Neutrino μ-τ reflection symmetry and its breaking in the minimal seesaw

In this paper, we attempt to implement the neutrino μ-τ reflection symmetry (which predicts θ23 = π/4 and δ = ±π/2 as well as trivial Majorana phases) in the minimal seesaw (which enables us to fix the neutrino masses). For some direct (the preliminary experimental hints towards θ23 ≠ π/4 and δ ≠ −π/2) and indirect (inclusion of the renormalization group equation effect and implementation of the leptogenesis mechanism) reasons, we particularly study the breakings of this symmetry and their phenomenological consequences.


Introduction
As is known, the discovery of neutrino oscillations indicates that neutrinos are massive and mixed [1]. On the one hand, the most popular way of generating neutrino masses is to invoke the seesaw mechanism which allows for a natural explanation of their smallness. In the type-I seesaw mechanism [2][3][4][5][6], one usually introduces three right-handed neutrino fields N i (for i = 1, 2, 3). They not only take part in Yukawa interactions with the lefthanded neutrino fields which lead to a Dirac mass matrix M D , but also have a Majorana mass matrix M N of themselves. Under the assumption of M N M D , the effective mass matrix for light neutrinos is obtained as [2][3][4][5][6] (1.1) The overall minus sign here is of no physical meaning and will be neglected in the following discussions. On the other hand, the neutrino mixing arises from a mismatch between their mass and flavor eigenstates and is described by a 3 × 3 unitary matrix U = U † l U ν [7,8] with U l and U ν being respectively the unitary matrix for diagonalizing the charged-lepton mass matrix M l and M ν . In the standard parametrization, U reads [1]  (1.2) Here P φ = Diag(e iφ 1 , e iφ 2 , e iφ 3 ) consists of three unphysical phases that can be removed via the rephasing of charged-lepton fields, while P ν = Diag(e iρ , e iσ , 1) contains two Majorana phases. Similar to the CKM matrix, V possesses a Dirac phase δ and three mixing angles θ ij (for ij = 12, 13, 23). For convenience, the abbreviations c ij = cos θ ij and s ij = sin θ ij -1 -
symmetry allows for an arbitrary θ 13 . Thanks to these interesting consequences, this symmetry has been attracting a lot of attention recently. 5 Nevertheless, the neutrino masses receive no clues from the µ-τ reflection symmetry. But they will be fixed if the smallest one (m 1 in the NH case, m 3 in the IH case) is to vanish, a possibility that is experimentally allowed and can be accommodated by the minimal seesaw 6 where only two right-handed neutrino fields N 1,2 are involved. In this paper, we just attempt to combine the µ-τ reflection symmetry with the minimal seesaw so that both the neutrino mixing and mass spectrum can be settled. The implications of such a marriage are discussed in the next section. In consideration of the preliminary experimental results towards θ 23 = π/4 7 and δ = −π/2, we particularly explore the possible symmetry breakings and their impacts on the mixing parameters in section 3. Section 4 is devoted to the specific symmetry breaking triggered by the renormalization group equation (RGE) effect. In section 5, the operation of leptogenesis in this framework will be studied. Finally, our main results are summarized in section 6.
2 µ-τ reflection symmetry in the minimal seesaw Conversely, one can reconstruct an M ν of the µ-τ reflection symmetry in terms of the U ν characterized by eq. (1.9) and neutrino masses by means of the relation In the situation of one neutrino mass being vanishing, the reconstructed M ν in such a way will take a form as [δ, σ] without affecting the physical results. And there is only one effective Majorana phase which we assign for m 2 (i.e., σ). Because of the condition Det(M ν ) = 0, 8 only five out of the six real components of these elements are independent. Taking the best-fit values for θ 13 , θ 12 , ∆m 2 21 and |∆m 2 31 | as input, we give the values of M αβ for various combinations of δ and σ (i.e., [δ, σ] = [π/2, 0], [−π/2, 0], [π/2, π/2] and [−π/2, π/2]) in the NH (table 1) and IH (table 2) cases. As is well known, the size of |M ee | which governs the rate of neutrino-less double beta decays 9 is much larger in the IH case than in the NH case. In the NH case, the elements exhibit a hierarchical structure as |M µµ | ∼ |M µτ | |M ee | ∼ |M eµ |, implying that they might have received contributions from different levels [67]. In the IH case, |M ee | (so does |M eµ | for σ = 0) becomes comparable to |M µµ | and |M µτ |. But for σ = π/2, M eµ will have a magnitude much smaller than the other ones as a result of the heavy cancellation between its two components respectively associated with m 1 and m 2 . By choosing the value of φ in such a way that the phase of M eµ or M µµ cancels out, one may convert M ν to a simpler form as given by tables 3-4.
