318
10. Loops and Renormalization I: The ABC Theory
[2]
However, equation (10.55) shows that the divergent part of Π is independent
C
of q
2 , or equivalently that the quantity (10.57) is finite. It follows that Z C is
finite in this theory. In other theories, quantities analogous to (10.55) might
contain a q
2 -dependent divergence, which would be formally absorbed in the
rescaling represented by Z C .
We may also analyse the vertex correction G
[2] of figure 10.6, and conclude
that it too is finite, because there are now three propagators giving six powers
of k in the denominator, with still only a four-dimensional d
4 k integration.
Once again, the analogous vertex correction in QED is divergent, as we shall
see in chapter 11; there too this divergence can be absorbed into a redefinition
of the physical charge. The ABC theory is, in fact, a ‘super-renormalizable’
one, meaning (loosely) that it has fewer divergences than might be expected.
We shall come back to the classification of theories (renormalizable, nonrenormalizable and super-renormalizable) at the end of the following chapter.
While it is not our purpose to present a full discussion of one-loop renormalization in the ABC theory (because it is not of any direct physical interest)
we will use it to introduce one more important idea before turning, in the next
chapter, to one-loop QED.
10.4 Bare and renormalized perturbation theory
10.4.1 Reorganizing perturbation theory
We have seen that, of the one-loop effects listed at the end of section 10.2, the
mass shifts given by equations such as (10.14) do involve formal divergences
as Λ → ∞, but the vertex correction and field strength renormalization are
finite in the ABC theory. We shall find that in QED the corresponding quantities are all divergent, so that the perturbative replacement of all Lagrangian
parameters by their ‘physical’ counterparts, together with field strength renormalizations, is mandatory in QED in order to get rid of ln Λ terms. However,
this process – of evaluating the connections between the two sets of parameters, and then inserting them into all the calculated amplitudes – is likely
to be very cumbersome. In this section, we shall introduce an alternative
formulation, which has both calculational and conceptual advantages.
By way of motivation, consider the QED analogue of the divergent part
of equation (10.7), which contributes a correction to the bare electron mass
of the form αm ln(Λ/m) where m is the electron mass. At Λ = 100 GeV the
magnitude of this is about 0.04 MeV (if we take m to have the physical value),
which is a shift of some 10%. The application of perturbation theory would
seem more plausible if this kind of correction were to be included from the
start, so that the ‘free’ part of the Hamiltonian (or Lagrangian) involved the
physical fields and parameters, rather than the (unobserved) ones appearing
10. Loops and Renormalization I: The ABC Theory
[2]
However, equation (10.55) shows that the divergent part of Π is independent
C
of q
2 , or equivalently that the quantity (10.57) is finite. It follows that Z C is
finite in this theory. In other theories, quantities analogous to (10.55) might
contain a q
2 -dependent divergence, which would be formally absorbed in the
rescaling represented by Z C .
We may also analyse the vertex correction G
[2] of figure 10.6, and conclude
that it too is finite, because there are now three propagators giving six powers
of k in the denominator, with still only a four-dimensional d
4 k integration.
Once again, the analogous vertex correction in QED is divergent, as we shall
see in chapter 11; there too this divergence can be absorbed into a redefinition
of the physical charge. The ABC theory is, in fact, a ‘super-renormalizable’
one, meaning (loosely) that it has fewer divergences than might be expected.
We shall come back to the classification of theories (renormalizable, nonrenormalizable and super-renormalizable) at the end of the following chapter.
While it is not our purpose to present a full discussion of one-loop renormalization in the ABC theory (because it is not of any direct physical interest)
we will use it to introduce one more important idea before turning, in the next
chapter, to one-loop QED.
10.4 Bare and renormalized perturbation theory
10.4.1 Reorganizing perturbation theory
We have seen that, of the one-loop effects listed at the end of section 10.2, the
mass shifts given by equations such as (10.14) do involve formal divergences
as Λ → ∞, but the vertex correction and field strength renormalization are
finite in the ABC theory. We shall find that in QED the corresponding quantities are all divergent, so that the perturbative replacement of all Lagrangian
parameters by their ‘physical’ counterparts, together with field strength renormalizations, is mandatory in QED in order to get rid of ln Λ terms. However,
this process – of evaluating the connections between the two sets of parameters, and then inserting them into all the calculated amplitudes – is likely
to be very cumbersome. In this section, we shall introduce an alternative
formulation, which has both calculational and conceptual advantages.
By way of motivation, consider the QED analogue of the divergent part
of equation (10.7), which contributes a correction to the bare electron mass
of the form αm ln(Λ/m) where m is the electron mass. At Λ = 100 GeV the
magnitude of this is about 0.04 MeV (if we take m to have the physical value),
which is a shift of some 10%. The application of perturbation theory would
seem more plausible if this kind of correction were to be included from the
start, so that the ‘free’ part of the Hamiltonian (or Lagrangian) involved the
physical fields and parameters, rather than the (unobserved) ones appearing