(\begin{array}{ccc}\cdot & 1 & \cdot \\ \cdot & -1 & \cdot \\ \cdot & -1 & \cdot \end{array}\right),\end{array}\end{eqnarray}$
whose predictions for the neutrino mixing parameters are in good agreement with the experimental measurements.In the minimal seesaw model, the TM1 mixing can be naturally realized from the following forms of ${M}_{{\rm{D}}}^{}$ [15]
$\begin{eqnarray}\begin{array}{ccc}{M}_{{\rm{D}}}^{} & = & \left(\begin{array}{cc}a\sqrt{{M}_{1}^{}} & c\sqrt{{M}_{2}^{}}\\ (-a-b)\sqrt{{M}_{1}^{}} & (-c-d)\sqrt{{M}_{2}^{}}\\ (-a+b)\sqrt{{M}_{1}^{}} & (-c+d)\sqrt{{M}_{2}^{}}\end{array}\right)\,{\rm{and}}\,\\ & & \left(\begin{array}{cc}2a\sqrt{{M}_{1}^{}} & b\sqrt{{M}_{2}^{}}\\ a\sqrt{{M}_{1}^{}} & (-b-c)\sqrt{{M}_{2}^{}}\\ a\sqrt{{M}_{1}^{}} & (-b+c)\sqrt{{M}_{2}^{}}\end{array}\right),\end{array}\end{eqnarray}$
where a, b, c and d are complex parameters. It is direct to verify that the resulting ${M}_{\nu }^{}$ from such forms of ${M}_{{\rm{D}}}^{}$ obeys the TM1 symmetry which is responsible for the TM1 mixing: $\begin{eqnarray}\begin{array}{rcl}{M}_{\nu }^{} & = & {R}_{\mathrm{TM}1}^{}{M}_{\nu }^{}{R}_{\mathrm{TM}1}^{}\qquad \mathrm{with}\qquad \\ {R}_{\mathrm{TM}1}^{} & = & -\displaystyle \frac{1}{3}\left(\begin{array}{ccc}1 & 2 & 2\\ 2 & -2 & 1\\ 2 & 1 & -2\end{array}\right).\end{array}\end{eqnarray}$
Now, we further impose the condition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$ on ${M}_{{\rm{D}}}^{}$ in equation (8 ) so that the 0ν2β decay will be hidden. For the second form of ${M}_{{\rm{D}}}^{}$ in equation (8 ), a further imposition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$ (i.e. b = 0) would lead one column of U to be proportional to (0, − 1, 1)T (corresponding to ${\theta }_{13}^{}=0$) which is not experimentally viable, so we will not consider it any more. For the first form of ${M}_{{\rm{D}}}^{}$ in equation (8 ), a further imposition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$ (i.e. c = 0) will lead to an ${M}_{\nu }^{}$ as11 ) according to the following formulas
$\begin{eqnarray}{M}_{\nu }^{}\simeq -\left(\begin{array}{ccc}{a}^{2} & -a(a+b) & -a(a-b)\\ -a(a+b) & {\left(a+b\right)}^{2}+{d}^{2} & {a}^{2}-{b}^{2}-{d}^{2}\\ -a(a-b) & {a}^{2}-{b}^{2}-{d}^{2} & {\left(a-b\right)}^{2}+{d}^{2}\end{array}\right).\end{eqnarray}$
Such an ${M}_{\nu }^{}$ can be diagonalized by the following unitary matrix $\begin{eqnarray}\begin{array}{ccc}U & = & {U}_{{\rm{TBM}}}^{}\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \cos \theta & \sin \theta {{\rm{e}}}^{-{\rm{i}}\varphi }\\ 0 & -\sin \theta {{\rm{e}}}^{{\rm{i}}\varphi } & \cos \theta \end{array}\right)\\ & & \times \,\left(\begin{array}{ccc}{{\rm{e}}}^{{\rm{i}}{\phi }_{1}^{}} & & \\ & {{\rm{e}}}^{{\rm{i}}{\phi }_{2}^{}} & \\ & & {{\rm{e}}}^{{\rm{i}}{\phi }_{3}^{}}\end{array}\right),\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}\tan 2\theta & = & \displaystyle \frac{2\sqrt{6}\left|3{a}^{* 2}{ab}+2{a}^{* }{b}^{* }({b}^{2}+{d}^{2})\right|}{4| {b}^{2}+{d}^{2}{| }^{2}-9| a{| }^{4}},\\ \varphi & = & \arg \left[3{a}^{* 2}{ab}+2{a}^{* }{b}^{* }({b}^{2}+{d}^{2})\right],\end{array}\end{eqnarray}$
