The background field method and critical vector models

We use the background field method to systematically derive CFT data for the critical $\phi^6$ vector model in three dimensions, and the Gross-Neveu model in dimensions $2\leq d \leq 4$. Specifically, we calculate the OPE coefficients and anomalous dimensions of various operators, up to next-to-leading order in the $1/N$ expansion.


Introduction
Due to their theoretical and experimental importance, conformal field theories (CFTs) have been extensively studied over the decades, through a broad range of techniques. One of the most prominent is the conformal bootstrap [1][2][3][4][5] which, rather than using a specific Lagrangian, works with general axiomatic parameters, such as the scaling dimensions of local operators and their operator product expansion (OPE) coefficients. The focus of this paper is inspired by the bootstrap program, but the CFT data is calculated using a complementary method: the diagrammatic 1/N expansion in the context of critical vector models [6,7].
The 1/N expansion is particularly useful when the theory is strongly coupled and the corresponding CFT cannot be studied perturbatively [8,9]. Thus, for instance, it provides an alternative to the classic ǫ-expansion, originally formulated for the scalar field theory with a quartic interaction in 4 − ǫ dimensions [10]. 1 Moreover, certain non-renormalizable quantum field theories which are not tractable within a traditional perturbative expansion become analytically controllable in the large N framework. These include the most wellknown vector models with a quartic interaction: the φ 4 O(N ) vector model for bosons [12], and the Gross-Neveu model for fermions [13,14]. Another interesting example in this class of models is the cubic scalar vector model in 6 − ǫ dimensions [15][16][17]. 2 A vector model which has been far less studied is the sextic φ 6 O(N ) model, in three dimensions [25]. The model has the notable feature -which motivated our study of it -of having a UV conformal fixed point, at large N , directly in three dimensions. An ǫ expansion, as for the Wilson-Fisher fixed point in the quartic model, is not required. In this paper we compute various anomalous dimensions and conformal three-point functions in the sextic O(N ) vector model, up to next-to-leading order in 1/N .
In the process of this study we found that, while the anomalous dimensions in critical vector models have been extensively studied, there has been much less study of the conformal three-point functions (the OPE data). We find that our method of computationin particular, applying the background field method in the context of large N conformal perturbation theory -provides a simple and coherent framework. We apply this method to the standard vector models, extending known results in the literature.
In section 2 we review the φ 6 vector model, the location of the UV fixed point at large N , and the anomalous dimensions of the field. Then in section 3 we compute several conformal three-point functions at next-to-leading order in 1/N . In section 4 we review the Gross-Neveu model, and in section 5 we compute several conformal three-point functions at next-to-leading order in 1/N . In Appendix A we perform similar computations in the context of the quartic O(N ) vector model. In Appendix B we discuss the φ 6 model, when the field φ is integrated out.
2 Review of the φ 6 model Consider the following three-dimensional O(N ) vector model with sextic interaction, The beta function of g 6 vanishes identically in the large N limit, and the model exhibits a line of fixed points, 0 ≤ g 6 ≤ (4π) 2 , where the boundaries of the conformal window are found by demanding stability of the effective potential [26], see also [27]. The 1/N corrections lift this degeneracy and only one isolated UV fixed point, which lies outside the stability region, survives at finite N [25]. The instability effect is, however, nonperturbative in 1/N . Moreover, it is not clear whether it will persist at any finite N , see [28] and references therein for a recent discussion. In this section we introduce our notation, review the derivation of the UV fixed point, and calculate the anomalous dimensions of φ and φ 2 . The model (2.1) can equivalently be expressed in terms of the auxiliary fields σ and ρ, with the action, The original action (2.1) is recovered from (2.2) by first integrating over the field σ along the purely imaginary direction, which results in the delta-functional δ(ρ − φ 2 / √ N ), and then integrating over ρ. Note that the fields ρ and φ are the only dynamical degrees of freedom in the large N limit, because the propagator of σ is suppressed by 1/N .
The propagator of the field φ in position space is, where the large N amplitude and scaling dimension are given by, The corresponding Feynman rule for the normalized field is, 3 Notice that the quadratic part of the action (2.2) is not diagonal, which leads to a nontrivial merging relation between the fields ρ and σ, ρ(x 1 )σ(x 2 ) = − δ (3) (x 12 ) = (2.5) Here the non-dynamical field σ is denoted by a dashed line. The Feynman rules for the cubic interactions in (2.2), φ 2 σ and ρ 3 , are shown in Fig.  1. They generate kinetic terms for σ and ρ, respectively. However, the O(ρ 2 ) term in the effective action is suppressed by 1/N relative to the O(σ 2 ). Indeed, the 1/N factor from the cubic vertices is cancelled by N fields φ running in the loop coming from σφ 2 , In our conventions, all propagators in the Feynman diagrams are normalized to unity, and therefore each Feynman graph should be multiplied by the corresponding amplitudes.
