Pictures Paths Particles Processes

November 7, 2013 119
statement that the photon propagator is free from infrared divergences 21 . This
is one example of a useful rule of thumb : when you encounter loop diagrams,
try to envisage the physics that is described by cutting them. In fact, the
cancellation can be pinpointed further ; the single statement that the single
diagram
is IR-finite means that the IR divergences in
∣
∣
2
⎛ ⎞ ⎛
and ⎝ ⎠ ⎝
⎞
∗
⎠ + (c.c.)
must cancel between them.
4.5 Some example calculations
4.5.1 The FEE model
As an example of an application of what we have learned so far, we shall investigate
at theory that contains two particle types, one of mass m, denoted by E,
and another denoted by F , of mass M. The Lagrangian density of this theory
is given by
L = 1 2 (∂µ ϕ E ) (∂ µ ϕ E ) − m2
2 ϕ E 2
+ 1 2 (∂µ ϕ F ) (∂ µ ϕ F ) − M 2
2 ϕ F 2 − mλ
2 ϕ F ϕ E 2 . (4.42)
There exists a single coupling between two E’s and one F . Note that the
Feynman rule for the vertex is given 22 by −imλ/¯h ; we have introduced a factor
m in order to ensure that
[ ] 1
dim[λ] = dim
¯h 1/2
with no length scale.
21 Two remarks are in order here. In the first place, the virtual-photon diagrams do contain
divergences related to the loop momentum going to infinity : these are ultraviolet (UV)
divergences. The photon propagator is therefore still ultraviolet divegrent, and this is cured
in the usual manner by renormalization. In the second place, the cancellation if IR divergences
takes place even when we restrict the phase space for the outgoing particles, provided that
zero-energy photons are admitted.
22 It is customary to leave out the (2π) 4 δ 4 () of momentum conservation, since it is present
in all vertex Feynman rules for translation-invariant interactions.