Large-D Behavior of the Feynman Amplitudes for a Just-Renormalizable Tensorial Group Field Theory

PHYSICAL REVIEW D 103, 085006 (2021)

Large-d behavior of the Feynman amplitudes for a just-renormalizable tensorial group field theory

† Vincent Lahoche1,* and Dine Ousmane Samary 1,2, 1Commissariatal ` ’Énergie Atomique (CEA, LIST), 8 Avenue de la Vauve, 91120 Palaiseau, France 2Facult´e des Sciences et Techniques (ICMPA-UNESCO Chair), Universit´ed’Abomey- Calavi, 072 BP 50, Benin

(Received 22 December 2020; accepted 22 March 2021; published 16 April 2021)

This paper aims at giving a novel approach to investigate the behavior of the renormalization group flow for tensorial group field theories to all order of the perturbation theory. From an appropriate choice of the kinetic kernel, we build an infinite family of just-renormalizable models, for tensor fields with arbitrary rank d. Investigating the large d-limit, we show that the self-energy melonic amplitude is decomposed as a product of loop-vertex functions depending only on dimensionless mass. The corresponding melonic amplitudes may be mapped as trees in the so-called Hubbard-Stratonivich representation, and we show that only trees with edges of different colors survive in the large d-limit. These two key features allow to resum the perturbative expansion for self energy, providing an explicit expression for arbitrary external momenta in terms of Lambert function. Finally, inserting this resummed solution into the Callan-Symanzik equations, and taking into account the strong relation between two and four point functions arising from melonic Ward- Takahashi identities, we then deduce an explicit expression for relevant and marginal β-functions, valid to all orders of the perturbative expansion. By investigating the solutions of the resulting flow, we conclude about the nonexistence of any fixed point in the investigated region of the full phase space.

DOI: 10.1103/PhysRevD.103.085006

I. INTRODUCTION excitations being interpreted as spin-network nodes, which can be combined to build LQG states. The fact that GFT Tensorial group field theories (TGFT) was born since a provides a field theoretical framework for spin foam theory, decade, from the merger between group field theories allowing to use standard field theoretical methods, re- (GFT) and colored tensor models, both appeared in the present a great progress in itself, explaining the success of quantum gravity context as peculiar field theoretical frame- the approach in the last decade. Among these success works [1–25]. highlighting the powerful of the field theoretical frame- GFTs on one hand, arise from loop quantum gravity work, the most important one at this day is undoubtedly the (LQG) and spin-foam theory, as a promising way to results obtained in the context of the quantum cosmology generate spin-foam amplitudes as Feynman amplitude [5–10], as a mathematical incarnation of the geometro- for a field theory defined over a group manifold, with genesis scenario, the space-time Universe is viewed as a specific nonlocal interactions. GFTs is introduced by the condensate of quantum gravity building blocks. Since the so-called Boulatov model for three dimensional gravity and first approaches describing an homogeneous Universe and arise as a way to implement simplicial decomposition for a recovering the classical Friedman equations in the classical pseudomanifold, including discrete connection through a limit; These recent results showed that inhomogeneous specific invariance called closure constraint [15–17]. More effects can be described as well in the same condensation recently, it was shown that GFTs may be viewed in a scenario, considering a multicondensate state, and the complementary way as a second quantized version of spin resulting evolution equations for perturbations are in strong network states of loop quantum gravity [4–12], quantum agreement with the classical results, so far from the Planck scale. There is no doubt that these attractive results are the *[email protected] beginning of a long history for quantum cosmology, where † [email protected] GFTs will be demonstrate their powerful. Despite the fact that they have been originally introduced Published by the American Physical Society under the terms of in the GFT context to cancel some pathologies as singular the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to topologies proliferation, colored tensor models (TM) on the the author(s) and the published article’s title, journal citation, other hand may be viewed as the generalization of random and DOI. Funded by SCOAP3. matrix which is the discrete approach to random geometry

2470-0010=2021=103(8)=085006(29) 085006-1 Published by the American Physical Society VINCENT LAHOCHE and DINE OUSMANE SAMARY PHYS. REV. D 103, 085006 (2021) for two dimensional manifolds [20,21]. The breakthrough In this paper, we address the question of the existence or of colored TM, and probably the reason of their success is not of such a fixed point in completely different point of certainly the existence of a tractable power counting, view, through an exploration of the large rank limit of a allowing to built a 1=N expansion like for matrix models just-renomalizable family of models. We show that in this [13,14]. The role played by the genus in the case of matrix limit, only a subfamily of melons survives, providing a well theory is now replaced by new quantity called the Gurau recurrence relation for Feynman amplitudes. Taking into degree. Despite the fact that the Gurau degree is not a account only the 1PI two point Feynman amplitudes, this topological invariant, in contrast to the genus, it provides a recurrence relation can be solved in terms of Lambert good definition of leading order graphs, which corresponds functions, leading to a explicit expression for two point to a vanish Gurau degree and leads to the so called melonic function to all orders of the perturbative expansion. From diagrams. In contrast to planar graphs, the melons obey to a this explicit solution, and taking into account the strong recursive definition allowing to map them as d-ary trees, relation between two and four point functions arising and then become easy to count. Beyond the vectors and from Ward identities in the melonic approximation, we matrices, tensors, and particularly the melonic diagrams solve the Callan-Symanzik (C-S) equation, and deduce may provide for the future developments, many different explicit expression for relevant and marginal β-functions. applications far from quantum gravity. Finally, investigating these solutions, we show that no-fixed TGFTs is built by merging some aspect of these two point occurs in the considered region of the full space of approaches (GFTs and TMs) such that: The fields remain couplings. defined on a group manifold, but the interactions inherit Note that the resummed two point function that we their structure from tensor models. A new notion of locality, obtain in our computation provides a solution, in a suitable usually referred as traciality replace the simplicial con- limit to the so-called closed melonic equation, first intro- straint from which GFT interactions are historically con- duced in [55–63] by direct inspirations of the Grosse structed. As for tensor models, this locality principle allows and Wulkenhaar works for noncommutative field theory to define a power counting, and then to address the question [56–60]. Up to the leading order melonic diagrams, this of renormalization see [26–34] and references therein. In equations is reputed to be very hard to solve. In the first standard quantum and statistical field theory context, the paper on this subject [55–63], the authors only addressed a canonical notions of scale and locality arise from the perturbative solution of a just-renormalizable model, up to background space-time itself. For GFT however, there order six. The same equation has been considered for a are no background to support these notions, and the scales, tensorial group field theory endowed with a specific gauge which are required for standard renormalization procedure invariance called closure constraint [34] on which only the have to be defined extrinsically as well. This is given in perturbative solution is also given. In the same reference practice through a modification of the kinetic action [35], paper, and from a BPHZ theorem the authors argued in the notion of scale arising from the spectrum of the kinetic favor of the existence of a solution to all order of the kernel, usually a linear combination of the identity and the perturbative expansion. Recently, a strong progress has Laplace-Beltrami operator, both defined over the group been achieved in [61], where an explicit solution has been manifold. Rigorous BPHZ theorems have been proved for found for a divergent free model. However, at this day, no such a kind of field theories, from which potentially such a solution exist for just-renormalizable models. In this interesting theory have been classified from their pertur- paper, we show that in a suitable large d limit, an explicit bative just-renormalizability [26–34]. Finally, nonpertur- analytic solution can be found for the melonic closed bative renomalization group aspects have been addressed equation. for these models through the popular Wetterich-Morris In detail, the outline of this paper is the following. In formalism, which is the most suitable to deal with the Sec. II we define the model, and introduce the useful specific locality of TGFTs in order to investigate the strong materials used in the rest of the paper from which more coupling regime [36–51]. Some non-Gaussian fixed points, information may be found in the Appendix and in the list of reminiscent of phase transitions have been obtained for all references cited above. Among the key results of this the investigated models; which have been pointed out to be section, we get a strong relation between four and two point in strong agreement with the phase condensation at the melonic functions, arising from Ward identity, which can heart of the geometrogenesis scenario [52–54]. More be translated as a local relation between β-function along recently, a series of papers [45–51] took into account the RG flow. In Sec. III we investigate the large rank Ward-Takahashi identities in the renormalization group behavior of the Feynman amplitude; first we provide the equations. Indeed, for the models without closure con- one and two loops computation in order to get the straint, the strong violations of Ward identities for the recurrence relation which could help to a generalization discovered fixed points particularly for marginal quartic at arbitrary n-loops. Then using the recurrence relation on interactions have been checked at the level of just-renor- the perturbative expansion, we derive the same result at all malizables interactions [46]. orders. Explicitly, we show that Feynman amplitudes for

085006-2 LARGE-D BEHAVIOR OF THE FEYNMAN AMPLITUDES FOR … PHYS. REV. D 103, 085006 (2021) two point graphs may be factorize as product of functions, the different choice of them by the subscript i. Note that whose, one depends on the external momentum. In the with our definition of the classical action all these compo- melonic sector, and for large d, each of these amplitudes nents are chosen with the same coupling. The only can be indexed by a planar rooted tree with edges of ðiÞ constraint over V ⃗ ⃗ ⃗ ⃗ comes from the tensoriality different colors rather than one obtained in ordinary p1;p2;p3;p4 criterion, ensuring that any index of a field T have to be Feynman graph. Then, by summing over all such trees, contracted with an index of a field T¯ . In this paper, we focus we get an explicit expression for self energy, from which on the quartic melonic model, for which the set of coupling we can deduce the β-functions. tensors write explicitly as: Y II. PRELIMINARIES VðiÞ ≔ δ δ δ δ : 2 p⃗1;p⃗2;p⃗3;p⃗4 p1ip4i p2ip3i p1jp2j p3jp4j ð Þ In this section we introduce the notations and the j≠i formalism that we will use for the rest of this paper, and All the interactions whose tensor couplings decompose in recall some important definitions and results (additional this way i.e., who do not factorize as product over subsets details could be found in standard references [47]). In a of indices for some T and T¯ , are called bubbles; and the second time, we build explicitly a just-renormalizable couplings defined from (2) are known as quartic melonic family of models for arbitrary rank d, some complementary bubbles. Bubbles may be fruitfully pictured as bipartite results about the proof of renormalizability could be found regular-colored graphs, a representation that we will use in Appendix A. Finally, we discuss the existence of a abundantly in the rest of this paper. The rule to build the nontrivial relation between four and two point functions, correspondence is the following. To each T and T¯ fields we holding to all orders in the perturbative expansion, and show explicitly that the information of the renormalization associate respectively black and white nodes; each of them group flow in the deep ultraviolet version reduces to the being hooked to d colored half edges. These d edges, information of the self-energy at zero momenta and its first corresponding to the d components of the tensor are then derivative. joined following the path provided by the interaction tensor, any half edge of color c starting from a black node being hooked to a half edge of the same color hooked to a A. Just-renormalizable Abelian TGFT in rank d white node. For melonic quartic couplings, we have: TGFT that we consider in this paper describes two fields φ and φ¯ , both defined on d copies of a compact Lie group manifold G: φ; φ¯ ∶ðGÞd → C. For our purpose we focus on the Abelian manifolds, choosing G ¼ Uð1Þ and the ð3Þ fields is then defined on the d-dimensional torus. From the trivial exponential map θ ↦ eiθ ∈ Uð1Þ, rather than functions of group elements, the fields can be understand as functions of the angle variables θ ∈ ½0; 2π½, and we denote ⃗ The interacting part of the classical action being fixed, let as φ θ1; …; θ ≡ φ θ the field arguments (same for the ð dÞ ð Þ us define the kinetic action. Renormalization requires that field φ¯ ). Moreover, instead to focus in the group (or Lie- the kinetic kernel Kðp⃗Þ must have a nontrivial spectrum in Algebra) representation, it is more convenient to use the order to provide a canonical notion of scale. The standard T T¯ Fourier representation, the Fourier components and of choice, motivated by radiative computation in GFTs [35], φ φ¯ tensors d and respectively, being formally of rank i.e., involves the Laplace-Beltrami operator over the group Zd C discrete maps from to . We denote their components as d 1 ¯ d manifold G .ForG ¼ Uð Þ, this Laplace-Beltrami oper- Tp⃗ and Tp⃗, with p⃗¼ðp1; …;pdÞ ∈ Z . At the classical ¯ ator is diagonal in the Fourier representation. Setting the level, tensors are described by the classical action S½T;T, kinetic kernel to be: which is assumed to be quartic for our purpose: X Kðp⃗Þ¼p⃗2 þ m2; ð4Þ ¯ ¯ S½T;T¼ Tp⃗Kðp⃗ÞTp⃗ p⃗ and the corresponding field theory, with quartic-melonic X X ðiÞ ¯ ¯ interactions have been stated to be just-renormalizable for þ λ V T ⃗ T ⃗ T ⃗ T ⃗ ; ð1Þ p⃗1;p⃗2;p⃗3;p⃗4 p1 p2 p3 p4 d ¼ 5 [26]. In this paper, we will relax the power of the i p⃗1;…;p⃗4 Laplacian term, and consider the slight generalization: where λ denotes the coupling constant and VðiÞ the K ⃗ ≔ ⃗2η 2η p⃗1;p⃗2;p⃗3;p⃗4 ðpÞ p þ m ; ð5Þ vertex tensor, i.e., a product of Kronecker deltas which dictates how the tensor indices are contracted together. This where η is chosen to be a positive half-integer, coupling tensor being obviously not unique, we distinguish η ¼ n=2; n ∈ N, and the notation p⃗2η simply means