Inverting the quadratic part of the effective action yields, where the large N amplitude and scaling dimension of the field ρ are given by, The corresponding Feynman rule for the normalized field is, In the next subsection, we are going to use the background field method to calculate the effective cubic vertex in the effective action, at next-to-leading order in the 1/N expansion. In preparation for this, here we establish an additional Feynman rule in the presence of a background fieldρ(x). This rule is based on the observation that the linear coupling between the fields ρ and σ induces a background fieldσ(x), which can be derived by considering the equation of motion operator for the field σ. It is given by, A quick consistency check can be carried out by contracting both sides of this expression withρ(x 2 ), and using (2.5), (2.6), (2.7), and the inverse propagator relation (d = 3, ∆ = 2), (2.9) There are simple diagrammatical rules for performing some of the integrals involved in the conformal perturbation theory calculations we will encounter. For instance, a simple loop diagram satisfies additivity (in position space), In addition, there is a propagator merging relation of the form, .
This relation can be represented diagrammatically as, 4 4 The middle vertex on the left-hand side is integrated over. Figure 2. Subleading correction to the cubic vertex of the effective action.ρ(x) is a fixed background field.
Finally, the star-triangle relation for the cubic scalar vertex is given by, (2.12) Diagrammatically it takes the form,

UV fixed point
The beta function of g 6 vanishes in the large N limit. However, the two diagrams shown in Fig. 2 induce a non-trivial RG flow at the next-to-leading order. These diagrams represent a 1/N correction to the cubic vertex of the effective action in the presence of the background fieldρ(x), 5 (2.13) 5 When calculating the first diagram in Fig. 2, we used expression (2.8) for the induced background fieldσ(x). The latter is coupled to φ 2 via a cubic vertex with amplitude −2/ √ N . When writing down the contributions of these diagrams to V , we accounted for the symmetry factors 1/2! and 1/3!, respectively.
The integral over x 4 in the first line can be carried out through the use of the star-triangle relation (2.12). 6 We get, (2.14) Naively, this vertex is conformally invariant, regardless of the value of g 6 . However, this is deceptive, since it exhibits a logarithmic UV divergence when the three fieldsρ(x) collide. This results in a non-trivial RG flow for g 6 . To derive the associated beta function, it is sufficient to set all the fields to be at the same point x 1 . Using the propagator merging relation (2.10) to integrate over x 2 , and introducing both a UV cut-off µ 0 and an IR cut-off µ to regularize the integral over x 3 yields, where ellipsis encode finite terms. Combining this result with the leading order cubic vertex, we deduce the following RG flow equation for the coupling, Equivalently, the beta function is, In particular, the model exhibits a UV stable fixed point at g * 6 = 192, as originally found in [25]. Note also that the 1/N correction, V , to the cubic vertex vanishes at the UV fixed point. We will use this fact later in section 3.2, when we derive the conformal three-point function ρρρ .

Anomalous dimensions of φ and φ 2
In this section we apply the background field method to derive the anomalous dimension of φ and of ρ ∼ φ 2 . 7 These two operators are special because, unlike composite operators with higher powers of φ, their anomalous dimensions vanish at leading and next-to-leading order in 1/N . One has to consider O(1/N 2 ) diagrams to find the first non-trivial correction.
For instance, the anomalous dimension of φ (denoted by γ φ ) is entirely fixed by the diagram in Fig. 3. It represents the leading order correction to the quadratic effective action of the scalar field. Using the Feynman rules gives, 6 The integral actually suffers from a power law divergence. However, such divergences depend on the choice of regularization scheme. For instance, they are absent within dimensional regularization. Therefore, we ignore them in what follows. Indeed, a similar remark applies to the integral identities, such as the propagator merging relation and the star-triangle relation. On the other hand, in general these relations are singular if the integral has a logarithmic divergence. 7 The identification ρ ∼ φ 2 follows from the equation of motion operator, i.e., vary (2.2) with respect to This term is singular in the vicinity of x 1 ∼ x 2 . The logarithmic divergence is associated with the anomalous dimension of φ, and we isolate it by expanding the background field φ(x 2 ) around x 1 , where ellipsis encode terms which do not contribute to γ φ . Introducing a spherical sharp cut-off µ, and substituting the fixed point value g * 6 = 192, yields (2.20) The value of γ φ can be obtained by comparing this expression with the full propagator of φ, where A φ is associated with higher order corrections to the leading order amplitude C φ . Hence, (2. 22) in full agreement with [25]. Next we consider the field ρ. Its full propagator can be written as, where γ ρ is the anomalous dimension, and A ρ is associated with 1/N corrections to the leading order amplitude C ρ . The two-point effective vertex at order 1/N in the presence of the background fieldρ(x) is shown in Fig. 4. It is finite and contributes to A ρ only; we will calculate it in the next section. Here our aim is to evaluate the anomalous dimension γ ρ . We therefore proceed to the O(1/N 2 ) diagrams which contribute to the effective two-point vertex. They can be obtained, for example, by dressing the vertices and propagators in Fig. 4. However, such diagrams are either finite or vanish, because the ρ propagator has no divergences at order 1/N , whereas, as we argued in the previous subsection, the next-to-leading order correction to the cubic vertex vanishes at the fixed point. Hence, the only divergent diagrams which contribute to γ ρ are shown in Fig. 5.
