#### Generalised geometrical CP violation in a T ′ lepton flavour model

Ivan Girardi
4
Aurora Meroni
2
3
4
S.T. Petcov
0
4
Martin Spinrath
1
4
0
IPMU,
University of Tokyo
, 5-1-5 Kashiwanoha, 277-8583 Kashiwa,
Japan
1
Institut fur Theoretische Teilchenphysik, Karlsruhe Institute of Technology
, Engesserstrae 7, D-76131 Karlsruhe,
Germany
2
Dipartimento di Matematica e Fisica,
Universit`a di Roma Tre
, Via della Vasca Navale 84, I-00146,
Rome
3
INFN, Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044 Frascati,
Italy
4
SISSA/INFN, Via Bonomea 265, I-34136 Trieste,
Italy
We analyse the interplay of generalised CP transformations and the nonAbelian discrete group T 0 and use the semi-direct product Gf = T 0 o HCP, as family symmetry acting in the lepton sector. The family symmetry is shown to be spontaneously broken in a geometrical manner. In the resulting flavour model, naturally small Majorana neutrino masses for the light active neutrinos are obtained through the type I see-saw mechanism. The known masses of the charged leptons, lepton mixing angles and the two neutrino mass squared differences are reproduced by the model with a good accuracy. The model allows for two neutrino mass spectra with normal ordering (NO) and one with inverted ordering (IO). For each of the three spectra the absolute scale of neutrino masses is predicted with relatively small uncertainty. The value of the Dirac CP violation (CPV) phase in the lepton mixing matrix is predicted to be = /2 or 3/2. Thus, the CP violating effects in neutrino oscillations are predicted to be maximal (given the values of the neutrino mixing angles) and experimentally observable. We present also predictions for the sum of the neutrino masses, for the Majorana CPV phases and for the effective Majorana mass in neutrinoless double beta decay. The predictions of the model can be tested in a variety of ongoing and future planned neutrino experiments.
Contents
1 Introduction
T 0 Symmetry and Generalised CP Transformations
2.1 The consistency conditions
2.2 Transformation properties under generalised CP
2.3 Conditions to violate physical CP The model
3.1 The flavon sector
3.2 The matter sector
3.2.1 The charged lepton sector
3.2.2 The neutrino sector
3.3 Comments about the r
3.4 Geometrical CP violation and residual symmetries
3.5 Predictions
3.5.1 Absolute neutrino mass scale
3.5.2 The mixing angles and Dirac CPV phase
3.5.3 The Majorana CPV phases
3.5.4 The neutrinoless double beta decay effective Majorana mass
3.5.5 Limiting cases
4 Summary and conclusions A Technicalities about T 0 B Messenger sector
Introduction
Understanding the origin of the patterns of neutrino masses and mixing, emerging from
the neutrino oscillation, 3H decay, cosmological, etc. data is one of the most challenging
problems in neutrino physics. It is part of the more general fundamental problem in particle
physics of understanding the origins of flavour, i.e., of the patterns of the quark, charged
lepton and neutrino masses and of the quark and lepton mixing.
At present we have compelling evidence for the existence of mixing of three light
massive neutrinos i, i = 1, 2, 3, in the weak charged lepton current (see, e.g., [1]). The
masses mi of the three light neutrinos i do not exceed approximately 1 eV, mi < 1 eV,
i.e., they are much smaller than the masses of the charged leptons and quarks. The three
light neutrino mixing is described (to a good approximation) by the Pontecorvo, Maki,
Nakagawa, Sakata (PMNS) 3 3 unitary mixing matrix, UPMNS. In the widely used
standard parametrisation [1], UPMNS is expressed in terms of the solar, atmospheric and
reactor neutrino mixing angles 12, 23 and 13, respectively, and one Dirac - , and two
Majorana [2] - 1 and 2, CP violation phases:
UPMNS U = V (12, 23, 13, ) Q(1, 2) ,
and we have used the standard notation cij cos ij , sij sin ij , 0 ij /2, 0
2. The matrix Q contains the two physical Majorana CP violation (CPV) phases:
Q = Diag e i 1/2, e i 2/2, 1 .
The parametrization of the phase matrix Q in eq. (1.3) differs from the standard one [1]
Q = Diag 1, ei 21/2, ei 31/2 . Obviously, one has 21 = (1 2) and 31 = 1. In the
case of the seesaw mechanism of neutrino mass generation, which we are going to employ,
the Majorana phases 1 and 2 (or 21 and 31) vary in the interval [3] 0 1,2 4.1 If
CP invariance holds, we have = 0, , 2, and [57] 1(2) = k(0) , k(0) = 0, 1, 2, 3, 4.
All compelling neutrino oscillation data can be described within the indicated 3-flavour
neutrino mixing scheme. These data allowed to determine the angles 12, 23 and 13 and
the two neutrino mass squared differences m221 and m231 (or m232), which drive the
observed oscillations involving the three active flavour neutrinos and antineutrinos, l and
l, l = e, , , with a relatively high precision [8, 9]. In table 1 we give the values of the
3flavour neutrino oscillation parameters as determined in the global analysis performed in [8].
An inspection of table 1 shows that although 13 6= 0, 23 6= /4 and 12 6= /4, the
deviations from these values are small, in fact we have sin 13 = 0.16 1, /4 23 =
0.11 and /4 12 = 0.20, where we have used the relevant best fit values in table 1.
The value of 13 and the magnitude of deviations of 23 and 12 from /4 suggest that
the observed values of 13, 23 and 12 might originate from certain symmetry values
which undergo relatively small (perturbative) corrections as a result of the corresponding
symmetry breaking. This idea was and continues to be widely explored in attempts to
understand the pattern of mixing in the lepton sector (see, e.g., [1032]). Given the fact
that the PMNS matrix is a product of two unitary matrices,
where Ue and U result respectively from the diagonalisation of the charged lepton and
neutrino mass matrices, it is usually assumed that U has a specific form dictated by a
1The interval beyond 2, 2 1,2 4, is relevant, e.g., in the calculations of the baryon
asymmetry within the leptogenesis scenario [3], in the calculation of the neutrinoless double beta decay effective
Majorana mass in the TeV scale version of the type I seesaw model of neutrino mass generation [4], etc.
|m231| (NO) 103 eV2
|m232| (IO) 103 eV2
7.54+00..2262
2.47+00..0160
2.46+00..0171
0.307+00..001186
0.386+00..002241
0.392+00..003292
0.0241+00..00002255
0.0244+00..00002235
symmetry which fixes the values of the three mixing angles in U that would differ, in
general, by perturbative corrections from those measured in the PMNS matrix, while Ue
(and symmetry breaking effects that we assume to be subleading) provide the requisite
corrections. A variety of potential symmetry forms of U , have been explored in the
literature on the subject (see, e.g., [33]). Many of the phenomenologically acceptable
symmetry forms of U , as the tribimaximal (TBM) [3438] and bimaximal (BM) [3942]
mixing, can be obtained using discrete flavour symmetries (see, e.g., the reviews [4345] and
the references quoted there in). Discrete symmetries combined with GUT symmetries have
been used also in attempts to construct realistic unified models of flavour (see, e.g., [43]).
