Towards phase transitions between discrete and continuum quantum spacetime from the Renormalization Group
TTowards phase transitions between discrete and continuum quantum spacetime fromthe Renormalization Group
Astrid Eichhorn and Tim Koslowski Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, Ontario, N2L 2Y5, CanadaE-mail: [email protected] Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, CanadaE-mail: [email protected]
We establish the functional Renormalization Group as an exploratory tool to investigate a pos-sible phase transition between a pre-geometric discrete phase and a geometric continuum phase inquantum gravity. In this paper, based on the analysis of [1], we study three new aspects of thedouble-scaling limit of matrix models as Renormalization Group fixed points: Firstly, we investi-gate multicritical fixed points, which are associated with quantum gravity coupled to conformalmatter. Secondly, we discuss an approximation that reduces the scheme dependence of our resultsas well as computational effort while giving good numerical results. This is a consequence of theapproximation being a solution to the unitary Ward-identity associated to the U(N) symmetry ofthe hermitian matrix model. Thirdly, we discuss a scenario that relates the double scaling limit tofixed points of continuum quantum gravity.
I. INTRODUCTION
The functional Renormalization Group (FRG) is anexploratory tool in the investigation of quantum gravitymodels. In [1] we showed how the FRG can be used torecover the double scaling limit of matrix models for 2DEuclidean quantum gravity as a Renormalization Group(RG) fixed point. The purpose of this paper is to furtherdevelop the use of the FRG as a tool in the investiga-tion of matrix models of 2D Euclidean quantum grav-ity. These models are the prototype of discrete models ofquantum gravity such as Euclidean and Causal Dynam-ical Triangulations [2, 3, 51], tensor models [5, 6], andgroup field theories [7–10]. The ultimate goal of this re-search is to provide a qualitatively well-understood andquantitatively precise analytical tool to study the contin-uum limit in four-dimensional models of discrete quan-tum spacetime.The particular results of this paper are the following:1. Besides the double scaling limit there exists a se-ries of multicritical points, which correspond to thecontinuum limit of pure quantum gravity coupledto conformal matter. We discover these multicrit-ical points in sec. III as fixed points of the FRGwith good matching of the critical exponents anddimensionless ratios of coupling constants.2. It turns out that the tadpole approximation to thebeta functions does not only simplify computations,but also leads to scheme independent results. Fur-ther, this approximation solves the tadpole approx-imation to the unitary Ward-identity. This leads toan improvement of the numerical results for criti-cal exponents compared to [1], where we found a50 % discrepancy of the critical exponent for thepure gravity model compared to the known ana-lytic value. This quantitative improvement arises,as the tadpole approximation is a good approxima- tion for fixed points with small critical values ofthe couplings. Besides, it provides a self-consistentimplementation of unitary symmetry.3. The continuum limit of matrix models is well un-derstood in terms of a lattice interpretation ofFeynman graphs of a matrix model, as a limit inwhich the lattice constant approaches zero. Thisstraightforward geometric picture is not availablein the FRG description. Nevertheless, there is alink between the FRG approach to matrix modelsand continuum quantum gravity (see sec. V).Let us now revisit the foundations of the approach of [1]before introducing their implementation in sec. II.
A. Renormalization Group and double-scaling limit
The idea behind the matrix- (resp. tensor-) model ap-proach to quantum gravity is to express the quantumgravity path integral as a sum over discrete triangula-tions or more generally tesselations. This discrete sumcan be translated into matrix or tensor models [11]. Intwo-dimensional quantum gravity, based on the Einstein-Hilbert action and a summation over topologies, thecorresponding matrix model is of the form Z = (cid:90) D ϕ e N ( − Tr ϕ + g Tr ϕ ) , (1)where ϕ is an N × N hermitian matrix, for reviews see,e.g., [12–16]. The continuum limit is obtained when Note that the restriction to the Einstein-Hilbert action is some-what arbitrary at the microscopic level, in particular from theFRG perspective. In fact, it could well be that the existence ofa continuum description of quantum spacetime requires higher-order operators. a r X i v : . [ g r- q c ] A ug N → ∞ (i.e., infinitely many degrees of freedom con-tribute), and g → g c . If these two limits are takenseparately, only spherical topologies contribute to thepartition function. To include all topologies, one shouldobserve that the partition function admits an expansionin Feynman graph topologies Z = (cid:88) h Z h N − h ) , (2)where h is the number of handles. It is possible to keepcontributions of all topologies, if the N suppression ofhigher topologies is compensated. This is possible, since Z h ≈ ( g c − g ) (1 − h )(2 − γ st ) as g → g c . Thus, if onetakes the double scaling limit [17–19]( g − g c ) − γ st2 N = C, (3)where C is a constant, while N → ∞ and g → g c , thenall Z h contribute to the large N -limit of Z . To under-stand the connection to the Renormalization Group, werealize that the double-scaling limit dictates a particularscaling of g with N . We can write Eq. (3) as g ( N ) = g c + (cid:18) NC (cid:19) − − γ st . (4)This is the structure of a solution to the linearized RGflow in the vicinity of a fixed point . Thus the doublescaling limit corresponds to an interacting fixed point ofthe Renormalization Group of the matrix model, in which N plays the role of an RG scale. The critical exponent θ = is related to the exponent γ st in the usual notionfor these models by θ = − γ st , i.e., γ st = − .Note that in the two-dimensional matrix-model casethe sum (2) is not summable, so the partition function Z exists only as a formal sum. In particular, while g c isthe radius of convergence for the perturbative expansionof the partition function at fixed topology, g c plays nosuch role in the sum over all topologies. Accessing thedouble scaling limit as an RG fixed point does not pro-vide a novel way to perform this sum. Instead, the FRGframework provides a way to determine whether thereexists a consistent scaling that relates g and N , suchthat a contribution of all topologies to the formal sumcan be retained in the large N -limit. The FRG allows usto derive a consistent scaling for the double-scaling limit.Whether or not such a scaling exists, and what the valueof the critical exponent is, is independent of the questionwhether the partition function converges. In particular, Given a beta function β g = µ∂ µ g ( µ ), we can linearize it arounda fixed point at g ∗ : ∂β g ∂g (cid:12)(cid:12)(cid:12) g = g ∗ ( g − g ∗ ) = 0. This is solved by g ( µ ) = g ∗ + c (cid:16) µµ (cid:17) − θ , where