  The first experimental investigation of the running of the top quark mass is
presented. The running, defined in the modified minimal subtraction
($\mathrm{\overline{MS}}$) renormalization scheme, is extracted from a
measurement of the differential top quark-antiquark ($\mathrm{t\bar{t}}$)
production cross section as a function of the invariant mass of the
$\mathrm{t\bar{t}}$ system using theoretical predictions at next-to-leading
order. The measurement is performed using proton-proton collisions at a
centre-of-mass energy of $13~\mathrm{TeV}$ recorded by the CMS detector at the
CERN LHC in 2016, corresponding to an integrated luminosity of
$35.9~\mathrm{fb^{-1}}$. Candidate $\mathrm{t\bar{t}}$ events are selected in
the final state with an electron and a muon of opposite charge, and the
differential cross section is determined at the parton level by means of a
maximum-likelihood fit to multidifferential distributions. The extracted
running is found to be compatible with the solution of the corresponding
renormalization group equation, up to a scale on the order of $1~\mathrm{TeV}$.