085006-3 VINCENT LAHOCHE and DINE OUSMANE SAMARY PHYS. REV. D 103, 085006 (2021) P ⃗2η ≔ d 2η p i¼1 jpij . The motivation for such a deformation with respect to the standard choice (4) arises from the power counting which will be discussed in detail below. The choice of η influence strongly the divergent degree, and may be fixed such that the model remains just- renormalizable for arbitrary rank d. Note that we introduced a η-dependent power on the mass term, in hope to get FIG. 1. A typical Feynman graph with three vertices and four external lines. The propagator lines hooked to black and white the same canonical dimension for a just-renormalizable nodes are dotted lines, and open dotted lines are external lines. theory. Finally let us remark that we do not consider the closure constraint in our models, even if it is considered as a crucial ingredient for GFTs. Renormalizability of the quartic melonic models has The statistical theory, introducing integration over been studied extensively, especially for η ¼ 1 and η ¼ 1=2, “thermal” fluctuations is defined from the classical action i.e., for linear and quadratic kinetic kernels [26,28,32].In 1)) by the path integral: particular, just-renormalizability has been proved for η ¼ 1 5 η Z and d ¼ . Here we just fix the parameter in different − ¯ manner, such that the model becomes just-renormalizable Z λ μ ¯ Sint½T;T ð Þ¼ d C½T;Te ; ð6Þ for arbitrary rank d. Such a fixation requires the knowledge of the power counting which takes place by the important where Sint designates the quartic part of the classical action, and useful techniques called multiscale analysis and may be and dμC is the normalized Gaussian measure for the help to establish the solid power-counting theorem. Note propagator C, defined as: that in practice, the presence of the parameter η does not change significantly the main steps of the proofs given in Z Θ Λ2η − ⃗2η the previous references. Some details was reproduced in μ ¯ ¯ ð p Þ δ d C½T;TTp⃗Tp⃗0 ¼ 2η 2η p⃗p⃗0 : ð7Þ Appendix A, and the result is the following: p⃗ þ m Proposition 1: The power counting ωðGÞ for a Feynman graph G with L G internal lines and F G where we introduced the step Heaviside function Θ to ð Þ ð Þ internal faces is given by: prevent UV divergences. Up to the standard permutation of sums and integrals (which in general is not well defined), ωðGÞ¼−2ηLðGÞþFðGÞ: ð9Þ the perturbative expansion in powers of the coupling λ organizes as a sum of amplitudes indexed of Feynman At this step, we choose η such that the leading order graphs, S graphs, such that the 1PI-connected N-point function N i.e., the melonic graphs appear in the renormalization (which depends on N external momenta) writes as: procedure and are just the relevant graphs for the computation of the beta functions in the UV sector. In Appendix A, X G ð−λÞVð N Þ we prove that a sufficient condition is: d>3, ensuring that S ¼ AG ; ð8Þ N G N G sð NÞ for any deviation from the melonic sector, the deleted faces N never compensate the variation over the number of internal where the sum run over the connected graphs with N (dotted) edges. Due to the recursive structure of the melonic external edges, VðG Þ designates the number of melonic graphs, as well recalled in Appendix A, we can prove the N following statement, which link together the number of vertices of the graph GN, and sðGNÞ is a combinatorial factor coming from the Wick theorem.1 Due to the specific internal dotted edges, vertices, and internal faces: combinatorial structure of the interactions, these Feynman FðGÞ¼ðd − 1ÞðLðGÞ − VðGÞþ1Þ: ð10Þ graphs GN are 2 simplex rather than ordinary graphs, i.e., have the sets of vertices, edges and faces. Such a typical Combining this expression with the following topological graph with four external edges is given on Figure 1 on relation arising from the valence of the quartic vertices: which the Wick contractions are pictured as dotted edges between black and white nodes, to whose we attribute the 4VðGÞ¼2LðGÞþNðGÞ; ð11Þ color 0, so that Feynman graphs become d þ 1 bipartite graphs. We recall that faces are bicolored cycles, including where NðGÞ designates the number of external lines, we necessarily the color 0, and may be open or closed, deduce that melonic diagrams diverges as: respectively for external and internal faces. d − 1 ωðGÞ¼½ðd − 1Þ − 4ηV þ ðd − 1Þ − − η N : 1Note that it does not reduce to the dimension of the auto- 2 morphism group of the considered graph, due to the absence of the factor 1=4 in front of λ in the classical action. ð12Þ

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A second relation comes from the first radiative correction for the 4-points function, see Fig. 2 (on right). denote by L2 the loop of length 2 involved on the diagram, we get the relation: ½λ¼2½λþ½L2. Now, observe that, L1 and L2 have the same number of internal faces, i.e., d − 1. There respective scaling then becomes: FIG. 2. Leading order contributions for 1PI 2 and 4 point functions. The figures have been drawn for d ¼ 4. ωðL1Þ¼−2η þðd − 1Þ; ωðL2Þ¼−4η þðd − 1Þ; ð16Þ For a just-renormalizable theory, the divergent degrees must have to be independent of the vertex number2 so that UV- as a result: divergences can be removed from a finite set of counterterms, even if the number of graphs is infinite. This condition ½m2η¼½λþðd − 1Þ − 2η; ½λ¼2½λþðd − 1Þ − 4η; fixes the value of η as: ð17Þ − 1 η d ¼ 4 : ð13Þ leading to:

2η For the standard field theories defined on the space-time, the ½λ¼4η − ðd − 1Þ; ½m ¼2η: ð18Þ just-renormalizability property is closely related to the dimension of the coupling, which has to vanish for just- Note that dimension of m2η is fixed to be 2η, as suggested by renormalizable theories. For TGFTs, there are no meaning to the notations. Moreover, if the theory is renormalizable, talk about dimension, because there are no space-time ½λ¼0, as expected from standard quantum field theory. background, and the sums over Zd are dimensionless. An Then we come to the following definition which ends intrinsic notion of dimension however emerges from the this section renormalization group flow itself, following the behavior of Definition 1 Boundary and heart vertices and faces: the renormalization group trajectories. In the vicinity of the (i) Any vertex hooked with an external edges is said to Gaussian point, the canonical dimension is then fixed from be a boundary vertex. Other vertices are called heart the behavior of the leading order Feynman amplitudes—the vertices dimension being fixed from the scaling of the leading order (ii) Any external faces running through a single external quantum corrections with respect to some UV cutoff. This vertex is said to be an boundary external faces. notion being of a great interest for the rest of this paper and we (iii) Any external faces running through at least one heart provide here a brief explanation of its origin. As an vertex is said to be an heart external face. illustration, let us consider the first quantum corrections for the mass parameter, which provides from the diagram B. Exact relation between effective (melonic) pictured on Fig. 2 (on left) below. If we denote by L1 the loop vertex and wave function involved on the diagram, the mass correction takes the form: Because of their recursive definition, there exist strong 2η relations between melonic diagrams with two, four or δm ¼ λK1L1; ð14Þ arbitrary number of external edges, such that the melonic where K1 is a numerical (cutoff independent) factor. sector is entirely determined by the knowledge of the Denoting by [x] the dimension of the quantity x,weget melonic self-energy. The aim of this section is to establish the first relation: the exact relation holding between two and four point functions, and the corresponding relations between 2η ½m ¼½λþ½L1: ð15Þ counter-terms. The melonic self energy Σðp⃗Þ, i.e., whose perturbative expansion keep only the melonic diagrams, is related to the 2 If the divergent degree decrease with the number of vertices, two point function Γð2Þðp⃗Þ as in ordinary field theory: the situation is still interesting, because it means that there are only a finite set of divergent graphs, which could be subtracted to 2 2η 2η rend the theory finite. This corresponds to a superrenormalizable Γð Þðp⃗Þ¼p⃗ þ m − Σðp⃗Þ: ð19Þ theory. In the other hand, the divergent degree increases with the number of vertices, and an infinite number of counterterms is Moreover, as a direct consequence of the recursive defi- required to make the theory well defined in the UV. Fixing an infinite number of counterterms, or equivalently an infinite nition of the melonic diagrams, the melonic self energy number of “initial conditions” break the predictivity of the obey to a closed equation, which as we announced in the theory, which is said to be nonrenormalizable. introduction is reputed to be very difficult to solve. The

085006-5 VINCENT LAHOCHE and DINE OUSMANE SAMARY PHYS. REV. D 103, 085006 (2021) proof of this closed equations and the main corollary ¯ ¯ U½T … ≔ ½U1 ½U2 ½U Tq ;q ;…;q ; statements can be found in [34,55,63]. To summarize: p1;p2; ;pd p1q1 p2q2 d pdqd 1 2 d Proposition 2 Closed equation for self-energy: Let ð26Þ Σðp⃗Þ be the melonic self energy, whose Feynman expan- P ¯ sion involves only melonic diagrams. Then, all the varia- where means complex conjugation. Obviously, p⃗Tp⃗Tp⃗ bles are completely decoupled and Σðp⃗Þ is a sum of d and any higher valence tensorial interactions are invariant independent terms, one per variables: under any such transformations. Then:

Xd U ½Sint¼Sint: ð27Þ Σðp⃗Þ ≕ τðpiÞ; ð20Þ i¼1 However, this is not the case for the kinetic term, due to the where the function τ∶Z → R has a single argument, and nontrivial propagator, which explicitly break the unitary invariance. Now, let us consider the two point function satisfies the closed equation: ¯ hTp⃗Tq⃗i. It is tempting to think that it transform like a X Θ Λ2η − ⃗2η representation of U ⊗ U. Indeed, even if the kinetic term τ ≔−2λ δ ð Pq Þ ðpÞ pq1 2η 2η : ð21Þ does not transform like a tensorial invariant, the integral: q⃗ þ m − d τðq Þ q⃗ i¼1 i Z ¯ ¯ −S ½T;T¯ hT ⃗T ⃗i ≔ dμ T ⃗T ⃗e int ; ð28Þ Note that τðpÞ only depends on p2. From the power p q C p q counting theorem, only the two and four point melonic diagrams diverge, and then require renormalization. does not depend on the broken symmetry transformation Moreover, in the deep UV limit, the knowledge of the of the kinetic term because of the formal translation invari- counterterms allows to compute the beta functions. As we ance of the Lebesgue integration measure, and in fact it has to be invariant under any unitary transformation. will see, the unitary symmetry of the action, explicitly ¯ broken by the kinetic kernel imply the existence of a strong Furthermore, hTp⃗Tq⃗i transforms like a trivial representation relation between four and two point functions through the of U×d ⊗ U×d. This can be translated in an infinitesimal standard Ward-Takahashi identity. More precisely, we have point of view considering an infinitesimal transformation † the following statement: U ¼ I þ iϵ, where ϵ ¼ ϵ is a Hermitian operator and I the Proposition 3 Zero-momenta Ward identity: Let γð4Þ ≔ identity operator. At the first order in ϵ,weget: 4 X Γð Þ be the zero momenta 1PI melonic 4-point function. 0⃗;0⃗;0⃗;0⃗ U ¼ I þ ϵ⃗; ð29Þ 4 i In the continuum limit, γð Þ is related to the first derivative i Γð2Þ ⃗ of the 2-point melonic function ðpÞ as: ⊗d ⊗ði−1Þ d−iþ1 where I ≔ I and ϵ⃗i ¼ I ⊗ ϵi ⊗ I . Then, the ¯ ¯ 1 invariance of hTp⃗Tq⃗i simply means that ϵ⃗i½hTp⃗Tq⃗i ¼ 0. γð4ÞLð1 þ ∂τÞ¼τ0ð0Þ; ð22Þ ϵ 2 Expanding this relation at the leading order in i, and due to ϵ⃗ 0 the symmetry i½Sint¼ , we get: τ0 0 ≔ ∂τ ∂ 2η L Z Z with the notation ð Þ = p1 jp1¼0, and the loop as ¯ −S ½T;T¯ ¯ −S ½T;T¯ the “boundary contribution” ∂τ are defined as: ϵ⃗i½dμCTp⃗Tq⃗e int þ dμCϵ⃗i½Tp⃗Tq⃗e int ¼ 0: X L ≔ δ Γð2Þ ⃗ −2 ð30Þ q10½ ðqÞ ; ð23Þ ⃗ q Each terms can be computed separately. The variation of the X 2η 2η ð2Þ −2 2η 2η covariance requires to be carefully derived, because the L∂τ ≔ δ 0ðq⃗ þ m Þ½Γ ðq⃗Þ δðΛ − q⃗ Þ: ð24Þ q1 propagator C is not invertible on Zd due to the Θ-function q⃗ ΘðΛ2η − p⃗2ηÞ. The computation of thevariation then requires 1 Θ Proof.—Let us consider the unitary transformations regularization of the infinite coming from = . The variation U ∈ U×d acting independently over each components of of the second term however can be computed straightforwardly. From: the tensors T and T¯ . U is a d-dimensional vector U ¼ Y Y U1;U2; …;U whose components U are unitary matri- ð dÞ i ¯ ¯ ¯ ϵ⃗½T ⃗T ⃗¼−ϵ 0 T ⃗0 δ 0 T ⃗þ T ⃗ϵ 0 T ⃗0 δ 0 ces acting on the indices of color i. The action of U on the i p q pip p pjpj q p qiqi q qjqj i ≠ ≠ two tensors is defined as (we sum over repeated indices): Yj i Y j i ¯ ¯ ¼ ϵ 0 T ⃗ δ 0 T ⃗0 − ϵ 0 T ⃗0 δ 0 T ⃗ qiqi p qjqj q pipi p pjpj q U T ≔ U1 U2 U T … j≠i j≠i ½ p1;p2;…;pd ½ p1q1 ½ p2q2 ½ dpdqd q1;q2; ;qd ¯ ¯ 0 ϵ 0 − 0 ϵ 0 ¼ Tp⃗Tq⃗⊥ ∪ q q q Tp⃗⊥ ∪ p Tq⃗ p p ; ð31Þ ð25Þ i f ig i i i f ig i i

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⃗ ≔ ⃗ ∈ Zd−1 occurring on the propagator. We use the well know relation where p⊥i p=fpig . Integrating with the measure − ¯ θ0 δ dμ e Sint½T;T, and after restrict our computation on the between Heaviside and Dirac function: ¼ , and the C ¯ Gaussian representation of the finite range δ-function: perturbative sector, it is obvious that hTp⃗Tq⃗i ∝ δp⃗q⃗ due to the momentum conservation along all the external faces. 1 2 2 Then, setting: δ ðxÞ ≔ pffiffiffi e−x =a ; ð34Þ a a π ¯ hTp⃗Tq⃗i¼Gðp⃗Þδp⃗q⃗; ð32Þ which goes to the standard Dirac function3 when a → 0. we get: Then we get the following limit: Z Z x − ¯ 1 − 2 2 μ ϵ⃗ ¯ Sint½T;T δ ⃗ − ⃗ ϵ θ ≔ ffiffiffi y =a θ Θ d C i½Tp⃗Tq⃗e ¼ p⃗⊥ q⃗⊥ ½GðpÞ GðqÞ qipi : aðxÞ p e dy; lim a ¼ : ð35Þ i i a π −∞ a→0 ð33Þ −1 2 2 Therefore, defining C0 ðp⃗Þ ≔ p⃗ þ m , our regularized ⃗ θ Λ2 − ⃗2 ⃗ Now let us focus on to the variation of the measure dμC.As propagator can be written CaðpÞ¼ að p ÞC0ðpÞ such explained before, we have to regularized the Θ-function that the Gaussian measure and its variation are written as:

P ¯ −1 X − T ⃗C ðp⃗ÞT ⃗ −1 μ ≔ p⃗ p a p ϵ⃗ μ −ϵ⃗ ¯ ⃗ μ d Ca e ; i½d Ca ¼ i Tp⃗Ca ðpÞTp⃗ d Ca : ð36Þ p⃗

The variation of the kinetic term then becomes: X X ϵ⃗ ¯ −1 ⃗ ϵ δ −1 ⃗ − −1 ⃗ ¯ i Tp⃗Ca ðpÞTp⃗ ¼ q p p⃗⊥ q⃗⊥ ½Ca ðpÞ Ca ðqÞTp⃗Tq⃗: ð37Þ i i i i p⃗ p;⃗q⃗

By considering the results (33) and (37), we get: X ϵ δ −1 ⃗ − −1 ⃗ ¯ ¯ r s r⃗⊥ s⃗⊥ ½Ca ðrÞ Ca ðsÞhTr⃗Ts⃗Tp⃗Tq⃗i i i i i ⃗⃗ r;s X δ ⃗ − ⃗ δ δ ϵ ¼ ½ p⃗⊥ q⃗ ðGðpÞ GðqÞÞ r q s p r s ; i ⊥i i i i i i i ri;si or, because of the arbitrariness of the infinitesimal transformation ϵ: X δ −1 ⃗ − −1 ⃗ ¯ ¯ δ ⃗ − ⃗ δ δ r⃗⊥ s⃗⊥ ½Ca ðrÞ Ca ðsÞhTr⃗Ts⃗Tp⃗Tq⃗i¼½ p⃗⊥ q⃗ ðGðpÞ GðqÞÞ r q s p : ð38Þ i i i ⊥i i i i i r⃗⊥ ;s⃗⊥ i i

4 Let Γð Þ be the 1PI four points function defined by the following relation p⃗1;p⃗2;p⃗3;p⃗4

¯ ¯ ≕ −Γð4Þ ⃗ ⃗ δ δ ⃗ ⃗ hTr⃗Ts⃗Tp⃗Tq⃗i ð r;⃗s;⃗p;⃗q⃗GðpÞGðqÞþ r⃗p⃗ s⃗q⃗ÞGðrÞGðsÞ: ð39Þ

Expression (38) becomes:

X δ −1 ⃗ − −1 ⃗ ⃗ ⃗ −Γð4Þ Γð2Þ ⃗Γð2Þ ⃗δ δ r⃗⊥ s⃗⊥ ½Ca ðrÞ Ca ðsÞGðrÞGðsÞ½ ⃗⃗⃗⃗þ ðpÞ ðqÞ r⃗p⃗ s⃗q⃗ i i r;s;p;q r⃗⊥ ;s⃗⊥ i i δ Γð2Þ ⃗ − Γð2Þ ⃗ δ δ ¼½ p⃗⊥ q⃗⊥ ð ðqÞ ðpÞÞ r q s p ; ð40Þ i i i i i i

3More precisely, it goes to the delta distribution on the space of test functions DðRÞ.

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Γð2Þ ⃗ ≔ 1 ⃗ θ Λ2η − ⃗2η with ðpÞ =GðpÞ. Using the proposition 8 in the ⃗ að p Þ GðpÞ¼ 2η 2η 2η 2η ; ð46Þ Appendix A and taking into account the leading order p⃗ þ m − θaðΛ − p⃗ ÞΣðp⃗Þ Γð4Þ contributions, r;⃗s;⃗p;⃗q⃗ is such that any diagrams in its Feynman expansion has two heart external faces of the and we get same color, running through the interior of the graph. As a Γð2Þ ⃗ θ−1 Λ2η − ⃗2η ⃗2η 2η − Σ ⃗ consequence, the leading contributions for Γð4Þ may be ðpÞ¼ a ð p Þðp þ m Þ ðpÞ: ð47Þ decomposed a sum indexed by a single color like the free energy Σ: We deduce that

∂Γð2Þ −1 ∂ Xd 0⃗ − dCa 0⃗ − Σ 0⃗ Γð4Þ ≔ Γð4Þ;i; ð41Þ 2η ð Þ 2η ð Þ¼ 2η ð Þ: ð48Þ ∂p1 dp1 ∂p1 i¼1 θ2 2 Moreover, from the same proposition 8, in addition to these Finally, taking into account the factor a coming from G , two heart external faces, we have (d − 1) boundary external and by choosing a to 0, the equation (43) writes as: faces of length 1 per external vertices (in the case when we 1 ∂ have only one vertex, it can be considered like an external L 1 ∂τ γð4Þ Σ 0⃗ 2 ð þ Þ × ¼ 2η ð Þ: ð49Þ vertex because external lines are hooked to him). Then, a ∂p1 moment of reflection show that the leading order 4-point P Σ ⃗ τ function must have the following structure: With the decomposition ðpÞ¼ i ðpiÞ the proof of the proposition is therefore completed. ▪ Γð4Þ;i γð4Þ VðiÞ ⃗ ↔ ⃗ Corollary 1 Exact relation between τ and γð4Þ: The ⃗ ⃗ ⃗ ⃗ ¼ p1ip3i ð ⃗ ⃗ ⃗ ⃗ þ p1 p3Þ p1;p2;p3;p4 p1;p2;p3;p4 zero-momenta melonic function γð4Þ and the loop L are 4 ≕ γð Þ SymVðiÞ ;; 42 p1ip3i p⃗1;p⃗2;p⃗3;p⃗4 ð Þ related as: 4λ where the last term comes from the Wick theorem. As a 4 γð Þ ¼ : ð50Þ result, only the component ΓðiÞ contributes significantly to 1 þ 2λL VðiÞ (40) at leading order. Also only a single term in Sym — have to be retained. Then setting q ¼ r and p ¼ s in a Proof. The proof is straightforward. From the closed i i i i equation for self-energy (21), we deduce an expression for p⃗ q⃗→ 0⃗ first time, and ¼ in a second time, (40) becomes at τ0 involving L: leading order: 2λLð1 þ ∂τÞ 1 X dC−1 τ0 0 2λL 1 − τ0 0 ∂τ → τ0 0 ; δ a ⃗ 2 ⃗ γð4Þ ð Þ¼ ð ð Þþ Þ ð Þ¼ 1 2λL r10 2η ðrÞG ðrÞ × þ 2 dr r⃗ 1 ð51Þ ∂ dC−1 − Γð2Þ 0⃗ a 0⃗ ¼ 2η ð Þþ 2η ð Þ: ð43Þ ∂p1 dp1 Then, inserting this equation in Eq. (22) of proposition 3, and after simplification of the factors ð1 þ ∂τÞ, we deduce 4 ð4Þ the corollary. ▪ where γð Þ ¼ 2γ . From the definition: 00 Now let us defined the functional action with the −1 0 counterterms which will be free for divergences. dCa 2η 2η θa 2η 2η −1 ⃗ 1 ⃗ Λ − ⃗ θ Denoting as Z, Z and Zλ the counterterms respectively 2η ðrÞ¼ þðr þ m Þ θ ð r Þ a m dr1 a for field strength, mass and coupling, such that the × ðΛ2η − r⃗2ηÞ: ð44Þ renormalized classical action, writing as: X ¯ ¯ 2η 2η The derivative on the right hand side requires the explicit S½T;T¼ Tp⃗ðZp⃗ þ Zmm ÞTp⃗ Γð2Þ p⃗ expression for . The effective propagator G is obtained X X from C and Σ as a geometric progression: ðiÞ ¯ ¯ a Zλλ V T ⃗ T ⃗ T ⃗ T ⃗ ; 52 þ p⃗1;p⃗2;p⃗3;p⃗4 p1 p2 p3 p4 ð Þ 1 i p⃗1;…;p⃗4 Σ Σ Σ G ¼ Ca þ Ca Ca þ Ca Ca Ca þ¼1 − Σ Ca; Ca The existence of such a set of counterterm is ensured by the ð45Þ renormalizability theorem. An other point of view is the behavior of effective vertex with the UV cutoff. To be more γð4Þ λ explicitly: precise, can be interpreted as an effective coupling eff :

085006-8 LARGE-D BEHAVIOR OF THE FEYNMAN AMPLITUDES FOR … PHYS. REV. D 103, 085006 (2021) X 1 1 2η 2η λ ¼ zλλ;zλ ≔ ; ð53Þ L∂τ ¼ δ 0δðΛ − q⃗ Þ: ð59Þ eff 1 þ 2λL Λ2η m2η q1 þ q⃗ Moreover, the relation between τð0Þ, τ0ð0Þ and the effective In the deep UV sector on which we focus in this paper, Λ mass and wave functions can be easily found from the ð2Þ becomes large, and the continuum limit can be considered definition of Γ . We have: Λ with variables xi ¼ qi= . We get: Z Γð2Þðp⃗Þ¼p⃗2η þ m2η − Σðp⃗Þ¼ð1 − τ0ð0ÞÞp⃗2η X δ δ Λ2η − ⃗2η ≈ 2d−1 xδ 1 − x2η 2η 2η q10 ð q Þ d ð Þ; ð60Þ − τ 0 O ⃗ Rþd−1 þðm d × ð ÞÞ þ ðp Þ; ð54Þ q⃗ from which we deduce the effective wave function Zeff and 2 and from Appendix B, the effective mass meff: L∂τ ≈ {ðdÞ 2λL 1 ∂τ 2η ; ð61Þ ≔ 1 − τ0 0 1 − ð þ Þ 1 − 2λL∂τ 1 þ m¯ Zeff ð Þ¼ 1 − 2λL ¼ zλð Þ; 2η ¯ Λ m ≔ m2η − d × τð0Þ: ð55Þ where we defined the dimensionless mass m ¼ m= such eff that: ∂τ One may expect that the boundary term introduce a {ðdÞ spurious dependence on the UV regularization, especially Z ¼ zλ 1 − 2λ : ð62Þ eff 1 þ m¯ 2η in regard to the limit a → 0. Indeed, the product of two distribution is not well defined in general. This is especially C. Melonic renormalization group equations the case of the product δΘ; and we have to be careful when we take the limit before of after the computation of the The renormalization group equations (RGE) are the integrals. In this case however, the limit can be well defined infinitesimal translation of a common feature of just- remembering that momenta are discrete variables, and that renormalizable theories. In the deep UV, and neglecting all the derivatives are formal continuum limit of finite the contributions of inessential couplings, any change of differences. Thus, taking the limit a → 0 before the fundamental cutoff, Λ → Λ0 may be exactly compensated continuum limit, we have to provide a sense for integrals by a change of field strength, relevant and marginal R — 1 Λ of the form − dxδð1 − xÞGðΘð1 − xÞÞ; which is the limit couplings up to corrections of order = . The infinitesi- for ϵ ∼ 1=Λ2η → 0 mal incarnation of this feature is the so-called C-S equation, of something that: – Z which writes as [64 67]: Θ 1 ϵ − − Θ 1 − Θ 1 − ∂ ∂ ∂ dx½ ð þ xÞ ð xÞGð ð xÞÞ ð56Þ β β − N γ ΓðNÞ 0 ∀ þ þ m 2η ⃗ ⃗ … ⃗ ¼ ; N; ∂t ∂λ ∂m 2 p1;p2; ;pN for some regular function GðyÞ. In the interval ð63Þ x ∈ ½1; 1 þ ϵ, Θð1 − xÞ must vanish. Therefore: Z Where ∂=∂t ≔ Λ∂=∂Λ; β and βm are beta functions for dx½Θð1 þ ϵ − xÞ − Θð1 − xÞGðΘð1 − xÞÞ quartic coupling and mass, and γ is the anomalous Z dimension. By considering the explicitly expression of ð2Þ ⃗ ð2Þ 2 ⃗ ð4Þ ð4Þ → ϵ 0 δ 1 − Γ ð0Þ, ∂Γ =∂p1ð0Þ and Γ ≡ γ , and taking into Gð Þ dx ð xÞ: ð57Þ 0⃗;0⃗;0⃗;0⃗ account the strong relation arising from the Ward identity, Note that the result depends on the convention used to we deduce the statement: compute the derivative. Indeed, with the convention Proposition 4: In the deep UV limit (Λ ≫ 1), and with Θð1 − xÞ − Θð1 − ϵ − xÞ—which is identical for ordinary boundary term given by equation (61), the β-functions β, β γ functions; we get the limit: m for coupling and mass; and the anomalous dimension Z are related as: dx Θ 1 − x − Θ 1 − ϵ − x G Θ 1 − x 2 ½ ð Þ ð Þ ð ð ÞÞ 2λ{ðdÞ 2λ {ðdÞ Z β ¼ γλ 1 − þ β ¯ ; ð64Þ 1 þ m¯ 2η ð1 þ m¯ 2ηÞ2 m → ϵGð1Þ dxδð1 − xÞ: ð58Þ which is valid in the interior of the region connected to the In this paper, we use of the first convention, because it offer Gaussian fixed point, below the singularity line of equation: the advantage to cancel the Σ dependence of the denom- 1 ¯ 2η − 2λ 0 inator of the boundary term, which simplify as: þ m {ðdÞ¼ ; ð65Þ

