  Anomalous large thermal conductivity has been observed numerically and
experimentally in one and two dimensional systems. All explicitly solvable
microscopic models proposed to date did not explain this phenomenon and there
is an open debate about the role of conservation of momentum. We introduce a
model whose thermal conductivity diverges in dimension 1 and 2 if momentum is
conserved, while it remains finite in dimension $d\ge 3$. We consider a system
of harmonic oscillators perturbed by a non-linear stochastic dynamics
conserving momentum and energy. We compute explicitly the time correlation
function of the energy current $C\_J(t)$, and we find that it behaves, for
large time, like $t^{-d/2}$ in the unpinned cases, and like $t^{-d/2-1}$ when
an on site harmonic potential is present. Consequently thermal conductivity is
finite if $d\ge 3$ or if an on-site potential is present, while it is infinite
in the other cases. This result clarifies the role of conservation of momentum
in the anomalous thermal conductivity in low dimensions.
