Why are tensor field theories asymptotically free?

Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris Sud - 91405 Orsay Cedex, France and Perimeter Institute for Theoretical Physics - 31 Caroline St. N, N2L 2Y5, Waterloo, ON, Canada

Received: 15 July 2015
Accepted: 21 September 2015

Abstract

In this pedagogic letter we explain the combinatorics underlying the generic asymptotic freedom of tensor field theories. We focus on simple combinatorial models with a propagator and quartic interactions and on the comparison between the intermediate field representations of the vector, matrix and tensor cases. The transition from asymptotic freedom (tensor case) to asymptotic safety (matrix case) is related to the crossing symmetry of the matrix vertex, whereas in the vector case, the lack of asymptotic freedom (“Landau ghost”), as in the ordinary scalar case, is simply due to the absence of any wave function renormalization at one loop.