PHYSICAL REVIEW D 102, 025003 (2020)
Vector model in various dimensions
† Mikhail Goykhman* and Michael Smolkin The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
(Received 28 March 2020; accepted 22 June 2020; published 6 July 2020)
We study behavior of the critical OðNÞ vector model with quartic interaction in 2 ≤ d ≤ 6 dimensions to the next-to-leading order in the large-N expansion. We derive and perform consistency checks that provide an evidence for the existence of a nontrivial fixed point and explore the corresponding conformal field theory (CFT). In particular, we use conformal techniques to calculate the multiloop diagrams up to and including 4 loops in general dimension. These results are used to calculate a new CFT data associated with the three-point function of the Hubbard-Stratonovich field. In 6 − ϵ dimensions our results match their counterparts obtained within a proposed alternative description of the model in terms of N þ 1 massless scalars with cubic interactions. In d ¼ 3 we find that the operator product expansion coefficient vanishes up to Oð1=N3=2Þ order.
DOI: 10.1103/PhysRevD.102.025003
I. INTRODUCTION requirement of renormalizability. The original argument was motivated by an attempt to avoid hitting the Landau In the past elementary particle theory was based on the pole along the RG flow by imposing a demand that the flow assumption that nature must be described by a renormaliz- of a QFT terminates at the UV fixed point. Asymptotically able quantum field theory. However, a dramatic progress in safe theories are not necessarily renormalizable in the usual the realm of critical phenomena revealed that any non- sense. However, they are interacting QFTs with no unphys- renormalizable theory will look as if it were renormalizable ical singularities at high energies. at sufficiently low energies. From this perspective renor- Just as the requirement of renormalizability leaves only malizable theories correspond to a subset of RG flow few possible interaction terms which one is allowed to trajectories for which all but a few of the couplings vanish. include in the Lagrangian, so does the requirement of In recent years it has become increasingly apparent that asymptotic safety imposes an infinite number of con- renormalizability is not a fundamental physical require- straints, limiting the number of physically acceptable ment, and any realistic quantum field theory may contain theories. Indeed, a UV fixed point in general has only nonrenormalizable as well as renormalizable interactions. finite number of relevant deformations, therefore any In fact, the experimental success of The Standard Model asymptotically safe field theory is entirely specified by a merely imposes a constraint on the characteristic energy finite number of parameters. Some of such theories are scale of any nonrenormalizable interaction. In particular, it represented by a renormalizable field theory at low ener- remains unclear what is the fundamental principle behind gies, while others might be nonrenormalizable. the choice of the trajectory corresponding to the real world Thus, for instance, a scalar field theory respecting from the infinite number of possible theories. ϕ → −ϕ symmetry rules out the possibility of any renor- One of the admissible ways to address this problem is to malizable interaction in five dimensions, yet it exhibits a demand that the Hamiltonian of the theory belongs to a nontrivial UV fixed point with just two relevant deforma- trajectory which terminates at a fixed point in UV. Theories tions in higher dimensions, and therefore it provides an with this property are called asymptotically safe. This example of asymptotically safe nonrenormalizable quan- concept was originally introduced by Weinberg [1,2] as tum field theory [1]. An immediate application of the an approach to select physically consistent quantum field asymptotic safety is related to the problem of quantum theories (QFT). It provided an alternative to the standard gravity [1–4] which is ongoing (see, e.g., [5] and references therein for recent work). *[email protected] Several techniques for studying UV properties of QFTs † [email protected] in particular and their RG flows in general have been developed. One of the approaches is to perform a dimen- Published by the American Physical Society under the terms of sional continuation, and subsequently apply a perturbative the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to expansion around the desired integer-valued physical the author(s) and the published article’s title, journal citation, space-time dimension. This is done with the hope that and DOI. Funded by SCOAP3. the result will have sufficiently accurate convergence
2470-0010=2020=102(2)=025003(28) 025003-1 Published by the American Physical Society MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) behavior to be applied to physically meaningful space-time Furthermore, it was shown that the large-N approach can dimensions, such as d ¼ 3, 4, 5. In this spirit the Wilson- meet the dimensional continuation method (see, e.g., [7] for Fisher fixed point has originally been derived in 4 − ϵ recent developments in this direction), thanks to the simple dimensions, in which case at the fixed point the coupling observation that the IR Wilson-Fisher fixed point in d ¼ constant is perturbative in ϵ [6]. In the context of asymp- 4 − ϵ dimensions can be analytically continued to obtain totic safety, gravity has been studied in 2 þ ϵ dimensions, perturbative UV fixed point in d ¼ 4 þ ϵ dimensions, see [2–4] for some of the early works. albeit the coupling constant at that fixed point is nega- a Recently the OðNÞ vector model of the scalar fields ϕ , tive-valued, and therefore the theory is unstable [1]. In fact, a ¼ 1; …;N, and σ with the relevant cubic interactions the work of [7] was partially motivated by an attempt to 3 ϕaϕaσ and σ in d ¼ 6 − ϵ dimensions has been exten- design a UV completion of the critical OðNÞ vector model sively studied perturbatively in ϵ, see e.g., [7,8]. This model in 4 025003-2 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) 2 counterterms and provide additional evidence in favor of g2Zϕ ¼ m þ δm; ð6Þ the asymptotic safety of the model. In Sec. IV we review diagrammatic and computational techniques for a CFT in 2 g4Zϕ ¼ g˜4 þ δ4; ð7Þ general d, and use them to evaluate various anomalous dimensions and amplitudes of the two-point functions. In where δm and δ4 are mass and quartic coupling counter- Sec. V we calculate the three-point functions associated terms respectively.3 with a CFT that emerges at the fixed point of the model in 2 ≤ ≤ 6 It turns out that all counterterms vanish to leading order d dimensions. We discuss our results in Sec. VI. in the 1=N expansion. Therefore to carry out calculations to the next-to-leading order, it is sufficient to linearize the full II. SETUP action with respect to various counterterms. The action (3) in terms of the renormalized parameters is thus given by In this paper we focus on a d-dimensional Euclidean Z vector model governed by the bare action 1 1 1 1 ¼ d ð∂ ϕ˜Þ2 þ 2ϕ˜2 þ δ ð∂ ϕ˜Þ2 þ δ ϕ˜2 Z S d x μ m ϕ μ m 1 1 2 2 2 2 d 2 2 g4 2 2 S ¼ d xð ∂μϕÞ þ g2ϕ þ ðϕ Þ ; ð1Þ 1 δ 2 2 N 4 2 − 1 þ 2δϕ þ δs − s˜ 4g˜4 g˜4 where the field ϕ has N components, but we suppress the 1 1 þ pffiffiffiffi 1 þ δ þ δ ˜ϕ˜2 ð Þ vector index for brevity. We keep the dimension d general 2 s ϕ s : 9 most of the time, however, some of the specific calculations N are carried out in d ¼ 5 and d ¼ 6 − ϵ. The behavior of the It is convenient to define the following combinations of the model at the fixed point is our main objective, but for now counterterms, we keep the mass parameter g2 to preserve generality. The straightforward and well-known approach (some- δ4 times referred to as the Hubbard-Stratonovich transforma- δˆ ¼ 2δ þ δ − ð Þ s ϕ s ˜ ; 10 tion) to study the model such as (1) is to