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1. The square amplitude of the process [math]q\bar q\to gg[/math] can be represented by the following classes of diagrams:

a) For each diagram, calculates the colour factor.

b) Adapt the answer for [math]e^+e^-\to\gamma\gamma[/math] to obtain the expression of the first two diagrams as a function of s, t and u, neglecting quark and lepton masses.

c) Calculates the last diagram, which is a square.

The answer can be checked by taking m=0 in equation (8.12) of the *Handbook of perturbative QCD* (see references).

d) Calculate the interference terms (3rd, 4th diagrams). Are there other similar interference terms?

e) GIve the expression of the differential cross section as a function of the solid angle, [math]d\sigma/d\Omega[/math].

2. Find the projector that extracts [math]W_1[/math] from [math]W_{\mu\nu}[/math].

3. Prove that in the naïve parton model [math]F_2-2xF_1[/math] is of order [math]1/\nu[/math]. What would it be in the QCD-improved parton model?

4. What is the consequence of momentum conservation on the splitting functions?

5. For small [math]x[/math] write the dominant terms of the DGLAP equation. Show that the gluon structure function is the dominant one.
Solve the DGLAP equation for gluons assuming that [math]g(x)=N/x[/math] at [math]Q^2=5 GeV^2[/math].