[

# [

Sylvain Carrozza
###### Abstract

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 , 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 the so-called melonic Feynman diagrams contribute. Linearizing around the Gaussian fixed point allows to recover the splitting between relevant, irrelevant, and marginal coupling constants. Pushing the perturbative expansion to second order for the marginal parameters, we are able to determine their behaviour in the vicinity of the Gaussian fixed point. Along the way, several technical tools are reviewed, including a discussion of combinatorial factors and of the Laplace approximation, which reduces the evaluation of the amplitudes in the UV limit to that of Gaussian integrals.

Discrete Renormalization Group for TGFT]Discrete Renormalization Group for Tensorial Group Field Theory

\@mkboth\shortauthors\shorttitle

[

\@xsect

Group Field Theory (GFT) [freidel_gft, daniele_rev2006, daniele_rev2011, Krajewski_rev] is an approach to Quantum Gravity which lies at the crossroad of Tensor Models [Ambjorn_tensors, gross, Sasakura:1990fs, razvan_jimmy_rev, tt2], Loop Quantum Gravity (LQG) [ashtekar_book, rovelli_book, thiemann_book, gambini_pullin_book, bojowald_book], and Spin Foam Models [perez_review2012, Baez:1997zt, oriti2001spacetime, Alexandrov:2011ab]. This research program aims at addressing questions which are notoriously difficult in loop quantum gravity, such as the construction of the dynamical sector of the theory and of its continuum limit [daniele_hydro], by means of quantum and statistical field theory techniques. A GFT is nothing but a field theory defined on a (compact) group manifold, with specific non-local interactions. The latter are chosen in such a way that the Feynman expansion generates cell complexes of a given dimension, interpreted as discrete space-time histories. In particular, the amplitudes of any spin foam model can be generated by a suitably constructed GFT, which is what triggered interest in the GFT formalism [dPFKR, GFT_rovelli_reisenberg]. Hence GFTs can be viewed as a natural way of completing the definition of spin foam models, which by themselves do not associate unambiguous amplitudes to boundary states. Note that alternatively to the summing strategy implemented in GFT, one can instead look for a definition of the refinement limit of spin foams [bianca_cyl, bianca_review, bahr2012]. A more direct route from canonical LQG to GFT has recently been proposed [daniele_2nd]: from this perspective, the GFT formalism defines a second-quantized version of LQG, and hence should be especially relevant to the analysis of the many-body sector of the theory (see [gfc_letter, gfc_full] for cosmological applications of these ideas).

Thanks to recent breakthroughs in the field of tensor models, triggered by the pioneering work of Gurau [razvan_colors], who found a generalization of the expansion of matrix models [RazvanN, RazvanVincentN, razvan_complete], standard field theory techniques are currently being developed and generalized to more and more complicated GFTs. The common ingredient to all the models studied so far is tensor invariance [r_vir, universality, uncoloring, tt2, tt3, tt4]: it provides a generalized notion of locality for GFTs, which as we have just mentioned are non-local in the ordinary sense. They then differ by: the rank of the tensor fields, identified with the space-time dimension; the space in which the indices of the tensor fields live; and the propagator. Theories based on tensors with indices in the integers\@xfootnote[0]Equivalently, such models can also be viewed as GFTs on the group . and trivial propagator are referred to as Tensor Models. A wealth of results about their expansions has now been accumulated [critical, dr_dually, v_revisiting, v_new, jr_branched], from single to multiple scalings in [wjd_double, stephane_double, bgr_double], and up to the rigorous non-perturbative level [razvan_beyond, delepouve_borel]. Similar models with non-trivial propagators are referred to as Tensorial Field Theories [tensor_4d, josephsamary, joseph_d2, joseph_etera]: they are indeed genuine field theories, for which full-fledge renormalization methods take over the large expansion. Finally, Tensorial Group Field Theories (TGFT) are Tensorial Field Theories based on (compact) Lie groups\@xfootnote[0]In view of the correspondence between simple GFTs and tensor models resulting from harmonic analysis, we have in mind a theory in which the group structure plays a significant enough role.. The only models of this type available in the literature so far are the so-called TGFTs with gauge invariance condition [cor_u1, cor_su2, samary_vignes, thesis]. This paper will focus on one such example, a rank- renormalizable model based on the gauge group . In addition to being technically relevant to spin foam models and LQG in general, it is expected to be related to Euclidean quantum gravity in three dimensions, though the full correspondence is unclear at present: while tensor invariance seems to provide a perfectly viable discretization prescription, the quantum gravity interpretation (if any) of the Laplace-type propagator has not been investigated in details.

