Minimal Modification To The Tri-bimaximal Neutrino Mixing

Current experimental data on neutrino oscillations are consistent with the tri-bimaximal mixing. If future experimental data will determine a non-zero $V_{e3}$ and/or find CP violations in neutrino oscillations, there is the need to modify the mixing pattern. We find that a simple neutrino mass matrix, resulting from $A_4$ family symmetry breaking with residual $Z_3$ and $Z_2$ discrete symmetries respectively for the Higgs sectors generating the charged lepton and neutrino mass matrices, can satisfy the required modifications. The neutrino mass matrix is minimally modified with just one additional complex parameter compared with the one producing the tri-bimaximal mixing. In this case, the CP violating Jarlskog factor $J$ has a simple form ($|J|=|V_{e1}V_{e3}|/2\sqrt{3}$ for real neutrino mass matrix), and also $V_{\mu i} = 1/\sqrt{3}$. We also discuss how this mixing matrix can be tested experimentally.


Introduction
The current neutrino mixing matrix from various experimental data [1,2] can be described by three neutrino mixing [3,4]. The mixing matrix V can be parameterized, using the Particle Data Group convention [2], by three mixing angles θ 12 , θ 23 and θ 13 , and one intrinsic CP violating phase δ for Dirac neutrinos. For Majorana neutrinos there are additional two independent Majorana CP violating phases. Present constraints on the mixing angles, at the 99% confidence level, are as the following [4] 30 • < θ 12 < 38 • , 36 • < θ 23 < 54 • , θ 13 < 10 • . (1) At present there is no experimental information about CP violating phases.
The above data can be well fitted by the tri-bimaximal mixing of the form With a suitable normalization of the signs for the matrix elements, the above tri-bimaximal mixing has θ 12 = sin −1 (1/ √ 3) = 35.2 • , θ 23 = 45 • and θ 13 = 0. Here we have omitted a possible diagonal Majorana phase matrix P = Diag(e iα 1 , e iα 2 , e iα 3 ) on the right. Since an overall phase does not play a role in any physical process, only two of the α 1,2,3 are physically independent.
The tri-bimaximal form for the mixing matrix was first proposed by Harrison, Perkins and Scott [5] and further studied by authors in Ref. [6]. Later, we independently arrived at the same Ansatz [7]. Many theoretical efforts have been made to produce such a mixing pattern [8,9,10,11]. Among them theories based on A 4 symmetry provide some interesting examples [9,10,11].
If future experimental data will find a non-zero value for V e3 , it is necessary to modify the mixing pattern. Another class of experimental data which may also lead to the requirement of modifying the tri-bimaximal mixing is the observation of CP violation in neutrino oscillations. CP violation in neutrino oscillations is proportional to the CP violating Jarlskog A non-zero J is related to the non-removable phase in V ("intrinsic" CP violation). This is different than the source of CP violation due to Majorana phases which do not show up in neutrino oscillations. The tri-bimaximal mixing leads to J = 0 and therefore has no intrinsic CP violation. In this note we analyze a simple mixing matrix [10,11], resulted from theories based on A 4 family symmetry breaking, satisfying the required modifications.
In the modified mixing matrix, V µi = 1/ √ 3 which are the same as those in the tribimaximal mixing. However, the matrix element V e3 is no longer zero, and is given by Here c = cos θ, s = sin θ with θ being a new mixing angle. The phase ρ is related to phases in the neutrino mass matrix. The detailed meaning will be given later.
The modified mixing matrix in various limits reduces to some of the forms considered in Refs. [13,14,15,16]. In the case c = s = 1/ √ 2 and ρ = 0, the mixing matrix reduces to the tri-bimaximal form. This mixing matrix has intrinsic CP violation with the Jarlskog factor J given by −(c 2 − s 2 )/6 √ 3. The two parameters θ and ρ can be determined by measuring |V e3 |, |V µ3 | and J. Therefore this mixing pattern can be tested experimentally in details.