introduce the g4 auxiliary field s which has no impact on the original path δˆ 1 integral, pffiffiffiffiffi4 ¼ δs þ δϕ; ð11Þ Z g˜4 2 1 1 d 2 2 g4 2 2 S ¼ d x ð∂μϕÞ þ g2ϕ þ ðϕ Þ 2 2 N in terms of which the action can be rewritten as follows 1 2 2 Z − − pgffiffiffiffi4 ϕ2 ð Þ 1 1 1 1 s : 2 ¼ d ð∂ ϕ˜Þ2 þ 2ϕ˜2 − ˜2 þ pffiffiffiffi ˜ϕ˜2 4g4 N S d x μ m s s 2 2 4g˜4 N 1 1 1 1 δˆ After a straightforward simplification this action takes the ˜ 2 ˜2 ˆ ˜2 pffiffiffiffi pffiffiffiffiffi4 ˜ ˜2 þ δϕð∂μϕÞ þ δmϕ − δss þ sϕ : form 2 2 4g˜4 N g˜4 Z 1 1 1 1 ð12Þ d 2 2 2 2 S ¼ d x ð∂μϕÞ þ g2ϕ − s þ pffiffiffiffi sϕ : 2 2 4g4 N Loop corrections along with renormalization conditions fix ð3Þ ˆ ˆ the counterterms δϕ, δm, δs, and δ4. Inverting (10), (11) one then obtains As usual, the model (3) has to be renormalized. ˜ To begin with, we introduce the renormalized fields ϕ, s˜ δˆ δ ¼ 2 pffiffiffiffiffi4 − δ ð Þ pffiffiffiffiffiffi pffiffiffiffiffi s ϕ ; 13 ˜ g˜4 ϕ ¼ Zϕϕ;s¼ Zss;˜ ð4Þ δ δˆ 4 pffiffiffiffiffi4 ˆ where the field strength renormalization constants Zϕ;s are ¼ 2 − δs: ð14Þ g˜4 g˜4 expressed in terms of the counterterms δϕ;s as 3 Zϕ ¼ 1 þ δϕ;Zs ¼ 1 þ δs: ð5Þ It follows from the definition (7) that the interaction term in the original bare action (1) can be expressed as the following sum Similarly, the renormalized mass m and coupling g˜4 are 2 2 ˜2 2 ˜2 2 defined by g4ðϕ Þ ¼ g˜4ðϕ Þ þ δ4ðϕ Þ : ð8Þ 025003-3 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) Finally, renormalization of the quartic coupling constant the Hubbard-Stratonovich auxiliary field s. Obviously, the can be read off (5), (7) and (14) matrix of propagators is diagonal, whereas each nontrivial interaction vertex carries two identical vector indices. δˆ pffiffiffiffiffi4 ˆ ˆ Hence, for simplicity we suppress the Kronecker delta g4 ¼ g˜4 1 þ 2 − 2δϕ − δs ¼ g˜4ð1 þ δs − δsÞ; g˜4 which explicitly emphasizes these facts. We also omit the ð Þ momentum conservation delta function at each vertex. 15 Vertices are denoted by a solid blob, whereas propagators are enclosed by black dots (absence of dots indicates an where in the last equality (13) was used. amputated external leg). The Callan-Symanzik equation can be used to find various relations between the counterterms of the theory. In particular, using it for the three-point function hϕϕsi, we argue in Appendix A that at the UV (IR) fixed point in 4 III. UV FIXED POINT IN d =5 A vast amount of literature, starting from the earlier works [1,24], is dedicated to studying the properties of the This is a geometric series which can be readily written in a ð Þ 4 UV conformal fixed point of the O N vector models with closed form. We denote it by GsðpÞ and represent it quartic interaction in 4 025003-4 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) To regularize the UV divergence we introduce a spherically symmetric sharp cutoff Λ Λ 1 p BðpÞ¼ − ð4m2 þp2Þtan−1 þ2jmjp : 12π3 64π2p 2jmj ð18Þ and the counterterm contribution The power law divergence can be eliminated by adjusting ˆ the counterterm δs. Unlike logarithmic divergences, the power law divergences depend on the details of regulari- zation scheme. For instance, they are absent in dimensional is finite. Introducing a sharp cutoff Λ and focusing on the regularization. Hence, we simply ignore such divergences logarithmic divergences only, yields5 in what follows to reduce clutter in the equations. In particular, in the large momentum limit which we are 1 64 Λ interested in for the purpose of finding the UV fixed point, δ ¼ − ð Þ ϕ 15π2 log μ ; 22 we obtain N where μ is an arbitrary renormalization scale. The anoma- p j ¼ − ð Þ lous dimension of the scalar field ϕ at the UV fixed point B p≫m 128π : 19 can be readily evaluated using the Callan-Symanzik equa- tion for the two-point function of ϕ˜, see Appendix A Using the asymptotic behavior (19) rather than the full expression (18), and thus focusing on the UV regime, one 1 ∂ 1 32 ð Þ γϕ ¼ δϕ ¼ : ð23Þ avoids passing through the pole of the propagator Gs p 2 ∂ log μ N 15π2 at some finite momentum [24]. Therefore in all of our ˜ ≫ 1 calculations we take the limit g4p to simplify the This result is in full agreement with [9,10,25,26]. propagator of the Hubbard-Stratonovich field (see, e.g., [7]), B. Renormalization of the interaction vertex 1 64π G ðpÞ¼− ¼ : ð20Þ There are two loop diagrams that contribute to renorm- s 2BðpÞ p alization of the interaction vertex at the next to leading order in the 1=N expansion. To calculate the corresponding ˜ ˆ By assumption, the beta function for g4 is expected to counterterm δ4 it is enough to set all external momenta to exhibit a UV fixed point. As pointed out in Appendix A, the zero. The first diagram takes the form counterterms must therefore satisfy Eq. (16). In particular, (16) serves as a nontrivial consistency check of the assumption about the UV behavior of the beta function. In order to explicitly verify this identity, we proceed to Oð1Þ calculation of the counterterms to N order. It is natural to set m2 ¼ 0 since we are ultimately interested in the UV behavior of the loop integrals. A. Scalar field two-point function To leading order in the 1=N expansion only GsðpÞ exhibits loop corrections. As argued above, there are no nontrivial UV divergences associated with loop diagrams at whereas the second diagram is given by this order, and all counterterms thus vanish. However, a nontrivial renormalization is induced at the next-to-leading 5If we keep the mass in the action (12), then there is an order in 1=N. additional counterterm of the form Oð1 Þ δ The N contribution to the counterterms ϕ;m are derived from the requirement that the sum of the loop 1 320m2 Λ δ ¼ − log : ð21Þ diagram m N 3π2 μ 025003-5 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) these diagrams with Ci, i ¼ 1; …; 5 and denote their external momentum by u. For simplicity, all the diagrams are amputated, i.e., the two external s propagators are factored out. Thus, for instance where we used (17) to get the following simple relation Z is given by 1 ∂BðqÞ d5p 1 − ¼ : ð24Þ 2 ∂ 2 ð2πÞ5 ð 2 þ 2Þ2ðð þ Þ2 þ 2Þ Z m p m p q m 1 2 4 d5r 64π C1 ¼ − pffiffiffiffi N 2 N ð2πÞ5 r In particular, the loop integral can be calculated by taking Z derivative of (18) with respect to the mass parameter. d5p 1 × : ð26Þ Finally, the tree level counterterm contribution can be ð2πÞ5 p2ðp þ uÞ2ðp þ rÞ2ðp þ r þ uÞ2 read off the Feynman rules we listed in the beginning of this section. Combining everything together and demanding cancellation of the divergences, we obtain This integral is somewhat laborious to evaluate in full generality. However, we are only interested in its behavior δˆ 1 320 Λ at large momenta pffiffiffiffiffi4 ¼ − ð Þ 2 log : 25 g˜4 N 3π μ Z Z 1024π d5p 1 d5r 1 C1 ¼ ð2πÞ5 2ð þ Þ2 ð2πÞ5 5 C. Auxiliary field propagator N p p u r 1 256 δˆ ¼ BðuÞ Λ þ …: ð Þ Next let us evaluate the counterterm s. To this end, we 3π2 log 27 perform renormalization of the Hubbard-Stratonovich N propagator. There are five loop diagrams which contribute at the next to leading order in the 1=N expansion. We tag Similarly, is given by Z Z 6 5 5 2 1 2 2 d q 64π 64π d p 1 C2 ¼ − pffiffiffiffi N : ð28Þ 2 N ð2πÞ5 q jq þ uj ð2πÞ5 p2ðp þ uÞ2ðp þ q þ uÞ2 Expanding it in the region of large momenta, gives 025003-6 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) 18 2 Z 5 Z 5 2 π d p 1 d r 1 An extra factor of two in C4 comes from the interchange of C2 ¼ N ð2πÞ5 p2ðp þ uÞ2 ð2πÞ5 ðr2Þ2 the ordinary vertex with the counterterm. More specifically, Z the integral expressions for these diagrams read d5q 1 × 5 2 2 2 Z ð2πÞ ðq Þ ðr þ qÞ 1 2 2 δˆ 5 1 ¼ 2 − pffiffiffiffi − pffiffiffiffi pffiffiffiffiffi4 d p 1 1024 C4 × N 5 2 2 ¼ ð Þ Λ þ ð Þ 2 g˜4 ð2πÞ p ðp þ uÞ 2 B u log : 29 N N N 3π ˆ δ4 ¼ 4 pffiffiffiffiffi BðuÞ; ð32Þ Next, we focus on g˜4 Z 2 5 2 2 d p −p δϕ C5 ¼ − pffiffiffiffi N ¼ −4δϕBðuÞ: N ð2πÞ5 ðp2Þ2ðp þ uÞ2 ð33Þ Combining the loop diagrams (27), (29), (31), (32), (33) results in the total loop correction to the hs˜ s˜i propagator7 1 2 X5 1 1 512 μ L ¼ − ¼ ð Þ Using Feynman rules results in the following integral s Ci 2 log : 34 2B B N 5π μ0 expression i¼1 Z 2 4 d5p 1 It satisfies C3 ¼ − pffiffiffiffi N N ð2πÞ5 ðp2Þ2ðp þ uÞ2 Z ∂ ∂ δˆ 5 64π pffiffiffiffiffi4 d q Bμ Ls ¼ −μ − δϕ : ð35Þ × : ð30Þ ∂μ ∂μ g˜4 ð2πÞ5 qðp þ qÞ2 L Performing integration over q and keeping the logarithmic Notice that s is finite (the logarithmically divergent terms divergence only,6 yields cancelled out), and therefore ˆ 1 256 δs ¼ 0: ð36Þ C3 ¼ − BðuÞ log Λ þ : ð31Þ N 15π2 Using the relation (13) we can rewrite (35) as The two remaining diagrams are ∂ μ ðδ þ 2 L Þ¼0 ð Þ ∂μ s B s : 37 Our calculations (36), (37) in d ¼ 5 agree with the general expression (16). As discussed at the end of Sec. II we interpret it as a consistency check of the assumption that the model is asymptotically safe. Finally, using the Callan-Symanzik equation for the and two-point function of s˜, see Appendix A, we derive the anomalous dimension of the auxiliary field s˜ to Oð1=NÞ order, 1 ∂ 1 512 γ ¼ μ ðδˆ þ 2BL Þ¼ : ð38Þ s 2 ∂μ s s N 5π2 This result agrees with the literature [9,10,25,26] and serves as a check of the calculation. 6Recall that we ignore scheme dependent power law diver- 7 gences. μ0 is an arbitrary IR scale. 025003-7 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) IV. POSITION SPACE CALCULATION IN A. Preliminary remarks GENERAL DIMENSION Performing the Fourier transform In the previous section we focused on providing an Z evidence for the existence of UV fixed point in the 5D ddk 1 22Δ−d ΓðΔÞ 1 eik·x ¼ ; ð44Þ vector model. Our next goal is to derive a new CFT data ð2πÞd 2 d−Δ d d j j2Δ ðk Þ2 π2 Γð2 − ΔÞ x associated with the model at criticality in 2 ≤ d ≤ 6. The stage is set in this section, whereas the new results are we find the coefficient Cϕ in (41) presented and derived in Sec. V. We start from calculating the next-to-leading Oð1 Þ order N 1 d corrections to the CFT propagators of the fundamental Cϕ ¼ Γ − 1 : ð45Þ d 2 scalar field ϕ and the auxiliary Hubbard-Stratonovich field 4π2 s, working in position space at the UV fixed point. Space- time dimension d will be assumed to be general 4 −2γ μ s h ð Þ ð Þi ¼ ð1 þ Þ ð Þ s x1 s x2 Cs As 2ðΔ þγ Þ ; 42 jx12j s s In particular, the calculation of each diagram starts from counting and writing down explicitly all the factors of Cϕ . where μ is an arbitrary renormalization scale, γϕ;s represent ;s The loop diagram in position space simply amounts to anomalous dimensions, whereas the symbols Δϕ;s stand for the scaling dimensions of the fields ϕ and s at the Gaussian adding two powers together. fixed point, Δ ¼ d − 1 Δ ¼ 2 2Δ þ Δ ¼ ð Þ ϕ 2 ; s ; ϕ s d: 43 While we have already derived the anomalous dimensions γϕ;s in Sec. II for the five dimensional model, in this section we calculate both the anomalous dimensions and the We also need the propagator splitting/merging relation – amplitudes Aϕ;s in general dimension d. Our findings in [25 27] this section are in full agreement with the known results in the literature, e.g., [33], and therefore the reader familiar 8It agrees with [7] after reconciling conventions for normali- with the subject can proceed to the next section. zation of the Hubbard-Stratonovich field. 025003-8 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) Z 1 d where the middle vertex on the l.h.s. is assumed to be d x3 2 a 2 b α ¼ −2 β ¼ −2 ðx3Þ ððx3 − x12Þ Þ integrated over, and we introduced d a3, d a2, 1 γ ¼ d − 2a1. Here we also accounted for the Feynman rule ¼ Uða; b; d − a − bÞ ; ð48Þ associated with the cubic vertex. 2 aþb−d ðx12Þ 2 where Γðd − ÞΓðd − ÞΓðd − Þ d 2 a 2 b 2 c Uða; b; cÞ¼π2 : ð49Þ ΓðaÞΓðbÞΓðcÞ Expression (48) can be graphically represented as B. ϕ propagator where the middle point on the left-hand side (l.h.s.) is an Let us consider the 1=N correction to the ϕ propagator. integrated over vertex. Additionally, we will make use of It is given by the following 1-loop diagram the following identity for a1 þ a2 þ a3 ¼ d, also known as the uniqueness relation [25–27,34–37] Z 1 ddx 2a1 2a2 2a3 jx1 − xj jx2 − xj jx3 − xj Uða1;a2;a3Þ ¼ ; ð50Þ d−2a3 d−2a2 d−2a1 jx12j jx13j jx23j where the function U is defined in (49). The uniqueness relation can be represented diagrammatically as [25–27] This diagram reveals a simplest application of the Feynman rules formulated in the previous subsection. It diverges in any d, and therefore we regularize it by adding δ=2 ≪ 1 to the scaling dimension of s [25–27]. In other words, we analytically continue the diagram in Δs rather than in d. Applying the merging relation (48) twice, yields Z 2 2 1 ð Þ¼ − pffiffiffiffi 3 μ−δ d P x1;x2 CϕCs d x3;4 2Δ 2Δ 2Δ þ2Δ þδ N jx13j ϕ jx24j ϕ jx34j ϕ s Z 4 δ δ 1 3 −δ d ¼ C C μ U Δϕ; Δϕ þ Δ þ ; − d x3 ϕ s s 2Δϕ dþδ N 2 2 jx13j jx23j 4 δ δ þ δ − δ 1 ¼ 3 μ−δ Δ Δ þ Δ þ − Δ d d − Δ ð Þ CϕCs U ϕ; ϕ s ; U ϕ; ; ϕ 2Δ þδ : 51 N 2 2 2 2 jx12j ϕ Combining with the free propagator (39) and expanding around δ ¼ 0, we obtain Cϕ 2γϕ 1 hϕð Þϕð Þi ¼ 1 þ þ þ OðδÞ ð Þ x1 x2 2Δ Aϕ δ : 52 jx12j ϕ δ ðjx12jμÞ Here 1 2d sinðπdÞΓðd−1Þ γ ¼ 2 2 ð Þ ϕ 3=2 d ; 53 N π ðd − 2ÞdΓð2 − 2Þ 025003-9 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) d 2 We introduced an extra power η to the internal lines Aϕ ¼ γϕ − : ð54Þ 2 − d d representing ϕ propagator, see e.g., [25,26,38,39]. Ulti- mately we are interested in the limit η, δ → 0. However, Divergence in the correlation function can be readily this limit can be reliably taken as long as η ¼ OðδÞ. Indeed, removed by the wave function renormalization, the resulting diagram is both symmetric with respect to η → −η and finite as η → 0, therefore its series expansion pffiffiffiffiffiffi 2γ η ¼ 0 ϕ ¼ ϕ˜ ¼ 1 þ ϕ ð Þ around takes the form [28] Zϕ ;Zϕ δ : 55 ˜ 2 4 C1 ¼ f0 þ f2η þ f4η þ ð59Þ As a result, the correlation function for the physical field ϕ˜ to the next-to-leading order in the 1=N expansion takes the Further expanding it around δ ¼ 0 and keeping in mind that form (41) with γϕ, Aϕ given by (53), (54). the coefficients fa have at most simple poles at δ ¼ 0,we Note that γϕ in d ¼ 5 agrees with (23), whereas in ˜ conclude that C1 ¼ f0 in the limit δ → 0, as long as general d our results match [9,10,25,26,33]. In particular, η ¼ OðδÞ. It is convenient to choose η ¼ δ=2. As a result, the anomalous dimension (53) and the amplitude shift (54) we obtain in d ¼ 6 − ϵ dimensions are given by Z ϵ 1 2 4 1 2 ˜ ¼ − pffiffiffiffi d γϕ ¼ þ Oðϵ Þ; ð56Þ C1 N d x3;6 2Δ N 2 ðjx13jjx26jÞ s Z N 1 11ϵ d 2 × d x5 δ δ Aϕ ¼ − þ Oðϵ Þ: ð57Þ 2Δϕþ 2Δϕþ jx35j 2jx56j 2 6N Z 1 d ð Þ × d x4 2Δ −δ 2Δ þδ 2Δ −δ : 60 j j ϕ 2j j s j j ϕ 2 C. Hubbard-Stratonovich propagator x34 x45 x46 1 Next we derive the =N correction to the propagator of Presence of η ¼ δ=2 makes it possible to use the unique- s the auxiliary field . As in Sec. III, we have three diagrams ness equation (50) to integrate over x4. Applying sub- C1;2;3. Furthermore, since the model is regulated by analytic sequently the propagator merging relation (48) to the continuation in the scaling dimension of the fields, there is integral over x5, yields no need to introduce explicitly the counterterm diagrams C4 5 of Sec. III. They are associated with the wave function ; ˜ 8 δ δ δ C1 ¼ U Δϕ − ; Δϕ − ; Δ þ renormalization and we merely implement it at the end of N 4 4 s 2 calculation. þ δ þ δ For each diagram we begin by counting and writing d d −δ × U 2 ; 2 ; down the prefactor associated with the amplitudes Cϕ;s.To Z this end, denote 1 1 d ð Þ × d x3;6 2Δ 2 −2Δ þδ : 61 ðj jj jÞ s j j d s 3 4 ˜ −δ x13 x26 x36 C1ðx1;x2Þ¼CsCϕC1ðx1;x2Þμ ; ð58Þ ˜ We postpone carrying out the remaining integrals over the where C1ðx1;x2Þ is given by two edge points x3;6. Similar integrals appear in the calculation of C2;3, and therefore it makes sense to evaluate them after we have assembled all the terms C1;2;3 together. The next diagram