θ = − ∂β g ∂g (cid:12)(cid:12)(cid:12) g = g ∗ , and c is a constantof integration and µ a reference scale. the RG will give meaningful results for the scaling expo-nent of the double-scaling limit even in cases where theexpression for the partition function does not converge .In higher dimensions, there are indications that the par-tition function is summable in the double-scaling limit[20–22] and contributions from higher orders in the 1 /N expansion [23] can be retained consistently. The FRGallows us to access tensor models corresponding to d = 4dimensions, where other methods that work successfullyin the matrix-model case, break down. The FRG willthus provide a method to derive the scaling exponent(s)of the double-scaling limit. Further, the FRG can alsobe applied to models with a matrix Laplacian, which areasymptotically free [24]. In these models, the FRG couldbe of considerable use to study the strongly-interacting“infrared” limit, where a phase transition could lead to a“condensed” phase of discrete building blocks, see, e.g.,[25].In this paper, we will further establish the FunctionalRenormalization Group as a useful tool to study matrixmodels, providing a starting point for research on d = 4dimensional tensor models. B. Renormalization Group scale in matrix models
Let us now expand on the use of N as an RG scale, asfirst proposed in [26], see also [27–29]. Applying Renor-malization Group tools in quantum gravity could seemfutile: The RG sorts quantum fluctuations according toa scale, separating large-scale from small-scale fluctua-tions, and integrating them out according to this orga-nizing principle. In quantum gravity, where all possi-ble geometries are included in the path-integral, no fixednotion of scale exists. Every configuration comes withits own notion of scale. It thus seems that the use ofRG techniques in quantum gravity requires to break thediffeomorphism invariance (the symmetry that ensuresbackground independence) of gravity by singling out onefield configuration to organize all other quantum fluc-tuations into large-scale or small-scale ones w.r.t. thispreferred field configuration. Hence, it seems that back-ground independence, a crucial requirement of quantumgravity, is incompatible with RG techniques.Different answers have been found to this apparentproblem: In continuum formulations, the backgroundfield method can be used in combination with diffeo-morphism Ward-identities to introduce a backgroundand fluctuation field [30–32], in such a way that back-ground independence is ensured. In these approaches, thetopology and dimensionality of quantum configurationsare fixed, and this background structure is sufficient tothen set up a useful background-field approach. WithinCausal Dynamical Triangulations, proposals for the RGflow were recently advanced [33, 34], based on equippingeach fundamental building block with a length scale a ,and then proceeding similarly to lattice field theory.In completely background-independent approaches,where not even fiducial background structure is al-lowed, topology is not fixed, and geometry is emergent,the notion of scale can clearly not be related to amomentum-scale as in standard quantum field theories.The only possible notion of scale is inherent to themodel, which “knows” nothing about momentum orlength scales. The notion of scale is related to thenumber of degrees of freedom that have been integratedin the path integral. Then, a distinction of UV andIR is possible: The model-inherent scale is infrared,if most degrees of freedom have been integrated out,and ultraviolet, if most degrees of freedom are not yetintegrated out. It is then obvious that the matrix size N is the only possible notion of scale in pure matrix/tensormodels. C. Multicritical matrix models and conformalmatter
In an extension of our earlier work [1], we will discussmodels beyond the pure gravity case here, which cor-respond to conformally coupled matter-gravity theories.The motivation to study these is clear: Our universe con-tains both matter and gravitational degrees of freedom,which are coupled. Thus matter degrees of freedom areimportant in the gravitational RG flow and vice-versa,see, e.g., [35] for the continuum case. It is thus highlyinteresting to study matrix models which correspond totesselations of surfaces including dynamical matter. Amodel for these theories was discovered in [36], wheremulticritical points for matrix models were found. If wegeneralize the matrix model to allow for a potential ofthe form V ( φ ) = 12 Tr φ − g φ − g φ + ..., (5)then this still corresponds to a model of random sur-faces, with the tesselations including squares, hexagons,octagons and so on. If we allow these couplings to changesign, then some configurations will come with a negativeweight, already suggesting that this model contains morethan just gravitational degrees of freedom. E.g. considera typical Feynman graph, which is a triangulation of aRiemann surface and consider a “contamination” withsquares. The additional squares can be viewed as a sol-dering of two triangles and this soldering can be phys-ically interpreted as a hard dimer [37]. In fact, thereexists a tower of multicritical models, which correspondto two-dimensional gravity coupled to conformal matter.They are specified by two integers ( p, q ), which deter-mine the central charge c = 1 − p − q ) / ( pq ). In thecase of multicritical matrix models, it turns out that( p, q ) = (2 , m − m = 2 , , ... . One can thenevaluate the critical exponent γ st in Liouville theory, andobtains the result γ st = − m + 3 /
2. For the matrix mod-els, the following pattern of critical points emerges: Set-ting g n = 0 for n > n max and letting the g n − , g n − etc. alternate in sign, with g <
0, leads to different univer-sality classes in the continuum limit. For the m th suchmulticritical point, n max = 2 m , which is characterizedby m − γ st = − m ,in agreement with the continuum result. (Note that dif-ferent conventions for the definition of γ are sometimesused in the literature. We follow [12].) Translated tothe critical exponent at an RG fixed point, this implies θ m = m +1 . The existence of further relevant directions[26], i.e., parameters that require tuning to reach thephase transition in the large N -limit, is suggestive of theexistence of further degrees of freedom. One can easilyconjecture that these additional degrees of freedom arematter [36]. As a new test of our Renormalization Groupmethod, we will search for the fixed points correspondingto the double-scaling limit at these multicritical points,see, e.g., [38], and compare the results for the criticalexponents with the exact values. II. FUNCTIONAL RENORMALIZATIONGROUP FOR MATRIX MODELS
