== 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).