Nontrivial directions for scalar fields.

We study the eigenvectors of the renormalization-group matrix for scalar fields at the Gaussian fixed point, and find that that there exist “relevant” directions in parameter space. They correspond to theories with exponential potentials that are nontrivial and asymptotically free. All other potentials, including polynomial potentials, are “irrelevant,” and lead to trivial theories. Away from the Gaussian fixed point, renormalization does not induce derivative couplings, but it generates non-local interactions.