The aim of this article is two-folds. The first objective is to pave the way towards general renormalization group techniques à la Wilson, which will allow to better understand the theory spaces of TGFTs, their flows and fixed points. Secondly, we want to determine the properties of the Gaussian fixed point of the specific model we are considering. Indeed, the examples worked out so far [josephaf, samary_beta] suggest that asymptotic freedom might be a reasonably generic property of Tensorial Field Theories. The occurrence of asymptotic freedom in such models is surprising at first, because they are not gauge theories. However, the non-locality of the interactions is responsible for wave-function renormalization terms which are absent from ordinary scalar field theories, and which typically dominate over the coupling constants renormalization terms. Hence the -functions can be negative despite the absence of non-Abelian gauge symmetries.

In section \@setrefsec:model, we introduce the model and the main results of [cor_su2] which are relevant to the present publication. In addition, we provide a detailed analysis of the symmetry factors appearing in the Feynman expansion, which significantly simplifies the calculations reported on in the later sections. In section \@setrefsec:rgflow we introduce a discrete version of Wilson’s renormalization group. It is based on the introduction of dimensionless coupling parameters, and differs in this sense from the analyses performed in previous works [josephaf, samary_beta, cor_su2]. A new notion of reducibility is also proposed, which is to some extent the correct generalization of -particle reducibility from ordinary local field theories to TGFTs. In section \@setrefsec:gaussian, we linearize the flow equations in the vicinity of the Gaussian fixed point. This allows to classify the coupling constants in terms of their relevance, and recover the fact that this model is renormalizable up to order six interactions. We then explicitly compute the eigendirections associated to this linear system, and deduce the functional relationship between the marginal coupling constants and the other renormalizable constants in the asymptotic UV region. In section \@setrefsec:as, the flow equations for the marginal coupling constants are pushed to second order in perturbation theory. Although the calculations underlying the results of this section are rather lengthy and technical, they are as far as we know the first of their kind in a non-Abelian model and are therefore included in full details. Finally, the qualitative properties of the flow equations in the vicinity of the Gaussian fixed point are investigated in section \@setrefsec:portrait. Relying on a continuous version of the discrete flow, we will be able to completely settle the question of asymptotic freedom when and have same signs, and in particular when they are strictly positive.

\@xsect

In this section, we introduce the TGFT model studied in [cor_su2], and summarize some of the key steps in the proof of its renormalizability. This paper was to a large extent based on a Bogolioubov recursion relation, which defined the renormalized series. The analysis was greatly simplified, thanks to a refined notion of graph connectedness, called face-connectedness. As already noted in [cor_su2], this structure is not appropriate in the effective Wilsonian language, where connected graphs in the ordinary sense\@xfootnote[0]This ordinary notion was called vertex-connectedness in [cor_su2]. must be summed over. Since the purpose of this paper is to investigate further the renormalization group flow equations of this model, we will only outline the definition of the effective series, in which the usual graph-theoretic notion of connectedness is at play.

\@xsect

We are interested in a group field theory for a field and its complex conjugate . The free theory is defined by a Gaussian measure , with covariance , that is to say:

 ∫dμC(φ,¯¯¯¯φ)φ(g1,g2,g3)¯¯¯¯φ(g′1,g′2,g′3)=C(g1,g2,g3;g′1,g′2,g′3). (\theequation)

This covariance can be perturbed by non-Gaussian interaction terms, encapsulated in an action , so as to define the (Euclidean) interacting partition function:

 Z≡∫dμC(φ,¯¯¯¯φ)exp(−S(φ,¯¯¯¯φ)). (\theequation)

In tensorial GFTs, is assumed to be a weighted sum of connected tensor invariants. By analogy with space-time based quantum field theories, this prescription is referred to as a locality principle, and is used as such for renormalization purposes. Connected tensor invariants in dimension are in one-to-one correspondence with -colored graphs (also called -bubbles), which are bipartite edge-colored closed graphs with fixed valency at each node. In our -dimensional context, a -colored graph is a connected graph with two types of nodes (black or white), and edges labeled by integers (the colors), in such a way that: a) any edge connects a white node to a black one; b) at any node, exactly three edges with distinct colors meet. Simple examples are provided in Figure \@setrefex_coloredgraphs.

