The neutrino mass problem in $SU(4) \times SU(2)_{L} \times SU(2)_{R}$ SUSY
GUT obtained from the compactification of $E_{8} \times E_{8}$ heterotic string
is analyzed. The estimated values of the neutrino masses and mixing angles can
explain the experimental data on solar neutrino flux.Comment: Latex file, 14 page

Asatrian, H. M.

Ioannisian, A. N.

English

arXiv

The neutrino mass problem in SU(4) \times SU(2)_{L} \times SU(2)_{R} SUSY GUT obtained from the compactification of E_{8} \times E_{8} heterotic string is analyzed. The estimated values of the neutrino masses and mixing angles can explain the experimental data on solar neutrino flux

Asatryan, H M

Ioannisian, A N

CERN Document Server

Preprint YERPHI-1450(20)-94
THE NEUTRINO MASSES IN SUSY GUT
H. M. Asatrian, A. N. Ioannisian.
Yerevan Physics Institute, Alikhanyan Br. 2, Yerevan, Armenia
e-mail: "hrach@uniphi.yerphi.am"
Abstract
The neutrino mass problem in SU(4)  SU(2)
L
 SU(2)
R
SUSY GUT
obtained from the compactication of E
8
 E
8
heterotic string is analyzed.
The estimated values of the neutrino masses and mixing angles can explain
the experimental data on solar neutrino ux.
Yerevan Physics Institute
Yerevan 1995
Recent solar neutrino experiments give an evidence for nonzero neutrino
masses. The solar neutrino decit can be explained in terms of the neutrino
resonant oscillations if the neutrino mass dierence is of order: m
2
 (0:3 
1:2) 10
 5
eV
2
or vacuum oscillations if m
2
 (0:5 1:1) 10
 10
eV
2
[1, 2, 3].
It is well-known, that in SO(10) GUT small neutrino masses can be
obtained via seesaw mechanism [4]. The neutrino mass matrix for three left

e
, 

, 

and three right 
c
e
, 
c

, 
c

neutrinos has the following form:
 
0 M
D
M
D
R
!
(1)
In (1) M is a Dirac type 3  3 mass matrix (usually it is equal to mass
matrix of u, c, t, quarks), R- is the right neutrino 3  3 mass matrix with
entries much greater than the electroweak symmetry breaking scale. After
the diagonalization of (1) one obtains three heavy Majorana states (their
masses practically coincide with the eigenvalues of matrix R) and three light
Majorana states with the mass matrix:
M

=
M
2
D
R
(2)
The scale of matrix R entries can be of order of the SO(10) subgroup G =
SU(4)  SU(2)
L
 SU(2)
R
breaking scale, if G is broken by the vacuum
expectation value (v.e.v.) of Higgs eld in the 126 representation of SO(10).
For the superstring inspired models, however, G can be broken only by the
v.e.v. of Higgs eld in 16 representation of SO(10). In this case the masses
of right neutrinos can arise only due to the radiative corrections [5]. For
the SUSY GUT, however, this mechanism does not work [6] and the right
neutrinos can obtain masses only due to the nonrenormalizable interactions
which can arise in the superstring models [7].
Our aim is consider the problem of the neutrino masses for the super-
symmetric model G = SU(4)  SU(2)
L
 SU(2)
R
, which may (or may not)
be considered as a subgroup of SO(10). The particle content of the model
is the following [8]: the sixteen fermions for each generation (including the
right neutrino) belong to the representations F and F, where
F = (4; 2; 1) = (u; d; ; e)
F = (4; 1; 2) = (u
c
; d
c
; 
c
; e
c
);
1
and the Higgs elds- to the representations: H = (4;1;2) and

H = (

4;1;2),
h = (1;2;2), D = (6;1;1). The vacuum expectation values (v.e.v.) of the
elds H and H are connected with the breaking of the group G and the v.e.v.
of the eld h -with the breaking of the group G
ew
= SU(2)
L
 U(1)
Y
. In
addition, there is a set of the G singlet scalar elds 
m
(m=1,2...). For the
supersymmetric models, derived from the E
8
E
8
heterotic string compacti-
cation over Calabi-Yau manifolds with SU(3) holonomy, the maximal gauge
group in the four dimension is E
8
 E
6
(by embedding the spin connection
of the manifold in the gauge group, E
8
 E
8
can be broken to E
8
 E
6
)
and chiral superelds belong to the 27;27;1 representations of E
6
[9].In this
case, the minimal set would consist of n
g
+ 1 SO(10) (or G) singlets, where
n
g
is the number of fermion generations. One of the singlets develops the
v.e.v. at the electroweak scale and generates the masses of the singlet elds
[8].
Then most general superpotential for the supersymmetric G = SU(4) 
SU(2)
L
 SU(2)
R
model has the form [7, 8] :
W = 
ij
1
F
i