085006-9 VINCENT LAHOCHE and DINE OUSMANE SAMARY PHYS. REV. D 103, 085006 (2021) β 2λ ∂ ∂ ∂ λ and where we introduced the -function for dimensionless −λγ β {ðdÞ β β 0 ¯ 2η ≔ 2ηΛ−2η Λ2ηβ ≔ β − 2η 2η þ þ þ þ m 2η 2η ¼ : mass m m , i.e., m¯ m m . 1 − 2λL∂τ ∂t ∂λ ∂m 1 þ m¯ ð2Þ ⃗ ð2Þ 2η ⃗ Proof.—From RGE for Γ ð0Þ, ∂Γ =∂p1 ð0Þ and 4 ð71Þ Γð Þ ≡ γð4Þ, and taking into account the relations 0⃗;0⃗;0⃗;0⃗ coming from Ward identities, we deduce the following Then after few simplifications, relations: 2λ{ðdÞ λ −γλ β β − β ¯ 0; ∂ ∂ ∂ þ þ 1 ¯ 2η − 2λ 1 ¯ 2η m ¼ β β − γ 2η − τ 0 0 þ m {ðdÞ þ m ∂ þ ∂λ þ m ∂ 2η ðm d × ð ÞÞ ¼ ð66Þ t m ð72Þ ∂ ∂ ∂ 2η β β − γ 1 − τ0 0 0 we then deduce the proposition assuming that 1 þ m¯ − þ þ m 2η ð ð ÞÞ ¼ ð67Þ ∂t ∂λ ∂m 2λ{ðdÞ ≠ 0 and m¯ 2η ≠−1. ▪ As direct consequences of this statement, and from ∂ ∂ ∂ β β − 2γ λ 0 inspection of the Eq. (72), we have: þ þ m 2η Zλ ¼ : ð68Þ ∂t ∂λ ∂m Corollary 2: In the deep UV limit, and in the melonic sector, any fixed point β ¼ βm ¼ 0 have to satisfy (at least) The two first equations are explicitly written as (we use the one of the two conditions: τ0 τ0 0 τ τ 0 notation for ð Þ and for ð Þ): (1) λ ¼ 0, (2) γ ¼ 0. ∂τ ∂τ ∂τ β − β 1 − γ 2η − τ 0 The first one corresponds to the Gaussian fixed point, d þ d m d 2η þ ðm d × Þ¼ ; ∂t ∂λ ∂m and has no real interest at this stage. We then expect that ð69Þ only the second one will be of relevant interest for non- Gaussian fixed point investigations. Moreover, from ∂τ0 ∂τ0 ∂τ0 Eq. (64), substituting β and solving the resulting equations þ β þ β þ γð1 − τ0Þ¼0: ð70Þ β γ ∂t ∂λ m ∂m2η for m and , we get: Corollary 3: The β-function for mass and the anoma- 0 0 2η For the third equations, we use the fact that Zλ ¼ð1 − τ Þ= lous dimensions be express only in terms of τ, τ , λ, m and ð1 − 2λL∂τÞ, from which it follows: Λ as:

2η 2η 2λ{ðdÞ ∂τ 2η m − dτ − γ m − d τ − λ 1 − 2η ð Þ ð ð ð 1þm¯ Þ ∂λÞÞ β ¯ ¼ − 2 ; ð73Þ m 2η 2η ∂τ 2λ {ðdÞ ∂τ Λ − d Λ 2η 2η ð ∂m þ 1þm¯ ∂λÞ

2ηðm2η − dτÞΩ γ ¼ ; ð74Þ 0 2λ{ðdÞ ∂τ0 2η 2λ{ðdÞ ∂τ 1 − τ λ 1 − 2η m − d τ − λ 1 − 2η Ω þ ð 1þm¯ Þ ∂λ þð ð ð 1þm¯ Þ ∂λÞÞ with: and then extend our results to all orders of the perturbative expansion, for two point and vacuum amplitudes. In a Λ2η ∂τ0 2λ2{ðdÞ ∂τ0 second time, we build an exact renormalization group ∂m2η þ 1þm¯ 2η ∂λ Ω ≔ 2 ð75Þ equation and show that no fixed point may be found in this 2η 2η ∂τ 2λ {ðdÞ ∂τ Λ − d Λ 2η 2η ð ∂m þ 1þm¯ ∂λÞ large d-limit. The result of this section therefore solve the closed equation of the two point correlation function at The set of three equations, (64), (73) and (74) shows large rank limit exploration. explicitly that the knowledge of τ and τ0 determine entirely the behavior of the RG flow in the deep UV. A. One and two-loops investigation

III. LARGE d BEHAVIOR OF THE 1. One-loop computation FEYNMAN AMPLITUDES A typical leading order contribution to the one-loop 1PI — In this section we investigate the large rank behavior of two point function has been drawn on Fig. 2 on left. Note the melonic Feynman amplitudes. We start with the that there are two Wick-contractions for this configuration, heuristic computation of the relevant quantities τ and τ0, meaning that for each melonic vertex, the contribution of

085006-10 LARGE-D BEHAVIOR OF THE FEYNMAN AMPLITUDES FOR … PHYS. REV. D 103, 085006 (2021) the diagram have to be counted twice. From the Feynman rules, we then deduce the one-loop self energy as: ð81Þ X 2η 2η 2η ΘðΛ − q⃗ − p Þ τð1Þðp Þ¼ i : ð76Þ i ⃗2η 2η 2η q⃗∈Zd−1 q þ pi þ m τ 0 τ0 0 where the subscript (1) refers to the number of loops. Due We recall that we need to compute only ð Þ and ð Þ to to the large Λ limit, we can simplify the computation taking build the renormalization group flow in the deep UV. From the continuum limit, and replacing the sum with an integral, the previous diagram, it is quite natural to split the without consequences on the leading order contributions. computation as the sum of two distinct contributions: We introduce the continuous variables x ≔ p =Λ. Then i i τð2Þ τð2Þk − 1 τð2Þ⊥ Eq. (76) becomes: ðpÞ¼ ðpÞþðd Þ ðpÞ: ð82Þ Z Θ 1 − x⃗2η τð2Þk ð1Þ d−1−2η ð Þ The first term, that we denoted as ðpÞ corresponds to τ Λx −2λΛ dxδΛ x1 − x ; ð Þ¼ ð Þ 2η 2η 2η the configuration where the two vertices are the same (same x⃗⊥ þ x þ m¯ color); and the second one corresponds to the case where ð77Þ the two vertices are different (different colors). We will compute each terms separately. From the Feynman rules, it where we introduced the dimensionless mass m¯ 2η ¼ 2η 2η follows: m =Λ , the δ-distribution of size 1=Λ: δΛðx1 − xÞ ≔ Q X Λδ , dx ≔ dx , and x⊥ ≡ x⊥1 ¼ðx2; …;x Þ. 2 2 1 pq1 i i d τð ÞkðpÞ¼s λ × τ τ ≔ τ Λ Λ2η k 2η 2η 2η 2 Defining the continuous function as ðxÞ ð xÞ= , ⃗ ðq⃗⊥ þ p þ m Þ q⃗ ;k⊥∈D p and because d − 1 − 2η ¼ 2η, we get finally: ⊥ ð Þ 1 Z × ; ð83Þ Θð1 − x⃗2ηÞ ⃗2η 2η 2η τð1Þ −2λ x k⊥ þ p þ m ðxÞ¼ d ⊥ 2η 2η 2η ; ð78Þ x⃗⊥ þ x þ m¯ where sk is a symmetry factor and where we have formally took the Λ → ∞ limit. This d−1 2η 2η 2η integral may be computed from the results given in DðpÞ ≔ fq⃗⊥ ∈ Z jq⃗⊥ ≤ Λ − p g: Appendix B, and we deduce the explicit formula: The symmetry factor receives two contributions. First we −1 2η d þ 1 d have a factor 1=2! coming from the expansion of the τð1Þðp Þ¼−2dλðΛ2η − p Þ Γ i i d − 1 exponential. Second, the number of allowed contractions 2η leading to such a melonic diagram can be given by the m2η þ p Λ2η þ m2η × 1 − i ln ; following. There are a first factor 2 coming from the choice Λ2η − 2η 2η 2η pi m þ pi of the vertex on which the external edges are hooked, and a second factor 2 coming from the orientation of the vertex. From which we get: Finally, a third factor 2 arise from the orientation of the second vertex (there are two different ways to hook this d þ 1 d−1 Σð1Þðp⃗¼ 0⃗Þ¼−2ddλΛ2η Γ vertex to the first one, and a single possibility to create the d − 1 last internal line of length one). As a result: 1 þ m¯ 2η × 1 − m¯ 2η ln ; ð79Þ 1 m¯ 2η 2 2 2 4 sk ¼ 2! × × × ¼ : ð84Þ and: At this stage we use the sums S1 and S2 defined in the 2 d þ 1 d−1 1 þ m¯ 2η Appendix B. Indeed, τð ÞkðpÞ can be factorized in two τð1Þ0ð0Þ¼2dλ Γ ð1 þ m¯ 2ηÞ ln : d − 1 m¯ 2η contributions corresponding to the two submelonic diagrams with two and four external points: ð80Þ X 1 τð2ÞkðpÞ¼4λ2 ⃗2η 2η 2η 2 2. Two-loops computation q⃗⊥∈D p ðq⊥ þ p þ m Þ ð Þ We now move on to the two loops computation of the X 1 2 × ; self energy τð ÞðpÞ. At the leading order in the deep UV, ⃗2η 2η 2η ⃗ k⊥ þ p þ m there are only one relevant diagram, which is k⊥∈DðpÞ

085006-11 VINCENT LAHOCHE and DINE OUSMANE SAMARY PHYS. REV. D 103, 085006 (2021)

Then using the sums, (B7) and (B10) in the Appendix B we get m2η þ p2η Λ2η þ m2η τð2ÞkðpÞ¼4½{ðdÞ2ðΛ2η − p2ηÞλ2 1 − ln Λ2η − p2η m2η þ p2η Λ2η þ m2η Λ2η − p2η × ln − : ð85Þ m2η þ p2η Λ2η þ m2η

The computation of the quantity τð2Þ⊥ may be given easily, due to the overlapped momentum between the two loops of the diagram. We get: X 1 1 ð2Þ⊥ 2 τ p s⊥λ × : 86 ð Þ¼ 2η 2η 2η 2 ⃗2η 2η 2η ð Þ ⃗ ðq⃗⊥ þ p þ m Þ k⊥ þ q2 þ m q⃗⊥∈DðpÞ;k⊥∈Dðq2Þ

By considering the Eq. (B5) and for simplicity, we will compute separately τð2Þ⊥ð0Þ and τð2Þ⊥0ðpÞ. We get, in the continuum limit: Z ð2Þ⊥ 2 2η ⃗ 2η ⃗2η 2η τ ð0Þ¼2s⊥λ Λ dq⃗⊥dk⊥Θð1 − q⃗⊥ ÞΘð1 − k⊥ − q2 Þ Z 1 u1δð1 − u1 − u2Þ × du1du2 2η 2η 2η ; ð87Þ 0 ⃗ ⃗ ¯ 2η 3 ðu1q⊥0 þ u2k⊥ þ q2 þ m Þ where we kept the notation q and k for continuous variables. Now we introduce the integral representation of the Heaviside Θ-functions, leading to: Z Z 1 ð2Þ⊥ 2 2η ⃗ 2η ⃗2η 2η τ ð0Þ¼2s⊥λ Λ dq⃗⊥dk⊥ dy1dy2δðy1 − q⃗⊥ Þδðy2 − k⊥ − q2 Þ Z 0 1 u1δð1 − u1 − u2Þ × du1du2 2η 2η 2η : ð88Þ 0 ⃗ ⃗ ¯ 2η 3 ðu1q⊥0 þ u2k⊥ þ q2 þ m Þ

Due to the properties of the δ-distribution this relation takes the simple form: Z Z 1 ð2Þ⊥ 2 2η ⃗ 2η ⃗2η 2η τ ð0Þ¼2s⊥λ Λ dq⃗⊥dk⊥ dy1dy2δðy1 − q⃗⊥ Þδðy2 − k⊥ − q2 Þ Z 0 1 u1δð1 − u1 − u2Þ × du1du2 2η 3 : ð89Þ 0 ðu1y1 þ u2y2 þ m¯ Þ

2η 2η ⃗2η 2η ⃗2η ð2Þ⊥ We make the change of variables: q⃗⊥ → y1q⃗⊥ , and k⊥ → ðy2 − y1q2 Þk⊥ ; splitting τ ð0Þ into two contributions:

ð2Þ⊥ 2 2η τ ð0Þ¼2s⊥λ Λ ½L1ðdÞ − L2ðdÞ; ð90Þ where L1ðdÞ and L2ðdÞ are defined as: Z Z 1 ⃗ 2η ⃗2η L1ðdÞ ≔ dq⃗⊥dk⊥δð1 − q⃗⊥ Þδð1 − k⊥ Þ × du1du2u1δð1 − u1 − u2Þ Z 0 1 y1y2 × dy1dy2 2η 3 ; ð91Þ 0 ðu1y1 þ u2y2 þ m¯ Þ Z Z 1 ⃗ 2η ⃗2η L2ðdÞ ≔ dq⃗⊥dk⊥δð1 − q⃗⊥ Þδð1 − k⊥ Þ × du1du2u1δð1 − u1 − u2Þ 0 Z 1 2 2η y1q2 × dy1dy2 2η 3 : ð92Þ 0 ðu1y1 þ u2y2 þ m¯ Þ

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Note that the role playing by the variable q2 is arbitrary. Then, we can sum over all choices of them, and finally dividing the result by d − 1 we get: Z Z 1 1 ⃗ 2η ⃗2η L2ðdÞ ≔ dq⃗⊥dk⊥δð1 − q⃗⊥ Þδð1 − k⊥ Þ × du1du2u1δð1 − u1 − u2Þ d − 1 0 Z 1 2 y1 × dy1dy2 2η 3 : 0 ðu1y1 þ u2y2 þ m¯ Þ

Interestingly, the d-dependence of the two loops integrals L1 and L2 can be factorized, (the same phenomena can be observed at one loop, as shown in the Appendix B). Moreover this factorization is a consequence of the role plays by our deformation parameter η. Then we choose η such that the theory remains just-renormalizable in any dimensions, i.e., the loop structure remains the same in any dimensions. Finally τð2Þ⊥ð0Þ takes the form: 1 ð2Þ⊥ 2 2 2η τ ð0Þ¼2s⊥λ ½{ðdÞ Λ R1 − R2 ; ð93Þ d − 1 where: Z Z 1 1 δ 1 − − y1y2 R1 ¼ du1du2u1 ð u1 u2Þ × dy1dy2 2η 3 ; ð94Þ 0 0 ðu1y1 þ u2y2 þ m¯ Þ and: Z Z 1 1 2 δ 1 − − y1 R2 ¼ du1du2u1 ð u1 u2Þ × dy1dy2 2η 3 : ð95Þ 0 0 ðu1y1 þ u2y2 þ m¯ Þ

