Interplay between $\mu$-$\tau$ reflection symmetry, four-zero texture and universal texture

In this paper, we consider exact $\mu-\tau$ reflection symmetries for quarks and leptons. By a bi-maximal transformation, the four-zero textures lead to a $\mu - \tau$ reflection symmetric form in a particular basis. In this basis, up- and down-type mass matrices from four-zero texture separately satisfy exact $\mu - \tau$ reflection symmetries, and it predicts the maximal CP violation in the Fritzsch -- Xing parameterization. In the lepton sector, in order to reconcile the $\mu - \tau$ reflection symmetries and observed $\sin \theta_{13}$, the maximal CP violation is discarded and it predicts $\delta^{PDG} \simeq \sin^{-1} [\sqrt{m_{e} \over m_{\mu}} {c_{13} s_{23} \over s_{13}}] \simeq 203^{\circ}$. Moreover, if the universal texture $(m_{f})_{11} = 0$ for $f=u,d,\nu,e$ is imposed, it predicts the lightest neutrino mass $|m_{1}| = 6.26$ or $2.54$ meV in the case of the normal hierarchy (NH), because the $\mu - \tau$ reflection symmetry restricts the Majorana phases to be $\alpha_{i} /2 = n \pi/2$.

In the lepton sector, in order to reconcile the µ − τ reflection symmetries and observed sin θ 13 , the maximal CP violation is discarded and it predicts δ P DG ≃ sin −1 [ me mµ c 13 s 23 This value is rather close to the best fit and in the 1σ region δ CP / • = 217 +40 −28 [24]. Moreover, if the universal texture (m f ) 11 = 0 for f = u, d, ν, e is imposed, it predicts the lightest neutrino mass |m 1 | = 6.26 or 2.54 meV in the case of the normal hierarchy (NH), because the µ − τ reflection symmetry restricts the Majorana phases to be α i /2 = nπ/2 [25,26].
This paper is organized as follows. In the next section, we review the four-zero texture [16] and µ − τ reflection symmetry. In Sec. 3, µ − τ symmetries are imposed on the lepton sector. The final section is devoted to conclusions.
2 Four-zero texture and µ − τ reflection symmetry In this section, we review the four-zero texture and its interplay of the µ − τ reflection symmetry [3,4]. First of all, the phenomenological mass matrices of the SM fermions f = u, d, e and neutrinos ν L are defined by Diagonalization of the mass matrices m f = U Lf m diag f U † Rf leads to the CKM and MNS mixing matrices Since both the matrix have large Dirac phase, the maximal CP violation have been discussed [16]. The CP phase depends on the basis of the fermions. In particular, the phase of the CKM matrix becomes almost maximal in the Fritzsch-Xing parameterization [23]: The best fit values are found to be [27] s u = 0.0863, s d = 0.212, s = 0.0423, δ F Z = 87.9 • .
Although the original Kobayashi-Maskawa parameterization [28] has similar result δ KM ≃ π/2 [29], the Fritzsch -Xing parameterization has reasonable physical view because it factorizes the large mixing 1-2 generations and the small one of 2-3 generations.
If we assume m u,d are Hermitian matrices, Eq. (3) strongly suggests that the mass matrices of quarks have "four-zero texture" or "modified Fritzsch texture" [16], For later convenience, the relative phase is pressed on the m u . In this case, the rotation matrices U u,d at leading order is written by the mass eigenvalues and a parameter r u,d .
Then, the CKM matrix It predicts V cb and V ts at leading order as follows r u ≃ r d = 81/32 ≃ 1.59 gives nice agreement between the prediction and the observation [30] . A bi-maximal transformation of the mass matrices by the following U BM , leads to These matrices (11), (12) separately satisfy exact µ − τ reflection symmetries: where The form of matrices (11), (12) has been indicated a study of universal texture [12]. However, they considered no symmetry and these forms were phenomenological description.

µ − τ reflection symmetry in the lepton sector
In this section, we impose the symmetry (13) on the lepton sector and research some predictions. If we anticipate the lepton mass matrices m ν,e have the four-zero textures, they are approximately diagonalized by the rotation matrices: with sin θ P DG 23 ≃ 1/ √ 2, sin δ P DG ≃ −π/2, sin θ P DG
In order to derive a proper sin θ 13 , we change the mixing matrix Eq. (16) in the following way.
The Majorana phases are omitted here and discussed later. Small 2-3 mixing of the V e can be absorbed to that of the V ν . Then, Here, we use sin θ P DG 13 = 0.150 from the latest global fit [24] and the following approximate values sin θ P DG 23 = cos θ P DG The sign ± corresponds to the sign of cos δ P DG . We adopt s 13 = −0.140 because the latest global fit found cos δ P DG < 0 [24].
All of the components is in the range of 3 σ.
On the analogy of quark masses, the lepton mass matrix m ν,e which predict the mixing matrix (19) will be the following forms These matrices have no symmetry at first glance. However, the bi-maximal transformation in Eq. (10) leads to These matrices (26), (27) also separately satisfy exact µ − τ reflection symmetries (13): Therefore, in this basis, we found quarks and leptons satisfy some universal µ−τ reflection symmetries. Note that the µ − τ symmetry is not imposed on m ν in the basis of four-zero texture (25).

Dirac phase δ P DG
In order to show the Dirac phase δ P DG , we evaluate the Jarskog invariant [31], The phase δ P DG vanishes in a limit of s e ≡ m e /m µ → 0. Then, the invariant can be evaluated from Eq. (20) treating m e /m µ ≃ 0.07 as a perturbation.
The value with (without) parentheses is induced from full calculation (only leading order). Since s 13 = −0.14 is small, Therefore This value is rather close to the best fit and in the 1σ region δ CP / • = 217 +40 −28 [24].
These values are rather small than other models because it vanishes in a limit of (m ν ) 11 = m e /m µ = 0. In particular, the phase factor +i in Eq. (38) generate destructive interference for α 2 = α 3 .

Conclusions
In this paper, we consider exact µ − τ reflectioin symmetries for quarks and leptons. By a bi-maximal transformation, the four-zero textures lead to a µ − τ reflection symmetric form in a particular basis. In this basis, up-and down-type mass matrices from four-zero texture separately satisfy exact µ − τ reflection symmetries, and it predicts the maximal CP violation in the Fritzsch -Xing parameterization.
In the lepton sector, in order to reconcile the µ − τ reflection symmetries and observed sin θ 13 , the maximal CP violation is discarded and it predicts δ P DG ≃ sin −1 [ me mµ c 13 s 23 Moreover, if the universal texture (m f ) 11 = 0 for f = u, d, ν, e is imposed, it predicts the lightest neutrino mass |m 1 | = 6.26 or 2.54 meV in the case of the normal hierarchy (NH), because the µ − τ reflection symmetry restricts the Majorana phases to be α i /2 = nπ/2.