Finite size scaling for O(N) φ4-theory at the upper critical dimension

A finite size scaling theory for the partition function zeroes and thermodynamic functions of O(N) φ4-theory in four dimensions is derived from renormalization group methods. The leading scaling behaviour is mean-field like with multiplicative logarithmic corrections which are linked to the triviality of the theory. These logarithmic corrections are independent of N for odd thermodynamic quantities and associated zeroes and are N dependent for the even ones. Thus a numerical study of finite size scaling in the Ising model serves as a non-perturbative test of triviality of φ44-theories for all N.