contributing to the s propagator at the Oð1=NÞ order is denoted by 4 6 ˜ −δ C2ðx1;x2Þ¼CsCϕC2ðx1;x2Þμ ; ð62Þ where 025003-10 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) Notice that we regularized this diagram by supplementing the internal s propagators with an additional power of δ=2 rather than δ. This is done in order to ensure the same total compensating power of μ on the right-hand side (r.h.s.) of (62) −δ as in C1, namely μ , and therefore such prescription guarantees consistency of regularization of the diagrams [28]. ˜ We have also modified some of the ϕ lines, relying on the technique we used in evaluating the diagram C1. Setting δ η ¼ 2 [38,39],seealso[28], allows us to carry out the integrals over x3;6 using the uniqueness relation (50), Z Z 1 2 6 1 1 ˜ ¼ − pffiffiffiffi 2 d d C2 N d x7;8 2Δ d x4;5 2Δ þδ 2Δ þδ 2 N ðjx17jjx28jÞ s jx48j ϕ 2jx57j ϕ 2 Z Z 1 1 d d ð Þ × d x3 2Δ −δ 2Δ þδ 2Δ d x6 2Δ −δ 2Δ þδ 2Δ : 63 jx37j ϕ 2jx34j s 2jx35j ϕ jx68j ϕ 2jx56j s 2jx46j ϕ Integrating over x3;6,gives Z 32 δ δ 2 1 ˜ ¼ Δ − Δ þ Δ d ˜ ð ;0Þ ð Þ C2 U ϕ ; s ; ϕ d x7;8 2Δ c2 x7;x8 ; 64 N 4 4 ðjx17jjx28jÞ s where Z 1 0 d ˜2ð 7 8; η Þ¼ 4 5 0 0 0 0 ð Þ c x ;x d x ; 2d−3Δ −η 2d−3Δ þη 2Δ þδ Δ −η Δ þη : 65 jx48j s jx57j s jx45j s jx47j s jx58j s Here we have introduced an additional exponent η0 [25,26,28] which has little impact on the final value of the diagram. Indeed, by performing the following change of integration variables x4 → x7 þ x8 − x5;x5 → x7 þ x8 − x4; ð66Þ 0 0 ˜ 0 one can see that c˜2ðx7;x8; η Þ¼c˜2ðx7;x8; −η Þ. Therefore, similarly to the calculation of C1, any choice of η ¼ OðδÞ 0 δ does not change the value of the diagram in the limit δ → 0. Specifically choosing η ¼ 2 makes it possible to apply the uniqueness relation (50) to the integral over x4 [28] followed by the propagator merging relation (48) to the integral over x5 32 δ δ 2 3Δ δ δ Δ δ ˜ s s C2 ¼ U Δϕ − ; Δ þ ; Δϕ U d − − ; Δ þ ; − N 4 s 4 2 4 s 2 2 4 Z d þ δ d þ δ 1 1 −δ d ð Þ × U ; ; d x7;8 2Δ 2d−2Δ þδ : 67 2 2 ðjx17jjx28jÞ s jx78j s Finally, the last diagram is given by 3 4 ˜ −δ C3ðx1;x2Þ¼Cs CϕC3ðx1;x2Þμ ; ð68Þ where 025003-11 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) As before, divergence in the correlation function is removed by the wave function renormalization, pffiffiffiffiffi 2γ ¼ ˜ ¼ 1 þ s ð Þ s Zss; Zs δ : 75 In particular, the correlation function for the physical field s˜ to the next-to-leading order in the 1=N expansion takes the form (42) with γs, As given by (73), (74). This diagram contains a subdiagram which was evalu- Notice that the wave function renormalization constants ated already, namely, the one-loop correction to the ϕ (55), (75) generate the following renormalization of the propagator. Hence, it can be readily simplified bare term in the action (3) 1 1 2γ þ γ 1 2 ˜2 ϕ s ˜2 ˜ 16 δ δ pffiffiffiffi ϕ s ¼ pffiffiffiffi ϕ s˜ þ pffiffiffiffi ϕ s;˜ ð76Þ C3 ¼ U Δϕ; Δϕ þ Δ þ ; − δ N s 2 2 N N N d þ δ d − δ where we keep Oð1=NÞ corrections only. Thus the counter- × U Δϕ; ; − Δϕ 2 2 term is given by Z 1 1 d ð Þ 2γϕ þ γ 1 × d x3;4 2Δ 2d−2Δ þδ : 69 c:t: s ˜2 ðjx13jjx24jÞ s jx34j s S ¼ pffiffiffiffi ϕ s:˜ ð77Þ int δ N Next we combine C1 2 3. To begin with, we note that ; ; To avoid clutter we suppress tilde above the renormalized (61), (67), (69) exhibit a similar structure, fields ϕ˜, s˜ in what follows. Of course, physical correlations Z 1 1 are associated with the renormalized fields. ˜ ¼ ˆ d ð Þ γ ¼ 5 Ci Ci d x3;4 2Δ 2d−2Δ þδ : 70 Before closing this section we note that s in d ðjx13jjx24jÞ s jx34j s agrees with (38), whereas in general d our results match [9,10,25,26,33]. In particular, the anomalous dimension Using (48) twice to perform the remaining integrals over (73) and the amplitude shift (74) in d ¼ 6 − ϵ dimensions x3 4, yields ; are given by δ δ þ δ − δ ˜ ¼ Δ − Δ þ − Δ d d − Δ 40ϵ Ci U s;d s 2 ; 2 U s; 2 ; 2 s γ ¼ þ Oðϵ2Þ; ð78Þ s N 1 ˆ ð Þ × Ci 2Δ þδ : 71 44 jx12j s A ¼ þ OðϵÞ: ð79Þ s N Summing up all three diagrams and expanding around δ ¼ 0 we obtain V. CONFORMAL THREE-POINT FUNCTIONS 2γ 1 In this section we calculate the next-to-leading order h ð Þ ð Þi ¼ Cs 1þ s þ þOðδÞ s x1 s x2 2Δ As δ ; correction to the OPE coefficients of the conformal three- j j s δ ðj μjÞ x12 x12 point functions hϕϕsi and hsssi associated with the funda- ð72Þ mental scalar field ϕ and the auxiliary Hubbard-Stratonovich field s. In a general d-dimensional OðNÞ symmetric CFT, with the results for hϕϕsi to the next-to-leading order and leading order hsssi correlator are known based on the conformal 1 4 sinðπdÞΓðdÞ bootstrap approach [9,10] (see also [40,41] for discussion in γ ¼ 2 ð Þ s d d ; 73 the context of OðNÞ sigma model). In contrast, we recover N πΓð2 þ 1ÞΓð2 − 1Þ these results in the critical ϕ4 model from the new per- dðd − 3Þþ4 πd spective which has not previously been given in the A ¼ 2γϕ H −3 þ π cot literature, and extend the calculations to derive a new s 4 − d d 2 OPE coefficient of the three-point function hsssi. 8 2 2 Furthermore, to facilitate and make progress in the þ þ þ − 2d − 1 ; ð74Þ ðd − 4Þ2 d − 2 d calculations we evaluate the 4-loop conformal triangle diagram, and the associated 3-loop trapezoid graph, as where Hn is the nth harmonic number. well as the 3-loop bellows diagram in general dimension d. 025003-12 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) To the best of our knowledge, these diagrams were not it agrees with [9,10].10 In fact, the only known complete evaluated in the literature before, and the corresponding derivation of the hϕϕsi three-point function to the next-to- analytic expressions were not displayed elsewhere. These leading order in the large-N expansion has been given in diagrams naturally appear in numerous conformal calcu- [9,10] by solving the consistency constraints in the boot- lations, and therefore might be useful in the future. strap approach. Therefore a direct diagrammatic technique, which is followed below, provides an independent deriva- hϕϕ i A. hϕϕsi tion of the result for the s three-point function. We begin by determining the dressed propagators and hϕϕ i The leading order result for the s correlator follows vertices, which will also be used later to calculate the next- directly from the tree-level diagram, to-leading order correction to the three-point function hsssi. The relevant diagrams are obtained by dressing ϕϕ hϕðx1Þϕðx2Þsðx3Þi the propagators and s interaction vertex of the tree Zleading level diagram. Let us denote the dressed propagators (41) 2 1 and (42) by a line and a grey blob, ¼ − pffiffiffiffi 2 d CϕCs d x 2Δ 2Δ 2Δ : N jx1 − xj ϕ jx2 − xj ϕ jx3 − xj s ð80Þ Using the uniqueness relation (50) we obtain the conformal three-point function Here the set of parameters ðC; A; Δ; γÞ stands either for Cϕϕ ð Δ γ Þ ϕ hϕð Þϕð Þ ð Þi ¼ s ð Þ Cϕ;Aϕ; ϕ; ϕ , when the -propagator is considered, or x1 x2 s x3 2Δ −Δ Δ Δ ; 81 j j ϕ s j j s j j s ð Δ γ Þ leading x12 x13 x23 for Cs;As; s; s , when the line represents s-propagator. The full three-point function can then be obtained by where the leading order coefficient is given by evaluating the following diagram 2 pffiffiffiffi 2 Cϕϕs ¼ − CϕCsUðΔϕ; Δϕ; ΔsÞ: ð82Þ N It is convenient to express the three-point functions in terms of normalized fields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ → ϕ Cϕð1 þ AϕÞ;s→ s Csð1 þ AsÞ; ð83Þ i.e., fields whose two-point functions are normalized to ˜ unity. The OPE coefficient, Cϕϕs, associated with the renormalized fields is given by9 Cϕϕ ˜ ¼ s ð Þ Cϕϕs 1 ; 84 2 CϕCs or equivalently [9,10], where the grey blob in the center represents a dressed 11 2 1 three-point vertex. By conformal invariance this diagram ˜ pffiffiffiffi 2 Cϕϕs ¼ − CϕCsUðΔϕ; Δϕ; ΔsÞ: ð85Þ N takes the form In particular, in d ¼ 6 − ϵ dimensions we obtain hϕðx1Þϕðx2Þsðx3Þi −2γ −γ rffiffiffiffiffi ð1 þ Þμ ϕ s ˜ Wϕϕs 6ϵ ¼ Cϕϕ ; ð87Þ ˜ s 2ðΔϕþγϕÞ−ðΔ þγ Þ Δ þγ Δ þγ Cϕϕ ¼ − þ OðϵÞ: ð86Þ jx12j s s jx13j s s jx23j s s s N Next we evaluate the 1=N correction to the three-point 10We thank Simone Giombi, Igor Klebanov and Gregory ˜ coefficient Cϕϕs in general dimension, and demonstrate that Tarnopolsky for attracting our attention to [10] and encouraging us to carry out the calculation in general dimension. 