The unique invariant associated to a given -bubble is constructed as follows: a) each white (resp. black) node represents a field (resp. a conjugate field ); b) a color- edge between the nodes and indicates a convolution with respect to the variables of the fields located at and respectively.

Example. The colored graph on the right side of Figure \@setrefex_coloredgraphs represents the following invariant integral:

 ∫[dg]6φ(g1,g2,g3)¯¯¯¯φ(g4,g2,g3)φ(g4,g5,g6)¯¯¯¯φ(g1,g5,g6). (\theequation)

With these definitions, the interaction part of the action can be written as

 S(φ,¯¯¯¯φ)=∑b∈Btbk(b)Ib(φ,¯¯¯¯φ), (\theequation)

where is the set of all bubbles, and is the coupling constant associated to . In order to simplify the counting of Feynman graphs, we divided each coupling constant by a combinatorial factor , defined as the number of automorphisms of the bubble graph .

###### Definition 1.

Let be a colored graph. An automorphism of is a permutation of its nodes\@xfootnote[0]This definition coincides with the more general concept of graph automorphism, even if a graph automorphism is usually thought of as a couple of permutations, respectively of the nodes and of the edges. When imposing compatibility with the colored structure, becomes redundant, hence our definition in terms of a single permutation., such that:

1. conserves the nature of the nodes;

2. if and are connected by an edge of color , then so do and .

At this stage, it is natural to assume that the colors have no physical role other than imposing combinatorial restrictions on the interactions, and we will therefore require to be invariant under color permutations. This can be formalized as follows. The group of permutations of the color set acts naturally on : for any , is the bubble obtained from by permutation of the color labels as dictated by . The invariance of under color permutation is simply the statement that:

 ∀b∈B,∀σ∈S3,tσ.b=tb. (\theequation)

The second ingredient of the model is the covariance . In [cor_u1, cor_su2], it was motivated from two basic requirements: first, it should impose the so-called gauge invariance condition of spin foam models; second, it should have a rich enough spectrum so as to provide an abstract notion of scale. The gauge invariance condition is an invariance of the field\@xfootnote[0]Note that there is no gauge symmetry involved at the field theory level, and that the nomenclature arises from the lattice gauge theory interpretation of the amplitudes. under an arbitrary simultaneous translation of its variables:

 ∀h∈SU(2),φ(g1h,g2h,g3h)=φ(g1,g2,g3). (\theequation)

In the GFT context we can however work with generic fields, and impose (\@setrefgauge_invariance) through the covariance. The latter should therefore contain the projector on the space of gauge invariant fields, defined by the kernel:

 P(g1,g2,g3;g′1,g′2,g′3)=∫dh3∏i=1δ(gihg′\tiny-1i). (\theequation)

In order to get a non-trivial spectrum, we combine it with the operator

 ˜C≡(m2−∑ℓΔℓ)−1, (\theequation)

where denotes the Laplace operator on acting on the color- variables. At this stage, this should be seen as a conservative natural choice, partially motivated by a study of the -point radiative corrections of the Boulatov-Ooguri models [Valentin_Joseph], and formal analogies with the Osterwalder-Schrader axioms [tt2, tt3]. The covariance is then defined as\@xfootnote[0]It is easily seen that and commute.

 C=˜CP=P˜C, (\theequation)

whose kernel can be written in the Schwinger representation:

 C(g1,g2,g3;g′1,g′2,g′3)=∫dαe−m2α∫dh3∏i=1Kα(gihg′\tiny-1i), (\theequation)

where is the heat kernel on at time . This covariance is ill-defined at coinciding points , and this can be regulated by cutting-off the contribution of the integral over in the neighborhood of . The divergences in this cut-off, generic in the perturbation expansion of the theory, can be analyzed in details.

