We give a preliminary report of a new quark mass matrix model basing on a
$SU(5)\otimes SU(5)\otimes Q_{12}$ symmetry embedding into a fully gauged
$SU(5)\otimes SU(5)\otimes SU(2)$.~\cite{fk} The two $SU(5)$'s contain the
standard SUSY $SU(5)$ as a diagonal subgroup, while the $Q_{12}$ or $SU(2)$ is
horizontal. Starting by assuming a judiciously-chosen set of chiral
supermultiplets, and a pattern of spontaneous symmetry breaking, we obtain the
low-energy chiral fermions together with a spectrum of superheavy fermions at
two different scales. The latter mediate Froggatt-Nielsen tree graphs that give
rise to a phenomenologically viable effective quark mass matrix texture. The
model is the first example of a nontrivial combination of supersymmetry without
R-parity, gauged nonabelian horizontal symmetry and
unification/anti-unification. It is expected to have some very interesting
features in SUSY-GUT phenomenology.Comment: missing title page restore

Kong, Otto C. W.

English

arXiv

We give a preliminary report of a new quark mass matrix model basing on a SU(5)\otimes SU(5)\otimes Q_{12} symmetry embedding into a fully gauged SU(5)\otimes SU(5)\otimes SU(2).~\cite{fk} The two SU(5)'s contain the standard SUSY SU(5) as a diagonal subgroup, while the Q_{12} or SU(2) is horizontal. Starting by assuming a judiciously-chosen set of chiral supermultiplets, and a pattern of spontaneous symmetry breaking, we obtain the low-energy chiral fermions together with a spectrum of superheavy fermions at two different scales. The latter mediate Froggatt-Nielsen tree graphs that give rise to a phenomenologically viable effective quark mass matrix texture. The model is the first example of a nontrivial combination of supersymmetry without R-parity, gauged nonabelian horizontal symmetry and unification/anti-unification. It is expected to have some very interesting features in SUSY-GUT phenomenology

Kong, O C W

CERN Document Server

arXiv:hep-ph/9507313 v2   17 Jul 1995
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1
9
9
5
.
1. The Light, the Heavy and the Superheavy
The smallness of most of the quark mass and mixing parameters and the strong
hierarchy among them is one of the most interesting puzzle in particle physics. Fla-
vor symmetry, especially a horizontal symmetry commuting with the Standard Model
group or the GUT group, paired with the Froggatt-Nielsen mechanism
2;3
, provide a
plausible explanation for the hierarchical texture pattern
4
of the quark mass matri-
ces. Following this popular approach, here we present a new model using a discrete
dicyclic subgroup
5
, Q
12
, of a gauged horizontal SU(2). The model is the rst example
of a nontrivial combination of supersymmetry without R-parity, gauged nonabelian
horizontal symmetry and unication/anti-unication
6
.
We recall the hierarchy in the quark sector parameters given in powers of   0:22,
at around the GUT scale:
3;4
jV
us
j  , jV
cb
j  
2
, jV
ub
j  
3
  
4
;
m
u
=m
c
 
3
  
4
, m
c
=m
t
 
3
  
4
,
m
d
=m
s
 
2
, m
s
=m
d
 
2
,
m
b
=m
t
 
3
, m
t
=hH
u
i  1.
To construct the light and heavy quark masses, the Froggatt-Nielsen mechanism
invokes a spectrum of superheavy fermions, which are essentially in vector-like pairs,
to communicate the eects of the avor symmetry breaking vevs to the low-energy
chiral fermions. Integrating out the superheavy particles then leaves us with eective
quark mass terms containing powers of small parameters of the form
<S>
M
, say  
or 
2
, where < S > is a symmetry breaking vev and M the superheavy fermion mass
allowed by the unbroken symmetry.
2. The SU(5)
 SU(5) 
 SU(2) Model
The model has a gauge symmetry given by SU(5)
SU(5)
 SU(2). For the two
SU(5)'s, they are the GUT groups for the third family and the lighter two families
respectively. The latter form a horizontal doublet.
2.1. Symmetry Breaking Pattern
The two SU(5)'s then break into a diagonal SU(5) which is identied as the
standard unication group, at energy scale M
0
. So below M
0
is the standard SUSY-
GUT story. We assume that the horizontal SU(2) is broken to Q
12
at energy scale
M
Q
12
with the latter subsequently totally broken at around M
0
. The symmetry
SU(5)
 SU(5) 
 SU(2)
SU(5)
 SU(5) 
Q
12
SU(5)
D
SU(3)
c

 SU(2)
L

 U(1)
Y
M
SU(2)
M
Q
12
M
GUT
M
SUSY
M
0
M
W
?
?
?
?
Fig. 1. The Symmetry Breaking Pattern of the Model
breaking pattern is summarised in Figure 1.
2.2. Fermion Content
The light, heavy and superheavy fermion contents of the model come from the
following list of SU(5) 
 SU(5) chiral supermultiplets:-
from SU(2) to Q
12
(10; 1) { 1  ! 1(T )
7  ! 1
0
+ 2
2
+ 2
4
+ 2
6
4  ! 2
1
+ 2
3
(

