On the finite amplitudes for open graphs in Abelian dynamical colored Boulatov-Ooguri models

TL;DR

This paper proves the convergence of amplitudes in a U(1)^3 Boulatov model with Laplacian dynamics, suggesting similar behavior for SU(2) models and discussing implications for higher-rank models.

Contribution

It demonstrates the convergence of open graph amplitudes in a linearized U(1)^3 Boulatov model with Laplacian dynamics, extending understanding of renormalization.

Findings

Amplitudes in the U(1)^3 model are convergent.

Conjecture that SU(2) models exhibit similar convergence.

Discussion of implications for higher-rank models.

Abstract

In the work [Int. J. Theor. Phys. 50, 2819 (2011)], it has been proved that the radiative corrections of the 2-point function in the SU(2) Boulatov tensor model generates a relevant (in the Renormalization Group sense) contribution of the form of a Laplacian. Such a term which was missing in the initial Boulatov model action should be added in that action before discussing the renormalization analysis of this model. In this work, by linearizing the group manifold, we prove that the amplitudes associated with Feynman graphs with external legs of the colored Boulatov model over U(1)^3 endowed with a Laplacian dynamics are all convergent. We conjecture that the same feature happens for the corresponding Boulatov model over SU(2). Higher rank models are also discussed.