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SEVEN
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The renormalization program of quantum field theory is presented and carried out to the order of one-loop diagrams in electrodynamics. It is then applied to the calculation of the magnetic moment anomaly, radiative corrections to Coulomb scattering (involving an analysis of infrared divergences), the atomic Lamb shift, and photon-photon scattering. We also include a discussion of induced electromagnetic long-range forces between neutral particles in the relativistic regime.
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7-1 ONE-LOOP RENORMALIZATION
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We undertake in this chapter the study of higher orders of perturbation theory. What appears, at first sight, as a straightforward exercise, requiring perhaps analytical skills, turns out to be a highly nontrivial problem due to the presence of ultraviolet divergences. The general presentation of the renormalization theory is postponed to a later chapter. In order to get some familiarity with the subject we concentrate here on the computation of radiative corrections to lowest order in quantum electrodynamics. This enables us to see how we extract sensible results from apparently ill-defined expressions, to compare them with experimental values, and to progressively introduce the concepts of renormalization. A serious drawback of this approach is that electrodynamics is in that respect a rather involved theory. We have to cope with gauge invariance and to disentangle infrared from ultraviolet divergences. Nevertheless, its amazing successes certainly make this effort worth while and justify inverting the logical order.
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The parameters such as masses and coupling constants which appear in the lagrangian are not directly measurable quantities. In the classical point-particle theory, for instance, we must add to the bare mass an electromagnetic contribution to obtain the physical inertial mass. The latter is, of course, finite while the former may well be infinite. We shall therefore give an operational definition to the fundamental parameters (finite in number). Renormalization theory will then show that the perturbative expressions for Green functions are finite when expressed in terms of these physical parameters. Masses will generally be defined as isolated poles of two-point functions. The corresponding residues, which appear as multiplicative constants of scattering amplitudes, will be absorbed into the definition of renormalized fields. Finally, coupling constants will be chosen by fixing the value of certain amplitudes at appropriate points in momentum space. In order to carry out this program it is better to deal first with well-defined finite quantities. The origin of divergences lies in the singular character of Green functions at short relative distances. Equivalently, in momentum space the Fourier transforms do not vanish fast enough at infinity. In an intermediate step we are then led to regularize the theory, i.e., to replace the original expressions by smoother ones such that the integrals become finite. We shall thus proceed in three steps: (1) regularize, (2) renormalize, and (3) eliminate the regularizing parameters. Renormalization will be successful if finite quantities are obtained as a result of this process.
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7-1-1 Vacuum Polarization
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Let us first take a look at the photon propagator in momentum space. To the free contribution
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(7-1)
we should add a correction which, according to the rules of Chap. 6, is given to lowest order by (see Fig. 7-1)
GW(k) = G~ j.(k)wP'V' (k)G~9J(k)
- PV(k) _ _ (_ . )2
d p (2n)4 tr y
(p p _ m + is Y i
(7-2)
p _ ~ _ m + is
The additional minus sign arises from the fermion loop. The integral seems quadratically divergent for large internal momentum p. To give it a meaning we use the Pauli-Villars regularization. This amounts to minimally coupling the
Figure 7-1 Photon propagator to lowest order.
QUANTUM FIELD THEORY
photons to additional spinor fields with a very large mass Asm. These fields might correspond to indefinite metric sectors of the Hilbert space. As far as the vacuum polarization tensor QjPV(k) is concerned, this prescription implies the replacement
QjPV(k, m) -+ QjPV(k, m)
CsQjPV(k, Asm)
(7-3)
s= 1
where the substitution is understood under the integral sign of Eq. (7-2). Were it not for the divergence of the integral we would recover the original value in the limit As -+ 00. The constants C s will be chosen in order to remove this divergence. The minimal coupling of the additional fields implies that gauge invariance is preserved in this regularization procedure. Denote collectively by the symbol A the large masses Asm and let QjPV(k, m, A) stand for the right-hand side of (7-3). We have 4 QjPV(k m A) = _e2 d p { tr "yPCp + m)yV(p - ~ + m) , , (2n)4 (p2 - m2 + ie)[(p - k)2 - m 2 + ieJ
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