where then (10.12) implies that the expansion functions must satisfy the follow ing equations:

Therefore, they are eigenfunctions of the operator O. Furthermore, since the fields satisfy the anticommutation relations (10.10), it follows that the expansion functions satisfy the following relation (the completeness condition):

from which it follows that any function (which satisfies the correct boundary conditions) can be expanded in terms of them. Hence all of the eigenfunctions of the operator O must be used in the field expansion (10.13).

Thus we see that the "free" (or more correctly, the unperturbed) field can be expanded in terms of the solutions of any Hamiltonian. In the study of relativistic

Note that the annihilation and creation operators for the proton (denoted by a subscript p, such as bp) commute with the annihilation and creation operators for

10.2 PHOTON PROPAGATOR: ep SCATTERING 287

the electron (denoted by a subscript e, such as be), so the order of the momenta in \Pi pi) does not matter.

The first term in (10.19) is particularly straightforward, giving

-i(PfPf\J dtHlNSTW PiPi^

= i—j_-__— [ + OC ¿tei(.Ef+£f~Ei-£i)t lMjEiEfEi (2tt)«7_00

* / Y~^~i(Pl~Piyr'e~i{P,~Pi)r [«C/bV*)] ["(P/h°«(Pi)],

where & = ^/m2 + P? and Ei = \/m2 + p? and the spins of the fermions have been suppressed for simplicity, i.e., u(p) = u(p, s). To reduce this, follow the now standard procedure and introduce

Then the integrals give ¿-functions, and the reduced Ai-matrix can be extracted,

47T J p x [u(Pf)y°u(Pi)] [u(pfh«u(p%)} . The integral over p is the familiar Yukawa integral, giving

The second term in (10.19) involves a contraction of the A% field. Most of the details are similar to the OPE calculation worked out in Sec. 9.8, except that the particles are no longer identical. Keeping only the e-p interaction term gives

^{Pj-PiYx.+i^-p^ ^(p^yu^)] [ü{p})^u(pi)] , (10.23)

where we anticipated the fact that (0|T (A1(xi)Aj (x2)) |0) is symmetric in xi,x2 and i,j. Again, introduce xi = X + \x

and use the fact that (0|T (^(xi)^^)) |0) depends only on x to obtain

Mrad = ie2 J d*x (0\T {A'(x1)A^ (x2)) |0) el" x x [u(l»/)7iu(l'i)] [uiPfh'u^)} . (10.25) Next, compute the transverse photon propagator, iDtr, which is defined to be iDH(x) = (0\T {A*(Xl)A3(X2))\0)

= (o\A«+Hx1)A^-\x2)e(h-t2) + A^(x2)A1^(x1)e(t2-tl)\o)

2uiL3

fc cfc

where A1^ are the positive and negative frequency parts of the vector field jperator A1 and the e's are the polarization vectors first introduced in Chapter 2. Recall that the e's are transverse; i.e., k • e - 0. Hence klW

Replacing the sums over k in (10.26) by integrals over k, using Eq. (4.77) to express the 0 functions as integrals over k0 ± w, and recalling that lo2 = k2 give iDli(x) = J

dAk (2vr)4

k ic

where the two poles in the denominator give contributions for xq > 0 and x0 < 0 as described in Eq. (9.45). Equation (10.28) for the transverse photon propagator is very similar to the expressions for the other propagators we have obtained previously; the only difference is the factor in square brackets. To obtain a more useful form, let us introduce the reference vector jf — (1,0). Then k'W

10.2 PHOTON PROPAGATOR: ep SCATTERING Hence the spin sum in Eq. (10.35) can be reduced to a trace u(Pf,Sfh» [«(Pi, SiMPi, 50] 7 vu(Pf, Sf)

S/Si

= £u0 (Pf, Sf) ua (Pf, Sf) [7" (M+ Ift) Y]a0 Sf

= (m+ If})0y(M+ If,) 7"]a/s = trace j If^j 7^ (M+ If,) 7" j . (10.38)

This is a general technique we will employ frequently from now on. It gives spin sums as traces of products of 7-matrices. There are tricks for evaluating such traces which will now be discussed.

Theorems for Computing Traces of Products of 7-Matrices Theorem 1: The trace of an odd number of 7-matrices is zero. Proof:

tr (i, i2...in)= tr jiv • • • in) = (-1)" tr <75 ■ ■ ■ K 75}

Theorem 2: The traces of zero, two, and four powers of 7-matrices are tr (1) = 4 (10.40a)

tr(4l>>t4)=4(a-bc-d-a-cb-d + a-db-c) . (10.40c)

Proof:

tr (if) = tr (P 4) = ±tr(4p + f>4) = a-6tr(l) =4a-6

= 8 a ■ b c ■ d - 2 a • c tr (p 4) + tr (p £ 4 4)

= 8a ■ b c ■ d — 8a ■ c b ■ d + 2a ■ dtr (J> fi) — tv(4 P fi 4)

= 8 (a - bc-d — a-cb-d + a- db-c)— tr(4fi£4) ■

The second interesting case is the ultra-relativistic limit when proton recoil becomes important. Here p' ~ E', p ~ E, m «: E, but E ^ E'. We have

Remarks

Electron-proton scattering is used to measure Ge and G m . These can be separated by measuring the differential cross section at two different scattering angles 9 for the same q2 (referred to as a Rosenbluth separation). Specifically, one measures da dSV

and separates A and B by plotting the ratio as a function of tan2 The structure functions are related to the form factors by

Forward scattering (scattering at small electron angles 6) is dominated by Ge and backward scattering (where 6 is near 180 deg) by Gm• At very high q2, Ge is hard to measure because r »1. At very small q2, r «1 and then Gm is hard to measure.

10.3 ANTIPARTICLES:

We now turn to another important illustration of the power of field theory. In addition to electron scattering, the same expressions also describe the production of pp pairs from e+e~ annihilation. Instead of describing pp production, we describe production, because the ¡i meson does not interact strongly, and therefore the lowest order EM result is more accurate. While the same mechanism works for pp production, it is modified by subsequent strong interactions in the final state, so that the QED result is not very reliable.

The relevant Feynman diagram is shown in Fig. 10.3. It must arise from r+oo

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