rapidities

Rapidity is a kinematic variable used in special relativity and high-energy physics to describe the component of a particle’s motion along a chosen axis, typically the beam (z) axis in collider experiments. It is defined for a particle with energy E and longitudinal momentum p_z by y = (1/2) ln((E + p_z)/(E − p_z)), in natural units where c = 1. Equivalently, tanh y = p_z / E, so y is the hyperbolic angle of the Lorentz boost that brings the particle from the laboratory frame to its rest frame along z.

Under a Lorentz boost along the z axis by rapidity Δy, momentum components transform such that y → y + Δy. Consequently, differences in rapidity between particles (Δy) are invariant under boosts along z. This makes rapidity a convenient variable for describing particle production in high-energy collisions.

Pseudorapidity η is defined from the polar angle θ relative to the beam axis by η = −ln[tan(θ/2)]. For massless particles (or in the high-energy limit), y and η coincide, y ≈ η. Pseudorapidity is often used in detector-related analyses because θ is directly accessible from the geometry of the apparatus.

Applications: Rapidities and pseudorapidities are used to characterize particle distributions, jets, and correlations in collider experiments. Distributions such as dN/dy or dN/dη summarize event structure, and separations in rapidity are frequently employed due to approximate boost invariance along the beam direction.