a r X i v : 0 8 0 2 . 1 6 9 3 v 1 [ h e p  p h ] 1 2 F e b 2 0 0 8
Embedding
A
4
into
SU
(3)
×
U
(1)
ﬂavor symmetry:Large neutrino mixing and fermion mass hierarchy in
SO
(10)
GUT
F. Bazzocchi
1
, S. Morisi
1
, M. Picariello
2
, E. TorrenteLujan
3
1
Instituto de F´ısica Corpuscular – C.S.I.C./Universitat de Val`encia E–46071 Val`encia, Spain
2
Dipartimento di Fisica  Universit`a del Salento
and
INFN Via Arnesano, ex collegio Fiorini, I–73100 Lecce, Italy and
3
University of Murcia  30100 Murcia  Spain
emails: bazzocchi@iﬁc.uv.es, morisi@iﬁc.uv.es, Marco.Picariello@le.infn.it, torrente@cern.chWe present a common explanation of the fermion mass hierarchy and the large lepton mixingangles in the context of a grand uniﬁed ﬂavor and gauge theory (GUTF). Our starting point is a(
SU
(3)
×
U
(1))
F
ﬂavor symmetry and a
SO
(10) GUT, a basic ingredient of our theory which playsa major role is that two diﬀerent breaking pattern of the ﬂavor symmetry are at work. On one side,the dynamical breaking of (
SU
(3)
×
U
(1))
F
ﬂavor symmetry into (
U
(2)
×
Z
3
)
F
explains why onefamily is much heavier than the others. On the other side, an explicit symmetry breaking of
SU
(3)
F
into a discrete ﬂavor symmetry leads to the observed tribimaximal mixing for the leptons. We writean explicit model where this discrete symmetry group is
A
4
. Naturalness of the charged fermionmass hierarchy appears as a consequence of the continuous
SU
(3)
F
symmetry. Moreover, the samediscrete
A
4
GUT invariant operators are the root of the large lepton mixing, small Cabibbo angle,and neutrino masses.
I. INTRODUCTION
Grand Uniﬁed Theory (GUT) [1, 2] are natural extensions of the Standard Model (SM) Indications toward GUT
are the tendency to unify for the gauge couplings, and the possibility to explain charge quantization and anomalycancellation. One of the main features of GUT is its potentiality to unify the particle representations and thefundamental parameters in a hopefully predictive framework.
SO
(10) is the smallest simple Lie group for which asingle anomalyfree irreducible representation (namely the spinor 16 representation) can accommodate the entire SMfermion content of each generation.Flavor physics appears as new extra horizontal symmetries. After the recent experimental evidences about neutrinophysics [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], within the experimental errors, the neutrino mixing matrix is compatible
with the so called tribimaximal matrix [15]
U
TB
=
−
2
/
√
6 1
/
√
3 01
/
√
6 1
/
√
3 1
/
√
21
/
√
6 1
/
√
3
−
1
/
√
2
.
(1)At this stage the parameters both the quark [16] and lepton [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] sectors are
known to a comparable level.To explain at the same moment the charged fermion mass hierarchy and the leptonquark mixing angle hierarchy isan unsolved problem, this is the ﬂavor puzzle. The problem of the mass hierarchy is often addressed by introducingcontinuous ﬂavor symmetries [29, 30]. On the other hand, discrete ﬂavor symmetry such as 23 [31, 32, 33],
S
3[34, 35, 36, 37],
A
4
[38, 39, 40, 41], or other symmetries [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], where introduced
to explain large lepton mixing angles, but in that case mass hierarchy remains unexplained.A milestone in these studies has been the discovery that mass hierarchies and mixing angles can be not directlycorrelated among them in the ﬂavor symmetry breaking [36, 54]. Fundamental steps in the realization of these ideas
are given in [38, 39]. These new ingredients allow us to escape from the nogo theorem [55] that seems to indicate
2that a maximal mixing angle
θ
23
