Particle Physics Booklet - Particle Data Group - Lawrence Berkeley ...

298 41. Cross-section formulae for specific processes
41.3. Hadroproduction of heavy quarks
For hadroproduction of heavy quarks Q = c, b, t, it is important to
include mass effects in the formulae. For q¯q → Q ¯ Q, one has
dσ
dΩ (q¯q → Q ¯ Q)= α2s 9s3 �
(m 2 Q − t)2 +(m 2 Q − u)2 +2m 2 Qs �
, (41.20)
while for gg → Q ¯ Q one has
dσ
dΩ (gg → Q ¯ Q)= α2 �
s 6
32s s2 (m2Q − t)(m2Q − u)
− m2 Q (s − 4m2 Q )
3(m2 Q − t)(m2 4 (m
+
Q − u) 3
2 Q − t)(m2 Q − u) − 2m2 Q (m2 Q + t)
(m2 Q − t)2
+ 4 (m
3
2 Q − t)(m2 Q − u) − 2m2 Q (m2 Q + u)
(m2 Q − u)2
�
− 3 (m2 Q − t)(m2 Q − u)+m2 Q (u − t)
s(m2 −3
Q − t)
(m2 Q − t)(m2 Q − u)+m2 Q (t − u)
s(m2 �
.
Q − u)
(41.21)
41.4. Production of Weak Gauge Bosons
41.4.1. W and Z resonant production :
Resonant production of a single W or Z is governed by the partial widths
√
2GF m
Γ(W → ℓiνi) =
3 W
(41.22)
√
12π
2GF |Vij|
Γ(W → qiqj)=3 2m3 W
(41.23)
√
12π
2GF m
Γ(Z → ff) =Nc
3 Z
�
6π
× (T3 − Qf sin 2 θW ) 2 +(Qf sin θW ) 2�
. (41.24)
The weak mixing angle is θW . The CKM matrix elements are Vij. Nc is 3
for qq and 1 for leptonic final states. These widths along with associated
branching fractions may be applied to the resonance production formula
of Sec. 41.1 to gain the total W or Z production cross section.
41.4.2. Production of pairs of weak gauge bosons :
The cross section for ff → W + W − is given in term of the couplings of the
left-handed and right-handed fermion f, ℓ =2(T3−QxW ), r = −2QxW ,
where T3 is the third component of weak isospin for the left-handed f, Q
is its electric charge (in units of the proton charge), and xW =sin2θW :
dσ 2πα2
=
dt Ncs2 �⎡� ⎣
ℓ + r s
Q +
4xW s − m2 �2 �
ℓ − r s
+
4x
Z
W s − m2 � ⎤
2
⎦ A(s, t, u)
Z
+ 1
�
Q +
2xW ℓ s
2xW s − m2 �
(Θ(−Q)I(s, t, u) − Θ(Q)I(s, u, t))
Z
+ 1
8x2 �
(Θ(−Q)E(s, t, u)+Θ(Q)E(s, u, t)) , (41.26)
W
where Θ(x) is1forx>0and0forx