The M ν given by eqs. (2.1)-(2.2) can be viewed as a result of the minimal seesaw: the Dirac mass matrix coupling N 1,2 with the left-handed neutrino fields is assumed to appear 8 Since the determinant of an M ν obeying the µ-τ reflection symmetry is always real, this condition only amounts to one constraint. 9 For a review with extensive references see refs. [63][64][65][66]. [δ, σ] with a 1,2 , b 1,2 , φ a,b and M 1,2 being real parameters. Apparently, its elements satisfy the conditions of M µi = M * τ i and M ei being real (for i = 1, 2). It is easy to see that these conditions still hold when the right-handed neutrino fields experience an orthogonal basis transformation. So, without loss of generality, we choose to work in the basis where the Majorana mass matrix for N 1,2 is diagonal M N = Diag(M 1 , M 2 ). By virtue of the seesaw formula in eq. (1.1), we arrive at an effective neutrino mass matrix

JHEP10(2017)102
Diagonalizing this M ν with a U ν characterized by eq. (1.9) yields the mixing parameters and neutrino masses where m 1 = m 1 e 2iρ , m 3 = m 3 e 2iγ (for ρ, γ = 0 or π/2) and For any given values of a 1,2 , b 1,2 and φ a,b , one mass will necessarily vanish as promised by the minimal seesaw. 10 The resulting θ 13 , φ, θ 12 and two non-zero masses can be calculated with the help of the other five equations.
If we are to derive the allowed values of a 1,2 , b 1,2 and φ a,b from the measured θ 13 , θ 23 , ∆m 2 21 and |∆m 2 31 |, one just needs to confront the M ν in eq. (2.4) with the results given by tables 1-2. Above all, it should be noted that both the ee and µτ elements of this M ν are positive. In order for them to fit in with the corresponding results in tables 1-2, one must have σ = π/2 (or 0) and M ν → −M ν (or + M ν ) in the NH (or IH) case. In light of the unphysical nature of φ, φ a,b − φ rather than φ a,b will be treated as effective independent parameters. Recall that only five real components of the neutrino mass matrix elements are independent, so the free parameters are more than the constraint equations by one. For this reason, in figure 1 we choose to present the results for a 2 , b 1,2 and φ a,b − φ as functions of a 1 (which stands in an equivalent position as b 1 ). In the numerical calculations here and in the following, the best-fit values for θ 13 , θ 12 , ∆m 2 21 and |∆m 2 31 | are input, whereas δ is specified as −π/2. We have only shown the results in the case of both a 1 and b 1 being positive. The results in the case of a 1 or (and) b 1 being negative can be obtained by simply making the replacement This is because the M ν in eq. (2.4) keeps invariant under this kind of transformations. It is interesting to find that the possibility of   3 Breaking of the µ-τ reflection symmetry In this section we study the possible breakings of µ-τ reflection symmetry and their impacts on the mixing parameters. 11 The most general perturbation to M D can be decomposed into a symmetry-conserving part and a symmetry-violating part: (3.2) And the full Dirac mass matrix can be reparameterized as For some discussions about the breakings of µ-τ interchange symmetry see refs. [69,70].  The dimensionless quantities 1,2,3,4 measure the strength of symmetry breaking. They should be small (e.g., | 1,2,3,4 | ≤ 0.1) in order for M D to assume an approximate µ-τ reflection symmetry. Since 2 and 4 play equivalent roles as 1 and 3 , they will be assumed to vanish in the following discussions. For the sake of simplicity, the hat symbols onâ 1,2 ,b 1,2 andφ a,b will also be neglected. Consequently, we are led to an effective neutrino mass matrix M ν of the form at the leading order. The unitary matrix U ν for diagonalizing M ν is expected to have some mixing parameters around the special values given by eq. (1.9), with the corresponding deviations being some small quantities. By making series expansions for these mixing-parameter deviations in the diagonalization process, at the leading order we acquire the following relations connecting them with the symmetry-breaking parameters 1,3 m 3 s 2 13 ∆δ + m 2 s 2 12 ∆σ = ∓ a 2 1 1 − a 2 1 + b 2 1 ∆φ 1 , where m 1 = 0 (or m 3 = 0) in the NH (or IH) case and the values of a 1,2 , b 1,2 and φ a,b − φ are the same as those presented in figure 1. The sign ∓ which takes − (or +) in the NH (or IH) case arises from the aforementioned fact that M ν might need an overall sign change so as to fit in with the numerical results. By solving these equations in a straightforward way, one will obtain the mixingparameter deviations as some linear functions of 1 , Re( 3 ) and Im

RGE triggered symmetry breaking