can never arise in the symmetric limit of whatever ﬂavor symmetry (global or local,continuous or discrete), provided that such a symmetry also explains the hierarchy among the fermion masses and isonly broken by small eﬀects, as we expect for a meaningful symmetry.In fact, in our theory, the mass hierarchy and large mixing angle are not srcinated at the same step in the symmetrybreaking pattern.Our ﬁnal aim would be the construction of a grand uniﬁed
SO
(10)like model where masses and mixing angles aregenerated by the ﬂavor and gauge symmetry breaking.We presented a viable
SO
(10) model with discrete ﬂavor symmetry in [38]. There we generated the observed leptonmixing but we ﬁtted the fermion masses by assuming the group
A
4
as ﬂavor symmetry and the “constrain” of assigningright and lefthanded fermion ﬁelds to the same representations. Indeed, we showed in [38] that the assignment of both lefthanded and righthanded SM ﬁelds to triplets of
A
4
, that is therefore compatible with
SO
(10), can lead tothe charged fermion textures proposed in [56] and given by
M
f
=
h
f
0
h
f
1
h
f
2
h
f
2
h
f
0
h
f
1
h
f
1
h
f
2
h
f
0
,
(2)with
h
f
0
, h
f
1
and
h
f
2
distinct parameters. In [38], in order to obtain a mass matrix of the form of
M
f
in eq. (2) withoutspoiling the predictions of the neutrino sector, we introduced higher order operators containing simultaneously a setof
SO
(10) representations
45
. The lepton mixing was naturally generated by the breaking pattern of
A
4
, while thefermion masses were obtained with a possible tuning in the ﬂavor parameters not constrained by the symmetries.We addressed the problem of the ﬁne tuning in [39] where the
A
F
4
ﬂavor discrete symmetry is embedded into(
SO
(3)
L
×
SO
(3)
R
)
F
. In that way we explicitly disentangled the mixing problem from the hierarchy one. We brokethe continuous ﬂavor (
SO
(3)
L
×
SO
(3)
R
)
F
symmetry both dynamically and explicitly. The two breaking termsproduced the charged fermion hierarchies on one hand and solved the leptonic mixing problem on the other hand. Inthis way not only a tribimaximal neutrino mixing was naturally generated but also the charged fermion hierarchiesby dynamically breaking of the continuous leftright ﬂavor symmetry. Finally the Cabibbo angle was obtained bytaking into account higher order operators. However the leftright ﬂavor group symmetry (
SO
(3)
L
×
SO
(3)
R
)
F
of [39] is not compatible with a grand uniﬁed gauge group, like
SO
(10), with all the fermions of one family in the samerepresentation, because in leftright ﬂavor symmetries the fermions of one family belong to diﬀerent representationsof the ﬂavor group.In this paper we merge all these ingredients together and we are able to construct a non renormalizable model withgrand uniﬁed gauge group
SO
(10) and with an extended ﬂavor symmetry (
SU
(3)
×
U
(1))
F
. In this new model boththe tribimaximal lepton mixing matrix and the hierarchy among the mass of the 3rd and the other fermion familiesnaturally appear from the symmetry breaking pattern. Our model is non renormalizable, however a renormalizableversion of it can be easily constructed because the particular structure of the operators introduced here. For thispurpose viable methods are well known, i.e. by integrated out given heavy extra ﬁelds [57].Our eﬀective
SO
(10) invariant Lagrangian is
L
=
L
SU
(3)
F
+
δ
L
A
4
,
(3)where
L
SU
(3)
F
is
SO
(10)
×
(
SU
(3)
×
U
(1))
F
invariant and
δ
L
A
4
is the explicit breaking term of the
SU
(3)
F
symmetrythat, at this level, leaves
SO
(10) unbroken. The charge assignment of the ﬁelds is such that the
SU
(3)
F
invariantoperator with lowest mass dimensions is only [38]
L
SU
(3)
F
=
h
0
161045
A
45
B
16
Φ
,
(4)where Φ, singlet of
SO
(10), transforms as
6
with respect to
SU
(3)
F
. The scalar ﬁelds
10
,
45
A
and
45
B
are singletsof
SU
(3)
F
. As noticed in [38], thanks to the two
45
s scalar ﬁelds, the operator in eq. (4) can give no contribution
