F d4k J 2TT4

The electron self-energy, denoted by £(p), is related to My. by

Fig. 11.4 Feynman diagram for the electron self-energy. Note that the loop momentum k cannot be constrained by momentum conservation.

Fig. 11.4 Feynman diagram for the electron self-energy. Note that the loop momentum k cannot be constrained by momentum conservation.

Fig. 11.5 First three of an infinite class of Feynman diagrams which defines the dressed electron propagator.

\/lass and Wave Function Renormalization

To understand the physical significance of the electron self-energy, consider Comp-ton scattering to higher order in e. There are an infinity of diagrams, but they can be organized into an infinite number of classes, with each class itself containing an infinite number of diagrams. The first three diagrams in one class are shown in Fig. 11.5. Recalling the definition of the electron propagator, Eq. (11.21b), these first three diagrams can be written

M6 = -ie2u{p2) iS(p) [-iE(p)] iS(p) [-t£(p)] iS(p) ^ u(px) ,

where in these diagrams £(p) is still given by Eq. (11.36) even though p2 is no longer equal to m2, and

11.3 ELECTRON SELF-ENERGY 333 This infinite class of diagrams can be summed using the geometric series m = £M2„

= -ie2u(p2) ¿f iS(p) [l + E(p)S(p) + [E(p)S(p)]2 = -ie2û(p2) ^ iS'(p) £ u(pi) , where the dressed propagator, S'(p), is iS'(p) = i S(p) [l + E(p)S(p) + [E(p)S(p)]2

Because E(p) is a matrix in Dirac space which transforms like a scalar and because pM is the only four-vector on which it can depend, E(p) must have the form

where A(p2) and B(p2) are scalar functions of p2. Hence E(p) commutes with iS(p), and

The effect of the self-energy E(p) is to modify both the mass and the normalization of the propagator. To see how this comes about, we regard E(p) as a function of j> [which is consistent with Eq. (11.41) because fi2= p2], and expand E(^) in a power series in the quantity (tf -m), where A(p2) and B(p2) are scalar functions of p2. Hence E(p) commutes with iS(p), and

E(j5) = E(m) + (ji -m)E'(m) + -m)2E"(m) + = E(m) + (;t -m)E'(m) + ,

where m is a constant to be chosen shortly and the coefficient of the second term can be found in the usual way, ji=m even though the matrix j>, which is constructed from the 7-matrices, can never equal m, which is a multiple of the identity. Note that the second line of (11.43) is exact because Er is simply the sum of all the higher order terms in the expansion, and therefore, by construction,

11.3 ELECTRON SELF-ENERGY 335

Fig. 11.6 The factor Z2 is removed by absorbing it into charges which occur at the end of each internal fermion line.

In principle, these equations can be solved for m and Z2, but in practice, Z2 is removed from the theory (discussed below) and m is fixed at the physical electron mass. An exception to this general rule occurs in the special case when the unrenormalized mass is zero. In this case, the renormalized mass will also be zero. This follows from Eq. (11.49) for the renormalized mass, which is and hence, in the absence of special conditions, Eq. (11.53) tells us that m — 0. In anticipation of Chapter 13, we point out now that in theories with spontaneous symmetry breaking (not QED), special conditions are established so that mass is spontaneously generated by the interaction. In this case the self-energy functions /. and B are calculated from Feynman diagrams using the anticipated m in place of m (which is zero, by assumption). Then the contribution from A is no longer zero, and Eq. (11.53) is replaced by a transcendental equation for m, known as the gap equation. We defer any further discussion of these ideas to Chapter 13.

The process by which Z2 is removed from the theory is referred to as wave function renormalization, and the change of m to m is referred to as mass renor-malization. It is important that this can be carried out, because A(fn2) and B(m2) are infinite, and if these infinities could not be removed, we could not obtain predictions from QED. After the renormalization is carried out, the remaining expressions are finite, and the theory makes meaningful predictions. In the end, the only thing lost in the renormalization process is the ability to calculate the mass shift of the electron and the change in its electric charge, and since the theory does not tell us how to calculate the electron mass and charge anyway, this does not reduce the predictive power of the original theory.

There are two steps which must be taken to remove the renormalization constant Z2. First, for internal electron lines, break Z2 — y/Z^ \fZ2 and absorb one factor of \[Z2 into each charge at each end of the electron line, as shown in Fig. 11.6. Hence the "bare" charge, eo, must be renormalized as follows:

Next, we must also multiply the wave function of each external electron by \/Zi so that the charge operator connected to each incoming and outgoing electron is similarly renormalized. The Feynman rule which incorporates this step is:

Rule 8: for each external fermion, a factor of \fZ2u, sfZ^v, or \fZ~2v, depending on whether or not the fermion is a particle or an antiparticle, and incoming or outgoing.

The origin of the external \[Z~2 factor is associated with renormalization of the free field functions in the presence of interactions.

The reader is warned that the charge will undergo further renormalization, so this is not what is called "charge renormalization."

Note that the dressed propagator (11.45) has been defined so that the Z2 multiplying Er can also be absorbed into the renormalization of the charges in Eft, so the renormalized dressed propagator, denoted by S, is iSW

Near the particle pole, the renormalization insures that the dressed propagator has the same form as the original undressed propagator.

1.4 VACUUM BUBBLES

Turn now to the last term in Eq. (11.30). Since it is fully contracted, it is a c-number and has a non-zero vacuum expectation value. It describes (see Fig. 11.7) a vacuum fluctuation in which an electron of four-momentum p, a photon of four-momentum k, and a positron of four-momentum -p—k spontaneously materialize from the vacuum at space-time point x\ and propagate to space-time point x2, where they annihilate. It is not zero because the particles are off-shell, so energy and momentum can be conserved.

However, it is not necessary to calculate such vacuum fluctuations; they can be shown to disappear from the theory. To prove this, note that the second order contribution from vacuum bubbles can be written

where it can be shown from (11.30) that c2 is real. This same matrix element also occurs (squared) in fourth order, where the four space-time points x\, x2, x-\.

338 LOOPS AND INTRODUCTION TO RENORMALIZATION the phase from bubble diagrams cancels,

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