The first integral may be straightforwardly computed: R1 is nothing but the same contribution in the final expression of τð2Þkðp ¼ 0Þ given in (85): 1 m2η Λ2η þ m2η Λ2η þ m2η Λ2η R1 ¼ 1 − ln × ln − ; ð96Þ 2 Λ2η m2η m2η Λ2η þ m2η 4 Moreover, it is easy to check that s⊥ ¼ . Indeed, with respect to the previous counting for sk, we lack a factor 2 coming from the exchange of the vertices, but we have an additional factor 2 arising from the expansion of the square of the interaction, which concerns only the contributions with vertices of different colors. Finally, the 2-loops contribution to τðp ¼ 0Þ writes as: 1 ð2Þ 2 2η 2 τ ðp ¼ 0Þ¼8d½{ðdÞ Λ λ R1 þ R2 : ð97Þ d − 1

The last term R2 can be interpreted as an overlapping effect, and in the large d limit this quantity disappears. The first derivative ∂τð2Þ⊥=∂p2η for zero momentum can be derived follows the same strategy. From expression (86),we get: Z ∂τð2Þ⊥ Θ 1 − ⃗2η Θ 1 − ⃗2η − 2η 2 ⃗ ð q⊥ Þ ð k⊥ q2 Þ ðp ¼ 0Þ¼−8λ × dq⃗⊥dk⊥ ∂ 2η 2η 2η 3 ⃗2η 2η 2η p ðq⃗⊥ þ m¯ Þ k⊥ þ q2 þ m¯ Z δ 1 − ⃗2η Θ 1 − ⃗2η − 2η 2 ⃗ ð q⊥ Þ ð k⊥ q2 Þ − 4λ × dq⃗⊥dk⊥ 2η 2η 2 ⃗2η 2η 2η ðq⃗⊥ þ m¯ Þ k⊥ þ q2 þ m¯

¼ P1 þ P2: ð98Þ where

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Z δ 1 − ⃗2η Θ 1 − ⃗2η − 2η 2 ⃗ ð q⊥ Þ ð k⊥ q2 Þ P2 ¼ −4λ × dq⃗⊥dk⊥ 1 ¯ 2η 2 ⃗2η 2η 2η ð þ m Þ k⊥ þ q2 þ m¯ Z Z 1 δ 1 − ⃗2η δ − ⃗2η − 2η 2 ⃗ ð q⊥ Þ ðy k⊥ q2 Þ ¼ −4λ × dy dq⃗⊥dk⊥ 1 ¯ 2η 2 ⃗2η 2η 2η 0 ð þ m Þ k⊥ þ q2 þ m¯ Z Z 4λ2 1 − 2η − ⃗ ⃗ y q2 δ 1 − ⃗2η δ 1 − ⃗2η ¼ 2η 2 × dq⊥dk⊥ dy 2η ð q⊥ Þ ð k⊥ Þ: ð99Þ ð1 þ m¯ Þ 0 y þ m¯

Then by summing all the possible choices of the variable q2 B. Structure of the n-loops graphs in large d limit − 1 and dividing by d , we get: In this section, we extend the result of the previous Z section to arbitrary large Feynman graphs in the large d 4λ2 1 y − 1 − 2 d−1 limit; providing the first hard statement of this paper. P2 ¼ 2η 2 ½{ðdÞ dy 2η : ð100Þ ð1 þ m¯ Þ 0 y þ m¯ Heuristically, if we discard the terms mixing coupling and mass, for n–loops, the expected the following behavior This expression correspond to the computation of the for τðpÞ: effective mass correction and in large d, and we retain: τðnÞðpÞ ∝ dn−1ð2{ðdÞÞn × λn: ð107Þ 4λ2 1 ¯ 2η → − 2 1 − ¯ 2η þ m P2 2η 2 ½{ðdÞ m ln 2η : This behavior can be proved recursively, but have proved d≫1 ð1 þ m¯ Þ m¯ for n ¼ 1 and n ¼ 2 which highlight the initial origin of the ð101Þ different factors in (107). For instance, a factor {ðdÞ seems to be associated to each loops. The origin of the factor d In the same manner: moreover is clear. As recalled in the Appendix A, the Z leading order contributions are trees in the so called 1 intermediate field representation, then, the typical graphs → − 8λ2 2 y1dy1 y2dy2 P1 ½{ðdÞ × 2η 3 2η ; ð102Þ τðpÞ τðpÞ0 d≫1 0 ðy1 þ m¯ Þ y2 þ m¯ contributing to and are trees with p colored edges. All the edges are color-free, except the color of the edge and after integration we get corresponding to the single boundary vertex. As a result, there are dp−1 different trees with the same uncolored τðpÞ τðpÞ0 1 1 m¯ 2η 1 1 combinatorial structure; and the cardinality of and → − 8λ2 2 − 1 p−1 P1 ½{ðdÞ × 2η 2η þ 2η is d times a purely combinatorial number depending d≫1 1 þ m¯ 2 1 þ m¯ 2 m¯ only on p. In this subsection, we will prove this intuition, 1 þ m¯ 2η × 1 − m¯ 2η ln : ð103Þ and investigate the structures and properties of higher-loops m¯ 2η diagrams in the large d limit. More precisely, we will prove the following statement: Note that, as the one-loop correction, the two loops Proposition 5: Let T n be a 2-points n-loops tree function is not perturbative in m¯ 2η. To summarize, in the contributing to τðnÞðpÞ and let r be its root loop vertex, large d limit, we get for two loops contributions to τ and τ0: at which the external colored edge is hooked. In the UV sector (Λ ≫ 1) and in the large dimension limit ð2Þ 2 2 2η 2η (d ≫ n ≥ 1), the perturbative n-loops amplitude AT τ ¼ −4d½{ðdÞ λ Λ ð1 þ lnðm¯ ÞÞ; ð104Þ n behaves in λ and d like: 1 2η 2η 2η ð2Þ0 2 2 A Λ − ¯ n τ ¼ 4d½{ðdÞ λ 1 − : ð105Þ T n ðpÞ¼ð p Þcnðm ;pÞð{ðdÞÞ ; ð108Þ m¯ 2η 2η where cnðm¯ ;pÞ includes a proper mass and external As remark, the infrared divergences that we observe at two momenta dependence: loops order occur in the computation at n-loops and these Y 2η divergences are increased with the number n of loops as ωðmðbÞÞðm¯ Þ c m¯ 2η −1 n−1 × A p ; Z nð Þ¼ð Þ − 1 ! rð Þ ∈T ½ðmðbÞ Þ Λ dp⃗ b n=r ≡ Λd−2n 2 n : ð106Þ 0 ðp⃗Þ ð109Þ

085006-14 LARGE-D BEHAVIOR OF THE FEYNMAN AMPLITUDES FOR … PHYS. REV. D 103, 085006 (2021) where T n ∈ T n and T n denotes the set of trees with n loop We can then consider F as the transformation sending any T ∈ T T ⊂ T − vertices and different colors on their edges, mðbÞ is the tree n n to a set F½ n nþ1 of cardinality ðd coordination number at the loop-vertex b, and ωðmðbÞÞðm¯ 2ηÞ nÞ × n whose elements are any trees with n þ 1 loop T is the mðbÞth derivative of ω defined as: vertices obtained from n by adding a leaf. Moreover, T ∩ T 0 ≠∅ we expect that F½ n F½ n in general, because any 2η T T 1 1 þ m¯ tree in nþ1 have more than one antecedent in n.Asan ωðm¯ 2ηÞ ≔ ðlnð1 þ m¯ 2ηÞþm¯ 2η − ðm¯ 2ηÞ2 ln 2 m¯ 2η illustration, the tree: Z Z m¯ 2η 1 y ≡ dx dy : ð110Þ 0 0 y þ x ð115Þ Finally, the root amplitude ArðpÞ sharing the external momenta dependence writes as:

mðrÞ−1 1 ∂ with n ¼ 4 has three antecedents, obtained from deletion ArðpÞ ≔ ðmðrÞ − 1Þ! ∂ðm¯ 2ηÞmðrÞ−1 of one among the three leafs hooked to the loop-vertex " !# ¯ 2η p2η 2η labeled 1: m þ Λ2η 1 þ m¯ × 1 − 2η ln 2η : ð111Þ 1 − p ¯ 2η p Λ2η m þ Λ2η

Proof.—The statement has been proved for n ¼ 1 and n ¼ 2. Let us then provide the general proof by recurrence, i.e., we assume that the proposition holds for n loops, and we will prove that the expected structure survives for n þ 1 loops. From proposition 3 (see Appendix A), the single ð116Þ colored self energy τðnÞðpÞ of order n, may be written as a sum of rooted trees with n loop-vertices in the intermediate field representation. The root being a colored edge hooked to one of the loop vertices, that we call external loop vertex, for instance:

ð112Þ ensuring the surjectivity of the map F. We have now all the material to build our recurrence. Let us consider a tree T n, with n loop-vertex, to which we add a leaf Lc of color c.We denote by T L the resulting tree with n 1 loop- is such a typical tree, with root of color red. n b c þ b Now, let T be the set of such a trees with n loop vertices, vertices; being the loop-vertex at which the leaf is n hooked. We assume that c ≠ 1, where 1 refers to the color and F be a surjective map from T to T 1: n nþ of the root. As we seen for the computation of the two- loops 2-point function, this restriction does not affect the F∶ T → T 1: ð113Þ n nþ large d limit, the color c being chosen among d − 1 colors rather than d. From Feynman rules, the amplitude The map F can be constructed explicitly. Indeed, for any AT L ðpÞ for the resulting n þ 1 graph can be written T ∈ T n c tree nþ1 nþ1 it is not hard to check that there exist a T T T explicitly as: single tree in n such that nþ1 may be obtained from n by adding one leaf: X A A 0 A T nbLc ðpÞ¼ T n ðp; qÞ Lc ðqÞ; ð117Þ q∈Z

ð114Þ A L where Lc ðqÞ is the Feynman amplitude for the leaf c and 0 T n the 2-root tree obtained from the single root tree T n by hooking an half edge of color c to the vertex b:

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ð118Þ

A A 0 A The relation between T n and T n can be obtained as follow. From Feynman rules, we get the explicit expression for T n as: Y X Y Y Y lðb0Þ AT p C p⃗ 0 δ δ δ ; n ð Þ¼ ð b Þ psðeÞptðeÞ psðeÞp psðeÞptðeÞ ð119Þ 0∈B ⃗ ∈∂ ∈F ∈∂ b fpb0 g e fer f e f where B and F are respectively the sets of loop vertices and colored edges hooked to b with color c0. The number of internal faces, and ∂f is the subset of monocolored edges manner to choose these mðbÞ edges is then: building the colored face f. Moreover sðeÞ and tðeÞ denote respectively the source and target loop vertices bounding mðbÞ! d! Q × ; ð120Þ the edge e, and fer is the external face running through the cðbÞ c b ! d − c b ! 0 m 0 ðbÞ! ð Þ ð ð ÞÞ root. A colored face on a tree corresponds to a colored and c c unclosed path, for internal as for external ordinary faces, where cðbÞ ≤ mðbÞ is the number of different colors for passing through loop vertices at which are hooked some edges hooked to b. For large d, and from standard Stirling connected components(see Fig. 3 below). formula, we get: Let fc be the external face of color c that we created on T 0 , starting at the loop-vertex b, and lðcÞ the correspond- n cðbÞ 0 d! 1 1 d ing colored path, having b as boundary. On T n, in addition ∼ − 1 cðbÞ!ðd − cðbÞÞ! cðbÞ! ð1 − cðbÞ=dÞd−cðbÞþ1 e to our external edge of color c, we have mðbÞ colored edges hooked to b. mðbÞ is the coordination number of the 1 d cðbÞ T ≤ → : vertex b on n; therefore mðbÞ n is a trivial bound. Each cðbÞ! e of these colored edges are hooked to connected compo- 2 nents, like on Fig. 3, where mðbÞ¼ . The number of The distribution is stitched for cðbÞ¼mðbÞ; and a little independent configurations, that is, the number of different deviation from this configuration receive a weight 1=d.For choices for the mðbÞ colors is nothing but the counting of instance, the first deviation: cðbÞ¼mðbÞ − 1 arise with the number of different manners to choice m b colors −1 −1 ð Þ relative weight: mðbÞðd=eÞ ≤ nðd=eÞ . Then in the among d. More precisely, let m 0 b is the number of c ð Þ limit d ≫ n ≫ 1, this term remains small, and crushed by the dominant configuration, ensuring that lðcÞ must have zero length for the dominant configurations. In other words, our added leaf on b open an internal ordinary face of length one or more is very small. As a result, the structure of T n that we have to consider is the following:

X YcðbÞ cðbÞ AT p C p⃗ AT p ; n ð Þ¼ ð bÞ pðmÞ ð cðmÞÞ ð121Þ 1 p⃗b m¼

where p⃗b is the internal momenta running through the loop vertex b, pðmÞ designates the order of the connected T component pðmÞ and cðmÞ the color of the edge m. Note that, because T n is a two point graph, one of the connected components have to be a four points graphs. FIG. 3. The colored path corresponding to an external blue- Note that to simplify the notations, we only indicate the face, and the connected components hooked to the loop vertices external variables for edges hooked to the vertex b. In the along the path. AT same way, the expression for nþ1 becomes:

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X X YcðbÞ cðbÞþ1 AT p C q⃗ δ C p⃗ AT p ; nþ1 ð Þ¼ ð Þ pcqc ð bÞ pðmÞ ð cðmÞÞ ð122Þ 1 q⃗ p⃗b m¼ or, more explicitly: X Θ Λ2η − ⃗2η X Θ Λ2η − ⃗2η YcðbÞ ð q Þ ð pb Þ AT ðpÞ¼ δ AT ðp Þ : nþ1 ⃗2η 2η pcqc 2η 2η cðbÞþ1 pðmÞ cðmÞ q þ m ðp⃗ þ m Þ 1 q⃗ p⃗b b m¼

In the continuum limit, this expression becomes: Z Z Θ 1 − ⃗2η Θ 1 − ⃗2η YcðbÞ 2η ð q Þ ð pb Þ ¯ AT p Λ δ × AT p ; nþ1 ð Þ¼ 2η 2η pcqc 2η 2η 1 pðmÞ ð cðmÞÞ ⃗ q⃗ m¯ ⃗ ⃗ ¯ cðbÞþ q þ pb ðpb þ m Þ m¼1 where we recall that x¯ means Λ− dimðxÞx. Using the integral parametric representation for the Heaviside Θ-functions, we get: Z Z Z δ − ⃗2η δ − ⃗2η cYðbÞ 2η ðy1 q Þ ðy2 pb Þ ¯ AT p Λ δ × AT p nþ1 ð Þ¼ 2η 2η pcqc 2η 2η 1 pðmÞ ð cðmÞÞ ⃗ q⃗ m¯ ⃗ ⃗ ¯ cðbÞþ y1;y2 q þ pb ðpb þ m Þ m¼1 Z Z Z δ − ⃗2η δ − ⃗2η YcðbÞ 2η ðy1 q Þ ðy2 pb Þ ¯ ¼ Λ δ × AT ðp Þ : 2η pcqc 2η c b 1 pðmÞ cðmÞ ⃗ y1 m¯ ⃗ ¯ ð Þþ y1;y2 q þ pb ðy2 þ m Þ m¼1

