The nontrivial fixed point discovered for $ϕ^4$-marginal couplings in tensorial group field theories have been showed to be incompatible with Ward-Takahashi identities. In this previous analysis we have stated that the case of models with interactions of order greater than four could probably lead to a fixed point compatible with local Ward's identities. In this paper we focus on a rank-4 Abelian $ϕ^6$-just renormalizable tensorial group field theory and describe the renormalization group flow over the sub-theory space where Ward constraint is satisfied along the flow, by using an improved version of the effective vertex expansion. We show that this model exhibit a nontrivial fixed points in this constrained subspace. Finally, the well-known asymptotically freedom of this model is highlighted.
The $D$-colored version of tensor models has been shown to admit a large $N$-limit expansion. The leading contributions result from so-called melonic graphs which are dual to the $D$-sphere. This is a note about the Schwinger-Dyson equations of the tensorial $φ^{4}_{5}$-model (with propagator $1/{\bf p}^{2}$) and their melonic approximation. We derive the master equations for two- and four-point correlation functions and discuss their solution.
Referring to recent works concerning the functional renormalization group for tensorial group fields theories [arXiv:1803.09902] and [arXiv:1809.00247], this paper gives in-depth explanation for, the ambiguity around the search of fixed points in the Wetterich flow equation and the Ward-Takahashi identities. We consider the $U(1)$-tensors models and discuss the non-existence of phase transition taking into account the Ward-identities as new constraint along the flow. We prove that the quartic melonic tensorial group fields theories without closure constraint are devoid with the fixed points an…
In recent years, some interesting investigations of the non-perturbative renormalization group equations for tensorial group field theories have been done in the truncation method, and completely discarding the Ward identities from their analysis. In this letter, in continuation of our recent series of papers, we present a new framework of investigation, namely the effective vertex expansion, allowing to consider infinite sectors rather than finite dimensional subspaces of the full theory space. We focus on the ultraviolet behavior, and provide a new and complete description of the renormaliza…
This paper, in addition to our recent works, intends to explore the behavior of the Wetterich flow equations in the portion of the theory space spanned by non-branching melons constrained with Ward-identities. We focus on a rank-5 just-renormalizable tensorial group field theory and consider a non-trivial extension of the local potential approximation namely effective vertex expansion for just-renormalizable quartic melonic interactions, disregarding effects coming from disconnected interactions. Investigating the dynamically constrained flow, we show explicitly that results weakly rely on the…
This paper aims at investigating the nonperturbative functional renormalization group for tensorial group field theories with nontrivial kinetic action and closure constraint. We consider the quartic melonic just-renormalizable $[U(1)]^6$ model and show that due to this closure constraint the first order Ward-Takahashi identity takes the trivial form as for the models with propagators proportional to identity. We then construct the new version of the effective vertex expansion applicable to this class of models, which in particular allows to close the hierarchical structure of the flow equatio…
This paper aims at giving a novel approach to investigate the behavior of the renormalization group flow for tensorial group field theories to all orders of the perturbation theory. From an appropriate choice of the kinetic kernel, we build an infinite family of just-renormalizable models, for tensor fields with arbitrary rank $d$. Investigating the large $ d$ limit, we show that the self-energy melonic amplitude is decomposed as a product of loop-vertex functions, depending only on dimensionless mass. The corresponding melonic amplitudes may be mapped as trees in the so-called Hubbard-Straton…
Renormalization group methods are an essential ingredient in the study of nonperturbative problems of quantum field theory. This paper deal with the symmetry constraints on the renormalization group flow for quartic melonic tensorial group field theories. Using the unitary invariance of the interactions, we provide a set of Ward-Takahashi identities which leads to relations between correlation functions. There are numerous reasons to consider such Ward identities in the functional renormalization group. Their compatibility along the flow provides a non-trivial constraint on the reliability of …