The Wetterich equation [39] is a functional differentialequation for the effective average action Γ k of a quan-tum field theory, which contains the effect of quantumfluctuations at momenta p > k , and encodes the effec-tive dynamics of low-energy effective fields. For generalreviews of the method, see [40, 41]. In [1] we adaptedthis equation to the setting of matrix models, where nomomentum scale exists. We thus introduce an infraredcutoff scale N , and write an infrared regulator R N ( a, b )as a function of the matrix indices a, b and the cutoffscale N , such thatlim a/N → ,b/N → R N ( a, b ) ab cd > , (6)lim N/a → ,N/b → R N ( a, b ) ab cd = 0 (7)lim N → Λ →∞ R N ( a, b ) ab cd → ∞ . (8)Since the quadratic term Tr( φ ) is dimensionless, i.e., itscales as N , we used a dimensionless regulator of theform R N ( a, b ) = (cid:18) Na + b − (cid:19) θ (cid:18) − a + b N (cid:19) , (9)in [1], that is modeled after Litim’s optimized cutoff forthe continuum case [42, 43].Including the IR-suppression term Tr( φ R N φ ) =∆ S N into the path integral Z N = (cid:90) Λ dϕ e − S [ ϕ ] − ∆ S N [ ϕ ]+ J · ϕ , (10)then allows us to define the effective average action by amodified Legendre transform.Γ N [ φ ] = sup J ( Jφ − ln Z N ) − ∆ S N [ φ ] . (11)The scale dependence of the effective average action isthen described by ∂ t Γ N = 12 tr (cid:16) Γ (2) N + R N (cid:17) − ∂ t R N , (12)where t = ln N . Herein Γ (2) N = ∂ ∂φ ab ∂φ cd Γ N . A. Symmetric theory space and truncation
To derive β functions from Eq. (12), we write theeffective action as a sum of local operators multi-plied by scale dependent couplings. Our model has a U ( N ) × Z symmetry, such that all operators of the formTr (cid:0) φ i (cid:1) . . . Tr (cid:0) φ i n (cid:1) with i + · · · + i n even are generatedand should be included in the effective action. The spaceof action functionals that are linear combinations of theseoperators is the Z × U ( N )-symmetric theory space. Onenow has to find a way to truncate this space to a fi-nite subspace that can be dealt with in practice withoutthrowing out those operators that are relevant for thephysical system we aim to describe. An organizing prin-ciple in this theory space is provided by the scaling di-mensionality. In our case, where no notion of momentumscales exist, the scaling dimensionality is related to thematrix size N : The requirement of a well-defined large- N limit of the matrix model allows us to derive a consistent(albeit not unique) canonical scaling of couplings: Forsingle-trace couplings ¯ g i of operators Tr φ i and their di-mensionaless version g i we obtained in [1], see also [17] g i = ¯ g i N i − Z i/ φ , (13)where Z φ is a wave-function renormalization, occurringas the prefactor of the quadratic term in the potential.Similarly we have that g i ...i n = ¯ g i ...i n N i ··· + in +( n − Z i ··· + in φ , (14)for multitrace couplings g i ...i n Tr (cid:0) φ i (cid:1) . . . Tr (cid:0) φ i n (cid:1) . Herewe have taken into account that couplings correspondingto multitrace-operators have a lower canonical dimen-sionality, to account for the additional traces.As only couplings with positive dimensionality are rel-evant and correspond to free parameters, we concludethat the theory has no free parameters at the Gaußianfixed point. At an interacting fixed point, such as thatcorresponding to the double-scaling limit, new operatorscan be shifted into relevance as interactions modify thescaling dimensions. The canonical dimensionality never-theless provides a useful organizing principle: If we as-sume that the contribution of quantum fluctuations tothe scaling dimensionality, η i , is bounded, this providesa guiding principle to set up a truncation: It should firstinclude single-trace operators up to a certain number of fields, and then double- and triple-trace operators up toa given canonical dimensionality. Note that similar con-siderations have been backed up by explicit evaluationsof scaling dimensions in continuum quantum gravity [44].We will use this reasoning to define useful truncations. B. Extended theory space with symmetry breakingoperators
An IR-suppression term that divides the matrix entries φ ab into IR and UV degrees of freedom necessarily breaksthe U ( N ) symmetry of the matrix model. To constructa regulator that depends on matrix indices, one needs tointroduce at least a constant matrix X with components X ab = a δ ab , (15)which allows one to introduce the operator (∆ φ ) ab =( a + b ) φ ab as ∆ φ := X φ + φ X. (16)This operator is closely related to the 2-dimensionalGrosse-Wulkenhaar Laplacian (∆ φ ) ab = ( a + b + 1) φ ab ,which can be used to set up an FRG approach to noncom-mutative scalar field theory [45]. Due to this analogy, wewill use the term “matrix Laplacian” for this operator.As a consequence of introducing the regulator, theRHS of the flow equation will also contain operators withinsertions of the constant matrix X , i.e., operators ofthe form Tr ( φ n X m ... ) ... Tr ( ... ) (with (cid:80) i n i even). Inother words, the RG flow has to be set up in the ex-tended theory space, which is the space of linear com-binations of these more general operators, because the U ( N )-symmetric theory space, which is a subspace ofthis full theory space, is not left invariant by the flow.The dimensionality of the operators with additional in-sertions of X can be inferred in two ways: From inspec-tion of Eq. (9), we realize that, as tr X ∼ N , this setsa dimensionality of 1. Alternatively, one can considerthe geometric picture that underlies the 2-dimensionalGrosse-Wulkenhaar model . There, the matrix size N is related to the noncommutative scale θ , which has di-mension l − , which is the same dimension as the Grosse-Wulkenhaar Laplacian ∆. Again, one concludes that X should carry one unit of dimension in N .This dimensionality of X (resp. ∆) has the useful effectof dimensionally suppressing operators with an insertionof X compared to the U ( N ) symmetric operators that areobtained by excising the X insertions. Thus, one expectsthat the operators with X insertions will be irrelevant. Note that the analogy with the Grosse-Wulkenhaar model canonly be made for the geometric structures, but not for φ , whichis dimensionless when treated as a 2D scalar field, whereas itis important to the matrix model approach to quantum gravitythat φ possesses dimension N . Once one realizes that the flow equation does not preserve the
C. Projection on a truncation
