1. Introduction
As we know, the phenomenon of neutrino oscillations indicates that neutrinos are massive and lepton flavors are mixed [1]. In order to accommodate such a beyond Standard Model (SM) phenomenon, one must extend the SM (where neutrinos are unable to get masses through the Higgs mechanism like other fermions, due to the absence of the right-handed neutrinos) in a proper way. In this connection, one of the most popular and natural ways of generating the finite but tiny neutrino masses is the type-I seesaw model in which a certain number of super heavy right-handed neutrinos ${N}_{I}^{}$ (for I = 1, 2, 3) are added on top of the SM [2]. These newly introduced particles can constitute the Yukawa coupling operators together with the left-handed neutrinos ${\nu }_{\alpha }^{}$ (which reside in the lepton doublets ${L}_{\alpha }^{}$, for α = e, μ, τ) and the Higgs field H: ${\left({Y}_{\nu }^{}\right)}_{\alpha I}^{}{\overline{L}}_{\alpha }^{}{{HN}}_{I}^{}$ with ${\left({Y}_{\nu }^{}\right)}_{\alpha I}^{}$ being the Yukawa coupling coefficients, which will contribute the Dirac neutrino masses $({M}_{{\rm{D}}}^{}{)}_{\alpha I}^{}=({Y}_{\nu }^{}{)}_{\alpha I}^{}v$ after the Higgs field acquires the non-vanishing vacuum expectation value v = 174 GeV. Remarkably, they can also have the Majorana masses of themselves. Here we will work in the basis of their Majorana mass matrix being diagonal ${M}_{{\rm{R}}}^{}=\mathrm{diag}({M}_{1}^{},{M}_{2}^{},{M}_{3}^{})$ (with ${M}_{I}^{}$ being three right-handed neutrino masses). Then, under the seesaw limit ${M}_{{\rm{R}}}^{}\gg {M}_{{\rm{D}}}^{}$, one will get an effective Majorana mass matrix ${M}_{\nu }^{}\simeq -{M}_{{\rm{D}}}^{}{M}_{{\rm{R}}}^{-1}{M}_{{\rm{D}}}^{T}$ for the light neutrinos by integrating the heavy right-handed neutrinos out. Thanks to such a seesaw formula, the smallness of the neutrino masses can be ascribed to the heaviness of the right-handed neutrinos.
When the flavor eigenstates of three charged leptons align with their mass eigenstates, the lepton flavor mixing matrix U arises as the unitary matrix for diagonalizing ${M}_{\nu }^{}$:
$\begin{eqnarray}{U}^{\dagger }{M}_{\nu }^{}{U}^{* }=\mathrm{diag}({m}_{1}^{},{m}_{2}^{},{m}_{3}^{}),\end{eqnarray}$
with ${m}_{i}^{}$ being three light neutrino masses. In the standard parametrization, U is expressed in terms of three lepton flavor mixing angles ${\theta }_{{ij}}^{}$ (for ij = 12, 13, 23), one Dirac CP phase δ and two Majorana CP phases ρ and σ as follows $\begin{eqnarray}\begin{array}{rcl}U & = & \left(\begin{array}{ccc}{c}_{12}^{}{c}_{13}^{} & {s}_{12}^{}{c}_{13}^{} & {s}_{13}^{}{{\rm{e}}}^{-{\rm{i}}\delta }\\ -{s}_{12}^{}{c}_{23}^{}-{c}_{12}^{}{s}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & {c}_{12}^{}{c}_{23}^{}-{s}_{12}^{}{s}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & {s}_{23}^{}{c}_{13}^{}\\ {s}_{12}^{}{s}_{23}^{}-{c}_{12}^{}{c}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & -{c}_{12}^{}{s}_{23}^{}-{s}_{12}^{}{c}_{23}^{}{s}_{13}^{}{{\rm{e}}}^{{\rm{i}}\delta } & {c}_{23}^{}{c}_{13}^{}\end{array}\right)\\ & & \times \left(\begin{array}{ccc}{{\rm{e}}}^{{\rm{i}}\rho } & & \\ & {{\rm{e}}}^{{\rm{i}}\sigma } & \\ & & 1\end{array}\right),\end{array}\end{eqnarray}$
where the abbreviations ${c}_{{ij}}^{}=\cos {\theta }_{{ij}}^{}$ and ${s}_{{ij}}^{}=\sin {\theta }_{{ij}}^{}$ have been employed. On the experimental side, neutrino oscillations are sensitive to the lepton flavor mixing angles, neutrino mass squared differences ${\rm{\Delta }}{m}_{{ij}}^{2}\equiv {m}_{i}^{2}-{m}_{j}^{2}$, and the Dirac CP phase δ. Several research groups have performed global analyses of the available neutrino oscillation data to extract the values of these parameters [3, 4]. For definiteness, we will use the results in [3] (copied in table 1 here) as reference values in the following numerical calculations. Note that the sign of ${\rm{\Delta }}{m}_{31}^{2}$ remains unclear, thereby allowing for two possible neutrino mass orderings: the normal ordering (NO) case ${m}_{1}^{}\lt {m}_{2}^{}\lt {m}_{3}^{}$ and inverted ordering (IO) case ${m}_{3}^{}\lt {m}_{1}^{}\lt {m}_{2}^{}$. And neutrino oscillations are completely insensitive to the absolute values of the neutrino masses and the Majorana CP phases. Their values can only be inferred from certain non-oscillatory experiments such as the neutrinoless double beta decay experiments [5]. Unfortunately, so far there has been no lower constraint on the value of 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 [3]. |