11By definition, the dressed vertex has no propagators attached 9 We ignore Aϕ;s in (83) to leading order in the 1=N expansion. to the external legs (amputated diagram). 025003-13 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) where the fields ϕ and s are normalized such that their two- Expansion of Uˆ in 1=N has the following form point function has unit amplitude. Our goal is to calculate 1 Wϕϕ to order Oð Þ. It represents a subleading correction to s N ˆ 1 ˜ U ¼ Nu1 þ u2 þ O : ð93Þ the OPE coefficient Cϕϕs. N The dressed 1PI vertex, Γðx; y; zÞ, in the above diagram Oð1 Þ was evaluated in [33] to =N order, see Eq. (4.18) there, Notice that it starts with the divergent in N → ∞ ZZZ piece Nu1, originating from the singular in that limit d γs Γðx1;x2;x3Þϕðx1Þϕðx2Þsðx3Þ factor Uð2 − γϕ − 2 ; Þ in (92). It is convenient to denote the ratio of the subleading to leading coefficients ZZZ −2γ −γ ˆ μ ϕ s ϕð Þϕð Þ ð Þ ¼ pZffiffiffiffi x1 x2 s x3 ð Þ in (93) by 2α 2α 2β ; 88 N jx3 − x1j jx3 − x2j jx1 − x2j u2 γ ðð26 − 3dÞd − 44Þþ2γϕððd − 6Þd þ 4Þ uˆ ¼ ¼ s : where in notations of [33] Nu1 2ðd − 4Þðd − 2Þ γ γ ð94Þ α ¼ Δ − s β ¼ Δ − γ þ s ð Þ ϕ 2 ; s ϕ 2 ; 89 Here the anomalous dimensions γϕ, γs are given by (53), (73). ˆ 12 and Z is a constant given by Eq. (A.4) in [33] Combining uˆ with the other next-to-leading order terms in (91),weobtain χ ð1Þ2 ðμ˜ − 2ÞΓðμ˜Þ ˆ ¼ − A A ð1 þ δ 0Þ Z 2 π2μ˜ V ; ¼ þ As þ δ ð Þ Wϕϕs Aϕ 2 V; 95 χ ¼ −ð2γϕ þ γsÞ; Γðd − ΔÞ d ðΔÞ¼ 2 μ˜ ¼ where A ΓðΔÞ ; 2 ; η ðμ˜ − 3Þð6μ˜2 − 9μ˜ þ 2Þ 0 δ 0 ¼ 1 δV ¼ uˆ þ δV V 2 ; N ðμ˜ − 2Þ −3 π −1 1 2d ðdðdð5d − 42Þþ116Þ − 96Þ sinð dÞΓðd Þ 4ð2 − μ˜ÞΓð2μ˜ − 2Þ ¼ 2 2 ; π3=2ð − 4Þð − 2ÞΓðd þ 1Þ η1 ≡ : ð Þ N d d 2 Γðμ˜ − 1Þ2Γð2 − μ˜ÞΓðμ˜ þ 1Þ 90 ð96Þ To get Wϕϕs one should simply attach the dressed propagators to Γðx; y; zÞ and apply the uniqueness for- which we can rewrite as mula (50) three times, because all 3 vertices x1;2;3 happen to be unique. Normalizing the external legs, yields 8 2 6 δV ¼ γϕ þ þ þ 5 ; ð97Þ ðd − 4Þ2 d − 2 d − 4 2 2 2 C ð1 þ AϕÞ C ð1 þ A Þ ˜ pffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϕ s s ˆ ˆ Cϕϕsð1 þ WϕϕsÞ¼− Z U; N 2 2 where γϕ is given by (53). Substituting (54), (74) and (97) into Cϕð1 þ AϕÞ Csð1 þ AsÞ (95) gives Wϕϕs in general dimension. It matches [9,10] ð91Þ where the generic OðNÞ symmetric CFTwas studied, e.g., see 13 Eq. (21) in [9]. In particular, the above Wϕϕs agrees with its where we defined counterpart in a CFT emerging at the IR fixed point of the OðNÞ vector model with cubic interactions in d ¼ 6 − ϵ γ γ ˆ ¼ Δ − s Δ − s Δ þ γ dimensions [7].Inthiscaseonegets U U ϕ 2 ; ϕ 2 ; s s γ γ 13Note that the leading terms in the large N expansion on the Δ − s Δ þ γ Δ − γ þ s × U ϕ 2 ; ϕ ϕ; s ϕ 2 l.h.s. and r.h.s. of (91) agree, as one can explicitly verify by direct calculation, Δ þ γ γ s s d s Δ þ γ − γ − ð Þ 2 × U ϕ ϕ; ; ϕ : 92 2 1 χ ð1Þ ðμ˜ − 2ÞΓðμ˜Þ 2 2 2 2 A A − pffiffiffiffi CϕCs − Nu1 N 2 π2μ˜ 2 1 12 pffiffiffiffi 2 ˜ Our notations differ from those of [33]. The precise relation ¼ − CϕCsUðΔϕ; Δϕ; ΔsÞ¼Cϕϕs: ð98Þ N is μ → μ˜, γψ → γs, HðΔÞ → AðΔÞ. 025003-14 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) 1 21ϵ 22 δ ¼ þ Oðϵ2Þ ¼ þ OðϵÞ ð Þ where the external legs are not normalized, and there- V ;Wϕϕs : 99 ˜ N 2 N fore Cϕϕs rather than Cϕϕs appears on the r.h.s. The term δV ∼ Oð1=NÞ equals Wϕϕs up to corrections Aϕ;s that we While this way of calculating Wϕϕs is straightforward, stripped off. In fact, δV is closely related to the sum of two it does not separate the effect of the anomalous dimen- loop diagrams, which have been studied in momentum sions intrinsic to the propagators from the contribution of space in Sec. III B. Notice that δV is rendered finite, the dressed vertex. Such a separation proves to be useful because besides those two loop diagrams there is a when we evaluate hsssi correlation function. In other contribution from the vertex counterterm (77). The last words, the impact of anomalous dimensions inherent to diagram on the r.h.s. simply subtracts the leading 1=N the propagators is singled out in our method of calculat- contribution from the second diagram ing the hsssi correlation function. Hence, for future purpose we define two additional Feynman rules. To begin with, we exclude Aϕ;s from the residue of the dressed propagators. These constants represent 1=N corrections to the amplitudes of the two-point functions and can be accounted at the very end. The dressed propagators without Aϕ;s will be denoted by a solid black blob14 where the last term within the parenthesis stands for the counterterm associated with the wave function renormal- ization (55) and (75), and we used regularized expressions for the dressed propagators obtained previously by the As before ðC; Δ; γÞ stand for ðCϕ ; Δϕ ; γϕ Þ, depending ;s ;s ;s direct calculation of the Feynman graphs. Note that the on the considered propagator. newly defined vertex with the bare propagators attached to Next we define a dressed vertex with the bare propa- it has logarithmic terms proportional to the anomalous gators attached to it, i.e., propagators without anomalous dimensions. Such terms emerge because of renormalization dimensions and Aϕ;s. Diagrammatically it is given by of the vertex, and are intrinsic to the vertex itself rather than to the propagators. We choose to keep track separately of the contributions associated with the dressed vertex correction δV and anomalous dimensions γϕ;s of the propagators. Hence, the above diagrams will be used extensively in what follows. This approach turns out to be useful when we evaluate the hsssi correlation function. B. hsssi In this subsection we calculate the three-point function h i Oð 1 Þ sss to N3=2 order. For the normalized field (83) it has the form −3γ ð1 þ 3 Þμ s Ws h ð Þ ð Þ ð Þi ¼ ˜ 3 s x1 s x2 s x3 Cs Δ þγ Δ þγ Δ þγ : jx12j s s jx13j s s jx23j s s ð101Þ ˜ We are going to calculate the leading coefficient C 3 and the ð100Þ s 1 3 =N correction Ws . The leading behavior is completely determined by the 14 Note that it is crucial to account for Aϕ;s, see e.g., (95). one-loop triangle diagram 025003-15 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) of diagrams, recall that the dressed ϕ- and s-propagators have a nontrivial 1=N correction due to the amplitudes Aϕ, As, as well as due to the anomalous dimensions γϕ, γs.We will account for the contribution related to Aϕ;s later in this subsection, while for now we focus on studying the effect associated with the dressed vertex (100) and the anomalous dimensions γϕ;s inherent to the dressed propagators. Quite surprisingly, it turns out that various diagrams obtained by dressing the leading hsssi triangle can be grouped in such a way that the integrals over their internal vertices can be carried out using the uniqueness relation. We illustrate how it works now. Dressing each of the three ϕϕs vertices of the leading hsssi triangle diagram gives Normalizing the external legs yields hsðx1Þsðx2Þsðx3Þi leading Z 2 3 C3 C3 1 ϕ s d ¼ N − pffiffiffiffi pffiffiffiffiffiffi d x4 5 3 ; 2Δ 2Δϕ 2Δ N C jx35j s jx45j jx24j s Z s 1 d ð Þ × d x6 2Δ 2Δ 2Δ : 102 jx16j s jx46j ϕ jx56j ϕ Applying uniqueness formula (50) to the integrals over x4;5;6, we arrive at the leading hsssi triangle expression Here the vertex blob is determined by (100). In addition, ˜ 3 each of the three internal ϕ-propagators and each of the h ð Þ ð Þ ð Þi ¼ Cs ð Þ s x1 s x2 s x3 Δ Δ Δ ; 103 s jx12j s jx23j s jx13j s three external -propagators in the leading triangle diagram leading need to be endowed with the anomalous dimensions. The corresponding diagram is given by where 8 3 Δ 3Δ ˜ pffiffiffiffi 3 2 2 s s C 3 ¼ − CϕCsUðΔϕ; Δϕ; ΔsÞ U ; Δs;d− : s N 2 2 ð104Þ One can readily show that for general d Δ 3Δ − 3 2 ðΔ Δ Δ Þ s Δ − s ¼ d CϕCsU ϕ; ϕ; s U 2 ; s;d 2 2 ; ð105Þ and therefore using (85) we obtain ˜ 3 ¼ 2ð − 3Þ ˜ ð Þ Cs d Cϕϕs; 106 where we used the black blob propagator notation intro- duced in Sec. VA. in agreement with [9,10]. To avoid overcounting the contribution of the leading Next we evaluate the subleading term Ws3 in (101).Itis order triangle diagram, it should be subtracted from the 3 2 Oð1 = Þ 3 ∼ Oð1 Þ determined by the =N diagrams with three external above graphs. The remainder contributes to Ws =N legs of type s. A large portion of them is obtained by which is the ultimate goal of our calculation. However, dressing the constituents of the leading order triangle to avoid clutter we do not carry out these subtractions diagram with 1=N corrections. Before studying this class explicitly. They are done by default in what follows. 