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In experimental
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, pseudorapidity, \eta, is a commonly used spatial
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
describing the angle of a particle relative to the beam axis. It is defined as :\eta \equiv -\ln\left tan\left(\frac\right)\right where \theta is the angle between the particle three-momentum \mathbf and the positive direction of the beam axis.Introduction to High-Energy Heavy-Ion Collisions, by Cheuk-Yin Wong, See page 24 for definition of rapidity. Inversely, :\theta = 2\arctan\left(e^\right). As a function of three-momentum \mathbf, pseudorapidity can be written as :\eta = \frac \ln \left(\frac\right) = \operatorname\left(\frac \right), where p_\text is the component of the momentum along the beam axis (i.e. the ''longitudinal'' momentum – using the conventional system of coordinates for
hadron collider A hadron collider is a very large particle accelerator built to test the predictions of various theories in particle physics, high-energy physics or nuclear physics by colliding hadrons. A hadron collider uses tunnels to accelerate, store, and colli ...
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 , \mathbf, \Rightarrow E \approx , \mathbf, \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 \ln \left(\frac\right) This differs slightly from the definition of
rapidity In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with di ...
in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
, which uses \left, \mathbf\ instead of p_\text. 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 , \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. 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 are
Lorentz invariant In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of v ...
under boosts along the longitudinal axis: they transform additively, similar to velocities in
Galilean relativity Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference. Galileo Galilei first described this principle in 1632 in his ''Dialogue Concerning the Two Chief World Systems'' using th ...
. A measurement of a rapidity difference \Delta y between particles (or \Delta\eta if the particles involved are massless) is hence not dependent on the longitudinal boost of the reference frame (such as the
laboratory frame In theoretical physics, a local reference frame (local frame) refers to a coordinate system or frame of reference that is only expected to function over a small region or a restricted region of space or spacetime. The term is most often used in t ...
). This is an important feature for hadron collider physics, where the colliding
partons In particle physics, the parton model is a model of hadrons, such as protons and neutrons, proposed by Richard Feynman. It is useful for interpreting the cascades of radiation (a parton shower) produced from quantum chromodynamics (QCD) processes ...
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 :y = \ln\left( \frac\right), where p_\text\equiv\sqrt is the transverse momentum (i.e. the component of the three-momentum perpendicular to the beam axis). Using a second-order Maclaurin expansion of y expressed in m/p_\text one can approximate rapidity by :y \approx \eta - \frac \left( \frac\right)^2 = \eta - \frac \left( \frac\right)^2 = \eta - \frac \left( \frac\right)^2, which makes it easy to see that for relativistic particles with p_\text \gg m, pseudorapidity becomes equal to (true) rapidity. Rapidity is used to define a measure of angular separation between particles commonly used in particle physics \Delta R \equiv \sqrt, 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: \Delta R \equiv \sqrt, which is Lorentz invariant if the involved particles are massless. 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 "transverse" ''x-y'' plane) orthogonal to the beam line.


Values

Here are some representative values: : Pseudorapidity is odd about \theta = 90^\circ. 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, polar angle in the transverse plane \phi and pseudorapidity \eta. To obtain cartesian momenta (p_\text, p_\text, p_\text) (with the z-axis defined as the beam axis), the following conversions are used: : p_\text = p_\text \cos \phi : p_\text = p_\text \sin \phi : p_\text = p_\text \sinh, which gives , \mathbf, = p_\text \cosh. Note that p_\text is the longitudinal momentum component, which is denoted p_\text in the text above (p_\text is the standard notation at hadron colliders). The equivalent relations to get the full
four momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
(in
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a Coherence (units of measurement), coherent unit of a quantity. For e ...
) using "true" rapidity y are: : p_\text = p_\text \cos \phi : p_\text = p_\text \sin \phi : p_\text = m_\text \sinh : E = m_\text \cosh, where m_\text \equiv \sqrt is the
transverse mass The transverse mass is a useful quantity to define for use in particle physics as it is invariant under Lorentz boost along the z direction. In natural units, it is: m_T^2 = m^2 + p_x^2 + p_y^2 = E^2 - p_z^2 *where the z-direction is along the bea ...
. A boost of velocity \beta_\text along the beam-axis of velocity corresponds to an additive change in rapidity of y_\text using the relation \beta_\text=\tanh. Under such a
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
, the rapidity of a particle will become y' = y + y_\text and the four momentum becomes : p'_\text = p_\text \cos \phi : p'_\text = p_\text \sin \phi : p'_\text = m_\text \sinh : E' = m_\text \cosh. 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 :y_\mathrm=\frac\ln\frac, where x_1 and x_2 are the momentum fraction of the colliding
partons In particle physics, the parton model is a model of hadrons, such as protons and neutrons, proposed by Richard Feynman. It is useful for interpreting the cascades of radiation (a parton shower) produced from quantum chromodynamics (QCD) processes ...
. When several particles are produced in the same collision, difference in rapidity \Delta y_=y_i-y_j between any two particles i and j will be invariant under any such boost along the beam axis. And if both particles are massless (m_i=m_j=0), this will also hold for pseudorapidity (\Delta \eta_).


References

{{Reflist * V. Chiochia (2010
Accelerators and Particle Detectors
from
University of Zurich The University of Zürich (UZH, german: Universität Zürich) is a public research university located in the city of Zürich, Switzerland. It is the largest university in Switzerland, with its 28,000 enrolled students. It was founded in 1833 f ...
Experimental particle physics