  It is generally believed that in spatial dimension d > 1 the leading
contribution to the entanglement entropy S = - tr rho_A log rho_A scales as the
area of the boundary of subsystem A. The coefficient of this "area law" is
non-universal. However, in the neighbourhood of a quantum critical point S is
believed to possess subleading universal corrections. In the present work, we
study the entanglement entropy in the quantum O(N) model in 1 < d < 3. We use
an expansion in epsilon = 3-d to evaluate i) the universal geometric correction
to S for an infinite cylinder divided along a circular boundary; ii) the
universal correction to S due to a finite correlation length. Both corrections
are different at the Wilson-Fisher and Gaussian fixed points, and the epsilon
-> 0 limit of the Wilson-Fisher fixed point is distinct from the Gaussian fixed
point. In addition, we compute the correlation length correction to the Renyi
entropy S_n = (log tr rho^n_A)/(1-n) in epsilon and large-N expansions. For N
-> infinity, this correction generally scales as N^2 rather than the naively
expected N. Moreover, the Renyi entropy has a phase transition as a function of
n for d close to 3.
