Beta functions of U(1 ) d gauge invariant just renormalizable tensor models
BBeta Functions of U (1) d Gauge Invariant Just Renormalizable Tensor Models
Dine Ousmane Samary ∗ International Chair in Mathematical Physics and ApplicationsICMPA-UNESCO Chair, 072 BP 50, Cotonou, B´enin (Dated: April 02, 2013)This manuscript reports the first order β -functions of recently proved just renormalizable randomtensor models endowed with a U (1) d gauge invariance [arXiv:1211. 2618]. The models that weconsider are polynomial Abelian ϕ and ϕ models. We show in this work that both models areasymptotically free in the UV. CONTENTS
I. Introduction 1II. Abelian TGFT with gauge invariance 2III. One-loop β -function of ϕ -model 4IV. Two-loop β -functions of the ϕ -model 6V. Acknowledgements 11Appendix I: Divergent series for ϕ -model 11Appendix II: Divergent series for ϕ -model 12References 15 I. INTRODUCTION
Many interesting physical systems can be represented mathematically as random matrix problems. In particular,matrix models, celebrated in the 80’s, provide a unique and well defined framework for addressing quantum gravity(QG) in two dimensions and its cortege of consequences on integrable systems [1]. The generalization of such modelsto higher dimensions is called random tensor models [2]. Recently, these tensor models have acknowledged a strongrevival thanks to the discovery by Gurau of the analogue of the t’Hooft 1 /N -expansion for the tensor situation [3]-[7]and of tensor renormalizable actions [8]-[11]. The tensor model framework begins to take a growing role in the problemof QG and raises as a true alternative to several known approaches [12–14].Tensorial group field theory (TGFT) [13, 14] is a recent proposal for the same problematic. It aims at providing acontent to a phase transition called geometrogenesis scenario by relating a discrete quantum pre-geometric phase of ourspacetime to the classical continuum limit consistent with Einstein gereral relativity. In short, within this approach,our spacetime and its geometry has to be reconstructed or must emerge from more fundamental and discrete degreesof freedom.Matrix models expand in graphs via ordinary perturbations of the Feynman path integral. These graphs can beseen as dual to triangulations of two dimensional surfaces. Here, the discrete degrees of freedom refer to matrices, ormore appropriately to their indices, or dually to triangles which glue to form a discrete version of a surface. In tensormodels, this idea generalizes. Feynman graphs in such tensor models are dual to triangulations of a D dimensionalobject. The tensor field possesses discrete indices and it is dually related to a basic D dimensional simplex whichshould be glued to others in order to form a discretization of a D dimensional manifold.As for any quantum field theory, the question of renormalizability of TGFT has been addressed and solved underspecific prescriptions [8]-[11]. Those conditions identify as the introduction of a Laplacian dynamics for the actionkinetic term [15] and the use of non local interaction of the tensor invariant form [16, 17]. Furthermore, as anotherimportant feature, the UV asymptotic freedom of some TGFTs has been proved in 3D [9] and 4D [18] (see also [19] for ∗ [email protected], [email protected] a r X i v : . [ h e p - t h ] J u l a shorter summary). This is strongly encouraging for the geometrogenesis scenario. Indeed, the asymptotic freedommeans that, after some scales towards the IR direction, the renormalized coupling constant of the theory starts to blowup and, certainly, this entails a phase transition towards new degrees of freedom. This is analogue of the asymptoticfreedom of non abelian Yang Mills theory leading to the better understanding of the quark confinement. However,the new degrees of freedom in TGFTs have been not yet investigated.New TGFT models, of the form of ϕ and ϕ theories, equipped with tensor fields obeying a gauge invariancecondition were recently shown just renormalizable at all orders of perturbation [11]. The gauge invariant conditionon tensor fields will help for the emergence of a well defined metric on the space after phase