In the present article we will exploit the approximate flavour symmetry based on the
group T 0, which is the double covering of the better known group A4 (see, e.g., [45]),
with the aim to explain the observed pattern of lepton (neutrino) mixing and to obtain
predictions for the CP violating phases in the PMNS matrix and possibly for the absolute
neutrino mass scale and the type of the neutrino mass spectrum. Flavour models based
on the discrete symmetry T 0 have been proposed by a number of authors [4653] before
the angle 13 was determined with a high precision in the Day Bay [54] and RENO [55]
experiments (see also [5659]). All these models predicted values of 13 which turned out
to be much smaller than the experimentally determined value.
In [52, 53], in particular, an attempt was made to construct a realistic unified
supersymmetric model of flavour, based on the group SU(5) T 0, which describes the quark
masses, the quark mixing and CP violation in the quark sector, the charged lepton masses
and the known mixing angles in the lepton sector, and predicts the angle 13 and possibly
the neutrino masses and the type of the neutrino mass spectrum as well as the values of
the CPV phases in the PMNS matrix. The light neutrino masses are generated in the
model by the type I seesaw mechanism [6065] and are naturally small. It was suggested
in [52, 53] that the complex Clebsch-Gordan (CG) coefficients of T 0 [66] might be a source
of CP violation and hence that the CP symmetry might be broken geometrically [67] in
models with approximate T 0 symmetry. Since the phases of the CG coefficients of T 0 are
fixed, this leads to specific predictions for the CPV phases in the quark and lepton mixing
matrices. Apart from the incorrect prediction for 13, the authors of [52, 53] did not
address the problem of vacuum alignment of the flavon vevs, i.e., of demonstrating that the
flavon vevs, needed for the correct description of the quark and lepton masses and of the
the mixing in both the quark and lepton sectors, can be derived from a flavon potential
and that the latter does not lead to additional arbitrary flavon vev phases which would
destroy the predictivity, e.g., of the leptonic CP violation of the model.
A SUSY SU(5) T 0 model of flavour, which reproduces the correct value of the lepton
mixing angle 13 was proposed in [68], where the problem of vacuum alignment of the
flavon vevs was also successfully addressed.2 In [68] it was assumed that the CP violation
in the quark and lepton sectors originates from the complexity of the CG coefficients of
T 0. This was possible by fixing the phases of the flavon vevs using the method of the
so-called discrete vacuum alignment, which was advocated in [70] and used in a variety
of other models with discrete flavour symmetries [7174]. The value of the angle 13 was
generated by charged lepton corrections to the TBM mixing using non-standard GUT
relations [16, 7578].
After the publication of [68] it was realised in [79, 80] that the requirement of CP
invariance in the context of theories with discrete flavour symmetries, imposed before the
breaking of the discrete symmetry leading to CP nonconservation and generation of the
masses of the matter fields of the theory, requires the introduction of the so-called
generalised CP transformations of the matter fields charged under the discrete symmetry. The
explicit form of the generalised CP transformations is dictated by the type of the discrete
symmetry. It was noticed in [79], in particular, that due to a subtle intimate relation
between CP symmetry and certain discrete family symmetries, like the one associated with
the group T 0, it can happen that the CP symmetry does not enforce the Yukawa type
couplings, which generate the matter field mass matrices after the symmetry breaking, to be
real but to have certain discrete phases predicted by the family symmetry in combination
with the generalised CP transformations. In the SU(5) T 0 model proposed in [68], these
phases, in principle, can change or modify completely the pattern of CP violation obtained
by exploiting the complexity of some of the T 0 CG coefficients.
In the present article we address the problem of the relation between the T 0 symmetry
and the CP symmetry in models of lepton flavour. After some general remarks about the
connection between the T 0 and CP symmetries in section 2, we present in section 3 a fully
consistent and explicit model of lepton flavour with a T 0 family symmetry and geometrical
CP violation. We show that the model reproduces correctly the charged lepton masses,
all leptonic mixing angles and neutrino mass squared differences and predicts the values
of the leptonic CP violating phases and the neutrino mass spectrum. We show also that
this model indeed exhibits geometrical CP violation. We clarify how the CP symmetry is
broken in the model by using the explicit form of the constructed flavon vacuum alignment
sector; without the knowledge of the flavon potential it is impossible to make conclusions
2A modified version of the model published in [52, 53], which predicts a correct value of the angle 13, was
constructed in [69], but the authors of [69] left open the issue of the vacuum alignment of the flavon vevs.
about the origin of CP symmetry breaking in flavour models with T 0 symmetry. In the
appendix we give some technical details about the group T 0 and present a UV completion
of the model, which is necessary in order to to select correctly certain T 0 contractions in
the relevant effective operators.
T 0 Symmetry and Generalised CP Transformations
In this section we would like to clarify the role of a generalised CP transformation combined
with the non-Abelian discrete symmetry group T 0. Let Gf = T 0 o HCP be the symmetry
group acting in the lepton sector such that both T 0 and HCP act on the lepton flavour
space. Motivated by this study we will present in the next section a model where Gf is
broken such that all lepton mixing angles and physical CP phases of the PMNS mixing
matrix can be predicted in terms of two mixing angles and two phases. The breaking of Gf
will be achieved through non zero vacuum expectation values (vevs) of some scalar fields,
the so-called flavons.
The consistency conditions
The discrete non-Abelian family symmetry group T 0 is the double covering of the
tetrahedral group A4 and its complete description in terms of generators, elements and
representations is given in appendix A. An interesting feature of this group is the fact that it
is the smallest group that admits 1-, 2-, and 3-dimensional representations and for which
the three representations can be related by the multiplication rule 2 2 = 3 1.3 T 0 has
seven different irreducible representations: the 1- and 3-dimensional representations 1, 10,
100, 3 are not faithful, i.e., not injective, while the doublet representations 2, 20 and 200 are
faithful. One interesting feature of the T 0 group is related to the tensor products involving
the 2-dimensional representation since the CG coefficients are complex.
We define now the transformation of a field (x) under the group T 0 and HCP
respectively as:
where r(g) is an irreducible representation r of the group element g T 0, x0 x0, ~x
and Xr is the unitary matrix representing the generalised CP transformation. In order
to introduce consistently the CP transformation for the family symmetry group T 0, the
matrix Xr should satisfy the consistency conditions [79, 80, 82]:
g, g0 T 0 .
Following the discussion given in [79, 80, 82] it is important to remark that the
consistency condition corresponds to a similarity transformation between the representation
r and CP. Since the structure of the group is preserved and an element g T 0 is
always mapped into an element g0 T 0, this map defines an automorphism of the group.
In general g and g0 might belong to different conjugacy classes: in this case the map defines
an outer automorphism.4
3The only other 24-element group that has representation of the same dimensions is the octahedral group
O (which is isomorphic to S4). In this case, however, the product of two doublet reps does not contain a
triplet [81].
4For details concerning the group of outer and inner automorphisms, Out(G) and Inn(G), see [79, 82].
It is worth noticing that the matrices Xr are defined up to an arbitrary global phase.
Indeed, without loss of generality, for each matrix Xr, one can define different phases
r for different irreducible representations and moreover one can define Xr up to a group
transformation (change of basis): in fact the consistency conditions in eq. (2.2) are invariant
under Xr ei r Xr and Xr r(g)Xr with g T 0.
