Perturbative critical behavior from spacetime dependent couplings

We find novel perturbative fixed points by introducing mildly spacetime dependent couplings into otherwise marginal terms. In four-dimensional quantum field theory, these are physical analogues of the small-epsilon Wilson-Fisher fixed point. Rather than considering 4 - epsilon dimensions, we stay in four dimensions but introduce couplings whose leading spacetime dependence is of the form lambda x(kappa)mu(kappa), with a small parameter kappa playing a role analogous to epsilon. We show, in phi(4) theory and in QED and QCD with massless flavors, that this leads to a critical theory under perturbative control over an exponentially wide window of spacetime positions x. The exact fixed point coupling lambda(*)(x) in our theory is identical to the running coupling of the translationally invariant theory, with the scale replaced by 1/x. Similar statements hold for three-dimensional phi(6) theories and two-dimensional sigma models with curved target spaces. We also describe strongly coupled examples using conformal perturbation theory.