where $\arg $ denotes the argument of the relevant quantity, giving an NO neutrino mass spectrum with ${m}_{1}^{}=0$ and $\begin{eqnarray}\begin{array}{rcl}{m}_{2}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{2}^{}} & = & -3{a}^{2}{\cos }^{2}\theta -2({b}^{2}+{d}^{2}){\sin }^{2}\theta {{\rm{e}}}^{2{\rm{i}}\varphi }\\ & & +\sqrt{6}{ab}\sin 2\theta {{\rm{e}}}^{-{\rm{i}}\varphi },\\ {m}_{3}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{3}^{}} & = & -2({b}^{2}+{d}^{2}){\cos }^{2}\theta -3{a}^{2}{\sin }^{2}\theta {{\rm{e}}}^{2{\rm{i}}\varphi }\\ & & -\sqrt{6}{ab}\sin 2\theta {{\rm{e}}}^{{\rm{i}}\varphi }.\end{array}\end{eqnarray}$
And the resulting neutrino mixing parameters can be extracted from U in equation ( $\begin{eqnarray}\begin{array}{rcl}{s}_{13}^{2} & = & \displaystyle \frac{1}{3}{\sin }^{2}\theta ,{s}_{12}^{2}=\displaystyle \frac{1}{3}-\displaystyle \frac{2{s}_{13}^{2}}{3-3{s}_{13}^{2}},\\ {s}_{23}^{2} & = & \displaystyle \frac{1}{2}+\displaystyle \frac{\sqrt{6}\sin 2\theta \cos \varphi }{6-2{\sin }^{2}\theta },\\ \cos \delta & = & -\displaystyle \frac{1-5{s}_{13}^{2}}{2\sqrt{2}{s}_{13}^{}\sqrt{1-3{s}_{13}^{2}}\tan 2{\theta }_{23}^{}},\\ \sigma & = & \varphi -\delta +{\phi }_{2}^{}-{\phi }_{3}^{}.\end{array}\end{eqnarray}$
Then, by confronting ${M}_{\nu }^{}$ in equation (10 ) against the experimental results for the neutrino masses and mixing parameters, we calculate the phenomenologically viable values of a, b and d. Before performing the calculation, we note that the phase of d can be taken out as the overall phase of ${M}_{\nu }^{}$ which is of no physical meaning, leaving us with ${\phi }_{{ad}}^{}\equiv \arg (a)-\arg (d)$ and ${\phi }_{{bd}}^{}\equiv \arg (b)-\arg (d)$ as the physical phases. Accordingly, figure 1(a) and (b) show the values of ∣a∣2, ∣b∣2, ${\phi }_{{ad}}^{}$ and ${\phi }_{{bd}}^{}$ versus ∣d∣2 for ${M}_{\nu }^{}$ in equation (10 ) to be consistent with the experimental results within the 3σ level. These results are obtained in the following way: for randomly selected values of ∣a∣2, ∣b∣2, ∣d∣2, ${\phi }_{{ad}}^{}$ and ${\phi }_{{bd}}^{}$, we check if the resulting values for the light neutrino masses and neutrino mixing parameters obtained from equations (12 )–(14 ) fall in their respective 3σ ranges [4]. If yes, then these values of ∣a∣2, ∣b∣2, ∣d∣2, ${\phi }_{{ad}}^{}$ and ${\phi }_{{bd}}^{}$ will be recorded. A repetition of such a procedure for enough times finally yields the results in figure 1. One can see that ∣b∣2 is close to (or much larger than) ∣a∣2 for ∣d∣2 → 0.02 eV (or ∣d∣2 → 0.05 eV). And there exists some parameter space where ${\phi }_{{ad}}^{}$ and ${\phi }_{{bd}}^{}$ can be equal to each other, in which case we would have only one effective phase. For these values of a, b and d, figure 1(c) shows the resulting values of δ and σ (also versus ∣d∣2). Here we have just shown the results for δ < 0, while the results for δ > 0 can be obtained simply