The practical use of the flow equation involves trun-cations of the theory space. Unfortunately, the RHS ofthe flow equation will in general not be of the form ofthe truncation. Thus one needs to find a prescription toproject the RHS of the flow equation onto the monomi-als in the truncation. In [1], we concerned ourselves onlywith the symmetric theory space and where then able todiscern the monomials in the truncation by inserting fieldconfigurations of the form φ = v X , after which we whereable to perform the operator traces as simple sums. It isobvious that this simple projection rule is not sufficientfor the full theory space. We thus refine the projectionrule. In a first reading, the following considerations canbe skipped, as the approximation that we will introducein sec. III A and the results in sec. IV can equally well beobtained with the simpler prescription in [1].The elementary operators Tr ( φ n X m ... ) ... Tr ( ... )form a basis of theory space, i.e., a general action func-tional can be expanded as a linear combination of theelementary operators . We thus have to find a projec-tion prescription that allows us to extract the coefficientsof the expansion of a general action functional in termsof elementary operators. This algorithm has to respectlinearity of the expansion and it has to satisfy the el-ementary projection property: If the action functionalis precisely one elementary operator then the expansioncoefficient of this operator is 1, while all other vanish. For practical calculations we will use the expansion (cid:16) Γ (2) N + R N (cid:17) − = (cid:80) ∞ n =0 P − (cid:0) − F [ φ ] P − (cid:1) n , where P U ( N )-symmetric theory space, it becomes a natural question toask what happens if one uses a standard regulator built from thematrix Laplacian ∆, e.g., a regulator of the Litim form R N = ( N − ∆) θ (cid:18) − ∆ N (cid:19) . (17)In the explicit calculations that we performed for this paper, itturned out that the flow generated using the dimensionful regula-tor (17) reproduces all the qualitative features of flow generatedusing the dimensionless regulator (9), and that the quantitativedifferences where small. It may at first seem surprising that thedimensionality of the regulator does not influence the qualita-tive features of the flow, but it is simply a consequence of thefact that we investigated the flow of a pure matrix model underthe change of matrix-size N and that both regulators are “smallmatrix”-suppression terms. The most general action functional is an arbitrary function ofthe matrix entries. However, since we use a vertex expansionto derive the RHS of the flow equation, we restrict ourselves toaction functionals that can be expressed as linear combinationsof the elementary operators. Notice also that this expansion isformal, i.e., we do not specify any norm in which this expansionis assumed to converge. The specification of such a norm is a verydelicate problem, similar to summability of perturbation series. This statement does of course require that elementary operatorsare linearly independent, which poses a restriction that we willdiscuss below. and F [ φ ] are fixed by Γ (2) N [ φ ] + R N =: P + F [ φ ] and F [ φ ≡
0] := 0. This leads to a significant simplification,since this ensures that the RHS of the flow equation is“polynomial” in φ whenever Γ N [ φ ] is. For polynomialaction functionals one can proceed as follows:Each term on the RHS of the flow equation is of theform T n [ φ ] = ( − n tr op (cid:16) P − ˙ R (cid:0) P − F [ φ ] (cid:1) n (cid:17) , where wedefined ˙ R = N ∂ N R N . We expand T n [ φ ] := ∞ (cid:88) k =1 V a ,b ,...,a k ,b k n,k φ a b ...φ a k b k , (18)where V a ,b ,...,a k ,b k n,k = 1(2 k )! δ k T n [ φ ] δφ a b ...φ a k b k (cid:12)(cid:12)(cid:12)(cid:12) φ ≡ . (19)For instance, starting from an action of the form Γ N =tr φ + g tr φ we will have V n,k ∼ g n δ k, n . We will nowuse the fact that δφ ab δφ cd = I , where I is the appropriate unitmatrix, e.g., I = ( δ ac δ bd + δ ad δ bc ) . Further, we choosean IR-suppression term that is an index dependent func-tion times the appropriate symmetrization of δ .. δ .. . Itthus follows that V a ,b ,...,a k ,b k n,k = (cid:88) i f a ,b ,...,a k ,b k n,k,i ( a , ..., b k )
2k factors (cid:122) (cid:125)(cid:124) (cid:123) δ ..,.. ... δ ..,.. , (20)where each upper index of any f a ,b ,...,a k ,b k n,k,i is con-tracted with an index of a Kronecker delta. We can thuswrite each f a ,b ,...,a k ,b k n,k,i ( a , ..., b k ) as a function of theindices g ( a , ..., b k ) times a contraction pattern (whichis given by the Kronecker-deltas). We now perform aTaylor-expansion of the g around vanishing index g ( i , ..., i k ) = g | (cid:126)i ≡ + ∂ g∂ i j (cid:12)(cid:12)(cid:12) (cid:126)i ≡ i j + ∂ g∂ i j ∂ i k (cid:12)(cid:12)(cid:12) (cid:126)i ≡ i j i k + ... = g + g i i + g i ,i i i + ... (21)The expansion coefficients g ki ,...,i k combined with thecontraction pattern δ i . ,i . ...δ i . ,i . can be identified as be-ing generated by a unique product of traces of matrixproducts of φ - and X -matrices: The contraction pattersdetermine how many φ ’s appear in each trace and the de-pendence of g ki ,...,i k on the indices i , ... tell us at whichpositions what power of the X -matrices have to be in-serted. To project onto the symmetric operators with no X -insertions, we would only consider the g term. The symmetrization is model dependent, e.g., an unconstrainedreal model will have no symmetrization, a real symmetric modelwill require symmetrization in a, b and complex matrix modelsrequire decomposition into real modes and subsequent (anti-)symmetrization if the model is Hermitian.
This procedure allows us to uniquely expand the RHSof the flow equation in terms of the elementary monomi-als Tr ( φ n X n ... ) ... Tr ( φ m X m ... ) that we use as thecoordinate basis for our theory space. Two subtletiesassociated with the use of this basis should be noted:1. Commutativity of the product of traces andcyclicity of the trace imply the monomi-als Tr ( φ n X n ... ) ... Tr ( φ m X m ... ) are notsimply labeled by the arrays of integers(( n , n , ... ) , ..., ( m , m , ... )), but by equiva-lence under all permutations of blocks and cyclicpermutations by an even number of steps ofnumbers within a block. This means that we haveto label the coordinate basis of the theory space,i.e., the coupling constants, by fixing a uniquerepresentative in each equivalence class.2. The regulator is not an analytic function of the in-dices due to the Heaviside function. This Heavisidefunction has however no observable effect when theeffective action is probed with “IR”-degrees of free-dom (i.e., matrices φ that have only the upper left N × N components nonvanishing). Thus, for effec-tive IR field theory, one can use the above identifi-cation of field monomials, since the Taylor expan-sion of the functions g around vanishing index is“blind” to the Heaviside function. III. RENORMALIZATION GROUP FLOW ANDGAUGE SYMMETRY
The above procedure would allow us to derive the RGflow in the extended, non-symmetric theory space. Sinceour model is symmetric under U ( N ), and the symmetry-breaking is only introduced by the regulator, there is aWard-identity that will impose a nontrivial constraint onthe RG flow in the extended theory space. The action ofa Hermitian pure matrix model is invariant under unitarytransformations which act on the field in the form φ (cid:55)→ O T φ O = φ + (cid:15) [ φ, A ] + O ( (cid:15) ) , (22)where A is the generator of an infinitesimal symmetrytransformation. The functional measure and the bareaction of a pure matrix model are invariant under unitarytransformations, but the change of the regulator term is G (cid:15) ∆ N S = (cid:15) Tr ( φ [ A, R N ] φ ) , (23)where we denoted the change of a functional F under aninfinitesimal gauge transformation by G (cid:15) F . Thus, the ef-fective average action satisfies the scale dependent Ward-Takahashi identity (WTI) W N Γ N = G (cid:15) Γ N − tr op (cid:32) [ A, R N ]Γ (2) N + R N (cid:33) = 0 . (24)It follows form the standard argument see, e.g., [41], thatthe RG-evolution of an initial condition that satisfies the initial WTI W N Γ N = 0 at an initial scale N , will satisfythe evolved WTI W N (cid:48) Γ N (cid:48) = 0 at a scale N (cid:48) . Hence, ifwe want to implement gauge symmetry, i.e., if we requirethe usual WTI Γ (cid:15) Γ = 0 to hold, then we have to imposethat satisfies Γ N the scale dependent WIT to ensure thatΓ = lim N → Γ N satisfies the usual WTI.It is important to notice that the scale dependentWTI can not be solved by a Γ N that respects the usualgauge symmetry G (cid:15) Γ N = 0, because the second term of(24) does not vanish unless N = 0. Hence, to imple-ment gauge symmetry in the flow, one is forced to “con-taminate” Γ N by turning on just the right amount ofcouplings for symmetry-breaking operators. Conversely,to implement gauge symmetry, we have to restrict thesearch for RG fixed points to solutions of the scale de-pendent WTI. A. Tadpole Approximation
