358
11. Loops and Renormalization II: QED
such as e (or α). In this case, since there are no mass factors (for good or ill)
to be associated with powers of α, as we go up in order of perturbation theory
it would seem plausible that the divergences get essentially no worse, and can
be cured by the counter terms which compensated those simplest divergences
which we examined in earlier sections – though for QED the proof is difficult,
and took many years to perfect.
Given any renormalizable theory, such as QED, it is always possible to
suppose that the ‘true’ theory contains additional non-renormalizable terms,
provided their mass scale is very much larger than the energy scale at which
the theory has been tested. For example, a term of the form (11.80) with
‘K/m’ replaced by some very large inverse mass M
−1 would be possible, and
would contribute an amount of order 4e/M to a lepton magnetic moment.
The present level of agreement between theory and experiment in the case of
the electron’s moment implies that M ≥ 4 × 10
9 GeV.
From this perspective, then, it may be less of a mystery why renormalizable theories are generally the relevant ones at presently posed energies.
Returning to the line of thought introduced in section 10.1.1, we may imagine that a ‘true’ theory exists at some enormously high energy Λ (the Planck
scale?) which, though not itself a local quantum field theory, can be written
in terms of all possible fields and their couplings, as allowed by certain symmetry principles. Our particular renormalizable subset of these theories then
emerges as a low-energy effective theory, due to the strong suppression of the
non-renormalizable terms. Of course, for this point of view to hold, we must
assume that the latter interactions do not have ‘unnaturally large’ couplings,
when expressed in terms of Λ.
This interpretation, if correct, deals rather neatly with what was, for many
physicists, an awkward aspect of renormalizable theories. On the one hand,
it was certainly an achievement to have rendered all perturbative calculations
finite as the cut-off went to infinity; but on the other, it was surely unreasonable to expect any such theory, established by confrontation with experiments
in currently accessible energy regimes, really to describe physics at arbitrarily
high energies. On the ‘low-energy effective field theory’ interpretation, we can
enjoy the calculational advantages of renormalizable field theories, while acknowledging – with no contradiction – the likelihood that at some scale ‘new
physics’ will enter.
Having thus argued that renormalizable theories emerge ‘naturally’ as lowenergy theories, we now seem to be faced with another puzzle: why were weak
interactions successfully describable, for many years, in terms of the nonrenormalizable four-fermion theory? The answer is that non-renormalizable
theories may be physically detectable at low energies if they contribute to
processes that would otherwise be forbidden. For example, the fact that (as
far as we know) neutrinos have neither electromagnetic nor strong interactions,
but only weak interactions, allowed the four-fermion theory to be detected –
but amplitudes were suppressed by powers of s/M
2 (relative to comparable
W
electromagnetic ones) and this was, indeed, why it was called ‘weak’ !
11. Loops and Renormalization II: QED
such as e (or α). In this case, since there are no mass factors (for good or ill)
to be associated with powers of α, as we go up in order of perturbation theory
it would seem plausible that the divergences get essentially no worse, and can
be cured by the counter terms which compensated those simplest divergences
which we examined in earlier sections – though for QED the proof is difficult,
and took many years to perfect.
Given any renormalizable theory, such as QED, it is always possible to
suppose that the ‘true’ theory contains additional non-renormalizable terms,
provided their mass scale is very much larger than the energy scale at which
the theory has been tested. For example, a term of the form (11.80) with
‘K/m’ replaced by some very large inverse mass M
−1 would be possible, and
would contribute an amount of order 4e/M to a lepton magnetic moment.
The present level of agreement between theory and experiment in the case of
the electron’s moment implies that M ≥ 4 × 10
9 GeV.
From this perspective, then, it may be less of a mystery why renormalizable theories are generally the relevant ones at presently posed energies.
Returning to the line of thought introduced in section 10.1.1, we may imagine that a ‘true’ theory exists at some enormously high energy Λ (the Planck
scale?) which, though not itself a local quantum field theory, can be written
in terms of all possible fields and their couplings, as allowed by certain symmetry principles. Our particular renormalizable subset of these theories then
emerges as a low-energy effective theory, due to the strong suppression of the
non-renormalizable terms. Of course, for this point of view to hold, we must
assume that the latter interactions do not have ‘unnaturally large’ couplings,
when expressed in terms of Λ.
This interpretation, if correct, deals rather neatly with what was, for many
physicists, an awkward aspect of renormalizable theories. On the one hand,
it was certainly an achievement to have rendered all perturbative calculations
finite as the cut-off went to infinity; but on the other, it was surely unreasonable to expect any such theory, established by confrontation with experiments
in currently accessible energy regimes, really to describe physics at arbitrarily
high energies. On the ‘low-energy effective field theory’ interpretation, we can
enjoy the calculational advantages of renormalizable field theories, while acknowledging – with no contradiction – the likelihood that at some scale ‘new
physics’ will enter.
Having thus argued that renormalizable theories emerge ‘naturally’ as lowenergy theories, we now seem to be faced with another puzzle: why were weak
interactions successfully describable, for many years, in terms of the nonrenormalizable four-fermion theory? The answer is that non-renormalizable
theories may be physically detectable at low energies if they contribute to
processes that would otherwise be forbidden. For example, the fact that (as
far as we know) neutrinos have neither electromagnetic nor strong interactions,
but only weak interactions, allowed the four-fermion theory to be detected –
but amplitudes were suppressed by powers of s/M
2 (relative to comparable
W
electromagnetic ones) and this was, indeed, why it was called ‘weak’ !