A flavor symmetry (FS) such as the µ-τ reflection symmetry under study is usually introduced at an energy scale Λ FS much higher than the electroweak (EW) one Λ EW ∼ 10 2 GeV [12,13]. Therefore, the RGE effects should be taken into account when the flavorsymmetry model is confronted with the low-energy experimental data [71]. During the evolution process, the significant difference between the Yukawa coupling of µ and that of τ may provide a unique source for symmetry breaking. This section is just devoted to such a specific breaking of the µ-τ reflection symmetry. At the one-loop level, the energy dependence of M ν is described by [72][73][74][75][76] where t ≡ ln (µ/µ 0 ) with µ denoting the renormalization scale, C and α read C = − 3 2 , α −3g 2 2 + 6y 2 t + λ , in the SM ; In the basis of M l being diagonal, the Yukawa coupling matrix for charged leptons is given by Y l = Diag(y e , y µ , y τ ). Due to y e y µ y τ , it is reasonable to neglect the contributions of y e and y µ . In eq. (4.1), the α-term is flavor universal and only contributes an overall rescaling factor I α for the neutrino mass matrix, while the other two terms are able to modify its structure. Given an M ν (Λ FS ) of the form given by eq. (2.4) at Λ FS , integration of eq. (4.1) enables us to derive the RGE-corrected neutrino mass matrix at Λ EW [77,78]    with I τ Diag(1, 1, 1 − ∆ τ ) and in the NH case, or (4.8) in the IH case. We subsequently show the dependence of these mixing-parameter deviations on the value of tan β (which varies from 10 to 50) in figure 3. One can see that the mixing parameters are pretty stable against the RGE corrections. Even for tan β = 50, one merely has |∆θ 23 | 0.017 (or 0.021), |∆δ| 0.028 (or 0.003) and |∆σ| 0.001 (or 0.007) in the NH (or IH) case.

Leptogenesis and the symmetry breaking
Besides accommodating the smallness of neutrino masses, the seesaw mechanism can also explain the observed baryon-antibaryon asymmetry of the Universe via the leptogenesis mechanism [79]: the CP-violating, lepton-number-violating and out-of-equilibrium decays of N i may generate a lepton-antilepton asymmetry which is eventually converted to the baryon-antibaryon asymmetry through the sphaleron process [80]. The essential CP violation is provided by the complex Yukawa couplings Y D = M D /v which couple N i with the left-handed neutrino fields, while the lepton-number violation originates from the Majorana mass terms of N i . And the departure from thermal equilibrium can occur if N i decay in a rate smaller than the expansion rate of the Universe when the temperature T drops to the mass scale of N i . The produced amount of lepton-antilepton asymmetry is crucially dependent on the CP-violating asymmetries between the decays of N i and their CP conjugate processes. In the minimal seesaw under study, we assume N 1,2 to have a strong mass hierarchy M 1 M 2 , in which case only the decay of N 1 is relevant for leptogenesis. The flavored CP-violating asymmetries ε α are given by [81] for α = e, µ, τ . A Y D corresponding to the M D in eq. (2.3) immediately yields [14] ε e = 0 , rendering the total CP-violating asymmetry ε = ε e + ε µ + ε τ vanishing. However, a successful leptogenesis is possible when the µ-τ reflection symmetry is broken [82] or (and) the flavor effects become relevant [83]. For the situation of M 1 > 10 12 GeV, the flavor effects are negligible. In order to achieve ε = 0, one has to break the µ-τ reflection symmetry. A Y D corresponding to the M D in eq. (3.3) (with 2,4 = 0 as assumed before) gives which means that ε is proportional to M 1 and a linear function of 1 , Re ( 3 ) and Im ( 3 ). The final baryon-to-entropy ratio can be written as [80] Here 12/37 is the efficiency factor of converting the lepton-antilepton asymmetry to the baryon-antibaryon asymmetry, whereas g * = 106.75 is the effective number of relativistic degrees of freedom at T = M 1 in the SM. In particular, κ is the washout factor and can be parametrized as [80] κ (2 ± 1) × 10 −2 × 0.01 eṼ m 1 In the present epoch (for s = 7.04n γ ), the baryon-to-photon ratio is given by which has an observed value of (6.08 ± 0.09) × 10 −10 [84]. If M 1 turns out to be smaller than 10 12 GeV, then the Yukawa interactions of charged leptons will enter in thermal equilibrium, making different lepton flavors distinguishable. In such a case, both the CP-violating asymmetries and washout factors associated with different flavors should be treated separately. Accordingly, the final baryon-to-entropy ratio can be rewritten as [85,86] Y B − 12 37 where the parameter K is defined as   For illustration, in figure 5 we show the η (as functions of a 1 ) arising from some example values of M 1 and symmetry-breaking parameters for the situation of 10 9 GeV < M 1 < 10 12 GeV. For comparison, the contributions from pure flavor effects (without symmetrybreaking effects) are also shown. It is easy to see that the flavor effects are more significant than the symmetry-breaking effects in the NH case, while the contrary is the case in the IH case. In the NH case the flavor effects themselves are competent for generating the observed η. But in the IH case the symmetry-breaking effects have to be invoked and the symmetry-breaking parameters should take some values at least O(0.1).
