Pictures Paths Particles Processes

November 7, 2013 185
= 4¯h2 Q e 2 Q µ
2
s 2
(
p2 α p 1 β + p 1 α p 2 β − (p 1 · p 2 )g αβ − m e 2 g αβ)
(
q1α q 2β + q 2α q 2β − (q 1 · q 2 )g αβ − m µ 2 g αβ
)
= 4¯h2 Q e 2 Q µ
2
s 2
(
2(p 1 · q 1 )(p 2 · q 2 ) + 2(p 1 · q 2 )(p 2 · q 1 )
− s(p 1 · p 2 ) − s(q 1 · q 2 ) + s 2 )
(7.35)
We shall work in the centre-of-mass frame of the colliding electron-positron
pairs. In that frame, we have
where
p 1,2 0 = q 1,2 0 = E , |⃗p 1,2 | = p , |⃗q 1,2 | = q , (7.36)
s = 4E 2 , p 2 = E 2 − m e
2
The various vector products are therefore given by
(p 1 · p 2 ) = s/2 − m e
2
, q 2 = E 2 − m µ 2 . (7.37)
, (q 1 · q 2 ) = s/2 − m µ 2 ,
(p 1 · q 1 ) = (p 2 · q 2 ) = s/4 − pq cos(θ)
(p 1 · q 2 ) = (p 2 · q 1 ) = s/4 + pq cos(θ) , (7.38)
where θ is the polar scattering angle, that is, the angle between ⃗p 1 and ⃗q 1 . We
also use the fact that Q µ and Q e are the negative of the unit charge, so that
Q µ Q e = 4πα/¯h.This leads to
〈
|M|
2 〉 = 16π2 α 2 (
s 2 s 2 (1 + cos(θ) 2 ) + 4s(m 2 e + m 2 µ ) sin(θ) 2
+ 16m e 2 m µ 2 cos(θ) 2 )
(7.39)
Using what we have already learned about the flux factor and the two-body
phase space, we can write the differential cross sction as
dσ = 1
64π 2 s
[ ] s −
2 1/2
4mµ 〈
|M|
2 〉 dΩ . (7.40)
s − 4m e
2
This cross section therefore only depends on s and the polar scattering angle:
there is, for unpolarized incoming beams, no azimuthal direction singled out
and there is therefore no azimuthal angle dependence 9 . The total cross section
is obtained by simple angular integration, and reads
σ =
4π α2
3 s
(1 + 2 m e 2 ) (1 + 2 m µ 2 ) [ ] s −
2 1/2
4mµ
. (7.41)
s
s s − 4m
2 e
9 This could be different, e.g. in the case of transversely polarized beams.