025003-16 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) Remarkably, when summing the Feynman diagrams with identical skeleton structure, the anomalous dimensions of the propagators behave additively at 1=N order. This tremendous simplification holds because γϕ;s ∼ 1=N, and therefore one can linearize a Feynman graph with respect to the anomalous dimensions. For instance, each of the three dressed vertices of the leading order triangle diagram contributes a 1=N term associated with the second diagram on the r.h.s. of (100). Combining these terms with the above Feynman graph gives Since this relation between the diagrams is reliable up to Oð1=NÞ order, we linearize over the γϕ in the internal ϕ-propagators and retain the next-to-leading corrections only. They are given by the sum of three diagrams which are identical up to a permutation of the external legs 2 2 2 where we integrated over the unique vertex with the lines Δϕ, Δϕ, Δ and used (82). The factor of ð− pffiffiffiÞ NCϕC is s N s associated with the Feynman rules for the vertices x4;5, the closed ϕ-loop, and the leading order amplitudes of the ϕ- and s- propagators. Furthermore, there is a contribution related to the conformal triangle on the r.h.s. of (100). There are three such terms, since there are three vertices in the leading order correlation function hsssi, 025003-17 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) As before only Oð1=NÞ terms are eventually retained, and therefore the last two diagrams can be combined by simply adding the anomalous dimensions of the corresponding propagators where the integrals over x4;5 were carried out using the uniqueness relation. Note that only terms up to Oð1=NÞ order are reliable, because we added the anomalous dimensions to combine the diagrams. Using (105), (106), the contribution to Ws3 takes the form Δ þγ 3Δ þγ γ γ UðΔ ; s s ;d− s sÞUðΔϕ þ s ; Δϕ − s ; Δ Þ 3δV þ fˆ ¼ 3 δV þ s 2 2 2 2 s − 1 Δs 3Δs UðΔs; 2 ;d− 2 ÞUðΔϕ; Δϕ; ΔsÞ πd 2 πd 6 sinð ÞΓðdÞð− þ π cotð ÞþH −4Þ ¼ 3δ þ 2 d−4 2 d þ Oð1 2Þ ð Þ V d d =N ; 107 NπΓð2 − 1ÞΓð2 þ 1Þ where Hn is the nth harmonic number. Expanding around d ¼ 6, yields 120 fˆ ¼ − þ Oðd − 6Þ: ð108Þ N 025003-18 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) d d Another contribution to Ws3 arises from the following Uð3 − 2 ; 2 − 1;d− 2Þ wˆ3 ¼ v3; ð110Þ diagram Uð1; 1;d− 2Þ where v3 is determined by the diagram Attaching propagator lines with powers 2d − 4 to x2;3, and integrating both sides of the obtained equation with respect Integrating over x4−9 through the use of uniqueness relation to x2;3, yields (50), and normalizing the external s legs according to (83), we obtain an additional contribution to Ws3 of the form ð3 − d d − 1 − 2Þ ˜ U 2 ; 2 ;d w3=C 3 with v3 ¼ v˜3ð0Þ; ð111Þ s Uð1; 1;d− 2Þ 1 2 9 pffiffiffiffi 3 9 6 3 w3 ¼ − N CϕCsUðΔϕ; Δϕ; ΔsÞ where v˜3ðδÞ is defined by the diagram 3=2 N Cs Δ 3Δ 3 s Δ − s ˆ × U 2 ; s;d 2 w3 3 64ð − 3Þ 3 d 3 2 ¼ − CϕCswˆ3; ð109Þ N3=2 where in the last line we used (105), and wˆ3 is defined by the diagram15 Here we have introduced an auxiliary regulator, δ, in order to apply the integration by parts relation (the diagram itself is finite in δ → 0 limit). We relegate the details to Appendix B. Combining (85), (106), (113), (110), (111) gives the result 2 ð3 − d d − 1 − 2Þ2 w3 16ðd − 3Þ 2 U 2 ; 2 ;d ¼ C C v˜3ð0Þ; ˜ ϕ s ð1 1 − 2Þ2 ðd−2 d−2 2Þ Cs3 N U ; ;d U 2 ; 2 ; ð112Þ where v˜3ð0Þ is given by (B9). Integrating both sides of this diagrammatic equation with Next, we evaluate the following contribution to Ws3 respect to x1 we obtain represented by the 3-loop “bellows”16 diagram17 15 From now on we explicitly use the values (43) for the scaling 16The name is based on the visual resemblance of the three- Δ ˆ dimensions ϕ;s. To get w3 from this diagram one needs to dimensional shape of this diagram to a bellows, see, e.g., https:// integrate over the three internal points. We already accountedp forffiffiffiffi en.wikipedia.org/wiki/Bellows. 17 the amplitudes Cϕ;s of the propagators and factors of −2= N We are grateful to anonymous referee at PRD who pointed coming from the interaction vertices. out this diagram to us. 025003-19 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) Normalizing the external s legs in accord with (83), and using the uniqueness relation (50) to integrate over the five 18 points x4−8, yields 4 w4 3 2 4 2 2 3 ⊃ ¼ − pffiffiffiffi ðΔ Δ Δ Þ Ws ˜ 2 NCϕCsU ϕ; ϕ; s v4; Cs3 N ð113Þ This is the so-called self-energy diagram. It can be reduced to the known ChTðα; βÞ graph, given by eq. (16) in [26].19 where v4 is defined by the following diagrammatic equa- pffiffiffiffi To this end we perform an inversion transformation on the tion (the multiplicative factors of −2= N and Cϕ in the ;s external point x and on both of the integrated vertices. This vertices and propagators of this diagram should be stripped gives v4 ¼ ChTð1; 1Þ. Combining all together, we arrive at off, because we already took them into account) 3dðd − 2Þðπ2 − 6ψ ð1Þðd − 1ÞÞ w4 ¼ γ 2 ð Þ ˜ ϕ 4ð − 4Þ ; 114 Cs3 d where ψ ð1Þ is the first derivative of the digamma function. In particular, 9 1 w4 ¼ − ð − 4Þ2ψ ð2Þð1ÞþOðð − 4Þ3Þ → 4 ˜ 2 d d ;d; Cs3 N ð115Þ 1 w4 ¼ −54 ð − 6ÞþOðð − 6Þ2Þ → 6 ð Þ ˜ d d ;d: 116 Cs3 N 2 6 Integrating both sides of this equation with respect to x3,we The plot of (114) for the range of values 025003-20 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) ˜ ˜ FIG. 2. Nw3=C 3 as a function of space-time dimension FIG. 1. Nw4=C 3 as a function of space-time dimension s s 2 6 2 3 w3 þ w4 3 ¼ 3δ þ ˆ þ þ 3 þ ð Þ Ws V f 2 As Aϕ ˜ : 117 Cs3 ˜ artifact of our normalization. Note that Cs3 has a simple ¼ 3 Substituting (95), yields zero in d . The theory becomes free at the fixed point in the limit 20 → 4 3 ð → 4Þ w3 þ w4 d , therefore Ws d must vanish. Indeed, based 3 ¼ 3 þ ˆ þ ð Þ Ws Wϕϕs f : 118 ð → 4Þ¼0 ˜ 3 on our (95), Wϕϕs d , in agreement with [9,10]. Cs w3þw4 From (118) it then remains to show that ˜ ðd → 4Þ¼0, C 3 ˜ s Here w3=Cs3 is given by (112), which we rewrite as ˜ which is indeed the case (in fact w3;4=Cs3 contributions vanish individually), as can be seen from (115), (119), w3 ¼ ˜ ð0Þ ð Þ ˜ b3v3 ; 119 (121), (123). Using (107) one can also verify that Cs3 fˆðd ¼ 4Þ¼0. where v˜3ð0Þ is calculated in Appendix B, and Moreover, it follows from (119), (122), (124) that in the vicinity of d ¼ 6,wehave 2 ð3 − d d − 1 − 2Þ2 16ðd − 3Þ 2 U 2 ; 2 ;d b3 ¼ C C : 216 N ϕ s ð1 1 − 2Þ2 ðd−2 d−2 2Þ w3 ¼ þ Oð − 6Þ ð Þ U ; ;d U 2 ; 2 ; ˜ d : 125 C 3 N ð120Þ s From (99), (108), (116), (118) and (125) we then obtain in ¼ 4 In particular, it simplifies in the vicinity of d ,6 d ¼ 6 − ϵ, 1 ð − 4Þ3 d 4 162 b3 ¼ þ Oððd − 4Þ Þ;d→ 4; ð121Þ π6 W 3 ¼ þ OðϵÞ: ð126Þ N s N 1 216ðd − 6Þ3 ¼ − þ Oðð − 6Þ4Þ → 6 ð Þ The same value for W 3 was obtained in [7,8] for the critical b3 9 d ;d: 122 s N π cubic model. This match between the OPE coefficients provides an additional nontrivial evidence for the equiv- Moreover, from (B9) we obtain alence between the models. In fact, it suggests existence of 6 a wide class of universal relations between the OðNÞ CFTs v˜3ð0Þ¼20π ζð5ÞþOðd − 4Þ;d→ 4; ð123Þ in general d, at least in the 1=N expansion. Some of these π9 relations have been established in [9,10] using the bootstrap ˜ ð0Þ¼− þ Oðð − 6Þ−2Þ → 6 ð Þ method and without considering a particular Lagrangian or v3 ð − 6Þ3 d ;d: 124 d space-time dimension. Our findings suggest that the boot- ˜ strap approach to the OðNÞ CFTs, initiated in [9,10], might The full expression for w3=Cs3 is displayed in (130) ψ ðnÞð Þ below, where x is nth derivative of the digamma 20 ψ ð0Þð Þ¼Γ0ð Þ Γð Þ γ We would like to thank Simone Giombi, Igor Klebanov and function x x = x , and is the Euler constant. Gregory Tarnopolsky for discussing with us the d ¼ 4 case. Their The plot of (130) for the range 2 ≤ d ≤ 6 is shown in Fig. 2. insight helped us to improve our previous version of the related There is an apparent singularity in d ¼ 3, which is an calculation. 