Using the Feynman rules yields, The integral over x 4 in the first expression within the parenthesis is power law divergent in the vicinity of x 4 ≈ x 3 . In principle, it could have a logarithmic divergence, but its coefficient vanishes; this can be seen by introducing a spherical sharp cut-off µ, and expanding The power law divergent terms do not contribute to γ ρ , while the finite part of the integral over x 4 can be evaluated by analytic continuation in the scaling dimension of the field ρ, or by dimensional regularization. In both regularization schemes, one can use the propagator merging relation and remove the regulator at the end. This procedure is sufficient to recover the contribution to γ ρ . The result is, Similarly, the integral over x 4 in the second expression within the parenthesis in (2.24) can be calculated by using the star-triangle relation (2.12), (2.26) Hence, 2∆ρσ(x2) + . . . , Figure 5. The O(1/N 2 ) correction to the quadratic effective action of ρ. The background field isσ ∼ρ, given by (2.8). All propagators on the outer circles of these diagrams correspond to the field φ (the label 2∆ φ on the lines is suppressed). Ellipsis represent O(1/N 2 ) graphs which do not contribute to the anomalous dimension γ ρ .
The remaining integrals over x 3 are logarithmically divergent. Using a spherical sharp cut-off µ yields, Thus, Or, equivalently, substituting (2.8) and using the inverse propagator relation (2.9), we obtain Comparing to (2.23), we get the anomalous dimension, This result does not match [25].

CFT data for the critical φ 6 model
In this section we use large N techniques to derive the OPE coefficients and anomalous dimensions of various primary operators in the critical φ 6 model. Our calculations extend known results in the literature [25].

Anomalous dimensions of (φ 2 ) n
In the large N limit there are two dynamical fields, φ and ρ = φ 2 / √ N , with scaling dimensions ∆ φ = 1/2 and ∆ ρ = 1, respectively. Their propagators are given by (2.3) and (2.6). In contrast, the dynamics of the field σ ∼ ρ 2 / √ N is suppressed by 1/N . As we saw in the previous section, higher order corrections induce anomalous dimensions to the leading order scalings of the fields. In this subsection, we calculate the anomalous dimension of the composite operators (φ 2 ) n . The subleading terms in the propagator, both the anomalous dimension and the amplitude corrections, also contribute to various next-to-leading order three-point functions, and will be needed for the computation of the OPE coefficients, to be performed in the next subsection.
The full propagator of the field ρ is given by (2.23). The amplitude correction A ρ at the next-to-leading order is entirely fixed by attaching two leading-order propagators (2.6) to the one-loop diagram shown earlier in Fig. 4. As a result, the O(1/N ) correction to (2.6) is given by, The integrals can be evaluated by using (2.9). At the critical point, g * 6 = 192, we obtain, While the anomalous dimension of ρ is highly suppressed, this is not the case for ρ n ∼ (φ 2 ) n with n ≥ 2. To leading order in 1/N we have where the black squares stand for insertions of ρ n , and dots represent additional lines (since n is general). The full propagator, which includes subleading corrections to (3.3), will take the form, where A n and γ n are the 1/N corrections to the amplitude and the anomalous dimension, respectively. There are four diagrams which contribute to the full propagator at order 1/N . Two of them are shown in Fig. 6. They are finite and contribute only to A n , The one-loop subdiagrams of C 1 and C 2 can be evaluated using (2.9), to get, Figure 6. Finite diagrams which contribute to the full correlator ρ n (x 3 )ρ n (0) at order 1/N . The black squares denote insertions of the composite operator ρ n . All propagators are propagators of ρ, (2.6). where A ρ is given by (3.2).
In addition, there are two divergent diagrams shown in Fig. 7. These diagrams are responsible for the anomalous dimension. Applying the Feynman rules gives, (3.7) These expressions can be combined after carrying out the integral over x 3 , through the use of the star-triangle relation (2.12), (3.8) The integral over x 2 diverges logarithmically. This divergence is associated with the anomalous dimension of ρ n . Using (2.28) yields, The anomalous dimensions for n = 1, 2 match [25], 8 while the anomalous dimensions γ n for n ≥ 3 are new. As an additional check of our result, we note that the anomalous dimension of φ 6 indeed satisfies γ 3 = ∂β ∂g 6 = − 288 π 2 N .