10; 1) { 7  ! 1
0
+ 2
2
+ 2
4
+ 2
6
4  ! 2
1
+ 2
3
(

5; 1) { 6  ! 2
1
+ 2
3
+ 2
5
(H
d
=b)
3  ! 1
0
+ 2
2
(5; 1) { 1  ! 1(H
u
)
4  ! 2
1
+ 2
3
3  ! 1
0
+ 2
2
(1; 10) { 2  ! 2
1
(Q)
1  ! 1
(1;

10) { 1  ! 1
(1;

5) { 2  ! 2
1
(D)
(A summary of the Q
12
representations is given in the appendix.)
The vector-like fermion pairs have Dirac masses of order M
SU(2)
where SU(2) is
a good symmetry. At M
Q
12
, the two (5; 1) doublets, 2
1
and 2
3
from the dimension 4
SU(2) representation married with the corresponding (

5; 1) doublets from the dimen-
sion 6 SU(2) representation to form Dirac fermions, leaving behind only the labelled
Q
12
singlets and doublets as GUT scale chiral particles. The list of low energy chiral
fermions is given by
 
u
d
!
L
 
c
s
!
L
9
>
>
=
>
>
>
;
Q(2
1
)
u
c
L
c
c
L
)
d
c
L
s
c
L
)
Q(2
1
)
D(2
1
)
 
t
b
!
L
T (1)
t
c
L
b
c
L
T (1)
H
d
=b(2
5
)
and the correspondent leptonic partners of the GUT multiplets, and the Higgsinos
from H
u
| 1 and H
d
=b | 2
5
. Recall that in the SU(5) language, H
u
is a 5, Q and
T are 10's while D and the interesting H
d
=b are

5's.
The horizontal doublet H
d
=b is denitely the most interesting element of the
model. It contains both the bottom-tau and the (down-sector) Higgs chiral multiplets.
The group properties of the representation 2
5
plays a very important role in the model,
as discussed below.
2.3. Supersymmetry without R-parity
The most interesting point to note about the assignment of the bottom-tau and
down-sector Higgs to a horizontal doublet is that it is incompatible with the standard
R-parity, which is put into MSSM by hand to avoid unacceptable B and L violation.
Here in our model, the group properties of the horizontal symmetry gives this required
matter parity feature
7
.
The only direct Yukawa couplings to Higgses are for the third family, giving rank
one quark mass matrices for both up and down sector at the rst order. In terms
of MSSM chiral superelds, the top get its mass from the term q^
^
h
1
^
u
c
contained in
the 10:5:10 coupling (Figure 2a). The bottom and the tau get their masses from the
terms q^
^
h
2
^
d
c
and
^
l
^
h
2
^
e
c
respectively, both contained in the 10:

5:

5 coupling (Figure3a).
The latter however does not give rise to the dangerous terms q^
^
l
^
d
c
,
^
l
^
l
^
e
c
, and
^
u
c
^
d
c
^
d
c
.
The secret is in the Q
12
product 2
5
 2
5
which contains the singlet 1 only in the
antisymmetric part, therefore always coupling the bottom-tau part to the Higgs part
but not to itself.
Moreover, no

5:5 term is allowed by the horizontal symmetry, hence both
^
l
^
h
1
and
^
h
2
^
h
1
are absent. In conclusion, to the rst order, the horizontal symmetry gives the
required matter parity feature and evades the -problem. Detailed properties of the
model in aspects like B and L violation, FCNC, and Higgsino masses and the role of
the horizontal symmetry in them is a very interesting question to be addressed.
2.4. The Quark Mass Matrices
All the above listed chiral supermultiplets are assumed to develop no vevs above
the M
GUT
. We need then the following vector-like multiplet with the specied vevs
to generate the eective quark mass matrices. They are, a < 2
4
> of (1; 1), a < 2
1
>
of (1; 24), a < 2
1
> and a < 2
2
> of (

10; 10) coupled only to 2nd family in Q or D, a
< 2
2
> of (5;

5) coupled only to 2nd family in Q or D, and a < 2
1
> of (5;

5) coupled
only to 1st family in Q or D.
The model is now completed. The lowest order eective quark mass term for the
up and down sector are given by the Froggatt-Nielsen tree graphs shown in Figure 2
and 3 respectively. Taking
< 2
k
> =M
Q
12
 , < 2
k
> =M
SU(2)
 
2
; < 2
k
>M
0
;
we arrive at the quark mass textures
M
u

0
B
@

8

6
0

6

4
0
0 0 1
1
C
A
; M
d

0
B
@
0 
5

5

4

3

3
0 
3
1
1
C
A
:
A comparison with the corresponding symmetric texture pattern from reference
4, as given by
M
u