The unnormalized -point functions can be expanded perturbatively in terms of Feynman amplitudes, which are indexed by generalized -colored graphs. The new type of lines, of color and represented as dashed lines, is introduced to represent the propagators; they connect the elementary bubble vertices generated by . These lines can also appear as external legs, contrary to the internal bubble lines, and this is the only respect in which the Feynman graphs differ from generic -colored graphs. Given a Feynman graph , we define as the number of external legs, and as the number of bubble vertices of type . Then

 ZSN=∑G|N(G)=N1k(G)(∏b∈B(−tb)nb(G))AG, (\theequation)

where is a symmetry factor associated to , and its amplitude. A simple counting shows that is nothing but the number of automorphisms of , which generalizes the definition of for bubbles (hence the notation).

###### Definition 2.

Let be a Feynman graph. An automorphism of is a permutation of its nodes, such that:

1. conserves the nature of the nodes;

2. if and are connected by an edge of color , then so do and ;

3. if a node (resp. ) is connected to an external leg , then (resp. ).

###### Proposition 1.

The symmetry factor associated to an arbitrary Feynman graph is the order of its group of automorphisms.

###### Proof.

Consider a Feynman graph , with labeled external legs, and bubbles of type for any . Define then the group

 G=∏b∈B|nb≠0Aut(b)nb×Snb, (\theequation)

where is the permutation group of elements, and is the group of automorphisms of . Its order is

 |G|=∏b∈Bk(b)nbnb! (\theequation)

One can fix an arbitrary labeling of all the nodes of all the vertices appearing in . Any Wick contraction with the same vertices appearing in the Feynman expansion is then represented by a set of pairs of labels. Call the set of all possible such pairings. acts naturally on , by permutation of identical bubbles and by automorphism on each individual bubble. The Feynman graph can be identified with the orbit of some , and the stabilizer to the group of automorphisms . Taking the factors coming from the perturbative expansion of the exponentials and from the definition of the action into account, we therefore obtain:

 1k(G) = (\theequation) = 1|Gx|=1|Aut(G)|.

It is immediate to remark that, because of the colored structure of the Feynman graphs, the symmetry factor of any connected and non-vacuum graph (i.e. with at least one external leg) is simply . This implies in particular that the connected and normalized Schwinger functions expand as:

 S(c)N=∑Gconnected|N(G)=N(∏b∈B(−tb)nb(G))AG. (\theequation)

Examples. The graphs , and represented in Figure \@setrefex_sym have different symmetry factors. is a connected and non-vacuum graph with external legs, hence . On the other hand, its amputated version admits one non-trivial automorphism, and therefore . Finally, the vacuum graph has a symmetry factor .

In order to write the explicit expression of , we need to introduce further graph-theoretic definitions and notations. We note the set of color- internal edges in , and the set of external legs, which we will sometimes simply call lines and legs. We can furthermore partition , where (resp. ) is the set of legs hooked to white (resp. black) nodes. A face of color is a non-empty subset which, upon addition of color- edges only, can be completed into a maximally connected subset of color- edges and (internal, color-) lines. Such a maximally connected subset may form a closed loop, we say that is internal (or closed) in this case, and is external (or open) otherwise. The set of internal faces and external faces of are respectively noted , and . We will also need to keep track of direct identifications of boundary variables through single colored lines. We call these empty external faces because they play a similar role as external faces; their set is denoted . When no confusion arises, we will use the same symbol to denote the cardinality of one of the sets defined so far and the set itself. We can fix an arbitrary orientation of the lines and faces , and encode their adjacency relations into a matrix or 0. However, in order to make the expressions more explicit, it is convenient to fix the orientations so that: a) is positively oriented from the white to the black end nodes it connects; b) if , and otherwise\@xfootnote[0]The fact that the faces can always be oriented in such a way is a particularity of tensorial GFTs.. This canonical orientation allows to define the source and target of an external face. The function maps to , where is the color of and is the external leg connected to its source. We define in a similar way. The bare amplitude of a graph is a function of external group elements , which formally writes:

