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Show that the maximal transverse momentum for the two-body interaction y*q --+ qg is given by
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In deriving (10.32), we integrated up to the maximum PT of the gluon and then used (10.33) to write log (Sj4) "" log Q2 in the large Q2 limit. The lower limit I-t on the transverse momentum is introduced as a cutoff to regularize the divergence when p} --+ O. Adding a( y*q --+ qg) to the parton model cross section, (10.14), we find QCD modifies (10.15) to
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where we have introduced the notation that the quark structure function q( y) ==
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tiY). The presence of the log Q2 factor means that the parton model scaling
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prediction for the structure functions should be violated. That is, in QCD, F2 is a function of Q2 as well as of x, but the variation with Q2 is only logarithmic. The violation of Bjorken scaling is a signature of gluon emission.
EXERCISE 10.6 Study the origin of the log Q2 term. Recall that the y*q --+ qg cross section, do/ dp}, is dominated by the forward peak. The i-channel quark propagator leads to a factor l/pj.. Show that helicity conservation at the gluon vertex weakens this singularity by introducing a factor p} in the numerator.
Equation (10.34) may be regarded as the first two terms in a power series in as; Q2 since as - (log Q2) - 1. But comparing the leading and next-order terms in (10.34), we find that the expansion parameter as is multiplied by log Q2. From (7.65), we know that a s(Q2) log(Q2//-t2) does not vanish at large Q2, and so (10.34) does not seem very useful as it stands. How should we proceed Can we absorb the log Q2 term into a modified quark probability distribution To this end, we rewrite (10.34) in the "parton-like" form
a s is a useful expansion parameter at large
+ tlq(X,Q2))
_ as dY Y tlq ( X, Q 2) = -2 log (Q2)f - q ( ) P qq (X) -2 7T /-t x Y Y
The quark densities q(x, Q2) now depend on Q2. We interpret this as arising from a photon with larger Q2 probing a wider range of p} within the proton. We can picture this as follows. As Q2 is increased to Q2 - Q~, say, the photon starts to "see" evidence for the point-like valence quarks within the proton; see Fig. 10.9a. If the quarks were noninteracting, no further structure would be resolved as Q2 was increased and exact scaling [described by q(x)] would set in, and the parton model would be satisfactory. However, QCD predicts that on increasing the resolution (Q2 Q~), we should "see" that each quark is itself surrounded by a cloud of partons. We have calculated one particular diagram, shown in Fig. 10.9b, but there are of course other diagrams with a greater number of partons. The number of resolved partons which share the proton's momentum increases with Q2. There is an increased probability of finding a quark at small X" and a decreased chance of finding one at high X, because high-momentum quarks lose momentum by radiating gluons.
10.5 Scaling Violations. The Altarelli-Parisi Equation
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Fig. 10.9 The quark structure of the proton as seen by a virtual photon as Q2
increases. The Q2 evolution of the quark densities is determined by QCD through (10.36). By considering the change in the quark density, t::.q( x, Q2), when one probes a further interval of t::.log Q2, (10.36) can be rewritten as an integro-differential equation for q(x, Q2):
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This is an "Altarelli- Parisi evolution equation." The equation mathematically expresses the fact that a quark with momentum fraction x [q(x, Q2) On the left-hand side] could have come from a parent quark with a larger momentum fraction y [q(y, Q2) On the right-hand side] which has radiated a gluon. The probability that this happens is proportional to aSPqq(x/y). The integral in (l0.37) is the sum over all possible momentum fractions y( > x) of the parent. To summarize: QCD predicts the breakdown of scaling and allows us to compute explicitly the dependence of the structure function on Q2. Given the quark structure function at some reference point q(x, Q~), we can compute it for any value of Q2 using the Altarelli-Parisi equation (l0.37). The experimental results for q(x, Q2), or, to be precise, F2(x, Q2), are displayed in Fig. 10.10. Moment analysis is often used to show that the Q2 variation of the structure function is described by the differential equation (l0.37). This procedure is purely technical and of nO interest to us (see, however, Exercise 10.16). The systematics of the Q2 dependence should be noted, however. Around x = 0.25 (w = 4), the structure function is found to scale, and Fig. 9.2 displays the absence of Q2 dependence at this particular x value. But for x :s 0.25, the structure function increases with Q2, while for x ~ 0.25, it decreases. Another way to state this result is that we resolve increasing numbers of "soft" quarks with increasing Q2.