We develop the functional renormalization group formalism for a tensorial group field theory with closure constraint, in the case of a just renormalizable model over , with quartic interactions. The method allows us to obtain a closed but non-autonomous system of differential equations which describe the renormalization group flow of the couplings beyond perturbation theory. The explicit dependence of the beta functions on the running scale is due to the existence of an external scale in the model, the radius of . We study the occurrence of fixed points and their critical properties in two different approximate regimes, corresponding to the deep UV and deep IR. Besides confirming the asymptotic freedom of the model, we find also a non-trivial fixed point, with one relevant direction. Our results are qualitatively similar to those found previously for a rank-3 model without closure constraint, and it is thus tempting to speculate that the presence of a Wilson-Fisher-like fixed point is a general feature of asymptotically free tensorial group field theories.

Functional Renormalization Group Approach for Tensorial Group Field Theory:

a Rank- Model with Closure Constraint

Dario and Vincent

Laboratoire de Physique Théorique, CNRS-UMR 8627, Université Paris-Sud 11, 91405 Orsay Cedex, France

1 Introduction

Among the many results obtained with the renormalization group, two of the most famous are certainly the asymptotic freedom of QCD and the non-trivial IR behavior of scalar field theories below the critical dimension, due to the presence of a Wilson-Fisher fixed point. However, the two phenomena are hard to find in the same system: the non-trivial IR behavior of QCD is due to confinement, not to an IR fixed point, and the marginal couplings of scalar field theories do not go to zero in the UV, running instead into Landau poles. One example of coexistence is provided by gauge theories with massless fermions, with close to the critical number above which asymptotic freedom is lost: in such case one finds the so-called Banks-Zaks fixed point [1]. However, such a fixed point is not a generic feature of gauge theories, it exists only for a restricted range of parameters. More recently, a new class of field theories has emerged which appears to enjoy asymptotic freedom in quite some generality, and for which we will show here another case of its coexistence with a Wilson-Fisher fixed point. Such theories are known as Tensorial Group Field Theories (TGFTs).

TGFTs are a particular case of Group Field Theories (GFTs) [2], a class of field theories defined on a group manifold, whose non-local interactions give a complex cellular structure to the Feynman’s graphs of the perturbation theory. Such a structure is particularly interesting for quantum gravity, because it provides a tool for summing over manifolds. In their simplest form, and historically their original one, they appear as generalizations of matrix models [3] to higher dimensions, in the form of tensor models [4]. The most fruitful form of these models are the colored tensor models, for which an expansion in , similar to the one for matrices, has been constructed by Gurau and his collaborators [5, 6, 7, 8, 9]. Such results boosted a fast expansion in this direction (e.g. [10] references therein). Proper GFT models can be seen as enrichments of tensor models with group-theoretic data. They originated in the context of topological field theory [11], and they were further developed because of their links with loop quantum gravity [12, 13] and spin-foam models [14]. Lastly, TGFTs can be seen either as an enrichment of the colored tensor models by the addition of group variables, or alternatively as GFTs whose interactions show the same structure and the same unitary invariance as the interactions of the colored tensor models. Such particular models will be studied in this paper, by means of a specific example.

From the quantum gravity point of view, one of the main challenges of GFTs is to understand how the quantum degrees of freedom organize to form a geometric structure which can be identified with a semi-classical space-time (a challenge that GFT share with loop quantum gravity [15]). In this quest, the physics of phase transitions seems to be a promising route. In the most widely admitted scenario [16], and according to recent works [17, 18], these phase transitions should correspond to the condensation of quanta corresponding to the fields involved in the GFTs. This phenomenon, very similar to the Bose-Einstein condensation, implies the spontaneous acquisition of a nonzero value by the mean field (something similar has been explored recently also in tensor models [19, 20]). In this approach, the help of the renormalization machinery, which perfectly matches with the GFTs formalism, is invaluable [21]. The renormalization of GFTs [22] and TGFTs [23, 24, 25, 26] has been studied for several years, and the TGFTs (with non-trivial propagator) are the only models found so far which allow to build rigorously a renormalization program, a success mainly due to the loosening of the ultra-locality of usual GFTs, and to the tensorial nature of the interactions. This leads to a new notion of locality, called “traciality”, which enables one to clearly define the contraction notion of a high graph [24]. For now, several approaches have been followed, especially perturbative ones, and some characteristics of these models start to emerge. One of them is their asymptotic freedom, which has been discovered by Ben Geloun and his collaborators by perturbative calculations [23], and which seems to be a generic characteristic of TGFTs [28]. Another aspect is the presence of non-trivial fixed points, which started being explored only recently by means of the Functional Renormalization Group (FRG) [29, 30] and the -expansion [31].