In this paper we consider the complete momentum-independent quartic order truncation for the effective average action of a real Abelian rank 3 tensorial group field theory. This complete truncation includes non-melonic as well as double-trace interactions. In the usual functional renormalization group perspective, the inclusion of more operators that belong to the underlying theory space corresponds to an improvement of the truncation of the effective average action. We show that the inclusion of non-melonic and double-trace operators in the truncation brings subtleties. In particular, we disc…
We address in this paper the issue of renormalizability for SU(2) Tensorial Group Field Theories (TGFT) with geometric Boulatov-type conditions in three dimensions. We prove that tensorial interactions up to degree 6 are just renormalizable without any anomaly. Our new models define the renormalizable TGFT version of the Boulatov model and provide therefore a new approach to quantum gravity in three dimensions. Among the many new technical results established in this paper are a general classification of just renormalizable models with gauge invariance condition, and in particular concerning p…
In this paper we continue our program of non-pertubative constructions of tensorial group field theories (TGFT). We prove analyticity and Borel summability in a suitable domain of the coupling constant of the simplest super-renormalizable TGFT which contains some ultraviolet divergencies, namely the color-symmetric quartic melonic rank-four model with Abelian $U(1)$ gauge invariance, nicknamed $U(1)-T^4_4$. We use a multiscale loop vertex expansion. It is an extension of the loop vertex expansion (the basic constructive technique for non-local theories) which is required for theories that invo…
In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. We derive the flow equation for the effective potential at any order restricting to a subclass of tensorial interactions called cyclic melonic and projecting to a constant field in group space. For a tensor field of rank $…
We study a just renormalizable tensorial group field theory of rank six with quartic melonic interactions and Abelian group U(1). We introduce the formalism of the intermediate field, which allows a precise characterization of the leading order Feynman graphs. We define the renormalization of the model, compute its (perturbative) renormalization group flow and write its expansion in terms of effective couplings. We then establish closed equations for the two point and four point functions at leading (melonic) order. Using the effective expansion and its uniform exponential bounds we prove that…
This manuscript aims at giving our new advance on the functional renormalization group applied to tensorial group field theory. It is based on a series of our three papers [arXiv:1803.09902], [arXiv:1809.00247] and [arXiv:1809.06081]. We consider the polynomial Abelian $U(1)^d$ models without closure constraint, especially we discuss the case of quartic melonic interaction. We present a new approach to solve the exact Wetterich flow equation called the effective vertex expansion method, and investigate the resulting flow equations, especially about the existence of non-Gaussian fixed points fo…
Nonperturbative renormalization group beyond melonic sector: The Effective Vertex Expansion method for group fields theories
Commissariat à l’Énergie Atomique (CEA, LIST)（法国原子能和替代能源委员会）；Faculté des Sciences et Techniques (ICMPA-UNESCO Chair), Université d’Abomey- Calavi（阿博米-卡拉维大学科学与技术学院）
We tackle the issue of renormalizability for Tensorial Group Field Theories (TGFT) including gauge invariance conditions, with the rigorous tool of multi-scale analysis, to prepare the ground for applications to quantum gravity models. In the process, we define the appropriate generalization of some key QFT notions, including: connectedness, locality and contraction of (high) subgraphs. We also define a new notion of Wick ordering, corresponding to the subtraction of (maximal) melonic tadpoles. We then consider the simplest examples of dynamical 4-dimensional TGFT with gauge invariance conditi…
In this paper we provide the closed equations that satisfy two-point correlation functions of the rank 3 and 4 tensorial group field theory. The formulation of the present problem extends the method used by Grosse and Wulkenhaar in [arXiv 0909.1389] to the tensor case. Ward-Takahashi identities and Schwinger-Dyson equations are combined to establish a nonlinear integral equation for the two-point functions. In the 3D case the solution of this equation is given perturbatively at second order of the coupling constant.
