Leptonic CP Violation and Wolfenstein Parametrization for Lepton Mixing

We investigate a general structure of lepton mixing matrix resulting from the SU$_F$(3) gauge family model with an appropriate vacuum structure of SU$_F$(3) symmetry breaking. It is shown that the lepton mixing matrix can be parametrized by using the Wolfenstein parametrization method to characterize its deviation from the tri-bimaximal mixing. A general analysis for the allowed leptonic CP-violating phase $\delta_e$ and the leptonic Wolfenstein parameters $\lambda_e$, $A_e$, $\rho_e$ is carried out based on the observed lepton mixing angles. We demonstrate how the leptonic CP violation correlates to the leptonic Wolfenstein parameters. It is found that the phase $\delta_e$ is strongly constrained and only a large or nearly maximal leptonic CP-violating phase $|\delta_e| \simeq 3\pi/4 \sim \pi/2$ is favorable when $\lambda_e>0.15 $. In particular, when taking $\lambda_e$ to be the Cabbibo angle $\gl_e\simeq \lambda \simeq 0.225$, a sensible result for leptonic Wolfenstein parameters and CP violation is obtained with $ A_e=1.40$, $\rho_e=0.20$, $\delta_{e}\sim 101.76\;^o$, which is compatible with the one in quark sector. An interesting correlation between leptons and quarks is observed, which indicates a possible common origin of masses and mixing for the charged-leptons and quarks.


I. INTRODUCTION
The standard model (SM) has been well established with the observation of the last particle predicted in the SM, i.e., Higgs particle, at the LHC experiment [1,2]. The neutrino oscillations with massive neutrinos [3][4][5][6][7][8][9][10][11][12][13] provide a strong evidence and a useful window for exploring new physics beyond the SM. In comparison with the quark masses and CKM quark mixing matrix [14] in the SM, the smallness of neutrino masses and large MNSP lepton mixing [15] have been a longterm puzzle to be understood as a possible indication for new physics. The greatest success of the SM is the gauge symmetry structure SU c (3) × SU L (2) × U Y (1) which characterizes three basic forces of strong and electroweak interactions. All the gauge symmetries are associated with the quantum numbers of quarks and leptons. SU c (3) characterizes the symmetry among three color quantum numbers of quarks, SU L (2) describes the symmetry between two isospin quantum numbers of quarks and leptons for each family, and U Y (1) is the symmetry corresponding to the hypercharge quantum number of quarks and leptons. The quark and lepton mixing matrices and CP violations reflect the properties of three family quarks and leptons. To understand the quark and lepton mixing matrices and CP violations, it is interesting to investigate the possible gauge symmetries among three family quantum numbers. Obviously, a non-abelian gauge family symmetry [16][17][18][19][20][21][22][23][24][25][26][27][28] for three families of quarks and leptons becomes natural as a simple extension of the SM gauge symmetry structure.
It has been shown in ref. [16] that the SU F (3) gauge family symmetry enables us to construct a simple gauge family model for understanding the mixing and masses of leptons. The SU F (3) gauge family symmetry was first introduced in early time for estimating the top quark mass [29].
It was found in ref. [16] that the model can provide a consistent prediction for the lepton mixing and neutrino masses when considering the appropriate vacuum structure of SU F (3) gauge symmetry breaking. Specifically, through appropriately making the SU F (3) gauge fixing condition with keeping a residual Z 2 -permutation symmetry in the neutrino sector, we can obtain in the neutrino sector the so-called tri-bimaximal mixing matrix [30][31][32][33][34] and largely degenerate neutrino masses, while the small mixing matrix in the charged-lepton sector is resulted by requiring the vacuum structure of spontaneous symmetry breaking to possess approximate global U(1) family symmetries [16]. Thus the deviation from tri-bimaximal mixing in the lepton mixing matrix is attributed to the small mixing in the charged-lepton sector, its smallness is protected by the mechanism of approximate global U(1) family symmetries [35][36][37][38]. As the spontaneously symmetry breaking CP-violating phases in the vacuum [39] are not restricted by the considered symmetries, they can in principle be large and maximal. The small masses of the neutrinos and charged leptons are simply ascribed to the usual seesaw mechanism. As a simple case, when applying the Wolfenstein parametrization [40] for the CKM quark mixing matrix to the charged-lepton mixing matrix with a similar hierarchy structure as the CKM quark mixing matrix, and making a naive ansatz that all the smallness due to the approximate global U(1) family symmetries is characterized by a single Wolfenstein parameter λ ≃ 0.22, we can obtain an interesting prediction for the lepton mixing matrices with a maximal spontaneous CP violation δ ≃ π/2 [16] which agrees with the current experimental data [41][42][43]. The corresponding Leptonic Jarlskog CP-violating invariant quantity [44] reaches the maximal value The resulting neutrino masses are largely degenerate with the value at the order m ν i ≃ O(λ 2 ) ≃ 0.04 ∼ 0.06 eV with a total mass m ν ∼ 0.15 eV, which is much larger than the minimal limit m ν ∼ 0.05 eV and is expected to be tested by the future experiments.
It is widely expected that the leptonic CP violation can be maximal or large enough to account for the observed matter-antimatter asymmetry in the universe via the leptogenesis mechanism as the CP violation in the SM is not enough to understand the baryogenesis. In this note, we are going to make a general analysis on the leptonic CP-violating phase and its correlation with the deviation from the tri-bimaximal neutrino mixing based on the current experimental results.

