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 there exist open-quote open-quote relevant close-quote close-quote directions in parameter space. They correspond to theories with exponential potentials that are nontrivial and asymptotically free. All other potentials, including polynomial potentials, are open-quote open-quote irrelevant,close-quote close-quote and lead to trivial theories. Away from the Gaussian fixed point, renormalization does not induce derivative couplings, but it generates nonlocal interactions. copyright 1996 The American Physical Society