 AG(gext(e,ℓ)) = ⎡⎣∏l∈L(G)∫dαle−m2αl∫dhl⎤⎦⎛⎝∏f∈F(G)Kα(f)⎛⎝−→∏l∈fhl⎞⎠⎞⎠ (\theequation) ∫[dg(e,l)]∏e∈N∙(G)C(gext(e,ℓ);g(e,ℓ))∏e∈N∘(G)C(g(e,ℓ);gext(e,ℓ)) ⎛⎝∏f∈Fext(G)Kα(f)⎛⎝gs(f)⎡⎣−→∏e∈fhe⎤⎦g\tiny-1% t(f)⎞⎠⎞⎠ ⎛⎜⎝∏f∈F∅ext(G)δ(gs(f)g\tiny-1t(f))⎞⎟⎠.

We clearly separated the contributions of the internal faces (first line), from the external propagators (second line\@xfootnote[0]We could simplify this expression by means of the symmetry , but we think that the present expression is better suited for what will come next.), and how these are connected to the holonomy variables in the bulk through the external faces (third and fourth line).

\@xsect

In [cor_su2], the scale ladder provided by the parameter was used as a basis for renormalization theory. Divergences from high scales (i.e. from the region ) generate counter-terms at lower scales (i.e. bigger parameters), and the theory is renormalizable if those can be reabsorbed into a finite number of coupling constants. The divergences can be most easily understood in the multiscale representation of the Feynman amplitudes. This consists in slicing the range of the Schwinger parameter , according to a geometric progression. To this effect, one fixes an arbitrary constant and define the propagator at scale by:

 Ci(g1,g2,g3;g′1,g′2,g′3)=∫M−2(i−1)M−2idαe−m2α∫dh3∏ℓ=1Kα(gℓhg′\tiny-1ℓ), (\theequation)

if and

 C0(g1,g2,g3;g′1,g′2,g′3)=∫+∞1dαe−m2α∫dh3∏ℓ=1Kα(gℓhg′\tiny-1ℓ). (\theequation)

This induces a decomposition of the amplitude of a graph as a sum indexed by internal scale attributions :

 AG=∑μAG,μ. (\theequation)

The amplitude at scale is simply obtained from (\@setrefampl_ab) by restricting the integrals to the slices . Note that the external legs are left untouched, and by convention they are attributed the scale label . The sum over is regulated thanks to the introduction of a cut-off on the scale labels , which can be removed only after renormalization. The main interest of the decomposition (\@setrefampl_multi) is that it allows to compute rigorous bounds on by a systematic optimization procedure, which yields simple bounds on as functions of .

Given a couple , one can construct a set of high subgraphs, defined as the maximally connected subgraphs with internal scales strictly smaller than the external scales. To this effect, let us define as the subgraph made of the lines of with scales higher or equal to . Its connected components\@xfootnote[0]As already mentioned before, in this paper we use the ordinary graph-theoretic notion of connectedness, which is referred to as vertex-connectedness in [cor_su2]. can be labeled , where is the number of connected components. The ’s are exactly the high subgraphs at scale : they are connected; their internal lines have scales higher or equal to ; and their external legs have scales strictly smaller than . An important property of the high subgraphs is that they form an inclusion tree. That is to say that two high subgraphs and are either line-disjoint, or one is included into the other; and furthermore all the high subgraphs are by definition included in , which is itself a high subgraph (at scale ). These high subgraphs are ultimately responsible for the nested structure of divergences, and when successively integrated out, make the coupling constants run with respect to . More precisely, only the divergent high subgraphs have to be taken into account in the renormalization group equations. They are determined by a precise power-counting theorem, and we refer the reader to [cor_u1, cor_su2] for details. For the purpose of this paper, we only need to know that the divergent high subgraphs are characterized by the inequality , where is the superficial degree of divergence, defined as [lin_GFT, cor_u1, cor_su2]:

 ω(H)=−2L(H)+3(F(H)−R(H)), (\theequation)

and is the rank of the incidence matrix of . Alternatively, as proven in [cor_su2], the divergence degree can be conveniently re-expressed in terms of the numbers of -valent vertices , the number of external legs and an integer :

 ω=3−N2+3ρ+∑k∈N∖{0}(k−3)n2k. (\theequation)

Note that in [cor_su2] the bare action was assumed to stop at -valent bubble interactions, in which case the last sum reduces to . The condition characterizes the class of melonic graphs\@xfootnote[0]In this paper, we always assume . When , may also be and does not imply that is melonic., which contains the divergent graphs as a subset when for (see Table \@setrefdiv).