025003-21 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) dðd − 3Þþ4 πd Wϕϕ ¼ γϕ H −3 þ π cot s 4 − d d 2 16 6 2 þ þ þ − 2d þ 3 ; ð128Þ ðd − 4Þ2 d − 4 d − 2 ˆ 6ðd − 1Þðd − 2Þ 2 πd f ¼ γϕ H −4 − þ π cot ; d − 4 d d − 4 2 ð129Þ 2 ð − 2Þð − 3Þ w3 ¼ γ d d d 6ψ ð1Þ d − 1 − ψ ð1Þð − 3Þ ˜ ϕ 2 d C 3 ðd − 4Þ 2 s 2 1 3 h i π π FIG. 3. The total =N correction Ws to sss as a function of − − þ 2π d ð Þ space-time dimension 2 025003-22 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) (see also [48] for an extensive discussion of the holographic perturbative approach, ϵ- and large-N expansions confirm three-point functions) argue that to leading order in the they exist in this model. Our calculations encompass the large N expansion, the higher-spin theory in AdS4 bulk next-to-leading order analysis in the 1=N expansion and implies that in d ¼ 3 extend previously known results in a few ways. We derive and perform consistency checks that provide hsssi¼0 þ Oð1=N3=2Þ: ð136Þ an additional evidence for the existence of a nontrivial fixed point. Moreover, we continue nonperturbative studies of This equation is in full agreement with the field theory dual, the emergent conformal field theory and use conformal which in fact has been known since [9,10], as we reviewed techniques to calculate a new CFT data associated with above in this section. the three-point functions of the fundamental scalar and The new result (133) for the next-to-leading order hsssi Hubbard-Stratonovich fields. This helps to expand our raises a natural question whether the holographic work of understanding of a CFT describing critical ϕ4 model in 3 2 [46,47] extends to order Oð1=N = Þ. Notice that [46] shows general dimension. Along the way we evaluate a number of that (136) is valid for the AdS4 dual of any OðNÞ CFT in conformal multiloop diagrams up to and including 4 loops d ¼ 3. In particular, no assumption was made that the AdS4 in general d. These diagrams are generic and have value in bulk is populated by the higher-spin fields of Vasiliev’s themselves since they are not restricted to the critical ϕ4 theory [46]. In fact, it may happen that (133) holds for a model, but rather inherent to various CFTs. ð Þ ¼ 3 large class of O N CFTs in d . Holography is a In [7,8] an alternative description of the critical OðNÞ possible tool to check this conjecture. In particular, one can model in terms of N þ 1 massless scalars with cubic try to extend the holographic argument of [46] to the next- interactions was proposed. It was explicitly shown that the 1 to-leading order in the =N expansion. A possible outcome scaling dimensions of various operators within the alter- 5=2 hsssi¼0 þ Oð1=N Þ of a generic bulk calculation native description match the known results for the critical would be a strong evidence in favor of the holographic OðNÞ theory. While our findings confirm the observations duality [43], and universality of the Ws3 for a broad class of made in [7,8], we find an additional agreement between ð Þ O N CFTs in general dimension. the next-to-leading coefficients of the hsssi three-point functions, previously unobserved in the literature. VI. DISCUSSION In fact, it can be shown that all higher order correlation h ð Þ ð Þi ¼ 4 5 … In this work we explore the OðNÞ critical vector model functions s x1 s xn , n ; ; in both models with quartic interaction in 2 ≤ d ≤ 6 dimensions. In higher match to leading order in the 1=N and ϵ expansion. In dimensions this model provides a nice illustration of the general, these correlators are entirely fixed by the 1PI asymptotically safe quantum field theory, whereas in lower vertices of the effective action. To leading order in the 1=N dimensions it is closely related to realistic systems such expansion these vertices are given by a single ϕ-loop 4 as the critical Ising model in 3D. While it is difficult to with s-legs attached to it. In the ϕ theory they have the prove the existence of the fixed point in full generality, following form Z 2 n Yn Γ1PIð … Þ¼ n n=2 − pffiffiffiffi d Ið … Þ ð Þ ð Þ ð Þ ϕ4 x1; xn NCϕCs d xi x1; ;xn s x1 s xn ; 137 N i¼1 where Iðx1; …;xnÞ is a space dependent structure representing the internal ϕ-loop, and the external s-legs are normalized according to (83). In the cubic model the same vertex equals to Z Yn Γ1PI ð … Þ¼ 3n=2ð− Þn d Ið … Þ ð Þ ð Þ ð Þ cubic x1; xn NCϕ g1 d xi x1; ;xn s x1 s xn ; 138 i¼1 where the fixed point value of the ϕϕs coupling constant is In (138) we took into account that the s field in the cubic given by [7] model is canonically normalized, and therefore the ampli- tude of its propagator in position space equals Cϕ. Note that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (137) and (138) share the same structure Iðx1; …;x Þ 6ϵð4πÞ3 1 n g1 ¼ 1 þ O ; ϵ : ð139Þ because the 1PI diagrams are identical up to an overall N N constant prefactor, which is explicitly written down in both 025003-23 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) cases. It then remains to show that these constants are factor problems in quantum cosmology. It would be identical. Indeed, interesting to explore these aspects in the context of critical ϕ4 model. 2 pffiffiffiffi Cϕ 1 Remarkably, the critical OðNÞ vector models exhibit g1 ¼ 1 þ O ϵ ð Þ N ; : 140 peculiar behavior when coupled to a thermal bath [51].We 2 Cs N hope that certain results presented in this paper might be The above matches support the equivalence suggested in of help toward understanding this behavior in lower 4 dimensions. Progress in this direction will be reported [7,8] between the critical ϕ vector model and the critical elsewhere [52]. cubic model in d ¼ 6 − ϵ dimensions. Based on these findings we propose that the new result for Ws3,which did not appear in the literature before our work, is in fact ACKNOWLEDGMENTS ð Þ universally applicable to a large class of O N CFTs. We thank Noam Chai, Soumangsu Chakraborty, Johan Therefore we interpret the match between the Ws3 coef- 4 Henriksson, Zohar Komargodski and Anastasios Petkou for ficients of the critical ϕ model and the critical cubic model helpful discussions and correspondence. We would like to in d ¼ 6 − ϵ dimensions as a particular manifestation of this express our special thanks of gratitude to Simone Giombi, universality. Moreover, we suspect that it might be possible Igor Klebanov and Gregory Tarnopolsky for numerous to systematically derive additional universal relations, at comments and stimulating correspondence which helped us least in the 1=N expansion, thereby replacing the equiv- to improve our results and their presentation. This work is 23 alence of [7,8] by a universally valid bootstrap statement. partially supported by the Binational Science Foundation Furthermore, we notice that the critical OðNÞ model in (Grant No. 2016186), the Israeli Science Foundation higher dimensions does have certain unphysical features. Center of Excellence (Grant No. 2289/18) and by the The well-known ϵ-expansion shows that the coupling Quantum Universe I-CORE program of the Israel Planning constant at the Wilson-Fisher fixed point is negative in and Budgeting Committee (Grant No. 1937/12). 