Pg Pq Fig. 11.10 The diagrams for y* Pq

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qqg showing the particle four-momenta.

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An Alternative Derivation of the e - e + --+ qqg Cross Section

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Substituting (11.33) into (11.31), we obtain for y*

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qqg (11.35)

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1':)11l2=N

+ x~ q (1 - x q )(l - x q )'

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where we have used (11.17) to eliminate xg.1f we "attach" the e-e+ pair, then, up to a factor (which changes N -> N'), (11.35) gives the cross section for e - e+ -> qqg (11.36)

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To verify that (11.36) is equivalent to our previous result (11.24), we must change the variable x q to x}. We have

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do dxqdx} do dX q dXijdx q dx} .

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(11.37)

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It is sufficient to use the small PT approximation inherent in the Altarelli-Parisi

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result (11.24). Using (11.19), we find

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dX21 --....I. dX q

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= 4x- (1 q,

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x-) q

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for x q

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(11.38)

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Thus, (11.36) may be written

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do dXijdx}

= N'

( 1 + x~ ) [ 1 ] 1 - x q 4(1 - x q )(l - xij)xij ,

(11.39)

where here again we have assumed x q = 1. In this limit, x q = (1 - x g ), and so the factor in square brackets is just X:;.2, as can be seen from (11.19). Thus, (11.39) becomes (11.40) where Pqq is given by (10.31). This is the same as the Altarelli-Parisi result of (11.24), and, indeed, we can identify the normalization coefficient to be

2a s N' = 3'IT o.

The exact O( aJ result is therefore 1

o dxqdx q

2a s x~ + x~ 3'IT (1 - x q )(l - x q )'

(11.41)

whereas (11.24) is the leading logarithmic approximation.

e + e - Annihilation and QeD

A Discussion of Three-Jet Events

The e - e +-> qqg events led to three hadronic jets in the final state. The distribution (11.30) can be written as

do _ a _1 log ( Q2 )

o dp}

4p}'

(11.42)

where PT is the transverse momentum between the q and q as a result of the emission of the gluon, recall Fig. 11.7. Only when the q (or q) recoils against g can its PT relative to the q (or q) be nonzero. For two-jet events, e-e+ -> qq, we have PT = O. Now (11.42) 'Shows that, for a fixed Pn the cross section ratio increases with increasing QL. That is, the number of q jets with a transverse momentum PT relative to the q jet increases with Q2. This is a result of the increased probability of emitting a gluon with a given PT value when the annihilation energy increases. The physics is identical to that in electroproduction. There, also, the cross section

10- 2

10 3 '-_L...---'-_.......l._---'-_--'-''''''----'-_...I.-_.J......_L...---l o 2 4 6 8 10

p~ [GeV 2 J

Fig. 11.11 The transverse momentum distribution d(J/dp~ of hadrons relative to the thrust axis for different e e' center-of-mass energies Q; (0) Q = 12 GeV; (e) 27.4:0; Q :0; 31.6 GeV; (X) 35.0:0; Q :0; 36.6 GeV. The curves are a QeD calculation. Data are from PETRA.

11.6 A Discussion of 11Iree-Jet Events

for producing jets with transverse momentum PT relative to the y*-direction increased as log Q2/p } [compare (10.30) and (11.24)]. The hadron fragments of the q jet will also have large PT relative to the quark direction since the PT distribution of these hadrons should follow the trend of the PT distribution of jets. The two distributions can be explicitly related using the D functions introduced in Section 11.2. The resulting PT and Q2 dependence of. hadrons relative to the thrust axis (whichever parton it refers to) is shown in Fig. 11.11. In some events, all three jets will be well separated despite the k T == 300 MeV of the daughter hadrons relative to their parent jet. One such event is shown in Fig. 11.12. The assumption that as is constant in (11.42) requires explanation. If we had taken as (Q 2) - 1/log Q 2, then the above discussion would be meaningless. The

Fig. 11.12 A three-jet event observed by the JADE detector at PETRA.

e + e - Annihilation and QeD

crucial point is that we are discussing a process with two momentum scales, p} and Q2 with p} Q2, a situation already encountered in Exercise 10.7. We argued there that as is in fact as< p}), but noted that keeping as constant gave the correct result to leading order.

EXERCISE 11.6 The x T distribution (11.30) can be translated into an "acollinearity" distribution do/dO, where is the angle between the q and q jet directions defined in (11.20) and Fig. 11.7. Show that for not too large,

An exact result can be obtained using (11.41) instead of (11.30). We have repeatedly drawn attention to the fact that hadrons fragmenting from a quark, or any other parton, form a cone around the direction Pq It is an experimental fact that the (k T ) for the hadron fragments is about 300 MeV, where T refers to the direction transverse to Pq We might anticipate that at higher energies the fragmentation cone would narrow,

(0) == (k T ) == 0.3 GeV Pq Q/2'

(11.43)

and the jets become narrow bundles of energetic particles when Q increases. This is not the case. Gluon radiation results in an increase of the (k T) of the hadrons which we associate with the original quark or antiquark jet. It is increasingly probable that in a very high-energy two-jet event, one of the observed jets is actually the fragmentation product of a qg or qg state as a result of gluon emission by the q or q. The experimentalist will recognize such a jet as being "fatter" or having increased (k T ). This dynamical broadening of (0) almost compensates for the kinematic narrowing given by (11.43). As a consequence, it turns out that the narrowing of jets with increasing Q2 is a logarithmic and not a linear effect, with 1 (0) - - - . (11.44) logQ2 This result has to be derived with care. We shall only briefly sketch how it comes about. By definition, Q do (11.45) (k T ) k T dk . o T Let us assume that the k T of the hadrons in the broadened jet just reflects the relative k T of the q or q and the emitted gluon. Then, do/dk T in (11.45) is nothing but the familiar transverse momentum distribution of gluon emission do _~ dk T kT '