pseudorapidity

Pseudorapidity, denoted by the symbol eta (η), is a spatial coordinate used in high-energy and collider physics to describe the angle of a particle relative to the beam (z) axis. It is defined as η = -ln[tan(θ/2)], where θ is the polar angle measured from the beam axis. The inverse relation is θ = 2 arctan(e^{-η}). Pseudorapidity is a purely angular variable and can take any real value; as θ approaches 0 or π, η tends to ±∞.

In the limit of massless or highly relativistic particles, pseudorapidity coincides with the rapidity y, defined by y = 1/2 ln[(E + p_z)/(E − p_z)]. Since rapidity has simple additive properties under boosts along the beam axis, y is convenient for describing particle production in collider frames. Consequently, for high-energy particles, η serves as a practical surrogate for y, and differences in η approximately reflect invariant separations along the beam direction.

Pseudorapidity is widely used because detector geometry is naturally organized in η–φ coordinates, where φ is the azimuthal angle around the beam axis. This makes η convenient for describing angular separations, jet shapes, and lepton isolations. The central region (η ≈ 0) corresponds to particles emitted perpendicular to the beam, while large |η| values describe particles emitted close to the beam direction (forward or backward regions). Typical detectors have tracking and calorimetry covering limited η ranges, e.g., |η| < 2.5 to 5, depending on subsystem.

Overall, η provides a convenient, energy-agnostic way to characterize particle direction in collider environments and is fundamental to event reconstruction and analysis.