F
j
+ 
im
2

F
i
H
m
+ 
3
HHD +
+ 
4

H

HD + 
m
5

m
hh+ 
mnl
6

m

n

l
+ (3)
+ 
ij
7
F
i
F
j
D + 
ij
8

F
i

F
j
+ 
m
9
DD
m
The 9  9 mass matrix of the three left, three right neutrinos and three
singlets has the following form
0
B
@
0 M
D
0
M
D
R M
G
0 M
G
M

1
C
A
; (4)
where M
G
is 3 3 matrix of right neutrino-singlets mixing, M

is 3 3 mass
matrix of singlets. As a result one obtains that (in the case of absence of
mixing) the light neutrino masses are proportional to m
2
q
m

=m
2
G
, where m
2
q
for q = u; c; t are the masses of u, c, t -quarks, m

is the typical singlet
mass M
W
(electroweak breaking scale), m
G
is of order of G breaking scale
 10
16
GeV [7] . This gives the ultralight neutrino masses: m

1
 10
 16
eV ,
m

2
 10
 12
eV , m

3
 10
 8
eV . The similar problems exist in SU(5) -
ipped model. The various possibilities to obtain more acceptable neutrino
masses are considered in [6, 7, 10]. Here we want to consider the problem of
the neutrino masses for the superstring derived models proposed by Witten
2
[11]. Witten [11] has shown that it is possible to construct stable, irreducible
and holomorphic SU(4) or SU(5) vector bundles over Calabi-Yau manifolds.
This means that one can obtain an SO(10) or SU(5) supersymmetric gauge
theories in four dimension by the embedding the structure group of the bundle
in E
8
. For the SU(4) vector bundle, when the maximal gauge group in the
four dimension is SO(10), the content of the chiral superelds is the following
[11, 12, 13, 14]:
n
g
16+ (16+ 16) + 10+ 1; (5)
where n
g
is the number of generations (n
g
=3), , ,  are the integer numbers
; ;   1. As for ordinary case of tangent bundle with the E
6
as a maximal
gauge group, in this case also it is possible to obtain models with the gauge
symmetries which are subgroups of SO(10) via Hosotani mechanism [11, 13,
15]. Let us consider the neutrino mass problem in such a model with gauge
symmetry G = SU(4)  SU(2)
L
 SU(2)
R
. For the simplest case  =  = 1
and  = 2 one has only two SO(10) singlets. One of these singlets developed
the v.e.v. of order of electroweak symmetry breaking scale. The second
singlet is mixed with the neutrinos. The 7  7 mass matrix for three left,
three right neutrinos and this singlet has the following form:
M =
0
B
@
M
L
M
D
0
M
D
R V
0 V
T
m

1
C
A
(6)
where M
L
, M
D
, R are 33 matrices, V-is a three dimensional column. The
elements of the left neutrino mass matrix M
L
can arise due to the nonrenor-
malizable interactions which are allowed in the string models [16] :

FFHHhh
M
3
;
where M is the typical scale connected with nonrenormalizable interactions,
it must be of order of Plank scale or string unication scale,  -some constant.
Then the matrix elements of M
L
are of order m
L

M
2
W
m
2
G
M
3
. The 3x3 matrix
R also arises due to the nonrenormalizable interactions [6, 7]:

0

F

FHH
M
;
3
The matrix elements of R are of order m
R

m
2
G
M
. The 3x3 neutrino Dirac
mass matrixM
D
(which we assume to be equal to the up quark mass matrix),
V
i
(i=1,2,3)- the mixing between singlet and right neutrinos and the mass
m

of the singlet arise from the usual interaction terms (3).
To obtain the estimates for the neutrino masses one must determine G
breaking scale m
G
. We will consider two cases: the symmetry G = SU(4)
SU(2)
L
 SU(2)
R
with and without the left-right discrete symmetry. The
renormalization group equations for one loop gives the following solutions for
the coupling constants 
1
; 
2
; 
3
[13]:

 1
i
(M
Z
) = 
 1
X
 
b
i
2
ln
M
S
M
Z
 
b
 
is
2
ln
M
R
M
S
 
^
b
is
2
ln
M
X
M
R
(7)
whereM
S
is the supersymmetry breaking scale (we assume that it is between
100GeV and 10000GeV), the values of b
i
; b
 
is
;
^
b
is
(i=1,2,3) are given in [13].
The more precise results can be obtained in two loop approximation. The
renormalization group equations analysis in two loop approximation gives
the following results for m
G
m
G
 (1:6  10
16
  2:2  10
17
)GeV (8)
for the case of the presence of the discrete left-right symmetry and
m
G
 (1:5  10
15
  2  10
16
)GeV (9)
for the case of absence of such a symmetry. The results of (8) and (9) are
obtained for the initial values of the electroweak coupling constants

3
(M
Z
) = 0:120  0:006
sin
2

W
= 0:2328  0:0007 (10)

 1
(M
Z
) = 128:8  0:9
Let as consider now the mass matrix (6). After the diagonalization of the
matrix (6) we obtain 7 Majorana neutrinos (three of which are light). One
can estimate the light neutrino mass values without specifying the exact
form of matrix (6). The only assumption we made, is the following: all
the matrix elements of the matrix M
L
are of the same order of magnitude
m
L

M
2
W
m
2
G
M
3
. The same statement must be valid for the 3  3 matrix R
4
and three dimensional column V separately: the elements of R are of order
m
R

m
2
G
M
and the elements of V are of order of m
G
. Then, it is easy to
estimate the value of the determinant of the matrix (6) and the sums of its
diagonal minors of 6x6, 5x5, 4x4 order:
detM  m
2
c
m
2
t
M
2
W
m
4
G
M
3
detM
6
 m
2
c
m
2
t
m
2
G
(11)
detM
5
 m
2
t
m
4
G
M
detM
4

m
6
G
M
2
Two eigenvalues of the matrix (6) are of order m
R
and the other two- of
order m
G
. One can obtain a simple formulae for the masses of three light
neutrinos for the case m

3
 m

2
 m

1
:
m

1

detM
detM
6

M
2
W
m
2
G
M
3
m

2

detM
6
detM
5
 r
 1
c
m
2
c
M
m
2
G
(12)
m

3

detM
5
detM
4
 r
 1
t
m
2
t
M
m
2
G
In (11), (12) r
c
; r
t
-are the factors connected with the quark mass renormal-
ization from the unication scale to ordinary energies. These factors depend
on the ratio of vacuum expectation values of scalar doublets connected with
the electroweak symmetry breaking (tg) and the t-quark mass. For the
t-quark mass m
t
= (175  15)GeV and 0:1 < tg < 10 one obtains:
20 > r
c
> 3
7 > r
t
> 1 (13)
In (13) the larger smaller of r
c
; r
t
correspond to the larger values of m
t
.
What about the mixing angles? To obtain the estimates for the mixing
angles between the left 
e
; 

; 

neutrinos one has to specify the form of Dirac
-type mass matrix M
D
. Let us consider the case when the mass matrix of
5
u, c, t- quarks (which we consider to be equal to M
D
) has a form, proposed
by Fritzsch [17]:
M
D
=
0
B
@
0 a 0
a 0 b
0 b c
1
C
A
(14)
where a =
p
m
u
m
c
; b =
p
m
c
m
t
; c = m
t
. We must made some assumptions
for the matrix R and M
L
also: let us consider the case when they have a
diagonal form:
R =
0
B
@
R
1
0 0
0 R
2
0
0 0 R
3
1
C
A
M
L
=
0
B
@
m
1
0 0
0 m
2
0
0 0 m
3
1
C
A
(15)
It is possible to estimate the mixing angles for three light neutrino states

1
; 
2
; 
3
by solving the equation for eigenvectors and eigenvalues of the ma-
trix M for the case m