It proves convenient to use the freedom associated with the arbitrary phases r to
define the generalised CP transformation for which the vev alignments of the flavon fields
can be chosen to be all real. We will show later on that the phases r are not physical and
therefore the results we present are independent from the specific values we assume. In the
context of the T 0 group this choice however helps us to extract a real flavon vev structure
which is a distinctive feature of some models proposed in the literature where the origin of
the physical CP violation arising in the lepton sector is tightly related to the combination
of real vevs, complex CGs5 and eventual phases arising from the requirement of invariance
of the superpotential under the generalised CP transformation.
Before going into details of the computations, let us comment that in the analysis
presented in [79] related to the group T 0, the CP transformations are defined as elements
of the outer automorphism group and are derived up to inner automorphisms of T 0 (up
to conjugacy transformations). In the present work we will consider instead all the
possible transformations including the inner automorphism group and we will discuss all the
convenient CP transformations which can be used to clarify the role of a generalised CP
symmetry in the context of the group T 0.
Transformation properties under generalised CP
We recall that we have the freedom to choose arbitrary phases r, so for instance in
the case of X1, X10 and X100 we are allowed to write the most general CP transformations
for the three inequivalent singlets of T 0 as
5This idea was pioneered in [52].
Differently from the case of the A4 family symmetry discussed in [82] in which one can
show that the generalised CP transformation can be represented as a group
transformation, in the case of T 0 we will show that this is true only for the singlet and the triplet
representations. For the doublets the action of the CP transformation cannot be written
as an action of a group element (i.e. @ g T 0 such that Xr = r(g) for r = 2, 20, 200).
We give a list of all the possible forms of Xr, which can be in general different for each
representation: the CP transformations on the singlets, X1,10,100 , are complex phases, as
mentioned above while the CP transformations on the doublets, X2,20,200 , and the triplets
X3, are given respectively in table 2 and 3. We stress that all the possible forms of Xr are
defined up to a phase, which can be in general different for each representation. Each CP
transformation we found generates a Z2 symmetry.
The generalised CP transformation HCP, acting on the lepton flavour space is given
by, see also [79],
This definition of the CP symmetry is particularly convenient because it acts on the 3- and
1- dimensional representations trivially. This particular transformation however is related
to any other possible CP transformation by a group transformation.
In other words, different choices of CP are related to each other by inner
automorphisms of the group i.e. the CP transformations listed in tables 2 and 3 are related to
each other through a conjugation with a group element. For example, another possible CP
transformation would be
3 conj(T 2)
S 7v S 7 T 2S3 T 2 1 = S2T 2ST ,
T 7v T 2 7conj(T2) T 2T 2 T 2 1 = T 2 .
Without loss of generality we choose as CP transformation the one defined through
eq. (2.6) and from eq. (2.1) using the results of table 2 and table 3 we can write the
representation of the CP transformation acting on the fields as
20 ,
where = ei 2/3, p = ei /12 and p5 = ei /4. Notice that we did not specify the values
of the phases r. Further we can check that the CP symmetry transformation chosen
generates a Z2 symmetry group. Indeed it is easy to show that u2 = E, therefore the
multiplication table of the group HCP = {E, u} is obviously equal to the multiplication
table of a Z2 group, from which we can write HCP = Z2.
200 ,
T S, S2T S
T ST 2, S2T ST 2
S2T ST , T ST
S3T S, ST S
S2T 2S, T 2S
T 2ST , S2T 2ST
1 0 0
0 2 0
0 0
1 0 0
0 1 0
0 0 1
1/3 2/3 2/3 2
2/3 1/3 2/3 2
2/3 2/3 1/3 2
1/3 2/3 2/3
2/3 1/3 2/3
2/3 2/3 1/3
1/3 2/3 2 2/3
2/3 1/3 2 2/3
2/3 2/3 2 1/3
1/3 2/3 2 2/3
2/3 2 1/3 2/3
2/3 2/3 1/3 2
1/3 2/3 2/3
22//332 21//332 2/13/32
1/3 2/3 2/3 2
2/3 2 1/3 2/3
2/3 2/3 2 1/3
42 + 4 + 1 22 2 + 4 22 2 + 4
19 22 + 4 2 42 + + 4 22 + 4 2
42 2 2 42 2 2 2 + 4 + 4
1/3 2/3 2/3 2
2/3 1/3 2 2/3
2/3 2 2/3 1/3
1/3 2/3 2 2/3
2/3 1/3 2/3 2
2/3 2 2/3 1/3
0
0 ei/4
2q5 !
Since we want to have real flavon vevs following the setup given in [68] it turns out
to be convenient to select the CP transformations with 1 = 10 = 100 = 3 = 0 and 200 =
20 = /4.6 With this choice the phases of the couplings of renormalisable operators is
fixed up to a sign by the CP symmetry. In fact, supposing one has a renormalisable operator
of the form O = (A B C) where is the coupling constant and A, B, C represent
the fields, then the generalised CP phase of the operator is defined as CP[O]/O. The
phase of is hence given by the equation = which is solved by
(arg() = arg()/2 or arg()/2
arg() = arg()/2 or arg()/2 +
10 10 ,
20 ,
100 100 ,
3 3 ,
200 ,
where we have again skipped the 2 representation because we will not need it later on.
6Since in our model later on we do not have fields in a 2 representation of T 0 the phase 2 is irrelevant
in our further discussion and we do not fix its value. A possible convenient choice might be 2 = 0 which
makes the mass term of a two-dimensional representation real.
[(200 200)3 3]1
[(2 20)3 3]1
[(3 3)1 1]1
[(3 3)10 100]1
[(3 3)100 10]1
[(1 1)1 1]1
[(10 10)100 10]1
[(100 100)10 100]1
[(10 100)1 1]1
Conditions to violate physical CP
In this section we try to clarify the origin of the phases entering the Lagrangian after T 0
breaking which are then responsible for physical CP violation. We will use the choice of
the r discussed in the previous section, i.e. 1 = 10 = 100 = 3 = 0 and 200 = 20 = /4.
We already know that the singlets and triplets do not introduce CP violation, see
also [79]. Therefore we only want to consider the doublets. Suppose we couple the doublet
flavons i to an operator Or containing matter fields and transforming in the representation
r of T 0. This means that the superpotential contains the operator
In order to obtain a singlet, the flavons (the doublets) have to be contracted to the
representation r which is the complex conjugate representation of r.
[20 (200 3)200 ]1
[200 (20 3)20 ]1
[(20 200)3 3]1
[(20 20)3 3]1
[(200 200)3 3]1
[(3 3)1 1]1
[(3 3)10 100]1
[(3 3)100 10]1
[(1 1)1 1]1
[(10 10)100 10]1
[(100 100)10 100]1
[(10 100)1 1]1
i
i
1
1
i
1
1
1
1
1
1
1
1
1
1
If the operator Or by itself conserves physical CP by which we mean that it does
not introduce any complex phases into the Lagrangian including the associated coupling
constant the only possible source of CP violation is coming from the doublet vevs and
the complex CG factors appearing in the contraction with the operator and the doublets.
For illustrative purpose we want to discuss this explicitly if we have two doublets 0 20
and 00 200.
For r = 1 there is only one possible combination using only 0 and 00 which is
(0 00)1. Using the tensor products of T 0 see for example [68] we find that the
combination is real if the vevs fulfill the following conditions
with X1, X2, Y1, Y2, and real parameters. For r = 10 and r = 100 the only possible
contractions (0 0)100 and (00 00)10 vanish due to the antisymmetry of the contraction.
For r = 1 there are three possible contractions. Either a flavon with itself or both
flavons together.
