Parametric representation of rank d tensorial group field theory: Abelian models with kinetic term ∑sps+μ

Abstract

We consider the parametric representation of the amplitudes of Abelian models in the so-called framework of rank d tensorial group field theory. These models are called Abelian because their fields live on copies of U(1)D. We concentrate on the case when these models are endowed with particular kinetic terms involving a linear power in momenta. A new dimensional regularization is introduced for particular models in this class: a rank 3 tensor model, an infinite tower of matrix models ϕ2n over U(1), and a matrix model over U(1)2. We prove that all divergent amplitudes are meromorphic functions in the complexified group dimension D ∈ ℂ. From this point, a standard subtraction program yielding analytic renormalized integrals could be applied. Furthermore, we identify and study in depth the Symanzik polynomials provided by the parametric amplitudes of generic rank d Abelian models. We find that these polynomials do not satisfy the ordinary Tutte’s rules (contraction/deletion). By scrutinizing the “face”-structure of these polynomials, we find a generalized polynomial which turns out to be stable only under contraction.