CHARACTERIZATION OF DEVIATION FROM TRI-BIMAXIMAL MIXING
Let us begin with the following general structure of MNSP lepton mixing matrix which has been expressed into the standard form with c ij = cosθ ij , s ij = sinθ ij [41]. An interesting symmetric parametrization was discussed in [45]. Where P β is a diagonal matrix of phase factors and can be rotated away by the redefinition of charged lepton fields, and φ i are the so-called Majorana phases for Majorana neutrinos. It is known that the lepton mixing matrix generally arises from two mixing matrices V e and V ν , they correspond to the charged-lepton and neutrino mixing matrices arising from diagonalizing the charged-lepton mass matrix and neutrino mass matrix respectively. When the charged-lepton mass matrix is Hermitian M e = P δ U e m E U † e P † δ = M † e with m E the diagonal mass matrix of charged leptons m E = diag.(m e , m µ , m τ ), the unitary chargedlepton mixing matrix V e can in general be written as where U e is a unitary matrix with CP-violating phase δ ′ e , and mixing angles c e ij = cosθ e ij , s e ij = sinθ e ij . P δ is a diagonal phase matrix with three phases δ i , while only two relative phases (δ i − δ j ) are physically observable CP-violating phases.
For the neutrino mixing matrix, when an appropriate Z 2 -symmetric neutrino mass matrix between the second and third neutrinos is considered to have three independent matrix elements, the resulting neutrino mixing matrix is completely determined to be which is the so-called tri-bimaximal neutrino mixing matrix.
It is noticed that when three phases δ m ij (i, j = 1, 2, 3, i < j) in the Hermitian mass matrix are not independent and they are related via δ m ij = δ i − δ j , one then has On the other hand, when θ e 13 is small in comparison with θ e 12 and θ e 23 , i.e., θ e 13 ≪ θ e 12 , θ e 23 , it is easily seen that δ ′ e will not be a dominant source of leptonic CP violation as the CP-violating phase δ ′ e is associated with the mixing angle θ e 13 . In this situation, we may neglect the effect of CP-violating phase δ ′ e and take a typical case δ ′ e ≃ 0 for simplicity of discussions. With these considerations, we may replace the unitary matrix U e by an orthogonal rotation matrix O e = U e (δ ′ e = 0) To investigate whether the leptonic CP violation can be maximally large with the present experimental measurements on the three mixing angles and the lepton mixing matrix can be characterized by the Wolfenstein parametrization method, we may make a sensible analysis by simply taking δ ′ e = 0. In ref. [46], the angle θ e 13 is assumed to be zero, thus the effect of δ ′ e automatically disappears. An alternative consideration was analyzed in [47], where the phase δ ′ e was assumed to be the only CP-violating source and the phases δ i are taken to be zero, i.e., δ i = 0.
The leptonic mixing angles θ ij and mass-square differences ∆m 2 ij have been measured by many experiments including the solar neutrino experiment, atmospheric neutrino experiment, accelerator experiment and reactor experiment. The best-fit results presented in PDG [41] are sin 2 2θ 12 = 0.857 ± 0.024, sin 2 2θ 13 = 0.095 ± 0.010, which slightly deviates from the tri-bimaximal neutrino mixing.
Note that the presently extracted mixing angles from experiments are not sensitive to the CPviolating phase due to the smallness of the effects concerning the CP violation. As the leptonic CP violation is strongly correlated to the non-zero θ 13 which characterizes the deviation from the tri-bimaximal neutrino mixing matrix, it is then interesting to investigate the leptonic CP-violating phase and its correlation with the deviation from the tri-bimaximal neutrino mixing based on the above structure of lepton mixing matrix and the current experimental results. It is seen that the deviation from tri-bimaximal neutrino mixing is described by the charged-lepton mixing matrix U e or orthogonal rotation matrix O e , the smallness of the mixing angle θ 13 indicates that s e 12 ∼ O(0.1), which motivates us to parameterize the rotation matrix O e via the Wolfenstein parametrization [40] with a hierarchy structure similar to the CKM quark mixing matrix. With the leptonic Wolfenstein parameter s e 12 ∼ λ e ∼ O(0.1), the charged-lepton mixing matrix can be written, to the order O(λ 3 e ), as the following form: where the phase matrix P * δ can be absorbed by the redefinitions of charged lepton fields. Note that there is no corresponding Wolfenstein parameter η e in the above parametrization as we have neglected the CP-violating phase δ ′ e . Thus the lepton mixing matrix is given by which shows that based on the tri-bimaximal neutrino mixing, the lepton mixing matrix can be parametrized by three leptonic Wolfenstein parameters: λ e , A e , ρ e , , and CP-violating phases As indicated from Z 2 symmetry of vacuum structure in the SU(3) F model [16], it is reasonable to assume that δ 2 ≃ δ 3 . When expressing the lepton mixing matrix V M N SP to be the standard form by requiring the matrix elements V 11 , V 12 , V 23 , V 33 be real with keeping two independent Majorana phases, we can read off the leptonic CP-violating phase from V 13 The Wolfenstein parametrization of lepton mixing matrix is simplified to be In terms of the standard form eq.(4), the lepton mixing matrix can be rewritten in terms of the leptonic Wolfenstein parameters as the following form where V i,j are the matrix elements of V M N SP via eq. (14). The two Majorana phases φ 1 , φ 2 turn out to be and the mixing angle θ ij can be expressed in terms of the leptonic Wolfenstein parameters as which shows that the leptonic Wolfenstein parameters λ e , A e , ρ e , and the CP-violating phase δ e characterize the lepton mixing with deviation from the tri-bimaximal mixing.
As an illustration, it is interesting to observe that by taking the leptonic Wolfenstein parameters to be the following typical values with a maximal CP-violating phase we obtain the predictions for the lepton mixing angles sin 2 2θ 12 ∼ 0.901, sin 2 2θ 13 ∼ 0.086, which are consistent with the PDG's best-fit results given in eq.
which is very close to the maximal CP-violating phase δ e ∼ 0.57π. The corresponding two Majorana phases with the input parameters as eq.(20) are yielded to be We shall make a general constraint on the leptonic Wolfenstein parameters λ e , A e , ρ e , and the CP-violating phase δ e by a detailed analysis below.

