  The running of the top quark mass is experimentally investigated for the
first time. The mass of the top quark in the modified minimal subtraction
($\mathrm{\overline{MS}}$) renormalization scheme is extracted from a
comparison of the differential top quark-antiquark ($\mathrm{t\overline{t}}$)
cross section as a function of the invariant mass of the
$\mathrm{t\overline{t}}$ system to next-to-leading-order theoretical
predictions. The differential cross section is determined at the parton level
by means of a maximum-likelihood fit to distributions of final-state
observables. The analysis is performed using $\mathrm{t\overline{t}}$ candidate
events in the e$^\pm\mu^\mp$ channel in proton-proton collision data at a
centre-of-mass energy of 13 TeV recorded by the CMS detector at the CERN LHC in
2016, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The
extracted running is found to be compatible with the scale dependence predicted
by the corresponding renormalization group equation. In this analysis, the
running is probed up to a scale of the order of 1 TeV.
