  The magnetic ground state of single crystalline RuO$_2$ was investigated by
the muon spin rotation/relaxation ($\mu$SR) experiment. The spin precession
signal due to the spontaneous internal magnetic field $B_{\rm loc}$, which is
expected in the magnetically ordered phase, was not observed in the temperature
range 5--400~K. Muon sites were evaluated by first-principles calculations
using dilute hydrogen simulating muon as pseudo-hydrogen, and $B_{\rm loc}$ was
simulated for the antiferromagnetic (AFM) structures with a Ru magnetic moment
$|{\bf m}_{\rm Ru}|\approx0.05\mu_\mathrm{B}$ suggested from diffraction
experiments. As a result, the possibility was ruled out that muons are
localized at sites where $B_{\rm loc}$ accidentally cancels. Conversely,
assuming that the slow relaxation observed in $\mu$SR spectra was part of the
precession signal, the upper limit for the magnitude of $|{\bf m}_{\rm Ru}|$
was estimated to be $4.8(2)\times10^{-4}\mu_{\rm B}$, which is significantly
less than $0.05\mu_\mathrm{B}$. These results indicate that the AFM order, as
reported, is unlikely to exist in the bulk crystal.