For the selfcontractions (0 0)3 and (00 00)3 the Lagrangian will not contain a
phase if the flavon fields 0 and 00 have the following structure
!
with X1, X2, Y1, Y2 real and = /4.
These alignments would conserve CP for sure only if the model contains only the
contractions (0 0)3, (00 00)3 and (0 00)1. Adding the contraction (0 00)3 would add
a phase to the Yukawa matrix resulting possibly in physical CP violation.
The model
work where we just want to discuss the connection of a T 0 family symmetry with CP and
illustrate it by a toy model which is nevertheless in full agreement with experimental data.
In this section we will only discuss the effective operators generated after
integrating out the heavy messenger fields. The full renormalisable superpotential including the
messenger fields is given in appendix B.
The flavon sector
We will start the discussion of the model with the flavon sector which is self-contained.
How the flavons couple to the matter sector will be discussed afterwards.
The model contains 14 flavon fields in 1-, 2- and 3-dimensional representations of T 0
and 5 auxiliary flavons in 1-dimensional representations. Before we will discuss the
superpotential which fixes the directions and phases of the flavon vevs we will first define them.
We have four flavons in the 3-dimensional representation of T 0 pointing in the directions
The first three flavons will be used in the charged lepton sector and the fourth one couples
only to the neutrino sector. These flavon vevs, like all the other flavon vevs, are real.
Further we introduce three doublets of T 0: 0 20, 00 200 and 00 200. We recall
that the doublets are the only representations of the family group T 0 which introduce
phases, due to the complexity of the Clebsh-Gordan coefficients. For the doublets we will
find the alignments
And finally, we introduce 7 flavon fields in one-dimensional representations of the family
group. In particular, we have (the primes indicate the types of singlet)
The and couple only to the neutrino sector while the other one-dimensional flavons
couple only to the charged lepton sector. Also the five auxiliary flavons i, i = 1, . . . , 5 get
real vevs which we do not label here explicitly.
The flavon quantum numbers are summarized in table 6. In this table we have also
included the five auxiliary flavon fields i which are only needed to fix the phases of the
other flavon vevs and all acquire real vevs by themselves.
We discuss now the superpotential in the flavon sector which aligns the flavon vevs.
We will use so-called F -term alignment where the vevs are determined from the F -term
conditions of the driving fields. The driving fields are listed with their quantum numbers
in table 7, where we have indicated for simplicity P = S, S, S and Si, with i = 1, . . . , 5,
because they have all the same quantum numbers under the whole symmetry group.
Z8
2
2
5
0
3
7
1
5
4
2
2
0
0
0
4
4
4
0
0
Z8
0
0
6
2
2
6
4
6
0
4
0
Z4
0
0
0
2
2
2
2
0
0
0
0
0
2
2
1
2
2
0
0
Z4
2
3
0
0
0
0
2
0
0
0
0
Z4
0
2
3
2
3
1
3
1
2
2
0
0
2
2
0
2
0
0
0
Z4
0
0
2
2
2
2
2
2
0
0
0
Z3
1
0
2
0
2
2
0
2
0
0
1
0
0
0
0
0
0
1
2
Z3
1
0
1
2
2
0
0
2
0
0
0
Z3
0
1
0
0
0
0
1
0
0
1
0
1
0
0
0
0
0
1
2
Z3
0
1
2
0
0
1
0
0
0
1
0
Z2
1
0
0
0
0
1
1
1
1
0
1
0
0
0
0
1
0
0
0
Z2
0
0
0
0
0
0
1
0
0
0
0
|FP |2 =
= 0.
which gives for the phase of the flavon vev
arg(h i) =
q = 1, . . . , n for in eq. (3.5),
q = 1, . . . , n for + in eq. (3.5).
This method will be used to fix the phases of the singlet and triplet flavon vevs (including
the i). Note that we have to introduce for every phase we fix in this way a P field and
only after a suitable choice of basis for this fields we end up with the simple structure we
show later, see also the appendix of [70]. For the directions of the triplets we use standard
expressions, cf. also the previous paper [68].
For the doublets, nevertheless, we use here a different method. Take for example the
term D h(00)2 0i. The F -term equations read
|FD1 | = (200)2 3 0 = 0 ,
|FD1 | = i(100)2 2 0 = 0 ,
|FD1 | = (1 i)100 200 1 0 = 0 .
+ D h 00 2 0i + D i 0 0 + 0 + D
+ S + + S21 14 M41
Hd
2
-1
1
0
7
2
2
0
1
0
We will not go through all the details and discuss each F -term condition but this potential
is minimized by the vacuum structure as in eqs. (3.1), (3.2) and (3.3). Finally, we want
to remark that the F -term equations do not fix the phase of the field . However, the
phase of this field will turn out to be unphysical because it can be canceled out through
an unphysical unitary transformation of the right-handed charged lepton fields as we will
show later explicitly.
The matter sector
Since we have discussed now the symmetry breaking flavon fields we will now proceed with
the discussion on how these fields couple to the matter sector and generate the Yukawa
couplings and right-handed Majorana neutrino masses.
The model contains three generations of lepton fields, the left-handed SU(2)L doublets
are organized in a triplet representation of T 0, the first two families of right-handed charged
lepton fields are organized in a two dimensional representation, 200, and the third family
sits in a 100. There are two Higgs doublets as usual in supersymmetric models. They are
both singlets, 1 under T 0. The model includes three heavy right-handed Majorana neutrino
fields N , which are organized in a triplet. The light active neutrino masses are generated
through the type I seesaw mechanism [6065]. At leading order tri-bimaximal mixing
(TBM) is predicted in the the neutrino sector which is corrected by the charged lepton
sector allowing a realistic fit of the measured parameters of the PMNS mixing matrix. The
quantum numbers of the matter fields are summarized in table 8.
In this work we use the right-left convention for the Yukawa matrices
L (Ye)ij eR i eL j Hd + H.c. ,
i.e. there exists a unitary matrix Ue which diagonalizes the product Ye Ye and contributes
to the physical PMNS mixing matrix.
The charged lepton sector
The Yukawa matrix Ye is generated after the flavons acquire their vevs and T 0 is broken.
The effective superpotential describing the couplings of the matter sector to the flavon
sector is given by
+ y2(e2) E 0 1 Hd (L )1 + y2(e13) E 0 1 Hd 0 00 3 L 1
2
a 0 0 a 0 0
Ye = i0b d +c i k 0e i0b eci 0e ,
from which it is clear that an eventual phase of 0 drops out in the physical combination
YeYe and we can choose the parameters in the Yukawa matrix to be real.
We remind that there are in principle three possible sources of complex phases which
can lead to physical CP violation: complex vevs, complex couplings whose phases are
determined by the invariance under the generalised CP symmetry and complex CG coefficients.
In our model all vevs are real due to our flavon alignment and the convenient choice of the
r phases.
Then the (physical) phases in Ye are completely induced by the complex couplings
and complex CG coefficients. In fact the insights we have gained before in section 2.3 can
be used here. The phase in the 1-1 element is unphysical (it drops out in the combination
Ye Ye. So the physical CP violation is to leading order given by the phases of the ratios
(Ye)21/(Ye)22 and (Ye)32/(Ye)33. Let us study for illustration the second ratio which has
two components, one with a non-trivial relative phase and one without. The real ratio d/e
is coming from the operators with the coefficients y3(e2) and y3(e3) and from the viewpoint of
T 0 o HCP there is not really any difference between the two because we have only added a
singlet which cannot break CP in our setup as we said before.
