Testable Deviation from Exact Tribimaximal Mixing

A simple relation U_{MNS}=V_{CKM}^\dagger U_{TB} between the lepton and quark mixing matrices (U_{MNS} and V_{CKM}) is speculated under an ansatz that U_{MNS} becomes an exact tribimaximal mixing U_{TB} in a limit V_{CKM}={\bf 1}. By using the observed CKM mixing parameters, possible values of neutrino oscillation parameters are estimated: \sin^2 \theta_{13}=0.024-0.028, \sin^2 2\theta_{23}=0.94-0.95 and \tan^2 \theta_{12}=0.24-1.00 depending on phase conventions of U_{TB}. Those values are testable soon by precision measurements in neutrino oscillation experiments.


Introduction
Recently, there has been considerable interest in the magnitude of the neutrino mixing angle θ 13 (ν e ↔ ν τ mixing angle), because it is a key value not only for checking neutrino mass matrix models, but also for searching CP -violation effects in the lepton sector. (For a review of models for θ 13 , see, for example, Ref. [1].) Recent observed neutrino oscillation data are in favor of the so-called "tribimaximal mixing" [2] which predicts θ 13 = 0, tan 2 θ 12 = 1/2 and sin 2 2θ 23 = 1, since the present data yield the values tan 2 θ 12 = 0.47 +0.06 −0.05 [3] and sin 2 2θ 23 = 1.00 −0.13 [4]. If the angle θ 13 is exactly zero or negligibly small, the observation of the CP -violation effects in the lepton sector will be hopeless even in future, as far as neutrino oscillation experiments are concerned. On the other hand, recently, Fogli et al. [5] have reported a sizable value sin 2 θ 13 = 0.016 ± 0.010 (1σ) from a global analysis of neutrino oscillation data.
The tribimaximal lepton mixing is given by the form Such a form with beautiful coefficients seems to be understood from a discrete symmetry of flavors [2]. In contrast to the lepton mixing matrix (Pontecorvo-Maki-Nakagawa-Sakata mixing matrix [6]) U P M N S , the observed Cabibbo-Kobayashi-Maskawa [7] (CKM) quark mixing matrix V CKM seems to have no beautiful form with Clebsch-Gordan-like coefficients, and V CKM , rather, looks like nearly V CKM ≃ 1. It is unlikely that a theory which exactly leads to the tribimaximal mixing (1) simultaneously gives the CKM mixing matrix with small and complicated mixing values. Therefore, it is interesting to consider a specific case that a theory of flavor symmetry gives V CKM = 1 in the limit of U P M N S = U T B . We consider that the observed form of the CKM matrix V CKM is due to some additional effects (e.g. symmetry breaking effects for the flavor symmetry). If this is true, then, the observed lepton mixing U P M N S will also deviate from the exact tribimaximal mixing U P M N S = U T B by additional effects which gives the deviation from V CKM = 1. (Also see, e.g., Ref. [8] for a possible deviation of U P M N S from a bimaximal mixing (not tribimaximal mixing) related to V CKM .) Recently, Datta [9] has investigated possible flavor changing neutral current processes using the same assumption that V CKM = 1 and U P M N S = U T B in a flavor symmetry limit. By using a specific mass matrix model, he have discussed realistic mixings V CKM and U P M N S caused by a small breaking of the flavor symmetry. Also, Plentinger and Rodejohann [10] have investigated possible deviations from tribimaximal mixing by assuming a special form of the neutrino mass matrix. Furthermore, there are many works which discuss specific mass matrix models from the point of the so-called "quark-lepton-complementarity" [11]. In this paper, however, we start only from putting a simple ansatz stated later (in Eqs. (9) and (10)), without referring to any mass matrix model explicitly.
For convenience of later discussions, we define the tribimaximal mixing by a form where P L = diag(e iα 1 , e iα 2 , e iα 3 ), by including freedom of the phase convention, although the tribimaximal mixing is conventionally expressed by the form (1). The purpose of the present paper is to speculate a possible form of the lepton mixing matrix U P M N S under the ansatz V CKM = 1 ↔ U P M N S = U T B . We show, as stated later, that a natural realization of this ansatz leads to a simple relation By using the observed CKM mixing parameters, we estimate values of the neutrino oscillation parameters sin 2 θ 13 , tan 2 θ 12 and sin 2 2θ 23 , which are defined by First, let us give conventions of the mass matrices: the quark and charged lepton mass matrices M f (f = u, d, e) are defined by the mass terms f L M f f R , so that those are diagonalized as Therefore, the quark and lepton mixing matrices, V CKM and U P M N S , are given by respectively. Hereafter, we refer to a flavor basis on which the mass matrix M f is diagonal (i.e. D f ) as "f -basis". For example, in the u-basis, up-quark, down-quark, charged-lepton and neutrino mass matrices are given by

Ansatz and speculation
Let us mention an ansatz which leads to the relation (4). We put the following ansatz: In the limit of U dL → 1, the matrix U eL also becomes a unit matrix 1, while the matrix U ν becomes the exact tribimaximal mixing U T B in the limit of U uL → 1. In other words, in the u-basis, the neutrino mass matrix M Here, we have supposed that, in a symmetry limit, i.e. when an origin which causes V CKM = 1 is switched off, the physical mass matrices M f become the diagonal forms D f , while the neutrino mass matrix M ν becomes a specific form M (u) ν defined by (9): In other words, we consider that a common origin in the down sector causes D d → M d and D e → M e , and a common origin in the up sector causes D u → M u and U T B D ν U T T B → M ν . Of course, this transformation (10) can not be realized by a flavor-basis transformation, because M f and D f are connected by Eqs. (6) and (7). It is well-known that physics at a low-energy is unchanged under any flavor-basis transformation.