| 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 → 0.02417 | ${0.02250}_{-0.00076}^{+0.00056}$ | 0.02039 → 0.02441 |
| 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 |
Owing to the particular values of the lepton flavor mixing angles (e.g. ${\sin }^{2}{\theta }_{12}^{}\sim 1/3$ and ${\sin }^{2}{\theta }_{23}^{}\sim 1/2$) and a preliminary experimental hint for δ ∼ −π/2 [6], the possibility that there may exist a certain flavor symmetry in the lepton sector has been receiving a lot of attention in the literature [7]. The flavor symmetries can help us reduce the free parameters of the neutrino mass matrices and thus enhance their predictive power. One popular candidate of the lepton flavor symmetry is the TM1 symmetry [8], under which ${M}_{\nu }^{}$ is required to obey the following condition
$\begin{eqnarray}\begin{array}{l}{R}_{\mathrm{TM}1}^{\dagger }{M}_{\nu }^{}{R}_{\mathrm{TM}1}^{* }={M}_{\nu }^{}\hspace{0.5cm}\mathrm{with}\\ {R}_{\mathrm{TM}1}^{}=-\displaystyle \frac{1}{3}\left(\begin{array}{ccc}1 & 2 & 2\\ 2 & -2 & 1\\ 2 & 1 & -2\end{array}\right).\end{array}\end{eqnarray}$
Such a symmetry will lead to a particular pattern of the lepton flavor mixing matrix (referred to as the TM1 mixing) as follows $\begin{eqnarray}{U}_{\mathrm{TM}1}^{}=\displaystyle \frac{1}{\sqrt{6}}\left(\begin{array}{ccc}2 & \cdot & \cdot \\ 1 & \cdot & \cdot \\ 1 & \cdot & \cdot \end{array}\right),\end{eqnarray}$
where the ‘·’ signs are used to indicate the unspecified matrix elements of ${U}_{\mathrm{TM}1}^{}$, which gives a ${\theta }_{12}^{}$ satisfying the observation of ${\sin }^{2}{\theta }_{12}^{}\sim 1/3$: $\begin{eqnarray}{\sin }^{2}{\theta }_{12}^{}=\displaystyle \frac{1}{3}-\displaystyle \frac{2{s}_{13}^{2}}{3-3{s}_{13}^{2}}\simeq 0.318.\end{eqnarray}$
Another popular candidate of the lepton flavor symmetry is the μ-τ reflection symmetry [9, 10], under which ${M}_{\nu }^{}$ is required to obey the following condition $\begin{eqnarray}\begin{array}{l}{R}_{\mu \tau }^{\dagger }{M}_{\nu }^{* }{R}_{\mu \tau }^{* }={M}_{\nu }^{}\qquad \mathrm{with}\\ {R}_{\mu \tau }^{}=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{array}\right).\end{array}\end{eqnarray}$
Such a symmetry will lead to the following interesting predictions for the lepton flavor mixing parameters $\begin{eqnarray}\begin{array}{rcl}{\theta }_{23}^{} & = & \displaystyle \frac{\pi }{4},\qquad \delta =\pm \displaystyle \frac{\pi }{2},\\ \qquad \rho & = & 0\ \mathrm{or}\ \displaystyle \frac{\pi }{2},\qquad \sigma =0\ \mathrm{or}\ \displaystyle \frac{\pi }{2}.\end{array}\end{eqnarray}$
In the literature, there are two other typical approaches to reducing the free parameters of the neutrino mass matrices and thus enhancing their predictive power. One is to reduce the number of the right-handed neutrinos to two (referred to as the minimal seesaw model [11, 12]), in which case the lightest neutrino mass remains to be vanishing (i.e. ${m}_{1}^{}=0$ in the NO case or ${m}_{3}^{}=0$ in the IO case) and only one Majorana CP phase is physically relevant (σ in the NO case or σ − ρ in the IO case). The other one is simply to impose texture zeros on the neutrino mass matrices (i.e. assuming some elements of the neutrino mass matrices to be vanishing) [11].
Motivated by the above facts, following the Occam’s razor principle, in this paper we put forward a very simple form of ${M}_{{\rm{D}}}^{}$ in the minimal seesaw model with the right-handed neutrino mass matrix being diagonal ${M}_{{\rm{R}}}^{}\,=\mathrm{diag}({M}_{1}^{},{M}_{2}^{})$: as will be seen in the next section, it has one texture zero and only contains three real parameters, and leads to an ${M}_{\nu }^{}$ that obeys the TM1 and μ-τ reflection symmetries simultaneously [13]. As we will see, such a model is highly restrictive and predictive. The rest of this paper is organized as follows. In the next section, we present the model and study its consequences for the lepton flavor mixing parameters and neutrino masses. Then, we study the implications of this model for leptogenesis in section 3 . Finally, a brief summary of our main results will be given in section 4 .
