602
M. Toharia
Fig. 1 Feynman diagram for
the production cross section
gg → h in a setup with new
vector-like fermions Q, T
and B
g m
g m
h
t, b, T, Q, B
The cross section for the process depicted in Fig. 1, is
σ gg→h =
α 2
s m 2
h
576π
|c
S
ggh |
2
+ |c
P
ggh |
2
δ(s − m
2
h )
(5)
where
c
S
ggh =
3
n=1
Re
y nn
m n
A
S
1/2 (τ f )
and
c
P
ggh =
3
n=1
I m
y nn
m n
A
P
1/2 (τ f ) (6)
with τ = m 2
h /4m 2
n and with the loop functions A S
1/2 (τ ) and A P
1/2 (τ ) as defined in
[9].
The amplitudes c S
ggh and c P
ggh can then be written in terms of traces involving the
fermion mass and Yukawa matrices involving top and vector-like up-quarks, and
bottom and any vector-like down quarks, M i and Y i with i = t, b, so that we obtain
c
S
ggh =
n
Re
y u
nn
m u
n
+
n
Re
y d
nn
m d
n
− Re
y b
m b
+ Re
y b
m b
A
S
1/2 (τ b ) (7)
where we have added and subtracted the bottom quark loop contribution in order to
keep the dependence in A 1/2 (τ b ), and with a similar expression holding for c P
ggh .
We evaluate exactly the sums in the top sector and find
n
y u
nn
m u
n
=
1
v
1 + 3ε Q t ε T
|Y t
2 |
|Y 0
t |
e
iθ t
2
1 − e
iθ t
1
|Y t
1 ||Y 0
t |
|Y Qt ||Y qT |
1 + ε Q t ε T
|Y t
2 |
|Y 0
t |
e
iθ t
2
1 − e
iθ t
1
|Y t
1 ||Y 0
t |
|Y Qt ||Y qT |
,
(8)
where we have defined the small parameters ε T =
v|Y qT |
|M T |
, ε Q t =
v|Y Qt |
|M Q |
, and
θ t
1 and θ t
2 are relative phases between the entries of the Yukawa matrix. Similar
expressions will hold in the bottom sector.