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).