2. The model and its consequences for the neutrino parameters
The model is given as follows: in the minimal seesaw model with the right-handed neutrino mass matrix being diagonal ${M}_{{\rm{R}}}^{}=\mathrm{diag}({M}_{1}^{},{M}_{2}^{})$, ${M}_{{\rm{D}}}^{}$ takes a very simple form as3 ), (6 )).
$\begin{eqnarray}{M}_{{\rm{D}}}^{}=\left(\begin{array}{cc}a\sqrt{{M}_{1}^{}} & 0\\ (-a-{\rm{i}}b)\sqrt{{M}_{1}^{}} & -c\sqrt{{M}_{2}^{}}\\ (-a+{\rm{i}}b)\sqrt{{M}_{1}^{}} & c\sqrt{{M}_{2}^{}}\end{array}\right){P}_{N}^{},\end{eqnarray}$
with a, b and c being real, and ${P}_{N}^{}=\mathrm{diag}(\sqrt{\eta },1)$ (for η = ±1). Then, the seesaw formula gives an ${M}_{\nu }^{}$ as $\begin{eqnarray}{M}_{\nu }^{}\simeq -\left(\begin{array}{ccc}\eta {a}^{2} & -\eta {a}^{2}-{\rm{i}}\eta {ab} & -\eta {a}^{2}+{\rm{i}}\eta {ab}\\ -\eta {a}^{2}-{\rm{i}}\eta {ab} & \eta ({a}^{2}-{b}^{2})+{c}^{2}+2{\rm{i}}\eta {ab} & \eta ({a}^{2}+{b}^{2})-{c}^{2}\\ -\eta {a}^{2}+{\rm{i}}\eta {ab} & \eta ({a}^{2}+{b}^{2})-{c}^{2} & \eta ({a}^{2}-{b}^{2})+{c}^{2}-2{\rm{i}}\eta {ab}\end{array}\right).\end{eqnarray}$
It is direct to verify that such an ${M}_{\nu }^{}$ does obey the TM1 and μ-τ reflection symmetries simultaneously (see equations (${M}_{\nu }^{}$ in equation (9 ) can be diagonalized by a unitary matrix of the following form (which is just the lepton flavor mixing matrix)
$\begin{eqnarray}\begin{array}{rcl}U & = & \frac{1}{\sqrt{6}}\left(\begin{array}{ccc}2 & \sqrt{2} & 0\\ 1 & -\sqrt{2} & -\sqrt{3}\\ 1 & -\sqrt{2} & \sqrt{3}\end{array}\right)\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & {\rm{i}}\end{array}\right)\\ & & \times \left(\begin{array}{ccc}1 & 0 & 0\\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{ccc}{{\rm{e}}}^{{\rm{i}}{\phi }_{1}^{}} & & \\ & {{\rm{e}}}^{{\rm{i}}{\phi }_{2}^{}} & \\ & & {{\rm{e}}}^{{\rm{i}}{\phi }_{3}^{}}\end{array}\right),\end{array}\end{eqnarray}$
with $\begin{eqnarray}\tan 2\theta =\displaystyle \frac{2\sqrt{6}\eta {ab}}{2(\eta {b}^{2}-{c}^{2})-3\eta {a}^{2}}.\end{eqnarray}$
And its diagonalization yields ${m}_{1}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{1}^{}}=0$ and $\begin{eqnarray}\begin{array}{l}{m}_{2}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{2}^{}}=\frac{1}{2}\left[-3\eta {a}^{2}-2(\eta {b}^{2}-{c}^{2})\right.\\ \quad \left.-\sqrt{{\left[3\eta {a}^{2}+2(\eta {b}^{2}-{c}^{2})\right]}^{2}+24\eta {a}^{2}{c}^{2}}\right],\\ {m}_{3}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{3}^{}}=\frac{1}{2}\left[-3\eta {a}^{2}-2(\eta {b}^{2}-{c}^{2})\right.\\ \quad \left.+\sqrt{{\left[3\eta {a}^{2}+2(\eta {b}^{2}-{c}^{2})\right]}^{2}+24\eta {a}^{2}{c}^{2}}\right],\end{array}\end{eqnarray}$
in the case of 3ηa2 + 2(ηb2 − c2) < 0, or $\begin{eqnarray}\begin{array}{l}{m}_{2}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{2}^{}}=\frac{1}{2}\left[-3\eta {a}^{2}-2(\eta {b}^{2}-{c}^{2})\right.\\ \quad \left.+\sqrt{{\left[3\eta {a}^{2}+2(\eta {b}^{2}-{c}^{2})\right]}^{2}+24\eta {a}^{2}{c}^{2}}\right],\\ {m}_{3}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{3}^{}}=\frac{1}{2}\left[-3\eta {a}^{2}-2(\eta {b}^{2}-{c}^{2})\right.\\ \quad \left.-\sqrt{{\left[3\eta {a}^{2}+2(\eta {b}^{2}-{c}^{2})\right]}^{2}+24\eta {a}^{2}{c}^{2}}\right],\end{array}\end{eqnarray}$
