where k = (Ek,k). Recall the matrix form of the Dirac spin sums, up(k, s)uy(k, s) = (m +

Fig. 10.10 Feynman diagrams for Compton scattering.

With these new additions to Rules 2-4, we have most of the basic rules needed for elementary calculations. We are still missing the rules associated with renor-malization (so far, all renormalization constants have been set to unity), closed loops, and some of the rules associated with isospin. We have also not discussed the electrodynamics of spin 0 and 1 bosons. We will cover some of these topics in the next chapter and later in Chapters 13 and 15.

Cross Section

We now calculate the unpolarized Compton scattering cross section in the LAB frame. To simplify the expression, we use

The last two conditions hold in the LAB frame, where pi = (m, 0) and t — (0, e), but would not hold in an arbitrary frame. (This is a feature of the Coulomb gauge, which requires that e° = 0, a condition which is not frame independent. Nevertheless, the total result is frame independent.) Also, (pt + ki)2 = m2 + 2mojt because kf = kj = 0. Taking tf to be real for simplicity, we have reduced M using 2p, • 6,- - and = mti^), etc.

Next, calculate the sum over spins of \M\2. Using the technique for carrying out spin sums which we worked out in the previous sections gives ffc • h = ef ■ kf = €i ■ pi = tf ■ pi = 0 .

This classical result can be obtained quite easily directly from the relativistic A^-matrix by letting k ~ k' —» 0 right from the start. A remarkable fact emerges from this calculation. If we use the identity (10.101)

the M -matrix can be written in the form s

[u(pf, Sf) O' u(pt +fcj,a)] [ulPi + ki,s)Ou(pi,si)}

[u(p},Sf)Q'vj-pj-kus)] [v{-pi-ki,s)Ou(pi,si)} 'f 2EPi+ki [EPi+ki + (EPi +wi)]

This is a convenient form for taking the classical limit, and the Thomson cross section can be easily computed in this way. However, it turns out that the classical limit comes entirely from the v v terms (the negative energy contribution). It is left as an exercise (Prob. 10.3) to work this out and discuss the results.

10.1 Calculate the differential cross section, in the one-photon exchange approximation, for the scattering of electrons from pions (pseudoscalar particles) initially at rest. First, write down the correct M-matrix using the Feynman rules (the form of the vertex for a spin zero boson is given in the Appendix). Then, square and calculate the unpolarized cross section. Finally, show that when the energy E of the incoming electron becomes very large,

10.2 Calculate the total cross section for the annihilation of electrons and positrons into muons and antimuons. (The muon is just like a heavy electron.) Do the calculation in the center of mass system, which is also the LAB system in a colliding beam accelerator where this experiment would usually be performed. When the energy E of the electron becomes very large, show that the unpolarized total cross section becomes

[You may use the results from Eq. (10.85) and (10.86).]

10.3 Show that the Thomson cross section can be obtained from the relativistic Feynman diagrams for Compton scattering using only the negative energy part of the virtual electron propagator. That is, using the decomposition show that the full result for the Thomson cross section comes from the second term in this decomposition, the first term giving a vanishingly small contribution. To make the calculation simple, carry out the calculation in the limit when all momenta are « m right from the start.

10.4 Suppose the muon could decay onto an electron and a photon through an electromagnetic-like term of the form ftint = ~9 [4>mu<m(x)la1pe(x) ^ Ve{x)-ya1l>mnon{x)] Aa{x) , where g is an unknown constant. (As far as we know, this process does not occur.) Calculate the total rate for this decay in terms of the unknown constant g.

10.5 Consider the annihilation of electron-positron pairs into two photons, i.e., e~ + e+ —» 27.

(a) Draw all of the Feynman diagrams which contribute to this process to order e2 in the electric charge. Let the momenta of the incoming electron be p-, of the incoming positron p+, and of the outgoing photons k 1 and k2. Label each diagram with these momenta and the momenta of any internal lines.

(b) Write down the correct Feynman amplitude for each diagram.

10.6 Assume two protons scatter by exchanging either a photon or a neutral 7r° meson.

(a) Draw clearly labeled Feynman diagrams showing the interactions to lowest order in the electric charge e or the nNN coupling constant g. Give the mathematical expression for the .M-matrix corresponding to each diagram.

(b) Give a rough estimate of the comparative size of the different diagrams when the scattering takes place at high energy and at non-forward angles (19 > 10°, for example). Which process is more important?

10.7 Consider electron-proton (ep) and positron-proton (ep) scattering in the framework of QED.

(a) Draw all the Feynman diagrams which contribute to ep and ep scattering in lowest order perturbation theory. Write the Al-matrix corresponding to each diagram.

(b) Calculate the difference in the cross sections for ep and ep scattering. It is sufficient to find

spins spins

Alternatively, you may be able to see the answer by examining Mcp and M ep directly.

(c) Using the insight gained in part (b), roughly how accurate an experiment would be required to distinguish between ep and ep scattering?

Fig. 10.11 Diagrams for irN scattering (Prob. 10.8).

10.8 Pión nucleón scattering.

(a) Calculate the total cross section for n+p and it~p scattering near threshold using only the two Feynman diagrams shown in Fig. 10.11. Assume the ■kNN coupling is — \/2 S75 for positively charged pions, 57s for 7r°pp, and -<775 for 7r°nn, where g2/Air = 14.0. Carry out the following steps:

(i) Write down the exact Feynman amplitudes, from the two diagrams in Fig. 10.11, for the following processes:

7T+p

(ii) Evaluate these amplitudes in the center of mass system in the limit when the momenta |p| of both the proton and pion is zero.

(iii) Calculate the total cross section for 7r+ +p —» anything and n~ +p —> anything and compare with the experimental results, which are

JT'Tp

47T ml 47r ml

where mv is the pion mass, (b) Redo the calculation of part (a) with the nNN coupling replaced by r i*t-**) a

2m n

75 for neutral pions (+ [ot ^ 2mN ' \- for 7r°nn, where p¡ is the outgoing nucleón four-momentum, p¿ is the incoming nucleón four-momentum, and to at is the nucleón mass. Do all three steps of this calculation just as you did for part (a) above. (If you are very careful, you will discover an important result in nN physics.)

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 11

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