by making a sign reversal. It turns out that δ is around − π/2, in agreement with the current experimental results. Furthermore, we study the resulting values of the mixing strengths $| {{\rm{\Theta }}}_{\alpha 1}^{}{| }^{2}\equiv | {\left({M}_{{\rm{D}}}^{}\right)}_{\alpha 1}^{}{| }^{2}/{M}_{1}^{2}$ of ${N}_{1}^{}$ with three left-handed neutrinos. The sizes of $| {{\rm{\Theta }}}_{\alpha 1}^{}{| }^{2}$ determine the discovery prospects of ${N}_{1}^{}$ in relevant experiments [16]. And the relative sizes of them determine which flavor-specific channel will be the most promising one for the discovery of ${N}_{1}^{}$. Figure 1(d) shows the resulting values of $| {{\rm{\Theta }}}_{\alpha 1}^{}{| }^{2}$ (also versus ∣d∣2). The results in figure 1(d) are obtained by taking ${M}_{1}^{}=1\,\mathrm{MeV}$ as a benchmark value. (Note that the results in figures 1(a)–(c) are independent of the value of ${M}_{1}^{}$ 
.) Given that $| {{\rm{\Theta }}}_{\alpha 1}^{}{| }^{2}$ are inversely proportional to ${M}_{1}^{}$, the results for other values of ${M}_{1}^{}$ can be obtained by rescaling the presented results proportionally. One can see that the values of $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\tau 1}^{}{| }^{2}$ are close to each other, which reflects the approximate μ–τ flavor symmetry in the neutrino sector [17, 18]. And they can reach $\simeq 4\times {10}^{-8}\ \mathrm{MeV}/{M}_{1}^{}$ at most, while $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$ can only reach $\simeq 4\times {10}^{-9}\ \mathrm{MeV}/{M}_{1}^{}$ at most. In the mass region of ${N}_{1}^{}$ relevant to our study (i.e. ${M}_{1}^{}\ll {{\rm{\Lambda }}}_{\beta }^{}\simeq 100$ MeV), the maximally allowed values of $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$, $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\tau 1}^{}{| }^{2}$ are smaller than the present experimental bounds for them by about two, two and five orders of magnitude (see figures 26–28 of [19]).
.) Given that $| {{\rm{\Theta }}}_{\alpha 1}^{}{| }^{2}$ are inversely proportional to ${M}_{1}^{}$, the results for other values of ${M}_{1}^{}$ can be obtained by rescaling the presented results proportionally. One can see that the values of $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\tau 1}^{}{| }^{2}$ are close to each other, which reflects the approximate μ–τ flavor symmetry in the neutrino sector [17, 18]. And they can reach $\simeq 4\times {10}^{-8}\ \mathrm{MeV}/{M}_{1}^{}$ at most, while $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$ can only reach $\simeq 4\times {10}^{-9}\ \mathrm{MeV}/{M}_{1}^{}$ at most. In the mass region of ${N}_{1}^{}$ relevant to our study (i.e. ${M}_{1}^{}\ll {{\rm{\Lambda }}}_{\beta }^{}\simeq 100$ MeV), the maximally allowed values of $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$, $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\tau 1}^{}{| }^{2}$ are smaller than the present experimental bounds for them by about two, two and five orders of magnitude (see figures 26–28 of [19]).