OPE coefficients
In this section we derive several OPE coefficients, up to the next-to-leading order in 1/N .
There are two Feynman diagrams which contribute to the leading order three-point function ρρρ , as shown in Fig. 8. Using the star-triangle relation (2.12) to integrate over the cubic vertex in U 1 yields, The other diagram, U 2 , is trivial to compute, as all three integrals are taken over delta functions (2.5), giving, Thus, at the fixed point g * 6 = 192, we get, It is standard to define the OPE coefficients to be for the fields which are normalized such that the amplitude of the two-point correlation function is unity. Based on (2.23), we therefore rescale the field ρ, and get, where the OPE coefficient is given bŷ The next-to-leading order correction to this OPE coefficient follows directly from our previous results. In particular, it is entirely determined by the 1/N corrections to the external propagators and to the cubic vertex. The former is encoded in the coefficient A ρ given by (3.2), whereas the latter, as argued in section 2.1, vanishes. Hence, for ρ normalized according to (3.14), we find that OPE coefficient up to order 1/N is, We now consider the three-point function φ φ φ 2 ∼ φφρ . To leading order in 1/N it is determined by a tree diagram of the form where we used the Feynman rules of section 2 to get the expression on the right hand side. The next-to-leading order correction is obtained by accounting for the 1/N corrections to the external propagators and the ρφ 2 vertex. The correction to the cubic vertex is represented by the one-loop diagram in Fig. 9. It vanishes, because the integrals over the delta functions (2.5) result in φ 2 = 0.
Furthermore, as argued in section 3.1, only the amplitude of the ρ propagator is modified at the next-to-leading order, whereas the anomalous dimension and the 1/N correction to the φ propagator are absent at this order. As a result, the three-point function φφρ up to order 1/N is given by, To get the OPE coefficient, we want to first properly normalize the fields: we rescale ρ, as was done in (3.14), and similarly rescale φ →

Review of the Gross-Neveu model
In this section we review some aspects of the Gross-Neveu model [13] which will be needed for section 5. Those who are familiar with the subject may proceed directly to the next section.
where ψ collectively denotes n Dirac fermions, andψ = ψ † denotes the standard Hermitian conjugation. We choose conventions with Hermitian γ-matrices, (γ µ ) † = γ µ , and, as is standard in large-N fermionic models, take N = n tr I where I is the identity matrix in the space of 2 [d/2] -dimensional Dirac spinors [14]. The standard trick for solving the Gross-Neveu model is to rewrite the model (4.1) using the Hubbard-Stratonovich field s, The coupling g is irrelevant at the Gaussian fixed point, and the model is therefore IR free in dimensions d > 2. However, the ǫ-expansion suggests that it has a non-trivial UV fixed point, and the corresponding CFT has been extensively studied [14,[29][30][31][32]. In strictly two dimensions, the model (4.2) is a renowned example of an asymptotically free field theory. As we will review in the next subsection, the large-N scaling dimension of s at the UV fixed point is equal to one, and so the corresponding CFT is unitary provided that d ≤ 4.

Feynman rules and useful identities
In momentum space the bare propagators of the fields in (4.2) are given by, 9 Since we are interested in the large-N limit of the model, we have to resum the so-called bubble diagrams to get the leading order propagator of the auxiliary field s [13], .
(4.4) 9 We will generally not keep track of the U (n) indices. Figure 10. Feynman rules in the presence of Dirac field.
In the last step we took the limit of large momentum, to ensure that the model is sitting at the UV fixed point. The fermionic bubble B(p) appearing above is given by, (4.5) Using the standard Fourier transform relation, we get the conformal propagator of the auxiliary field s in position space [14] (see also [32,33]), Similarly, the propagator of the fermion field ψ is, We will use position space Feynman rules, as we did in the previous sections for the scalar field theory. Since we now have fermions, a Dirac propagator will have an arrow, and there is no symmetry factor of 2 in theψψs vertex, see Fig. 10. The amplitudes of the propagators are normalized to unity, and therefore one needs to multiply each Feynman diagram by an appropriate power of C ψ,s .
The integral identities that were used in scalar conformal perturbation theory can be generalized to accommodate fermions [29] (see also [34] for a recent review), e.g., Moreover, the propagator merging relations with fermions take the form, where the insertion point of the middle vertex on the left-hand side is integrated over, A(∆) is defined in (4.6), and . (4.9) In the case of a Yukawa vertex, the star-triangle relation (2.12) becomes (∆ 1 +∆ 2 +∆ 3 = d), 10) or diagrammatically,

Propagators at the next-to-leading order
We now review the formalism to reproduce the leading order anomalous dimensions of the fields s and ψ, as well as the 1/N corrections to the amplitudes of their propagators [29,33]. These results will be used extensively in the next section. For the Dirac fermion the relevant conformal graph is, This diagram diverges; we regularized it by introducing a slight shift δ in the scaling dimension of s and an arbitrary scale µ, to maintain correct dimensionality [1,35]. The integrals over internal vertices x 3,4 can be done by using the propagator merging relations reviewed above. Expanding around δ = 0 yields, where [29,33], Similarly, one can derive the regularized s field propagator [33] s(x)s(0) = C s 1 |x| 2 with where H n is the n'th harmonic number.
