In the beam pipe, the particles are accelerated to velocities very close to the speed of light. At such velocities, we have to take into account the relativistic effects such as Lorentz transforms. The angle theta between the particle being considered and the undeflected beam lies in the laboratory frame of reference. In the particle frame of reference, this angle has no meaning due to the Lorentz transformation. We then use the pseudorapidity to represent the angle theta in the particle’s frame of reference. The pseudorapidity is an angular variable defined by:
h = - ln tan ( q / 2 );
form there, the angle theta is defined by:
q = 2 tan-1 (e-h ).
Below is a table of the relation between theta and the pseudorapidity for some round values.
We can see in figure 1 above how pseudorapidity replaces the angle theta in the particle’s frame of reference. With the azimuthal angle f, it is enough to describe any directions in CMS.