transition [10, 12]. Therenormalization of the model followed from a multi-scale analysis and a generalized locality principle leading to apower-counting theorem [20].In the present work, we calculate the first order β -function of both models and prove that these models areasymptotically free in the UV regime. This paper also emphasizes that this asymptotic freedom could be a genericfeature of all TGFTs for model with and without gauge invariance [14]. Such a feature will strengthen the status ofTGFTs as pertinent candidates for gravity emergent scenario.The paper is organized as follows. We recall in section 2 the main results concerning the renormalizability of ϕ and ϕ -tensor models as proved in [11]. Section 3 is devoted to the study of the one-loop β -function of the ϕ -model andsection 4 addresses the computation of the same quantity, this time at higher order loops, for the ϕ -model. Finally,an appendix gathers technical points useful for the proof of our statements. II. ABELIAN TGFT WITH GAUGE INVARIANCE
This section addresses a summary of the results obtained in [11]. We mainly present the model and its renormal-ization.TGFTs over a group G are defined by a complex field ϕ over d copies of group G , i.e. ϕ : G d −→ C ( g , · · · , g d ) (cid:55)−→ ϕ ( g , · · · , g d ) . (1)The gauge invariance condition [12] is achieved by imposing that the fields obey the relation ϕ ( hg , . . . , hg d ) = ϕ ( g , . . . , g d ) , ∀ h ∈ G . (2)For Abelian TGFTs, one fixes the group G = U (1). In the momentum representation, the field writes ϕ ( g , · · · , g d ) = (cid:88) p ϕ [ p ] e ip θ e ip θ · · · e ip d θ d , θ k ∈ [0 , π ) , where we denote ϕ [ p ] = ϕ ··· d := ϕ ( p , p , · · · , p d ), with p k ∈ Z and g k = e iθ k ∈ U (1).The generalized locality principle of the TGFTs considered in [11] requires to define the interactions as the sum oftensor invariants [3]. From now, we will focus on d = 6 , , and define two models described by S [ ¯ ϕ, ϕ ] = (cid:88) p , ··· ,p ¯ ϕ δ ( (cid:88) i p i )( p + m ) ϕ + λ (4)4 , V , , (3) S [ ¯ ϕ, ϕ ] = (cid:88) p , ··· ,p ¯ ϕ δ ( (cid:88) i p i )( p + m ) ϕ + λ (6)4 , V , + λ , V , + λ , V , + λ , V , , (4)where δ ( (cid:80) di p i ) should be understood as a Kronecker symbol δ (cid:80) di p i , and p = (cid:80) di p i , d = 6 , , respectively, andwhere the interactions are of the form given by V , = (cid:88) Z ¯ ϕ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) + permutations , (5) V , = (cid:88) Z ¯ ϕ ϕ (cid:48) (cid:48) (cid:48) (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) + permutations , (6) V , = (cid:0) (cid:88) Z ¯ ϕ ϕ (cid:1) , (7)
11 2356653244
FIG. 1. Vertex representation of ϕ -model A = V , B = V , C = V ,
11 23455432 D = V , FIG. 2. Vertex representation of ϕ -model V , = (cid:88) Z ¯ ϕ ϕ (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) (cid:48) (cid:48) (cid:48) ¯ ϕ ϕ + permuts , (8) V , = (cid:88) Z ¯ ϕ ϕ (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) ¯ ϕ ϕ (cid:48) (cid:48) (cid:48) + permutations. (9)The “permutations” are performed on the color indices. The vertices are graphically represented in fig. 1 and fig. 2.As one notices, there is two kinds of lines in the vertices. The first type are parametrized by 1 , , . . . , d and oneexternal half-line without any number. Call by 0 the color of this half-line.The propagator of each model reads: C ([ p ]) = 1 (cid:80) di =1 p i + m δ ( d (cid:88) i =1 p i ) , d = 6 , , (10)and it is represented graphically as a line with d strands, see fig. 3.A Feynman graph is a graph composed with lines of color 0 (propagators) and vertices. Hence, whenever we referto a line in the following it will be always a 0-color line and G is an uncolored tensor graph in the sense of [3] and [17]which have d -strand lines of color 0.Let L and F be the sets of internal lines and faces of the graph G . The multi-scale analysis shows that the divergencedegree of the amplitude of a graph associated with both models can be written ω d ( G ) = 2 L − F + R (11)where L = |L| , F = |F| and R is the rank of matrix (cid:0) (cid:15) lf , l ∈ L , f ∈ F (cid:1) , defined by (cid:15) lf ( G ) = l ∈ f and their orientation match, − l ∈ f and their orientation do not match,0 otherwise. (12) d FIG. 3. Propagator of d -dimensional tensor model The following statement holds [11]:
