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 $Q=M$.

Fig.1