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 func- tions 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;T ¼ 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 intro- duced 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;T e ; 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;T Tp⃗Tp⃗0 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 compu- tation 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 − 4η V 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: ½λ ¼2½λ þ½L2 . 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 counter- terms, even if the number of graphs is infinite. This condition ½m2η ¼½λ þðd − 1Þ − 2η; ½λ ¼2½λ þðd − 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 ½λ ¼4η − 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 ½λ ¼0, 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 ¼½λ þ½L1 : 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 ½Sint ¼Sint: 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 i½Sint ¼ , 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: ϵ⃗i½dμC Tp⃗Tq⃗e int þ dμCϵ⃗i½Tp⃗Tq⃗ e 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 straightfor- wardly. 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 d pdqd 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 i½Tp⃗Tq⃗ e 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 ðqÞ Tp⃗Tq⃗: 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 ðsÞ hTr⃗Ts⃗Tp⃗Tq⃗i 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 ðsÞ hTr⃗Ts⃗Tp⃗Tq⃗i¼½ 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 ðsÞ GðrÞGðsÞ½ ⃗⃗⃗⃗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.
085006-7 VINCENT LAHOCHE and DINE OUSMANE SAMARY PHYS. REV. D 103, 085006 (2021)
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;T ¼ 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 þ ¼1 − 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 xÞ Gð 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 − xÞ GðΘð1 − 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 dia- grams 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{ðdÞ 2ðΛ2η − 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 n bLc 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:
085006-15 VINCENT LAHOCHE and DINE OUSMANE SAMARY PHYS. REV. D 103, 085006 (2021)
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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