Theorem II.1
The models ϕ defined by S and ϕ defined by S are perturbatively renormalizable at all orders. The proof of this statement rests on a power counting theorem which can be summarized by the following table givingthe list of primitively divergent graphs (for precisions and notations, see [11]):
N ω ( G ) ω ( ∂ G ) C ∂ G − ω d ( G ) ϕ ϕ N the number of external fields, ω ( G ) the degree of the colored extension of the graph G , ω ( ∂ G ) the degree ofthe colored extension of the boundary ∂ G of the graph G , C ∂ G the number of connected component of the boundarygraph ∂ G .Using this table, we are now in position to compute renormalized coupling equations. III. ONE-LOOP β -FUNCTION OF ϕ -MODEL This section is devoted to the one-loop evaluation of the β -function of ϕ . To proceed, we enlarge the space ofcoupling constants so that (3) becomes S [ ¯ ϕ, ϕ ] (cid:88) p , ··· p ¯ ϕ δ ( (cid:88) i p i )( p + m ) ϕ + (cid:80) ρ =1 λ , ρ V , ρ . (14)Only at the end we will perform a merging of all coupling at the same value λ , ρ = λ , . Thus by introducing adistinction between the colors, ρ = 1 , , . . . ,
6, the combinatorics becomes less involved.We have the following theorem:
Theorem III.1
At one-loop, the renormalized coupling constant associated with λ is given by λ ren4 = λ + 19 π √ λ I + O ( λ ) , with I = (cid:90) ∞ e − αm α dα (15) such that the β -function of the model with single wave-function renormalization and single coupling constant is givenby β = − π √ . We now prove Theorem III.1. Let Z be the wave function renormalization which writes: Z = 1 − ∂ ∂b ρ Σ (cid:12)(cid:12)(cid:12) [ b ]=0 , ρ = 1 , , · · · , , (16)where Σ is called the self-energy or the sum of all amputated one-particle irreducible (1PI) two-point functions whichmust be evaluated at one-loop. The derivative on Σ is with respect to an external argument. The β -function of themodel ϕ is encoded by the following quotient λ ren4 = − Γ (0) Z (17)where Γ is the sum of all amputated 1PI four-point functions computated at one-loop and at low external momentathat we symbolize by a unique argument (0). Self-energy and wave function renormalization.
Having a look on (16) only is relevant the dependance in somecolor ρ of Σ. We will evaluate only this part in the self-energy at one-loop. For two sets of external arguments [ b ] and[ b (cid:48) ], one has Σ([ b ] , [ b (cid:48) ]) = < ¯ ϕ [ b ] ϕ [ b (cid:48) ] > t P I = (cid:88) G K G A G ([ b ] , [ b (cid:48) ]) (18)where K G is a combinatorial factor and A G is the amplitude of the graph G . Let S ( b ) = (cid:88) p , ··· ,p (cid:2)(cid:0) (cid:88) k =1 p k (cid:1) + (cid:0) (cid:88) k =1 p k (cid:1) + 2 b (cid:88) k =1 p k + 2 b + m (cid:3) − (19) S (cid:48) ( b ) = (cid:88) p , ··· ,p (cid:2)(cid:0) (cid:88) k =1 p k (cid:1) + (cid:0) (cid:88) k =1 p k (cid:1) + 2 b (cid:88) k =1 p k + 2 b + m (cid:3) − (20) K ( b ) = (cid:88) p , ··· ,p (cid:0) (cid:80) k =1 p k + 2 b (cid:1) (cid:2) (cid:80) k =1 p k + (cid:0) (cid:80) k =1 p k (cid:1) + 2 b (cid:80) k =1 p k + 2 b + m (cid:3) . (21)At one-loop, there exist six tadpole graphs T ρ , ρ = 1 , · · · , , that contribute to the relation (18). For instance T isrepresented in fig. 4. The amplitude associated to the tadpole T ρ is given by FIG. 4. Tadpole graphs T A T ρ = − λ ,ρ S ( b ρ ) . (22)The combinatorial weight of these graphs T ρ is K T ρ = 2. Then (18) is re-expressed asΣ([ b ]) = − (cid:88) ρ =1 λ ,ρ S ( b ρ ) . (23)We have the following relation (see Appendix I for details): S (cid:48) (0) = π √ I , K (0) = π √ I , I = (cid:90) ∞ dα e − αm α , (24)then ∂ Σ[ b ] ∂b ρ (cid:12)(cid:12)(cid:12) [ b ]=0 = 4 λ ,ρ (cid:16) S (cid:48) ( b ρ ) − K ( b ρ ) (cid:17)(cid:12)(cid:12)(cid:12) [ b ]=0 = 12 π √ λ ,ρ I . (25)Using the fact that the tadpole amplitudes are symmetric with respect to the external variables, we reduce all couplingconstants to the same value i.e. λ ,ρ = λ , and get the wave function renormalization as Z = 1 − π √ λ I + O ( λ ) . (26) Four-point functions.