Restricting the search for RG-fixed points to solutionsof the scale dependent WTI will be dealt with in futurework. For the present paper, we make the following im-portant observation: If we assume that the sought-forfixed point lies at small values of the couplings, then wecan approximate the β functions by the first order inthe vertex expansion: Assuming that combinatorial fac-tors in the loop diagrams are O (1), and the fixed-pointvalue of all couplings is ∼ (cid:15) <
1, then the n -vertex dia-gram is suppressed by a factor (cid:15) n − in comparison to thetadpole diagram. Accordingly it is a self-consistent ap-proximation to take into account only tadpole diagrams,if the corresponding fixed-point values indeed turn outto satisfy our requirement. We now make the follow-ing central observation: As all vertices arising within atruncation consisting of U ( N ) invariants are themselves U ( N ) invariant, symmetry-breaking operators cannot begenerated by the tadpole diagram. Any nontrivial index-dependence on the RHS of the flow equation can alwaysbe shifted away, as there is no nontrivial index depen-dence in the vertex. This is completely analogous to thecase of, e.g., standard λφ theory: As the vertex pro-portional to λ is momentum-independent, the tadpolediagram cannot generate a momentum-dependent opera-tor. Thus no non-trivial wave-function renormalization isgenerated from the tadpole diagram ∼ λ . In our case, anon-trivial index-dependence is analogous to a non-trivialmomentum dependence.This reasoning can be applied to the vertex expan-sion of the RHS of the flow equation as well as the ver-tex expansion of the second term of the scale dependentWTI (24). We conclude that the scale-dependent WTI issolved by a U ( N )-symmetric Γ N in the tadpole approx-imation and, conversely, that the tadpole approximationto the RG flow preserves U ( N )-symmetry.One might now wonder whether the functional Renor-malization Group will be of use to uncover the double-scaling limit in higher-dimensional tensor models. If thecritical value of the coupling would lie at a large value,the tadpole approximation would not be applicable. Hereit is crucial that the critical value of the coupling corre-sponds to the radius of convergence of the perturbativeexpansion, and as such is guaranteed to lie at values muchsmaller than one. Accordingly, the use of the tadpoleapproximation is justified to explore the double-scalinglimit in higher-dimensional tensor models. An addedbenefit lies in the fact that the evaluation of the β func-tions in a large truncation is simplified considerably, ifwe restrict ourselves to the tadpole approximation. Thuswe are confident that our method will also allow us tosuccessfully tackle higher-dimensional tensor models. IV. β -FUNCTIONS AND FIXED POINTS The considerations of the previous section suggest thatthe tadpole approximation to the β -functions will im-prove the results of [1] for fixed points with small valuesof the couplings. We now confirm this suggestion andin this course also uncover the multicritical fixed pointswith the FRG for the first time. A. Single-trace approximation
As a first step, let us reconsider the single-trace trun-cation studied in [1], which isΓ k = Z φ Tr φ + n max (cid:88) n =2 g n n Tr (cid:0) φ n (cid:1) , (25) where we take n max = 7 in accordance with [1]. Em-ploying eq. 63 in [1], restricting ourselves to tadpole dia-grams, we then obtain a set of beta functions as follows: η = 2 g x, (26) β n = (( n −
1) + n η ) g n − n x g n +1) , (27)where the first term in the beta functions arises fromthe canonical dimensionality of the couplings and η = − N ∂ N ln Z φ . We have set [ ˙ RP − ] = x in order to studythe scheme dependence of our results.Here, we also neglect the term ∼ η , that is generatedby ∂ N R N on the right-hand-side of the flow equation.We then obtain a set of fixed points and critical expo-nents listed in tab. I. g g g g g g θ θ θ θ θ θ − x − -1 − -2 - − x x − − -1 − − x x − x
34 12 − − − − x x − x x
45 35 25 − − − x x − x x − x
56 23 12 13 − − x x − x x − x x
67 57 47 37 27
TABLE I: We show fixed points and critical exponents that we obtain in a single-trace truncation including all couplings upto g . We include only tadpole diagrams, and parameterize ˙ RP = x . The first fixed point is the Gaußian fixed point, wherethe critical exponents equal the canonical scaling dimen-sionality of the couplings.The second fixed point corresponds to the well-knowndouble-scaling limit, with one relevant direction. As in[1], this first approximation yields a critical exponent θ = 1, instead of the analytically known exact value θ = . Multicritical Points
All other fixed points correspond to multicritical pointsof increasing order m . They show the well-known pat-tern of alternating signs for the couplings, correspondingto stable/unstable potentials. As expected for the m thmulticritical point, m − ? ]. The next critical exponents are expected to be θ m − = m + . We obtain, similarly to [26], θ m − = m .The confirmation of the existence of the multicriticalpoints within the FRG approach to matrix models is oneof the main new findings of this paper.Moreover, one can find analytic expressions for themulticritical points in the tadpole approximation to thesingle-trace truncation as follows: assume that the g n vanish for all n > m , so the β n vanish for all n > m ,while β m = 0 and η = 2 x g imply η = 1 − mm , g = 1 − m m x . (28)The vanishing of the remaining β n give the linear recur-sion relation g n +2 = (cid:0) n − n − m − m (cid:1) g n x . For the initialcondition (28) the solution is, in agreement with tab. I,given by g n = (1 − m ) n − ( n − m x ) − n , (29)where we used the Pochhammer symbol ( a ) n = a ( a +1) ... ( a + n ).After inserting the k -th multicritical value η = − kk into the tadpole approximation to the vertex expansion,one finds that the Jacobian ∂ β n ∂g m = δ n,m (cid:16) ( n − − nm ( m − (cid:17) − δ n +1 ,m n x (30)is triangular and independent of the couplings g n . Wecan thus recover the positive critical exponents from thediagonal entries of the Jacobian θ ( m ) n = nm , where: n = 2 , ..., m (31)in agreement with tab. I. (Notice that, for simplicity ofpresentation, we treated η as a constant and not as afunction of g , when we calculated the critical exponents,which turns out not to have an effect on the result in thesingle trace approximation.) Universality
