Pictures Paths Particles Processes

108 November 7, 2013
Note that the connected blobs may themselves contain many different individual
diagrams. By separating the blobs A and B we indicate that the unstable
particles is actually quite long-lived so that the place where it is produced and
that where it decays tend to be clearly separated.
Now, we shall assume that we have somehow solved the problem of how to
go from connected Green’s function to amplitude, and that we have applied this
procedure to the above process. We then have for the amplitude the form
M = [A]
i¯h
p 2 − m 2 [B] , (4.5)
+ imΓ
where p = q 1 + · · · + q n is the momentum of the (internal!) line corresponding
with the unstable particle, and p 2 = p·p. The unstable particle’s mass is m, and
its total decay width is Γ. The symbols [A] and [B] stand for the processed connected
Green’s functions for the ‘production’ process A and the ‘decay’ process
B, but with the Feynman factors for the unstable particle removed. Assuming,
for simplicity, that F symm = 1, we then have for the differential cross section the
form
dσ = Φ σ |[A]| 2 |[B]| 2 ¯h 2
(p 2 − m 2 ) 2 + m 2 Γ 2 dV (P ; k 1, . . . , k j , q 1 , . . . , q n ) , (4.6)
where P = p a + p b . In order to emphasize that p is the sum of the q’s, we may
write this also as
dσ = Φ σ |[A]| 2 |[B]| 2 dV (P ; k 1 , . . . , k j , q 1 , . . . , q n )
¯h 2
(p 2 − m 2 ) 2 + m 2 Γ 2 d 4 p
(2π) 4 (2π)4 δ 4 (p − Σq) , (4.7)
with obvious notation for the sum over the wavevectors q.
Now, we let the unstable particle approach stability, so that the location
where it decays becomes widely separated from that where it is produced. That
is, we examine the case that Γ becomes very, very small, and we may approximate
7 1
(p 2 − m 2 ) 2 → π
+ m 2 Γ 2 mΓ δ(p2 − m 2 ) . (4.8)
We can then use this to rewrite
dV (P ; k 1 , . . . , k j , q 1 , . . . , q n )
(p 2 − m 2 ) 2 + m 2 Γ 2 d 4 p
(2π) 4 (2π)4 δ 4 (p − Σq) (4.9)
7 This follows from the well-known representation of the Dirac delta function as
1
δ(x) = lim
z→0 π
which has unit integral and vanishes for every x ≠ 0.
z
x 2 + z 2 ,