The 1PI four-point function amplitudes Γ ,ρ , ρ = 1 , , · · · , , are given byΓ ,ρ ([ b ] , [ b (cid:48) ]) = < ¯ ϕ [ b ] ϕ [ b ] ¯ ϕ [ b (cid:48) ] ϕ [ b (cid:48) ] > t P I = (cid:88) G K G A G ([ b ] , [ b (cid:48) ]) , (27)where [ b ] j , [ b (cid:48) ] j , j = 1 , , are the external strand indices. Using the cyclic permutation over the six indices ρ , thefour-point functions are explicitly given byΓ , ( b , · · · , b , b (cid:48) · · · , b (cid:48) ) = < ¯ ϕ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) > t P I (28)Γ , ( b , · · · , b , b (cid:48) · · · , b (cid:48) ) = < ¯ ϕ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) > t P I (29)Γ , ( b , · · · , b , b (cid:48) · · · , b (cid:48) ) = < ¯ ϕ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) > t P I (30)Γ , ( b , · · · , b , b (cid:48) · · · , b (cid:48) ) = < ¯ ϕ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) > t P I (31)Γ , ( b , · · · , b , b (cid:48) · · · , b (cid:48) ) = < ¯ ϕ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) > t P I (32)Γ , ( b , · · · , b , b (cid:48) · · · , b (cid:48) ) = < ¯ ϕ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ¯ ϕ (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) ϕ (cid:48) > t P I (33)At one-loop, there is a unique graph contributing to Γ ρ . It is of the form given by fig. 5. The combinatorial factor FIG. 5. The melonic one-loop four-point graph of this graph is always K G = 2 · ·
2. The amplitude associated of this graph is A G ,(cid:15) ( b ) = λ ,(cid:15) . S (cid:48) ( b ) . (34)We obtain Γ (0) = − λ + λ S (cid:48) (0) + O ( λ ) = − λ + π √ λ I + O ( λ ) . (35)The renormalizable coupling constant is finally given by λ ren4 = − Γ (0 , Z = λ + 19 π √ λ I + O ( λ ) . (36)This result shows that the ϕ model is asymptotically free in the UV regime. The β -function at one-loop of the modelreads from (36): β = − π √ . (37) IV. TWO-LOOP β -FUNCTIONS OF THE ϕ -MODEL V , V , FIG. 6. New graphical representation of vertices V , and V , In the ϕ -model, there are two types of coupling constants and so we must evaluate two renormalized couplingequations. In order to compute the β -functions of the ϕ model it is important to note that the vertices of the type V , are parametrized by five indices ρ = 1 , , · · · ,
5, and the vertices contributing to V , are parametrized by tenindices ρρ (cid:48) = 1 . , . , . , . , . , . , . , . , . , .
5. The couple ρρ (cid:48) will be totally symmetric i.e., ρρ (cid:48) = ρ (cid:48) ρ . Forsimplicity, the graphs of fig. 6 represent henceforth the vertices of ϕ model. For the same combinatorial reasonsevoked above, we enlarge again the space of coupling and write (4) as S [ ¯ ϕ, ϕ ] = (cid:88) p , ··· ,p ¯ ϕ δ ( (cid:88) i p i )( p + m ) ϕ + 13 (cid:88) ρ λ , ρ V , ρ + (cid:88) ρρ (cid:48) λ , ρ V , ρρ (cid:48) + 12 (cid:88) ρ λ , ρ V , ρ + 12 (cid:88) ρ λ , V , . (38)We have the following theorem: Theorem IV.1
The renormalized coupling constants λ ren6 , and λ ren6 , satisfy the equations λ ren6 , = λ , + 9 π λ , I (cid:48) + 12 (cid:0) √
31 + 58 (cid:1) π λ , λ , I (cid:48) + O ( λ p , λ − p , ) , (39) and λ ren6 , = λ , + 4 (cid:16) √
31 + 118 (cid:17) π λ , I (cid:48) + 11 π λ , λ , I (cid:48) + O ( λ p , λ − p , ) , (40) p = 0 , , , . Self-energy and wave function renormalization.
