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

TL;DR

This paper derives correction terms for the invariant mass formula in collider physics, quantifying deviations due to finite particle mass and confirming the robustness of the standard approximations.

Contribution

It provides explicit correction expressions up to order (m/E)^4 for two, three, and four-particle systems, extending the standard invariant mass formula.

Findings

Corrections are quadratic in m/E and are often suppressed due to cancellations.

Leading correction terms are of order (m/E)^4, indicating high robustness of standard assumptions.

Zeroth-order invariant mass formulas are simpler than previously appreciated.

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.