To remove the 1/δ poles in the above propagators, one has to renormalize the fields, As a result, the physical correlators take the form, 10  Notice that the field strength renormalization (4.17) induces a counterterm, which at the next-to-leading order in 1/N is given by, Or, equivalently, (4.21)

Gross-Neveu-Yukawa model
Before closing this section, we briefly review the relation between the Gross-Neveu and Gross-Neveu-Yukawa (GNY) models. The results reviewed here are used to check our calculations in the next section. The GNY model is defined by the action, 10 In order to avoid cluttering the notation, we do not distinguish between physical and bare fields.
This model is manifestly renormalizable in d ≤ 4, because the couplings g 1 and g 2 are relevant at the Gaussian fixed point in d < 4 and marginally irrelevant in d = 4. Below four dimensions they flow to an interacting fixed point in the IR limit. The critical values of the couplings can be calculated by using the standard ǫ-expansion in d = 4 − ǫ dimensions [14], Furthermore, the scaling dimensions of the fields at the fixed point were calculated in [14] (see [36] for a comprehensive review), This formula shows that in the large N limit ∆ (4−ǫ) s = 1 + O(1/N ), up to linear order in ǫ. In fact, a general argument based on the 1/N expansion shows that this relation is true for any 2 ≤ d ≤ 4, see, e.g., [36]. In particular, the operators (∂s) 2 and s 4 are irrelevant in the large N limit below four dimensions, because their scaling dimension approaches 4 in this limit. Therefore, the effective action of the GNY model in the deep IR matches that of the GN model in the deep UV, and the two models are described by the same CFT at their respective fixed points [14,16,37,38].
As was pointed out in [14], the equivalence between the GN and the GNY models is analogous to the equivalence between the critical φ 4 vector model and the non-linear sigma model. A similar equivalence, albeit one that is slightly less exhaustive, also holds between the critical φ 4 vector model in 4 < d ≤ 6 and the model of N + 1 massless scalars with cubic interaction [16,17]. 11 As an illustration of the equivalence between the critical GN and GNY models, let us consider the 1PI vertices for the field s. To leading order in the 1/N expansion, they are represented diagrammatically by a single fermionic loop with a given number of s legs attached to it. Because of the equivalence, vertices with the same number and type of legs in both models should match, 12 i.e., for n external fields s, we have 26) where the two sides of this equation represent the same vertex in the GN and GNY models respectively, I n (x 1 , . . . , x n ) corresponds to the internal ψ-loop, and the external fields s are normalized such that the amplitude of their propagators equals unity. In particular,C s = 11 It was argued in these works that the equivalence between the models in d = 6 − ǫ dimensions holds up to O(ǫ 3 ) order. 12 A similar correspondence holds between the critical φ 4 vector model and the model of N + 1 massless scalars with cubic interaction [39].
, because the field s in the GNY action (4.22) is canonically normalized. Using (4.7) and (4.23), one can verify that the above identity indeed holds to leading order in the ǫ and 1/N expansion.
Moreover, the symmetry pattern of the models match. The actions (4.2) and (4.22) are invariant under the discrete symmetry [13,14,36] s → −s , ψ → γ µ ψ ,ψ → −ψγ µ , Notice that such a transformation leaves the fermionic kinetic term invariant while it transformsψψ → −ψψ, in analogy with the chiral transformation in even dimensions and parity in odd dimensions. Thus, for instance, any correlator with an odd number of fields s must vanish. The three point function sss is the simplest instance where one can directly see that the correlator vanishes. To leading order in the large-N expansion, it is given by, where the last equality is true in any integer dimension d = 2, 3, 4, whereas in general 2 < d < 4 it holds by analytic continuation [32]. 13 In contrast, the three-point function ψ ψs is nontrivial. To leading order in ǫ it can be derived using the tree level cubic interaction and the star-triangle relation (4.10), where the OPE coefficient is given bỹ Note that, by definition of the OPE coefficient, all the fields are normalized so that their two-point functions have unit amplitude.

CFT data for the critical Gross-Neveu model
In this section we use the background field method to derive some CFT data for the critical Gross-Neveu model. Specifically, we calculate the three-point function ψ (x 1 )ψ(x 2 )s(x 3 ) and the associated conformal effective vertex, up to next-to-leading order in the 1/N expansion. In addition, we calculate the leading order OPE coefficient determined by the correlation function ψ (x 1 )ψ(x 2 )s 2 (x 3 ) . Our calculations hold in general 2 ≤ d ≤ 4, and the final results agree with their counterparts obtained through the ǫ-expansion of the critical Gross-Neveu-Yukawa model in the vicinity of d = 4. The latter serves as a check of our calculations, since the two critical models are equivalent [14].