We observe that, although the g n ∗ depend on x , thecritical exponents do not. Thus these values are uni-versal, i.e., independent of the choice of regularization scheme. The dimensionful couplings themselves are notuniversal, i.e. g n ∗ = g n ∗ ( x ). However, we can alsoform universal (dimensionless) ratios of couplings, suchas g /g , which accordingly are independent of x . Forinstance, the second multicritical point, where g = 0,has g /g = 4, which is in reasonable agreement with theexact g /g = , and in fact corresponds exactly to thevalue in [26].Note that to obtain the value g ∗ = g c = − for m = 2,we would have to set x = 3, which clearly shows thatfixed-point values are non-universal. This is expected asin our setting they carry a non-trivial scaling dimension-ality with N . In the same way that fixed-point values fordimensionful couplings cannot be universal in standardquantum field theories, no such universality is expectedin our case.A subtle difference to standard quantum field theo-ries arises, as the notion of scale that we introduce heredisappears, once the integration over all quantum fluctu-ations has been completed: In the limit N → ∞ , thereis no other quantity left in the matrix models that wouldstill set a scale. This is a major difference to standardquantum field theories: There, two distinct notions ofscale exist: One is a Renormalization Group scale µ ,that decides which quantum fluctuations have been in-tegrated out. The other is a model-inherent momentumscale, which enters the operators of the model, such as,e.g., the kinetic term. Since both scales are momentumscales, the dimensionality assigned to couplings using themodel-inherent momentum, or the ”external” RG scale,agrees. In particular, a nontrivial notion of dimension-ality remains in the limit µ → ∞ . This is different inthe case of the matrix model, where non-trivial scalingdimensions do not exist once N → ∞ . The couplingsbecome dimensionless in that limit. This observationshould explain, why our fixed-point values are nonuni-versal, whereas other methods that are used to derivethe double-scaling limit in matrix models yield a univer-sal number for g c . B. Multi-trace truncation
We now investigate a truncation that takes into ac-count all operators up to a fixed dimensionality. In par-ticular, we include g , . . . , g , g , . . . , g , g , g and g , g . Again, we restrict ourselves to tadpole dia-grams. According to our reasoning in sec. II, the canoni-cal dimensionality provides a useful organizing principlefor the couplings in theory space. We thus expect thatthis multitrace truncation should show improved resultsover the single-trace truncation, as it will consistentlytake into account all operators up to a given dimension-ality.The interesting double- and triple trace operators arecontained in the following truncationΓ N = Z φ Tr (cid:0) φ (cid:1) + (cid:80) n ≥ g n Tr (cid:0) φ n (cid:1) + (cid:80) n ≥ g , n Tr (cid:0) φ (cid:1) Tr (cid:0) φ n (cid:1) + (cid:80) n ≥ g , n Tr (cid:0) φ (cid:1) Tr (cid:0) φ n (cid:1) + (cid:80) n ≥ g , , n Tr (cid:0) φ (cid:1) Tr (cid:0) φ n (cid:1) . (32)The tadpole approximation to the beta functions in this truncation is η = 2 x ( g + g , ) (33) β n = (( n −
1) + n η ) g n − n x (2 g n +2 + g , n ) (34) β , = 2 (1 + η ) g , − x ( g + 4 g , + 3 g , , ) (35) β , = 3 (1 + η ) g , − x ( g + 3 g , + 4 g , + g , , ) (36) β , n = ( n + 1) (1 + η ) g , n − x ( g n +4 + ( n + 1) g , n +2 + 2 g , n + g , , n ) , n ≥ β , = 4 (1 + η ) g , − x ( g + 6 g , ) (38) β , n = ( n + 2) (1 + η ) g , n − x (2 g n +6 + (2 n + 2) g , n +2 ) , n ≥ β , , = (4 + 3 η ) g , , − x (3 g , + 4 g , , ) (40) β , , = (5 + 4 η ) g , , − x ( g , + 3 g , + 6 g , , ) (41) β , , n = (( n + 3) + ( n + 2) η ) g , , n − x ((2 n + 4) g , n +4 + (2 n + 2) g , , n +2 ) , (42)(43)where the exceptions arise due to additional symmetryfactors (e.g., the derivative of Tr φ Tr φ has an additionalfactor of 2 when coupling into g , , than the derivativeof Tr φ Tr φ when coupling into g , . )The general structure has already been discussed in [1],and is as follows, cf. fig. 1: At leading order in 1 /N , onlya tadpole diagram of g ,i ,...,i n can couple into g i ,...,i n .Our truncation to tadpole diagrams is therefore not atruncation, but already the full result, when it comes tothe back-coupling of higher orders in the number of tracesinto the lower-order beta functions. level number of fields g g g g … g g g g … g g … … g g g … g … … FIG. 1: We schematically show the structure of the flowequation. We indicate tapole diagrams by arrows. Clearlyonly neighbouring levels couple into each other’s beta func-tions. Note that at higher order in the traces, the levels getmore complicated, e.g., there exist two-trace terms of the form g ,i , g ,i , g ,i ... . Furthermore, an n -trace operator can only generate an( n + 1)-trace operator through a tadpole diagram, as thecontraction on the right-hand side of the flow equationdoes not generate more than one additional traces: Atmost, the structure of Γ (2) and the subsequent contrac-tion permits to split one of the Tr φ i -terms into two new,Tr φ i − j Tr φ j terms.We thus have a structure where only neighbouring“levels” (i.e., numbers of traces) are coupled in the betafunctions. Thus, already the three-trace terms affect thesingle-trace terms only indirectly. Additionally, manycontributions that could be possible if one only countsthe number of fields, are disallowed because of the trace-structure of the flow equation: For instance, just count-ing the number of fields, one could expect a coupling of g , into β g , , . This contribution does not exist, as thesecond derivative of Tr φ Tr φ cannot generate a termthat separates into 3 traces with two fields each uponcontraction with the propagator.Beyond the tadpole level, two-vertex diagrams can“span” a larger number of levels, as combinations of ver-tices with i traces and vertices with i + 2 traces can alsocouple into operators with i + 2 traces.For our explicit solution of the fixed-point equations,we only include up to g . For consistency, this impliesthat we have to take into account up to three-trace op-erators. We thus expect to find the first 5 multicriticalpoints. It turns out, that a subset of the multitrace cou-plings is also nonvanishing at these multicritical points,and that they have an important effect on the criticalexponents, cf. tab. II.0 g g g g g g g g g g g g g θ θ θ θ θ − x − x x x − x x − x x − x − x x − x x x − x x x x RP − = x . A number of further fixed points that we obtain is not shown, as they donot correspond to any known analytical solution and are thus most probably artifacts of the truncation. Most importantly, the largest critical exponent whichcorresponds to the pure-gravity critical exponent turnsout to be θ ∼ . θ = 0 .