Let us conclude this section by recalling how melonic graphs are defined.

###### Definition 3.

Let be a graph. For any integer such that , a -dipole is a line of linking two nodes and which are connected by exactly additional colored lines.

###### Definition 4.

Let be a graph. The contraction of a -dipole is an operation consisting in:

1. deleting the two nodes and linked by , together with the lines that connect them;

2. reconnecting the resulting pairs of open legs according to their colors.

We call the resulting graph.

See Figure \@setrefdipoles for examples of -dipoles and their contractions.

###### Definition 5.

We call contraction of a subgraph the successive contractions of all the lines of . The resulting graph is independent of the order in which the lines of are contracted, and is noted .

Given a graph , and some lines , we will denote by the minimal subgraph of containing the lines .

###### Definition 6.

A melopole is a single-vertex graph such that there is at least one ordering of its lines as such that is a -dipole for . See Figure \@setrefex_melop.

###### Definition 7.

A melonic graph is a connected\@xfootnote[0]This definition slightly differs from that of [cor_su2], in the sense that melonic graphs were defined to be face-connected there, which is a stronger condition. graph containing at least one maximal tree such that is a melopole.

A simple example of melonic graph is provided in Figure \@setrefex_melonic.

\@xsect

In the multiscale representation of the amplitudes, the divergences can be extracted by the action of so-called contraction operators. In order to define them, let us introduce generalized amplitudes

 AG(gext(e,ℓ);t) ≡ ⎡⎣∏l∈L(G)∫dαle−m2αl∫dhl⎤⎦⎛⎝∏f∈F(G)Kα(f)⎛⎝−→∏l∈fhl⎞⎠⎞⎠ (\theequation) ∫[dg(e,l)]∏e∈N∙(G)C(gext(e,ℓ);g(e,ℓ)(t))∏e∈N∘(G)C(g(e,ℓ);gext(e,ℓ)) ⎛⎝∏f∈Fext(G)Kα(f)⎛⎝gs(f)⎡⎣−→∏e∈fhe⎤⎦g\tiny-1% t(f)⎞⎠⎞⎠ ⎛⎜⎝∏f∈F∅ext(G)δ(gs(f)g\tiny-1t(f))⎞⎟⎠.

where and\@xfootnote[0]For any , denotes the Lie algebra element with the smallest norm such that .

 ∀f∈Fext(G),gs(f)(t) ≡ (\theequation) ∀f∈F∅ext(G),gs(f)(t) ≡ gs(f). (\theequation)

We can then express any amplitude as a Taylor expansion with respect to the parameter :

 AG,μ=AG,μ(⋅;1) = AG,μ(⋅;0)+ω(G)∑k=11k!A(k)G,μ(⋅;0) (\theequation) +∫10dt(1−t)ω(G)ω(G)!A(ω(G)+1)G,μ(⋅;t).

The interest of such an expression is that the remainder can be shown to be power-counting convergent [cor_su2], and can therefore be dispensed with. Moreover, one shows that

 AG,μ(gext(e,ℓ);0)∝AG/{l1,…,lk},μ(gext(e,ℓ);0), (\theequation)

where are the lines of . We can therefore implicitly define the contraction operator by the equation:

 AG,μ(gext(e,ℓ);0)=[τAG,μ]AG/{l1,…,lk},μ(gext(e,ℓ);0). (\theequation)

More explicitly, the heat-kernels associated to the external faces can be integrated out, yielding:

 [τAG,μ]=⎡⎣∏l∈L(G)∫dαle−m2αl∫dhl⎤⎦⎛⎝∏f∈F(G)Kα(f)⎛⎝−→∏l∈fhl⎞⎠⎞⎠. (\theequation)