3
 m

1
;m

2
. Using the following formulae for the
light neutrino masses:
m

3
'
detM
5
detM
4
m

2
m

1
'
detM
detM
5
(16)
m

2
+m

1
'
detM
6
detM
5
and calculating the determinant of matrix M and sums of its main minors
of 6, 5, and 4 order
detM '  m
1
m
2
c
m
2
t
M
2
1
+m
1
m
2
m
2
t
(R
1
M
2
2
+R
2
M
2
1
)
  2m
1
p
m
u
m
c
m
c
m
2
t
M
1
M
2
  2m
1
m
2
p
m
t
m
c
m
t
R
1
M
2
M
3
 m
1
m
u
m
c
m
2
t
M
2
2
 m
2
m
u
m
c
m
2
t
M
2
2
detM
6
'  m
2
c
m
2
t
M
2
1
+ (m
1
+m
2
)m
2
t
(R
1
M
2
2
+R
2
M
2
1
) (17)
  2
p
m
u
m
c
m
c
m
2
t
M
1
M
2
  2(m
1
+m
2
)
p
m
t
m
c
m
t
R
1
M
2
M
3
detM
5
' m
2
t
(R
1
M
2
2
+R
2
M
2
1
)  2
p
m
c
m
t
m
t
R
1
M
2
M
3
detM
4
' M
2
1
R
2
R
3
+M
2
2
R
1
R
3
+M
2
3
R
1
R
2
(whereM
1
;M
2
;M
3
are the elements inM connected with the mixing of right
neutrinos and singlet: V
T
= (M
1
;M
2
;M
3
)) one can obtain the estimates for
6
the light neutrino masses. In (17) we omit the terms not relevant for our
consideration. We will consider two alternatives for the 
1
; 
2
:
m

1
 m

2
(18)
m

1
 m

2
(19)
The conditions (18), (19) are equivalent to the conditions
m
G

s
m
c
M
W
M (20)
m
G

s
m
c
M
W
M (21)
For the rst case (18), (20) one obtains the same formulae for neutrino masses
as previously
m

1
 m
L
m

2

m
2
c
r
c
R
m

3

m
2
t
r
t
R
(22)
where we assume that R  R
1
 R
2
 R
3
and m
1
 m
2
 m
3
. The three
eigenstates 
i
, i=1,2,3 of matrix (6) with three lightest masses one can express
via weak eigenstates 

,  = e; ;  by means of unitary transformation:

i
= a
i


, where a
i
is the unitary matrix. For the mixing angles between

e
and 

() and 
e
and 

(
0
) one obtains:
tan 
a
1
a
1e

s
m
u
m
c
tan
0

a
1
a
1e

s
m
u
m
t
(23)
Taking into account the results (8), (9) for the G symmetry breaking scale
one obtains the following estimates for the three light neutrino masses:
m

1
 (10
 12
  2  10
 10
)

M
P l
M

3
eV
m

2
 (4  10
 8
  10
 5
)
M
M
P l
eV (24)
m

3
 (8  10
 3
  0:6)
M
M
P l
eV
7
for case of the presence of discrete left-right (case (a)) symmetry and
m

1
 (10
 14
  10
 12
)

M
P l
M

3
eV
m

2
 (6  10
 6
  2:5  10
 3
)
M
M
P l
eV (25)
m

3
 (0:3  150)
M
M
P l
eV
for the case of the absence of left-right discrete symmetry (case (b)). In (24),
(25) M
P l
= 1:2  10
19
GeV is the Plank mass. Of course, all the estimates
(24), (25) are correct with accuracy of order of magnitude.
To explain the solar neutrino decit via resonant oscillations the neutrino
mass dierence and mixing angles must be of order
p
m
2
 (1:7  3:5)10
 3
eV
sin
2
2  (0:6   1:4)10
 2
or (0:65  0:85) (26)
Then for the case (a) it is possible to obtain such a mass dierence between
third and second and rst neutrinos if the fraction
M
M
Pl
is of order  (1 0:01)
which is reasonable value [13]. The problem arises with mixing angle: the
formula (23) gives small value for sin
2
2  10
 4
.
For the case (b) such a mass dierence it is possible to obtain between the
second and the rst neutrinos if the fraction
M
M
Pl
is order of unity. In this case
the condition (20) is valid. The mixing angle is of order sin
2
  1:3  10
 2
which is a reasonable value. Thus, in this case obtained values for the mixing
angles and neutrino mass dierence allow one to explain the solar neutrino
decit via resonant oscillations in the sun.
To explain the solar neutrino decit via longwave vacuum oscillations the
neutrino mass dierence and the mixing angles must be of order
p
m
2
 (0:5  1:1)10
 5
eV
sin
2
2 > 0:75 (27)
It is clear that it is impossible for the considering case as our mixing angles
are too small.
Let as proceed to the case (19), (21). For the case (a) the condition (21)
gives for M:
M
M
P l
= (0:018   0:22) (28)
8
For such a values of M the light neutrino masses are of order
m

