  A semi-inclusive measurement of charged hadron multiplicities in deep
inelastic muon scattering off an isoscalar target was performed using data
collected by the COMPASS Collaboration at CERN. The following kinematic domain
is covered by the data: photon virtuality $Q^{2}>1$ (GeV/$c$)$^2$, invariant
mass of the hadronic system $W > 5$ GeV/$c^2$, Bjorken scaling variable in the
range $0.003 < x < 0.4$, fraction of the virtual photon energy carried by the
hadron in the range $0.2 < z < 0.8$, square of the hadron transverse momentum
with respect to the virtual photon direction in the range 0.02 (GeV/$c)^2 <
P_{\rm{hT}}^{2} < 3$ (GeV/$c$)$^2$. The multiplicities are presented as a
function of $P_{\rm{hT}}^{2}$ in three-dimensional bins of $x$, $Q^2$, $z$ and
compared to previous semi-inclusive measurements. We explore the
small-$P_{\rm{hT}}^{2}$ region, i.e. $P_{\rm{hT}}^{2} < 1$ (GeV/$c$)$^2$, where
hadron transverse momenta are expected to arise from non-perturbative effects,
and also the domain of larger $P_{\rm{hT}}^{2}$, where contributions from
higher-order perturbative QCD are expected to dominate. The multiplicities are
fitted using a single-exponential function at small $P_{\rm{hT}}^{2}$ to study
the dependence of the average transverse momentum $\langle
P_{\rm{hT}}^{2}\rangle$ on $x$, $Q^2$ and $z$. The power-law behaviour of the
multiplicities at large $P_{\rm{hT}}^{2}$ is investigated using various
functional forms. The fits describe the data reasonably well over the full
measured range.
