== Basic kinematics in hadron hadron collisions ==

The rapidity %$y$% and pseudo-rapidity %$\eta$% are defined as:



where the %$z$% direction is that of the colliding beams.
==== 1. ====

Verify that for a particle of mass %$m$% :



==== 2. ====

Prove that %$\tanh \eta=\cos \theta$%.

==== 3. ====

Consider a set of particles produced uniformly in longitudinal phase space



Find the distribution in %$\eta$%.

==== 4. ====

Prove that rapidity equals pseudo-rapidity, %$\eta=y$% for a relativistic particle %$E\gg m$%.

==== 5. ====

Prove that for Lorentz transformation (boost) in the beam (%$z$%) directions, the rapidity %$y$% of every particle is shifted by a constant %$y_0$%, related to the boost velocity. Find the relation between %$\beta$% and %$y_0$% for a generic boost:



==== 6. ====

Consider a generic particle %$X$% of mass %$M$% (such as a Z boson or a Higgs) produced on shell at the LHC , with zero transverse momentum, %$pp \to X$%. Find the relevant values of %$x_1,x_2$% of the initial partons that can be accessed by producing such a particle. Compare your results with that of Fig.1, considering the scale %$Q=M$%.

<img width="559" alt="lhcgridx.png" src="%ATTACHURLPATH%/lhcgridx.png" height="796" />

Fig.1