Pseudorapidity

In experimental particle physics, Pseudorapidity, $eta$, is a commonly used spatial coordinate describing the angle of a particle relative to the beam axis. It is defined as (For example, see [Introduction to High-Energy Heavy-Ion Collisions, by Cheuk-Yin Wong, See page 24 for definition of rapidity. ] )

::$eta\; =\; -lnleft\; [\; anleft(frac\{\; heta\}\{2\}\; ight)\; ight]\; ,$

where $heta\; ,$ is the angle between the particle momentum $vec\; p$ and the beam axis. In terms of the momentum, the pseudorapidity variable can be written as

::$eta\; =\; frac\{1\}\{2\}\; ln\; left(frac\{left|vec\; p\; ight|+p\_L\}\{left|vec\; p\; ight|-p\_L\}\; ight),$

In the limit where the particle is travelling close to the speed of light, or in the approximation that the mass of the particle is nearly zero, it is numerically close to the rapidity, "y", defined in Special Relativity as

::$y\; =\; frac\{1\}\{2\}\; ln\; left(frac\{E+p\_L\}\{E-p\_L\}\; ight)$

when the particle is relativistic. Here, $p\_L$ is the component of the momentum along the beam direction. However, pseudorapidity depends only on the polar angle of its trajectory, and not on the energy of the particle.

In hadron collider physics, the rapidity (or pseudorapidity) is preferred over the polar angle $heta$ because, loosely speaking, particle production is constant as a function of rapidity. One speaks of the "forward" direction in a hadron collider experiment, which refers to regions of the detector that are close to the beam axis, at high $|eta|,$.

The difference in the rapidity of two particles is independent of Lorentz boosts along the beam axis.

Here are some representative values::

Pseudorapidity is odd about $heta\; =\; 90$ degrees. In other words, $eta$ at $180-\; heta$ is equal to $-eta$ at $heta$.