QCD Radiation from heavy quarks: $e^{+ e}- \to Q \bar Q g$

1.

Compute the differential cross section for the case of massive final state using a program for symbolic calculations (such as Mathematica+FeynCalc) and compare your result with

$ \frac{1}{\sigma^{{LO}} \frac{d}2\sigma}{dx_1dx_2}= \frac{1}{\beta} C_F \frac{\alpha_S}{2\pi} \left[ \frac{2(x_1+x_2-1-\rho/2)}{(1-x_1)(1-x_2)} -\frac{\rho}{2} \left( \frac{1}{(1-x_1)^{2}+ \frac{1}{(1-x_2)}2}\right) \left. + \frac{1}{1+\rho/2} \frac{(1-x_1)^2+(1-x_2)2}{(1-x_1)(1-x_2)}\right]\, P, \right. $ (1)

where

$ \rho=\frac{4 m^{2}{s}\le 1\_{\qquad \beta=\sqrt{1-\rho} $,}}

and

$\sigma^{{LO}= N_c (\sum_{f} Q_f}2) 4 \pi \alpha^{2/(3s) $.}

2.

Verify that the massless limit of Eq. 1 corresponds to

3.

Study the soft and collinear limits. Is the collinear divergence still there? Write the soft and collinear approximation of the amplitude in the case the gluon is close to the quark:

where $z=2 E_g/\sqrt{s}$ is the energy fraction of the gluon and $\theta$ the angle between the gluon and the quark. Plot the behaviour of the matrix element in the massless and massive cases and compare with the Fig.1 below. Explain this behaviour in terms of angular momentum conservation.

4.

Use MadGraph/MadEvent and verify that there are no collinear divergences to be regulated and the cross section is finite with just a minimum cut on the energy of the gluon. Plot the behaviour of the cross sections as a function of the quark mass and verify that it has a logarithmic behaviour.

Fig. 1 : Dead cone

-- Main.FabioMaltoni - 29 Aug 2007