Abstract In Giombi et al (2018 Phys. Rev. D 98 105005), a prismatic tensor model was introduced. We study here the diagrammatics and the double scaling limit of this model, using the intermediate field method. We explicitly exhibit the next-to-leading order Feynman graphs in the 1 / N expansion. Using appropriate combinatorial tools, we further study the general term of the 1 / N expansion and we compute the two-point function in the double scaling limit. We find that the double scaling limit mechanism implemented here is similar to the one implemented for various quartic tensor models.
This thesis focuses on renormalization of quantum field theories. Its first part considers three tensor models in three dimensions, a Fermionic quartic with tensors of rank-3 and two Bosonic sextic, of ranks 3 and 5. We rely upon the large-
N melonic expansion of tensor models. For the first model, invariant under
U(N)3, we obtain the RG flow of the two melonic couplings and the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. The discrete chiral symmetry breaks spontaneously and we compare with the three-dimensional Gross-Neveu model. Beyond the massless
U(N)3 symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into
U(N2)×U(N/2)×U(N/2). A matrix model invariant under
U(N)×U(N2), with close properties, is also studied. For the other models, with symmetry groups
U(N)3 and
O(N)5, a non-melonic coupling (the "wheel") with an optimal scaling in
N drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-
N, the RG flows of the sextic couplings up to four loops. Only the rank-3 model displays non-trivial fixed points (two real Wilson-Fisher-like in the short-range case and a line of fixed points in the other). We finally obtain the real conformal dimensions of the primary bilinear operators. In the second part, we establish the first results of perturbative multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At criticality, an emergent infinite spine provides a space of effective dimension
4/3 on which to compute averaged correlation fonctions. This approach formalizes the notion of a QFT on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph reviewed in detail.
Abstract In the first section we give a briefly presentation of the U(N)D-invariant tensor models (N being again the size of the tensor, and D being the dimension). The next section is then dedicated to the analysis of the Dyson–Schwinger equations (DSE) in the large N limit. These results are essential to implement the double scaling limit mechanism of the DSEs, which is done in the third section. The main result of this chapter is the doubly-scaled 2-point function for a model with generic melonic interactions. However, several assumptions on the large N scaling of cumulants are made along the way. They are proved using various combinatorial methods.
The definition of a double scaling limit represents an important goal in the development of tensor models. We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the colored tensor models. We show that the radius of convergence of the nlo series coincides with that of the leading order melonic sector. Meanwhile, the value of the susceptibility exponent, γ nlo = 3/2, signals a departure from the leading order behavior. Both pieces of information provide clues for a non-trivial double scaling limit, for which we put forward some precise conjecture.
This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with
O(N)3 symmetry and long-range propagator. The existence of a non-perturbative fixed point in any
d at large
N is established. We found four lines of fixed points parametrized by the so-called tetrahedral coupling. One of them is infrared attractive, strongly interacting and gives rise to a new kind of CFT, called melonic CFTs which are then studied in more details. We first compute dimensions of bilinears and OPE coefficients at the fixed point which are consistent with a unitary CFT at large
N. We then compute
1/N corrections. At next-to-leading order, the line of fixed points collapses to one fixed point. However, the corrections are complex and unitarity is broken at NLO. Finally, we show that this model respects the
F-theorem. The next part of the thesis investigates sextic tensor field theories in rank
3 and
5. In rank
3, we found two IR stable real fixed points in short range and a line of IR stable real fixed points in long range. Surprisingly, the only fixed point in rank
5 is the Gaussian one. For the rank
3 model, in the short-range case, we still find two IR stable fixed points at NLO. However, in the long-range case, the corrections to the fixed points are non-perturbative and hence unreliable: we found no precursor of the large
N fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large
N limit. We prove that models with tensors in an irreducible representation of
O(N) or
Sp(N) in rank
5 indeed admit a large
N limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.