3to the neutrino sector for some set of the
45
s vev (i.e. in the explicit model of [38] one vev is proportional to therighthanded isospin
T
3
R
and the other one to the hypercharge
Y
, here the directions
A
and
B
are diﬀerent but stillwith the same property).When the scalar ﬁeld Φ develops a vev in the direction
Φ
ij
= 1(
∀
i,j
) we obtain a democratic mass matrices forall the charged fermions, that gives a massive 3rd family and two massless families. The democratic structure of thecharged fermion mass matrices avoids the ﬁne tuning needed to explain the mass hierarchy between the 3rd and theother two families that is usually needed in presence of charged fermion mass matrices of the form of eq. (2). Thedemocratic mass matrices preserves the (
U
(2)
×
Z
3
)
F
subgroup of (
SU
(3)
×
U
(1))
F
that leaves invariant the (1
,
1
,
1)vector in the ﬂavor space. Therefore at this stage only one mixing angle can be generated.The neutrino mass matrix and the ﬁrst and second families masses arise when we switch on the explicitly breakingterms of
SU
(3)
F
into
A
4
. If we neglect the ordering problem of the
45
s and the possibility to have more than oneﬂavon for each operator, the most general Lagrangian invariant under the ﬂavor structure of the theory is
δ
L
A
4
=
h
ijk
φ
k
16
i
10 45
A
45
B
45
C
45
D
16
j
+
h
′
ijk
φ
k
16
i
45
C
45
D
10 45
A
45
B
16
j
+ (5)
h
′′
ijk
˜
φ
k
16
i
10 45
C
45
D
16
j
+
g
16
i
126 16
i
ζ
S
+
g
′
ijk
16
i
126 16
j
ζ
kT
where the indices
{
i,j,k,l
}
are
A
4
, subgroup of
SU
(3), indeces and the sum over the gauge indices is understood.The scalar ﬁeld
126
is a singlet
1
′
of
A
4
, while the
45
C
, and
45
D
are other scalars that transform as
45
of
SO
(10),and are singlets of
A
4
. The ﬂavon ﬁelds
φ
, ˜
φ
,
ζ
T
are triplets under
A
4
, while
ϕ
and
ζ
S
are singlets.As found in [38] the terms in the second line of
δ
L
A
4
generates the light neutrino mass matrix. The terms in theﬁrst two lines in
δ
L
A
4
gives a contribution to the mass matrices that has the nice properties to commute with theleading order term obtained from eq. (3).After the breaking of
A
4
, it generates the ﬁrst and second family masses and ﬁx the mixing matrix in the leptonsector to be tribimaximal.The plan of the paper is as follow. First, in sec.
II
we introduce the basic ingredient of the model, i.e. thegeneral structure of the symmetry breaking, all the involved ﬁelds and how they transform under the gauge and ﬂavorsymmetries. Then in sec.
III
we show how the 3rd family masses are generated via the breaking (
SU
(3)
×
U
(1))
F
into (
U
(2)
×
Z
3
)
F
, how the 1st and 2nd family masses are generated together with maximal mixings in the leptonsector, and how the neutrino masses are generated with a resulting tribimaximal mixing matrix in the lepton sector.Finally we show how the Cabibbo angle is naturally generated without the introduction of new operators. Finally insec.
V
we report our conclusions.
II. BASIC INGREDIENTS
Let us ﬁrst investigate the ﬁeld content of the theory and the ﬂavor charges. We report the ﬁeld content of ourmodel in Table (I). With our charge assignment, the only allowed operators of lower mass dimensions are given ineqs. (45), if we neglect the ordering problem of the
45
and the possibility to have more than one ﬂavon for eachoperator. In this sense our Lagrangian is the most general one invariant under the ﬂavor structure of the theory.Moreover, independently from the fact that nature prefer a dominant seesaw of type I (i.e. heavy Majorana righthanded neutrino mass and intermediate Dirac neutrino mass) or of type II (i.e. light Majorana lefthanded neutrinomass) or a mixed scenario, the transformation properties of the
ζ
S
must be assumed to be
1
′
, as we will explain insec.
