Pictures Paths Particles Processes

November 7, 2013 121
where the Källén function crops up again. In the P µ rest frame, the phase space
integration element is therefore given by
dV (P ; q 1 , q 2 ) = 1 (
32π 2 λ 1, m 1 2
s , m 2 2 ) 1/2
dΩ . (4.52)
s
4.5.3 A decay process
As a first application, we shall assume that M > 2m so that the F particle can
decay into a pair of E’s:
F (P ) → E(q 1 ) E(q 2 ) .
In lowest order, its single Feynman graph is given by
P
q
q
2
1
The corresponding matrix element is quite simple :
M = −i mλ
¯h
(√¯h
) 3
= −imλ
√¯h , (4.53)
so that it has dimensionality dim[1/L] as it should. The decay width is therefore
dΓ(F → EE) =
1
2M |M|2 dV (P ; q 1 , q 2 ) 1
√
2!
= m2 λ 2¯h
128π 2 M
1 − 4m2 dΩ . (4.54)
M
2
Note the occurrence of the symmetry factor 1/2! arising from the fact that the
two final-state E particles are indistinguishable. The angular integration is of
course trivial in this simple case, and we immediately find the total width
Γ(F → EE) = m2 λ 2¯h
32πM
with the correct dimensionality dim[Γ] = dim[1/L].
4.5.4 A scattering process
√
1 − 4m2
M 2 , (4.55)
As a second application, we take the mass M of the F particle to be zero. We
now have an extremely primitive picture of the electron-photon system, where
E is the electron and F the photon. We consider the process of ‘Compton
scattering’ :
E(p 1 ) F (p 2 ) → E(q 1 ) F (q 2 )