Abstract
We study $3d$ $O(N)$ symmetric scalar field theories using Polchinski's renormalisation group. In the infinite $N$ limit the model is solved exactly including at strong coupling. At short distances the theory is described by a line of asymptotically safe ultraviolet fixed points bounded by asymptotic freedom at weak, and the Bardeen-Moshe-Bander phenomenon at strong sextic coupling. The Wilson-Fisher fixed point arises as an isolated low-energy fixed point. Further results include the conformal window for asymptotic safety, convergence-limiting poles in the complex field plane, and the phase diagram with regions of first and second order phase transitions. We substantiate a duality between Polchinski's and Wetterich's versions of the functional renormalisation group, also showing that that eigenperturbations are identical at any fixed point. At a critical sextic coupling, the duality is worked out in detail to explain the spontaneous breaking of scale symmetry responsible for the generation of a light dilaton. Implications for asymptotic safety in other theories are indicated.
Highlights
Fixed points of the renormalization group (RG) play a fundamental role in quantum field theory and statistical physics [1]
Further results include the conformal window for asymptotic safety, convergence-limiting poles in the complex field plane, and the phase diagram with regions of first and second order phase transitions
We substantiate a duality between Polchinski’s and Wetterich’s versions of the functional renormalization group, showing that eigenperturbations are identical at any fixed point
Summary
Fixed points of the renormalization group (RG) play a fundamental role in quantum field theory and statistical physics [1]. Interacting UV fixed points ensure that the renormalization group evolution of couplings remains finite even at shortest distances, opening new opportunities to define fundamental theories including quantum gravity [5,6]. It has been conjectured that asymptotic safety may exist at strong coupling [17,18,19], In 3d, the interest in fixed points of relativistic quantum fields is motivated by thermal and quantum phase transitions. We study interacting fixed points and asymptotic safety of 3d OðNÞ symmetric field theories in the large-N limit. The main technical novelty of this work is the use of Polchinski’s renormalization group, originally employed for proofs of renormalizability [61] It is based on a Wilsonian UV cutoff, and allows a complete.
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