4 þ ϵ dimensions [1]. This means that the potential is unbounded from below for large values of the field. However, perturbative calculations in 1=N do not reveal APPENDIX A: CALLAN-SYMANZIK EQUATIONS any sign of instability at the level of the correlation In this appendix we use the Callan-Symanzik equation to functions. Thus, for instance, unitarity bounds are satisfied. ˆ ˆ derive various relations between the counterterms δϕ, δ , δ4, Of course, this argument only shows that the vacuum state s and the anomalous dimensions γϕ, γs. In particular we of the theory is metastable if the instability observed within δ ϵ-expansion persists in the ϵ → 1 limit. establish the identity (16) satisfied by the counterterms s δˆ Indeed, in the recent work [11] the authors provide a and s. The calculation is carried out for the model (12) ¼ 0 nonperturbative argument in favor of instability of the with mass set to zero m . Depending on the dimension model. Their analysis rests on the observation that the path d the model is assumed to sit either at the UV or IR integral in the large-N limit is entirely dominated by the fixed point. saddle point. The corresponding saddle point equation is Let us start with the two-point function for the renor- ϕ Weyl invariant at the fixed point, and therefore admits a malized field , family of solutions parametrized by size and location. This ˜ ˜ type of large-N instantons was previously observed in the hϕðpÞϕðqÞi ϕ6 2 critical model in three dimensions [50]. In particular, it 1 −p δϕ ¼ð2πÞdδð þ Þ þ þ was argued that the instantons and associated instability of p q 2 loop corrections 2 2 : p ðp Þ the critical ϕ6 model can be used as a toy model toward holographic resolution of the singularity and conformal ðA1Þ 23 It satisfies the Callan-Symanzik equation Another reason to believe in the universality of the critical OðNÞ vector models in general d is due to the observation related 3 ¼ 8 ∂ to the behavior of Ws coefficient when extrapolated to d . μ þ 2γ hϕ˜ð Þϕ˜ð Þi ¼ 0 ð Þ While the vector model in d ¼ 8 is manifestly nonunitary, since ∂μ ϕ p q ; A2 the scaling dimension of the operator s is below the unitarity bound, we can nevertheless formally compare our result 3 ð ¼ 8Þ¼−950 NWs d with a similar calculation in the exotic which leads to critical model in 8 − ϵ dimensions [49]. We find a precise match with NW 3 calculated in [49]. We are grateful to Simone Giombi, s 1 ∂ 2 Igor Klebanov and Gregory Tarnopolsky for letting us know γϕ ¼ μ δϕ þ Oð1=N Þ: ðA3Þ about [49] and suggesting to compare the results. 2 ∂μ 025003-24 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) Similarly, the Callan-Symanzik equation for the Hubbard- where B is given by (46). At the order Oð1=NÞ the Stratonovich field reads loop corrections to the hs˜ s˜i propagator in (A5) contain dependence on the renormalization scale μ, due to con- ∂ tributions from the ϕϕs interaction vertex counterterm, μ þ 2γ hs˜ðpÞs˜ðqÞi ¼ 0; ðA4Þ ∂μ s and the ϕ field strength renormalization counterterm. Denoting the associated terms with LsðμÞ and using Here we have (A4) we obtain hs˜ðpÞs˜ðqÞi 1 ∂ ˆ 2 1 δˆ γ ¼ μ ðδ þ 2BL ÞþOð1=N Þ: ðA6Þ s s 2 ∂μ s s ¼ð2πÞdδðpþqÞ − þloop correctionsþ ; 2BðpÞ 2BðpÞ ð Þ A5 Finally, for the three-point function 1 1 −1 2 2 δˆ 2 hϕ˜ð Þϕ˜ð Þ ˜ð Þi¼ð2πÞdδð þ þ Þ − pffiffiffiffi þ − pffiffiffiffi pffiffiffiffiffi4 þð−2δ − δˆ Þ − pffiffiffiffi p1 p2 s q p1 p2 q 2 2 loop corrections ϕ s ; p1 p2 2BðqÞ N N g˜4 N ðA7Þ the Callan-Symanzik equation at the fixed point takes the ˜ ð0Þ¼ Uð1;2;d−3Þ suggests the following relation v3 ðd−1 d−1 2Þ2 × form U 2 ;2 ; limλ0→0ð4.346Þ. The direct calculation of v˜3ð0Þ in this ∂ Appendix matches this result. μ þ 2γ þ γ hϕ˜ð Þϕ˜ð Þ ˜ð Þi ¼ 0 ð Þ ∂μ ϕ s p1 p2 s q : A8 To calculate the trapezoid graph we use the following relation obtained by integration by parts [53], see also As a result, we obtain [36,37] ∂ μ ðδ − δˆ þ 2 L Þ¼0 þ Oð1 2Þ ð Þ ∂μ s s B s =N ; A9 where we substituted (A3), (A6) and used (13). APPENDIX B: CALCULATION OF THE TRAPEZOID DIAGRAM In this appendix we calculate the trapezoid diagram v˜3ðδÞ defined by (111). For completeness we reproduce here the defining diagram It turns out that this diagram is quite ubiquitous, because various conformal graphs can be reduced to the trapezoid form through the use of conformal techniques, see for instance Eqs. (4.345) and (4.346) in [30]. Although we did not find an explicit analytic expression for this diagram in the literature, the diagrammatic identity (4.345) in [30] 025003-25 MIKHAIL GOYKHMAN and MICHAEL SMOLKIN PHYS. REV. D 102, 025003 (2020) We apply this relation to the top left vertex of the v˜3 diagram (with α1 ¼ 1, α2 ¼ d − 3 þ δ, α3 ¼ 1), getting a sum of four terms. Then we use the propagator merging relation (48) to integrate over one of the vertices in the first three of those terms. As a result, the first three terms acquire the form of a self-energy diagram, and the fourth terms acquires the form of a self- energy diagram with a line attached to the left-hand vertex. These self-energy diagrams can be brought to a similar form by an inversion transformation around the right-hand vertex. In the end we obtain 1 ˜ ðδÞ¼− ð ð − 3 þ δ 2 1 − δÞþð − 3 þ δÞ ð − 2 þ δ 1 1 − δÞÞ v3 δ U d ; ; d U d ; ; d1 δ δ δ δ − 2 1 − − 3 þ − ð − 3 þ δÞ − 2 þ δ 1 − 1 − ð Þ U ; 2 ;d 2 d2 d U d ; 2 ; 2 d3 ; B1 where we denoted δ ¼ d − 1 þ δ d − 1 − ð Þ d1 F 2 ; 2 2 ; B2 δ ¼ − 3 þ δ d − 1 − ð Þ d2 F d ; 2 2 ; B3 δ ¼ 1 − 3 þ ð Þ d3 F ;d 2 B4 for the function Fðα; βÞ defined by the self-energy diagram To calculate this diagram we can again use the integration by parts relation for the top vertex, with α1 ¼ 1, α2 ¼ α, α3 ¼ β, after which all the remaining integrals can be straightforwardly taken using (48). As a result, we obtain Uð1; 1;d− 2Þ d 3d Fðα; βÞ¼ α Uðα þ 1; β;d− α − β − 1Þ − U α þ 1; β þ 2 − ; − α − β − 3 d − 2 − α − β 2 2 3 þ β ðα β þ 1 − α − β − 1Þ − α þ 2 − d β þ 1 d − α − β − 3 ð Þ U ; ;d U 2 ; ; 2 : B5 This diagram is dual by Fourier transform to the ChT diagram given by Eq. (16) in [26], as can be checked explicitly Að1Þ3AðαÞAðβÞ d d Fðα; βÞ¼ ChT − α; − β ; ðB6Þ Að3 þ α þ β − dÞ 2 2 where d Að2−αÞAðαÞ Aðαþβ−1ÞAð−α−βþ3Þ Að2−βÞAðβÞ π Aðd − 2Þ ð1−βÞðαþβ−2Þ þ ðα−1Þðβ−1Þ þ ð1−αÞðαþβ−2Þ ðα βÞ¼ ð Þ ChT ; d ; B7 Γð2 − 1Þ Γðd − ΔÞ ðΔÞ¼ 2 ð Þ A ΓðΔÞ : B8 025003-26 VECTOR MODEL IN VARIOUS DIMENSIONS PHYS. REV. D 102, 025003 (2020) As a result, from (B1) we obtain 4−d 3dþ5 2 −2 π 2 2 ð1Þ ð1Þ d πd π v˜3ð0Þ¼ ψ ðd − 3Þ − 6ψ − 1 þ H −4 H −4 þ 2π cot þ : ðB9Þ 2 πd d−3 2 d d 2 6 ðd − 4Þ sin ð 2 ÞΓð 2 ÞΓðd − 2Þ The ϵ-expansion of the trapezoid diagram in d ¼ 4 − 2ϵ dimensions was carried out in [36] (see Eq. (19) therein) and [37] (see Eq. (2.24) therein) to Oðϵ2Þ order.24 Hence, we can test our result (B9) by substituting d ¼ 4 − 2ϵ, expanding in ϵ and comparing with [36,37].25 Performing the comparison we obtain a precise match to all available orders [i.e., up to Oðϵ2Þ]of the ϵ-expansion. 24 Conventions of these references are slightly different from ours. One would need to set a1 ¼ a4 ¼ −2, a2 ¼ a3 ¼ a5 ¼ a6 ¼ a7 ¼ 0, and keep in mind that those references put the label equal to a half of the exponent on top of the propagator lines. In −d conventions of [36,37] there is also a factor of π 2 in each vertex. 25We thank Simone Giombi, Igor Klebanov and Gregory Tarnopolsky for the related discussion. [1] S. Weinberg, Critical phenomena for field theorists, in [15] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Understanding the Fundamental Constituents of Matter, Math. Phys. 2, 253 (1998). edited by A. Zichichi, The Subnuclear Series Vol. 14 [16] S. S. Gubser, I. R. Klebanov, and A. M. 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