in the case of 3ηa2 + 2(ηb2 − c2) > 0. There are two immediate consequences of ${m}_{1}^{}=0$: the neutrino masses is predicted to be of the NO case, and their values will be pinned down (i.e. ${m}_{2}^{}\simeq 8.7$ meV and ${m}_{2}^{}\simeq 50.6$ meV) with the help of the experimental results for the neutrino mass squared differences; ${\phi }_{1}^{}$ is undeterminable and thus of no physical meaning. As for ${\phi }_{2}^{}$ and ${\phi }_{3}^{}$, they serve to ensure the positiveness of ${m}_{2}^{}$ and ${m}_{3}^{}$, and will simply take the value of 0 or π/2 due to the realness of ${m}_{2}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{2}^{}}$ and ${m}_{3}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{3}^{}}$.With the help of the freedom of redefining the unphysical phases [to be specific, we have made the transformation $U\to {P}_{l}^{}U$ with ${P}_{l}^{}={\mathrm{ie}}^{-{\rm{i}}{\phi }_{3}^{}}\mathrm{diag}(1,{{\rm{e}}}^{{\rm{i}}\phi },{{\rm{e}}}^{-{\rm{i}}\phi })$ for ${e}^{{\rm{i}}\phi }\,=(-\sqrt{3}\cos \theta -{\rm{i}}\sqrt{2}\sin \theta )/\sqrt{3-{\sin }^{2}\theta }$], U in equation (10 ) can be converted into the following standard-parametrization form2 ), one not only can retrieve the predictions of the TM1 and μ-τ reflection symmetries for the lepton flavor mixing parameters in equations (5 ), (7 ), but further arrives at
$\begin{eqnarray}\begin{array}{rcl}U & = & \frac{1}{\sqrt{6}}\left(\begin{array}{ccc}2 & \sqrt{2}\cos \theta & -\sqrt{2}\mathrm{isin}\theta \\ \frac{-\sqrt{3}\cos \theta -{\rm{i}}\sqrt{2}\sin \theta }{\sqrt{3-{\sin }^{2}\theta }} & \frac{\sqrt{6}-\mathrm{icos}\theta \sin \theta }{\sqrt{3-{\sin }^{2}\theta }} & \sqrt{3-{\sin }^{2}\theta }\\ \frac{\sqrt{3}\cos \theta -{\rm{i}}\sqrt{2}\sin \theta }{\sqrt{3-{\sin }^{2}\theta }} & \frac{-\sqrt{6}-\mathrm{icos}\theta \sin \theta }{\sqrt{3-{\sin }^{2}\theta }} & \sqrt{3-{\sin }^{2}\theta }\end{array}\right)\\ & & \times \left(\begin{array}{ccc}{\mathrm{ie}}^{{\rm{i}}({\phi }_{1}^{}-{\phi }_{3}^{})} & & \\ & {\mathrm{ie}}^{{\rm{i}}({\phi }_{2}^{}-{\phi }_{3}^{})} & \\ & & 1\end{array}\right).\end{array}\end{eqnarray}$
In obtaining this result, we have taken into account that the periodicity of the Majorana CP phases is π. By making a direct comparison between this U with that in equation ( $\begin{eqnarray}{s}_{13}^{2}=\displaystyle \frac{1}{3}{\sin }^{2}\theta ,\qquad \delta =\mathrm{sign}(\sin \theta )\displaystyle \frac{\pi }{2},\end{eqnarray}$
and σ = 0 (or π/2) for $| {\phi }_{2}^{}-{\phi }_{3}^{}| =\pi /2$ (or 0). Note that, given ${m}_{1}^{}=0$, the other Majorana CP phase ρ has no physical meaning.Then, we study the concrete consequences of the cases of 3ηa2 + 2(ηb2 − c2) < 0 and >0, respectively. Let us first consider the case of 3ηa2 + 2(ηb2 − c2) < 0. In this case, with the help of equations (11 ), (12 ) and ${\sin }^{2}\theta =3{s}_{13}^{2}$, one can fit the values of a2, b2 and c2 by inputting the measured values of ${s}_{13}^{2}$, ${\rm{\Delta }}{m}_{21}^{2}$ and ${\rm{\Delta }}{m}_{31}^{2}$. (1) For η = + 1, one obtains11 ), (15 ), (16 ). From the results in equation (16 ), the effective Majorana neutrino mass $| ({M}_{\nu }^{}{)}_{{ee}}^{}| $ that controls the rate of the neutrinoless double beta decay can also be immediately read as $| ({M}_{\nu }^{}{)}_{{ee}}^{}| ={a}^{2}\simeq 1.6$ meV. Furthermore, as can be seen from equation (12 ), for η = 1 one has ${m}_{2}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{2}^{}}\lt 0$ and ${m}_{3}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{3}^{}}\gt 0$ (corresponding to ${\phi }_{2}^{}=\pi /2$ and ${\phi }_{3}^{}=0$) and thus σ = 0. (2) For η = − 1, one obtains
$\begin{eqnarray}\begin{array}{rcl}{a}^{2} & \simeq & 1.6\ \mathrm{meV},\qquad {b}^{2}\simeq 23.2\ \mathrm{meV},\\ {c}^{2} & \simeq & 46.6\ \mathrm{meV}.\end{array}\end{eqnarray}$
From these results, the values of a, b and c themselves can be directly obtained as $\pm \sqrt{{a}^{2}}$, $\pm \sqrt{{b}^{2}}$ and $\pm \sqrt{{c}^{2}}$. But it should be noted that one needs to ensure ab > 0 (or <0) in order to have δ = −π/2 (or π/2), which can be easily understood with the help of equations ( $\begin{eqnarray}\begin{array}{rcl}{a}^{2} & \simeq & 3.8\ \mathrm{meV},\qquad {b}^{2}\simeq 4.8\ \mathrm{meV},\\ {c}^{2} & \simeq & 19.1\ \mathrm{meV}.\end{array}\end{eqnarray}$
