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(That z is indeed the fractional momentum of the quark after gluon radiation can be easily seen by adding its momentum zp; to that of the photon and noting the mass-shell condition of the resulting outgoing quark, (q + Zp;)2 = O. This shows that z is given by (10.29), our previous definition.) In (l0.57') we see that the O(O:O:s) cross section factors into the 0(0:) parton model cross section (e}ao ) and the probability Yqq that the quark radiates a gluon with fraction 1 - z of its momentum and with transverse momentum Pro How is this possible Cross sections are calculated from probability amplitudes, where the different amplitudes for the process are added and then the square modulus of the sum is taken. What we have found is that, in the limit that the gluons have a PT that is not too large, the calculation of (l0.57') can be viewed as two sequential events, that is, provided we retain only the singular part, lip}, of the full PT distribution as we did in deriving (10.30). The probabilities for the interaction e;ao and the gluon emission Yqq can then be calculated separately and multiplied. That is, the probabilistic parton picture thus applies to gluon emission.

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10.9 The Weizsacker-WiUiams Fonnula

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This technique is not special to quarks and gluons (QCD). In fact, it was known, since the work of Weizsacker and Williams in 1934, that it applies equally well to leptons and photons (QED). Consider, for example, the process ep --+ eX. The cross section may be written

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(10.60)

where z and PT are, respectively, the momentum fraction and transverse momentum of the outgoing electron, and

(10.61)

see, for example, Chen and Zerwas (1975) Phys. Rev. D12, 187. Equation (10.60) follows from (10.58) after taking account of the factor 2 mismatch in the definitions of a and as' namely, as --+ 2a, see (10.19). Pee(z) is just Pq/z) as given by (10.31) but without the color factor. Equation (10.60) is known in QED as the equivalent photon distribution. Similarly, one can define the QED equivalent of Pqg given by (10.41):

Pe-y(z) = z2 + (1 - z)2.

(10.62)

In the next chapter, we further illustrate the power of this technique by calculating the cross section for the process e+e- --+ qqg, which we view as e+e---+ qq followed by the emission of a gluon from the quark (or antiquark). Yq/z, p}) has been computed once and for all and can just be substituted into other diagrams.

e + e - Annihilation and QeD

In the previous chapter, we studied QCD in the framework of a truly historic type of experiment, namdy, the deep inelastic scattering of leptons by hadrons. The large-Q2, short-wavelength, virtual photon, prepared by the inelastic scattering of the lepton (see Fig. 11.Ia), probes the proton, revealing its constituents ( 9) and their color interactions ( 10). The resulting picture is easy to interpret, because the short-distance (small as) nature of the quark-gluon interactions allows us to confront the experimental results with quantitative perturbative calculations. High-resolution photons can also be prepared by colliding high-energy electron and positron beams head-on (see Fig. 11.Ib). The exceptional power of this experimental technique is illustrated by the gallery of diagrams in Fig. 11.2: e+ecolliders can be used to study QED, weak interactions, quarks, and gluons and also to study or search for heavy quarks and leptons. Moreover, e+ e- annihilation is a "clean" process in the sense that leptons (rather than hadrons, which are complex structures made of partons) appear in the initial state. For these reasons, we choose e+e - processes as our main working example to illustrate how the ideas and techniques of s 9 and 10 carryover to other experimental situations.