2η 1=2η Finally, rescaling the qi variables as qi → ðy1 − qc Þ qi ∀ i ≠ c, Z Z 2η 2η YcðbÞ δ y2 − p⃗ A Λ2η y1 1 − qc ð b Þ A¯ T n 1 ðpÞ¼ {ðdÞ 2η 2η 1 × T p m ðpcðmÞÞ : þ y1 m¯ ⃗ y1 ¯ cðbÞþ ð Þ y1 þ y2;pb ðy2 þ m Þ m¼1

We have d − cðbÞ different choices for the color c leaving this expression unchanged; then, we can use the same trick as for the computation of P2 for the two-loop contribution. By summing all the possible choices, and taking into account the 2η properties of the corresponding delta function, we generate a factor 1=ðd − cðbÞÞ; such that the term qc =y1 can be discarded in the large d limit. As a result the amplitude becomes:

Z 2η cYðbÞ −1 ∂ δ y2 − p⃗ A Λ2η ω0 ¯ 2η ð b Þ A¯ T 1 ðpÞ¼ {ðdÞ ðm Þ 2η 2η T ðpcðmÞÞ : nþ m b ∂m¯ ⃗ ¯ mðbÞ pðmÞ ð Þ y2;pb ðy2 þ m Þ m¼1

The procedure can be continued from external leafs to fðλÞ¼ln ZðλÞ; ð123Þ the root vertex. In large d, we seen that all the faces have length one with a very large probability, such that only where, in contrast to τ, f admits a Feynman expansion with the root vertex shares the external momenta. Finally, P P amplitudes labeled with vacuum diagrams. In contrast with ðmðbÞ−1Þ −1 2 − 1 −1 b −1 n two point diagrams considered in Sec. III B, vacuum because b mðbÞ¼ n , ð Þ ¼ð Þ , which ends the proof of the proposition. ▪ diagrams in the intermediate field representation have no Definition 2: We will denote by τ⋆ðpÞ the part of two roots: point function expanding only in terms of the melonics two X X 1 n points amplitudes, keeping only the relevant ones in the fðλÞ¼ ð−λÞ AG ; ð124Þ G n large d limit. sð nÞ n Gn Note that the strategy for nonvacuum two point diagrams can be done for vacuum diagrams as well, allowing to where Gn are vacuum Feynman diagrams. Except the compute perturbative contribution of the free energy in the absence of rooted loop-vertex, the proof of proposition 5 ⋆ same way as the two point function τ . The free energy with may be repeated step by step for vacuum diagrams; vanish source is therefore, we must have:

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n Corollary 4: Let Gn be a melonic vacuum diagram with contribution 2 arise from the two possible orientations for n-loops and T n the corresponding tree in the intermediate each original quartic vertices, building the edges of the tree. field representation. In the large-d limit (d ≫ n), the The second factor is a purely combinatorial number relevant contribution may be decomposed as: counting the number of color arrangements. More precisely, we must have a factor ðn − 1Þ! counting the number A ¯ 2η nþ1 T n ¼ vnðm Þð{ðdÞÞ ; ð125Þ of different permutation of the internal (original) vertices, arising from Wick contractions. Another factor arise from 2η where vnðm¯ Þ depends only on mass, and is explicitly the expansion of the exponential itself. Indeed, denoting as given by: ai for i running from 1 to d the quartic interaction involved in the action, we must have a combinatorial factor counting Y 2η ωðmðbÞÞðm¯ Þ the number N ðn; dÞ of the way to build an arrangement of c m¯ 2η −1 n : 126 nð Þ¼ð Þ − 1 ! ð Þ n quartic vertex of different colors (but including the color ∈T ½ðmðbÞ Þ b n n 1) among ða1 þ a2 þþadÞ . It is not hard to check that this number must be equal to: C. Formal summation theorems ðd − 1Þ! N ðn; dÞ¼n ðn − 1Þ!: ð129Þ In this section we provide the last two relevant results of ðd − nÞ!ðn − 1Þ! this paper, i.e., the resummation theorem, leading to an ⋆ explicit expression for τ ðpÞ. To make the proof clearer, we The first factor count the n different ways to choose the root divide it into two steps, computing τ⋆ð0Þ as a first step, ⋆ vertex of color 1. The central factor, on the other hand count from which we will deduce τ ðpÞ in a second time. the number of way to choose n − 1 different colors among the remaining d − 1, and finally the last factor ðn − 1Þ! 1. Resummation for τð0Þ count the different arrangements for a given selection of The perturbative expansion for τð0Þ may be written as a n − 1 colors. Taking into account all these contributions, 1 ˜0 T sum over amplitudes indexed by 1PI Feynman diagrams we define =s ð 1;nÞ such that: G , with external vertex of color i: i ! 2n ! 1 n ≕ ! d X∞ X n 0 : ð130Þ 1 s˜ T 1 d d − n ! s˜ T 1 ⋆ n ð ;nÞ ð Þ ð ;nÞ τ ð0Þ¼ ð−λÞ AG ; ð127Þ G i;n 1 G sð i;nÞ n¼ i;n The interest to extract this factor comes from the explicit expression of the leading order amplitudes in large d. The ! G Where the symmetry factor n =sð 1;nÞ count the number of amplitude in fact, does not depends on the selected set of independent Wick contractions leading to the same graph colors for the n edges, so that the sum can be reduced on the G1;n, and where the last sum run over Feynman diagrams set Tn of planar rooted trees with n vertices: with n vertices, and external vertex of color 1. For the rest of this section, we fix i ¼ 1. Moreover, we focus on the X 1 2n d! X 1 AT AT ; 131 melonic diagrams only, and in the large rank limit is ˜ T ¼ − ! ˜0 T ð Þ T ∈T sð Þ d ðd nÞ T ∈T s ð Þ restricted on the melonic diagrams having vertices of n;d n different colors and we denote by M this set. As recalled n;d τ 0 in Appendix A, melonic diagrams correspond to trees in the and the zero-momentum two point function ð Þ can be HS representation. Moreover, as in the proposition 5, the written as: amplitudes AG depends only on the coordination vertex 1;n X∞ n X ⋆ ð−2λÞ d! 1 numbers of the corresponding tree, and may be naturally τ 0 AT : 132 ð Þ¼ − ! ˜0 T ð Þ 1 d ðd nÞ s ð Þ indexed by tree rather than melon diagram. In an abusive n¼ T ∈Tn A ≡ A notation, we must have G1;n T 1;n . Our final aim is then The remaining factor 1=s˜0 T depends only on the com- to rewrite the previous sum over Mn;d as a sum over rooted ð Þ tress, with root of color 1 and edges of different colors. binatorial structure of trees, but not on the specificities of the model. In fact, the same factor have to be occurs for More precisely, denoting as Tn;d the corresponding set of models with trivial propagator and a single melonic trees, we have to find s˜ðT 1;nÞ such that: interaction. For such a model, AT ¼ 1, and the computa- X 1 X 1 tion have be done explicitly in the melonic sector using ≕ : ð128Þ Schwinger-Dyson equation [63,68–71]. The result is sðGi;nÞ s˜ðT 1;nÞ Gi;n∈Mn;d T 1;n∈Tn;d pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − 1 8λ X∞ ! ˜ T Σ þ −2λ n To compute n =sð 1;nÞ, we first remark that this factor ¼ 2 ¼ ð Þ Cn−1; ð133Þ must be a product of three distinct contributions. The first n¼1

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X∞ X Yn where fC g denote the Catalan numbers. Therefore, C −1 ⋆ 1 1 ðn − 1Þ! n n τ¯ 0 − 2 λ n Q ω0 ð{bÞ ð Þ¼ ð d{ðdÞ Þ ! ! ð Þ : is precisely the number of planar rooted trees with n loop d n … {b n¼1 Pi1; ;in b b¼1 vertices: −1 {b¼n X b Cn−1 ≡ ; ð134Þ ð141Þ T ∈T n This expression provides a first important intermediate 0 ensuring s˜ ðT Þ¼1. The Catalan numbers Cn are defined statement. Indeed, we see that each term of the sum as: involves increasing the power of d{ðdÞ. Therefore the existence of an interesting large d limit imply the existence 1 2 of an appropriate rescaling of the coupling constant, C ¼ C n; ð135Þ n n þ 1 n ensuring that each leading order terms in 1=d receives the same weight. The rescaling can be read directly from Cn Cn where p denotes the usual binomial coefficients p ¼ the previous expression, and we summarize this result as an n!=p!ðn − pÞ!. The amplitude AT depending only on the intermediate statement: coordination numbers of the tree, it could be suitable to Lemma 1: In the melonic sector, the d → ∞ limit exist convert the sum over trees as a sum over modified for the classical action with rescaled coupling λ → g=d{ðdÞ: coordination numbers { ≔ mðbÞ − 1, satisfying the hard X b ¯ ¯ constraint: S½T;T¼ Tp⃗Kðp⃗ÞTp⃗ X p⃗ X X {b ¼ n − 1: ð136Þ g ðiÞ ¯ ¯ þ V T ⃗ T ⃗ T ⃗ T ⃗ : b d{ðdÞ p⃗1;p⃗2;p⃗3;p⃗4 p1 p2 p3 p4 i p⃗1;…;p⃗4 Moreover, it is not hard to prove that ð142Þ X 1 X 2n−2 Now, we move on to the main statement of this section, the ¼ C ⇒ C −1 ¼ : ð137Þ n−1 n n resummation theorem, providing an explicit expression for Pi1;…;in Pi1;…;in −1 −1 the large d melonic two point function. The trick to resum {b¼n {b¼n b b the complicated expression given by (141) use the generalized Leibniz formula: Then, Cn−1 being the sum over trees, the previous decomposition is nothing but the desired result, a sum over the n X ! Y d n ðkiÞ f1f2 f f ; 143 trees rewritten as a sum over coordination numbers. With n ð mÞ¼ ! ! ! i ð Þ dx … k1 k2 km this respect, τð0Þ becomes: kP1; ;km i k ¼n i i X∞ X Yn 1 d! 1 ω0 ð{bÞ τ⋆ 0 −Λ2η 2 λ n ð Þ ð Þ¼ ð {ðdÞ Þ such that (141) can be rewritten as: d ðd − nÞ! n {b! n¼1 Pi1;…;in b¼1 { ¼n−1 ∞ −1 b b X 1 n ⋆ n d 0 2η n −dτ¯ ð0Þ¼ ð2gÞ ðω ðm¯ þ xÞÞ j 0: ð144Þ ð138Þ n! n−1 x¼ n¼1 dx where we took into account the proposition 5. In the This expression leads to a transcendental equation thanks large-d limit, we may use the standard Stirling formula pffiffiffiffiffiffiffiffi to the well-known Lagrange inversion theorem, which n −n n! ∼ 2πnn e . Now due to the fact that state that: Theorem 1 Lagrange inversion theorem: Let f be a C∞ − 1− 1− − − ðd − nÞd n ¼ edð n=dÞðlnðdÞþlnð n=dÞÞ ¼ dd ne n þ Oðn=dÞ; function and z be a function of the variables x, y and f as: ð139Þ z ¼ x þ yfðzÞ: ð145Þ we must have: Therefore, for any C∞ function h, we must have: n −d d! d e X∞ −1 O n O yk dk ¼ d−n −d n þ ðn=dÞ¼d þ ðn=dÞ k 0 d − n ! − þ hðzÞ¼hðxÞþ −1 ððfðxÞÞ h ðxÞÞ: ð146Þ ð Þ ðd nÞ e k! k k¼1 dx ð140Þ Applying this result where h being the identity function, and the previous expression (138) becomes, introducing the from Eq. (144) we get straightforwardly that z ≔−dτ¯⋆ð0Þ dimensionless function τ¯⋆ð0Þ¼τ⋆ð0Þ=Λ2η: must satisfy the transcendental closed equation:

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2η 1 þ x z ¼ð2gÞω0ðm¯ þ zÞ; ω0ðxÞ¼1 − x ln : ð147Þ m¯ 2η þ 2g x ≡ − 2η ð154Þ ¯ 2η −1þm¯ þ2g 2η 2gWð−ðm þ 1Þe 2g Þþm¯ þ 2g It is not hard to recover the one and two loop computations, 2g Eqs. (79) and (97). Indeed, expanding z in power of g as 1 2 This asymptotic formula have to completed with some z ¼ zð Þ þ zð Þ þ, we get: important remarks. The Lambert function WðxÞ is muti- 1 2η 2 2 2η 2η valued in the interval −1=e ≤ x ≤ 0. The first branch, zð Þ ¼ 2gω0ðm¯ Þ;zð Þ ¼ð2gÞ ω0ðm¯ Þω00ðm¯ Þ; usually called W0ðxÞ is defined on R for x ≥−1=e, ð148Þ whereas the second branch, W−1ðxÞ is defined on −1=e ≤ x ≤ 0. In both cases, the physical region have to which coincide respectively with (79) and (97). Introducing be bounded by the condition Δ ≥−1=e, to ensure the the effective mass u ≔ m¯ 2η þ z, the closed equation can be reality of the resummed solution. If we only consider rewritten as: positive definite coupling, to ensure integrability of the partition function, the choice of the solution must be 1 þ u 1 ¯ 2η 2 ≥ 0 u ¼ðm¯ 2η þ 2gÞ − ð2gÞu ln ; ð149Þ depends on the sign of mass. For þ m = g ,itis u not hard to check that only W−1 admits a perturbative → 0 → 0− −1 −1 expansion around g . Indeed, for x , or, defining t ≔ 1 þ u : ⋆ 2η W−1ðxÞ ≈ lnð−xÞ, and u → m¯ , which is the result we 1 ¯ 2η 2 0 2η 2η 2η 2η have expected. In the opposite, for þ m = g< , the −1þm¯ þ2g −1þm¯ þ2g −1þm¯ þ2g em¯ e 2g t ln ðe 2g tÞþe 2g þ 1 ¼ 0: solution in agreement with the perturbative expansion is 2g W0. Indeed, for large x, W0ðxÞ ≈ lnðxÞ, and, for g → 0: ð150Þ 2η ⋆ jm¯ j u → 2η This equation can be formally solved in terms of Lambert ¯ 2η jm¯ þ1j−1=2g ¯ 2η 2g ln m 1 e 2g − 2g m 1 functions WðxÞ, defined with the following simple relation: ðj 2g þ j Þ j 2g þ j ¼ −jm¯ 2ηjþOðgÞ: WðxÞeWðxÞ ¼ x; ð151Þ To summarize, in the positive region 1 þ m¯ 2η=2g ≥ 0, and we get: Δ ≥−1=e, the two solutions W0 and W−1 coexist, but 2η only the second one admits the good limit for g → 0. In the 1 þ m¯ þ 2g 2η t ¼ exp WðΔÞþ ; negative region 1 þ m¯ =2g<0, Δ ≥−1=e however, 2g only the solution W0 exist, and admits the expected limit 2η 2η m¯ −1þm¯ þ2g for vanishing coupling. We have then two branches of Δ ≔− þ 1 e 2g : ð152Þ 2g solutions, and a strong discontinuity along the line 2g þ m¯ 2η ¼ 0. Note that, however, the two solutions are Strictly speaking, this formula hold in the very large d limit, continuous at the point g ¼ 0, as the previous computation i.e., for d → þ∞. Indeed, the sum over n being for arbitrary has showed explicitly. We will continue this discussion in large n, we expect that the condition d ≫ n must be the last section on which we will extend our solution to violated for large n. Then, to sum over large n,wehave arbitrary momenta. to assume the convergence of the series a` priori, and then discard the 1=d contributions as subleading order. In other 2. Solution for arbitrary momentum words, the formula (152) must be viewed as an asymptotic ⋆ formula, to which the exact two point function must For τ ðpÞ, from proposition (5), the expression (141) converge in the limit d → þ∞. To summarize, we have must be replaced by: then proved the following statement: X∞ X Y Proposition 6: In the very large d limit, the melonic d ⋆ 1 ðn − 1Þ! − τ¯ ðxÞ¼ ð2gÞn Q ðω0Þð{bÞ effective mass for the rescaled quartic melonic model given 1 − x n! ≠ { ! ⋆ n¼1 Pi1;…;in b r b b≠r by (142) goes asymptotically toward u , given by: { ¼n−1 b b 2η A 1 ¯ 2η 2 × rðxÞ; ð155Þ ⋆ m¯ − þm þ g u ≔ exp −W − þ 1 e 2g 2g τ⋆ 2η where we took implicitly into account that ðpÞ depends 1 þ m¯ þ 2g −1 2η − − 1 ð153Þ only on p , and introduce the dimensionless variable 2g x ≔ p2η=Λ2η. Moreover, it is easy to check that:

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1 ∂{r ∂ A ω˜ ¯ 2η D. Solving C-S equations in the large d limit rðxÞ¼ 2η 2η ðm ;xÞ; ð156Þ ! ∂ ¯ {r ∂ ¯ {r ðm Þ m We now move on two the last topic of this section. What we can learn from the previous formula about the global with: renormalization group flow? The explicit expression for all Z the beta functions can be obtained directly from proposi- ∂ 1 tions 6 and 7, merged with corollary 2 and Eq. (72). ω˜ ¯ 2η y 2η ðm ;xÞ¼ dy ¯ 2η : ð157Þ However, due to the complicated structure of the previous ∂m¯ 0 y þ m þx 1−x expression, we keep the deep analysis for another work. To conclude this part we focus on the existence of non- As a result, the expansion (155) can be rewritten is a more Gaussian fixed points. In Sec. II, we showed that, in the suggesting form as: melonic sector, all non-Gaussian fixed points have to verify the strong condition γ ¼ 0, γ being the anomalous dimen- 1 X∞ 1 d ⋆ n sion. From this condition, we can investigate the possibility − τ¯ ðxÞ¼− ð1 − xÞ ð2gÞ β β β γ 0 1 − x d n! that the -functions and m¯ both vanish when ¼ .To n¼1 X Y this end, let us consider the C-S equation for effective mass, ðn − 1Þ! ¯ 2η − τ 0 × Q ðω˜ 0Þð{rÞ ðω0Þð{bÞ : (73), replacing the effective mass m d ð Þ by the { ! 2η ⋆ 2η Pi1;…;in b b b≠r asymptotic solution for large d, Λ u ðg; m¯ Þ: { ¼n−1 b b ∂ ∂ ∂ ð158Þ 2η ⋆ 2η Λ þ βðgÞ þð2η þ β ¯ Þ Λ u ðg; m¯ Þ¼0: ∂Λ ∂g m ∂m¯ 2η where the “prime” designates derivative with respect to ð163Þ m¯ 2η. Once again, from the generalized Leibniz formula, each term may be rewritten as a single derivative of order Then, computing each derivative, we get straightforwardly: n − 1 acting on a product of functions: ∂ ⋆ ∂ ⋆ 2η ⋆ ¯ 2η β u β u 0 ∞ u ðg; m Þþ ðgÞ þ m¯ 2η ¼ : ð164Þ d X 1 ∂n−1 ∂g ∂m¯ − τ¯⋆ðxÞ¼ ð2gÞn ððω0ÞnΞ0Þ; ð159Þ 1 − x n! ∂ðm¯ 2ηÞn−1 n¼1 Now, let us consider the relation (72) between β-functions. This relation is a consequence of the Ward identities, and where we introduced Ξ0 defined as Ξ0 ≔ ω˜ 0=ω0. Therefore, hold for any dimension. Setting γ ¼ 0, and up to the 2η fixing the arbitrary integration constant such that Ξ0ðm¯ þ replacement βðλÞ → βðgÞ and λ → g=d{ðdÞ, we get: 0 y; xÞ vanish for y ¼ ; the Lagrange inversion theorem 1 must be applied, leading to: 1 1 2g g βðgÞþ βðgÞ − β ¯ d{ðdÞ d2{ðdÞ 1 þ m¯ 2η − 2g=d 1 þ m¯ 2η m d − τ¯⋆ðxÞ¼Ξðm¯ 2η þ z; xÞ; ð160Þ ¼ 0: ð165Þ 1 − x

It is not hard to check that β must be of order 1 whereas β ¯ 2 ω0 ¯ 2η m where z must be defined as z ¼ð gÞ ðm þ zÞ, which is must be of order d. Indeed, in contrast with the mass, the − τ 0 nothing but d ð Þ given in the last section. Therefore, the radiative corrections for couplings require to fix one color. τ¯⋆ full asymptotic function ðxÞ is essentially the one loop The same conclusion may be deduced directly from the function, where the bare mass is replaced by the effec- previous expression. Note that, due to the mass dimension, tive mass: 2η 2η the expansion of βm¯ have to start with −2ηm¯ ≈−dm¯ =2. Proposition 7: In the large d limit, and in the melonic Setting d arbitrary large, we get: sector, the momentum depends two point function τðpÞ goes toward the asymptotic behavior: 2 1 2g βðgÞ¼ β ¯ : ð166Þ d ð1 þ m¯ 2ηÞ2 m 1 − x τ¯⋆ðxÞ¼− Ξðm¯ 2η − dτð0Þ;xÞ; ð161Þ d Then, from Eq. (164), we deduce straightforwardly: where the function Ξ is defined as: 2η ⋆ ¯ 2η −2η β − u ðg; m Þ O 1 m¯ ¼ 2 ⋆ ⋆ ¼ ⋆ þ ð =dÞ: 1 2g ∂u ∂u ∂ 2η Z 2η 2 2η m¯ lnðu Þ y ω˜ 0 ¯ 2η d ð1þm¯ Þ ∂g þ ∂m¯ Ξ ≔ ðm þ t; xÞ ðy; xÞ dt 2η : ð162Þ 0 ω0ðm¯ þ tÞ ð167Þ

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Therefore, to get a nontrivial fixed point, we must have 1.4 β 0 m¯ ¼ . Investigating this condition requires some alge- 1.2 braic manipulations. Computing the derivative of the logarithm using proposition 6, we get: 1.0

0.8 ∂ 2η Δ ∂ 2η ⋆ m¯ Wð Þþ m¯ a ∂ 2η m¯ lnðu Þ¼ −W Δ −a ; e ð Þ − 1 0.6

2η where a ≔ ð1 þ m¯ þ 2gÞ=2g. From proposition 6, we 0.4 then deduce that: 0.2

Δ WðΔÞ ∂ 2η Δ þe m¯ þ 2 ⋆ ⋆ 0 g 0.4 0.2 0.2 0.4 ∂ ¯ 2η lnðu Þ¼u ðW ðΔÞ∂ ¯ 2η Δ þ 1=2gÞ¼ ; m m Δ þ eWðΔÞ −W x ð168Þ FIG. 4. Numerical plot of the function fðxÞ¼xe ð Þ þ 1. where we used the well-known formula for the derivative of We are now in position to investigate the existence of the Lambert function: non-Gaussian fixed point. From the elementary properties Δ Δ Δ 1 of the Lambert-W function, þ Wð Þ vanish only for ¼ W0ðxÞ¼ : ð169Þ −1=e (see Fig. 4 below). This point, however as been x þ eWðxÞ pointed out to be the boundary of the analytically region, beyond it the Lambert function takes complex values and β Therefore, the expression for m¯ becomes: the resummation break down. Moreover, the condition Δ ¼ −1=e is a global condition on the boundary and not an 2η Δ þ eWðΔÞ β ¯ − isolated point. Therefore: m ¼ ⋆ Δ WðΔÞ : ð170Þ u ∂ 2η Δ þe m¯ þ 2g Claim 1: In the large d limit, and in the melonic sector, there are no isolated fixed point in the interior of the The derivative of Δ can be easily computed, leading to: perturbative region Δ > −1=e. 2η 2η 1 m¯ 1 − 1 m¯ − IV. DISCUSSION AND CONCLUSION ∂ 2η Δ − 1 a a m¯ ¼ þ þ e ¼ e ; 2g 2g 2g 2g 2g In this paper we investigated a new family of tensorial ð171Þ group field theories, just renormalizable for arbitrary ranks d. From Ward Takahashi identities, we showed that, for any and Eq. (170) becomes: d, a strong relation exists between β-functions in the deep UV limit, using C-S equation, from which we deduce that 4ηg Δ eWðΔÞ 4ηg Δ eWðΔÞ any melonic non-Gaussian fixed point must have vanishing β − þ þ −WðΔÞ m¯ ¼ ⋆ − Δ ¼ ⋆ − Δ − e : anomalous dimension (the C-S equation is used here u −e a þ eWð Þ u e Wð Þ a − 1 because of its flexibility without a choice of a certain ð172Þ regulator, but the same result may be obtained by using the Wetterich flow equation or other RG methods [72,73] − Δ − Now, from proposition (6), the denominator e Wð Þ a − 1 is which is deserved to forthcoming investigation). Similar ⋆ nothing but 1=u . Then, we finally deduce the following relations have been deduced recently in the functional corollary: renormalization framework, and all tentative to merge Corollary 5: To any fixed point in the deep UV region together this constraint with approximate melonic solutions Λ ≫ 1 β β ( ), the melonic beta functions ðgÞ and m¯ have to of the exact renormalization group flow equations lead to satisfy asymptotically, for very large d: the same conclusions: the disappearance of the non- Gaussian melonic fixed point [46–51]. This result was −WðΔÞ βm¯ ¼ 4ηgð1 þ Δe Þ¼4ηgð1 þ WðΔÞÞ; ð173Þ remains a claim, due to the necessary to use approximations to solve the renormalization group equations. In all cited and papers, the principal approximation are given by the necessity to use the derivative expansion to obtain a 3 2g tractable parametrization of the theory space. The difficulty, βðgÞ¼ ð1 þ WðΔÞÞ: ð174Þ ð1 þ m¯ 2ηÞ2 as mentioned in [47] comes essentially, in the melonic sector and may be translated as the difficulty to solve the where Δ given by Eq. (152). closed equation satisfied by the two point function. In this