The FRG is a widely used formulation of the Wilsonian RG, with a vast range of applications [32, 33, 34, 35, 36, 37, 38]. For many purposes its most convenient formulation is in terms of the effective average action, which satisfies what is often called the Wetterich equation [39, 40]. The main advantage of the FRG is that it allows to explore different approximation methods that do not necessarily rely on having a small coupling constant, and thus are particularly interesting in the search of non-trivial fixed points. It is precisely for such reason that we will employ here the FRG machinery, following the path opened by its application to matrix models [41] and TGFTs [29, 30].

In this paper, we will further develop the FRG approach to TGFTs, by studying the case of a just renormalizable Abelian TGFT [26]. After briefly reviewing TGFTs in section 2, and the FRG construction at the beginning of section 3, we will obtain, by the end of section 3, a closed system of differential equations which describe the renormalization group flow of the model. In sections 4 and 5, we will study the occurrence of fixed points and phase transitions in two different approximate regimes, corresponding to the deep UV and deep IR, or to the limit in which the radius of the group manifold goes to infinite and zero, respectively. We will summarize and discuss our findings in section 6, leaving the connection to the perturbative calculations for the appendices.

2 Tensorial Group Field Theories

2.1 Generalities

A tensorial group field theory on copies of the compact group is built along the lines of uncolored tensor models of rank [9]. The theory is defined by a choice of Gaussian measure and of interaction in the generating function


where refers to the covariance of the free theory. The fields and , as well as the sources and are functions on that take values in ,

and the notation means


where stands for the Haar measure on the group being considered. Obviously, if has dimension , the TGFT is a field theory in dimension . We denote by and the set of fields and , respectively. The true peculiarity of TGFT models lies in the non-local form of the interactions contained in , which are built as follows. The convolution of the fields follows the outline of index contractions from the interactions of the uncolored tensor models [9]: group elements are contracted (i.e. identified and integrated over the group) pairwise, such that an element appearing as the -th argument of a is contracted with an element appearing as the -th argument of a . A pattern of contracted fields is sometimes referred to as a bubble. What makes the bubbles non-local is both the restriction to pairwise identification and the fact that a given field can contract different arguments to different fields. While the interactions are non-local on the group, the pairwise contraction allows the definition of a new locality principle, known as traciality, essential for the renormalization. In the lattice gauge field theory language, traciality means essentially that, for all the leading order graphs, flat holonomies imply necessarily trivial connection [24]. In the case that is the abelian group , this amounts to imposing invariance under a formal unitary symmetry, the limit for going to infinity of the uncolored tensor models’ invariance. The interactions are formed by all the possible index contractions compatible with the invariance, i.e. they are formed by the pairwise contraction described above, with no insertion of external functions or operators on the group. We refer to these interactions as invariant traces:


where labels each contraction possible, involving an arbitrary number of fields, and is associated to the coupling . Those contractions are graphically represented by bipartite colored graphs, whose vertices, black or white, are associated to the fields and , respectively, and whose lines correspond to the contracted field indices. For example, the graph of figure 1 below,

Figure 1: Example of invariant bubble in dimension 6

corresponds to the following rank-6 interaction:


Other examples of bubbles are given on figure 2, for various ranks. There is obviously only one bubble constructible out of two fields, represented in the last graph of figure 2. This is the only local interaction in the usual sense, it corresponds to a mass term, and it is often included directly in the covariance. All the other interactions arise from the multiplication of fields at different points in .

