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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 QFT, these are physical analogues of the small-$ε$ Wilson-Fisher fixed point. Rather than considering $4-ε$ dimensions, we stay in four dimensions but introduce couplings whose leading spacetime dependence is of the form $λx^κμ^κ$, with a small parameter $κ$ playing a role analogous to $ε$. We show, in $ϕ^4$ 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 $λ_*(x)$ 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 $ϕ^6$ theories and two-dimensional sigma models with curved target spaces. We also describe strongly coupled examples using conformal perturbation theory.