== 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).