Figure 1. (a) and (b): The values of ∣a∣2, ∣b∣2, ${\phi }_{{ad}}^{}$ and ${\phi }_{{bd}}^{}$ versus ∣d∣2 for ${M}_{\nu }^{}$ in equation ( |
Finally, we point out that the scenario that ${M}_{{\rm{D}}}^{}$ of the AIT model further obeys the TM2 symmetry which is responsible for the TM2 mixing is not experimentally viable: in the minimal seesaw model, the TM2 mixing can be naturally realized from the following form of ${M}_{{\rm{D}}}^{}$ [15]
$\begin{eqnarray}{M}_{{\rm{D}}}^{}=\left(\begin{array}{cc}a\sqrt{{M}_{1}^{}} & 2b\sqrt{{M}_{2}^{}}\\ -a\sqrt{{M}_{1}^{}} & (b-c)\sqrt{{M}_{2}^{}}\\ -a\sqrt{{M}_{1}^{}} & (b+c)\sqrt{{M}_{2}^{}}\end{array}\right).\end{eqnarray}$
For such a form of ${M}_{{\rm{D}}}^{}$, a further imposition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$ (i.e. b = 0) would also lead one column of U to be proportional to ${\left(0,-\mathrm{1,1}\right)}^{T}$ which is not experimentally viable.3. Scenario of ${M}_{{\rm{D}}}^{}$ obeying the μ–τ reflection symmetry
In this section, we study the scenario that ${M}_{{\rm{D}}}^{}$ of the AIT model further obeys the μ–τ reflection symmetry.
The μ–τ interchange symmetry [17, 18] (under which the neutrino mass matrix keeps invariant with respect to the ${\nu }_{\mu }^{}\leftrightarrow {\nu }_{\tau }^{}$ interchange) was ever very popular for its predictions ${\theta }_{23}^{}=\pi /4$ and ${\theta }_{13}^{}=0$. After the experimental observation of a relatively large ${\theta }_{13}^{}$ and a preliminary experimental hint for δ ∼ −π/2, the μ–τ reflection symmetry [20] has become increasingly popular, under which the neutrino mass matrix keeps invariant with respect to the following transformations of three left-handed neutrino fields
$\begin{eqnarray}{\nu }_{e}^{}\leftrightarrow {\nu }_{e}^{c},{\nu }_{\mu }^{}\leftrightarrow {\nu }_{\tau }^{c},{\nu }_{\tau }^{}\leftrightarrow {\nu }_{\mu }^{c},\end{eqnarray}$
with the superscript ‘c’ denoting the charge conjugation of relevant neutrino fields. Such a symmetry leads to the following interesting predictions for the neutrino mixing parameters $\begin{eqnarray}{\theta }_{23}^{}=\displaystyle \frac{\pi }{4},\delta =\pm \displaystyle \frac{\pi }{2},\rho ,\sigma =0\ \mathrm{or}\ \displaystyle \frac{\pi }{2}.\end{eqnarray}$
In the following we will take δ = −π/2 (which is more favored by the experimental results) out of ±π/2.In the minimal seesaw model, the μ–τ reflection symmetry can be naturally realized from the following form of ${M}_{{\rm{D}}}^{}$ [21]
$\begin{eqnarray}{M}_{{\rm{D}}}^{}=\left(\begin{array}{cc}a\sqrt{{M}_{1}^{}} & c\sqrt{{M}_{2}^{}}\\ b\sqrt{{M}_{1}^{}} & d\sqrt{{M}_{2}^{}}\\ {b}^{* }\sqrt{{M}_{1}^{}} & {d}^{* }\sqrt{{M}_{2}^{}}\end{array}\right){P}_{N}^{},\end{eqnarray}$
