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Dynamical Selection of Critical Exponents

In renormalized field theories there are in general one or few fixed points which are accessible by the renormalization-group flow. They can be identified from the fixed-point equations. Exceptionally, an infinite family of fixed points exists, parameterized by a scaling exponent $ζ$, itself function of a non-renormalizing parameter. Here we report a different scenario with an infinite family of fixed points of which seemingly only one is chosen by the renormalization-group flow. This dynamical selection takes place in systems with an attractive interaction ${\cal V}(ϕ)$, as in standard $ϕ^4$ theory, but where the potential $\cal V$ at large $ϕ$ goes to zero, as e.g. the attraction by a defect.