8. Thisis a major step forward from the investigations in [1].Already for the pure-gravity fixed point, our result con-stitutes an improvement over our previous result in amulti-trace truncation, as well as the considerable im-provement over the perturbative calculation in [27].For the fixed-point values, we observe that universalcombinations such as g g depart further from the exactresult than in the single-trace approximation. We at-tribute this to the fact that multi-trace operators at thesame order of the fields are nonvanishing, e.g., g , (cid:54) = 0at the m = 3 multicritical point. Analytic Solution
If the initial condition to the flow is such that all cou-plings that correspond to operators with more than 2 k fields vanish, then the tadpole approximation to the RGflow will preserve this condition, because a tadpole dia-gram from a truncation with 2 k fields will generate oper-ators with at most 2 k − k fields vanish. For such a fixed point search, one can em-ploy the following strategy:1. There will be a finite number (the number of dis-tinct integer partitions of k ) of beta functions forthe operators with 2 k fields, which are of the form β a = (dim( a ) + k η ) g a . These imply that eitherthat η = − dim( a ) k for one a and all other couplingswith 2 k fields vanish just as in tab. II, or that all g ... vanish. The second case corresponds to a fixedpoint with at most 2 k − k . 2. The remaining beta functions are of the form β b =(dim( b ) − dim( a ) nk ) g (2 n ) b + x (linear in g (2 n +2) ),where the superscript bracket denotes the numberof fields in the corresponding operator. Vanishingof the beta functions thus gives the recursion rela-tion g (2 n ) b = dim( a ) nk − dim( b ) x (linear in g (2 n +2) .3. One now sets g a = α and uses the recursion rela-tions to derive all coupling constants with 2 k − k − α . This provides in particular g ( α ) and g , ( α ).Notice that the recursion relation is linear, whichimplies that g b ( α ) = α g b (1).4. The anomalous dimension then implies that2 α x ( g (1) + g , (1)) = − dim( a ) k , which shows that α = − dim( a )2 k x ( g (1)+ g , (1)) .A technical difference between the single- and multitracetruncation is that it is easy to find the solution of the sin-gletrace recursion relation as a function of k (see Eq. (29)above), while we where unable to express the solution ofthe multitrace recursion relation as a function of k . C. Two-vertex contributions
The next logical step is to include two-vertex contri-butions to the beta functions, which provide the leadingorder corrections to the tadpole approximation for fixedpoints with small couplings. These contributions do, aswe explained above, imply that the solution to the Wardidentity requires the inclusion of non-symmetric opera-tors. In the explicit symmetric truncations that we con-sidered so far, it turns out that the inclusion of two-vertexcontributions has far-reaching effects: We observe, thatnone of the multicritical points can be found after theinclusion of these additional terms, and only the pure-gravity double-scaling limit remains. One of course ex-pects that the inclusion of two-vertex diagrams will leadto an improvement once one restricts the flow to a consis-tent approximation to the solution of the Ward identity.The test of this expectation goes beyond the scope of thispaper and will be investigated in a future paper.1
V. CONNECTION TO CONTINUUM β FUNCTIONS FOR GRAVITY AND THEASYMPTOTIC SAFETY SCENARIO
In research on quantum gravity, many approaches existin parallel and it is a priori unclear whether they are inany way related. In particular, the continuum approachbased on a quantum field theory for the metric, knownas the asymptotic safety scenario [31, 46, 47], for reviewssee, e.g., [48], and approaches based on a discretization ofgeometry, such as matrix or tensor models, differ in manyaspects. The fundamental variables are taken to be dif-ferent (metric versus matrices/tensors), the symmetriesdiffer (diffeomorphism symmetry versus U ( N ) symme-try), and the Renormalization Group flow is formulatedwith respect to two completely different notions of scale(defined with respect to a fiducial background metric ver-sus matrix/tensor size). Nevertheless, one could expectthe following scenario, see, e.g., [25, 49] and also a re-lated discussion in [50], where the interacting fixed pointunderlying asymptotic safety is related to a phase tran-sition from the “pregeometric” to the geometric phase ofa tensor model/group field theory.To be more specific, the tentative non-trivial contin-uum limit in matrix/tensor models, signaled by an in-teracting fixed point, is characterized by a set of criticalexponents. In a more physical sense, that limit can alsobe interpreted as a phase transition from a pre-geometricphase, where no notion of the metric exists, to a geomet-ric phase with a non-vanishing expectation value of themetric. The approach to this continuous (i.e., second or-der or higher) phase transition is described by the criticalexponents see, e.g., [51] for a recent example in CausalDynamical Triangulations.In contrast, the continuum approach known as asymp-totic safety is based on the existence of a non-vanishingmetric, and cannot easily describe the phase with van-ishing expectation value of the metric, see, however, [52].At very high momentum scales, the scaling of operatorsis determined by critical exponents of an interacting fixedpoint of the Renormalization Group flow.These two scenarios can be interpreted as two sidesof the same picture, where the same phase transition –from a pre-geometric phase to a geometric phase – isapproached from the two sides: Matrix/tensor modelsare extremely well-adapted to describe the pregeomet-ric phase, and the approach to that phase transition. Onthe other hand, the physics of the geometric phase is thenmore straightforwardly accessed in the continuum quan-tum field theory setting. For this connection between thetwo settings to hold, the critical exponents calculated onboth sides should agree. Further evidence for such a con-nection between continuum and discrete quantum gravitycould be provided by observables, such as, e.g., the spec-tral dimension, which has been found to equal two in theUV in both discrete [53] as well as continuum settings[54, 55].In the following, we will review explicitly, how the fixed point in matrix models and the corresponding critical ex-ponent is related to the beta function of 2+ (cid:15) dimensionalcontinuum quantum gravity [47, 56]. This provides a sim-ple example of the relationship between the continuumsetting (i.e. the asymptotic safety scenario) and the ma-trix/tensor model setting. Whether a similar relationshipexists in higher dimensions, remains to be investigated.The well-known geometric picture that underlies thematrix model approach to quantum gravity is the discreteapproximation of the geometry of a compact Riemannsurface by tesselations with indistinguishable buildingblocks, e.g., be equilateral triangles or squares. A con-tinuum geometry is attained in the limit in which thenumber N of elementary building blocks diverges whiletheir individual size is rescaled by scaling the lattice con-stant a as a/N , so the total area A of the Riemannsurface remains unchanged.The discrete approximation at finite N defines a reg-ularized measure for the functional integral of two-dimensional Euclidean quantum gravity as the sum overall tesselations. For this one observes that the Einstein-Hilbert action in two dimensions consists of a cosmologi-cal constant term proportional to the total area A and atopological term proportional to the Euler characteristic χ . Both quantities possess simple expressions in termsof the tesselation: the total area is N a and the Eu-ler characteristic is number of vertices V - edges E +faces F . The aim of this approach is to define the func-tional measure by taking the continuum limit a → a/N = const. .The discrete partition function at finite N can be rep-resented as a matrix model. The underlying observationis that the Feynman graphs of a matrix model possess aninterpretation in terms of tesselations, which arise as thedual to the ribbon graphs of the matrix model. Startingfrom the matrix- model action S = N Tr( − φ + gφ ),one sees that each closed loop contributes a factor N dueto the summation of a free index, while the Feynmanrules assign each vertex a factor N and each propagatora factor N − . Thus, we obtain a factor N V − E + F = N χ for each Feynman diagram.It follows that in matrix models, the matrix size N isrelated to the bare Newton coupling by N = e G , (44)as G is the prefactor of χ in the action.Furthermore, the relation g = e − Λ a (45)holds for the dynamical triangulation, see, e.g., [16]. Thisfollows from the fact that each configuration is weightedby e − Λ A = e − Λ a n , where n is the number of squares. Strictly speaking we demand compact without boundary. g n . Wethen translate to a dimensionless cosmological constant, λ = a − Λ.Accordingly, the double-scaling-limit for pure gravitytranslates into the requirement :( g − g c ) / N = (cid:0) e − λ − e − λ c (cid:1) / e G = const . (46)This establishes the correspondence between the doublescaling limit and the continuum limit. We see that takingthe lattice spacing a →
