Pseudorapidity

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Pseudorapidity: a simple guide

Pseudorapidity η is a way to describe the direction of a particle relative to the beam axis in collider experiments. It is defined from the polar angle θ (the angle between the particle’s momentum and the beam direction) as:
- η = -ln(tan(θ/2))

Key ideas to know

- Why use η? It depends only on direction, not on the particle’s energy. It’s convenient because particle production is often studied as a function of rapidity, and differences in rapidity are simple to handle when boosting along the beam axis.

- Relation to rapidity: For very energetic (nearly massless) particles, η ≈ y, where y = 0.5 ln((E+pz)/(E-pz)) is the true rapidity. This makes η a good stand-in when mass is small.

- Forward direction: Large |η| values mean the particle is traveling close to the beam (forward or backward regions). η = 0 corresponds to motion perpendicular to the beam (θ = 90°). The relation is antisymmetric: η(θ) = -η(180° − θ).

- Momentum in terms of η: If pT is the transverse momentum and φ is the azimuthal angle in the transverse plane, then
- pT = sqrt(px^2 + py^2)
- |p| = pT cosh(η)
- pz = pT sinh(η)

- Angular separation: The separation between two particles is measured by
- ΔR = sqrt((Δη)^2 + (Δφ)^2)
This is Lorentz invariant under boosts along the beam axis for massless particles, and it is computed using purely angular quantities (η and φ).

- Practical use: In experiments, we usually measure pT, φ, and η. These allow us to reconstruct the momentum and study how particles are produced and distributed, while remaining robust under changes of the reference frame along the beam direction.

In short, pseudorapidity is a simple, energy-free way to describe particle directions, closely linked to the true rapidity for fast particles, and it helps physicists compare events across different frames along the collider axis.