Pseudorapidity

In experimental particle physics, pseudorapidity, , is a commonly used spatial coordinate describing the angle of a particle relative to the beam axis. It is defined as

where is the angle between the particle three-momentum and the positive direction of the beam axis. Inversely,

As a function of three-momentum , pseudorapidity can be written as

where is the component of the momentum along the beam axis (i.e. the longitudinal momentum – using the conventional system of coordinates for hadron collider physics, this is also commonly denoted ). In the limit where the particle is travelling close to the speed of light, or equivalently in the approximation that the mass of the particle is negligible, one can make the substitution (i.e. in this limit, the particle's only energy is its momentum-energy, similar to the case of the photon), and hence the pseudorapidity converges to the definition of rapidity used in experimental particle physics:

This differs slightly from the definition of rapidity in special relativity, which uses instead of . However, pseudorapidity depends only on the polar angle of the particle's trajectory, and not on the energy of the particle. 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 ; in contexts where the distinction between "forward" and "backward" is relevant, the former refers to the positive z-direction and the latter to the negative z-direction.