348
11. Loops and Renormalization II: QED
This relation is true to all orders (Z 1 = Z 2 ), provided a gauge-invariant
regularization is used. It is a very significant relation, as already indicated
after (11.8). It shows, first, that the gauge principle survives renormalization
provided the regularization is gauge invariant. More physically, it tells us that
the bare and renormalized charges are related simply by (cf (11.6))
1/2
e = e 0 Z .
(11.78)
3
In other words, the interaction-dependent rescaling of the bare charge is due
solely to vacuum polarization effects in the photon propagator, which are
the same for all charged particles interacting with the photon. By contrast,
both Z 1 and Z 2 do depend on the specific type of the interacting charged
particle, since these quantities involve the particle masses. The ratio of bare
to renormalized charge is independent of particle type. Hence if a set of bare
charges are all equal (or ‘universal’), the renormalized ones will be too. But
we saw in section 2.6 how just such a notion of universality was present in
theories constructed according to the (electromagnetic) gauge principle. We
now see how the universality survives renormalization. In volume 2 we shall
find that a similar universality holds, empirically, in the case of the weak
interaction, giving a strong indication that this force too should be described
by a renormalizable gauge theory.
11.7 The anomalous magnetic moment and tests of QED
[2]
Returning now to Γ μ , just as in section 11.5.2 we regarded the vacuum po[2]
¯
larization correction 1 +
1 Π γ as a contribution to the fermion’s charge form
2
factor F 1 (q
2 ), so we may expect that the vertex correction will also contribute
to the form factor. Indeed, let us recall the general form of the electromagnetic
vertex for a spin1 particle (cf (8.208)):
2
[
]
F 2 (q
2 )
′
ν
−ieu ¯(p , s
′ ) F 1 (q
2 )γ μ + iκ
σ μν q u(p, s)
(11.79)
2m
where κ is the ‘anomalous’ part of the magnetic moment, i.e. the magnetic
moment is (eħ/2m)(1 + κ), the ‘1’ being the Dirac value calculated in sec2
tion 3.5. In (11.79), F 1 and F 2 are each normalized to 1 at q = 0. Our
[2]
vertex Γ μ contributes to both the charge and the magnetic moment form
[2]
[2]
factors; let us call the contributions F 1 and κF 2 . Now the Z 1 counter term
[2]
multiplies γ μ , and therefore clearly cancels a divergence in F . Is there also,
1
[2]
we may ask, a divergence in κF ?
2
[2]
Actually, κF is convergent, and this is highly significant to the physics of
2
renormalization. Had it been divergent, we would either have had to abandon
11. Loops and Renormalization II: QED
This relation is true to all orders (Z 1 = Z 2 ), provided a gauge-invariant
regularization is used. It is a very significant relation, as already indicated
after (11.8). It shows, first, that the gauge principle survives renormalization
provided the regularization is gauge invariant. More physically, it tells us that
the bare and renormalized charges are related simply by (cf (11.6))
1/2
e = e 0 Z .
(11.78)
3
In other words, the interaction-dependent rescaling of the bare charge is due
solely to vacuum polarization effects in the photon propagator, which are
the same for all charged particles interacting with the photon. By contrast,
both Z 1 and Z 2 do depend on the specific type of the interacting charged
particle, since these quantities involve the particle masses. The ratio of bare
to renormalized charge is independent of particle type. Hence if a set of bare
charges are all equal (or ‘universal’), the renormalized ones will be too. But
we saw in section 2.6 how just such a notion of universality was present in
theories constructed according to the (electromagnetic) gauge principle. We
now see how the universality survives renormalization. In volume 2 we shall
find that a similar universality holds, empirically, in the case of the weak
interaction, giving a strong indication that this force too should be described
by a renormalizable gauge theory.
11.7 The anomalous magnetic moment and tests of QED
[2]
Returning now to Γ μ , just as in section 11.5.2 we regarded the vacuum po[2]
¯
larization correction 1 +
1 Π γ as a contribution to the fermion’s charge form
2
factor F 1 (q
2 ), so we may expect that the vertex correction will also contribute
to the form factor. Indeed, let us recall the general form of the electromagnetic
vertex for a spin1 particle (cf (8.208)):
2
[
]
F 2 (q
2 )
′
ν
−ieu ¯(p , s
′ ) F 1 (q
2 )γ μ + iκ
σ μν q u(p, s)
(11.79)
2m
where κ is the ‘anomalous’ part of the magnetic moment, i.e. the magnetic
moment is (eħ/2m)(1 + κ), the ‘1’ being the Dirac value calculated in sec2
tion 3.5. In (11.79), F 1 and F 2 are each normalized to 1 at q = 0. Our
[2]
vertex Γ μ contributes to both the charge and the magnetic moment form
[2]
[2]
factors; let us call the contributions F 1 and κF 2 . Now the Z 1 counter term
[2]
multiplies γ μ , and therefore clearly cancels a divergence in F . Is there also,
1
[2]
we may ask, a divergence in κF ?
2
[2]
Actually, κF is convergent, and this is highly significant to the physics of
2
renormalization. Had it been divergent, we would either have had to abandon