This time, when obtaining the values of a, b and c themselves, one needs to ensure ab > 0 (or <0) in order to have δ = π/2 (or −π/2). And the effective Majorana neutrino mass for the neutrinoless double beta decay can be read as $| ({M}_{\nu }^{}{)}_{{ee}}^{}| \simeq 3.8$ meV. Finally, for η = −1 one has ${m}_{2}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{2}^{}}\gt 0$ and ${m}_{3}^{}{{\rm{e}}}^{2{\rm{i}}{\phi }_{3}^{}}\gt 0$ (corresponding to ${\phi }_{2}^{}={\phi }_{3}^{}=\pi /2$) and thus σ = π/2.For the case of 3ηa2 + 2(ηb2 − c2) > 0, the values of a2, b2 and c2 can be fitted with the help of equations (11 ), (13 ) and ${\sin }^{2}\theta =3{s}_{13}^{2}$. It is found that, in this case, there are no phenomenologically viable values of a2, b2 and c2.
3. Implications for leptogenesis
In this section, we study the implications of the model for leptogenesis. As we know, the seesaw model also provides an attractive explanation (via the leptogenesis mechanism [14, 15]) for the baryon–antibaryon asymmetry of the Universe [16]
$\begin{eqnarray}{Y}_{{\rm{B}}}^{}\equiv \displaystyle \frac{{n}_{{\rm{B}}}^{}-{n}_{\bar{{\rm{B}}}}^{}}{s}\simeq (8.69\pm 0.04)\times {10}^{-11},\end{eqnarray}$
where ${n}_{{\rm{B}}}^{}$ (${n}_{\bar{{\rm{B}}}}^{}$) is the baryon (antibaryon) number density and s the entropy density. The leptogenesis mechanism works in a way as follows: a lepton–antilepton asymmetry is firstly generated from the decays of the right-handed neutrinos and then partly converted into the baryon–antibaryon asymmetry via the sphaleron processes.Depending on the temperature ranges where leptogenesis takes effect (approximately the right-handed neutrino mass scale), there are three distinct leptogenesis regimes in terms of the relevant flavor effects [17]. (1) Unflavored regime: in the temperature range above 1012 GeV where the charged lepton Yukawa ${y}_{\alpha }^{}$ interactions have not yet entered thermal equilibrium, three lepton flavors are indistinguishable and should be treated in a universal manner. In this regime, the final baryon asymmetry can be calculated according to
$\begin{eqnarray}{Y}_{{\rm{B}}}^{}=-{cr}{\varepsilon }_{I}^{}\kappa ({\widetilde{m}}_{I}^{}),\end{eqnarray}$
where c = 28/79 describes the transition efficiency from the lepton asymmetry to the baryon asymmetry via the sphaleron processes, and r ≃ 4 × 10−3 measures the ratio of the equilibrium ${N}_{I}^{}$ number density to the entropy density. And ${\varepsilon }_{I}^{}$ is the total asymmetry between the decay rates of ${N}_{I}^{}\to {L}_{\alpha }^{}+H$ and their CP-conjugate processes ${N}_{I}^{}\to {\overline{L}}_{\alpha }^{}+\overline{H}$ $\begin{eqnarray}\begin{array}{l}{\varepsilon }_{I}^{}=\displaystyle \frac{1}{8\pi {\left({M}_{{\rm{D}}}^{\dagger }{M}_{{\rm{D}}}^{}\right)}_{{II}}^{}{v}^{2}}\\ \quad \times \displaystyle \sum _{J\ne I}^{}\mathrm{Im}\left[{\left({M}_{{\rm{D}}}^{\dagger }{M}_{{\rm{D}}}^{}\right)}_{{IJ}}^{2}\right]{ \mathcal F }\left(\displaystyle \frac{{M}_{J}^{2}}{{M}_{I}^{2}}\right),\end{array}\end{eqnarray}$
with ${ \mathcal F }(x)=\sqrt{x}\{(2-x)/(1-x)+(1+x)\mathrm{ln}[x/(1+x)]\}$. It is a sum (over three lepton flavors) of the flavor-specific CP asymmetries $\begin{eqnarray}\begin{array}{l}{\varepsilon }_{I\alpha }^{}=\displaystyle \frac{1}{8\pi {\left({M}_{{\rm{D}}}^{\dagger }{M}_{{\rm{D}}}^{}\right)}_{{II}}^{}{v}^{2}}\displaystyle \sum _{J\ne I}^{}\\ \quad \times \left\{\mathrm{Im}\left[{\left({M}_{{\rm{D}}}^{* }\right)}_{\alpha I}^{}{\left({M}_{{\rm{D}}}^{}\right)}_{\alpha J}^{}{\left({M}_{{\rm{D}}}^{\dagger }{M}_{{\rm{D}}}^{}\right)}_{{IJ}}^{}\right]{ \mathcal F }\left(\displaystyle \frac{{M}_{J}^{2}}{{M}_{I}^{2}}\right)\right.\\ \quad +\left.\mathrm{Im}\left[{\left({M}_{{\rm{D}}}^{* }\right)}_{\alpha I}^{}{\left({M}_{{\rm{D}}}^{}\right)}_{\alpha J}^{}{\left({M}_{{\rm{D}}}^{\dagger }{M}_{{\rm{D}}}^{}\right)}_{{IJ}}^{* }\right]{ \mathcal G }\left(\displaystyle \frac{{M}_{J}^{2}}{{M}_{I}^{2}}\right)\right\},\end{array}\end{eqnarray}$
