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R2

From IFPA wiki

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

Qqgg.png

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].

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