Hence, the S-matrix may be calculated from U'j, which has no vacuum bubbles. The removal of the vacuum bubbles does not change any physics because vacuum bubbles contribute only to an overall phase, which is unobservable.

Now consider the fifth term in Eq. (11.30). This contributes to the self-energy of a photon and is referred to as the vacuum polarization. The relevant Feynman diagram is given in Fig. 11.8. From our experience so far, we expect the Feynman diagram to give the following integral:

' d*p 7j.q.e3/ [m+ j>]a,a 7la0t\ [m+ j> -(27r)4 [m - p2 - ie] [m2 - (p - q)2 - ie]

d4p tr

This result is almost correct. The correct result includes an extra minus sign which is associated with every closed fermion loop and gives us another Feynman rule:

Rule 6: for each closed fermion loop, a minus sign.

To derive this result, with the correct sign, compute the matrix element of the fifth term in Eq. (11.30):

P2-ie la0

The photon matrix element gives (q'€f\:AHxl)Ai(x2):\q€i) =

s/2iTI^L3

The two terms come from the two possible pairings of annihilation and creation operators in : A A: . The rest of the integrand is symmetric under i <-> j, x\ <-> x2, and p <-> p', because the trace can be cyclically rearranged. Hence the two terms

11.5 VACUUM POLARIZATION 341

Fig. 11.9 Three diagrams which contribute to the dressed photon propagator.

Fig. 11.9 Three diagrams which contribute to the dressed photon propagator.

Hence Z3 removes the infinity contained in 11^(0), and its removal from the theory will eliminate the infinities associated with vacuum polarization.

The constant Z3 is absorbed into the charge, just as was done with Z2 for the electron. There must be one charge at the end of each photon line, and hence sfZl is absorbed into each charge (in this case it is only because only one photon is connected to each charge). Therefore the result [Eq. (11.54)] for charge renormalization gets extended to

However, we have still not finished with charge renormalization!

Finally, external photons must be renormalized in the same way as electrons, giving us an addition to Rule 8:

Rule 8: for each external photon, a factor eM or \fZ~z e'1 * depending on whether or not the photon is incoming or outgoing.

We close this discussion with a final observation. For j<j2 small, after removal of Z3, the dressed photon propagator becomes (see the next section)

For electron scattering, the momentum transfers q2 are negative, and hence we see that the effective force between charged particles which are scattering increases with higher energy (momentum transfer) corresponding to an increase in the force

at shorter distances. This can be restated by saying that the effective charge at short distances (high momentum transfer) grows. To get a quantitative estimate of the importance of this effect on atomic systems, expand ( 11.75) for q2 - -q2,

157Tjn2

Fourier transforming this to momentum space gives the familiar Coulomb potential plus the Uehling term,

-Ze2

Anr 607T27n2

Note that this affects S-states only, and contributes to the Lamb shift which we estimated in Chapter 3. It is of the opposite sign, contributing about -27 MHz to the overall shift of about 1058 MHz. Since the Lamb shift is known to about 0.01 MHz, this effect makes a small but important contribution to the total, and the overall agreement between theory and experiment confirms the correctness of this estimate.

In QCD, other terms due to gluon self-interactions contribute to the gluon self-energy. These terms change the sign of the corresponding n-function, giving the result that the effective coupling constant decreases at short distances (high momentum transfer). This leads to the remarkable property of QCD known as asymptotic freedom, in which the forces go to zero at high energy. It also suggests that the forces will increase at high distances (low energy) and hence suggests onfinement. These very interesting subjects will be taken up in Chapter 17.

We now discuss the evaluation of the loop integral (11.67).

In this section we develop a general method which can be used to evaluate any one-loop Feynman integral. The method will be extended in Sec. 16.2 to the evaluation of Feynman integrals with more than one loop, and with these techniques we will be able to evaluate all Feynman diagrams. All of the formulae needed are summarized in Appendix C.

A general one-loop Feynman integral (in four space-time dimensions) is of the following form:

where the Ai are the denominators of Feynman propagators [cf. Eq. (11.36) for the electron self-energy and Eq. (11.67) for the vacuum polarization] and N is a numerator function which is a polynomial in the loop momentum k^. The calculation of this Feynman integral is carried out in two steps.

11.6 LOOP INTEGRALS AND DIMENSIONAL REGULARIZATION 343

• The different denominators are combined into a single denominator, and the combined denominator is reduced to standard form by translating, or shifting, the internal loop momentum.

• The integral is then evaluated using an integral identity. The first step makes use of identities of the form which are easily proved by direct integration. The integration variables Zi are referred to as Feynman parameters. The two identities (11.79) are the only two we will need in this chapter, but a completely general identity which covers any case which might be encountered is proven in Sec. 16.2, and given in Appendix

To complete the reduction to standard form (the first step), it is necessary to observe that the combined denominator D always has the form where k is the internal loop momentum and Q is a vector function of the external momenta and the Feynman parameters. This form follows from the observation that each of the individual propagators in the loop is itself of the form At — m2 + k ■ qi - k2, so that when they are combined as in Eq. (11.80), the k2 term has the coefficient z\ 4- z2 + (1 - Z\ - z2) = 1. This holds for a loop with any number of propagators. Thus the square of the denominator can always be completed by shifting k = k' + Q, which gives

This shift must also be carried out in the numerator N, which assumes the general form

N = No + k'^ + k'^N^+k^k'XNr+k'tik'X^NrX + - ■ ■ . (11-82)

where the N' are tensors which do not depend on k'. Since the denominator is even in k' (in fact, it depends on k'2 only), all of the odd terms reduce to zero and the even ones can be simplified using identities we will introduce shortly.