In order to see whether there exists a choice of basic parameters which can lead to sizable ∆δ, ∆θ 23 and successful leptogenesis, in figures 6-7 we present the resulting ∆δ and ∆θ 23 from the parameter choices shown in figures 4-5 that can lead to successful leptogenesis. From figure 6 one finds that for the situation of M 1 > 10 12 GeV a sizable ∆δ can be generated in association with a realistic η from 1 in the NH case (see the sub-figure labelled as (a)), Re( 3 ) in the IH case (see the sub-figure labelled as (d)) or Im( 3 ) in the IH case (see the sub-figure labelled as (f)), but a sizable ∆θ 23 has no chance to arise along with a successful leptogenesis. The results in figure 7 tell us that for the situation of 10 9 GeV < M 1 < 10 12 GeV all the parameter choices except for that shown in the subfigure labelled as (b) may give rise to a sizable ∆δ, while only the parameter choice shown in the sub-figure labelled as (c) is capable of producing a sizable ∆θ 23 .

Summary
The µ-τ reflection symmetry is powerful in shaping the neutrino mixing: it leads to the interesting results θ 23 = π/4 and δ = ±π/2 (which are close to the current experimental results) as well as trivial Majorana phases. On the other hand, the minimal seesaw has strong predictive power in settling the neutrino mass spectrum: it enforces the condition of m 1 = 0 or m 3 = 0. In this paper, an attempt of implementing the µ-τ reflection symmetry in the minimal seesaw has been made. Such a marriage results in σ = π/2 (or 0) and thus |M ee | 0.37 (or 4.86) eV in the NH (or IH) case. Motivated by the preliminary experimental hints towards θ 23 = π/4 and δ = −π/2, we particularly explore the possible symmetry breakings and their phenomenological consequences. Some parameters characterizing the breaking of µ-τ refection symmetry are first defined and their implications for the mixing-parameter deviations then analyzed. It is found that 1 is difficult to produce considerable mixing-parameter deviations, while Re( 3 ) is relatively easy. Inversely, a considerable ∆θ 23 can be attributed to Re( 3 ), while a considerable ∆δ may arise from any symmetry-breaking parameter.
As a unique example, the symmetry breaking triggered by the RGE effects is studied. It turns out that the mixing parameters are rather stable against the RGE corrections. Even for tan β 50 in the MSSM, the RGE-induced mixing-parameter deviations are only of O(0.01). Finally, the operation of leptogenesis in the framework under study is discussed. For the situation of M 1 > 10 12 GeV where one has a vanishing ε, the µτ reflection symmetry must be broken to make the leptogenesis mechanism work. For illustration, we give some example values of M 1 and symmetry-breaking parameters that       may give rise to the observed value of η. For the situation of 10 9 GeV < M 1 < 10 12 GeV, it is possible that the flavor effects themselves (without symmetry-breaking effects) are sufficient for producing the observed η in the NH case. But in the IH case, one has to turn to the symmetry-breaking effects for help. For both situations, it is easier to achieve a realistic value of η in the NH case than in the IH case. Last but not least, we point out that the mixing-parameter deviations can be connected to the implementation of leptogenesis, considering that they may originate from the same symmetry breaking. The results show that a sizable ∆δ can be generated in association with the observed η in many cases, but a sizable ∆θ 23 can only arise along with a successful leptogenesis from Re( 3 ) in the NH case for the situation of 10 9 GeV < M 1 < 10 12 GeV.