Intrinsic Deviation from the Tri-bimaximal Neutrino Mixing in a Class of A_4 Flavor Models

It is well known that the tri-bimaximal neutrino mixing pattern V_0 can be derived from a class of flavor models with the non-Abelian A_4 symmetry. We point out that small corrections to V_0, which are inherent in the A_4 models and arise from both the charged-lepton and neutrino sectors, have been omitted in the previous works. We show that such corrections may lead the 3 \times 3 neutrino mixing matrix V to a non-unitary deviation from V_0, but they cannot result in a nonzero value of \theta_13 or any new CP-violating phases. Current experimental constraints on the unitarity of V allow us to constrain the model parameters to some extent.

A more realistic form of the MNSP matrix V is expected to slightly deviate from V 0 due to some nontrivial perturbations 1 , such that both nonzero θ 13 and CP violation can emerge.
It is possible to derive the tri-bimaximal mixing pattern V 0 from some neutrino mass models with certain flavor symmetries [9]. In this connection the earliest and most popular application is the non-Abelian discrete A 4 symmetry (see, e.g., Refs. [10][11][12]). But the neutrino mixing matrix derived from a specific A 4 model is in general not equal to V 0 1 For instance, a possible interrelation with the quark-lepton complementarity is discussed in Ref. [8].
unless some approximations are made. In other words, small corrections to V 0 are generally inherent in the A 4 models and can arise both from the charged-lepton sector and from the neutrino sector. This observation is particularly interesting for an A 4 model built in the vicinity of the TeV scale, because the resultant corrections to V 0 may not be strongly suppressed. We show that such corrections can lead the 3 × 3 neutrino mixing matrix V to a non-unitary deviation from V 0 , although they cannot give rise to a nonzero value of θ 13 or any new CP-violating phases. We find that current experimental constraints on the unitarity of V allow us to constrain the parameters of an A 4 model to some extent.
The remaining part of this paper is organized as follows. In section II we first outline the salient features of a typical A 4 model and then diagonalize the 6 × 6 mass matrices of charged leptons and neutrinos. We show that both U ω and U ν in Eq. (2) get modified in this framework. In section III we work out the non-unitary departure of the resultant 3 ×3 MNSP matrix V from the tri-bimaximal mixing pattern V 0 = U T ω U * ν . We also constrain the model parameters to some extent by taking account of current experimental constraints on the unitarity of V . Section IV is devoted to a summary and some concluding remarks.

II. CORRECTIONS TO U ω AND U ν IN A TYPICAL A 4 MODEL
Let us consider a simple but typical A 4 model proposed by Babu and He in Ref. [12].
The model is an extension of the standard electroweak SU(2) L × U(1) Y model with some additional particles, and it is supersymmetric and A 4 × Z 4 × Z 3 -invariant. The particle content and charge assignments are summarized in Table I. The discrete symmetries force the superpotentials of quarks and leptons to have the following forms: works as an R-parity such that the superpotentials possess two units of charge. Thanks to the supersymmetry and new scalars in Eq. (3), it is possible to obtain the vacuum expectation values [12] S a = 0 , where Thus the A 4 symmetry is broken after χ and χ ′ develop their vacuum expectation values.
In the basis of (e, E) versus (e c , E c ) T , we obtain the 6 × 6 mass matrix of charged leptons from Eqs. (3) and (4): where 1 denotes the 3 × 3 identity matrix, and Note that f ℓ , M E and h α (for α = e, µ, τ ) can all be arranged to be real in a suitable phase convention, and the mass scale M E is assumed to be extremely large in comparison as a good approximation. The masses of three standard charged leptons turn out to be where α runs over e, µ and τ . Eq. (7) shows that U ω receives a small correction: It is actually U ′ ω that characterizes the contribution of charged leptons to the lepton flavor mixing in this A 4 model. Now we turn to the neutrino sector. The type-I seesaw mechanism [13] is implemented in the A 4 model under consideration, and thus the overall neutrino mass matrix is a symmetric 6 × 6 matrix: where M R takes the form The symmetric neutrino mass matrix in Eq. (10) can be diagonalized via the orthogonal to a good degree of accuracy. In this expression U ν has been given in Eq. (2), P ν denotes a diagonal phase matrix [12], and U R is a unitary matrix responsible for the diagonalization of M R . The masses of three light (active) neutrinos turn out to be m 1 ≃ |m 0 (1 + x)|, Because both m 0 and x are complex, it is possible to adjust their magnitudes and phases such that the resultant values of m i (for i = 1, 2, 3) satisfy current experimental data on the neutrino mass spectrum [12]. Eq. (12) shows that U ν P ν , which signifies the contribution of neutrinos to the lepton flavor mixing, receives a small correction: In other words, U ′ ν is not exactly unitary and its departure from U ν is in general an unavoidable consequence in the type-I seesaw mechanism [14].