The key point in obtaining the tri-bimaximal mixing pattern in theories based on A 4 family symmetry is to first get the matrices U l and U ν , which diagonalze the charged lepton mass matrix M l and neutrino mass matrix M ν , U † l M l U r = D l and U T ν M ν U ν = D ν (assuming Majorana neutrinos), to have the following forms [9,10,11] where ω 3 = 1 and 1 + ω + ω 2 = 0. Then using the definition for the mixing matrix V = U † l U ν to obtain where For U ν , we recognize that it is just a rotation through 45 o in the (1 − 3) plane. Recalling

Tri-bimaximal mass matrix and modifications
Let us now briefly discuss how tri-bimaximal mixing can arise and how it is minimally modified in A 4 models following Ref. [10]. The basic issue is that A 4 symmetry is broken down to two different discrete subgroups upon charged leptons and upon the neutrinos acquiring masses, namely Z 3 and Z 2 respectively. The clash between these two different subgroups was called the "sequestering problem" [11]. To explain the clash, let us be more specific with Higgs mechanism supplying the lepton masses. We emphasize, however, that the results of this paper are not dependent on any specific model.
The two forms for U l and U ν in eq.(3) are very different. In A 4 theories, this requires at least two separate Higgs sectors. We consider a case with three Higgs fields [10,11], Φ, φ (standard model doublet) and χ (standard model singlet). Under the A 4 , Φ and χ both transform as 3, and φ as 1. The standard left-handed leptons l L , right-handed charged leptons (l 1 R , l 2 R , l 3 R ), and right-handed neutrinos ν R transform as 3, (1 ′′ , 1, 1 ′ ) and 3, respectively. We refer the readers for more details on A 4 group properties to Refs. [9,10,11,17]. The Lagrangian responsible for the lepton mass matrix is If the vev structure is of the form < Φ 1,2,3 >= v Φ , < χ 1,3 >= 0, < χ 2 >= v χ , and < φ >= v φ , one would obtain the charged lepton mass term as From the above, we can identify the charged lepton mass to be √ 3λ i v Φ , and U l to have the "magic" form in eq. (3). U r is a unit matrix.
The neutrino mass matrix has the see-saw form with where M D = Diag(1, 1, 1)λ ν v φ , and m χ = λ χ v χ . From this one obtains the light neutrino mass matrix M ν of the form given in eq. (5). One therefore has a model for the tri-bimaximal mixing.
The vev structure of the Higgs fields breaks A 4 , but left some residual symmetries. The Higgs doublet Φ i with equal vacuum expectation values breaks A 4 down to a Z 3 generated by {I, c, a}, and the vev of only the χ 2 component to be non-zero in χ breaks A 4 down to a Z 2 generated by {1, r 2 }. Here a, c, r 2 are A 4 group elements defined in Ref. [10].
We note that the charged lepton mass matrix and the neutrino mass matrix are related to two separate Higgs sectors, Φ, and, χ and φ, respectively. If there is no communication between the two Higgs sectors, the residual Z 3 and Z 2 symmetries will be maintained. In general Φ and χ mix in the Higgs potential, it is not possible to keep the vev structure for Φ and χ discussed above [10,11]. One needs to separate them from communicating in the Higgs potential and therefore the sequestering problem. This sequestering problem will complicate the situation. However, models realizing such separation have been constructed with additional symmetries [11]. For our purpose here, we will assume that the sequestering problem is solved and study the consequences.
As long as the Z 3 symmetry is not broken, i.e. equal vev for Φ i , the form of U l obtained in the above is stable against higher order corrections. Also if the Z 2 symmetry is not broken, the "12", "21", "23" and "32" entries in M D and M R and therefore M ν are prevented from getting non-zero values. However it does not protect the "11" and "22" entries be equal [10,11]. Therefore after symmetry breaking a more general form of the light neutrino mass matrix M ν will emerge with rather than the M ν in (5). The above neutrino mass matrix has been obtained previously in Refs. [10,11].