Figure 2: Examples of interaction bubbles in dimension 5, 4 and 6

In the TGFT models, a specific symmetry is often imposed at the level of the field variables, known as the closure constraint, which reads


This gauge symmetry is part of the physical enrichments added to purely tensorial models in the GFT context and it can be interpreted as geometrical data. Its origin can be found in the spin-foam foundations and LQG requirements of group field theories [2, 24], and recent works have shown that it implies some nice properties in a purely field theoretical context. In particular, it has been pointed out that it might favor asymptotic freedom, see [28].

The choice of the covariance must respect this invariance and must enforce it at the level of the Feynman graphs. The usual choice is the following, given here in Schwinger (or parametric) representation,


where is the mass parameter of the theory and is the heat kernel on G, verifying


being the Laplacian operator on .

As in the standard quantum field theories, the calculation of perturbations to the Gaussian model can be evaluated thanks to the Wick theorem, and any -points Schwinger function, , can be evaluated as a sum over Feynman graphs. These graphs, of which figure 3 is an example, are formed by all possible Wick contractions, represented by dashed lines, between all couples .

Figure 3: A typical Feynman graph in with four external lines

A -points Schwinger function can thus be written as


where is the dimension of the automorphism group of the graph (the symmetry factor), label the vertices of and is the graph amplitude. The expression of the amplitude can be obtained by using the elementary properties of the heat kernel. We find:


In this formula, are the source and terminus of the oriented edge , designates the incidence matrix, being 0, +1 or -1 given that the line belongs or not to the boundary of , the sign depending on the relative orientation, and and are respectively the sets of lines, internal and external faces. Note that in the language of lattice gauge theory, the integrals over the group variables in the amplitude (9) can be understood as a sum over discrete connections on the cellular complex defined by the Feynman graph . From this point of view, the product is an holonomy around the face , and the symmetry of the Feynman amplitudes (where is the number of vertices in ) coming from the invariance under the transformations


is a discrete version of gauge invariance.

In this paper we study the case and , that is, a six-dimensional Abelian model, with interactions limited to the degree 4 melonic bubbles of the same form as that of figure 1. All the interactions of that kind, for each possible choice of the solitary line color, are included in the model with the same coupling constant. In this model, the heat kernel has a simple expression in momentum representation


and the integration on the group in the propagator’s definition (6) shows a , so that the expression of our model’s propagator is written, in momentum representation


where stands for a Kronecker delta. The interaction part can be written as:


where designates the bubble of figure 1 whose solitary line is of color and is a super-tensor of order four, invariant under unitary transformations.333That is, for a fixed size of a tensor, As mentioned before, these interactions are chosen to be melonic, meaning that their Gurau degree [7] vanishes: . In momentum space we have:


As explained in [9], the Gurau degree characterizes the colored graphs associated to the interaction bubble, and interestingly, it plays a role similar to the genus for the matrix models in the expansion of the colored tensor models. Melons appear as the leading order graphs in this framework, and it can be proved that they are dual to topological spheres. Their names comes from the leading order graphs of the colored tensorial models, called melons in [8]. Our model is uncolored, in the sense of [9], but our Feynman graphs admit a natural colored extension, and one can show that their corresponding Gurau degree vanishes for the melonic graphs.

Because of the non-local structure of the interactions, the renormalization becomes quite non trivial. However, recent works have proved that a renormalization scheme is possible, thanks to the traciality property of the divergent graphs (see [24]), coming from the tensorial structure of the interactions. The power counting of our model was carried out using multi-scale analysis, and it has been proven [24] that the divergent degree for an amplitude is given by:


where and designate respectively the number of lines and faces of the graph mentioned, and the rank of the incidence matrix defined below equation (9). It can be checked that our model is just-renormalizable, with divergent graphs of degree and requiring only mass, field and coupling renormalization. More interestingly, it can be proved that all the divergent graphs are melonic, in the sense explained before. Hence, for our model, the melonic sector contains all the graphs that need to be renormalized.