The following proposition holds:
Proposition IV.1
The wave function renormalization of the model is given by Z = 1 − π λ , I (cid:48) − (cid:0) √
31 + 58 (cid:1) π λ , I (cid:48) + O ( λ p , λ − p , ) , (41) p = 0 , , , and where I (cid:48) writes I (cid:48) = (cid:90) ∞ (cid:90) ∞ d α e − αm α . (42) proof: Let us consider the following series S ( b ) = (cid:88) p ,p ,p q ,q ,q (cid:110)(cid:2) (cid:88) k =1 p k + ( (cid:88) k =1 p k ) + 2 b (cid:88) k =1 p k + 2 b + m (cid:3) − × (cid:2) (cid:88) k =1 q k + ( (cid:88) k =1 q k ) + 2 b (cid:88) k =1 q k + 2 b + m (cid:3) − (cid:111) , (43) S ( b ) = (cid:88) p ,p ,p ,p q ,q (cid:110)(cid:0) (cid:88) k =1 p k + ( (cid:88) k =1 p k ) + m (cid:1) − (cid:0) (cid:88) k =1 q k + ( (cid:88) k =1 q k ) + 2 b (cid:88) k =1 q k + 2( (cid:88) k =1 p k ) − b (cid:88) k =1 p k − (cid:88) k =1 p k (cid:88) k =1 q k + 2 b + m (cid:1) − (cid:111) , (44)and S ( b, b (cid:48) ) = (cid:88) p ,p ,p q ,q ,q (cid:110)(cid:2) (cid:88) k =1 p k + ( (cid:88) k =1 p k ) + 2 b (cid:88) k =1 p k + 2 b + m (cid:3) − × (cid:2) (cid:88) k =1 q k + ( (cid:88) k =1 q k ) + 2 b (cid:48) (cid:88) k =1 q k + 2 b (cid:48) + m (cid:3) − (cid:111) . (45)The graphs contributing to the self-energy are of the form listed in fig. 7. The amplitude corresponding to the T ρ T ρρ (cid:48) T ρρ (cid:48) FIG. 7. Divergent tadpoles graphs of ϕ -model tadpoles graphs T ρ is given by the following relation A T ρ ( b ρ ) = − λ , ρ S ( b ρ ) . (46)In the above expression b ρ is an external strand index. Using the combinatorial number associated to the tadpolegraph T ρ given by K T ρ = 3, the sum of 1PI two-point functions are given byΩ , ( b ρ ) = 3 A T ρ ( b ρ ) . (47)Similarly, the amplitude corresponding to the tadpole graphs T ρ and T ρ are respectively given by relations A T ρ ( b ) = − λ , ρ S ( b ) , (48)and A T ρ (cid:48) ( b , b ρ ) = − λ , ρ S ( b , b ρ ) . (49)The combinatorial factors are K T ρ = 1 and K T ρ = 1. Therefore the sum of these contribution yieldsΩ , ( b , b ρ ) = A T ρ ( b ) + A T , ρ ( b , b ρ ) . (50)Combining the relations (47) and (50), we get a part of the self-energy involving the variable b Σ ( b , b ρ ) = 3 A T ( b ) + (cid:88) ρ (cid:104) A T ρ ( b ) + A T ρ ( b , b ρ ) (cid:105) + O ( λ p , λ − p , ) . (51)The wave function renormalization of the model is given by Z = 1 − ∂ ∂b Σ ( b , b ρ ) (cid:12)(cid:12) b = b ρ =0 . (52)Using appendix V, we have the following relations: ∂ ∂b Ω , ( b ) (cid:12)(cid:12) b =0 = 5 π λ , I (cid:48) , I (cid:48) = (cid:90) ∞ (cid:90) ∞ d α e − αm α (53) ∂ ∂b Ω , ( b , b ρ ) (cid:12)(cid:12) b = b ρ =0 = (cid:0) √
31 + 58 (cid:1) π λ , , ρ I (cid:48) . (54)We restrict from now the coupling constants in each sector such that λ , ρ = λ , and λ , ρρ (cid:48) = λ , so that, thewave function renormalization is Z = 1 − π λ , I (cid:48) − (cid:0) √
31 + 58 (cid:1) π λ , I (cid:48) + O ( λ p , λ − p , ) , (55) p = 0 , , (cid:3) Six-point functions.