0, while physical quantities, suchas the renormalized cosmological constant, Λ R , are heldfixed, i.e., Λ R = a − ( λ − λ c ) is equivalent to taking thedouble scaling limit of the matrix model.For λ − λ c (cid:28)
1, we then obtain that (cid:0) Λ R a (cid:1) / e G = const . (47)Taking the lattice spacing to zero then requires us toadjust the bare Newton coupling G appropriately, i.e., G = G ( a ). It is then straightforward to derive thescale-dependence of G from Eq. (47): − a∂ a G ( a ) = − G . (48)Since a∂ a = − µ∂ µ where µ is a Renormalization Groupmomentum scale, we finally obtain β G = µ∂ µ G = − G . (49)Note that the numerical value of the coefficient dependsof course on whether the action is defined as RG , or R πG .Let us comment on the connection with continuum fieldtheory: In 2+ (cid:15) dimensions, where quantum gravity basedon the Einstein-Hilbert action on a fixed topology is nolonger trivial, one can obtain a nontrivial beta functionform a standard quantum field theory calculation [56].Its main feature is a term ∼ G with a negative sign.Together with the term arising from a nontrivial dimen-sionality in d = 2 + (cid:15) , this term is responsible for theexistence of a UV attractive interacting fixed point.Here, we have a similar result, with a term ∼ G witha negative sign, which corresponds to asymptotic free-dom. As a difference to the results in [47, 56], it arisesin d = 2. Crucially, it does not follow from a simple scal-ing limit, where g → g c . Instead, it originates fromthe double-scaling limit, where all topologies with higherEuler character contribute. One could thus interpret theexistence of asymptotic freedom in d = 2 dimensionalquantum gravity as stemming from topological fluctua-tions. Most interestingly, going to d = 2 + (cid:15) , a term (cid:15) G will arise from the canonical dimensionality of the cou-pling. The non-trivial term ∼ G will then again inducea nontrivial fixed point, making quantum gravity in 2 + (cid:15) dimensions asymptotically safe. The following results were communicated to us by Jan Ambjorn.
This result exemplifies a tentative scenario in whichthe double-scaling limit in tensor models describes thesame phase transition as a non-Gaußian fixed point inthe asymptotic safety scenario in continuum gravity: I.e.,both could be different sides of the same picture, as in-dicated by the possibility to derive a beta-function for G featuring an asymptotically safe fixed point from thedouble-scaling limit.An even more interesting scenario would be if bothmatter and gravitational degrees of freedom could beencoded in the dynamics of a tensor model. The ex-istence of multicritical points in matrix models, corre-sponding to conformal matter coupled to gravity, showsthat such a scenario works in two dimensions. To under-stand whether a similar scenario could work in four di-mensions, one should compare the critical exponents ob-tained on the tensor model side, to those obtained withinasymptotic safety under the coupling to matter. On thecontinuum side, some critical exponents for gravity in thepresence of matter degrees of freedom are known [35], andalso critical exponents corresponding to matter opera-tors at the interacting fixed points are (partially) known.While currently a complete catalogue of all relevant oper-ators at the interacting fixed point of gravity and matteris still work in progress, its availability would open thedoor to investigate a scenario where matter and gravitydegrees of freedom are both encoded in a tensor model. VI. CONCLUSIONS
In this paper we advance the functional Renormaliza-tion Group as an exploratory tool for the continuum limitin matrix and tensor models for quantum gravity.In particular, we develop a new, self-consistent approx-imation which allows to obtain useful results: The use ofthe matrix size N as a Renormalization Group scale im-plies a breaking of the U ( N ) invariance of the matrixmodel, which is encoded in the Ward identity Eq. (24).We find indications that the systematic deviation of therelevant critical exponent we found in [1] is due to break-ing of this Ward identity. The Ward identity tells us thatin order to obtain a U ( N ) symmetric continuum limit,we have to consider truncations that include a fine-tunedamount of symmetry breaking operators.A crucial new observation in this paper is that a re-striction to tadpole diagrams allows to solve the Wardidentity in a self-consistent approximation with a trun-cation that contains only symmetric operators. Thus theeffect of symmetry-breaking terms can be consistentlyneglected when approximating the Wetterich equationby its tadpole part. Most importantly, this approxima-tion is well-adapted to discover the double-scaling limitin matrix and tensor models, since the fixed-point valueof the couplings corresponds to the radius of convergenceof the perturbative expansion of the partition function,and therefore lies at small values. At the correspondingRG fixed point, all couplings are thus much smaller than3one, and loop diagrams with a higher number of verticesare therefore suppressed.As an added bonus, the restriction to the tadpole ap-proximation very significantly reduces the computationalcomplexity, in particular at the level of multiple-trace op-erators.Moreover, the restriction to the tadpole approximationenables us for the first time to confirm the existence ofmulticritical points for matrix models within the FRGframework, both numerically and by analytic considera-tions. These RG fixed points correspond to continuumlimits of quantum gravity coupled to conformal matterdegrees of freedom. It is particularly reassuring that,within the tadpole approximation, we find the numericalvalue of the leading critical exponent within about 1%of its analytical value 4 / ∼ G with a negativecoefficient, which, in d = 2 + (cid:15) , induces an interactingfixed point for G . This is precisely the result that canbe obtained explicitly by calculating the beta function inthe continuum, and thus provides a hint that the double-scaling limit – or more generally continuum limit – ofmatrix models could be related to asymptotic safety. Acknowledgements
We acknowledge helpful discussionswith R. Gurau and V. Rivasseau on tensor models. Wewould also like to thank J. Ambjorn for extensive discus-sions on the relation of the matrix model double scalinglimit to the continuum beta function. This research wassupported in part by Perimeter Institute for TheoreticalPhysics and in part by the National Science and ResearchCouncil of Canada through a grant to the University ofNew Brunswick. Research at Perimeter Institute is sup-ported by the Government of Canada through IndustryCanada and by the Province of Ontario through the Min-istry of Research and Innovation. T.K. is grateful forhospitality at the Perimeter Institute where part of thework for this paper was completed. [1] A. Eichhorn and T. Koslowski, Phys. Rev. D , 084016(2013) [arXiv:1309.1690 [gr-qc]].[2] J. Ambjorn and J. Jurkiewicz, Phys. Lett. B , 42(1992).[3] J. Ambjorn, A. Goerlich, J. Jurkiewicz and R. Loll, Phys.Rept. , 127 (2012) [arXiv:1203.3591 [hep-th]].[4] J. Ambjorn, S. 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