# Pseudorapidity

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 polar angle approaches zero, pseudorapidity tends towards 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

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

where $\theta$ is the angle between the particle three-momentum $\mathbf{p}$ and the positive direction of the beam axis.[1] Inversely,

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

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

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

where $p_\text{L}$ 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 $p_z$). 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 $m \ll p \Rightarrow E \approx p \Rightarrow \eta \approx y$ (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:

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

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 the particle's 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, and because differences in rapidity and pseudorapidity are Lorentz invariant (because they transform additively, similar to velocities in Galilean relativity), unlike differences in $\theta$ – this means that a measurement of $\Delta y$ or $\Delta\eta$ between particles is not dependent on specifying a reference frame, such as the rest frame of a particle or the laboratory frame. 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 $|\eta|$; 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.

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 $p_\text{T}\equiv\sqrt{p_{x}^{2}+p_{y}^{2}}$ is the transverse momentum (i.e. the component of the three-momentum perpendicular to the beam axis).

Pseudorapidity can also be used to define a Lorentz-invariant measure of angular separation between particles, $\left(\Delta R\right)^{2} \equiv \left(\Delta \eta\right)^{2} + \left(\Delta \phi\right)^{2}$ – the difference in azimuthal angle, $\Delta\phi$, is invariant under Lorentz boosts along the beam line (z-axis) because it is measured in a plane (i.e. the x-y plane) orthogonal to the beam line.

## Values

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

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

## References

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