Choose collider coordinates without losing the mass

Choose the beam as the z axis and define $p_T=\sqrt{p_x^2+p_y^2}$. For an on-shell particle of mass m, define its single-particle transverse mass $m_T=\sqrt{m^2+p_T^2}$. The mass shell becomes $E^2-p_z^2=m_T^2$, a hyperbola. Hyperbolic coordinates write E and longitudinal momentum using a rapidity y. This single-particle transverse mass is a coordinate quantity; later the module defines a different two-system transverse mass used to bound a parent mass. State which definition is being used.

A passive boost to a frame moving along positive z with rapidity $\zeta$, where $\beta=\tanh\zeta$, leaves transverse momentum unchanged and sends $y$ to $y-\zeta$. This follows immediately by applying the boost to $E+p_z$ and $E-p_z$, which acquire opposite exponential factors. Differences of true rapidities therefore survive a common longitudinal boost. The simple shift applies to a fixed axis and a pure longitudinal boost, not to arbitrary rotations or transverse boosts.

Pseudorapidity is defined from the polar angle alone: $\eta=-\log\tan(\theta/2)$. Equivalently it replaces E by the three-momentum magnitude in the rapidity logarithm. For a massless particle with nonzero transverse momentum, E equals that magnitude and the two coordinates agree exactly. For a massive particle, E is larger, so at positive longitudinal momentum true rapidity is smaller than pseudorapidity. A direction measurement alone cannot recover the missing mass-dependent energy.

The approximation becomes good when mass is negligible compared with the relevant momentum scale, and it can fail for soft heavy particles. A direction very close to the beam gives a large pseudorapidity because a small polar angle is being mapped to an unbounded coordinate. This does not mean the particle's speed exceeds light or that its energy has been measured. A particle exactly on the beam makes the angle coordinate singular, so finite-acceptance statements should specify the transverse and angular region rather than treating infinite pseudorapidity as an ordinary detector bin.

A histogram density depends on its coordinate measure. Uniform events per unit solid angle do not give a uniform pseudorapidity distribution; changing variables introduces a Jacobian. Detector acceptance is also commonly specified by pseudorapidity because it tracks geometry, while a theoretical boost-invariant quantity may use true rapidity. When comparing them, propagate the mass and momentum dependence rather than simply relabeling an axis. For reconstructed composite objects, their invariant masses can make the distinction relevant even when individual constituents are nearly massless.

Azimuthal angle and transverse momentum retain their values under a longitudinal boost. Together with true rapidity differences, they provide useful descriptions of collider configurations. A distance defined with pseudorapidity differences is approximately boost stable for ultrarelativistic objects, but that approximation has a domain. Also distinguish a boost of the complete event from an unknown change in the constituents' collision energy: an invariant mass remains fixed in the first operation and can vary in the second. Coordinate convenience does not remove the need to reconstruct or model the relevant four-vectors.