496
E. Rabinovici
sense.
ˆ
f L =
ij k
λ ij k ab ˆ
L
a
i
ˆ
L
b
j
ˆ
E
c
k + λ
ij k ab ˆ
L
a
i
ˆ
Q
b
j
ˆ
D
c
k
+
i
μ
i ab ˆ
L
a
i
ˆ
H
b
u .
(8.121)
ˆ
f B =
ij k
λ
"
ij k
ˆ
U
c
i
ˆ
D
c
j
ˆ
D
c
k .
(8.122)
The B and L symmetries are in any case not respected by non-perturbative effects.
The above mentioned terms can also be forbidden by imposing a different global
symmetry called R-parity which was already mentioned in the context of SUSY
breaking. It is defined here as:
R = (−)
3(B−L)+2s ,
(8.123)
s denoting the spin of the particle. The standard model particles are even
under the R-parity, while their superpartners are odd under it. In MSSM the
bosonic partners of the standard model matter fermions carry non-zero L and B
quantum numbers. The (non)conservation of B and L symmetries are not in general
correlated. They happen to coincide on the terms disallowed above. For example,
the terms below both respect R-parity but the first violates L number conservation
and the second does not respect B number conservation.
ab ˆ
L
a ˆ
H
b
u cd ˆ
L
c ˆ
H
d
u
(8.124)
and
ˆ
U c ˆ
U c ˆ
D c ˆ
E c
(8.125)
For the record we write down all the dimension three and dimension two
operators which softly break SUSY in the MSSM model. i and j run over the
generations and are summed over as are the doublet SU(2) indices a and b.
L sof t = −
˜
Q
†
i m 2
Q ij
˜
Q j + ˜
d
†
R i
m 2
D ij
˜
d R j + ˜
u
†
R i
m 2
U ij
˜
u R j
+ ˜
L
†
i m 2
L ij
˜
L j + ˜
e
†
R i
m 2
E ij
˜
e R j + m 2
H u
|H u | 2 + m 2
H d
|H d | 2
−
1
2
M 1 λ 0 λ 0 + M 2 λ A λ A + M 3 ˜
g B ˜
g B
−
i
2
M
1 λ 0 γ 5 λ 0 + M
2 λ A γ 5 λ A + M
3 ˜
g B γ 5 ˜
g B
+
(a u ) ij ab ˜
Q a
i H b
u ˜
u
†
R j
+ (a d ) ij ˜
Q a
i H da ˜
d
†
R j
+ (a e ) ij ˜
L a
i H da ˜
e
†
R j
+ h.c.
+
(c u ) ij ab ˜
Q
a
i H
∗b
d ˜
u
†
R j
+ (c d ) ij ˜
Q
a
i H ∗
ua
˜
d
†
R j
+ (c e ) ij ˜
L
a
i H ∗
ua ˜
e
†
R j
+ h.c.
+
bH a
d H da + h.c.
.
(8.126)
E. Rabinovici
sense.
ˆ
f L =
ij k
λ ij k ab ˆ
L
a
i
ˆ
L
b
j
ˆ
E
c
k + λ
ij k ab ˆ
L
a
i
ˆ
Q
b
j
ˆ
D
c
k
+
i
μ
i ab ˆ
L
a
i
ˆ
H
b
u .
(8.121)
ˆ
f B =
ij k
λ
"
ij k
ˆ
U
c
i
ˆ
D
c
j
ˆ
D
c
k .
(8.122)
The B and L symmetries are in any case not respected by non-perturbative effects.
The above mentioned terms can also be forbidden by imposing a different global
symmetry called R-parity which was already mentioned in the context of SUSY
breaking. It is defined here as:
R = (−)
3(B−L)+2s ,
(8.123)
s denoting the spin of the particle. The standard model particles are even
under the R-parity, while their superpartners are odd under it. In MSSM the
bosonic partners of the standard model matter fermions carry non-zero L and B
quantum numbers. The (non)conservation of B and L symmetries are not in general
correlated. They happen to coincide on the terms disallowed above. For example,
the terms below both respect R-parity but the first violates L number conservation
and the second does not respect B number conservation.
ab ˆ
L
a ˆ
H
b
u cd ˆ
L
c ˆ
H
d
u
(8.124)
and
ˆ
U c ˆ
U c ˆ
D c ˆ
E c
(8.125)
For the record we write down all the dimension three and dimension two
operators which softly break SUSY in the MSSM model. i and j run over the
generations and are summed over as are the doublet SU(2) indices a and b.
L sof t = −
˜
Q
†
i m 2
Q ij
˜
Q j + ˜
d
†
R i
m 2
D ij
˜
d R j + ˜
u
†
R i
m 2
U ij
˜
u R j
+ ˜
L
†
i m 2
L ij
˜
L j + ˜
e
†
R i
m 2
E ij
˜
e R j + m 2
H u
|H u | 2 + m 2
H d
|H d | 2
−
1
2
M 1 λ 0 λ 0 + M 2 λ A λ A + M 3 ˜
g B ˜
g B
−
i
2
M
1 λ 0 γ 5 λ 0 + M
2 λ A γ 5 λ A + M
3 ˜
g B γ 5 ˜
g B
+
(a u ) ij ab ˜
Q a
i H b
u ˜
u
†
R j
+ (a d ) ij ˜
Q a
i H da ˜
d
†
R j
+ (a e ) ij ˜
L a
i H da ˜
e
†
R j
+ h.c.
+
(c u ) ij ab ˜
Q
a
i H
∗b
d ˜
u
†
R j
+ (c d ) ij ˜
Q
a
i H ∗
ua
˜
d
†
R j
+ (c e ) ij ˜
L
a
i H ∗
ua ˜
e
†
R j
+ h.c.
+
bH a
d H da + h.c.
.
(8.126)