Two Jet Kinematics

At the LHC, partons in the incoming beams (beam energy $E_b$=7 TeV) collide with a momementum fraction $x_{1,2}$ and produce two jets with negligible mass, transverse momentum $p_T$ and rapidities $y_{3,4}$.

1.

Show that $x_1 = \frac{p_T}{\sqrt{s}} ( e^{{y_3}+ e}{y_4}), x_2 = \frac{p_T}{\sqrt{s}} ( e^{{-y_3}+ e}{-y_4})\,. $

2.

Show that the invariant mass of the jet-jet system is

and the centre-of-mass scattering angle is given by

3.

Discuss the regions of $x_{1,2}, M_{JJ}$ and $\theta^{*$ that can be studied at the LHC with a jet trigger of $p_T > 35 $ GeV and $}

y_{3,4}

<3$.

4.

Use MadGraph/MadEvent to obtain the relative contribution of $gg,qg+\bar q g,qq+q\bar q$ initial states to the jet $E_T$ distribution as function of the $E_T$ $(10<E_T<E_{\rm max}/4)$ at the Tevatron Run II ($p\bar p$ collisions at 1.96 TeV) and the LHC ($pp$ collisions at 14 TeV).