The ansatz (9) states that the mixing matrix U νL in the neutrino sector, which is defined by Therefore, the observed lepton mixing matrix U P M N S is given by where U ed is a flavor-basis transformation matrix defined by (The relation (12) is also derived by using relations U .) According to this notation, the CKM mixing matrix V CKM is expressed as V CKM = U ud . Since U ed = U † ue U ud = U † ue V CKM , if we consider U ue = 1, we obtain U ed = V CKM , so that we will obtain U P M N S = U T B from the relation (11). However, such a case U eu = 1 is unlikely under our ansatz U eL → 1 in the limit of U dL → 1. Generally speaking, U ue can vary from U ue = 1 to U ue = V CKM , so that U ed varies from U ed = V CKM to U ed = 1 and Eq.(12) varies from U P M N S = U T B to U P M N S = V † CKM U T B . (Here, we have considered that U ue does, at least, not take a large mixing more than V CKM and a rotation to an opposite direction, V † CKM .) Therefore, we can consider that the relation (4) describes a maximal deviation of U P M N S from U T B . In spite of such a general consideration, we think that the case U ed = 1 (or highly U ed ≃ 1) is a most natural realization of our ansatz (10), because it means U eL → 1 in the limit U dL → 1. Therefore, in this paper, we adopt the case U ed = 1, and investigate possible numerical values of the neutrino oscillation parameters sin 2 θ 13 , tan 2 θ 12 and sin 2 2θ 23 under the relation (4).
By the way, we are also interested in whether those values are dependent on the phase parameters α i and γ i defined in Eq. (3). The relation (12) is invariant under the rephasing under the rephasing (note that U νL does not have such a freedom of rephasing). Therefore, the phase matrices P L and P R originate in the mass matrix M (u) ν as shown in Eq. (9). Then, Eq. (9) can be rewritten as where Since the matrix U 0 T B is orthogonal, the mass matrix M (u) ν has to be real. In other words, the phase matrix P L is determined from the form M (u) ν so that M (u) ν is real. On the other hand, the phase matrix P R is fixed so that D ν P 2 R is real. Then, we find that the numerical results for |(U P M N S ) ij | are independent of the phases γ i in P R , because U P M N S is expressed by U P M N S = U P R =1 P M N S P R , so that the quantities |(U P M N S ) ij | = |(U P R =1 P M N S ) ij e iγ j | are independent of the phase parameters γ j . The results are only dependent on the phase parameters α i in P L . Hereafter, for simplicity, we put P R = 1.
Let us show that the neutrino oscillation parameters sin 2 θ 13 and sin 2 2θ 23 are only dependent on a relative phase parameter α ≡ α 3 − α 2 . Since (U P M N S ) i3 is expressed as the values |(U P M N S ) i3 | are dependent only on the parameter α. We illustrate the behaviors of sin 2 θ 13 and sin 2 2θ 23 versus α in Fig.1 and Fig.2, respectively. Here, for numerical evaluation, we have used the Wolfenstein parameterization [12] of V CKM and the best-fit values [13] λ = 0.2272, A = 0.818, ρ = 0.221 and η = 0.340. We find that the values sin 2 θ 13 and sin 2 2θ 23 are almost insensitive to the value α, and those take sin 2 θ 13 = 0.024 − 0.028 and sin 2 2θ 23 = 0.94 − 0.95. Those values are consistent with the present experimental data. As shown in Fig.1, if we take the result sin 2 θ 13 = 0.016 ± 0.010 (1σ) obtained from a global analysis of neutrino oscillation data by Fogli et al. [5], we can obtain allowed bounds for α. The sizable value sin 2 θ 13 is within a reach of forthcoming neutrino experiments planning by Double Chooz, Daya Bay, RENO, OPERA, and so on. The value sin 2 2θ 23 = 0.94 − 0.95 is consistent with the present observed value [4] sin 2 2θ 23 = 1.00 −0.13 , and the predicted value will also be testable soon by precision measurements in solar and reactor neutrino experiments. Previously, Plentinger and Rodejohann [10] have predicted possible deviations from tribimaximal mixing by assuming a specific form of the neutrino mass matrix and by assuming a CKM-like hierarchy of the mixing angles (θ e 12 = λ, θ e 23 = Aλ 2 , θ e 13 = Bλ 3 ) in the charged lepton sector. Furthermore, they have assumed the quark-lepton-complementarity (QLC) [11], and put an ad hoc relation θ e 12 = θ C (θ C is the Cabibbo mixing angle). Then, they have obtained a relation Their result (17) agrees with our result sin 2 θ 13 = 0.024 − 0.028, because from Eq. (16).