Finally, if is obtain from by amputation of its external legs, we define:

 [τA¯¯¯G,μ]≡[τAG,μ]. (\theequation)

Let us now consider higher order terms. To begin with, the first order always identically vanishes:

 A(1)G,μ(⋅;0)=0. (\theequation)

The model studied in the present paper generates quadratically divergent graphs at most, which means we will not need to go beyond second order. Moreover, all the quadratically divergent graphs have external legs and are melonic. The amplitudes we need to expand to second order have therefore the following structure:

 AG,μ(gextℓ,¯¯¯gextℓ;t) = ⎡⎣∏l∈L(G)∫dαle−m2αl∫dhl⎤⎦ (\theequation) ⎛⎝∏f∈F(G)Kα(f)⎛⎝−→∏l∈fhl⎞⎠⎞⎠ ∫[dgℓ][d¯¯¯gℓ]C(gextℓ;gℓ(t))C(¯¯¯gℓ;¯¯¯gextℓ) ⎛⎝∏f∈Fext(G)Kα(f)⎛⎝gs(f)⎡⎣−→∏e∈fhe⎤⎦g\tiny-1% t(f)⎞⎠⎞⎠ ⎛⎜⎝∏f∈F∅ext(G)δ(gs(f)g\tiny-1t(f))⎞⎟⎠

where has three elements. Suppose for instance that , with of color . Then on can prove that [cor_su2]:

 (\theequation)

where we have defined

 τ(2)AG,μ ≡ 16⎡⎣∏l∈L(G)∫dαle−m2αl∫dhl⎤⎦⎛⎝∏f∈F(G)Kα(f)⎛⎝−→∏l∈fhl⎞⎠⎞⎠ (\theequation) ∫dg|Xg|2Kα(f1)⎛⎝g−−→∏e∈f1he⎞⎠.

By use of the Leibniz rule, together with the fact that terms containing first derivatives vanish, the last two equations can be generalized to

 12A(2)G,μ(gextℓ,¯¯¯gextℓ;0)=∫[dgextℓ]3C(gextℓ;gℓ)(∑ℓ[τ(2)ℓAG,μ]Δgℓ)C(gℓ;¯¯¯gextℓ), (\theequation)

where

 τ(2)ℓAG,μ = 16⎡⎣∏l∈L(G)∫dαle−m2αl∫dhl⎤⎦⎛⎝∏f∈F(G)Kα(f)⎛⎝−→∏l∈fhl⎞⎠⎞⎠ (\theequation) ∫dg|Xg|2Kα(fℓ)⎛⎝g−−→∏e∈fℓhe⎞⎠

if there exists a of color , and

 τ(2)ℓAG,μ=0 (\theequation)

otherwise. Again, this definition makes no reference to external legs, and can therefore be generalized to amputated graphs: if is obtained by amputation of the external legs of , then

 τ(2)ℓA¯¯¯G,μ≡τ(2)ℓAG,μ. (\theequation)
\@xsect

In this section, we define general flow equations for Wilson’s effective action. They involve: a) a step by step integration of UV slices which retains only the tensor invariant contributions of high divergent graphs, as already outlined above; and b) a large scale approximation () of the coefficients entering the equations. Because we are working with a field theory defined on a compact group rather than Euclidean space, finite size effects make the renormalization group non-autonomous\@xfootnote[0]I thank Daniele Oriti for discussions on this point. i.e. the flow equations have an explicit dependence in the cut-off. However, such effects can be neglected in the deep UV regime, where only the local structure of is probed.

