4tt2

This value was first calculated by Schwinger in 1948 [Sc 48].

The current agreement between theory and experiment represents an impressive confirmation of the correctness of QED.* The magnetic moment is often expressed in terms of the gyromagnetic ratio g related to the magnetic moment by where, as we saw in Chapter 5, the value predicted by the Dirac equation is g = 2. It turns out that the departure from this value, usually expressed in terms of the

"For a recent account, see [KL 90], anomalous moment k = g/2-1, can be measured directly and has been measured recently with very high accuracy [VS 87]:

Kexpt = .0011596521884(43) , where the numbers in parentheses are an estimate of the error (in the last digits given). The theoretical value has been calculated to eighth order [KL 90]:

Kih = - 0.328478966 + 1.17611(42) - 1.434(138) = .001 159652140(28) , where the theoretical error is mainly do to the uncertainty in a (which is currently determined from the quantized Hall effect) but also includes the uncertainty in the numerical evaluation of the sixth and eighth order integrals. Hence the difference (experiment - theory) is 0.000000000048(28), giving agreement (within 1.7 standard deviations) for the value of g to a part in 1012!

As we saw, the vertex correction to the electron current diverges as k —> oo. However, the divergence is localized entirely in the F\ term which multiplies and hence only affects the charge. We can renormalize it by subtracting the value of the vertex at q2 = 0, which guarantees that the remainder term is zero at q2 = 0, and hence does not affect the charge. We write

A^P,q)=[Fl(q2)-F1(0)]^ + F2(q2)t-^^+Fl(0)^ . (11.127)

The infinite constant Fi (0) will be a new renormalization constant which we will define to be

We can now fully discuss the renormalization of the charge.

First, note that the sum of all the Feynman diagrams which describe interactions "near" a single charge can be organized into four classes as shown in Fig. 11.12. The central circle represents "proper" vertex corrections, illustrated by the diagram (1) in the upper right corner of the figure. Proper vertex corrections are those which cannot be separated into two disconnected pieces by cutting one electron or one photon line. One says that they are one particle irreducible. The other three diagrams shown in the figure are one particle reducible, or "improper" contributions to the vertex function; they can all be separated into two parts by cutting a single line which connects them to the vertex. The vacuum polarization contribution (2) can be separated from the vertex by cutting the photon line (the dark dashed line shown in the figure is the "cut"), and both of the electron self-energy contributions, (3) and (4), can be separated by cutting an electron line, as shown.

11.9 CHARGE RENORMALIZATION 359

The significance of this analysis is that the sum of all Feynman diagrams \>hich contribute to the vertex (both proper and improper) is the product of all of the diagrams in each of the four separate classes. This is the justification for considering dressed propagators and (proper) vertex corrections separately. The full vertex, P'M, can therefore be expressed in terms of the proper vertex, r" = 7m + A*\ through the following relation:

S(/)r^(p',p)5(/5)A(p' -p) = S'(t')r>1(p\p)S'(i)A'{p' - p) , (11.129)

where S' and A' are dressed propagators and S and A are bare, undressed prop-

Remembering that the renormalization of the propagator must be shared equally between the two charges at either end, we can use (11.129) to obtain the following final result for the renormalization of the electric charge:

If the charge is renormalized in the above fashion, the three renormalization constants Z\, Z2, and Zj, will all be removed from the theory. In Chapter 16 we will discuss how it can be shown that this procedure works to all orders.

agators.

11.9 CHARGE RENORMALIZATION 361

• The contributions from diagrams (2) - (6) are multiplied by \ because only | of these contributions are identified with the charge under study (the other half go with other charges not under consideration).

