![](https://csdnimg.cn/release/download_crawler_static/12284151/bg5.jpg)
the extra φ
2
scalar doublet, the σ
2
, ρ
2
, η electrically neutral scalar singlets, and the q
jL
,
u
jR
(j = 1, 2, 3) quark fields are nontrivially charged. It is worth mentioning that the fields
u
nR
(n = 1, 2), ρ
2
, ρ
3
and η are charged under the spontaneously broken Z
(1)
2
symmetry,
whereas the remaining fields previously introduced are neutral under this symmetry. As we
will be shown below, the σ
2
, ρ
2
, η electrically neutral scalar singlets will also mediate the
one and two level radiative seesaw mechanisms that give masses to the second and third
family of SM down type quarks and charged leptons as well as to the first family of SM
charged fermions. Some of these scalars singlets will also participate in the three loop level
radiative seesaw mechanism that produces the light active neutrino masses. Furthermore,
the SU
2L
singlet heavy quarks T
L
, T
R
with electric charges equal to 2/3 have to be added
to the fermion spectrum in order to implement the one-loop radiative seesaw mechanism
that gives rise to the charm quark mass. Besides, an electrically neutral weak-singlet scalar
σ
1
is needed in the scalar spectrum to provide a tree-level mass for the exotic T quark and
to close the one-loop Feynman diagram that generates the charm quark mass. Let us note
that σ
1
and ρ
3
are the only scalars neutral under the unbroken Z
(2)
2
symmetry, and thus
they conveniently acquire vacuum expectation values (VEVs) that break the U
1X
gauge
symmetry and the Z
(1)
2
discrete symmetry, respectively.
In addition, we assume that the q
nL
(n = 1, 2) fields have U
1X
charges that are different
from the charge of q
3L
, while the SU
2L
scalar doublets φ
1
and φ
2
have different U
1X
charges
as well. Thus, the third row of the up-type quark mass matrix is generated at tree level,
whereas the first and second row emerge at one-loop level. Note that, since there is only one
heavy exotic T quark mediating the one-loop radiative seesaw mechanism that generates
the first and second row of the up-type quark mass matrix, the determinant of this matrix
is equal to zero. Therefore, the up quark is massless at one-loop level, and in order to
generate an up quark mass at two-loop level, the following operators are required:
q
nL
φ
2
B
3R
, B
4L
ϕ
−
1
u
nR
, B
3L
σ
∗
2
B
4R
, m
B
k
B
kL
B
kR
, n = 1, 2,
σ
1
σ
2
σ
∗
3
ρ
3
, ε
ab
φ
†
1
a
φ
†
2
b
ϕ
+
1
σ
3
, a, b = 1, 2 , , k = 3, 4, (2.4)
where the extra SU
2L
singlet heavy quarks B
3L
, B
3R
, B
4L
, B
4R
have electric charges
equal to −1/3, and σ
3
and ϕ
−
1
are electrically neutral and charged weak-singlet scalars,
respectively. The scalar fields σ
3
and ϕ
−
1
are charged under the spontaneously broken Z
(1)
2
symmetry. The operators given above enable to generate the two-loop contributions to the
first and second rows of the up-type quark mass matrix, and these contributions yield a
nonvanishing determinant for the up-type quark mass matrix, giving rise to a suppressed
two-loop up quark mass. It is worth mentioning that the nonvanishing parts of the SM up
type quark mass matrix top quark are the (3, 3) entry, which appears at tree level and the
upper left 2 ×2 block which receive one and two loop level contributions. Let us note that
in order that the two loop contribution to the upper left 2 × 2 block of the SM up type
quark mass matrix gives a nonvanishing two loop level up quark mass, the one and two
loop level contributions should not have common vertices. If they have a common vertex,
the two loop level contribution can be treated as an effective 1-loop one having the same
the left and (or) right handed fermionic field in the internal line as in the former one loop
– 4 –