III. NON-UNITARY CORRECTIONS TO V 0
With the help of the results obtained in Eqs. (9) and (14), we are able to calculate the MNSP matrix V = U ′ ω T (U ′ ν P ν ) * and demonstrate its non-unitary deviation from the tri-bimaximal mixing pattern V 0 . We find where and Q l and Q ν have been given below Eq. (2). In obtaining Eq. (15) (15) might not be negligibly small. These two terms, which are inherent in the model itself, measure the non-unitary contribution to V or the departure of V from V ′ 0 P * ν . This observation makes sense since it indicates that the exact tri-bimaximal neutrino mixing pattern V 0 is not an exact consequence of a class of A 4 flavor models.
One may parametrize the analytical result obtained in Eq. (15) as follows: where the Hermitian matrix η signifies the non-unitary deviation of V from V 0 P * ν . Note that the diagonal phase matrix Q l in V can always be rotated away through a redefinition of the phases of three charged leptons, and the diagonal phase matrices Q ν and P * ν in V only provide us with the Majorana phases which have nothing to do with leptonic CP violation in neutrino oscillations. Note also that η itself is real in this A 4 model, as one can easily see from Eq. (15), and thus the unitarity violation of V does not give rise to any new CP-violating phases. Moreover, it is impossible to obtain nonzero V e3 or θ 13 from this typical A 4 model, simply because η eµ = −η eτ holds. Such a disappointing observation implies that the residual flavor symmetry remains powerful to keep V e3 or θ 13 vanishing and forbid CP violation, even though the MNSP matrix V is not exactly unitary.
Current experimental data allow us to constrain the matrix elements of η and then constrain the model parameters to some extent. A recent analysis yields [15] |η| < In view of Eqs. (15) and (16), we immediately obtain where ∆m 2 21 ≡ m 2 2 − m 2 1 ≃ 7.6 × 10 −5 eV 2 and ∆m 2 31 ≡ m 2 3 − m 2 1 ≃ m 2 3 − m 2 2 ≡ ∆m 2 32 ≃ ±2.4 × 10 −3 eV 2 [2]. Eq. (19) leads us to a simple but instructive relation for three off-diagonal matrix elements of η: Therefore, |η eµ |/|η µτ | = |η eτ |/|η µτ | ≃ 2.1 × 10 −2 . Comparing this prediction with Eq. (18), one may self-consistently get |η eµ | = |η eτ | < 2.3 × 10 −5 by taking |η µτ | < 1.1 × 10 −3 . So it is more appropriate to use the upper bound of |η µτ | to constrain the lower bound of |f ν | by means of Eq. (19). We arrive at This result, which depends on the value of tan β in the supersymmetric A 4 model, implies that the Yukawa coupling of neutrinos should not be too small in order to preserve the unitarity of V at an experimentally-allowed level. It clearly indicates that an arbitrary choice of f ν in the neglect of small unitarity violation of V is inappropriate for model building, because the correlation between f ν and the deviation of V from the tri-bimaximal mixing pattern is an intrinsic property of a class of A 4 models.
The diagonal matrix elements of η consist of the contributions from both the chargedlepton sector and the neutrino sector, as shown in Eq. (15). Their competition depends on the sizes of f ℓ , f ν and tan β. For simplicity, here we assume that the charged-lepton contribution to η αα (for α = e, µ, τ ) is dominant. Then it is straightforward to obtain As a result, η ee : η µµ : η τ τ ≃ m 2 e : m 2 µ : m 2 τ ≃ 1 : 44566 : 12880040 , where we have input the central values of three charged-lepton masses at the electroweak scale [16]. Comparing this prediction with Eq. (18), one may self-consistently arrive at |η ee | < 2.1 × 10 −10 and |η µµ | < 9.3 × 10 −6 by taking |η τ τ | < 2.7 × 10 −3 . It is therefore more appropriate to use the upper bound of |η τ τ | to constrain the lower bound of |f ℓ | with the help of Eq. (22). We find where m τ ≃ 1746.24 MeV has been input at the electroweak scale [16]. This result, which also depends on the value of tan β in the supersymmetric A 4 model, shows that the Yukawa coupling of charged leptons should be relatively large in order to preserve the unitarity of V as constrained by current measurements. We stress that an arbitrary choice of either f ℓ or f ν in the neglect of small unitarity violation of V might be problematic for model building, simply because they receive constraints both from the model itself and from the experimental data. In this sense one must be cautious to claim that an A 4 flavor model can predict the tri-bimaximal neutrino mixing pattern whose matrix elements are constant and thus have nothing to do with the model parameters [17]. In fact, the slight (non-unitary) deviation of V from the tri-bimaximal mixing pattern is likely to impose a strong restriction on some model parameters like f ℓ , f ν and tan β.

IV. SUMMARY
We have examined a class of A 4 flavor models to see whether the tri-bimaximal neutrino mixing pattern V 0 is an exact consequence of such models. We find that small corrections to V 0 are actually inherent in the A 4 models and may arise from both the charged-lepton and neutrino sectors. We have demonstrated that such corrections may lead the MNSP matrix V to a non-unitary deviation from V 0 , but they cannot result in a nonzero V e3 (or θ 13 ) or any new CP-violating phases. In particular, the slight unitarity violation of V is sensitive to several model parameters, including the Yukawa couplings of charged leptons and neutrinos. We have shown that current experimental constraints on the unitarity of V allow us to constrain the model parameters to some extent.
We stress that the departure of V from V 0 explored in this work is an intrinsic property of a class of flavor models with the non-Abelian A 4 symmetry. Different departures may result either from the vacuum-expectation-value misalignments in a certain A 4 model or from some purely phenomenological perturbations [18]. The non-unitary deviation of V from V 0 is in some sense more interesting because it might give rise to new CP-violating effects in a variety of long-baseline neutrino oscillation experiments [19]. Since a lot of attention has been paid to how to derive the tri-bimaximal neutrino mixing pattern V 0 , the points revealed in our paper should be taken into account when one attempts to build specific flavor models with discrete family symmetries.