with a and c being real parameters, b and d being complex parameters, and ${P}_{N}^{}=\mathrm{diag}(\sqrt{\eta },1)$ (for η = ±1). It is direct to verify that the resulting ${M}_{\nu }^{}$ from such an ${M}_{{\rm{D}}}^{}$ obeys the μ–τ reflection symmetry: $\begin{eqnarray}\begin{array}{rcl}{M}_{\nu }^{} & = & {R}_{\mu \tau }^{}{M}_{\nu }^{* }{R}_{\mu \tau }^{}\ \mathrm{with}\\ {R}_{\mu \tau }^{} & = & \left(\begin{array}{ccc}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{array}\right).\end{array}\end{eqnarray}$
For the form of ${M}_{{\rm{D}}}^{}$ in equation (18 ), a further imposition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$ (i.e. c = 0) will lead to an ${M}_{\nu }^{}$ as
$\begin{eqnarray}{M}_{\nu }^{}\simeq -\left(\begin{array}{ccc}\eta {a}^{2} & \eta {ab} & \eta {{ab}}^{* }\\ \eta {ab} & \eta {b}^{2}+{d}^{2} & \eta | b{| }^{2}+| d{| }^{2}\\ \eta {{ab}}^{* } & \eta | b{| }^{2}+| d{| }^{2} & \eta {b}^{* 2}+{d}^{* 2}\end{array}\right).\end{eqnarray}$
Such an ${M}_{\nu }^{}$ can be diagonalized by the following unitary matrix $\begin{eqnarray}\begin{array}{rcl}U & = & \displaystyle \frac{1}{\sqrt{2}}\left(\begin{array}{ccc}\sqrt{2}{c}_{12}^{}{c}_{13}^{} & \sqrt{2}{s}_{12}^{}{c}_{13}^{} & {\rm{i}}\sqrt{2}{s}_{13}^{}\\ -{s}_{12}^{}+{\rm{i}}{c}_{12}^{}{s}_{13}^{} & {c}_{12}^{}+{\rm{i}}{s}_{12}^{}{s}_{13}^{} & {c}_{13}^{}\\ -{s}_{12}^{}-{\rm{i}}{c}_{12}^{}{s}_{13}^{} & {c}_{12}^{}-{\rm{i}}{s}_{12}^{}{s}_{13}^{} & -{c}_{13}^{}\end{array}\right)\\ & & \times \left(\begin{array}{ccc}{{\rm{e}}}^{{\rm{i}}{\phi }_{1}^{}} & & \\ & {{\rm{e}}}^{{\rm{i}}{\phi }_{2}^{}} & \\ & & {{\rm{e}}}^{{\rm{i}}{\phi }_{3}^{}}\end{array}\right),\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}\tan {\theta }_{13}^{} & = & \displaystyle \frac{\mathrm{Im}(\eta {b}^{2}+{d}^{2})}{\sqrt{2}\eta a\mathrm{Re}(b)},\\ \tan 2{\theta }_{13}^{} & = & \displaystyle \frac{2\sqrt{2}\eta a\mathrm{Im}(b)}{\eta {a}^{2}+\mathrm{Re}(\eta {b}^{2}+{d}^{2})-(\eta | b{| }^{2}+| d{| }^{2})},\\ \tan 2{\theta }_{12}^{} & = & \displaystyle \frac{2B}{C-A},\\ A & = & -{c}_{13}^{2}\eta {a}^{2}+{s}_{13}^{2}\left[\mathrm{Re}(\eta {b}^{2}+{d}^{2})-(\eta | b{| }^{2}+| d{| }^{2})\right]\\ & & -2\sqrt{2}{c}_{13}^{}{s}_{13}^{}\eta a\mathrm{Im}(b),\\ B & = & -\sqrt{2}{c}_{13}^{}\eta a\mathrm{Re}(b)-{s}_{13}^{}\mathrm{Im}(\eta {b}^{2}+{d}^{2}),\\ C & = & -\mathrm{Re}(\eta {b}^{2}+{d}^{2})-(\eta | b{| }^{2}+| d{| }^{2}),\end{array}\end{eqnarray}$
