  We investigate the shear viscosity of massless classical scalar fields in the
$\phi^4$ theory on a lattice by using the Green-Kubo formula. Based on the
scaling property of the classical field, the shear viscosity is represented
using a scaling function. Equilibrium expectation value of the time-correlation
function of the energy-momentum tensor is evaluated as the ensemble average of
the classical field configurations, whose time evolution is obtained by solving
the classical equation of motion starting from the initial condition in thermal
equilibrium. It is found that there are two distinct damping time scales in the
time-correlation function, which is found to show damped oscillation behavior
in the early stage around a slow monotonous decay with an exponential form, and
the slow decay part is found to dominate the shear viscosity in the massless
classical field theory. This kind of slow decay is also known to exist in the
molecular dynamics simulation, then it may be a generic feature of dense
matter.