In what follows, the amplitudes of the two-point functions are normalized to unity, i.e., we rescale the fields, The leading order three-point function for the normalized fields, which can be derived by using the tree level cubic vertex and the star-triangle relation, reviewed in section 4.1, is, where the leading order OPE coefficient is given by, 3) Loops are associated with higher order terms in 1/N , which modify (5.2). However, conformal invariance completely fixes the structure of the full three-point function. In our case, it takes the form, where the anomalous dimensions and Wψ ψs encompass loop corrections to the three-point function (5.2). The anomalous dimensions γ s and γ ψ were evaluated in the previous section; in this section our goal is to determine Wψ ψs . To this end, we need to dress the tree level diagram with 1/N corrections. As we will see below, Wψ ψs receives contributions from the 1/N corrections to the amplitudes of external propagators, as well as from the 1/N corrections to the cubic vertex itself. The former corrections were evaluated in the previous section (A s and A ψ ), so we only need to evaluate the latter terms.
The full cubic vertex of the effective action is non-local. By conformal invariance its form is fixed up to an overall constant prefactor, and can be represented diagrammatically as a conformal triangle [40], where the unknown constantẐ is closely related to Wψ ψs , while α and β are determined by noticing that each vertex, denoted by a solid dot in the above graph, should be conformal after attaching to it an appropriate full propagator (4.18) or (4.19). Hence, We now focus on calculating the only remaining unknown,Ẑ. By definition, it is determined by the following diagrammatic equation, which represents the perturbative expansion of the three-point function ψ ( where the external legs are not amputated; they correspond to the full propagators (4.18) and (4.19). The first graph on the right-hand side represents the cubic interaction of (4.2). The rest of the terms reveal the 1/N terms. Specifically, the two diagrams on the second line of (5.7) account for the loop corrections to the leading order cubic vertex. In fact, the last diagram is trivial, because it contains a vanishing sss sub-graph, discussed in section 4.3. On the other hand, the first diagram on the second line of (5.7) results in a non-trivial next-to-leading order contribution to the cubic vertex. 14 This diagram diverges and can be regularized by a small shift δ in the scaling dimension of s. This is the same regulator δ used in section 4.2 to calculate the 1/N corrections to the propagators of the fields ψ and s.
While the couplings are not renormalized at the fixed point, there is still renormalization of the fields. In particular, the second diagram on the right-hand side of (5.7) is associated with the counterterm (4.21) induced by wave function renormalization. In the limit δ → 0, this counterterm cancels the divergences of the loop diagrams.
The diagrammatic equation ( Zµ −2γ ψ +γs Note that the amplitudes of the external fermionic propagators (4.18) cancel on both sides of this equation, and we ignore them in what follows. Each diagram in (5.8) can be calculated by repeatedly applying the fermion-fermion and fermion-scalar propagator merging relations reviewed in section 4.1. The calculation of each diagram starts by integrating over the insertion point of the vertex with a constant backgrounds. In particular, the left-hand side of (5.8) takes the form, The two last graphs of (5.7) are analogous to the corresponding vertex diagrams in the O(N ) vector model. However, unlike the case studied here, in the critical O(N ) vector model both diagrams give rise to a non-trivial contribution, see, e.g., [39] for a recent discussion. In appendix A we recoverẐ in the vector model using the background field method.
where δZ ∼ O(1/N ), and expanding in the small parameters γ s , γ ψ ∼ O(1/N ), yields l.h.s. of (5.8) = 8π On the other hand, this expression is equal to the sum of the three diagrams on the right-hand side of (5.8), which we denote by, 15 The tree level term is given by, It contributes to the leading and subleading orders in 1/N , (5.15) Finally, the loop diagram takes the form 15 Recall that the last diagram in (5.8) vanishes, because it has a subdiagram sss which does not respect s → −s symmetry.