On the other hand, for the value tan 2 θ 12 , there is no simple situation (one-parameter dependency). The values (U P M N S ) 11 and (U P M N S ) 12 are given by so that the values |(U P M N S ) 11 | and |(U P M N S ) 12 | depend not only on β ≡ α 2 − α 1 but also on α ≡ α 3 − α 2 . However, since the observed CKM matrix parameters show 1 ≫ |(V CKM ) cd | 2 ≫ |(V CKM ) td | 2 , we can neglect the terms (V CKM ) * 31 e −iα 3 compared with (V CKM ) * 11 e −iα 1 and (V CKM ) * 21 e −iα 2 , so that the value tan 2 θ 12 approximately depends on only the parameter β. We illustrate the behavior of tan 2 θ 12 versus β ≡ α 2 − α 1 in Fig.3, in which we take typical values of α such as α = 0 and α = −2π/3. We can see that tan 2 θ 12 is, in fact, insensitive to the parameter α. In contrast to the cases of sin 2 θ 13 and sin 2 2θ 23 , the value of tan 2 θ 12 are highly sensitive to the parameter β as shown by from Eq. (20). The similar result has been obtained by Plentinger and Rodejohann [10]. The value of tan 2 θ 12 takes from 0.24 to 1.00 according to the variation in β. In order to fit the observed value [3] tan 2 θ 12 ≃ 0.5, we must take β ≃ ±π/2. This will put a constraint on scenarios which give a tribimaximal mixing. Note that, from the relation (4), we can obtain a CP violating observable as well as in a model given in Ref. [10]. Therefore, if we require a maximal CP violation in the lepton sector, we obtain β ≃ ±π/2 as pointed out in Ref. [10], which is compatible with the constraint from the observed value tan 2 θ 12 ≃ 0.5. [14] 3 Summary In conclusion, under the ansatz "U P M N S → U T B in the limit of V CKM → 1", we have speculated a simple relation U P M N S = V † CKM U T B . We have not referred an explicit mechanism (model) which gives such a CKM mixing V CKM = 1 in the limit of U P M N S = U T B . For example, a model [10] by Plentinger and Rodejohann is one of mass matrix models which explicitly realize our ansatz because they have put an ad hoc assumption sin θ e 12 = sin θ C . A model [9] by Datta is also one of such models. However, such a model-building is not a purpose of the present paper. We have started our investigation by admitting the relation U P M N S → U T B as V CKM → 1 as an ansatz. The relation U P M N S = V † CKM U T B is widely valid for all models which are consistent with our ansatz.
By using the observed CKM matrix parameters, we have estimated the lepton mixing parameters sin 2 θ 13 , sin 2 2θ 23 and tan 2 θ 12 . The values of sin 2 2θ 23 and sin 2 θ 13 are almost independent of the phase convention, and they take values sin 2 θ 13 = 0.024 − 0.028 and sin 2 2θ 23 = 0.94 − 0.95. The sizable value of sin 2 θ 13 is within a reach of forthcoming neutrino experiments planning by Double Chooz, Daya Bay, RENO, OPERA, and so on. The value of sin 2 2θ 23 is also testable soon by precision measurements in solar and reactor neutrino experiments. On the other hand, the value of tan 2 θ 12 has highly depended on the phase convention of the tribimaximal mixing, and the value has been in a range 0.24 < tan 2 θ 12 < 1.00. Note that the phase matrix P L cannot be absorbed into the rephasing of V CKM , although it seems to be possible from the expression (4). Since the present observed value of tan 2 θ 12 is tan 2 θ 12 ≃ 0.5, the phase parameter β is constrained as β ≃ ±π/2. This put a strong constraint on models which lead to the exact tribimaximal mixing (2). The requirement of a maximal CP violation in the lepton sector is interestingly related to the observed value tan 2 θ 12 ≃ 0.5.
If the predicted values sin 2 θ 13 = 0.024 − 0.028 and sin 2 2θ 23 ≃ 0.94 − 0.95 are denied by forthcoming neutrino oscillation experiments, it means a denial of the simple view that the lepton mixing U P M N S becomes the exact tribimaximal mixing U T B in the limit of V CKM → 1. We will be compelled to consider that the view stated above is oversimplified and the situation of quark and lepton flavor mixings is more complicated. The observed values of neutrino oscillation parameters will provide us a promising clue to a possible structure of U ed , although we simply assumed U ed = 1 in the expression (12). This will shortly become clear by forthcoming experiments. Fig. 1 Behavior of sin 2 θ 13 versus α = α 3 − α 2 . The horizontal dashed and dotted lines denote the analysis sin 2 θ 13 = 0.016 ± 0.010 (1σ) by Fogli et al. [5].