The neutrino sector
The neutrino sector is constructed using a superpotential similar to that used in [68]: the
light neutrino masses are generated through the type I see-saw mechanism, i.e. introducing
right-handed heavy Majorana states which are accommodated in a triplet under T 0. We
have the effective superpotential
The right-handed neutrino mass matrix MR is diagonalised by the TBM matrix [3438]
Since X and Z are real parameters, the phases 1, 2 and 3 take values 0 or . A light
neutrino Majorana mass term is generated after electroweak symmetry breaking via the
type I see-saw mechanism:
M = MDT MR1 MD = U Diag (m1, m2, m3) U ,
U = i UTBM Diag ei 1/2, ei 2/2, ei 3/2
Diag ei 1/2, ei 2/2, ei 3/2 ,
= Diag M1 ei 1 , M2 ei 2 , M3 ei 3 , M1,2,3 > 0 .
and m1,2,3 > 0 are the light neutrino masses,
mi =
, i = 1, 2, 3 .
The phase factor i in eq. (3.21) corresponds to an unphysical phase and we will drop it in
what follows. Note also that one of the phases k, say 1, is physically irrelevant since it
can be considered as a common phase of the neutrino mixing matrix. In the following we
will always set 1 = 0. This corresponds to the choice (X + 3Z) > 0.
At this point we want to comment on the role of the phases r appearing in the definition of
the CP transformation in eq. (2.11). These phases are arbitrary and hence they should not
contribute to physical observables. This means, for instance, that these arbitrary phases
must not appear in the Yukawa matrices after T 0 is broken. However it is not enough to
look at the Yukawa couplings alone but one also has to study the flavon vacuum alignment
sector. We want to show next a simple example for which, as expected, these phases turn
out to be unphysical.
In order to show this we consider as example (Ye)22 and (Ye)21 respectively generated
by the following operators:
The fields together with their charges have been defined before in table 6. We will now
be more explicit and consider all the possible phases arising in each of the given operators
under the CP transformation of eq. (2.9) where the r were included explicitly. For each
flavon vev in the operators we will denote the arising phase with a bar correspondingly,
i.e. for the vev of the flavon we will have ei 0 where 0 is the modulus of the vev.
Then using the transformations in eq. (2.9) and table 2.9 we get
arg ((Ye)22 (Ye)21) = 0 + (1 10)/2 .
arg ((Ye)22 (Ye)21) = ,
the phases 1, 10 and 3 cancel out.
This shows how the r cancel out in a complete model and become unphysical.
Including them only in one sector, for instance, in the Yukawa sector they might appear to be
physical and only after considering also the flavon alignment sector it can be shown that
they are unphysical which is nevertheless quite cumbersome in a realistic model due to the
many fields and couplings involved.
Geometrical CP violation and residual symmetries
In this section we want to provide a better understanding of the quality of symmetry
breaking our model exhibits. To be more precise we will argue that our model breaks CP in
a geometrical fashion and then we will discuss the residual symmetries of the mass matrices.
Geometrical CP violation was first defined in [67] and there it is tightly related to the
so-called calculable phases which are phases of flavon vevs which do not depend on the
parameters of the potential but only on the geometry of the potential. This applies also
to our model. All complex phases are determined in the end by the (discrete) symmetry
group of our model. In particular the symmetries T 0 o HCP and the Zn factors play a
crucial role here. For the singlets and triplets in fact the Zn symmetries (in combination
with CP) make the phases calculable using the discrete vacuum alignment technique [70].
For the doublets then the symmetry T 0 o HCP enters via fixing the phases of the couplings
and fixing relative phases between different components of the multiplets. In particular,
all flavon vevs are left invariant under the generalised CP symmetry and hence protected
by it. However the calculable phases are necessary but not sufficient for geometrical CP
violation. For this we have to see if CP is broken or not.
For this we will have a look at the residual symmetries of the mass matrices after
T 0 o HCP is broken. First of all, we observe that the vev structure mentioned in section 3
gives a breaking pattern which is different in the neutrino and in the charged lepton sector,
i.e. the residual groups G and Ge are different.
In the charged lepton sector the group T 0 is fully broken by the singlet, doublet and
triplet vevs. If it exists, the residual group in the charged lepton sector is defined through
the elements which leave invariant the flavon vevs and satisfy
Xe =
with gei Ge < T 0 o HCP ,
Xe Ge < T 0 o HCP .
The first condition is the ordinary condition to study residual symmetries while the second
one is relevant only for models with spontaneous CP violation.
In our model this conditions are not satisfied for any (g) or Xe. Hence there is
no residual symmetry group in the charged lepton sector and even more CP is broken
spontaneously. Together with the fact that all our phases are determined by symmetries
(up to signs and discrete choices) we have demonstrated now that our model exhibits
geometrical CP violation.
In the neutrino sector we can write similar relations that take into account the
symmetrical structure of the Majorana mass matrix, and in particular as before the residual
symmetry is defined through the elements which leave invariant the flavon vevs and satisfy
In our model M is a real matrix and therefore (gi ) and X are defined through the same
conditions. Defining O as the orthogonal matrix which diagonalizes the real symmetric
(O XT OT ) Mdiag (O X OT ) = Mdiag
D =
1/3 1/3 1/3
O = 0 1/2 1/2 ,
p2/3 1/6 1/6
1 0 0
T S T 2 = OT 0 1 0 O ,
0 0 1
which also leaves invariant the vev structure. This symmetry is a Z2 symmetry. In summary
the residual symmetry in the neutrino sector is a Klein group K4 = Z2 Z2, in which one
Z2 comes from HCP and the other one from T 0. HCP is conserved because in the neutrino
sector X can be chosen as the identity matrix and M is real.
Combining the two we find
Gf T 0 o HCP T 0 o Z2
(Ge = ,
so that T 0 o HCP is completely broken and there is no residual symmetry left.
Absolute neutrino mass scale
Before we consider the mixing angles and phases in the PMNS matrix we first will discuss
the neutrino spectra predicted by the model. We get the same results as in [68] because
our neutrino mass matrix has exactly the same structure. The forms of the Dirac and
Majorana mass terms given in eq. (3.16) imply that in the model considered by us both
light neutrino mass spectra with normal ordering (NO) and with inverted ordering (IO)
are allowed (see also [68]). In total three different spectra for the light active neutrinos
are possible. They correspond to the different choices of the values of the phases i in
eq. (3.21). More specifically, the cases 1 = 2 = 3 = 0 and 1 = 2 = 0 and 3 =
correspond to NO spectra of the type A and B, respectively. For 1 = 2 = 0 and 3 =
also IO spectrum is possible. The neutrino masses in cases of the three spectra are given by:
NO spectrum B : (m1, m2, m3) = (5.87, 10.48, 48.88) 103 eV ,
IO spectrum : (m1, m2, m3) = (51.53, 52.26, 17.34) 103 eV ,
where we have used the best fit values of m221 and |m231(32)| given in table 1. Employing
the 3 allowed ranges of values of the two neutrino mass squared differences quoted in
table 1, we find the intervals in which m1,2,3 can vary:
NO spectrum A:
NO spectrum B:
IO spectrum:
Correspondingly, we get for the sum of the neutrino masses:
where we have given the predictions using the best fit values and the 3 intervals of the
allowed values of m1, m2 and m3 quoted above.