with ${ \mathcal G }(x)=1/(1-x)$. Finally, $\kappa ({\widetilde{m}}_{I}^{})$ is the efficiency factor taking into account the washout effects. Its value depends on the washout mass parameter $\begin{eqnarray}{\widetilde{m}}_{I}^{}=\displaystyle \sum _{\alpha }^{}{\widetilde{m}}_{I\alpha }^{}=\displaystyle \sum _{\alpha }^{}\displaystyle \frac{| {\left({M}_{{\rm{D}}}^{}\right)}_{\alpha I}^{}{| }^{2}}{{M}_{I}^{}},\end{eqnarray}$
and the dependence can be well described by the following analytical fits [18] $\begin{eqnarray}\displaystyle \frac{1}{\kappa (x)}\simeq \displaystyle \frac{3.3\times {10}^{-3}\,\mathrm{eV}}{x}+{\left(\displaystyle \frac{x}{5.5\times {10}^{-4}\,\mathrm{eV}}\right)}^{1.16}.\end{eqnarray}$
(2) Two-flavor regime: in the temperature range 109—1012 GeV where the ${y}_{\tau }^{}$-related (${y}_{\mu }^{}$- and ${y}_{e}^{}$-related) interactions have (not) entered thermal equilibrium, the τ flavor is distinguishable from the other two flavors which remain indistinguishable and thus should be treated separately. In this regime, the final baryon asymmetry can be calculated according to $\begin{eqnarray}{Y}_{{\rm{B}}}^{}=-{cr}\left[{\varepsilon }_{I\gamma }^{}\kappa \left(\displaystyle \frac{417}{589}{\widetilde{m}}_{I\gamma }^{}\right)+{\varepsilon }_{I\tau }^{}\kappa \left(\displaystyle \frac{390}{589}{\widetilde{m}}_{I\tau }^{}\right)\right],\end{eqnarray}$
with ${\varepsilon }_{I\gamma }^{}={\varepsilon }_{{Ie}}^{}+{\varepsilon }_{I\mu }^{}$ and ${\widetilde{m}}_{I\gamma }^{}={\widetilde{m}}_{{Ie}}^{}+{\widetilde{m}}_{I\mu }^{}$. (3) Three-flavor regime: in the temperature range below 109 GeV where the ${y}_{\mu }^{}$-related interactions also enter thermal equilibrium, all three flavors are distinguishable and should be treated separately. In this regime, the final baryon asymmetry can be calculated according to $\begin{eqnarray}\begin{array}{l}{Y}_{{\rm{B}}}^{}=-{cr}\left[{\varepsilon }_{I\tau }^{}\kappa \left(\displaystyle \frac{344}{537}{\widetilde{m}}_{I\tau }^{}\right)+{\varepsilon }_{I\mu }^{}\kappa \left(\displaystyle \frac{344}{537}{\widetilde{m}}_{I\mu }^{}\right)\right.\\ \quad \left.+{\varepsilon }_{{Ie}}^{}\kappa \left(\displaystyle \frac{453}{537}{\widetilde{m}}_{{Ie}}^{}\right)\right].\end{array}\end{eqnarray}$
For the model studied in the present paper, as can be seen from equations (8 ), (20 )–(22 ), we have19 ), (25 )], meaning that leptogenesis cannot work successfully in these two regimes. Fortunately, in the two-flavor regime, a successful leptogenesis may become possible [19]: owing to the relations in equations (26 ), (24 ) becomes
$\begin{eqnarray}\begin{array}{l}{\varepsilon }_{I}^{}=0,\qquad {\varepsilon }_{{Ie}}^{}=0,\\ {\varepsilon }_{I\mu }^{}=-{\varepsilon }_{I\tau }^{},\qquad {\widetilde{m}}_{I\mu }^{}={\widetilde{m}}_{I\tau }^{}.\end{array}\end{eqnarray}$
As an immediate result, one would arrive at ${Y}_{{\rm{B}}}^{}=0$ in the unflavored and three-flavor regimes [see equations ( $\begin{eqnarray}{Y}_{{\rm{B}}}^{}=-{cr}{\varepsilon }_{I\mu }^{}\left[\kappa \left(\displaystyle \frac{417}{589}{\widetilde{m}}_{I\gamma }^{}\right)-\kappa \left(\displaystyle \frac{390}{589}{\widetilde{m}}_{I\tau }^{}\right)\right].\end{eqnarray}$