\@xsect

In ordinary quantum field theories, Wilson’s effective action is best described in terms of dimensionless coupling constants. In GFT, and more generally in quantum gravity, such a notion is a priori empty since all the fields and coupling constants are already strictly speaking dimensionless. Dimensionful quantities should only appear in the spectra of quantum observables such as the length, area or volume operators. For example, in canonical loop quantum gravity, the area of a surface punctured by a single spin-network link with spin is where is the (dimensionless) Immirzi parameter and is the Planck length. In this respect, the spins themselves can be attributed a canonical dimension. In our context, let us attribute a unit canonical dimension

 [j]≡1 (\theequation)

to spin variables. Since the propagator decays quadratically in the spins, one immediately infers the dimension of the mass: . As for the canonical dimensions of all the other coupling constants, they can be deduced from the power-counting. For example, Table \@setrefdiv shows that four-valent coupling constants will receive linearly divergent contributions of order at scale . Hence they are to be thought of as coupling constants with unit canonical dimension. More generally, we are lead in this way to define

 [tb]=3−Nb2≡db, (\theequation)

for an arbitrary coupling constant , where is the valency of the bubble . Note that such a definition of canonical dimension has also recently been introduced in matrix and tensor models, for similar purposes [astrid_tim]. We define the effective action at scale as a sum over all possible bubbles

 Si(φ,¯¯¯¯φ) = ∑btb,iIb(φ,¯¯¯¯φ)k(b) (\theequation) = ∑bub,iMdbiIb(φ,¯¯¯¯φ)k(b), (\theequation)

where (resp. ) are the dimensionful (resp. dimensionless) coupling constants. We furthermore again assume color permutation invariance, and set up . The latter is consistent provided that we allow the mass parameter of the covariance to vary with the scale. We will denote by the covariance in the slice with mass , and by the full covariance with cut-off . The dimensionless mass coupling at scale is

 u2,i≡m2iM2i. (\theequation)

In the following, we will use perturbative expansions with respect to the dimensionless coupling constants. The degree of divergence

 ω=3−N2+∑k∈N(3−k)n2k+3ρ, (\theequation)

tells us how Feynman amplitudes diverge in an expansion with respect to the dimensionful parameters . Taking into account the additional scalings appearing in equation (\@setrefS_dl), we immediately infer a modified degree of divergence

 ¯¯¯ω=3−N2+3ρ (\theequation)

for the new expansion in the ’s. From now on, (a non-vacuum) graph will be said to be divergent whenever , that is whenever and . The new classification of divergent graphs is summarized in Table \@setrefdiv2.

\@xsect

In order to determine the effective action and the mass coupling from and , we proceed in two steps. The first step consists in integrating out fluctuations at scale to deduce the effective action before wave-function renormalization:

 Ki−1exp(−˜Si−1(Φ,¯¯¯¯Φ)+Ri−1(Φ,¯¯¯¯Φ))≡∫dμCi,mi(φ,¯¯¯¯φ)exp(−Si(Φ+φ,¯¯¯¯Φ+¯¯¯¯φ)), (\theequation)

where the rest term is a sum of contributions which are suppressed at large , and is a possibly large constant due to vacuum divergences. contains in particular the Feynman graphs with , and the convergent Taylor remainders associated to the non-tensorial parts of the non-vacuum divergent graphs. may be written as

 ˜Si−1=CTφ,i−1Sφ+CTm,i−1S2+∑b|Nb≠2~ub,i−1Mdb(i−1)Ibk(b), (\theequation)

in terms of intermediate dimensionless coupling constants . The additional wave-function and mass terms

 Sφ(φ,¯¯¯¯φ) = ∫[dg]3φ(g1,g2,g3)(−3∑l=1Δℓ)¯¯¯¯φ(g1,g2,g3), (\theequation) S2(φ,¯¯¯¯φ) = ∫[dg]3φ(g1,g2,g3)¯¯¯¯φ(g1,g2,g3), (\theequation)

respectively parameterized by a dimensionless constant and a dimension constant , are generated by the -valent divergent graphs. They need to be reabsorbed into the covariance via a field renormalization, which is the second step of the procedure. To this effect, let us define the operator

 Mi−1=−CTm,i−1+CTφ,i−1∑ℓΔℓ, (\theequation)

and call the covariance of the measure:

 dμCi−1mi(Φ,¯¯¯¯Φ)exp(∫[dgℓ][dg′ℓ]Φ(g1,g2,g3)Mi−1(gℓ;g′ℓ)¯¯¯¯Φ(g′1,g′2,g′3)). (\theequation)

It can be computed by summing over connected -point functions, in the following way:

 ˜C = Ci−1mi+Ci−1miMi−1Ci−1mi+Ci−1m