1
 m

2
 4  (10
 8
  10
 5
)
M
M
P l
eV  (7:1  10
 10
  2:2  10
 6
)eV
m

3
 (8  10
 3
  0:6)
M
M
P l
eV  (1:4  10
 4
  0:13)eV (29)
It is clear that the mass dierence of second and rst family is too small.
The mass dierence between third and rst family can achieve acceptable
value but the mixing angle as in previous case will be small sin
2
2  10
 4
.
Let us consider the case (b). From (21) one obtains for M
M
M
P l
= (0:0014   0:02) (30)
For such a values of M the light neutrino masses are of order
m

1
 m

2
 (6  10
 6
  2:5  10
 3
)
M
M
P l
eV  (8:4  10
 9
  5  10
 5
)eV
m

3
 (0:3  150)
M
M
P l
eV  (4:2  10
 3
  3:0)eV (31)
The value (0:5  1:1)10
 5
eV which is necessary to explain the solar neutrino
decit via longwave vacuum oscillations is achieved for the mass dierence
of rst two neutrinos for the following values of M:
M
M
P l
= (0:02  0:002) (32)
What about the mixing angles for the considering case? As the masses of
two lightest neutrinos are close to each other, the mixing angle between them
can be relatively large.
The angle relevant for the neutrino oscillations is given by (if we neglect
the mixing with third light neutrino):
tan  =
a
1
a
1e
'
s
m
c
m
u
m

1
 m
1
m

1
 m
2
(33)
where
m

1
= m
1
+
1
2
m
 
1 +
m
1
 m
2
+m
0
m
1
 m
2
 m
0
!
9
m
2
= m
2
+m
0
+
1
2
m
 
1  
m
1
 m
2
+m
0
m
1
 m
2
 m
0
!
m =  
m
u
m
c
m
2
t
M
1
M
2
detM
5
(34)
m
0
=  
m
2
c
m
2
t
M
2
1
+m
u
m
c
m
2
t
M
2
2
+ 2
p
m
u
m
c
m
c
m
2
t
M
1
M
2
detM
5
To explain the solar neutrino decit via vacuum longwave oscillations tan
must satisfy the condition
0:58 < tan < 1:75 (35)
(as it follows from (27)). This gives :
(m
1
 m
2
 m
0
)
2
 mm
0
(36)
or, with the same accuracy:
(m

1
 m

2
)
2
 mm
0
(37)
Thus, it is possible to obtain large (35) mixing angles in this case if the
condition (36) takes place. This means that the masses of rst two light
neutrinos must be very close to each other: the dierence of masses must be
 100 times smaller than the masses.
Thus, our analyze shows that it is possible to explain the solar neutrino
decit for the considering theory for the case of a unication group G =
SU(4)  SU(2)
L
 SU(2)
R
without the left-right discrete symmetry can be
explained in two ways. 1)Via resonant oscillations in the sun if the value of
which is characteristic scale of nonrenormalizable interactions is of order of
Plank mass M
P l
= 1:2  10
19
GeV . Generally speaking, it is more natural,
that such a interactions arises in string scale M
s
=
g
string
M
Pl
p
8
= 1:7  10
18
GeV
[18, 19] however the dierence of  7 can be explained by means of coupling
constants , 
0
in nonrenormalizable terms:

FFHHhh
M
3
; 
0

F

FHH
M
The mixing angle in this case  0:06 which is the value we need. 2)Via
longwave vacuum oscillations: in this case the masses of two light neutrinos
are very close to each other and the mixing angle is large as needed.
10
Authors thank H. H. Asatryan for help in most of calculations.
The research described in this publication was made possible in part by
Grant N MVU000 from the International Science Foundation.
11
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12


The Neutrino Masses in SUSY GUT

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