Reaching the Planck scale with muon lifetime measurements

TL;DR

This paper explores how modified dispersion relations from quantum gravity theories affect muon lifetime measurements, proposing a method to potentially reach Planck scale sensitivity in collider experiments.

Contribution

It derives a Finslerian length measure and time dilation formula for various quantum gravity-inspired dispersion relations, linking them to observable muon lifetime effects.

Findings

Finsler length measure constructed for first-order perturbations

Time dilation formulas derived for $ppa$-Poincare9 and string theory models

Potential to constrain quantum gravity effects with collider muon lifetime measurements

Abstract

Planck scale modified dispersion relations are one way how to capture the influence of quantum gravity on the propagation of fundamental point particles effectively. We derive the time dilation between an observer's or particle's proper time, given by a Finslerian length measure induced from a modified dispersion relation, and a reference laboratory time. To do so, the Finsler length measure for general first order perturbations of the general relativistic dispersion relation is constructed explicitly. From this we then derive the time dilation formula for the $\kappa$-Poincar\'e dispersion relation in several momentum space bases, as well as for modified dispersion relations considered in the context of string theory and loop quantum gravity. Most interestingly we find that the momentum Lorentz factor in the present and future colliders can, in principle, become large enough to constrain the Finsler realization of the $\kappa$-Poincar\'e dispersion relation in the bicrossproduct basis as well as a string theory inspired modified dispersion relation, at Planck scale sensitivity with the help of the muon's lifetime.