356
11. Loops and Renormalization II: QED
and our amplitude (11.100) is, in fact,
[2]
[2]
−iG F − iG F [G (s) − G (s 0 )].
(11.102)
l
l
In (11.102), we see the familiar outcome of such renormalization – the
appearance of subtractions of the divergent amplitude (cf (10.74), (11.11),
[2]
(11.33) and (11.70)). In fact, because dG /ds is also divergent, we need a
l
second subtraction – and correspondingly, a new counter term, not present in
the original Lagrangian, of the form
¯ ˆ
¯ ˆ
G d ψ n ∂ /ψ ˆ n ψ νe ∂ /ψ ˆ νe
for example; there will also be others, but we are concerned only with the general idea. The occurrence of such a new counter term is characteristic of a nonrenormalizable theory, but at this stage of the proceedings the only penalty
we pay is the need to import another constant from experiment, namely the
[2]
value D of dG /ds at some fixed s, say s = s 0 ; D will be related to the
l
renormalized value of G d . We will then write our renormalized amplitude, up
to 0(G
2 ), as
F
−iG F [1 + D(s − s 0 ) + G ¯ [2] (s)]
(11.103)
l
where G ¯ [2] (s) is finite, and vanishes along with its first derivative at s = s 0 ;
l
[2]
s 0 )
2
¯
that is, G (s) contributes calculable terms of order (s −
if expanded
l
about s = s 0 .
The moral of the story so far, then, is that we can perform a one-loop
renormalization of this theory, at the cost of taking additional parameters
from experiments and introducing new terms in the Lagrangian. What about
the next order? Figure 11.13 shows a two-loop diagram in our theory, which is
of order G
3 . Writing the amplitude as −iG F G
[3] (s), the ultraviolet behaviour
k 4
F
l
of G (s) is given by
[3]
l
(−iG F )
2
∫ d
4 k 1 d
4 k 2
(11.104)
where k is a linear function of k 1 and k 2 . This has a leading ultraviolet
divergence ∼ Λ
4 , even worse than that of G
[2] . As suggested earlier, it is
l
indeed the case that, the higher we go in perturbation theory in this model,
the worse the divergences become. We can, of course, eliminate this divergence
[3]
in G by performing a further subtraction, requiring the provision of more
l
parameters from experiment. By now the pattern should be becoming clear:
new counter terms will have to be introduced at each order of perturbation
theory, and ultimately we shall need an infinite number of them, and hence
an infinite number of parameters determined from experiment – and we shall
have zero predictive capacity.
Does this imply that the theory is useless? We have learned that G ¯ [2] (s)
produces a calculable term of order G
2 (s − s 0 )
2 when expanded about s = s 0 ;
F
l
11. Loops and Renormalization II: QED
and our amplitude (11.100) is, in fact,
[2]
[2]
−iG F − iG F [G (s) − G (s 0 )].
(11.102)
l
l
In (11.102), we see the familiar outcome of such renormalization – the
appearance of subtractions of the divergent amplitude (cf (10.74), (11.11),
[2]
(11.33) and (11.70)). In fact, because dG /ds is also divergent, we need a
l
second subtraction – and correspondingly, a new counter term, not present in
the original Lagrangian, of the form
¯ ˆ
¯ ˆ
G d ψ n ∂ /ψ ˆ n ψ νe ∂ /ψ ˆ νe
for example; there will also be others, but we are concerned only with the general idea. The occurrence of such a new counter term is characteristic of a nonrenormalizable theory, but at this stage of the proceedings the only penalty
we pay is the need to import another constant from experiment, namely the
[2]
value D of dG /ds at some fixed s, say s = s 0 ; D will be related to the
l
renormalized value of G d . We will then write our renormalized amplitude, up
to 0(G
2 ), as
F
−iG F [1 + D(s − s 0 ) + G ¯ [2] (s)]
(11.103)
l
where G ¯ [2] (s) is finite, and vanishes along with its first derivative at s = s 0 ;
l
[2]
s 0 )
2
¯
that is, G (s) contributes calculable terms of order (s −
if expanded
l
about s = s 0 .
The moral of the story so far, then, is that we can perform a one-loop
renormalization of this theory, at the cost of taking additional parameters
from experiments and introducing new terms in the Lagrangian. What about
the next order? Figure 11.13 shows a two-loop diagram in our theory, which is
of order G
3 . Writing the amplitude as −iG F G
[3] (s), the ultraviolet behaviour
k 4
F
l
of G (s) is given by
[3]
l
(−iG F )
2
∫ d
4 k 1 d
4 k 2
(11.104)
where k is a linear function of k 1 and k 2 . This has a leading ultraviolet
divergence ∼ Λ
4 , even worse than that of G
[2] . As suggested earlier, it is
l
indeed the case that, the higher we go in perturbation theory in this model,
the worse the divergences become. We can, of course, eliminate this divergence
[3]
in G by performing a further subtraction, requiring the provision of more
l
parameters from experiment. By now the pattern should be becoming clear:
new counter terms will have to be introduced at each order of perturbation
theory, and ultimately we shall need an infinite number of them, and hence
an infinite number of parameters determined from experiment – and we shall
have zero predictive capacity.
Does this imply that the theory is useless? We have learned that G ¯ [2] (s)
produces a calculable term of order G
2 (s − s 0 )
2 when expanded about s = s 0 ;
F
l