Pseudorapidity
In experimental particle physics, pseudorapidity, formula_1, is a commonly used spatial coordinate describing the angle of a particle relative to the beam axis. It is defined as where formula_3 is the angle between the particle three-momentum formula_4 and the positive direction of the beam axis. Inversely, As a function of three-momentum formula_4, pseudorapidity can be written as where formula_8 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 formula_9). 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 formula_10 (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 formula_12 instead of formula_8. 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 formula_14; 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. In hadron collider physics, the rapidity (or pseudorapidity) is preferred over the polar angle formula_15 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. A measurement of a rapidity difference formula_16 between particles (or formula_17 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 where formula_19 is the transverse momentum (i.e. the component of the three-momentum perpendicular to the beam axis). Using a  second-order Maclaurin expansion of formula_20 expressed in formula_21 one can approximate rapidity by which makes it easy to see that for relativistic particles with formula_23, pseudorapidity becomes equal to (true) rapidity. Rapidity is used to define a measure of angular separation between particles commonly used in particle physics formula_24, which is Lorentz invariant under a boost along the longitudinal (beam) direction. Often, the rapidity term in this expression is replaced by pseudorapidity, yielding a definition with purely angular quantities: formula_25, which is Lorentz invariant if the involved particles are massless. The difference in azimuthal angle, formula_26, 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: Pseudorapidity is odd about formula_27 degrees. In other words, formula_28. Conversion to Cartesian momenta. Hadron colliders measure physical momenta in terms of transverse momentum formula_29, polar angle in the transverse plane formula_30 and pseudorapidity formula_1. To obtain cartesian momenta formula_32 (with the formula_33-axis defined as the beam axis), the following conversions are used: which gives formula_37. Note that formula_38 is the longitudinal momentum component, which is denoted formula_8  in the text above (formula_38 is the standard notation at hadron colliders). The equivalent relations to get the full four momentum (in natural units) using "true" rapidity formula_20 are: where formula_46 is the transverse mass. A boost of velocity formula_47 along the beam-axis of velocity corresponds to an additive change in rapidity of formula_48 using the relation formula_49. Under such a Lorentz transformation, the rapidity of a particle will become formula_50 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, the corresponding rapidity will be where formula_56 and formula_57 are the momentum fraction of the colliding partons. When several particles are produced in the same collision, difference in rapidity formula_58 between any two particles formula_59 and formula_60 will be invariant under any such boost along the beam axis. And if both particles are massless (formula_61), this will also hold for pseudorapidity (formula_62).