0
B
@
0 
6
0

6

4
0
0 0 1
1
C
A
; M
d

0
B
@
0 
4
0

4

3

3
0 
3
1
1
C
A
:
is suÆciently convincing that it is phenomenologically viable.
Possible alternative formulation of the model under a SU(5)
U(1)
SU(2)
1
and
comparison with the simple Q
2N
models built under the same approach
8
are skipped
here.
3. Acknowledgements
The author want to thank his thesis adviser, P.H. Frampton, for helpful guidance,
constant encouragement and enjoyable collaboration. This work was supported in
part by the U.S. Department of Energy under Grant DE-FG05-85ER-40219, Task B.
4. Appendix on Q
2N
Representations
5;8
May be a dihedral group D
N
as the symmetry of a N-sided planar polygon is more
familiar to physicists. The dicyclic group Q
2N
is a double cover of D
N
, as the parent
continous group SU(2) being a double cover of the rotation group SO(3). Q
2N
is of
order 4N ; Q
12
, the N = 6 candidate in our model has order 24.
Irreducible representations of Q
2N
are given by 4 singlets 1; 1
0
; 1
00
; 1
000
and (N   1)
doublets 2
k
(with 1  k  (N 1)). Most important for our purposes are the product
formulae: -
1
0
 1
0
= 1 (1)
1
0
 2
k
= 2
k
(2)
2
k
 2
l
= 2
(jk lj)
+ 2
(minfk+l;2N k lg)
(3)
where, in a generalized notation, 2
0
 1 + 1
0
and 2
N
 1
00
+ 1
000
.
5. References
1. P.H. Frampton and O.C.W. Kong, UNC-Chapel Hill Report IFP-715-UNC
(manuscript in preparation).
2. C.D. Froggatt and H.B. Nielsen, Nucl.Phys.B147 (1979) 277.
3. M. Leurer, Y. Nir and N. Seiberg, Nucl.Phys.B398 (1993) 319; ibid. B420
(1994) 468.
4. P. Ramond, R.G. Roberts and G.G. Ross Nucl.Phys.B406 (1993) 19.
5. P.H. Frampton and T.W. Kephart, Phys.Rev.D51 (1995) R1, and UNC-Chapel
Hill Report IFP-702-UNC.
6. C. Froggatt, G. Lowe and H. Nielsen, Phys.Lett.B311 (1993) 163;
R. Barbieri, G. Dvali and A. Strumia, Nucl.Phys.B435 (1995) 102.
7. S.Dimopoulos and H. Georgi, Nucl.Phys.B193 (1981) 150;
M.C. Bento, L. Hall and G.G. Ross, Nucl.Phys.B292 (1987) 400.
8. P.H. Frampton and O.C.W. Kong, Quark Mass Textures within a Finite Non-
abelian Dicyclic Group, to be published in Phys.Rev.Lett.75 (1995), UNC-
Chapel Hill Report IFP-713-UNC, hep-ph/9502395.
T   1
H
u
  1
Q  2
1
(a): (M
u
)
33
-


hH
u
i
T (t)T (t)
(b): (M
u
)
22
-




Q(c)
hH
u
i
2
2
h2
1
i
Q(c)
--
h2
1
i
2
2
(c): (M
u
)
21
=(M
u
)
12
-




Q(c)
hH
u
i
2
2
h2
2
i
--
h2
1
i
2
2


Q(u)
1
h2
1
i
(d): (M
u
)
11
-




Q(u)
hH
u
i
2
2
h2
2
i
--

--

2
2
1
h2
2
ih2
1
i 

Q(u)
1
h2
1
i
Fig. 2. Froggatt-Nielsen tree graphs for M
u
.*
T   1
H
d
=b  2
5
Q  2
1
D   2
1
(a): (M
d
)
33
-


hH
d
i
bT (b)
(b): (M
d
)
32
-




T (b)
hH
d
i
2
1
h2
2
i
D(s)
-
h2
4
i
2
4
(c): (M
d
)
23
-




Q(s)
hH
d
i
2
3
h2
4
i
b
--
h2
1
i
2
2
(d): (M
d
)
13
-




Q(d)
hH
d
i
2
3
h2
4
i
--

--

2
2
1
h2
1
i h2
2
i
b
Fig. 3. Froggatt-Nielsen tree graphs for M
d
.*
(e): (M
d
)
22
--




Q(s)
hH
d
i
2
3
h2
2
i
D(s)
-
h2
1
i
2
2
(f): (M
d
)
21
-




Q(s)
hH
d
i
2
2
h2
1
i
D(d)
--
h2
2
i
2
3
(g): (M
d
)
12
-




Q(s)
hH
d
i
2
3
h2
2
i
--

--

2
2
1
h2
1
i h2
2
i
D(d)
Fig. 3. Froggatt-Nielsen tree graphs for M
d
:*
* In both Figure 2 and 3, the superheavy fermions and scalar vevs are indicated by their represen-
tations in terms of the Q
12
symmetry, while the low energy chiral particles are indicated by their
label. All vertical dash-lines represen
t scalar vevs. In Figure 2, all horizontal lines represent fermions; those shown by arrows are (10; 1)'s
while those shown by double-arrows are (1; 10)'s under SU (5)
SU (5). Likewise in Figure 3, except
that those left pointing arrows and do
uble-arrows (fermions to the right of H
d
) are here (

5; 1)'s and (1;

5)'s respectively. The full repre-
sentations for the scalars can then be gured out easily.


The Light, the Heavy and the Superheavy --- a Nonabelian Flavor Symmetry for the Full Hierarchy

Abstract

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