The mixing angles and Dirac CPV phase
We will derive next expressions for the mixing angles and the CPV phases in the standard
parametrisation of the PMNS matrix in terms of the parameters of the model. The
expression for the charged lepton mass matrix Ye given in eq. (3.13) contains altogether seven
parameters: five real parameters and two phases, one of which is equal to /2. Three
(combinations of) parameters are determined by the three charged lepton masses. The
remaining two real parameters and two phases are related to two angles and two phases
in the matrix Ue which diagonalises the product Ye Ye and enters into the expression of
the PMNS matrix: UPMNS = UeU , where U is of TBM form (see eq. (3.19)), while
Ue R23R12, R23 and R12 are orthogonal matrices describing rotations in the 2-3 and 1-2
planes, respectively. It proves convenient to adopt for the matrices Ue and U the notation
used in [17]:
(Ue = e R231 (2e3) R121 (1e2)
R23 (2e3) = 00 cosisn2e32e3 csoins 2e2e33 .
Using the expression for the charged lepton mass matrix Ye given in eq. (3.13) and
comparing the right and the left sides of the equation
Ye Ye = Uediag me2, m2, m2 Ue ,
we find that me2 = a2, m2 = c2 and m2 = e2. For Ue given in eq. (3.41) this equality holds
only under the condition that sin 1e2 and sin 2e3 are sufficiently small. Using the leading
terms in powers of the small parameters sin 1e2 and sin 2e3 we get the approximate relations:
sin 1e2 ei e ' i cb , sin 2e3 ei(ee) = c2 ee2 ei ' e ei e , (3.44)
where e = /2, e = e e, e [0, 2], 1e2 ' |b/c| and 2e3 ' |/e|.
In the discussion that follows 1e2, 2e3, e and e are treated as arbitrary angles and
phases, i.e., no assumption about their magnitude is made.
The lepton mixing we obtain in the model we have constructed, including the Dirac
CPV phase but not the Majorana CPV phases, was investigated in detail on
general phenomenological grounds in ref. [17] and we will use the results obtained in [17].
The three angles 12, 23 and 13 and the Dirac and Majorana CPV phases and
1 and 2 (see eqs. (1.1)(1.3)), of the PMNS mixing matrix UPMNS = UeU =
R12(1e2)R23(2e3)eR23(23)R12(12) , can be expressed as functions of the two real
angles, 1e2 and 2e3, and the two phases, e and e present in Ue. However, as was shown
in [17], the three angles 12, 23 and 13 and the Dirac phase are expressed in terms of
the angle 1e2, an angle 23 and just one phase , where
R23 (2e3) e R23 (23) = P R23 23 Q .
= arg e i e cos 2e3 + e i e sin 2e3 ,
= arg e i e cos 2e3 + e i e sin 2e3 . (3.48)
The phase is unphysical (it can be absorbed in the lepton field). The phase contributes
to the matrix of physical Majorana phases, which now is equal to Q = Q . The PMNS
matrix takes the form:
UPMNS = R12(1e2) () R23 23 R12(12) Q ,
where 12 = sin1 1/3 . Thus, the four observables 12, 23, 13 and are functions of
three parameters 1e2, 23 and . As a consequence, the Dirac phase can be expressed
as a function of the three PMNS angles 12, 23 and 13, leading to a new sum rule
relating and 12, 23 and 13 [17]. Using the measured values of 12, 23 and 13, the
authors of [17] obtained predictions for the values of and of the rephasing invariant
JCP = Im Ue1U3Ue3U1 , which controls the magnitude of CP violating effects in neutrino
oscillations [84], as well as for the 2 and 3 ranges of allowed values of sin 12, sin 23 and
sin 13. These predictions are valid also in the model under discussion.
To be more specific, using eq. (3.49) we get for the angles 12, 23 and 13 of the
standard parametrisation of UPMNS [17]:
sin2 23 sin2 13 , cos2 23 = cos2 23
1 sin2 13 1 sin2 13
2 + 2 sin 223 sin 13 cos sin2 23 !
1 cos2 23 cos2 13
where the first relation sin 13 = sin 1e2 sin 23 was used in order to obtain the expressions
for sin2 23 and sin2 12. Clearly, the angle 23 differs little from the atmospheric neutrino
mixing angle 23. For sin2 13 = 0.024 and sin2 23 = 0.39 we have sin 1e2 = 0.2. Comparing
the imaginary and real parts of Ue1U3Ue3U1, obtained using eq. (3.49) and the standard
parametrisation of UPMNS, one gets the following relation between the phase and the
Dirac phase [17]:
These values correspond to
Thus, our model predicts ' /2 or 3/2. The fact that the value of the Dirac CPV
phase is determined (up to an ambiguity of the sign of sin ) by the values of the three
mixing angles 12, 23 and 13 of the PMNS matrix, (3.53), is the most striking prediction
of the model considered. For the best fit values of 12, 23 and 13 we get = /2 or
3/2. These result implies also that in the model under discussion, the JCP factor, which
determines the magnitude of CP violation in neutrino oscillations, is also a function of the
three angles 12, 23 and 13 of the PMNS matrix:
This allows to obtain predictions for the range of possible values of JCP using the current
data on sin2 12, sin2 23 and sin2 13. For the best fit values of these parameters (see table 1)
we find: JCP ' 0.034.
The quoted results on and JCP were obtained first on the basis of a phenomenological
analysis in [17]. Here they are obtained for the first time within a selfconsistent model of
lepton flavour based on the T 0 family symmetry.
In [17] the authors performed a detailed statistical analysis which permitted to
determine the ranges of allowed values of sin2 12, sin2 23, sin 13, and JCP at a given
confidence level. We quote below some of the results obtained in [17], which are valid also
in the model constructed by us.
Most importantly, the CP conserving values of = 0; ; 2 are excluded with respect to
the best fit CP violating values = /2; 3/2 at more than 4. Correspondingly, JCP = 0
is also excluded with respect to the best-fit values JCP ' (0.034) and JCP ' 0.034 at
more than 4. Further, the 3 allowed ranges of values of both and JCP form rather
narrow intervals. In the case of the best fit value = 3/2, for instance, we have in the
cases of NO and IO spectra:
NO : JCP = 0.034 ,
IO : JCP = 0.034 ,
0.028 < JCP < 0.039 ,
0.039 < JCP < 0.028 ,
0.027 < JCP < 0.039 ,
0.039 < JCP < 0.026 ,
where we have quoted the best fit value of JCP as well. The positive values are related to
the 2 minimum at = /2.
7Due to the slight difference between the best fit values of sin2 23 and sin 13 in the cases of NO and
IO spectra (see table 1), the values we obtain for cos in the two cases differ somewhat. However, this
difference is equal to 104 in absolute value and we will neglect it in what follows.
