Eur. Phys. J. C (2015) 75 :434 Page 5 of 26 434
is identified with U
PMNS
with the CP-phases set to zero and
neutrino mixing angles set to the center of their experimental
values.
One can observe that m
eff
remains unchanged by consis-
tent changes on the scales of M
N
and Y
ν
. This is no longer
correct for the off-diagonal entries in the slepton-mass matri-
ces (parameterized by slepton δ
FAB
ij
, see the next subsec-
tion). These quantities have a quadratic dependence on Y
ν
and a logarithmic dependence on M
N
;seeEq.(9). Therefore
larger values of M
N
imply larger LFV effects. By setting
M
N
= 10
14
GeV, the largest values of Y
ν
are of about 0.29,
this implies an important restriction on the parameters space
arising from the BR(μ → eγ)as will be discussed in Sects.
3 and 4. An example of models with almost degenerate ν
R
can be found in [50]. For our numerical analysis we tested
several scenarios and we found that the one defined here is
the simplest and also the one with larger LFV prediction.
2.3 Scalar fermion sector with flavor mixing
In this section we give a brief description about how we
parameterize flavor mixing at the EW scale. We are using
the same notation as in [44–46,57,58]. However, while in
this section we give a general description, in our analysis
below, contrary to our previous analyses [57], this time we
concentrate on the origin of the flavor mixing as discussed in
the previous sections.
The most general hypothesis for flavor mixing assumes
a mass matrix that is not diagonal in flavor space, both for
squarks and sleptons. In the squarks sector and charged slep-
ton sector we have 6 × 6 mass matrices, based on the corre-
sponding six electroweak interaction eigenstates,
˜
U
L,R
with
U = u, c, t for up-type squarks,
˜
D
L,R
with D = d, s, b for
down-type squarks and
˜
L
L,R
with L = e,μ,τ for charged
sleptons. For the sneutrinos we have a 3×3 mass matrix, since
within the MSSM even with type I seesaw (right-handed
neutrinos decouple below their respective mass scale) we
have only three electroweak interaction eigenstates, ˜ν
L
with
ν = ν
e
,ν
μ
,ν
τ
.
The non-diagonal entries in this 6 × 6 general matrix for
sfermions can be described in terms of a set of dimensionless
parameters δ
FAB
ij
(F = Q, U, D, L , E; A, B = L, R; i, j =
1, 2, 3, i = j) where F identifies the sfermion type, L , R
refer to the “left-” and “right-handed” SUSY partners of the
corresponding fermionic degrees of freedom, and i, j indices
run over the three generations. (Non-zero values for the δ
FAB
ij
are generated via the processes discussed in the previous
subsections.)
One usually writes the 6 × 6 non-diagonal mass matri-
ces, M
2
˜u
and M
2
˜
d
, referred to the Super-CKM basis, being
ordered, respectively, as ( ˜u
L
, ˜c
L
,
˜
t
L
, ˜u
R
, ˜c
R
,
˜
t
R
), (
˜
d
L
, ˜s
L
,
˜
b
L
,
˜
d
R
, ˜s
R
,
˜
b
R
) and M
2
˜
l
referred to the Super-PMNS basis,
being ordered as (˜e
L
, ˜μ
L
, ˜τ
L
, ˜e
R
, ˜μ
R
, ˜τ
R
), and write them
in terms of left- and right-handed blocks M
2
˜qAB
, M
2
˜
lAB
(q = u, d, A, B = L, R), which are non-diagonal 3 × 3
matrices,
M
2
˜q
=
⎛
⎝
M
2
˜qLL
M
2
˜qLR
M
2†
˜qLR
M
2
˜qRR
⎞
⎠
, ˜q =˜u,
˜
d, (13)
where
M
2
˜uLLij
= m
2
˜
U
L
ij
+ (m
2
u
i
+ (T
u
3
− Q
u
s
2
w
)M
2
Z
cos 2β)δ
ij
,
M
2
˜uRRij
= m
2
˜
U
R
ij
+ (m
2
u
i
+ Q
u
s
2
w
M
2
Z
cos 2β)δ
ij
,
M
2
˜uLRij
=H
0
2
A
u
ij
− m
u
i
μ cot βδ
ij
, (14)
M
2
˜
dLLij
= m
2
˜
D
L
ij
+ (m
2
d
i
+ (T
d
3
− Q
d
s
2
w
)M
2
Z
cos 2β)δ
ij
,
M
2
˜
dRRij
= m
2
˜
D
R
ij
+ (m
2
d
i
+ Q
d
s
2
w
M
2
Z
cos 2β)δ
ij
,
M
2
˜
dLRij
=H
0
1
A
d
ij
− m
d
i
μ tan βδ
ij
,
and
M
2
˜
l
=
⎛
⎝
M
2
˜
lLL
M
2
˜
lLR
M
2†
˜
lLR
M
2
˜
lRR
⎞
⎠
, (15)
where
M
2
˜
lLLij
= m
2
˜
Lij
+
m
2
l
i
+
−
1
2
+ s
2
w
M
2
Z
cos 2β
δ
ij
,
M
2
˜
lRRij
= m
2
˜
Eij
+ (m
2
l
i
− s
2
w
M
2
Z
cos 2β)δ
ij
, (16)
M
2
˜
lLRij
=H
0
1
A
l
ij
− m
l
i
μ tan βδ
ij
,
with, i, j = 1, 2, 3, Q
u
= 2/3, Q
d
=−1/3, T
u
3
= 1/2, and
T
d
3
=−1/2. The M
Z,W
denote the Z and W boson masses,
with s
2
w
= 1 − M
2
W
/M
2
Z
= 1 − c
2
w
, and (m
u
1
, m
u
2
, m
u
3
) =
(m
u
, m
c
, m
t
), (m
d
1
, m
d
2
, m
d
3
) = (m
d
, m
s
, m
b
) are the
quark masses and (m
l
1
, m
l
2
, m
l
3
) = (m
e
, m
μ
, m
τ
) are the
lepton masses. μ is the Higgsino mass term and tan β =
v
2
/v
1
with v
1
=H
0
1
and v
2
=H
0
2
being the two vac-
uum expectation values of the corresponding neutral Higgs
boson in the Higgs SU(2)
L
doublets, H
1
= (H
0
1
H
−
1
) and
H
2
= (H
+
2
H
0
2
).
It should be noted that the non-diagonality in flavor comes
exclusively from the soft SUSY-breaking parameters, that
could be non-vanishing for i = j , namely: the masses
m
˜
Qij
and m
˜
Lij
for the sfermion SU(2) doublets, the masses
m
2
˜
U
L
ij
, m
2
˜
U
R
ij
, m
2
˜
D
L
ij
, m
2
˜
D
R
ij
, m
˜
Eij
for the sfermion SU(2)
singlets and the trilinear couplings, A
f
ij
.
In the sneutrino sector there is, correspondingly, a one-
block 3×3 mass matrix, that is referred to the ( ˜ν
eL
, ˜ν
μL
, ˜ν
τ L
)
electroweak interaction basis:
123