Pictures Paths Particles Processes

November 7, 2013 117
identical mass) and Lagrangian
L =
9∑
n=1
( 1
2 (∂µ ϕ n )(∂ µ ϕ n ) − 1 2 m2 ϕ 2 n)
− V ,
V = λ 123 ϕ 1 ϕ 2 ϕ 3 + λ 245 ϕ 2 ϕ 4 ϕ 5 + λ 349 ϕ 3 ϕ 4 ϕ 9
+λ 567 ϕ 5 ϕ 6 ϕ 7 + λ 789 ϕ 7 ϕ 8 ϕ 9 . (4.38)
Nothing forbids us to assign to the various ϕϕϕ couplings precisely the value λ.
Now, it is easily seen that, in order λ 123 λ 245 λ 349 λ 567 λ 789 , the diagram (4.37) is
the only diagram that can contribute in this theory 16 ! We can do even more :
by inspection of all possibilities, we can simply realize that the only final states
k in the unitarity condition (4.34) must be precisely k = {2, 3}, {5, 9}, {2, 4, 9}
or {3, 4, 5}, if we want to end up with the right order in perturbation theory 17 .
In other words,
+ +
+ + + = 0 , (4.39)
where we have omitted the line labellings : indeed, the same identity must
hold for the original diagram (4.35)! This establishes the so-called cutting rules
(also called the Cutkoski rules), which can be most simply expressed in words:
take a diagram and move the cutting line through it from right to left in all
possible manners, making sure that the two halves in which the diagram is cut
remain connected and that neither the inital state or the final state is dissected.
The particles described by internal lines through which the cut runs must be
assumed to be on their mass shell 18 . The sum of all the possible contributions
then vanishes 19 .
4.4.4 Infrared cancellations in QED
As an illustration of how the cutting rules may be applied we shall make a
slight jump ahead and consider quantum electrodynamics, that is the theory
16 The secret resides in the fact that in V the external fields 1,6 and 8 occur precisely once,
and the other fields precisely twice.
17 Note that, for instance, the choice k = {5, 7, 8} would result in the right-hand half of the
diagram being disconnected ; the choice k = {2, 4, 7} is inconsistent since both 6 and 8 are in
the final state.
18 This may mean that the situation thus described fails to meet the restrictions of momentum/energy
conservation ; then, that contribution vanishes.
19 You might object that in a theory with many different fields the symmetry factors of
the diagrams will, in general, be different from those of a theory with only a single field,
and this is true : however, in the summation over the ‘intermediate states’ k we must of
course also include the ‘indentical-particle’ symmetry factor F symm, which precisely repairs
the correspondence — another illustration of the crucial rôle of the symmetry factors !