In a recently proposed renormalizable model of neutrino mixing using the non-Abelian discrete symmetry in the context of a supersymmetric extension of the Standard Model with gauged , a correlation was obtained between and in the case where all four parameters are real. Here we consider one parameter to be complex, thus allowing for one Dirac CP phase and two Majorana CP phases . We find a slight modification to this correlation as a function of . For a given set of input values of , , , and , we obtain and (the effective Majorana neutrino mass in neutrinoless double beta decay) as functions of . We find that the structure of this model always yields small .
CP Phases of Neutrino Mixing in a Supersymmetric
Gauge Model with Lepton Flavor Symmetry
Hajime Ishimori, Shaaban Khalil and Ernest Ma
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Centre for Theoretical Physics, Zewail City of Science and Technology,
Sheikh Zayed, 6 October City, 12588, Giza, Egypt
Department of Mathematics, Ain Shams University,
Faculty of Science, Cairo 11566, Egypt
Department of Physics and Astronomy, University of California,
Riverside, California 92521, USA
Institute for Advanced Study, Hong Kong University of Science and Technology,
Hong Kong, China
Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa 277-8583, Japan
The most general Majorana neutrino mass matrix has six complex entries, i.e. twelve parameters. Three are overall phases of the mass eigenstates which are unobservable. The nine others are three masses, three mixing angles, and three phases: one Dirac phase , i.e. the analog of the one complex phase of the quark mixing matrix, and two relative Majorana phases for two of the three mass eigenstates. The existence of nonzero or means that conservation is violated. It is one of most important issues of neutrino physics yet to be explored experimentally.
The application of the non-Abelian discrete symetry  (and others) to neutrino mixing has been successful in explaining tribimaximal mixing, i.e. , , and . In particular, a generic three-parameter model  predicts all of the above with , leaving the three neutrino masses arbitrary. Recently, the first evidence that has been published  by the T2K Collaboration, i.e.
for and normal (inverted) hierarchy of neutrino masses. Slightly different but similar ranges are obtained for nonzero values of . More recently, the Double Chooz Collaboration has also reported  a measurement of
Their best fit is obtained by minimizing its as a function of . However, all values are allowed within one standard deviation. One month ago, the first precise measurement of was announced by the Daya Bay Collaboration :
based only on a rate analysis, resulting in a 5.2 effect. It has been followed by the announcement this month of the RENO Collaboration :
again based only on a rate analysis, resulting in a 6.3 effect.
To account for , the original proposal has to be modified . Similarly, the original supersymmetric gauge model with lepton flavor symmetry  (which obtained tribimaximal mixing) has to be replaced as well . In that latter paper, it is shown that a neutrino mass matrix with four parameters allow a nonzero . Assuming that all four parameters are real, thus requiring two conditions among the six observables, i.e. the three masses and three mixing angles, the prediction
is obtained. This scenario applies of course only to the case . Here we consider instead the case where one parameter is complex. We then have five real parameters to describe the three masses, the three mixing angles, and the three phases. Given five inputs, we should then be able to predict the other four parameters.
The tetrahedral group (12 elements) is the smallest group with a real 3 representation. The Frobenius group (21 elements) is the smallest group with a pair of complex 3 and 3 representations. It is generated by
where , so that , , and . The character table of (with ) is given below.
The group multiplication rules of include
Note that has two invariants and has one invariant.
We now follow Ref.  in deriving the neutrino mass matrix. Under , let , , , and . The Yukawa couplings generate the charged-lepton mass matrix
if , as in the original proposal .
Let , then the Yukawa couplings are allowed, with
for which is necessary for consistency since has already been assumed for . Note that and have , and both are necessary because of supersymmetry. However, the analysis of neutrino mixing does not involve these extra supersymmetric partners.
Now add the neutral electroweak Higgs singlets and , both with . Then there are two Yukawa invariants: and (which has to be symmetric in ). Note that is not the same as because they have different . This means that both and the complexity of the and representations in are required for this scenario. The heavy Majorana mass matrix for is then
where , , , and have been assumed. This means that the residual symmetry in the singlet Higgs sector is , whereas that in the doublet Higgs sector is . This choice allows nonzero , whereas the choice of Ref.  enforces .
The seesaw neutrino mass matrix is now
where det. Redefining the parameters to absorb the overall constant, we obtain the following neutrino mass matrix in the tribimaximal basis:
This is obtained by first rotating with the unitary matrix of Eq. (9), which converts it to the basis, then by Eq. (14) below. Note that for and , this matrix becomes diagonal: , which is the tribimaximal limit. Normal hierarchy of neutrino masses is obtained if and inverted hierarchy is obtained if .
The neutrino mixing matrix has 4 parameters: and . We choose the convention to conform with that of the tribimaximal mixing matrix
where are the physical neutrino masses, with
If and are known, then all are functions only of .
In Ref. , the parameters are assumed to be real, hence and are zero. We now consider to be complex. Thus are real and are complex. Since is in the tribimaximal basis, it can be diagonalized by an approximately diagonal unitary matrix. To first order, let
we obtain , , and in terms of . Using the four measured values , , , , and varying , we then obtain , the physical relative Majorana phases in Eq. (15), and the effective Majorana neutrino mass in neutrinoless double beta decay, i.e.
Because of the structure of Eq. (13) from the symmetry, even though the phase of the complex parameter may be large, i.e. large, cannot be too large, because in the limit , there can be no violation, so any violating effect has to be proportional to where sets the neutrino mass scale. This is typically less than one because measures the deviation of from the tribimaximal limit of 1/2.
The unitary matrix has entries
To obtain , we rotate the phases of the and rows so that is real and negative, and is real and positive. These phases are absorbed by the and leptons and are unobservable. We then rotate the columns so that and , where and are real and positive. The physical relative Majorana phases of are then . We now extract the three angles as well as as follows.
To see the approximate dependence of on the parameters , we assume normal hierarchy and let
Expanding in over , we then have
Using Eq. (26), and neglecting , we obtain
Assuming inverted hierarchy, we let
and is determined by
For our numerical analysis, we set
We then diagonalize Eq. (15) exactly and scan for solutions satisfying the above experimental inputs. Assuming normal hierarchy, we find to range from 0.9501 for to 0.9505 for , as shown in Fig. 1. This is an imperceptible change, so our model prediction for is basically unchanged from the real case. We show the absolute values , , , and as functions of in Fig. 2, and versus in Fig. 3. As expected, may be large, but remains small. We then plot the three physical neutrino masses as well as as functions of in Fig. 4, and the Majorana phases versus in Fig. 5. For inverted hierarchy, we show in Figs. 6 to 10 the corresponding plots. We note again that is small, but now are much larger. This can be seen from Eq. (40) versus Eq. (32).
In conclusion, we have studied how the model of Ref.  allows violation in the neutrino mixing matrix. There are three real parameters and one complex parameter , from which nine physical observables may be derived. Given the experimental inputs , , , and the recently measured , the remaining five observables depend on only one variable which we choose to be . Because of the structure of the neutrino mass matrix constrained by , even if has a large phase, i.e. is large, remains small. For and , we find to be essentially fixed at 0.95 as changes from 0.0 to 0.2. The Majorana phases are comparable to in magnitude for normal hierarchy, but are much larger for inverted hierarchy.
Acknowledgments: The work of H.I. is supported by Grant-in-Aid for Scientific Research, No. 23.696, from the Japan Society of Promotion of Science. The work of S.K. is supported in part by ICTP Grant AC-80. The work of E.M. is supported in part by the U. S. Department of Energy under Grant No. DE-AC02-06CH11357.
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