D - A\z\ + A2z2 + ^3(1 - zi - z2) = B2 + 2k-Q - k2 ,

After step one has been completed, we are confronted with an integral of the following form:

where n is an integer and we will assume for now that the numerator N is independent of k. We will assume that there are no zeros in the denominator for finite fc2, and consider the convergence of the integral at large k. If n > 2, the integral will converge, but we will encounter many cases when n < 2, and the integral is divergent (the vacuum polarization and electron self-energies are examples where n = 2). The general method for treating these divergent integrals is to imagine that we are evaluating them in a number of space-time dimensions d < 4. In this case, the volume integration goes like ddk ~ kd, but the denominator still goes like fc2n, so the integral will converge as long as d < 2n. As d —» 4, the singularity returns, but it is easily identified and isolated into a renormalization constant, as we have already discussed briefly, and the finite part of the integral is then clearly defined. The process of separating the integral into its finite and infinite parts is referred to as regularization and must be done before the infinity can be removed by absorbing it into the coupling constants of the theory, a process referred to as renormalization. The general procedure for renormalizing theories is discussed in some detail in Chapter 16; in this chapter we introduce these ideas using second order QED as an example.

The integral (11.83) can be evaluated using the following identity:

where d is the number of dimensions (as discussed above) and T(a) is the familiar generalization of the factorial function with T(q) = (a - l)r(a - 1) and r(l) = 1. For a = n, an integer, r(n) = (n - 1)!, but T(q) is also defined for noninteger values of a. A convenient integral representation for T which we will use frequently is*

After step one has been completed, we are confronted with an integral of the following form:

We will use this representation to prove (11.84).

Proof: We begin with the observation that

A good reference for special functions is Abramowitz and Stegun (1964).

A good reference for special functions is Abramowitz and Stegun (1964).

11.6 LOOP INTEGRALS AND DIMENSIONAL REGULARIZATION This identity is easily proved by direct integration:

Jo dz e

Note the crucial role played by the "ie" prescription; it defines the integral in (11.86) by providing convergence for large z and plays a similar role by defining the function at any singular points D2 = 0.

The identity (11.86) is now generalized by differentiating both sides n - 1 times with respect to C2:

Next, we integrate (11.87) over k using the following identities, which hold because 2 > 0, dk I

These integrals may be evaluated using well-known methods for integrating functions in the complex plane. Initially, the integrals are along the real axis in the complex k0 (or k\) plane. To evaluate the first integral, rotate the k0 contour through a positive angle 0. Then ko —► re!c?, and k^ = r2 e21^ = r2 (cos 20 + ¿sin20), so that the integral converges as long as 7r/2 > 0 > 0 (and the contribution from the arc at ko — oo is zero). At 0 = 7r/4 = 45° we have optimal convergence:

,ik0z in/4

2TU/Z

For the dfcj integral, convergence requires rotating by 0 = — 7t/4, giving the opposite sign for i.

For an integral with one time dimension and d — 1 space dimensions, the combined effect of the identities (11.88) is stated in the following identity:

ddk (27r)d

Jk z

To prove this identity, note that in d dimensions, k2 — - and ddk =

dk0 nti dki. Hence the integral factors into d terms,

which proves (11.90).

Finally, combining the results (11.87) and (11.90) gives the result

Scaling this integral by substituting t — iz(C2 - it) gives f ** i _ - w1 yd/2 rtfrMf-.

However, the integral over t is just the integral representation for T(n - ci/2), Eq. (11.85), and hence the identity (11.84) has been proved. I

Before returning to our discussion of vacuum polarization, observe that the integral over the vector components of k can be quickly reduced using the results we have previously obtained. We will show that

_ jgV r(n - 1 - d/2) /J_\ n-l~d'2 __2(4tr)d/2 r (n) VC2 J

To prove this, first note that terms with ^ v are zero because they are odd under changing kM —* -fcM (or kv —* -k"). For the ^ = v terms, assume that C2 is real and positive, and note that the singularities in the k0 complex plane are therefore in the second and fourth quadrants:

Hence we may rotate the k0 integration contour as we did above by letting k0 = re1* and changing (j> continuously from 0 to 7t/2. This changes —» -kfi, and the resulting integral is transformed from a d-dimensional Minkowski space* to a

* A Minkowski space is one with an indefinite metric (in our example the diagonal elements of the metric are +1,-1,-1,-1 in d = 4 dimensions). The rotation of the fco contour has the effect of changing the metric to a Euclidean form: -1,-1,—1,-1.

11.6 LOOP INTEGRALS AND DIMENSIONAL REGULARIZARON

(¿-dimensional Euclidean space, where the integrand is completely symmetric in all components of k, and from this symmetry we can conclude that k^k" —► k2/d. Rotating back to Minkowski space changes the sign of the term on both sides, giving the first line of the identity (11.93).

To get the second, we use the properties of the T-function, as follows

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