Question of an ultraviolet zero of the beta function of the λ (φ → 2)42 theory

We investigate the possibility of an ultraviolet (UV) zero in the n-loop beta function of a λ(φ→2)2 field theory with an N-component scalar field, φ→, in four spacetime dimensions, up to the level of n=5 loops. Although the two-loop beta function has a UV zero, we find that the n-loop beta function for n=3, 4, 5 either does not have a UV zero or does not have one at a value of λ in approximate agreement with the two-loop calculation. Similar results are obtained after application of scheme transformations to the beta function and via calculation of Padé approximants. We thus conclude that in the range of λ where the perturbative calculation of the n-loop beta function is reliable, the theory does not exhibit robust evidence of a UV zero up to the level of n=5 loops.