We study the functional renormalization group of a three-dimensional tensorial Group Field Theory (GFT) with gauge group SU(2). This model generates (generalized) lattice gauge theory amplitudes, and is known to be perturbatively renormalizable up to order 6 melonic interactions. We consider a series of truncations of the exact Wetterich--Morris equation, which retain increasingly many perturbatively irrelevant melonic interactions. This tensorial analogue of the ordinary local potential approximation allows to investigate the existence of non-perturbative fixed points of the renormalization g…
Superconformal Ward identities are derived for the the four point functions of chiral primary BPS operators for $\N=2,4$ superconformal symmetry in four dimensions. Manipulations of arbitrary tensorial fields are simplified by introducing a null vector so that the four point functions depend on two internal $R$-symmetry invariants as well as two conformal invariants. The solutions of these identities are interpreted in terms of the operator product expansion and are shown to accommodate long supermultiplets with free scale dimensions and also short and semi-short multiplets with protected dime…
This article provides a Wilsonian description of the perturbatively renormalizable Tensorial Group Field Theory introduced in arXiv:1303.6772 [hep-th] (Commun. Math. Phys. 330, 581-637). It is a rank-3 model based on the gauge group SU(2), and as such is expected to be related to Euclidean quantum gravity in three dimensions. By means of a power-counting argument, we introduce a notion of dimensionality of the free parameters defining the action. General flow equations for the dimensionless bare coupling constants can then be derived, in terms of a discretely varying cut-off, and in which all …
Ward-Takahashi identities are nonperturbative relations between correlation functions and arising from symmetries in quantum and statistical fields theories, as Noether currents conservation for classical theories. Since their historical origin, these identities were considered to prove the exact relation between counter-terms to all order of the perturbative expansion. Recently they have been considered in relation with nonperturbative renormalization group aspects for some classes of quantum field theories namely tensorial group field theories and matrix models, both characterized by a speci…
Continuum spacetime is expected to emerge via phase transition in discrete approaches to quantum gravity. A promising example is tensorial group field theory but its phase diagram remains an open issue. The results of recent attempts in terms of the functional renormalization group method remain inconclusive since they are restricted to truncations of low order. We overcome this barrier with a local-potential approximation for $\textrm{U}(1)$ tensor fields at arbitrary rank $r$ focusing on a specific class of so-called cyclic-melonic interactions. Projecting onto constant field configurations,…
Colored tensor models (CTM) is a random geometrical approach to quantum gravity. We scrutinize the structure of the connected correlation functions of general CTM-interactions and organize them by boundaries of Feynman graphs. For rank-$D$ interactions including, but not restricted to, all melonic $φ^4$-vertices---to wit, solely those quartic vertices that can lead to dominant spherical contributions in the large-$N$ expansion---the aforementioned boundary graphs are shown to be precisely all (possibly disconnected) vertex-bipartite regularly edge-$D$-colored graphs. The concept of CTM-compati…
We discuss specific hypergeometric solutions of the conformal Ward identities (CWI's) of scalar 4-point functions of primary fields in momentum space, in $d$ spacetime dimensions. We determine such solutions using various dual conformal ansätze (DCA's). We start from a generic dual conformal correlator, and require it to be conformally covariant in coordinate space. The two requirements constrain such solutions to take a unique hypergeometric form. They describe correlators which are at the same time conformal and dual conformal in any dimension. These specific ansätze also show the existence …
Stochastic dynamics for group field theories II: Methods for nonequilibrium renormalization group
Université Paris Saclay（巴黎萨克雷大学）；Cea（法国原子能和替代能源委员会）；Faculté des Sciences et Techniques (ICMPA-UNESCO Chair) Université d’Abomey-Calavi（阿博米-卡拉维大学科学与技术学院）
In the first part, we concentrate on CFTs in coordinate space. We lay the foundations of Conformal Field Theory and we also demonstrate a method where by using the embedding formalism we can derive up to n-point scalar conformal correlators. We proceed with our analysis in momentum space and we illustrate the theory of the conformal anomalies. We move on to analyse the renormalization of the correlators through counterterms followed by a discussion of the anomaly action. In the second part of the thesis. we derive and analyze the conformal Ward identities (CWIs) of a tensorial 4-point function…
