Dp 6P6342 bfc di0 0dpbpy3b4dd2bidl0

The two terms arise because there are now two qualitatively different ways the matrix element can be non-zero. To see this, recall that the full normal ordered expansion of the fermion product in the Hamiltonian density is

: — b\ h [u(q2)^u(qi)} + b\ d\ feb'Mfi)] --„-' --*-'

term 1 term 2

Fig. 10.6 Illustration showing how the exchange diagram can be obtained from the annihilation diagram by antisymmetrization of the two "outgoing" fermions.

where the minus sign in front of term 4 comes from the interchange of the d and d* operators required by the normal ordering. For e+e- —► , only term 3 occurred (paired with term 2 from the muon matrix element), but there were two terms which give rise to it. In e+e~ —» e+e~, the requirement that we have precisely one of each of the b, b>, d, d^ can occur four ways, by the following pairings:

term 2 x term 3 + term 3 x term 2 = 2 x [term 2 x term 3] term 1 x term 4 + term 4 x term 1 = 2 x [term 1 x term 4]

'iowever, as the brackets in Eq. (10.83) show, these lead to the two diagrams given in Fig. 10.5 with a relative minus sign. Furthermore, the two terms differ only in the exchange of — k+ <-» p_, as has already been discussed.

Cross Section

The next task is to calculate the total cross section for the production of pairs in e+e~ collisions. These experiments are normally carried out in colliding beam accelerators, where the LAB system is the same as the CM system. Hence the calculation will be carried out in the CM system. In this system, the energies of the initial and final particles are the same, but the magnitude of the momenta are different. Introduce a = 4£2 = (k+ + it_)2 = 2 m2 + 2 k+ ■ fc_

where m = me is the electron mass and M — mM is the muon mass. The unpolarized cross section from Eq. (9.54), is spins spins

where [recall the discussion following Eq. (10.35)]

where the projection operators for u and v spinors, Eqs. (5.137) and (5.141), have been used.

It is left as an exercise (Prob. 10.2) to complete the calculation of the cross section. If the energy E » M or m, the total cross section reduces to

,-ig. do,r, where, as always, a = is the fine structure constant. This is an interesting result; it leads to a discussion of the total cross section for the production of strongly interacting particles (hadrons) and to a discussion of the evidence for quarks.


As an application of the ideas developed so far, consider e+e~ annihilation into hadrons at very high energy. We can compute the total cross section for this process if we borrow two facts from high energy physics:

• All hadrons are made of spin \ quarks, which are charged, and spin 1 gluons, which are neutral.

• The strong coupling constant, ga, is small at very high energies. In particular, the strong fine structure constant, as = g2/Air, is a function of q2, the square of the momentum transfer, and ag(q2) decreases as q2 increases. For high q2 in the range of 10's to 100's of GeV2, a3(q2) < 0.3, and we can use perturbation theory.

The first of these ideas was already used in Sec. 6.2.

Using these facts, the production of hadrons from a virtual photon (created by the e+e~ annihilation) must proceed first through the creation of a qq pair. The diagram is identical in structure to the production of a up, pair and is shown in Fig. 10.7. Since gluons are neutral, the first correction to this diagram is single gluon emission by a quark, which is described by the two Feynman diagrams shown in Fig. 10.8. In Feynman diagrams, it is customary to represent

Table 10.1 The six flavors and three generations of quarks.









~ 5 - 7 MeV


l 3




~ 1.5 GeV





~150 MeV




not seen





~ 4.5 GeV


the scale of the hadronic cross section.

Since the qq production diagram (10.7) differs from the production diagram (10.3) only by the charge of the quark, we see immediately that

where Qi is the charge of the ith quark (in units of e) and the sum is over all quarks which can be produced at the energy y/s. This is because the cross section is independent of mass, etc., and depends only on (Qje)2. Thus, only Qf enters the ratio.

The known and conjectured, quarks are grouped into three families (or generations), as shown in Table 10.1. (See Appendix D for more discussion.) According to QCD, each flavor of quark comes in three colors (an internal quantum number analogous to spin), so that we have the following predictions:

y + 3 (è) = T for 9 < VK < 2mt0p . The corrections to these simple predictions are about +10%, as described above.

A compilation of the data for R is shown in Fig. 10.9 [taken from RP 92]. Added to the upper figure are solid lines corresponding to the predictions given in Eq. (10.90). The solid lines in the lower figure are the prediction R = ^ theoretical corrections from electroweak and higher order QCD processes. Note that:

• The ratio R does increase at the c and b thresholds and is more or less constant between thresholds.

As a final example, we treat Compton scattering, which is the scattering of photons from electrons 7 + e —» 7 + e. This introduces two new Feynman rules: (i) the treatment of (real) 7's in the initial and final state and (ii) the use of a fermion propagator (in this case an electron) describing the propagation of a virtual, offmass shell spin | particle.

The first (lowest order) non-zero contribution to non-forward Compton scattering comes from the second order contribution of the radiation part of the Hamil-tonian:

S = {-i)2\{ksPj\ J d4xld4x2T(HRAD(xi)HRAD(x2))\klPi) , (10.91)

where kt and kj are the momenta of the incident and outgoing photons, and px and pf are the momenta of the electron. (Why doesn't the Hinst term contribute in this case?) Interchanging xx and x2 permits us to eliminate T in favor of a 9{t 1 — t2) function and a factor of 2, giving

S = —e2(0|fcP/ an, Jd*xi d4x2 9(h - t2): i/>e(n)7 • A{xi)ipe(xi):

where the a's are the annihilation operators for the A field. Next, recall that Tpe(x) = tpi+\x) +ipi~\x), where ~ b and ipi~ d. Hence, to "balance" the b and tf of the final and initial states, we need precisely one and one The other ip and i/> operators must be left to balance each other, i.e., their b, M, d, d+ must pair off so that they give a non-zero vacuum expectation value. Such internal pairing is referred to as contraction, and we have seen it before whenever a propagator arose. This case is different only because the contraction is between the same field (electron) which also occurs in the initial and final state. Since contractions cannot occur between fields in a single H^ad (because they are normal ordered and all vacuum expectation values give zero), there are precisely two terms which contribute, corresponding to the two different possible contractions:

x j: $+>(n)7 • A(x1)j>e(x1y.: i>e{x2)l ■ A(x2)4+)(x2)-

+ : Mxl)1-A(xl)^(x1y.:i>{e+)(x2)1 ■ A(x2)j>e(x2): Jaj^ |0) ,

where the contractions are shown by the horizontal brackets. Each of these terms generates two more terms corresponding to different pairings of each A with ak.


or . Hence we obtain

J {2-K)6y/16wiojfEiEf x tV (J eHkfXi-k.-Xi) fj *ei ^(kfX-2-kiXi) J t f i

X (OlMxi^faMlisUsfayterti-p.-'*)

where the minus sign in front of the second fermion term comes from the fact that this term requires an odd number of interchanges of anticommuting Fermi fields to get i/>(+)(xi) to the right and ip(-+){x2) to the left. Both terms in { } can be combined if we interchange i <-> j and x\ «-► x2 in the second term. The combined result can be written where

d4xi d4x2 (27r)6v/16Wi ujjE.Ef uQ(pfhla0 iSPy(x1,x2)^sub(pl) ,

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