Abstract
We discuss walking behavior in gauge theories and weak first-order phase transitions in statistical physics. Despite appearing in very different systems (QCD below the conformal window, the Potts model, deconfined criticality) these two phenomena both imply approximate scale invariance in a range of energies and have the same RG interpretation: a flow passing between pairs of fixed point at complex coupling. We discuss what distinguishes a real theory from a complex theory and call these fixed points complex CFTs. By using conformal perturbation theory we show how observables of the walking theory are computable by perturbing the complex CFTs. This paper discusses the general mechanism while a companion paper [1] will treat a specific and computable example: the two-dimensional Q-state Potts model with Q > 4. Concerning walking in 4d gauge theories, we also comment on the (un)likelihood of the light pseudo-dilaton, and on non-minimal scenarios of the conformal window termination.
Highlights
Walking is a somewhat mysterious behavior which can conjecturally be exhibited by some four-dimensional (4d) gauge theories
We introduced a new type of conformal field theories that we call complex since they correspond to fixed points of renormalization group (RG) flows that exist at complex values of coupling constants
We argued that these theories can be well defined, and that one can work with them in the same way as one does with usual real conformal field theories (CFTs)
Summary
Walking is a somewhat mysterious behavior which can conjecturally be exhibited by some four-dimensional (4d) gauge theories. We will first improve understanding of walking by drawing intuition from a much simpler example of this behavior, belonging to the realm of statistical physics: the Q-state Potts model in 2d. This model is known to have a conformal phase for Q < 4, and a first-order phase transition at Q > 4. We present a computational paradigm, a kind of conformal perturbation theory, which allows to compute certain properties of walking RG flows in terms of CFT data of complex CFTs. In this paper we only discuss general features of this paradigm. Appendix D discusses features of conformal window and walking in 4d gauge theories arising in the large N limit
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