As expected, the sum of the three terms (5.14), (5.15) and (5.17) is finite in the limit δ → 0 r.h.s. of (5. Comparing (5.11) to (5.18), we getẐ (5.10), The same result was recently obtained in [33], using a different method. Finally, calculating the three-point function ψ (x 1 )ψ(x 2 )s(x 3 ) simply involves attaching the full propagators (4.18), (4.19) to the effective cubic vertex (5.5), and integrating over the internal vertices through the use of the star-triangle relation (4.10). After normalizing the external fields as in (5.1), the final result reads, In the last line we expanded in the small anomalous dimensions. Expanding the right-hand side of (5.21), up to next-to-leading order in 1/N , and comparing to (5.4), we deduce that the leading term reproduces (5.3), while the next-to-leading order correction is given by, 16 It is instructive to check our result for Wψ ψs . To this end, we substitute (5.3) and (5.23) into (5.4), and expand the three-point function ψ ψs in ǫ = 4 − d. The leading order term in ǫ matches (4.29), in accord with the equivalence between the critical Gross-Neveu and Gross-Neveu-Yukawa models. Moreover, Wψ ψs vanishes in d = 2, and therefore the OPE coefficient equals −1, up to order 1/N . This result is expected, because the critical GN model is free in two dimensions and ψ ψs ∼ ψ ψ(ψψ) , since s ∼ψψ/ √ N , by the equations of motion. In particular, it follows from the standard Wick contraction applied to a free field theory (with properly normalized fields) that the OPE coefficient is indeed −1, to all orders in 1/N . Figure 11 displays Wψ ψs for dimensions 2 ≤ d ≤ 4.
Let us close this section by deriving the three-point function ψ ψs 2 between the Dirac fields and the composite operator s 2 . To leading order in the 1/N expansion, this correlator is not affected by the anomalous dimensions of ψ and s 2 , On the other hand, it is determined by the diagram where insertion of s 2 is denoted by a black square. Using the star-triangle relation (4.10) to integrate over the internal vertices, and normalizing s 2 and the Dirac fields according to s 2 → √ 2 C s s 2 and (5.1), respectively, yields This result is in full agreement with the GNY model. 17 Moreover, the three point function (5.24) respects discrete symmetry (4.27). Indeed,

Discussion
In this paper we used the background field method as a simple way to calculate conformal cubic vertices in critical O(N ) vector models, up to next-to-leading order in 1/N . To the best of our knowledge, this method of calculation has not been previously employed in large-N conformal perturbation theory. We explicitly studied two critical vector models: the O(N ) sextic φ 6 model in dimension d = 3, and the U (n) Gross-Neveu model in dimensions 2 ≤ d ≤ 4. For the φ 6 model, we derived the leading order anomalous dimensions, (2.31) and (3.10), of the composite operators ρ n ∼ (φ 2 ) n , as well as the OPE coefficients associated with the three-point functions φ 2 φ 2 φ 2 and φφφ 2 , up to the next-to-leading order in 1/N , see (3.17) and (3.19). For the critical Gross-Neveu model, we derived the leading order and next-toleading order OPE coefficient of the three-point functions ψ ψs and ψ ψs 2 , see (5.3), (5.23) and (5.25). Our results agree with the literature [25,33] and extend it in several ways.
There are a number of motivations for studying critical vector models. The Gross-Neveu model, specifically, is asymptotically safe despite being non-renormalizable in the vicinity of the Gaussian fixed point above two dimensions. Its RG flow is therefore completely determined by a finite number of parameters, and its UV fixed point has been extensively studied in the large-N limit [14,[29][30][31][32][33]37]. In particular, it was shown that the UV CFT of the model is equivalent to an IR CFT of the renormalizable Gross-Neveu-Yukawa model [14]. From this perspective, the Gross-Neveu model provides a laboratory where theoretical questions about the structure of RG flows and mechanisms of UV completion can be explicitly addressed. In two dimensions it presents a simple example of an asymptotically free field theory [13].
In contrast, the φ 6 vector model is manifestly renormalizable in three dimensions, yet it also possesses a UV fixed point which is interesting in its own way. It turns out that the UV physics of the model is closely related to the study of big crunch singularities in asymptotically AdS 4 spacetimes [42,43], see also [44]. In the latter context, one studies a marginal triple trace deformation of ABJM theory [45], where the deformation corresponds to adding a potential which is unbounded from below. The UV fixed point of the deformed ABJM theory in the limit of zero 't Hooft coupling is described by the critical φ 6 model studied in this work. It would be interesting to systematically explore the effects of the non-perturbative instability in 1/N [26] on the CFT data of the critical φ 6 model and understand the cosmological implications of our findings in the holographic context studied in [42,43]. 17 In the GNY model ψ ψs 2 is determined by a similar diagram. To explicitly verify that it matches the GN counterpart, it is sufficient to use the leading order relation (g ⋆ 1 ) 2C s/Cs = 1+O(ǫ, 1/N ), see section 4.3.
It is tempting to study the critical φ 6 vector model away from d = 3. By continuity, the UV fixed point does not disappear as we move towards other dimensions, provided that N is sufficiently large. Thus, we expect to generate a family of CFTs parameterized by d, and a natural question is whether there is a range for d such that the corresponding CFTs are non-perturbatively stable.
Finally, it turns out that critical vector models exhibit peculiar behavior when coupled to a thermal bath. For instance, it was shown recently that there are O(N ) theories which have some of their internal symmetries broken at arbitrary finite temperature [46]. The results of this paper might be of help in understanding this behavior in three dimensions.