III. CONSTRAINTS ON LEPTONIC WOLFENSTEIN PARAMETERS
As it is shown in previous section that the deviation from tri-bimaximal lepton mixing matrix can be described by three leptonic Wolfenstein parameters λ e , A e , ρ e . It is seen from eq.(17) and eq.(18) that sin θ 13 depends on λ e and A e ρ e , while sin θ 23 relies on λ e and A e . In this section, we shall take the mixing angles θ 13 and θ 23 indicated from the measurements as the input to provide a general constraint on leptonic Wolfenstein parameters λ e , A e , ρ e .

A. Constraints From θ 13
The precise measurements on θ 13 have been carried out by DayaBay Collaboration group [42] and RENO Collaboration group [43]. These two experiments measured the disappearance of ν e from the reactor. The ∆m 2 31 dominated amplitude is given by P (ν e → ν e ) = 1 − sin 2 2θ 13 sin 2 ∆m 2 31 L 4E .
which shows that the measured results do not sensitively correlate to the values of other two mixing angles θ 12 , θ 23 and CP-violating phase δ e . From eq.(17), it is seen that θ 13 is also insensitive to the CP-violating phase δ e . Thus we may use the experimental data on θ 13 to make constraints on the leptonic Wolfenstein parameters.
Before doing that, it is noticed that when keeping the expansion of lepton mixing matrix to the order O(λ e ), we arrive at the following simple relation where we have used the best fit values sin 2 θ 13 ∼ 0.0225 and sin 2 θ 23 ∼ 0.42 [41]  sin θ c = λ ≃ 0.225 in quark sector [41]. This observation checks the consistence of the assumption that λ e ∼ O(10 −1 ).
Let us now turn to make a general analysis by adopting the precisely measured mixing angle θ 13 [41] sin 2 2θ 13 = 0.096 ± 0.013(±0.040) at 1σ (3σ), which enables us to constrain the allowed region of the combined leptonic Wolfenstein parameters A e ρ e for a given λ e .
The contour plot for the input sin 2 2θ 13 is shown in figure1. It is seen from figure 1 that A e ρ e λ 2 e < 0 only occur for small values of λ e . In the plot, we have restricted the region to be in the range −1 ≤ A e ρ e λ 2 e ≤ 0.5, so that it satisfies the perturbative expanding of Wolfenstein parametrization. It also leads to a reasonable region for the parameter λ e λ e ≃ 0.11 ∼ 0.40.
which will be taken to be a possible allowed region when considering constraints from other two mixing angles θ 12 and θ 23 .
From eq. (17), it is seen that sin 2 θ 13 is an even function of λ e . Thus for the region with λ e < 0, the contours of sin 2 θ 13 are just the mirror images of figure 1, which is omitted here.

B. Constraints From θ 23 and θ 13
For tri-bimaximal mixing, there is a maximal mixing sin 2 θ 23 = 1/2. The small deviation to the maximal mixing indicates that the leptonic Wolfenstein parameters should be small, which may still be consistent with the current data within the experimental errors. While the recent global fitting results appear to indicate a quite large deviation from the maximal mixing with sin 2 θ 23 = 0.386 +0.024 −0.021 [48], and sin 2 θ 23 = 0.41 +0.037 −0.025 [49]. For a general discussion, we may consider a constraint from a wide range of θ 23 by covering over different global fitting results, i.e., sin 2 θ 23 ≃ 0.365 ∼ 0.450. The resulting constraint is shown in the left panel of figure 2 for parameter A e as a function of λ e .