085006-22 LARGE-D BEHAVIOR OF THE FEYNMAN AMPLITUDES FOR … PHYS. REV. D 103, 085006 (2021) paper, considering the large d limit, we were able to obtain APPENDIX A: POWER COUNTING AND an asymptotic formula for the bare two point function, RENORMALIZABILITY solving the closed equation in the same limit by construc- In this section we provide the power counting for our tion. Then, using the C-S formalism, we deduced the deformed family of model, and we will recall some explicit expressions of the β functions, to all orders of basic properties of the leading order graphs, the so called the perturbation theory at the points where the anomalous melons. The proofs are standard, and we will give only the dimension vanish. From an analytic expression, it is clear relevant details for the unfamiliar reader. For more details that in the considered limit, no isolated fixed points occurs see [26,32]. in the analytic region where the perturbative expansion can be resummed as a real function. Obviously, the results discussed in this introductory 1. Multiscale analysis paper do not exhaust this novel topic. First of all, we do We start by fixing our notations. First, we introduce an not investigate the behavior of the full RG flow, we focused integer ρ and a positive real number M so that Λ ¼ Mρ. only on the regions with vanishing anomalous dimension. Then we define the sharp momentum cutoff χ≤ρðp⃗Þ, equal Indeed, the disappearance of isolated fixed points is not the to 1 if p⃗2η ≤ M2ηρ and zero otherwise which is nothing but end of the history; and a rigorous analysis of the RG the Heaviside step function. The theory with sharp “cutoff behavior have to be done in the future. Related to this point, ρ” is defined using the covariance the method used to obtain the β-function, using the Callan- Symanzik equation is crudely rudimentary, and a more ρ C ðp⃗Þ¼Cðp⃗Þχ≤ρðp⃗Þ: ðA1Þ sophisticated approach exist to build the RG flow. The solution for the two point function, for instance, can be Then, the key strategy of the multiscale analysis is to slice used to improve the truncations abundantly used to solve the theory according to: the functional RG equations. Other approaches, using discrete slicing in the momentum space have to be Xρ ρ 2η considered for TGFT these last years, and could be C ðp⃗Þ¼ Ciðp⃗Þ;Ciðp⃗Þ¼Cðp⃗Þχiðp⃗ ÞðA2Þ investigated beyond the one-loop order using the large-d i¼1 limit. The nature of the limit in itself should be studied ⃗2η 2η carefully. Indeed, our resummed formula provide only the where χ1 is 1 if p ≤ M and zero otherwise and for i ≥ 2 χ 2ηði−1Þ ⃗2η ≤ 2ηi asymptotic behavior of the correlation function in the large i is 1 if M < p M and zero otherwise. rank limit; but we have no control over the neglected Now, we need to define the notion of subgraph. A subgraph contributions of order 1=d, which can become relevant S ⊂ G in an initial Feynman graph is a certain subset of when n, the order of the perturbative expansion, and d are dotted edges (propagators C) with the vertices hooked to commensurable. This situation is reminiscent to what them; the half-edges attached to the vertices of S (whether occurs for large N expansions for matrix and tensor models external lines of G or half-internal lines of G which do not around the critical point, leading to the double scaling limit belong to S) form the external edges of G. investigations. Then, this question, and more generally the Decomposing each propagator into slices, multiscale existence of a true 1=d expansion have to be addressed for decomposition attributes a scale to each line l ∈ LðGÞ of an incoming work. Finally, the existence of two branches of any amplitude AG associated to the Feynman graph G. Let solution for the resummed two point function has to be us start by establishing multiscale power counting. investigated as well. In fact, as we will see in Sec. III, the The amplitude of a graph G, AG, with fixed external two branches are continuous at the Gaussian fixed point, momenta, is thus divided into the sum of all the scale 2η but a finite gap exist for finite jm¯ j. Moreover, the fact that attributions μ ¼fil; l ∈ LðGÞg, where il is the scale of the the two solutions coexist in the region Δ ≤ 0 seems two momentum p of line l: indicate that along a certain curve passing through the X origin, we pass continuously from a picture with two AðGÞ¼ AμðGÞ: ðA3Þ vacuum to a picture with one vacuum state, but with a μ strong discontinuity for other points along the line g ¼ 0. μ This qualitative picture is reminiscent to a first order phase At fixed scale attribution , we can identify the power transition, the Gaussian fixed point playing the role of a counting as the powers of M. The essential role is played by G G critical point. The possible existence of such a transition the subgraph i built as the subset of dotted edges of with have been discussed in some recent papers [48], using scales higher than i. From the momentum conservation rule approximates solutions for the RG flow. However at this along any loop vertex, this subgraph is automatically a PI stage, it is too early to view our result as a definitive subgraph which decomposes into kðiÞ connected PI com- G ∪kðiÞ GðkÞ statement, which needs to be confirmed by exten- ponents: i ¼ k¼1 i . Note that the inclusion relations sive works. between these connected components indexed by the pair

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ω Gk ði; kÞ build the tree which is called Gallavotti-Nicolò tree. Identifying the exponent with ð i Þ for each connected We have: Gk ▪ components i , we conclude the proof. Theorem 2: The amplitude AμðGÞ is bounded by: 2. Leading order graphs Y YkðiÞ LðGÞ ωðGkÞ In order to discuss the leading order sector, we introduce jAμðGÞj ≤ K M i ;K>0; ðA4Þ i k¼1 an alternative representation of the theory, called intermediate field representation, in which the properties of the and the divergence degree ωðHÞ of a connected subgraph leading sector become very nice. Usually, intermediate field H is given by: representation is introduced as a “trick” coming from the properties of the Gaussian integration, and allowing to ωðHÞ¼−2ηLðHÞþFðHÞ; ðA5Þ break a quartic interaction for a single field as a three body interaction for two fields. To simplify the presentation, we H H introduce the intermediate field decomposition as a one-to- where Lð Þ and Fð Þ are respectively the number of lines – and internal faces of the subgraph H. one correspondence between Feynman graphs [74 77] see 2η also [78–80] and references therein. In this section more- Proof.—First we have the trivial bounds (for K ¼ M ): over, we only focus our discussion to the vacuum graphs. −2ηi First, to each vertex of type i, we associate an edge of the jC ðp⃗Þj ≤ KM χ≤ ðp⃗Þ: ðA6Þ i i same color. Second, to each loop made with a cycle of doted edges, we associate a black node, whose the number Then, fixing the external momenta for all external faces, the of corners corresponds to the length of the loop (Fig. 5 Feynman amplitude (in this momentum representation) is provides some illustrations). To distinguish this represen- bounded by tation with the standard Feynman one, we call colored Y Y X Y edges the edges of the Feynman graphs in the intermediate G ≤ −2ηil χ ⃗ field representation, and loop-vertices their nodes. jAμð Þj KM ≤il ðpÞ; ∈Z l∈∂ The main statement is then the following: l∈LðGÞ f∈FintðGÞ pf f Theorem 3: The 1PI leading order vacuum graphs are ðA7Þ trees in the intermediate field representation. Moreover, 4η must be equal to d − 1 for a just renormalizable theory. We which is deduced straightforwardly from the standard call melonic diagrams these trees. Feynman rules. Then, as a first step, we distribute the Proof.—First of all, consider the case of a 1PI vacuum GðkÞ powers of M to all the i connectedQ components. To graph. If it is a tree made with n loop vertices, it must have i −1 i 2 − 1 − 1 1 this end, we note that: M ¼ M 0 M, implying: c ¼ ðn Þ corners, and F ¼ðd Þn þ faces, since Q Q Q j¼ ω −2ηi 2η il −2η each colored edge glues two faces. As a result, ¼ l∈ M l ¼ M l∈ M . Then, inverting LðGÞ LðGÞ i¼0 −4ηðn − 1Þþðd − 1Þn þ 1 ¼ the order of the double product leads to ½d − 1 − 4ηn þð1 þ 4ηÞ. Then consider a graph with q colored edges, which is not Y Y Y Y YkðiÞ Y necessarily a tree. For q ¼ 1 there are two typical con- −2ηil −2η −2η M ¼ M ¼ M figurations: l∈ G kðiÞ 1 k Lð Þ i l∈Lð∪ GkÞ i k¼ l∈LðG Þ Y k¼1 i i −2ηLðGkÞ ¼ M i : ðA8Þ i;k

The final step is to optimize the weight of the sum over the momenta pf of the internal faces. Summing over pf with a ⃗ i factor χ≤iðpÞ leads to a factor KM , hence we should sum with the smallest values iðfÞ of slices i for the lines l ∈ ∂f along the face f. This is exactly the value at which, starting from i large and going down toward i ¼ 0 the face becomes Gk first internal forQ some i .P Hence in this way we could bound the sums by f∈FintðGÞ pf∈Z

Y YkðiÞ FðGkÞ M i : ðA9Þ FIG. 5. Correspondence between original representation (on i k¼1 left) and intermediate field representation (on right).

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different ways to build a graph with q þ 1 colored edges. ðA10Þ From the typical tree and so one for each choices of colors for the intermediate field edges. From direct computation, the divergent degrees ðA11Þ are respectively, from left to right: ωL ¼ 2½d − 1 − 4ηþ ð1 þ 4ηÞ and ωR ¼ ωL − ðd − 2Þ; then the leading order graph is the one on the left, which is a tree. Now, starting with a tree for arbitrary q, we have to investigate all the we have four possible moves:

ðA12Þ

where the moves are pictured with dotted edges. The two diagram. However, it is clear that such a cutting delete d moves one the right preserve the tree structure, then, the internal faces, except if the chosen tadpole is on the path of power counting is the expected one for such a tree: ωT ¼ the opened heart external face. Indeed, in this case, the −4ηðn − 1Þþðd− 1Þn þ 1 ¼½d − 1 − 4ηðn þ 1Þþð1 þ 4ηÞ. cutting d − 1 faces (which become boundary external The two moves on left however both introduce a loop. For faces) for the same in dotted lines, and the power counting the first one, we create at least a single face and two is clearly optimal. Moreover, due to the deletion, we created corners. The variation for power counting is then optimally another heart external face, obviously of the same color as :δω ¼ −4η þ 2. Obviously this bounds hold for the second the one for the original two point diagram. Recursively, we move on the left which creates a tadpole edge. Then for deduce the following proposition: these two moves, we have the bound: Proposition 8: A 1PI melonic diagram with 2N external lines has Nðd − 1Þ external faces of length 1 shared by ω ≤ ½d − 1 − 4ηðn þ 1Þþð1 þ 4ηÞ − ðd − 1 − 4ηÞþδω external vertices and N heart external faces of the same color running through the internal vertices and/or internal ω − − 3 ¼ T ðd Þ: ðA13Þ lines (i.e., through the heart graph). To complete these definitions, and of interest for our As a result, the power counting is bounded by trees for incoming results, we have the following proposition: d>3. Finally, if we need to have a just-renormalizable Proposition 9: All the divergences are contained in the leading sector, the divergent degrees does not increase melonic sector. with the number of loop vertices. We then require Pointing out that all the counterterms in the perturbative d − 1 − 4η ¼ 0, implying η > 1=2. ▪ renormalization are fixed from the melonic diagrams only. The leading order nonvacuum graphs can be obtained A proof may be found in [27]. Finally, we can add an following a recursive procedure. To this end, we have to important remark about melonic diagrams: Their divergent keep in mind the definitions 1 of Sec. II B, that we complete degrees depend only on the number of external edges, as with the following: expected for a just-renormalizable theory. To be more Definition 3: The heart graph of a melonic 1PI precise, note that the number of dotted edges is related G 2 4 − Feynman graph is the subset of vertices and lines to the number of vertices as L ¼ V Next, where Next obtained from deletion of the external vertices. denotes the number of external dotted edges. Moreover, it is Now, consider a vacuum melonic diagram. We obtain a easy to see, from the recursive definition of melons that two point graph cutting one of the dotted edges. Due to the F ¼ðd − 1ÞðL − V þ 1Þ. Indeed, starting from a Feynman structure of melonic diagrams, it is clear that if we cut an graph in the original representation, it is obvious that edge which is not a tadpole (i.e., an edge in a loop of length contracting a tree (dotted) edge does not change the upper than one), we obtain a 1PR diagram. Cutting a first divergent degree and the number of faces. Then, con- tadpole edge, we delete d faces. d − 1 of them become tracting all the edges over such a spanning tree, we get boundary external faces while the other one becomes an L − V þ 1 remaining edges, hooked on a single vertex, heart external face. We have then obtained a 1PI two points building a rosette. Now, we delete the edges optimally, melonic diagram. Then to obtain a four points melonic following successive (d − 1)-dipole contractions. We recall diagram, we have to cut another tadpole edge on this that a k-dipole is made with two black and white nodes (in

085006-25 VINCENT LAHOCHE and DINE OUSMANE SAMARY PHYS. REV. D 103, 085006 (2021) the original representation), wished together with one we get an effective loop of length p, on which we can dotted edge and k colored edges. In the intermediate field contract p − 1 edges to get a new tadpole, that we can representation, we can then start from a vacuum diagram, contract, and so one. Repeating the same procedure for and proceed both with the dipole and tree contractions. all loop-vertex, we get F ¼ðd − 1ÞðL − V þ 1Þþ1.For Starting from a leaf, hooked to an effective vertex b with a nonvacuum graph with 2N external edges, creating p external edges hooked to him, we can contract the leaf, them cost d − 1 faces per deleted tadpole, except for the discarding (d − 1) faces and 1 dotted edge. We may first one, which cost d faces, and the desired result assume that only leafs are hooked to b, except for one follows. From this counting for faces the divergence colored edge. Using the same procedure for all the leafs, degree becomes

ω ¼ −2ηL þ F ¼ −2ηð2V − N =2Þþðd − 1ÞðV − N =2 þ 1Þ ext ext − 1 − 1 − 4η − 1 − d − η ¼½ðd Þ V þ ðd Þ 2 Next : ðA14Þ which is nothing but the relation (12).

APPENDIX B: THE KEY SUMS WITH SHARP REGULATOR In this section we derive the important sums that arises in the computation of the loop expansion of the two point correlation function. Consider the following sum:

X 2η 2η ΘðΛ − jq⃗ jÞ S1ðp; a; bÞ¼ ⃗2η : ðB1Þ −1 ajq jþb q⃗∈Zd

In the large Λ limit and by introducing the continuous variable x ¼ q=Λ we get the following integral representation Z Θ 1 − x⃗2η ≈ 2d−1Λ2η d−1 ð Þ S1ðp; a; bÞ I1ðp; a; bÞ¼ d x 2η 0 Rþd−1 ax⃗ þ b Z Z 1 δ y − x⃗2η 2d−1Λ2η d−1 ð Þ ¼ dy d x 2η 0 ; ðB2Þ 0 Rþd−1 ax⃗ þ b with b0 ¼ b=Λ2η. Using the properties of the delta distribution, we find: Z Z 1 ydy 2d−1Λ2η d−1 δ 1 − ⃗2η I1ðp; a; bÞ¼ 0 d x ð x Þ 0 ay þ b Rþd−1 1 b0 a þ b0 ¼ Λ2η{ðdÞ 1 − ln ðB3Þ a a b0 with: Z Z Z 1 1 d−1 d−1 2η d−1 2η {ðdÞ ≔ 2 d xδð1 − x⃗ Þ¼2 dx1 dxd−1δð1 − x⃗ Þ: ðB4Þ Rd−1 0 0

This integral can be computed using Feynman parameters formula (ℜðαÞ > 0): Z Q 1 Γ − 1 α 1 X α−1 ððd Þ Þ δ 1 − P iui α α ¼ d−1 du1 dud−1 ui −1 −1 ; ðB5Þ Γ α 0 d ðd Þ A1 Ad−1 ½ ð Þ i ð i¼1 AiuiÞ with Ai ¼ 1 ∀ i: d þ 1 d−1 {ðdÞ¼2d−1 Γ : ðB6Þ d − 1

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Then: and 1 d−1 1 d d−1 2η d þ I1ðp; a; bÞ¼2 Λ Γ S2ðp; a; bÞ¼− S1ðp; a; bÞ; ðB9Þ d − 1 a db b0 a þ b0 providing the integral approximation: × 1 − ln : ðB7Þ a b0 1 d−1 1 d−1 d þ I2ðp; a; bÞ¼2 Γ In the same way we define: d − 1 a2 X Θ Λ2η − ⃗2η a þ b0 a ð jq jÞ × ln − : ðB10Þ S2ðp; a; bÞ¼ ⃗2η 2 ; ðB8Þ b0 a þ b0 −1 ½ajq jþb q⃗∈Zd

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