We see that unless $390{\widetilde{m}}_{I\tau }^{}$ coincides with $417{\widetilde{m}}_{I\gamma }^{}$, ${Y}_{{\rm{B}}}^{}$ will be non-vanishing. For this reason, we will work in the two-flavor regime in the following discussions.Unless the right-handed neutrino masses are nearly degenerate, which is beyond the scope of the present paper, the contribution to leptogenesis mainly comes from the lightest right-handed neutrino because those from the heavier ones suffer from its subsequent washout effects. For completeness, here we will study the scenarios of ${N}_{1}^{}$ and ${N}_{2}^{}$ being the lighter ones, respectively. Let us first consider the scenario of ${N}_{1}^{}$ being lighter than ${N}_{2}^{}$ (i.e. ${M}_{1}^{}\lt {M}_{2}^{}$). For this scenario, the final baryon asymmetry can be calculated according to equation (27 ) by taking I = 1 and16 ), (17 ) and the discussions below them], we are left with only two free parameters (i.e. ${M}_{1}^{}$ and ${M}_{2}^{}$) to calculate the allowed values of ${Y}_{{\rm{B}}}^{}$. In fact, the dependence of ${Y}_{{\rm{B}}}^{}$ on the concrete value of the right-handed neutrino mass ratio is fairly weak provided that they are hierarchical. Hence we take ${M}_{2}^{}/{M}_{1}^{}=3$ as a benchmark value in the following calculations. In figure 1, the allowed values of ${Y}_{{\rm{B}}}^{}$ are shown as functions of ${M}_{1}^{}$ for δ = −π/2 (which is more experimentally favored than δ = π/2), while the results for δ = π/2 can be obtained by making a sign reversal for ${Y}_{{\rm{B}}}^{}$. We see that ${Y}_{{\rm{B}}}^{}$ has no chance to reach its observed value in the case of η = +1, but can do so at ${M}_{1}^{}\simeq 1.2\times {10}^{11}$ GeV in the case of η = −1. This means that the requirement of leptogenesis being successful will help us exclude the case of σ = 0 (corresponding to η = +1).
$\begin{eqnarray}\begin{array}{rcl}{\varepsilon }_{1\mu }^{} & = & \displaystyle \frac{{{abc}}^{2}{M}_{2}^{}}{4\pi {v}^{2}(3{a}^{2}+2{b}^{2})}\left[{ \mathcal G }\left(\displaystyle \frac{{M}_{2}^{2}}{{M}_{1}^{2}}\right)-\eta { \mathcal F }\left(\displaystyle \frac{{M}_{2}^{2}}{{M}_{1}^{2}}\right)\right],\\ {\widetilde{m}}_{1\gamma }^{} & = & 2{a}^{2}+{b}^{2},\qquad {\widetilde{m}}_{1\tau }^{}={a}^{2}+{b}^{2}.\end{array}\end{eqnarray}$
Given that the values of a, b and c have been determined from the measured values of ${s}_{13}^{2}$, ${\rm{\Delta }}{m}_{21}^{2}$ and ${\rm{\Delta }}{m}_{31}^{2}$ [see equations (
Figure 1. In the scenario of ${M}_{1}^{}\lt {M}_{2}^{}$, the allowed values of ${Y}_{{\rm{B}}}^{}$ as functions of ${M}_{1}^{}$ for δ = −π/2. The red line - the results for the η = +1 case. The blue line: the results for the η = −1 case. The green line - the observed value of ${Y}_{{\rm{B}}}^{}$. |
Then, we consider the scenario of ${N}_{2}^{}$ being lighter than ${N}_{1}^{}$ instead (i.e. ${M}_{2}^{}\lt {M}_{1}^{}$). For this scenario, the final baryon asymmetry can be calculated according to equation (27 ) by taking I = 2 and
$\begin{eqnarray}\begin{array}{rcl}{\varepsilon }_{2\mu }^{} & = & \displaystyle \frac{{{abM}}_{1}^{}}{8\pi {v}^{2}}\left[\eta { \mathcal F }\left(\displaystyle \frac{{M}_{1}^{2}}{{M}_{2}^{2}}\right)-{ \mathcal G }\left(\displaystyle \frac{{M}_{1}^{2}}{{M}_{2}^{2}}\right)\right],\\ {\widetilde{m}}_{2\gamma }^{} & = & {\widetilde{m}}_{1\tau }^{}={c}^{2}.\end{array}\end{eqnarray}$
The numerical calculations show that, if one still takes δ = −π/2, ${Y}_{{\rm{B}}}^{}$ will have a negative value, thereby falsifying leptogenesis.4. Summary
As we know, one of the most popular and natural ways of generating finite but tiny neutrino masses is the type-I seesaw model, which also provides an attractive explanation for the baryon asymmetry of the Universe via the leptogenesis mechanism. But the seesaw model with three right-handed neutrinos has too many free parameters to give definite predictions for the neutrino parameters and leptogenesis. In the literature, there are several typical approaches to reducing its free parameters and thus enhancing its predictive power; one is to constrain its flavor structure by employing some flavor symmetry such as the TM1 and μ-τ reflection symmetries, which are experimentally motivated and have interesting phenomenological consequences; one is to reduce the number of the right-handed neutrinos to two (i.e. the minimal seesaw model); another one is to impose texture zeros on the neutrino mass matrices.