By combining the constraint from θ 13 and θ 23 , we are able to obtain the constraint for the allowed region of ρ e as a function of λ e . As shown in the right panel of figure2, by taking the central value of sin 2 θ 13 given in eq.(26), we can obtain the allowed region for ρ e as a function of λ e from the given values of sin 2 θ 23 . It is seen that a wide region swept by the curve when sin 2 θ 23 increasing from 0.365 to 0.450 is allowed.
Note that there is a special situation that for λ e = √ 2 sin θ 13 , then A e ρ e = 0, namely ρ e = 0 for A e = 0. As a consequence, four curves intersect with each other at this point, as indicated in figure2.

IV. LEPTONIC CP VIOLATION AND LEPTON-QUARK CORRELATION
In this section, we should make a general analysis on the leptonic CP-violating phase and its correlation to the deviation from the tri-bimaximal neutrino mixing, which is characterized by the leptonic Wolfenstein parameters as discussed in the previous section.

A. Constraints From θ 12 and Leptonic CP Violation
It is seen from eq. (19) that sin θ 12 depends on CP-violating phase δ e , λ e ,A e ρ e . Here A e ρ e can be constrained from θ 13 for a given λ e .
The mixing angle θ 12 is well determined from solar neutrino oscillation experiments. The measured value of θ 12 generally correlates to the value of θ 23 . It is convenient to obtain the values of θ 12 by setting sin 2 2θ 23 = 1. The global fitting results have provided us with both values of θ 23 and θ 12 . Although a non-maximal θ 23 is hinted [48], there is no tension among different global fitting results on θ 12 . For instance, sin 2 θ 12 = 0.307 +0.018 −0.016 [48], sin 2 θ 12 = 0.311 ± 0.013 [49], and sin 2 θ 12 = 0.320 +0.016 −0.017 [50]. Here we take the result sin 2 θ 12 = 0.312 +0.018 −0.015 given in PDG [41] to make constraints on the CP-violating phase δ e as a function of λ e . The value of A e ρ e is constrained from θ 13 . The allowed region for CP-violating phase δ e is given as a function of λ e in figure3, where we have taken the central value sin 2 2θ 13 = 0.096 to yield the value of A e ρ e .
It is seen from figure 3 that there are two special regions: for 0.11 ≤ λ e ≤ 0.15, the values of sin 2 θ 12 is insensitive to δ e , the allowed region of δ e ranges from ±π to ±π/2. While for λ e ≥ 0.2, the constraint on δ e becomes very strong, the resulting CP-violating phase is near maximal δ e ∼ ±π/2. In this region, sin 2 θ 12 is insensitive to the values of λ e . Thus the leptonic CP violation favors a maximal CP violation for a large range of leptonic Wolfenstein parameter λ e . Note that a minimal CP-violating phase δ e ∼ 1.08π was obtained in a global fit [48] when the atmospheric neutrino data are included, while such a fitting result corresponds to a special region in the parameter space, which does not exclude a large or nearly maximal CP violation.