2.2 Canonical dimension of a tensorial bubble

In order to investigate the influence of the dimension on the occurrence of fixed points in the FRG analysis, we need to work with dimensionless quantities, as in usual quantum or statistical field theories. In the present case, there exists an external scale (the radius of ) which we have set to 1, thus implicitly making all quantities dimensionless in the usual sense. In fact, as pointed out in [29], and as usual in the presence of an external scale [42], this is the reason why we will obtain non-autonomous RG flow equations. However, what matters for our purposes is how such quantities scale with the cutoff, which we refer to as scaling dimension. This dimensional notion appears quite naturally in the perturbative calculations. In Schwinger regularization, which introduces a cutoff in the lower end of the integral over appearing in (9), we get, at the dominant order in and in dimension :


where and are two numerical constants independent of or (see appendix B). The exponent of in the above expressions is in general a universal number, meaning that it does not depend on the chosen renormalization (for example, we find the same exponent by regularizing the theory by means of a cutoff on the momentum). We therefore define the (canonical444At a non-trivial fixed point the scaling dimension will in general be anomalous, i.e. different from the canonical one.) scaling dimension of a quantity in such a way that by redefining we obtain a homogeneous expression in for the renormalization of . The equation (17) then shows that the scaling dimension of is , so that both summands have the same dimension. By using this result in (16), we find , implying that , since gives the radiative corrections of the mass term.

We can easily check the coherence of these definitions at all orders, noting that the exponent of appearing in (16) and  (17) is nothing else but the divergence degree of the corresponding graph. The divergent degree is given by (15), and it can easily checked [24] that the most divergent graphs, the so-called melons, verify:


where is the number of vertices in the graph. The equation (18), together with the combinatorial relation , leads to


We limit ourselves to the interactions that we can find in the initial theory. A 1PI graph with four external legs has, a priori the same dimension as the coupling constant . That is to say:

implying . The same argument applied to an 1PI function with two external legs justifies :


3 FRG approach for TGFT

In this section we apply the Functional Renormalization Group formalism to the TGFTs introduced above. First we derive the Wetterich equation [39, 40] in this context, and later we use it to extract the flow equations in a simple approximation. For general reviews and applications of the FRG we refer to [32, 33, 34, 35, 36, 37, 38].

3.1 Effective average action for TGFTs

Starting from the generating functional defined by the equation (1), we define the following deformation:


where we have added to the action an IR cutoff or “momentum-dependent mass term” , chosen ”ultralocal” in the momentum representation, and formally defined as:


where is the Fourier transform (or Peter-Weyl decomposition) of the field: . As usual, in order to make sense of the partition function we assume that a UV regulator is also present, e.g. a sharp cutoff on the momenta (for the usual norm: ). In practice, one often works in the limiting case , because the Wetterich equation is well defined in that limit (although not its path integral origin).

The cutoff function is a positive definite function chosen so that:

for all and .

, implying:


This condition ensures that the original model is in the family (21). Physically, it means that the original model is recovered when all the fluctuations are integrated out.

, ensuring that all the fluctuations are frozen when . As a consequence, the bare action will be represented by the initial condition for the flow at .

For , the cutoff is chosen so that


a condition ensuring that the UV modes are almost unaffected by the additional cutoff term, while , or , will guarantee that the IR modes are decoupled.

, for all and , which means that high modes should not be suppressed more than low modes.

As it stands, (21) defines an infinite-dimensional deformation of the original partition function. However, the role of the precise cutoff function, chosen to satisfy the above requirements, is secondary with respect to the role of the parametric dependence on . From a Wilsonian point of view, the former corresponds to a choice of coarse graining scheme, while the latter corresponds to the coarse graining scale. We are primarily interested on the scale dependence of the theory, and therefore we take the point of view that a specific cutoff function has been chosen, and view (21) as a one-parameter family of theories.555In principle, physical quantities, such as critical exponents, are independent of the coarse graining scheme (see for example the universality of the one-loop beta function for marginal couplings, which we discuss in appendix A), but approximations generally spoil this property. Scheme dependence is thus an important issue, which has been greatly developed into the art of optimization [43], but we will not discuss it further here. We thus obtain a one-parameter family of free energies,


and by their Legendre transform, a one-parameter family of effective actions, collectively called effective average action. To be more precise, the effective average action is defined as:


where the source is to be expressed as a function of the effective mean field via the solution of


The previous properties concerning the cutoff term mean, at the effective average action level, that:

, so that when all the fluctuations are frozen, the effective average action coincides with the initial “microscopic” action.

