Critical behavior of a nonlocal φ4 field theory and asymptotic freedom

Abstract

The critical behavior of a nonlocal scalar field theory is studied. This theory has a nonlocal kinetic term which involves a real power 1-2α of the Laplacian. The interaction term is the usual local φ4 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 this nonlocal 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 ν and η. In four dimensions for α<0 this beta function presents asymptotic freedom in the UV. This is remarkable since no non-Abelian vector fields are included. However, this comes at the expense of losing reflection positivity.