The initial calculation of the six-point functions shows that they prolifer quickly [18]. However,in the present gauge invariant model which is more constrained, several of these should be not renormalized becauseeither are convergent (pay attention to the fact that gauge invariant models are less divergent than the ordinary one)or turn out to violate the face-connectedness condition (see discussion below and fig. 9) [10, 11].At the end, we will focus on the six-point functions which are face-connected graphs of type V , − V , and V , − V , , see fig. 8. This will be used for the calculation of the sum of 1PI six-point functions Γ , ρ and Γ , ρρ (cid:48) .The renormalized coupling constant equations for λ ren6 , and λ ren6 , ρ (cid:48) are defined by λ ren6 , ρ = − Γ , ρ (0 , Z , λ ren6 , ρρ (cid:48) = − Γ , ρρ (cid:48) (0 , Z . (56) G ρ , G ρρ (cid:48) G (cid:48) ρρ (cid:48) , G ρρ (cid:48) FIG. 8. Face-connected divergent six-point graphs of ϕ -model Proof of Theorem IV.1
The first part of this proof is about the evaluation of amplitudes of various graphs of fig. 8. We introduce someformal sums: S = (cid:88) p ,p ,p q ,q ,q (cid:0) (cid:88) k =1 p k + ( (cid:88) k =1 p k ) + m (cid:1) − (cid:0) (cid:88) k =1 q k + ( (cid:88) k =1 q k ) + m (cid:1) − S = (cid:88) p ,p ,p q ,q ,q (cid:0) (cid:88) k =1 p k + ( (cid:88) k =1 p k ) + m (cid:1) − (cid:0) (cid:88) k =1 q k + ( (cid:88) k =1 ( p k − q k )) + ( (cid:88) k =1 p k ) + m (cid:1) − = (cid:88) p ,p q ,q ,q ,q (cid:0) (cid:88) k =1 p k + ( (cid:88) k =1 q k − (cid:88) k =1 p k ) + ( (cid:88) k =1 q k ) + m (cid:1) − (cid:0) (cid:88) k =1 q k + ( (cid:88) k =1 q k ) + m (cid:1) − . (57)A calculation yields, at low external momenta, A G ρ (0 , . . . ,
0) = λ , ρ . K G ρ S = 3 · .λ , ρ S , (58) A G ρρ (cid:48) (0 , . . . ,
0) = 13 λ , ρ (cid:88) ρ (cid:48) λ , ρρ (cid:48) K G ρρ (cid:48) S = 3 λ , ρ [ (cid:88) ρ (cid:48) (cid:54) = ρ λ , ρρ (cid:48) ] S , (59) A G (cid:48) ρρ (cid:48) (0 , . . . ,
0) = 13 ( λ , ρ + λ , ρ (cid:48) ) λ , ρρ (cid:48) K G (cid:48) ρρ (cid:48) S = 2( λ , ρ + λ , ρ (cid:48) ) λ , ρρ (cid:48) S , (60) A G ρρ (cid:48) (0 , . . . ,
0) = λ , ρρ (cid:48) (cid:2) (cid:88) ˜ ρ (cid:54) = ρ λ , ρ ˜ ρ + (cid:88) ˜ ρ (cid:54) = ρ (cid:48) λ , ρ (cid:48) ˜ ρ (cid:3) ( S + S ) , (61) K G ρ = 3 · , K G ρρ (cid:48) = 3 · , K G (cid:48) ρρ (cid:48) = 3 · . (62)The contributions to Γ , ρ are obtained from G ρ and G ρρ (cid:48) . Using these, we getΓ , ρ (0 , . . . ,
0) = − λ , ρ + λ , ρ (cid:104) λ , ρ S + 3[ (cid:88) ρ (cid:48) (cid:54) = ρ λ , ρρ (cid:48) ] S (cid:105) + O ( λ p , λ − p , ) . (63)The contributions to Γ , ρρ (cid:48) are obtained from G (cid:48) ρρ (cid:48) and G ρρ (cid:48) . One findsΓ , ρρ (cid:48) (0 , . . . ,
0) = − λ , ρρ (cid:48) + 2( λ , ρ + λ , ρ (cid:48) ) λ , ρρ (cid:48) S + λ , ρρ (cid:48) (cid:2) (cid:88) ˜ ρ (cid:54) = ρ λ , ρ ˜ ρ + (cid:88) ˜ ρ (cid:54) = ρ (cid:48) λ , ρ (cid:48) ˜ ρ (cid:3) ( S + S ) + O ( λ p , λ − p , ) (64)Reducing to the smaller space of couplings λ , ρ = λ , and λ , ρρ (cid:48) = λ , , we getΓ , (0 , . . . ,
0) = − λ , + 6 λ , S + 12 λ , λ , S + O ( λ p , λ − p , ) , , (0 , . . . ,
0) = − λ , + 8 λ ( S + S ) + 4 λ , λ , S + O ( λ p , λ − p , ) . (65)Asymptotically, we can obtain the relation S = π I (cid:48) , S = π √ I (cid:48) (66)(see Appendix II for more detail). At one-loop the renormalized coupling constant λ ren6 , and λ ren6 , are given by λ ren6 , = λ , + 9 π λ , I (cid:48) + 12 (cid:0) √