, meaning that when all the fluctuations are integrated out, the effective average action coincides with the full effective action.

The definition (21) of the modified action can be usually interpreted as a modification of the propagator. For a standard Gaussian measure, with usual kinetic action, this property is obvious, but it is not so obvious for our model, for which the covariance has been defined via equation (6) rather than by an explicit kinetic action.666Alternatively, we can define our covariance in its momentum representation, directly via equation (12). In this case, we see that because of the projector , the definition of the inverse , which usually appears in the kinetic action, is slightly subtle and we prefer to present a more formal derivation involving only . To prove this point, we use the Wick-theorem. Consider the following covariance:


and the two Gaussian integrations:


where is the normalized Gaussian integration: . In a first step, we will prove that . This comes obviously from the Wick theorem. Using the derivative representation of a Gaussian integral:


where the elements of the (super-) matrix are defined as . Next, consider the following Gaussian integral:


Using the Wick theorem, and the previous result, we obtain:

where is the permutation group of elements. Now, using the Wick theorem, the big parenthesis can be written as:


where the brackets mean the Gaussian contractions with covariance , and with subscript select only the “non-vacuum” contractions (i.e. involving the external fields only). The Cauchy decomposition formula allows to rewrite the double sums as:

The first term is nothing but the trivial Gaussian integral, i.e. the normalization of the Gaussian measure :

implying finally:

Reporting this result into the equation (3.1) for leads to:


Finally, because of the fact that the factor is nothing but the vacuum contribution of the measure , we find:


Not surprisingly, this result is exactly what is expected for a standard Gaussian integration with well defined kinetic action. Hence, it follows from the properties of the Gaussian integration that the definition (21), with modification of the action, can be interpreted as a modification of the covariance, i.e.


In fact, the flow equations for (21) and (36) differ for vacuum terms due to the implicit in the measure of (36) (see (35)), but as usual such vacuum terms are completely unimportant.

Because of the gauge invariance of our model, it is useful to define the projector into the gauge invariant field as:


and the invariant field subspace as . Interestingly, because of the gauge constraint in the propagator, the gauge constraint is implemented, at the graph level, at each colored line hooked to a black or white vertex. Hence, the mean field , defined by (27) lives in . The gauge symmetry is then dynamically implemented by the propagator, and contaminates each -points functions : .

For the 1-point function , this property can be proved as follow. From the general expression (8), can be expanded as:


where is a Feynman amplitude of order with one external point. A typical graph contributing to this expansion is depicted on the Figure 4 below, and its explicit expression has the following structure:


where for convenience we have used of the propagator defined by (28) unlike of the propagator . From its definition, the propagator lives in , because it verifies :

Hence, it follows that the three vector indices of the (super-)tensor are projected into the gauge invariant space, and this tensor is equivalent to :


where . The consequences are clearer in Fourier components:


The index structure of the bubble vertex implies that , and . But, because of the Kronecker delta, , implying and finally . Hence satisfies the closure constraint: , and the last term of the previous equation can then be rewritten as:


meaning that , or that , thus proving our claim.

Figure 4: Typical graph contributing to

Using the Legendre transform (26) and the definition (21), we obtain formally the so-called “Wetterich equation”, describing the evolution of the effective average action. We have:

Proposition 1

(Wetterich equation)
For a given cutoff , the effective average action satisfies the following partial differential equation:


where is the ”super-trace”, meaning the trace over the block-indices, and . The presence of restricts the sums over the subspace of which we denote by and define by : .

Proof: The proof is rather standard [39, 40], but it is useful to review it because of the TGFT context and the appearance of the projector in the equation. Applying the operator on the two members of the equation (26), we have


The term can be easily computed from the definitions (21) and (25):


Using (27), many terms cancel in (44), and we are left with


Deriving (27) with respect to we find


while deriving (26) with respect to and then gives


Therefore we obtain, in matrix notation:


The second functional derivative is the -points function which, as the effective mean field