RG flows and fixed points of O(N)r models

Abstract

By means of ϵ and large N expansions, we study generalizations of the O(N) model where the fundamental fields are tensors of rank r rather than vectors, and where the global symmetry (up to additional discrete symmetries and quotients) is O(N)r, focusing on the cases r ≤ 5. Owing to the distinct ways of performing index contractions, these theories contain multiple quartic operators, which mix under the RG flow. At all large N fixed points, melonic operators are absent and the leading Feynman diagrams are bubble diagrams, so that all perturbative fixed points can be readily matched to full large N solutions obtained from Hubbard-Stratonovich transformations. The family of fixed points we uncover extend to arbitrary higher values of r, and as their number grows superexponentially with r, these theories offer a vast generalization of the critical O(N) model.We also study sextic O(N)r theories, whose large N limits are obscured by the fact that the dominant Feynman diagrams are not restricted to melonic or bubble diagrams. For these theories the large N dynamics differ qualitatively across different values of r, and we demonstrate that the RG flows possess a numerous and diverse set of perturbative fixed points beginning at rank four.

Classifying large N limits of multiscalar theories by algebra

“…An algebraic structure that can rein in the rapidly increasing number of Feynman diagrams would be immensely useful at higher orders. For example, since large N limits of vectorial type, like the O(N ) model, are one-loop exact, the one-loop algebra completely characterises the leading behaviour in such limits [17]. While a more general higher order picture is the ultimate motivation for our detailed exploration of the algebraic structure, it is beyond the scope of the present study.…”