Motivated by these facts, following the Occam’s razor principle, in this paper we have put forward a very simple form of ${M}_{{\rm{D}}}^{}$ in the minimal seesaw model with the right-handed neutrino mass matrix being diagonal; as shown in equation (8 ), it has one texture zero and only contains three real parameters a, b and c. It turns out that, as can be seen from equation (9 ), such a model leads to an ${M}_{\nu }^{}$ that obeys the TM1 and μ-τ reflection symmetries simultaneously. Then, we have derived the consequences of this model for the lepton flavor mixing parameters and neutrino masses. On the one hand, all the lepton flavor mixing parameters except for ${\theta }_{13}^{}$ are predicted: the value of ${\theta }_{12}^{}$ is predicted by the TM1 symmetry [see equation (5 )], while those of ${\theta }_{23}^{}$, δ, ρ and σ by the μ-τ reflection symmetry [see equation (7 )]. On the other hand, the model predicts the neutrino masses to be of the NO case with ${m}_{1}^{}=0$, for which all the three light neutrino masses will be pinned down (i.e. ${m}_{2}^{}\simeq 8.7$ meV and ${m}_{2}^{}\simeq 50.6$ meV) with the help of the experimental results for the neutrino mass squared differences. Furthermore, the values of a, b and c have been determined from the measured values of ${s}_{13}^{2}$, ${\rm{\Delta }}{m}_{21}^{2}$ and ${\rm{\Delta }}{m}_{31}^{2}$ (see equations (16 ), (17 )). For these results, the effective Majorana neutrino mass $| {\left({M}_{\nu }^{}\right)}_{{ee}}^{}| $ that controls the rate of the neutrinoless double beta decay is predicted to be 1.6 or 3.8 meV in the case of η = +1 or −1 (corresponding to σ = 0 or π/2).
We have then studied the implications of the model for leptogenesis. It turns out that only in the two-flavor leptogenesis regime (which holds in the temperature range 109−1012 GeV) can leptogenesis have a chance to be successful. Given that the contribution to leptogenesis mainly comes from the lightest right-handed neutrino in the case of the right-handed neutrino masses being hierarchical, we have studied the scenarios of ${N}_{1}^{}$ and ${N}_{2}^{}$ being the lighter ones, respectively. The results show that, if one takes δ = −π/2, which is more experimentally favored than δ = π/2, leptogenesis can be successful in the scenario of ${M}_{1}^{}\lt {M}_{2}^{}$, but not in the scenario of ${M}_{2}^{}\lt {M}_{1}^{}$. And in the scenario of ${M}_{1}^{}\lt {M}_{2}^{}$, a successful leptogenesis can be achieved at ${M}_{1}^{}\simeq 1.2\times {10}^{11}$ GeV in the case of η = −1, but not in the case of η = +1. In this way, the requirement of leptogenesis being successful will help us exclude the case of σ = 0 (corresponding to η = +1).
Before closing this paper, we would like to give a final remark: the ideal values of the lepton flavor mixing parameters (e.g. ${\theta }_{23}^{}$ and δ) predicted by this model may considerably deviate from their realistic values. In this situation, to make these predictions more consistent with the experimental results, the exact TM1 and μ-τ reflection symmetries should be broken. This can be achieved by including the renormalization group evolution effects [20]. It should also be noted that, with the breaking of the μ-τ reflection symmetry, leptogenesis has a chance to be successful even in the unflavored regime. For more details about the latter point, we refer the interested reader to [21].