IVB
.In our opinion, the ordering problem can be related to a deeper structure of the theory, for example its version as arenormalizable model, and we will not investigate further it here. However the fact that will not be possible to expressthe directions
A
and
B
as rational combinations of
C
and
D
, together with the fact that
45
appears only as couples
4(
45
A
,
45
B
) and (
45
C
,
45
D
) seems to us to indicate that the right representations to introduce are the irreducible partof the
2025
that can get a vev diagonal over the
16
matter ﬁelds with charges
AB
and
CD
. If this is the case we arereally including all the allowed operator and there is not any more an ordering problem.After symmetry breaking, once the Higgs acquire vevs, the quadratic part for the fermions of the Lagrangian ineqs. (45) can be rewritten in a compact form, i.e. with an abuse of notation in the
SO
(10) contractions, as
L
Dirac
=
h
0
(
16
1
16
′
1
+
16
2
16
′
2
+
16
3
16
′
3
)
v
10
+ (6a)+
h
1
(
16
1
16
′′′
2
+
16
2
16
′′′
3
+
16
3
16
′′′
1
) +
h
2
(
16
1
16
′′′
3
+
16
2
16
′′′
1
+
16
3
16
′′′
2
)
v
10
v
φ
(6b)+
h
′
1
(
16
′′
1
16
′
2
+
16
′′
2
16
′
3
+
16
′′
3
16
′
1
) +
h
′
2
(
16
′′
1
16
′
3
+
16
′′
2
16
′
1
+
16
′′
3
16
′
2
)
v
10
v
φ
(6c)+
h
′′
1
(
16
1
16
′′
2
+
16
2
16
′′
3
+
16
3
16
′′
1
) +
h
′′
2
(
16
1
16
′′
3
+
16
2
16
′′
1
+
16
3
16
′′
2
)
v
10
v
˜
φ
(6d)+
g
(
16
1
16
1
+
16
2
16
2
+
16
3
16
3
)
v
126
v
ζ
S
+
g
′
1
16
1
16
2
+
g
′
2
16
2
16
1
v
126
v
ζ
T
(6e)where we have assumed that the two
A
4

3
plets
φ
and ˜
φ
acquire vev in the (1
,
1
,
1) direction of
A
4
, while the
ζ
T
vevis in the direction (0
,
0
,
1). In eqs. (6) we introduced
16
′′′
i
≡
v
45
A
v
45
B
v
45
C
v
45
D
16
i
,
16
′′
i
≡
v
45
C
v
45
D
16
i
,
(7)
16
′
i
≡
v
45
A
v
45
B
16
i
,
with
i
= 1
,
2
,
3
.
We obtain the following expression by absorbing the vevs of the
45
s into the coupling constants
16
′
=
x
′
Q
Q, x
′
U
U
c
, x
′
D
D
c
, x
′
L
L, x
′
E
E
c
, x
′
N
N
c
,
(8a)
16
′′
=
x
′′
Q
Q, x
′′
U
U
c
, x
′′
D
D
c
, x
′′
L
L, x
′′
E
E
c
, x
′′
N
N
c
,
(8b)
16
′′′
=
x
′′′
Q
Q, x
′′′
U
U
c
, x
′′′
D
D
c
, x
′′′
L
L, x
′′′
E
E
c
, x
′′′
N
N
c
(8c)where
x
′
f
,
x
′′
f
, and
x
′′′
f
are the quantum numbers respectively of the product of the charges
A
and
B
, of the productof the charge
C
and
D
, and of the product of the charges
A
,
B
,
C
, and
D
. In particular we notice that
x
′′′
f
=
x
′′
f
x
′
f
.
(9)We report the charges of each fermion in Table (II).
III. DYNAMICAL BREAKINGA.
(
SU
(3)
×
U
(1))
F
→
(
U
(2)
×
Z
3
)
F
gives the charged fermion 3rd family masses
We assume that the Φ
SU
(3)
F

6
plet ﬁeld acquire a vev
Φ
ij
=
v
Φ
(
∀
i,j
). In this case the charged fermion massmatrix obtained is the socalled democratic mass matrix [58] given by
M
0
f
=
m
f
3
3
1 1 11 1 11 1 1
,
(10)where
m
U
3
=
v
Φ
(
x
′
U
+
x
′
Q
)
v
U
10
h
0
,
(11a)
m
D
3
=
v
Φ
(
x
′
D
+
x
′
Q
)
v
D
10
h
0
,
(11b)
m
N
3
=
v
Φ
(
x
′
N
+
x
′
L
)
v
U
10
h
0
,
(11c)
m
E
3
=
v
Φ
(
x
′
E
+
x
′
L
)
v
D
10
h
0
.
