Multi-Particle Invariant Mass -- Standard Expressions and Corrections to Order $(m/E)^4$

TL;DR

The paper revisits the standard, Lorentz-invariant expressions for the invariant mass of two to four particles in collider kinematics and systematically derives corrections in powers of $m/E$ up to $O((m/E)^4)$. It identifies two correction sources—the use of pseudorapidity in place of true rapidity and the surd in the energy–momentum relation—and shows that the leading $O((m/E)^2)$ terms partially cancel across two- and multi-particle cases, with the next-to-leading $O((m/E)^4)$ corrections generally smaller for more particles. The leading corrections to energy and momentum components have distinct angular dependences, but the $O((m/E)^2)$ terms to the invariant masses are largely θ- and η-insensitive at leading order, ensuring robustness of the zeroth-order expressions. Practically, for LHC-scale energies where $E \gg m$, these corrections are small, but the results provide a principled bound on deviations from the standard formulas and a clear framework for including finite-mass effects when needed.

Abstract

In collider-based particle physics, $invariant\ mass$ refers to the magnitude of the total-momentum 4-vector of a system of particles. An expression for the invariant mass of a 2-particle system is well known; it assumes that both the total energy $E$ and the transverse momentum $p_\mathrm{T}$ of each particle in the system greatly exceed its mass $m$. This note explores these assumptions by computing correction terms in powers of $m/E$ up to order $(m/E)^4$. The assumptions are found to be robust: not only is the leading correction quadratic in $m/E$, but also cancellations reduce its coefficient and that of the next-to-leading correction, which is of order $(m/E)^4$. Three- and four-particle systems are also treated and the generalisation to larger numbers of particles indicated. The zeroth-order expressions for these multi-particle systems are remarkably simple; they deserve to be better known.