The preceding results and discussion are illustrated qualitatively in figure 1, where
we show the correlation between the value of sin and JCP for the 1 and 2 ranges of
allowed values of sin2 12, sin2 23 and sin2 13, which were taken from table 1. The figure
was produced assuming flat distribution of the values of sin2 12, sin2 23 and sin2 13 in the
quoted intervals around the corresponding best fit values. As can be seen from figure 1,
the predicted values of both sin and JCP thus obtained form rather narrow intervals.8
As it follows from table 1, the angle 23 is determined using the current neutrino
oscillation data with largest uncertainty. We give next the values of the Dirac phase for
two values of sin2 23 from its 3 allowed range, sin2 23 = 0.50 and 0.60, and for the best
fit values of sin2 12 and sin2 13:
sin2 23 = 0.50 : cos = 0.123 , = 97.09
or 262.91 ;
These results show that | sin |, which determines the magnitude of the CP violation effects
in neutrino oscillations, exhibits very weak dependence on the value of sin2 23: for any
value of sin2 23 from the interval 0.39 sin2 23 0.60 we get | sin | 0.98.
The predictions of the model for and JCP will be tested in the experiments searching
for CP violation in neutrino oscillations, which will provide information on the value of the
Dirac phase .
The Majorana CPV phases
Using the expressions for the angles 1e2 and 23 and for cos in terms of sin 13, sin 12 and
sin 23 and the best fit values of sin 13, sin 12 and sin 23, we can calculate the numerical
form of UPMNS from which we can extract the values of the physical CPV Majorana phases.
We follow the procedure described in [85]. Obviously, there are two such forms of UPMNS
corresponding to the two possible values of . In the case of = 266.02 and ' 102.55
we find:
0.517
Recasting this expression in the form of the standard parametrisation of UPMNS we get:
0.822 0.547
UPMNS = P 0.436 ei 169.41 0.658 ei 4.65
0.365 ei 16.04 0.517 ei 172.53
8The 2 ranges of allowed values of sin and JCP shown in figure 1 match approximately the 3 ranges
of allowed values of sin and JCP obtained in [17] by performing a more rigorous statistical (2) analysis.
0.517
0.155 e i 93.98
where P The
phases in the matrices P and P can be absorbed by the charged lepton fields and are
unphysical. In contrast, the phases in the matrices Q2 and Q2 contribute to the physical
Majorana phases. We can finally write the Majorana phase matrix in the parametrization
given in (1.1) (1 = 0):
21 = 16.04 23 ,
22 = 7.47 3 2 2 , sin = 0.976 .
It is clear from eq. (3.71) that the value of cos can be determined knowing the values of
cos and sin 23, independently of the values of 2e3 and e. This, obviously, allows to find
also | sin |, but not the sign of sin . In the case of sin 2e3 sin e 1 of interest, eq. (3.72)
allows to correlate the sign of sin with the sign of sin and thus to determine for a
given : we have sin < 0 if sin > 0, and sin > 0 for sin < 0. Thus, for = 102.5530
(corresponding to = 266.02) we find = 105.4118 and = = 207.9648 =
(180 + 27.9648), while for = 102.5530 (corresponding to = 93.98) we obtain
= + 105.4118 and = +207.96 = + (180 + 27.96).
The results thus derived allow us to calculate numerically the Majorana CPV phases.
For the best fit values of the neutrino mixing angles we get:
for = 102.55( = 93.98) ; (3.73)
1 = (23.84 360 + 3) = (23.84 + 360 + 3) ,
2 = (70.883602 +3) = (70.88+3602 +3) for = 102.55( = 266.02) , (3.74)
where we have used the fact that 1(2) and 1(2) + 4 lead to the same physical results. In
the cases of the three types of neutrino mass spectrum allowed by the model, which are
characterised, in particular, by specific values of the 2 and 3 we find:
NO A spectrum, i.e., 2 = 3 = 0:
1 = (23.84 + 360), 2 = (70.88 + 360) for = 102.55( = 93.98) ,
1 = (23.84 + 360), 2 = (70.88 + 360) for = 102.55( = 266.02) ; (3.75)
NO B spectrum, i.e., 2 = 0 and 3 = :
IO spectrum, 3 = 0 and 2 = :
The neutrinoless double beta decay effective Majorana mass
Knowing the values of the neutrino masses and the Majorana and Dirac CPV phases we
can derive predictions for the neutrinoless double beta (()0 -) decay effective Majorana
mass |hmi| (see, e.g., [86, 87]). Since |hmi| depends only on the cosines of the CPV phases,
we get the same result for = + 102.55 ( = 266.02) and = 102.55 ( = 93.98).
Thus, for = 102.55, using the best fit values of the neutrino mixing angles,
we obtain:
|hmi| = 4.88 103 eV ,
|hmi| = 7.30 103 eV ,
|hmi| = 26.34 103 eV ,
NO A spectrum ;
NO B spectrum ;
IO spectrum .
In figure 2 we show the general phenomenologically allowed 3 range of values of |hmi|
for the NO (blue area) and IO (red area) neutrino mass spectra as a function of the lightest
neutrino mass. The values of |hmi| quoted above and corresponding to the three types of
neutrino mass spectrum (NO A, NO B and IO), predicted by the model constructed in the
present article, are indicated with black crosses. The vertical lines in figure 2 correspond
to mmin = 8.6 104 eV and 1.0 102 eV; for a given value of mmin from the interval
determined by these two values, [8.6 104, 1.0 102] eV, one can have |hmi| = 0 for
specific values of the Majorana CPV phases.
Limiting cases
i) sin2 12 = 13 + 31 sin2 13 31
1 1 1
ii) sin2 23 = 2 2 sin2 13 2
where we have written the corrections in terms of 13. Both cases could be realised by
choosing a certain set of messengers. If we remove the messenger pair 2A0 , 2A00 , our model
would correspond to the case i), while if we remove the messenger pair 1C00 , 1C0 , the model
would correspond to the case ii). The model we have constructed, which includes both
messenger pairs, gives a somewhat better description of the current data on the neutrino
mixing angles. This brief discussion shows how important the messenger sector can be for
getting meaningful predictions.
Summary and conclusions
In this work we have analyzed the presence of a generalised CP symmetry, HCP, combined
with the non-Abelian discrete group T 0 in the lepton flavour space, i.e. the possibility of
the existence of a symmetry group Gf = T 0 o HCP acting among the three generations of
charged leptons and neutrinos. The phenomenological implications of the breaking of such
a symmetry group both in the charged lepton and neutrino sectors are thus explored
especially in connection with the CP violation appearing in the leptonic mixing matrix, UPMNS.
First of all we have derived in section 2 all the possible generalised CP transformations
for all the representations of the T 0 group i.e. we found all possible outer automorphisms
of the group T 0 following the consistency conditions given in [79, 80, 82]. We have
chosen as generalised CP symmetry the transformation u : (T, S) (T 2, S2T 2S T ) which
corresponds to a Z2 symmetry and it is defined up to an inner automorphism. The
transformation u is particularly convenient since, in the basis chosen for the generators S and
T , for the 1 and 3-dimensional representations it is trivially defined as the identity up to
a global unphysical phase r where the index r refers to the representation. More
importantly we found that, given this specific generalised CP symmetry combined with T 0, it is
possible to fix the vevs of the flavon fields to real values in such a way that no complex
phases, and thus no physical CP violation, stem from the vevs themselves.
Moreover, for a list of possible renormalisable operators, namely O = (A B C)
where is the coupling constant and A, B, C are fields, we derived the constraints on
the phase under the assumption of invariance under the generalised CP transformation.
This list of possible operators can be used to construct a CP-conserving renormalisable
superpotential for the flavon sector and therefore can be used in order to show that real
vev structures can be achieved.
