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