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Pseudorapidity values shown on a polar plot. In particle physics, an angle of zero is usually along the beam axis, and thus particles with high pseudorapidity values are generally lost, escaping through the space in the detector along with the beam.
As angle increases from zero, pseudorapidity decreases from infinity.

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

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

where \theta is the angle between the particle momentum \mathbf{p} and the beam axis.[1] Inversely,

\theta = 2\arctan\left(e^{-\eta}\right).

In terms of the momentum, the pseudorapidity variable can be written as

\eta = \frac{1}{2} \ln \left(\frac{\left|\mathbf{p}\right|+p_\text{L}}{\left|\mathbf{p}\right|-p_\text{L}}\right),

where p_\text{L} is the component of the momentum along the beam axis. 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, pseudorapidity is numerically close to the experimental particle physicist's definition of rapidity,

y = \frac{1}{2} \ln \left(\frac{E+p_\text{L}}{E-p_\text{L}}\right)

This differs slightly from the definition of rapidity in special relativity, which uses \left|\mathbf{p}\right| instead of p_\text{L}. 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 \theta 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 rapidity as a function of pseudorapidity is given by

y = \ln\left( \frac{\sqrt{m^2 + p_T^2 \cosh^2 \eta} + p_T \sinh \eta}{\sqrt{m^2 + p_T^2}}\right),

where p_\text{T} is the transverse momentum (the momentum perpendicular to the beam axis). The difference in the rapidity of two particles is independent of Lorentz boosts along the beam axis.


A plot of polar angle vs. pseudorapidity.

Here are some representative values:

\theta \eta \theta \eta
180° −∞
0.1° 7.04 179.9° −7.04
0.5° 5.43 179.5° −5.43
4.74 179° −4.74
4.05 178° −4.05
3.13 175° −3.13
10° 2.44 170° −2.44
20° 1.74 160° −1.74
30° 1.32 150° −1.32
45° 0.88 135° −0.88
60° 0.55 120° −0.55
80° 0.175 100° −0.175
90° 0

Pseudorapidity is odd about \theta = 90 degrees. In other words, \eta(\theta)=-\eta(180^\circ-\theta).

Conversion to Cartesian Momenta[edit]

Hadron colliders measure physical momenta in terms of transverse momentum p_\text{T}, polar angle in the transverse plane \phi and pseudorapidity \eta. To obtain cartesian momenta (p_x, p_y, p_z) (with the z-axis defined as the beam axis), the following conversions are used:

 p_x = p_\text{T} \cos \phi
 p_y = p_\text{T} \sin \phi
 p_z = p_\text{T} \sinh{\eta}.

Therefore,  |p|= p_\text{T} \cosh{\eta}.


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