Motivated by this preliminary study we constructed in section 3 a supersymmetric
flavour model able to describe the observed patterns and mixing for three generations of
charged lepton fields and the three light active neutrinos.
We have constructed an effective superpotential with operators up to mass dimension
six giving the charged lepton and neutrino Yukawa couplings and the Majorana mass term
for the RH neutrinos. Naturally small neutrino masses are generated by the type I see-saw
mechanism. At leading order, the mixing in the neutrino sector is described by the
tribimaximal mixing, which is then perturbed by additional contributions coming from the
charged lepton sector. The latter are responsible for the compatibility of the predictions
on the mixing angles with the experimental values and, in particular, with the non-zero
value of the reactor mixing angle 13.
Similarly to what was found in [68], we find that both types of neutrino mass spectrum
with normal ordering (NO) and inverted ordering (IO) are possible within the model
and that the NO spectrum can be of two varieties, A and B. They differ by the value of the
lightest neutrino mass. Only one spectrum of the IO type is compatible with the model. For
each of the three neutrino mass spectra, NO A, NO B and IO, the absolute scale of neutrino
masses is predicted with relatively small uncertainty. This allows us to predict the value
of the sum of the neutrino masses for the three spectra. The Dirac phase is predicted to
be approximately = /2 or 3/2. More concretely, for the best fit values of the neutrino
mixing angles quoted in table 1 we get = 93.98 or = 266.02. The deviations of from
the values 90 and 270 are correlated with the deviation of atmospheric neutrino mixing
angle 23 from /4. Thus, the CP violating effects in neutrino oscillations are predicted
to be nearly maximal (given the values of the neutrino mixing angles) and experimentally
observable. The values of the Majorana CPV phases are also predicted by the model. This
allows us to predict the neutrinoless double beta decay effective Majorana mass in each of
the three cases of neutrino mass spectrum allowed by the model, NO A, NO B and IO.
The predictions of the model can be tested in ongoing and future planned i) accelerators
experiments searching for CP violation in neutrino oscillations (T2K, NOA, etc.), ii)
experiments aiming to determine the absolute neutrino mass scale, and iii) experiments
searching for neutrinoless double beta decay.
It is important to comment that in this model the physical CP violation emerging
in the PMNS mixing matrix stems only from the charged lepton sector. Indeed, in the
neutrino sector the Majorana mass matrix and the Dirac Yukawa couplings are real and
the CP violation is caused by the complex CP violating phases arising in the charged lepton
sector. The presence of the latter is a consequence of the requirement of invariance of the
theory under the generalised CP symmetry at the fundamental level and of the complex
CGs of the T 0 group.
We also found that the residual group in the charged lepton sector is trivial i.e. Ge =
and since the phases of the flavon vevs are completely independent of the coupling constants
of the flavon superpotential, the CP symmetry is broken geometrically (according to the
definition of geometrical CP violation given in [67]). In the neutrino sector, the residual
subgroup is instead a Klein group, G = K4 = Z2 Z2 with one Z2 coming from the
generalised CP symmetry HCP.
Concluding, we have shown that the spontaneous breaking of a symmetry group Gf =
T 0 o HCP in the leptonic sector through a real flavon vev structure is possible and, at
the same time, CP violation in the leptonic sector can take place. In this scenario the
appearance of the CP violating phases in the PMNS mixing matrix can be traced to two
factors: i) the requirement of invariance of the Lagrangian of the theory under HCP at
the fundamental level, and ii) the complex CGs of the T 0 group. The model we have
constructed allows for two neutrino mass spectra with normal ordering (NO) and one
with inverted ordering (IO). For each of the three spectra the absolute scale of neutrino
masses is predicted with relatively small uncertainty. The value of the Dirac CP violation
(CPV) phase in the lepton mixing matrix is predicted to be = /2 or 3/2. Thus,
the CP violating effects in neutrino oscillations are predicted to be nearly maximal and
experimentally observable. We present also predictions for the sum of the neutrino masses,
for the Majorana CPV phases and for the effective Majorana mass in neutrinoless double
beta decay. The predictions of the model can be tested in a variety of ongoing and future
planned neutrino experiments.
Note added. After the submission of our article to the arXiv, an update of the global fits
to the neutrino oscillation data appeared [90]. The results reported in [90] are in agreement
with the predictions of our model. More specifically, the authors of [90] find that the best
fit value of is 3/2, which is one of the two possible values predicted by in our model.
Similar results on were obtained in the global analysis of the neutrino oscillation data
performed in [91].
The work of M. Spinrath was partially supported by the ERC Advanced Grant no. 267985
DaMESyFla, by the EU Marie Curie ITN UNILHC (PITN-GA-2009-237920).
A. Meroni acknowledges MIUR (Italy) for financial support under the program Futuro
in Ricerca 2010 (RBFR10O36O). This work was supported in part also by the European
Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442-INVISIBLES),
by the INFN program on Astroparticle Physics(A.M., S.T.P.) and by the World Premier
International Research Center Initiative (WPI Initiative), MEXT, Japan (S.T.P.).
S2 = R
R2 = T 3 = (ST )3 = E
RT = T R .
The number of the unitary irreducible representations of a discrete group is equal to the
number of the conjugacy classes. For T 0 they are seven, which are classified given the
elements T , S, because R S2, we summarize them as
1 C1 : {E} ,
4 C3 : T, S3T S, ST, T S 10 C2 : S2
,
400 C6 : S2T, ST S, S3T, S2T S
,
6 C4 : S, S3, T ST 2, T 2ST, S2T ST 2, S2T 2ST .
The representations of T 0 can be expressed as
40 C3 : T 2, S2T ST, S2T 2S, S3T 2
4000 C6 : S2T 2, T ST, T 2S, ST 2
2 : T =
20 : T =
200 : T =
R =
R =
R =
1 0 0 1 0 0
3 : T = 0 0 , R = 0 1 0 ,
0 0 2 0 0 1
We use the definition of the representation of T 0 given in [4651] in which and p are fixed
to be respectively = e 2 3i and p = e i12 . Finally T 0 has n = 13 subgroups excluding the
whole group:
Trivial subgroup
E = {E} ;
Z2 subgroup
Z4 subgroups
Z6 subgroups
A complete table of the CGs coefficients can be found in [68].
Messenger sector
The effective model we have considered so far contains only non-renormalisable operators
allowed by the symmetry group Gf Z2 Z32 Z42 Z8 U(1)R. But in fact using only this
symmetry there would be more effective operators allowed which might spoil our model
predictions.
Therefore we discuss in in this section we a so-called ultraviolet completion defining
a renormalisable theory which gives the effective model described in the previous sections
after integrating out the heavy messenger superfields. In this way we can justify why we
have chosen only a certain subset of the effective operators allowed by the symmetries. The
quantum numbers of the messenger fields are given in table 9. We label them with ,
and for the charged lepton, neutrino and flavon sector respectively.
For the charged lepton sector we find the renormalisable superpotential Weren
which through the diagrams of figure 3 generates at low energy the non-renormalisable
superpotential WYe of eq. (3.12).
For the neutrino and the flavon sector we obtained similarly to the previous case
Wren = N 2 + N 2 + N 2 + LN 1 + Hu1 + Hu1 ,
Wflavon = D 3B + 3 3B + D 00 3B + D 3C + 1 3C + D 0 3C
ren
The corresponding diagrams that generate the effective operators in the neutrino and flavon
sector in our model are given in figures 4 and 5.
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