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 U (1) D . We concentrate on the case when these models are endowed with particular kinetic terms involving a linear power in momenta. New dimensional regularization and renormalization schemes are introduced for particular models in this class: a rank 3 tensor model, an infinite tower of matrix models φ 2n over U (1), and a matrix model over U (1) 2 . For all divergent amplitudes, we identify a domain of meromorphicity in a strip determined by the real part of the group dimension D. From this point, the ordinary subtraction program is applied and leads to convergent and analytic renormalized integrals. 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.

Functional renormalization group analysis of rank-3 tensorial group field theory: The full quartic invariant truncation

“…Hence, those terms do not belong to the proposed truncation (10) and we disregard them. Having (23), (24) and (25) we can perform explicit computations.…”

Flowing in Group Field Theory Space: a Review

“…and ρ(G) = 0 if and only if G is a melonic graph. 21 When ω(G) = 0 the divergences are logarithmic. 22 The variable ρ of this equation is a combinatorial quantity associated to a graph, and has nothing to do with the cut-off appearing in equation ( We refer the reader to, e.g., [19,35] for a more precise combinatorial characterization of melonic graphs in this context.…”

Renormalizable enhanced tensor field theory: The quartic melonic case

“…These models can be regarded as a rightful extension of matrix field theories like the Grosse and Wulkenhaar model [119,120,121], an asymptotically safe non-local quantum field theory stemming from noncommutative geometry. The parametric representation and the ensuing dimensional regularization have been extended to tensor field theory with the emergence of new Symanzik polynomial invariants for tensor graphs [61]. Moreover, the computations of the perturbative β-functions for φ 4 -and φ 6 -like models were achieved in the UV [88,86,89,91,96,95,97].…”