31 + 58 (cid:1) π λ , λ , I (cid:48) + O ( λ p , λ − p , ) , (67)and λ ren6 , = λ , + 4 (cid:16) √
31 + 118 (cid:17) π λ , I (cid:48) + 11 π λ , λ , I (cid:48) + O ( λ p , λ − p , ) . (68) (cid:3) Discussion: • Let us come back on the subtle issue about the notion of connectedness in this theory. The correct notionof connectedness should be the one of face-connectedness. Several graphs which a priori are divergent should notrenormalize any coupling constant. For instance, graphs of the form given in fig. 9 are face-disconnected divergentsix-point graphs. They do not contribute to the 1PI six-point functions. The amplitudes of the graphs are A G (cid:48)(cid:48) ρρ (cid:48) (0 , . . . ,
0) = 13 λ , ρ (cid:88) ρ (cid:48) λ , ρρ (cid:48) K G (cid:48)(cid:48) ρρ (cid:48) S = 3 λ , ρ [ (cid:88) ρ (cid:48) (cid:54) = ρ λ , ρρ (cid:48) ] S (69) K G (cid:48)(cid:48) ρρ (cid:48) = 3 · G (cid:48)(cid:48) ρρ (cid:48) FIG. 9. Face-disconnected and divergent six-point graphs of ϕ -model • We now discuss the results of Theorem IV.1. Equation (67) can be re-expressed as λ ren6 , = λ , − β λ , I (cid:48) − β λ , λ , I (cid:48) + O ( λ p , λ − p , ) , (71)where, at this order of perturbation, the β -function splits into coefficients β and β given by β = − π , β = − (cid:0) √
31 + 58 (cid:1) π . (72)This clearly shows that λ ren6 , ≥ λ , proving that this sector is asymptotically free, provided all coupling are positive.In the same way, equation (68) can be re-expressed as λ ren6 , = λ , − β λ , I (cid:48) − β λ , λ , I (cid:48) + O ( λ p , λ − p , ) (73)where the β -functions β and β are given by β = − (cid:16) √
31 + 118 (cid:17) π , β = − π . (74)The same conclusion holds for the sector λ , which is asymptotically free. Both relations (72) and (74) show thatthe model with both interactions is asymptotically free in the U V regime. Hence, gauge invariant TGFT modelsof the form present here make a sense at arbitrary small scales yielding, far in the UV, a theory of non interactingspheres. Indeed, according to [3], all interactions presented here (called melonic) are nothing simplicial complexeswith the sphere topology. The present results also show that both models might experience a phase transition whenthe renormalized coupling constants become larger and larger in the IR. This feature deserves full investigation.1 • We will now discuss a renormalized coupling constants λ ren4 , and λ ren4 , . We have already shown that, at high scale,the bare values of coupling constants λ , and λ , vanish. Further the divergent four-point functions must not havemore than one vertex type V , , or V , , the only divergent graphs are those couples with V , or V , . Using relations(67) and (68) we come to the conclusion that λ ren4 , = λ , + O ( λ p , λ − p ,k ) , k = 1 or k = 2 , (75) λ ren4 , = λ , + O ( λ p , λ − p ,k ) , k = 1 or k = 2 . (76)Then the ϕ sector is safe at all loops and the β -functions are given by β , = β , = 0 . (77) V. ACKNOWLEDGEMENTS
The author is grateful to Vincent Rivasseau, Joseph Ben Geloun and Fabien Vignes-Tourneret for useful commentsthat allowed to improve the paper. This work is partially supported by the Abdus Salam International Centre forTheoretical Physics (ICTP, Trieste, Italy) through the Office of External Activities (OEA) - Prj-15. The ICMPA isalso in partnership with the Daniel Iagolnitzer Foundation (DIF), France.
