11.8. Which theories are renormalizable – and does it matter?
353
11.8 Which theories are renormalizable – and does it
matter?
In the course of our travels thus far, we have met theories which exhibit
three different types of ultraviolet behaviour. In the ABC theory at one-loop
order, we found that both the field strength renormalizations and the vertex
correction were finite; only the mass shifts diverged as Λ → ∞. The theory was
called ‘super-renormalizable’. In QED, we needed divergent renormalization
constants Z i as well as an infinite mass shift – but (although we did not
attempt to explain why) these counter terms were enough to cure divergences
systematically to all orders and the theory was renormalizable. Finally, we
asserted that the anomalous coupling (11.80) was non-renormalizable. In the
final section of this volume we shall try to shed more light on these distinctions
and their significance.
Is there some way of telling which of these ultraviolet behaviours a given
Lagrangian is going to exhibit, without going through the calculations? The
answer is yes (nearly), and the test is surprisingly simple. It has to do with the
dimensionality of a theory’s coupling constant. We have seen (section 6.3.1)
that the dimensionality of ‘g’ in the ABC theory is M
1 (using mass as the
remaining dimension when ħ = c = 1), that of e in QED is M
0 (section 7.4)
¯
and that of the coefficient of the anomalous coupling ˆ
ψ ˆ F ˆ μν in (11.80)
ψσ μν
is M
−1 . These couplings have positive, zero and negative mass dimension,
respectively. It is no accident that the three theories, with different dimensions
for their couplings, have different ultraviolet behaviour and hence different
renormalizability.
That coupling constant dimensionality and ultraviolet behaviour are related can be understood by simple dimensional considerations. Compare, for
example, the vertex corrections in the ABC theory (figure 10.6) and in QED
(figure 11.8). These amplitudes behave essentially as
∫ d
4 k
G
[2]
∼ g
2
(11.94)
ph
k 2 k 2 k 2
and
∫ d
4 k
Γ
[2]
∼ e
2
(11.95)
k 2 k /k /
respectively, for large k. Both are dimensionless: but in (11.94) the positive
2
(mass)
2 dimension of g ph is compensated by two additional factors of k
2 in
the denominator of the integral, as compared with (11.95), with the result
that (11.94) is ultraviolet convergent but (11.95) is not. The analysis can be
extended to higher-order diagrams: for the ABC theory, the more powers of
g ph which are involved, the more denominator factors are necessary, and hence
the better the convergence is. Indeed, in this kind of ‘super-renormalizable’
353
11.8 Which theories are renormalizable – and does it
matter?
In the course of our travels thus far, we have met theories which exhibit
three different types of ultraviolet behaviour. In the ABC theory at one-loop
order, we found that both the field strength renormalizations and the vertex
correction were finite; only the mass shifts diverged as Λ → ∞. The theory was
called ‘super-renormalizable’. In QED, we needed divergent renormalization
constants Z i as well as an infinite mass shift – but (although we did not
attempt to explain why) these counter terms were enough to cure divergences
systematically to all orders and the theory was renormalizable. Finally, we
asserted that the anomalous coupling (11.80) was non-renormalizable. In the
final section of this volume we shall try to shed more light on these distinctions
and their significance.
Is there some way of telling which of these ultraviolet behaviours a given
Lagrangian is going to exhibit, without going through the calculations? The
answer is yes (nearly), and the test is surprisingly simple. It has to do with the
dimensionality of a theory’s coupling constant. We have seen (section 6.3.1)
that the dimensionality of ‘g’ in the ABC theory is M
1 (using mass as the
remaining dimension when ħ = c = 1), that of e in QED is M
0 (section 7.4)
¯
and that of the coefficient of the anomalous coupling ˆ
ψ ˆ F ˆ μν in (11.80)
ψσ μν
is M
−1 . These couplings have positive, zero and negative mass dimension,
respectively. It is no accident that the three theories, with different dimensions
for their couplings, have different ultraviolet behaviour and hence different
renormalizability.
That coupling constant dimensionality and ultraviolet behaviour are related can be understood by simple dimensional considerations. Compare, for
example, the vertex corrections in the ABC theory (figure 10.6) and in QED
(figure 11.8). These amplitudes behave essentially as
∫ d
4 k
G
[2]
∼ g
2
(11.94)
ph
k 2 k 2 k 2
and
∫ d
4 k
Γ
[2]
∼ e
2
(11.95)
k 2 k /k /
respectively, for large k. Both are dimensionless: but in (11.94) the positive
2
(mass)
2 dimension of g ph is compensated by two additional factors of k
2 in
the denominator of the integral, as compared with (11.95), with the result
that (11.94) is ultraviolet convergent but (11.95) is not. The analysis can be
extended to higher-order diagrams: for the ABC theory, the more powers of
g ph which are involved, the more denominator factors are necessary, and hence
the better the convergence is. Indeed, in this kind of ‘super-renormalizable’