B. Combination of All Constraints and Lepton-quark Correlation
It is useful to combine all the constraints obtained from sin 2 θ 13 , sin 2 θ 23 and sin 2 θ 12 and plot them together in the same figure4, so that it is easily seen the allowed values of A e , ρ e and δ e for a given λ e . A big uncertainty arises from whether sin 2 θ 23 is largely deviate from the maximal mixing, which makes the allowed values of A e and ρ e become large.
A global fitting result cited in [48] corresponds to a solution of the small λ e with δ e ∼ π. From the results given in [48] for the normal hierarchy: sin 2 θ 23 = 0.386 +0.024 −0.021 , sin 2 θ 13 = 0.0241 ± 0.0025, sin 2 θ 12 = 0.307 +0.018 −0.016 , and δ e = 1.08π, one can easily read from Fig.4 With the central values, two Majorana phases are found to be very small For the case of inverted hierarchy, the result is very close to the above one, we shall omit it here.
where the CP-violating phase δ e is strongly constrained, only a large or nearly maximal CP violation is favorable.
It is interesting to observe from To show manifestly such an interesting situation, it is useful to plot the leptonic Wolfenstein parameters in a parameter space as shown in Fig.5 . It is easily seen that when fixing the parameter λ e ≃ λ ≃ 0.225, the whole parameters space for A e , ρ e and δ e is almost located on two planes with δ e ∼ ±π/2. The above analysis implies that only a large or nearly maximal leptonic CP violation is favorable in a large region of parameter space when λ e > 0.15.
The above results indicate a strong correlation between charged-leptons and quarks. An assumption that V e ≃ V CKM and V M N SP ≃ V † CKM V T B was discussed early in [51][52][53].

V. CONCLUSIONS AND REMARKS
We have shown that the lepton mixing can be parametrized by the Wolfenstein parametrization method based on a general structure of lepton mixing matrix, where the mixing matrix from neutrino sector is a tri-bimaximal mixing and the mixing matrix from charged-lepton has small mixing. Such a structure of lepton mixing has been shown to be resulted from the SU F (3) gauge family model [16] when considering the appropriate vacuum structure of SU F (3) gauge symmetry breaking. Where the tri-bimaximal mixing can be yielded from the residual Z 2 -permutation symmetry in the neutrino sector and the small mixing in the charged-lepton sector is led by requiring the vacuum structure of spontaneous symmetry breaking to possess approximate global U (1) family symmetries. We have demonstrated that the small mixing matrix in the charged-lepton sector characterizes the deviation from tri-bimaximal mixing in the lepton mixing matrix, and can be parametrized by the Wolfenstein parametrization method. As the spontaneous CP-violating phases in the vacuum are in general not restricted by the considered symmetries, so that they can in principle be large and maximal. phase δ e is not well constrained, its allowed region can range from |δ e | ∼ π to |δ e | ∼ π/2. While for the large values of λ e ≃ 0.15 ∼ 0.40, the CP-violating phase δ e has strongly been constrained, only a large or nearly maximal leptonic CP violation with |δ e | ≃ 3π/4 ∼ π/2 is allowed.
It has been demonstrated that when taking the leptonic Wolfenstein parameter λ e to be the Cabbibo angle in quark sector, λ e ≃ λ ≃ 0.225, we are able to obtain a sensible result with In conclusion, the lepton mixing matrix can well be characterized by leptonic Wolfenstein parameters in the basis of tri-bimaximal neutrino mixing. The leptonic CP violation has a strong correlation to the leptonic Wolfenstein parameters, a large or nearly maximal leptonic CP violation is favorable in a large region of parameters. More precise measurements for the lepton mixing angles are very helpful. It is essential to have a direct measurement for the leptonic CP violation in near future.