APPENDIX I: DIVERGENT SERIES FOR ϕ -MODEL Proposition V.1
Let I = (cid:82) ∞ dα e − αm α be a logarithmically divergent quantity in the UV regime. The series S (cid:48) (0) and K (0) asymptotically write as S (cid:48) (0) = π √ I , K (0) = π √ I . (78)The rest of this section is devoted to the proof of this proposition. Let us recall the Schwinger formula: Let A be apositive define operator and n is an integer then we get1 A n +1 = 1 n ! (cid:90) ∞ dα α n e − αA . (79)For A = (cid:80) k =1 p k + (cid:0) (cid:80) k =1 p k (cid:1) + m , we arrive at expression (cid:88) [ p ] ∈ Z A = lim Λ → lim Λ (cid:48) → (cid:88) [ p ] (cid:90) ∞ Λ (cid:48) dα α n e − αA = lim Λ (cid:48) → (cid:90) ∞ Λ (cid:48) dα α n lim Λ → (cid:88) [ p ] e − αA = (cid:90) ∞ dα α e − αm (cid:88) [ p ] ∈ Z e − α [ | p | + (cid:80) i =1 , i Lemma V.1 Let −∞ < p < ∞ . For n → ∞ , uniformaly in any finite interval of positive β , we get ∞ (cid:88) p = −∞ e − βn p = (cid:114) nπβ . (81) Proof V.1 The proof of this lemma is given in [21]. Noting that in the previous lemma βn → α = M − i → 0. Then (cid:80) ∞ p = −∞ e − αp = (cid:112) πα . Then (cid:88) [ p ] ∈ Z e − α [ | p | + (cid:80) i =1 , i Let −∞ < p < ∞ . For α → uniformly in any finite interval of constant c , we get ∞ (cid:88) p = −∞ p e − αp +2 cp = cα (cid:114) πα e α c , ∞ (cid:88) p = −∞ p n e − αp +2 cp = 12 n − α (cid:114) πα ddc (cid:16) ce α c (cid:17) . (88)Using this lemma we get easily X = 12 (cid:90) ∞ dα α e − αm (cid:16) − π α √ (cid:17) = − π √ I . (89)Therefore K (0) = π √ I . APPENDIX II: DIVERGENT SERIES FOR ϕ -MODEL In this section, we will focus on the divergent terms of the ϕ -model. Let us consider the functions Ω , ( b ) andΩ , ( b, b (cid:48) ). The second order partial derivative respect to external strand b participated to the expression of the wavefunction. The goal of this part is the proof of the following proposition Proposition V.2 Let I (cid:48) = (cid:82) ∞ (cid:82) ∞ d α e − αm α be a logarithmically divergent quantity in the UV regime. The partialderivative of Ω , ( b ) and Ω , ( b, b (cid:48) ) are respectively given by ∂ ∂b Ω , ( b ) (cid:12)(cid:12) b =0 = 5 π λ , I (cid:48) , (90) ∂ ∂b Ω , ( b, b (cid:48) ) (cid:12)(cid:12) b = b (cid:48) =0 = (cid:2) √