RG flows with global symmetry breaking and bounds from chaos

Abstract

We discuss general aspects of renormalization group (RG) flows between two conformal fixed points in 4D with a broken continuous global symmetry in the UV. Every such RG flow can be described in terms of the dynamics of Nambu-Goldstone bosons of broken conformal and global symmetries. We derive the low-energy effective action that describes this class of RG flows from basic symmetry principles. We view the theory of Nambu-Goldstone bosons as a theory in anti--de Sitter space with the flat space limit. This enables an equivalent 3D CFT (${\mathrm{CFT}}_{3}$) formulation of these 4D RG flows in terms of spectral deformations of a generalized free ${\mathrm{CFT}}_{3}$. We utilize this dual description to impose further constraints on the low-energy effective action associated with unitary RG flows in 4D by invoking the chaos bound in 3D. This approach naturally provides a set of independent monotonically decreasing $C$-functions for 4D RG flows with global symmetry breaking by explicitly relating 4D $C$-functions with certain out-of-time-order correlators that diagnose chaos in 3D. We also comment on a more general connection between RG and chaos in QFT.