Failure of universal finite-size scaling above the upper critical dimension

Previous theories have predicted that O(n) symmetric systems in a finite cubic geometry with periodic boundary conditions have universal finite-size scaling functions near criticality in d>4 dimensions. On the basis of exact results for the O(n) symmetric 4 model in the large n limit we show that universal finite-size scaling does not hold in the predicted form because of significant cut-off and lattice effects for d>4. It is shown that finite-size scaling is valid with two reference lengths which turn out to be identical with the amplitudes of the bulk correlation length. For the 4 field theory the finite-size scaling functions are shown to be non-universal, i.e., to depend explicitly on the cut-off and on the bare four-point coupling constant, whereas for a 4 lattice model the finite-size scaling functions have a different form that is independent of the lattice spacing and the four-point coupling constant