dd¯ → W 0+W 0−
¯ 2 ) → W 0 − (q− ) + W 0 + (q+ ) goes through a t-channel exchange
The subprocess d(p1 ) + d(p
of the virtual top. The amplitude involves the vertex −g 0 W 0 µ t̄γ µ (gR PR )d, where a pure
right-handed coupling is assumed.
M = −(g 0 gR )2 v̄(p2 )γ ν
p2t
6 pt
γ µ PR u(p1 )ε∗− µ ε∗+ ν
2
− mt
The fermion spin sum over probability gives a numerator factor
0
0
(g 0 gR )4 Tr (6 pt γ µ 6 p1 γ µ 6 pt γ ν 6 p2 γ ν PR ), where pt = p1 − q− = q+ − p2 .
The vector boson polarization sum is done by the formula ε∗ µ εµ0 = −gµµ0 + qµ qµ0 /m0 2W .
X
g 0 4 gR4
|M| = 2
(pt − m2t )2
2
"
t2
2+
2m4W 0
!
t2
Tr (6 pt 6 p1 6 pt 6 p2 ) + 2 2 Tr (6 p2 6 p1 )
mW 0
#
o
g 0 4 gR4 n
2
4
2
2
2
2
(4
+
t
/m
=
0 )[−(t − mW 0 ) − ts] + 4st /mW 0
W
(t − m2t )2
i
g 0 4 gR4 h
2
4
4
2
2
(4
+
t
/m
=
0 )(ut − mW 0 ) + 4st /mW 0
W
(t − m2t )2
The cross section becomes
!
1
1 X
1
β dΩ
dσ =
|M|2
4
3
2s
8π 4π
| {z } | {z } | {z }
|
{z
}
spin
color
flux
phasespace
1
where β = (1 − 4m2W 0 /s) 2 . The above formula agrees well with that by [1] (their E(s, t, u)
in Eq.(3.10)).
uū → Z 0 Z 0 a la Murayam et al.
We can rescale Eq.(3.5) from [1] to approach the Z 0 Z 0 production via Murayama’s scenario
of a off-diagonal new vertex −g 0 Z 0 µ t̄γ µ (gR PR )u. The transition probability becomes
X
T =
+
M = g 04 gR4 × T
h
i
1
2
4
4
2
2
(4
+
t
/m
)(ut
−
m
)
+
4st
/m
Z0
Z0
Z0
(t − m2t )2
h
i
1
2
4
4
2
2
(4
+
u
/m
)(ut
−
m
)
+
4su
/m
0
0
0
Z
Z
Z
(u − m2t )2
"
4m2Z 0 s
tu
1
2s
4ut
−
+ − 2
+
2
2
4
(t − mt )(u − mt )
ut
2mZ 0 2 mZ 0
#
The first and second lines are from the direct squares of the t and u channels. The third
line from the interferece is obtained from rescaling those in Brown and Mikaelian[1]. The
phase space integral has to be divided 2! because of the identical particle effect.
1
References
[1] R. W. Brown and K. O. Mikaelian, Phys. Rev. D 19, 922 (1979).
2