A Interaction vertex in the critical φ 4 model
In this appendix we derive the effective interaction vertex in the critical φ 4 vector model in 2 ≤ d ≤ 6 dimensions. The φ 4 vector model has an IR fixed point in dimensions 2 ≤ d ≤ 4, and a UV fixed point in dimensions 4 < d ≤ 6. For a recent discussion of this model see [39] and references therein, including [6,7,41,[47][48][49][50][51][52][53]. Our goal is to demonstrate that the background field method, used in section 5 to calculate the effectiveψψs vertex in the Gross-Neveu model, is also applicable to the O(N ) vector model. Our result for the cubic vertex in the φ 4 model fully agrees with [41], where it was evaluated using a different method. Since, to the best of our knowledge, the background field method has not been used before in the context of large-N conformal perturbation theory, we felt some researchers may find the current discussion useful. The discussion in this appendix parallels the discussion in section 5.
The critical φ 4 vector model is described by the action, where φ is a real N -component multiplet of scalar fields in the fundamental representation of O(N ), s is the auxiliary Hubbard-Stratonovich field, δ is a UV regulator, and γ φ,s are the anomalous dimensions associated with the loop corrections to the large-N scaling dimensions ∆ φ = d/2 − 1 and ∆ s = 2 of φ and s, respectively. Note that the Hubbard-Stratonovich field is dynamical; its propagator is built from an infinite series of φ bubbles. 18 18 The effective action for s includes an infinite tower of non-local n-vertices suppressed by 1/N n 2 −1 .
The UV divergences of the theory are regulated by adding a small shift to the scaling dimension of the internal s-lines in the Feynman graphs, i.e., 2∆ s → 2∆ s +δ. In particular, the last term in (A.1) is the counterterm generated by wave function renormalization of the fields φ and s. Other possible counterterms vanish at the conformal fixed point.
The cubic term in the effective action will be non-local. By conformal invariance, its form is fixed up to an overall constantẐ, and can be represented diagrammatically as a conformal triangle [40], Using the above cubic vertex, one can calculate the corresponding conformal threepoint function by attaching to it the full φ and s propagators, which are given by, where the normalizations are, and the anomalous dimensions are [6,7,39,41,[47][48][49][50][51][52][53], Note that the integration vertices x 1 , x 2 and x 3 become conformal and can be integrated over using the star-triangle relation (2.12). We now calculateẐ, up to next-to-leading order in the 1/N expansion. To this end, we notice that the full cubic vertex (A. 2β The first term on the right-hand side of (A.7) corresponds to the leading order tree level vertex in (A.1), while the rest of the terms represent O(1/N 3 2 ) corrections. Specifically, the last two diagrams account for the loop corrections to the tree level vertex. The scaling dimension of s in these diagrams undergoes a slight shift of δ/2, in order to regularize divergent loops. The last graph on the first line of (A.7) is associated with the counterterm, see (A.1); in the limit δ → 0, it cancels the divergences of the loop diagrams. The external legs are associated with the fields φ and s, as indicated by the corresponding labels in the figure.
The diagrammatic equation (A.7) definesẐ up to order O(1/N 3 2 ). It turns out that the background field method simplifies this equation, and getting an explicit expression for Z boils down to a relatively short calculation.
To begin with, we decompose the Hubbard-Stratonovich field into a constant backgrounds and fluctuating component s, i.e. we replace s →s + s. Next, we substitute this decomposition into (A.7) and retain only the linear terms ins. Now, replacing the external fields φ with the full propagators (A.4) defines a linear (ins) correction to the two-point function φ(x 1 )φ(x 2 ) s , in the presence of a constant backgrounds, sss ss 2α 2α 2β .8) and the second loop diagram is given by, B The φ 6 model in terms of auxiliary fields In this appendix we give a slight variant of the derivation of some of the results for the sextic model that were presented in the main body of the paper. Recall that the action for the φ 6 model is given by (2.2), Let us explicitly integrate out the φ fields, which results in the effective action, We now expand the logarithm to next-to-leading order in 1/N . Making use of the matrix element, x|(∂ 2 ) −1 σ(x)|y = − C φ |x − y| σ(y) , and discarding the constant and linear in σ terms (which are removed by appropriate counterterms) we obtain, In order to diagonalize the quadratic part, we define a new field s in terms of σ, Using the star-triangle relation, as well as the inverse propagator relation, we obtain the effective action, up to next-to-leading order,  When all three fields ρ are located at different points, we can combine two O(ρ 3 ) vertices into one, with the total amplitude 64−g 6 √ N . In particular, we recover the leading order threepoint function (3.15).
(B. 13) in agreement with (2.14). 20 This vertex vanishes at the fixed point g ⋆ 6 = 192. 20 Recall that effective vertex and the corresponding diagram are related by the symmetry factor of −1/3!