giving the neutrino masses as $\begin{eqnarray}\begin{array}{rcl}{m}_{1}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{1}^{}} & = & {c}_{12}^{2}A+{s}_{12}^{2}C-2{c}_{12}^{}{s}_{12}^{}B,\\ {m}_{2}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{2}^{}} & = & {s}_{12}^{2}A+{c}_{12}^{2}C+2{c}_{12}^{}{s}_{12}^{}B,\\ {m}_{3}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{3}^{}} & = & {c}_{13}^{2}[-\mathrm{Re}(\eta {b}^{2}+{d}^{2})+(\eta | b{| }^{2}+| d{| }^{2})]\\ & & +{s}_{13}^{2}\eta {a}^{2}-2\sqrt{2}{c}_{13}^{}{s}_{13}^{}\eta a\mathrm{Im}(b).\end{array}\end{eqnarray}$
Obviously, due to the reality of A, B and C, one has ${\phi }_{1}^{},{\phi }_{2}^{},{\phi }_{3}^{}=0$ or π/2. From these results ρ and σ can be further obtained as $\rho ={\phi }_{1}^{}-{\phi }_{3}^{}$ and $\sigma ={\phi }_{2}^{}-{\phi }_{3}^{}$.Then, by confronting ${M}_{\nu }^{}$ in equation (20 ) against the experimental results for the neutrino masses and mixing parameters, we calculate the phenomenologically viable values of a, b and d. Before performing the calculation, we note that a proper redefinition of the unphysical phases of three left-handed neutrino fields can transform d to real20 ) to be consistent with the experimental results within the 3σ level. It is found that, in order to get viable results, η needs to be 1 (−1) in the case of σ = 0 (π/2). Figure 2(c) shows the allowed values of $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}=| {{\rm{\Theta }}}_{\tau 1}^{}{| }^{2}$ (which is a natural consequence of the μ–τ reflection symmetry) versus a2, which are also obtained by taking ${M}_{1}^{}=1\,\mathrm{MeV}$ as a benchmark value. In the case of σ = 0, $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$ can reach $\simeq 2\times {10}^{-9}\ \mathrm{MeV}/{M}_{1}^{}$ and $\simeq 4\times {10}^{-8}\ \mathrm{MeV}/{M}_{1}^{}$ at most, which are smaller than the present experimental bounds for them by about two orders of magnitude. In the case of σ = π/2, $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$ can reach $\simeq 5\times {10}^{-9}\ \mathrm{MeV}/{M}_{1}^{}$ and $\simeq 1\times {10}^{-8}\ \mathrm{MeV}/{M}_{1}^{}$ at most. figures 2(d)–(f) are the counterparts of figures 2(a)–(c) in the IO case. In this case, in order to get viable results, η needs to be 1 (−1) in the case of σ − ρ = 0 (π/2).6(Note that only the difference of ρ and σ is of physical meaning in the case of ${m}_{3}^{}=0$.) In the case of σ − ρ = 0, $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$ can reach $\simeq 5\times {10}^{-8}\ \mathrm{MeV}/{M}_{1}^{}$ and $\simeq 5\times {10}^{-10}\ \mathrm{MeV}/{M}_{1}^{}$ at most, which are smaller than the present experimental bounds for them by about one and four orders of magnitude. In the case of σ − ρ = π/2, $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$ can reach $\simeq 2\times {10}^{-9}\ \mathrm{MeV}/{M}_{1}^{}$ and $\simeq 6\times {10}^{-9}\ \mathrm{MeV}/{M}_{1}^{}$ at most.