(11d)
5This matrix has only one eigenvalue diﬀerent from zero,
m
f
3
, and can be assumed to be the mass of the 3rd family.To avoid any non diagonal contribution to the Dirac neutrino mass matrix we impose
x
′
N
+
x
′
L
= 0
.
(12a)To have the bottomtau uniﬁcation, we must impose also
x
′
L
+
x
′
E
=
x
′
Q
+
x
′
D
.
(12b)The unitary matrix
U
that diagonalizes the symmetric matrix
M
0
f
has one angle and the three phases undeterminated.One possible parametrization is given by [39]
U
= 1
√
3
√
2cos
θe
iα
√
2sin
θe
i
(
β
+
γ
)
1
−
e
iα
(
cos
θ
√
2
+
32
sin
θe
−
iγ
)
e
iβ
(
32
cos
θ
−
1
√
2
sin
θe
iγ
) 1
−
e
iα
(
cos
θ
√
2
−
32
sin
θe
−
iγ
)
−
e
iβ
(
32
cos
θ
+
1
√
2
sin
θe
iγ
) 1
.
(13)The freedom in the
U
matrix shows the remaining ﬂavor symmetry
U
(2)
F
. The unknow angle and phases are ﬁxedonly after breaking the democratic structure of
M
0
f
, i.e. the
U
(2)
F
ﬂavor symmetry, with a small perturbation
δM
f
,i.e.
M
f
=
M
0
f
+
δM
f
.
The eﬀect of
δM
f
is to give a small mass to the ﬁrst and second family and to ﬁx the mixing angles. To have thatthe mixing matrix diagonalizing the full
M
f
belongs to the families of matrix of eq. (13), we must require that
δM
f
commute with
M
0
f
. This has the nice consequence that we have automatically the selection of the breaking patternof
A
4
into
Z
3
.
IV. EXPLICITLY BREAKING
SU
(3)
F
→
A
4
We will assume the presence of an hidden scalar sector that breaks spontaneously the continuous
SU
(3)
F
into thediscrete
A
4
. Under this hypothesis it is quite natural to assume that the explicit breaking terms to be added to theLagrangian are small.
A.
A
4
→
Z
3
generates the charged fermions 1st and 2nd family masses and mixing
When the
φ
and ˜
φ A
4

3
plets take vev as
˜
φ
∝
φ
=
v
φ
(1
,
1
,
1) we have new contributions to the mass matrices.I.e., for the charged leptons we get the operator
δ
E ijk
ǫ
αβ
H
αd
L
βi
E
j
φ
k
→
ǫ
αβ
H
αd
δ
E
1
(
L
β
2
E
3
+
L
β
3
E
1
+
L
β
1
E
2
) +
δ
E
2
(
L
β
3
E
2
+
L
β
1
E
3
+
L
β
2
E
1
)
v
φ
,
(14)where the two
δ
E i
arise by the two diﬀerent contractions of
A
4
. The value of
δ
E
s can be read from the Lagrangian ineq. (6) and is
δ
E
1
= (
h
1
x
′′′
E
+
h
2
x
′′′
L
) + (
h
′
1
x
′′
L
x
′
E
+
h
′
2
x
′
L
x
′′
E
) + (
h
′′
1
x
′′
E
+
h
′′
2
x
′′
L
)
,
(15)
δ
E
2
= (
h
2
x
′′′
E
+
h
1
x
′′′
L
) + (
h
′
2
x
′′
L
x
′
E
+
h
′
1
x
′
L
x
′′
E
) + (
h
′′
2
x
′′
E
+
h
′′
1
x
′′
L
)
.
(16)Because the
SO
(10) uniﬁcation, the operators in the up, down and neutrino sectors have similar expressions. Inparticular for the contributions to the Dirac neutrino mass matrix we have
δ
N
1
= (
h
1
x
′′′
N
+
h
2
x
′′′
L
) + (
h
′
1
x
′′
L
x
′
N
+
h
′
2
x
′
L
x
′′
N
) + (
h
′′
1
x
′′
N
+
h
′′
2
x
′′
L
)
,
(17)
δ
N
2
= (
h
2
x
′′′
N
+
h
1
x
′′′
L
) + (
h
′
2
x
′′
L
x
′
N
+
h
′
1
x
′
L
x
′′
N
) + (
h
′′
2
x
′′
N
+
h
′′
1
x
′′
L
)
.
(18)