Reaching the Planck scale with muon lifetime measurements

Planck scale modified dispersion relations are one way 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-Poincare 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-Poincare 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.

Automated summary

This study investigates the effects of Planck scale modified dispersion relations on the propagation of fundamental point particles. By constructing Finsler length measurements, time dilation formulas are derived. The research suggests that momentum Lorentz factors in present and future colliders may be large enough to constrain the realization of dispersion relations.