Note that this is zero at q2 = 0, as expected. For small |g|2 « m2, n(<?2) can be approximated by expanding the logarithm n(92) — -4 f dxx2{l-x)2 = ^£-2 , (11.101) |qa|«m2 n m2 J0 157T m2

which is the result we anticipated in (11.77).

Note that the subtracted self-energy n(g2) is complex if q2 > 4m2. To see nis, note that the maximum value of x(l - x) in the interval [0,1] is and hence when q2/4m2 > 1, the argument of the log in Eq. (11.100) becomes negative at some point in the region of integration, and the log becomes complex. As this is an example of a general property of Feynman diagrams which is of great importance, we will discuss it in more detail in the next section.

We begin our discussion of dispersion relations by considering the self-energy of a neutral particle in the symmetric <f>3 theory. The self-energy in second order comes from the matrix element

(p'| J d4xld4x2T (:4>a(xi): :<*>3(x2): ) |p) . (11.102)

Using Wick's theorem, four of these fields must be contracted into propagators, leaving the remaining two to balance to annihilation and creation operators contained in the final and initial states. However, since all of the fields are identical, the two contractions can be made in many ways. There are 3 x 3 = 9 possible

Next, change integration variables from x to s = n2/x(l - x). This gives finally where p(y/s; (j,, n) is the two-body phase space factor defined in Chapter 9, Eq. (9.21).

Equation (11.107) is an example of a dispersion integral. It expresses the amplitude as an integral over the region where it is singular. In addition to displaying the singularities of the Feynman amplitude explicitly, this integral representation defines the amplitude as an analytic function, so that we may study it using the powerful mathematics of complex analysis. If the location of its singularities is known, the mere knowledge that a dispersion relation exists can sometimes be used to estimate the behavior of an amplitude. But the real power of dispersion theory rests in three facts:

• All Feynman diagrams satisfy dispersion relations, and hence the exact amplitudes probably do also.

• There will be a singularity, or a cut, in an amplitude whenever the external variables have values for which it is possible for all the particles in an intermediate state to be on-mass-shell, i.e., to be physical.

• The imaginary part of the amplitude (referred to as its absorptive part) along any of its cuts can be determined from unitarity.

The last two observations give dispersion theory an element of predictive power, and the first means that it is a very general technique for the study of relativistic interactions. In the 1960's, before the advent of gauge theories, it was believed by some that dispersion theory might be the best method for the study of the strong interactions. This did not turn out to be true, but these methods still belong in the arsenal of the well-equipped physicist.*

Our task here is to use the self-energy (11.107) and the vacuum polarization to illustrate the last two of the above general facts about dispersion theory [the first is already illustrated by (11.107)]. The second is illustrated in Fig. 11.10. For the <jp self-energy, the two intermediate particles can be physical whenever the total energy in their center of mass is greater that 2/x, and since q2 is the square of this energy, the cut runs from oo. The same is true of the vacuum polarization diagram; the production of physical e+ e~ pairs is possible whenever the energy of a virtual photon at rest is greater than 2m, or when q2 > 4m2.

*For a review of dispersion methods see, for example. Barton (1965). For details about the singularities of Feynman amplitudes, see Todorov (1971).

Next, change integration variables from x to s = n2/x(l - x). This gives finally

Ai = m2 - (p' - k)2 - it = 2p' • k A2 = m2-{p - k)2 A3 = \2 -k2 -it . The combined denominator becomes

D = 2(21 p' + z2p) ■ k + A2(l -zx- z2) -k2 -it Next, shift k so as to complete the square of the denominator:

k = k' + z\p' + z2p ■ The shifted denominator reduces to

= (21 + z2)2 m2 - ziz2 q2 + A2 (1 - 2i - z2) - k'2 - it , which shows that D is also symmetric in z\ and z2.

This shift in k —> k' must also be carried out in the numerator, where it gives

N" =7" [m+ J (1 - Zl) - 22 i> - 7m [m+ f (1 - z2)~ f' Zl- ft'] 7„ =Y [m+ ft (1 - zi) - z2 i] [m+ t> (1 - z2) - zi i>] 7„

+ iv $ r $ 7, where all terms linear in k' have been dropped (because they will integrate to zero). Using the identities

7"fiH 7m = -2 tU enables us to further reduce the numerator:

N" = - 2m27M + 4m{p'M(l - Zl) - z2pM +p"{ 1 - z2) - zip'»}

- 2( i> (1 - z2) - 2l ¿V( it (1 - 20 - 22 i>) - 2 Y $

i'-i t+t = - 2m27M + 4m {(p' + p)"( 1 - Zl - z2) + (p' - p)fl(z2 - 2X)}

- 2m(l - 2! - z2){r i (1 - zi)- 4 7m(1 - 22)] + 2(1-22X1-20 £7^ -4*'"*' +27"/c'2

=7"(-2m2- 2m2(l - 2X - 22)2 - 2(1 - 22)(1 - 2i)92 + 2fc'2) - 4*:'" ff' + 4m(l - 2x - 22)(p' +p)^ - 2m(l - 2! - z2) (l - \{Zi + *2)) [7^], where Eq. (11.115) was used in the second step, z2-z\ terms have been dropped because they integrate to zero, and we used the Dirac equation to reduce > 0. Next, use the identity (11.93) to replace k^k" by g^k2 in the numerator:

=7m (-2m2 [1 + (1 - 2i - z2)2} - 2q2{\ - 22)(1 - zx) + fc'2) + 4m(1 - z\ - z2)(p' + pY + 4m(l - - z2) (l - \{zx + z2)) ia^qv, where Wl^^A] = cr^qv Finally, use the Gordon decomposition (see Prob. 11.1)

to get

7VM =7M|—2m2 [1 - 4(1 - zx - z2) + (1 - 21 - z2)2} -2q2(l-zi)(l-z2) + k2} - i2m a^q» (1 - zi - z2)(zi + z2) .

Our calculation has shown that the first correction to the "bare" electromagnetic coupling generates a correction to the " M term and a new term of the form

Specifically, the form of the electromagnetic vertex function is h»{P,q) = F,{q2)Y + F2{q2)

icr^qu 2m

where the functions F\ and F2 are scalar functions of q2, the 7M term is the .amiliar Dirac current, and the o^q" term is the induced anomalous (or Pauli) current discussed in Prob. 5.6 and Sec. 10.2. While we have not shown it, (11.120) is the most general form which AM can take. Higher order corrections will not contribute any new operators; they will only add to the scalar functions F\ and F2. To second order, these functions are

[D(k'2)}J where the numerator Nx is

Ari = k2 - 2q2{\ - zi)(l - z2) - 2 m2 [l - 4(1 - Zi - z2) + (1 -

Note that Fx diverges (because of the k'2 term in the numerator), but F2 is finite. We will return to a discussion of Fx later. For now, we evaluate F2 at q2 = 0. The value of F2 at this point is the anomalous magnetic moment of the electron, which we denote by k.

2 if

(2tt)4 [m2(2l + Z2)2 + A2(1 _ Zl _ X2) _ k>3 _ ¿e]3 ' We first do the integration using the identity (11.84) with d = 4 and n = 3, d4k 1 1

32tr2B2

This reduces the integral to

32tt2

Jo Jo

The singularity at z\ + z2 — 0 is only a point in a two-dimensional space and hence is integrable. Let A2 —> 0 and change variables from 21 and z2 to £ and 17, where

Then the volume element transforms to r-l rl-Z! pi A

Jo Jo Jo J - j and the anomalous moment is quickly calculated, giving the famous result

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