These factors of \ are the perturbative equivalent of the square root encountered in Sees. 11.2 and 11.5 and in Eq. (11.130) above. In the discussion of self-energies presented in these sections, we summed contributions to all orders in perturbation theory. For the vacuum polarization discussed in Sec. 11.5, the self-energy, as q2 —> 0 where the renormalization is defined, had the form II —» q2(Z3 1 - 1), so the infinite sum of powers of the self-energy multiplied by the propagator given in Eq. (11.71) becomes

Since JZ; is associated with each charge, the relevant expansion for each charge is

which explains the factor of ^ for the second order term. Note that these arguments make use of the fact that Z^1 - 1 is of leading order e2 and is considered small, even though the integral which defines it is divergent. A similar argument holds for the electron self-energies.

However, a significant difference between the electron self-energy terms and the vacuum polarization is that the electron mass is shifted by the self-energy. The infinite electron sum analogous to (11.132), as p2 —» m2, becomes

Since m ^ m, the pole at p2 = m2 is not canceled as p2 —> m2. However, this change in mass is clearly unphysical, because, to each order in perturbation theory, the mass is to be fixed to the observed electron mass. Thus we must add a counterterm to cancel this mass shift; this is easily done by subtracting the term 6m = m — m from Then the sum (11.134) is changed to w + gmy =

giving only the renormalization factor Z2. The mass counterterm therefore keeps track of the mass shift and cancels out, order-by-order, any shift which the calculation produces. This explains our last Feynman rule:

where the range of the d3k integral in the multiplicative factor \C\2 is determined by the experimental conditions, to be discussed shortly.

11.10 BREMSSTRAHLUNG AND RADIATIVE CORRECTIONS 367

scattered electron soft y (not seen)

Fig. 11.16 Soft, forward going photons will always be present in the detector.

scattered electron soft y (not seen)

Fig. 11.16 Soft, forward going photons will always be present in the detector.

in which case k || p, since this is where the cross section is largest. Hence, k cannot be bigger than AB; if it were, it would change the energy of the electron so much that it would either not be seen by the detector at all or it would be recognized as not coming from elastic scattering. (For example, if k were 2AE, and the detector were set to measure scattered electrons of energy E and angle 8, the scattered electron would have to have an energy, before emitting the photon, of at least E + | AE, and such an electron would be traveling at the wrong angle to be confused with an elastically scattered electron.)

One of the central problems in the computation of radiative corrections is that there is no lower limit on the bremsstrahlung photon energy, k, so that the integral diverges at the lower limit. If we choose an arbitrary lower limit, km¡n, the measured cross section becomes

As fcmin —> 0, the correction factor becomes infinite. How does QED control this effect and give finite radiative corrections?

To get finite results, we need to treat corrections to elastic electron scattering of order e4. It turns out that the interference of such corrections with the lowest order process (of order e2) is of order e6 (the same order as bremsstrahlung) and give infinities which precisely cancel those which arise from bremsstrahlung. Diagrammatically, the situation is shown in Fig. 11.17.

There are three types of terms: A (order e2) and B (order e4) are contributions to elastic ep scattering which can interfere and C (order e3) are bremsstrahlung contributions which are added incoherently. Hence, the differential cross section has the form

radiative correction factor

radiative correction factor

lowest order elastic interference elastic bremsstrahlung

lowest order elastic interference elastic bremsstrahlung

We will show that the infinities in the Re (AB*) and |C|2 terms cancel.

Since large k2 contributions are finite, we will track only those terms which diverge as k —> 0. Since we integrate over k in the bremsstrahlung contributions, it does not matter that we also integrate over k in the vertex and self-energy corrections.

The infinite terms at small k arise from the photon pole. Recall Eq. (11.116) for the vertex correction, AM[|(p/ + pl), <j],

A" [\{pj + pt), q] =^¡0)4 Qj) 7" (m+ is - *) V (™+ ti - *) -y*'

where, with A2 = 0, the demoninator D can be written in a factored form which displays its dependence on the virtual photon energy, k0,

D=(k - k0 - ie)(k + k0 - ie)(EPf-k - E¡ + k0 - it)

x (£p/_fc + Ef - k0 - ie)(Ep,_fc - £¿ + k0 - it){EPt-k + Et - k0 - it).

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 12

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