$\begin{eqnarray}\begin{array}{l}{M}_{\nu }^{}\to {P}_{l}^{\dagger }{M}_{\nu }^{}{P}_{l}^{* }\ \mathrm{with}\\ {P}_{l}^{}=\mathrm{diag}\left[1,{{\rm{e}}}^{\mathrm{iarg}(d)},{{\rm{e}}}^{-\mathrm{iarg}(d)}\right],\end{array}\end{eqnarray}$
leaving us with ${\phi }_{{bd}}^{}\equiv \arg (b)-\arg (d)$ as the only physical phase. Accordingly, figures 2(a) and (b) show, in the NO case, the values of ∣b∣2, ∣d∣2 and ${\phi }_{{bd}}^{}$ versus a2 for ${M}_{\nu }^{}$ in equation (
Figure 2. (a) and (b): In the NO case, the values of ∣b∣2, ∣d∣2 and ${\phi }_{{bd}}^{}$ versus a2 for ${M}_{\nu }^{}$ in equation ( |
4. Summary
In this paper, we have made a further study of the AIT model. In the AIT model, the two right-handed neutrinos of the minimal seesaw model have a hierarchical mass structure: the lighter one ${N}_{1}^{}$ is lighter enough than ${{\rm{\Lambda }}}_{\beta }^{}$ while the heavier one ${N}_{2}^{}$ is sufficiently heavy to decouple from the 0νββ decay. Under the condition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$, the 0νββ decay will be hidden: the direct and indirect contributions of ${N}_{1}^{}$ to ${m}_{\beta \beta }^{}$ exactly cancel out each other, while ${N}_{2}^{}$ has neither direct nor indirect contribution to ${m}_{\beta \beta }^{}$.
In this paper, on top of the AIT model, we study the interesting scenario that ${M}_{{\rm{D}}}^{}$ further obeys the TM1 symmetry or μ–τ reflection symmetry which are well motivated by the experimental results for the neutrino mixing parameters. These two scenarios can be realized by a further imposition of ${\left({M}_{{\rm{D}}}^{}\right)}_{e2}^{}=0$ on the particular forms of ${M}_{{\rm{D}}}^{}$ that can naturally realize these two symmetries.
For the scenario that ${M}_{{\rm{D}}}^{}$ of the AIT model further obeys the TM1 symmetry (which gives an ${M}_{\nu }^{}$ as in equation (10 )), only ∣a∣2, ∣b∣2, ∣d∣2, ${\phi }_{{ad}}^{}$ and ${\phi }_{{bd}}^{}$ are physically relevant. We have calculated their phenomenologically viable values. It is found that the resulting neutrino masses are of the NO case. And δ is around −π/2, in agreement with the current experimental results. Furthermore, we have obtained the possible values of the mixing strengths $| {{\rm{\Theta }}}_{\alpha 1}^{}{| }^{2}$ of ${N}_{1}^{}$ with three left-handed neutrinos, which determine the discovery prospects of ${N}_{1}^{}$ in relevant experiments. It is found that $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$ and $| {{\rm{\Theta }}}_{\tau 1}^{}{| }^{2}$ are close to each other. And their maximally allowed values are much larger than those of $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$.
For the scenario that ${M}_{{\rm{D}}}^{}$ of the AIT model further obeys the μ–τ reflection symmetry (which gives an ${M}_{\nu }^{}$ as in equation (20 )), only a2, ∣b∣2, ∣d∣2 and ${\phi }_{{bd}}^{}$ are physically relevant. We have calculated their phenomenologically viable values. In this scenario, the resulting neutrino masses can be either of the NO case or of the IO case. In the NO case with σ = 0, one has $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}\ll | {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$. In the IO case with σ − ρ = 0, $| {{\rm{\Theta }}}_{e1}^{}{| }^{2}$ becomes much larger than $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}$. Note that one has exactly $| {{\rm{\Theta }}}_{\mu 1}^{}{| }^{2}=| {{\rm{\Theta }}}_{\tau 1}^{}{| }^{2}$ under the μ–τ reflection symmetry.
For the above two scenarios, in most of the parameter space, the maximally allowed values of $| {{\rm{\Theta }}}_{\alpha 1}^{}{| }^{2}$ are smaller than the present experimental bounds for them by about two orders of magnitude. We note that the mixing strengths between the left- and right-handed neutrinos can be naturally enhanced in the seesaw models invoking an approximate lepton number conservation (e.g. the inverse seesaw model [22]).
Finally, we point out that the AIT model is incapable of explaining the baryon–antibaryon asymmetry of the Universe via the leptogenesis mechanism [23] because the mass of the lighter right-handed neutrino is far below the Davidson–Ibarra lower bound (∼1010 GeV) on the right-handed neutrino masses from the requirement of leptogenesis being viable in the case of the right-handed neutrino masses being hierarchical [24].