In this basis, the neutrino masses m 1,2,3 have Majorana phases −2α 1,2,3 with The masses are given by Properties of the modified mixing matrix One clearly sees that the new mixing matrix can be very different from the tri-bimaximal, but the entries V µi = 1/ √ 3 are the same as those of the tri-bimaximal mixing. This can be tested in the near future by experiments. There are of course many new features. Two important qualitative differences are: (a). V e3 is not zero any more. We have |V e3 | = |(ce iρ − s)|/ √ 3. In the real neutrino mass matrix case, for small ε [10], (b). There are intrinsic CP violation. This can be easily checked by evaluating the Jarlskog factor, we obtain [11] It is surprising to note that the CP violating Jarlskog factor J does not contain the phase ρ implying that even if the parameters α, β and ε are real (or ρ = 0) there is intrinsic CP violation. In this case the value of J is equal to −iV e1 V e3 /2 √ 3 whose size can be as large as 0.04. This is sizeable enough to be measured in future experiments.
Note that the mixing matrix, apart from the Majorana phases α i , is a two-parameter, ρ and θ, matrix. They can be completely determined experimentally.
One can always choose a convention with both s and c be positive. We then have sin 2θ = 1 − (cos 2θ) 2 .
The phase factor ρ can be determined by additional measurements of V e3 and V µ3 . We Combining the sign of sin ρ can also be determined. The consistency of the above two equations can provide tests for the mixing matrix proposed.
We comment that the mixing matrix contains some of the cases studied by Xing [13], Bjorken,Harrison and Scott[14], Friedberg and Lee [15], and Xing, Zhang and Zhou [16] in various limiting cases. We find the following two limiting cases interesting.
(1). c = s = 1/ √ 2. In this case there is no intrinsic CP violation (no CP violation can be observed in neutrino oscillations). We have from eq. (12), This is the same mixing matrix, up to some Majorana phases, in eq. (1.18)

obtained in
Ref. [15]. From this we also see that the phase ρ indeed does not play the role of a Dirac phase which causes intrinsic CP violation.
(2). ρ = 0, in this case there is intrinsic CP violation. We have and J = −iV e1 V e3 /2 √ 3. This mixing matrix looks similar to that in eq. (1.18) of Ref. [15], but the appearance of "i" makes it CP violating.
It would be interesting to see how these two limiting cases can be experimentally distin- guished. An obvious way to tell the difference is to determine whether J is zero or not by measuring CP violation in neutrino oscillations. If J turns out to be non-zero, case (1) has to be abandoned. Before J can be measured, precise measurements of |V µ3 | and |V e3 | can also tell the difference since for case (1), one has where "+" should be taken if cos(ρ/2) and sin(ρ/2) have the same sign. Otherwise "−" should be taken. While for case (2), one has Since |V e3 | is small, |V µ3 | in case (2) has a weaker dependence on |V e3 | compared with case (1).
Measurements on quantities related to neutrino masses can also determine the parameters in the neutrino mass matrix. We list a few of them in the below.
i) Neutrinoless double beta decay measurement determines the 11-element m ee of the neutrino mass matrix in the basis where the charged lepton mass matrix has been diagonalized.
We finally comment that to rule out the mixing proposed here it is necessary to have precise measurement of |V e2 |. If future experiments will determine a |V e2 | significantly deviate from 1/ √ 3, one has to further modify the model. In the A 4 based model discussed earlier, this implies that a further break down of the residual Z 3 and Z 2 must happen. If just Z 3 is broken, the vev of the components < Φ i > will not be equal, this will affect the "magic" form U † l in eq. (3), whereas if Z 2 is broken, the zero entries in eq.(9) will become non-zero. In general both Z 3 and Z 2 may be broken at the same time. The form of the mixing will become the most general one with corrections to all elements [18]. We have to wait more precise data to tell us if the simple mixing proposed here need to be further modified.