Critical behaviour of a non-local \phi^{4} field theory and asymptotic freedom

The critical behaviour of a non-local scalar field theory is studied. This theory has a non-local kinetic term which involves a real power 1-2\alpha of the Laplacian. The interaction term is the usual local \phi^{4} interaction. The lowest order Feynman diagrams corresponding to coupling constant renormalization, mass renormalization and field renormalization are computed. Particular features appearing in the renormalization of these non-local theory that differ from the case of local theories are studied. The previous calculations lead to the perturbative computation of the coupling constant beta function and critical exponents \nu and \eta. In four dimensions for \alpha<0 this beta function presents asymptotic freedom in the UV. This is remarcable since no non-abelian vector fields are included. However this comes at the expense of loosing reflection positivity.