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Wikipedia / Pseudorapidity

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

and the positive direction of the beam axis. Inversely,

, pseudorapidity can be written as

is the component of the momentum along the beam axis (i.e. the longitudinal momentum using the conventional system of coordinates for hadron collider). In the limit where the particle is travelling close to the speed of light (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; 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.

because, loosely speaking, particle production is constant as a function of rapidity, and because differences in rapidity are Lorentz invariant under boosts along the longitudinal axis: they transform additively, similar to velocities in Galilean relativity if the particles involved are massless) is hence not dependent on the longitudinal boost of the reference frame (such as the laboratory frame). This is an important feature for hadron collider physics, where the colliding partons carry different longitudinal momentum fractions x, which means that the rest frames of the parton-parton collisions will have different longitudinal boosts.

The rapidity as a function of pseudorapidity is given by

is the transverse momentum (i.e. the component of the three-momentum perpendicular to the beam axis).

Using a second-order Maclaurin expansion one can approximate rapidity by

, pseudorapidity becomes equal to (true) rapidity.

, is invariant under Lorentz boosts along the beam line (z-axis) because it is measured in a plane (i.e. the "transverse" x-y plane) orthogonal to the beam line.

Values

Here are some representative values:

  1. {| class=wikitable

|- | 0° | ∞ | 180° | −∞ |- | 0.1° | 7.04 | 179.9° | −7.04 |- | 0.5° | 5.43 | 179.5° | −5.43 |- | 1° | 4.74 | 179° | −4.74 |- | 2° | 4.05 | 178° | −4.05 |- | 5° | 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 |colspan=2| |}

.

Conversion to Cartesian momenta

-axis defined as the beam axis), the following conversions are used:

is the standard notation at hadron colliders).

The equivalent relations to get the full four-momentum (in natural units are:

is the transverse mass.

. Under such a Lorentz transformation and the four-momentum becomes

This sort of transformation is common in hadron colliders. For example, if two hadrons of identical type undergo an inelastic collision along the beam axis with the same speed, then the corresponding rapidity will be

are the momentum fraction of the colliding partons).

References

Bibliography