# Pseudorapidity: Wikis

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# Encyclopedia

As angle increases from zero, pseudorapidity decreases from infinity. In particle physics, an angle of zero is usually along the beam axis.

In experimental particle physics, pseudorapidity, η, is a commonly used spatial coordinate describing the angle of a particle relative to the beam axis. It is defined as (For example, see [1])

$\eta = -\ln\left[\tan\left(\frac{\theta}{2}\right)\right],$

where θ 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

$\eta = \frac{1}{2} \ln \left(\frac{\left|\vec p\right|+p_L}{\left|\vec p\right|-p_L}\right),$

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, numerically close to the experimental particle physicist's definition of rapidity,

$y = \frac{1}{2} \ln \left(\frac{E+p_L}{E-p_L}\right)$

Here, pL is the component of the momentum along the beam direction. (This differs slightly from the definition of rapidity in special relativity, which uses $\left|\vec p\right|$ instead of pL.) 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 θ 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:

θ (degrees) η
0 infinite
5 3.13
10 2.44
20 1.74
30 1.31
45 0.88
60 0.55
80 0.175
90 0

Pseudorapidity is odd about θ = 90 degrees. In other words, η at 180 − θ is equal to − η at θ.

## References

1. ^ Introduction to High-Energy Heavy-Ion Collisions, by Cheuk-Yin Wong, See page 24 for definition of rapidity.