Abstract
We study classical and quantum (at large-$N$) field equations of bosonic tensor models with quartic interactions and $O(N)^3$ symmetry. Among various possible patterns of spontaneous symmetry breaking we highlight an $SO(3)$ invariant solution, with the tensor field expressed in terms of the Wigner $3jm$ symbol. We argue that such solution has a special role in the large-$N$ limit, as in particular its scaling in $N$ can provide an on-shell justification for the melonic large-$N$ limit of the two-particle irreducible effective action in a broken phase.
Highlights
Spontaneous symmetry breaking is an extensively studied subject in statistical and quantum field theory, as it is a paradigm of phase transitions, and it offers the basis for many theoretical developments and phenomenological models
We will consider a very concrete example in this paper: we will take N 1⁄4 N3, and we will choose a potential with OðNÞ3 symmetry, which is much smaller than OðN3Þ: the number of generators of OðNÞ3 is 3NðN − 1Þ=2, which acts as an upper bound on the number of Goldstone bosons, well below their number in the OðN3Þ case
In this paper we have identified an SOð3Þ-invariant solution to the field equations of the bosonic OðNÞ3invariant tensor model with quartic interactions
Summary
Spontaneous symmetry breaking is an extensively studied subject in statistical and quantum field theory, as it is a paradigm of phase transitions, and it offers the basis for many theoretical developments and phenomenological models. For d ≥ 2, they represent a novel class of quantum field theories for which we can hope to control nonperturbative aspects via the large-N limit, and in particular discover new interacting conformal field theories [12–19] It is mainly their applications for d ≥ 1 that motivate our study of spontaneous symmetry breaking (SSB) in tensor models. SSB in tensor models has been mostly unexplored, with the only exception being the work of Diaz and Rosabal [22], which as we will argue in the following, did not provide interesting symmetry breaking solutions from the point of view of the large-N limit. In Appendix A we collect useful formulas and notations, while in Appendix B we discuss other solutions of the classical field equations, which are less interesting from the large-N point of view, as they either become trivial or they are only sensitive to one of the three interaction terms
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