Flow equation for the scalar model in the large N expansion and its applications

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . We study the flow equation of the O(N) φ model in d dimensions at the next-to-leading order (NLO) in the 1/N expansion. Using the Schwinger-Dyson equation, we derive 2-pt and 4-pt functions of flowed fields. As the first application of the NLO calculations, we study the running coupling defined from the connected 4-pt function of flowed fields in the d+ 1 dimensional theory. We show in particular that this running coupling has not only the UV fixed point but also an IR fixed point (Wilson-Fisher fixed point) in the 3 dimensional massless scalar theory. As the second application, we calculate the NLO correction to the induced metric in d+ 1 dimensions with d = 3 in the massless limit. While the induced metric describes a 4dimensional Euclidean Anti-de-Sitter (AdS) space at the leading order as shown in the previous paper, the NLO corrections make the space asymptotically AdS only in UV and IR limits. Remarkably, while the AdS radius does not receive a NLO correction in the UV limit, the AdS radius decreases at the NLO in the IR limit, which corresponds to the Wilson-Fisher fixed point in the original scalar model in 3 dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index B30, B32,B35,B37


Introduction
In the previous paper [1], the present authors studied the proposal [2] that a d + 1 dimensional induced metric can be constructed from a d dimensional field theory using gradient flow [3][4][5][6], applying the method to the O(N) ϕ 4 model. We have shown that in the large N limit the induced metric becomes classical and describes Euclidean Anti-de-Sitter (AdS) space in both ultra-violet (UV) and infra-red (IR) limits of the flow direction. The method proposed in Ref. [2] may provide an alternative way to understand the AdS/CFT (or more generally Gravity/Gauge theory) correspondence [7], and the result in Ref. [1] might be related to the correspondence between O(N) vector models in d-dimensions and (generalized) gravity theories in d + 1 dimensions [8].
To further investigate a possible connection between Ref. [1] and Ref. [8] at the quantum level, one must calculate, for example, the anomalous dimension of the O(N) invariant operator φ 2 (x), which requires the next-to-leading order (NLO) of the 1/N expansion for the flow equation to evaluate necessary quantum corrections. Since the method employed in Refs. [1,2] is a specific one adopted for the large N limit, some systematic way to solve the flow equation in the 1/N expansion is needed.
In this paper, we employ the Schwinger-Dyson equation ( As the first application of the NLO calculations, we define a running coupling from the connected 4-pt function of flowed fields, which runs with the flow time t such that t = 0 corresponds to the UV limit while t = ∞ is the IR limit. This property establishes that the flow equation can be interpreted as a renormalization group transformation. In particular at d = 3, we show that the running coupling so defined has not only the asymptotic free UV fixed point but also a Wilson-Fisher IR fixed point for the massless case. As the second application, we investigate the NLO correction to the induced metric in 3 + 1 dimensions from the massless scalar model in 3 dimensions. In the massless limit, the whole 4-dimensional space becomes AdS at the leading order, as shown in Ref. [1]. The NLO corrections give a small perturbation to the metric, which makes the space asymptotically AdS in UV (t = 0) and IR (t = ∞) limits only. A remarkable thing is that, while the NLO corrections do not change the AdS radius in the UV limit, the AdS radius is reduced by the NLO correction in the IR limit, which corresponds to the Wilson-Fisher IR fixed point of the original theory. In other words, a nontrivial fixed point in the field theory leads to a change of the AdS radius in the geometry at the NLO. The induced metric at NLO describes a 4-dimensional space connecting one asymptotically AdS space at UV to an other asymptotically AdS space at IR, which have different radii. This paper is organized as follows. In Sec. 2, we first introduce the O(N) invariant ϕ 4 model in d dimensions. We then formulate the Schwinger-Dyson equation (SDE) for the flowed fields, and solve it to derive 2-pt and 4-pt functions of flowed fields at the NLO. In Sec. 3, we define a running coupling from the connected 4-pt function of flowed fields and investigate its behavior as a function of the flow time t. In Sec. 4, we study the induced metric from the 3 dimensional massless scalar model at the NLO. We finally give a summary of this paper in Sec. 5. We collect all technical details in appendices. In appendix A, using the SDE, we present results at the NLO in the 1/N expansion of the d dimensional theory necessary for the main text. We also perform the renormalization of the d dimensional theory at the NLO, and explicitly calculate renormalization constants for various d. In appendix B, we give detailed derivations of solutions to the SDE for the flow fields at the NLO. We explicitly evaluate 2-pt and 4-pt functions of the flowed field in appendix C while we derive the induced metric in appendix D, for the massless scalar theory in 3 dimensions.
where ϕ a (x) is an N component scalar field, ( · ) indicates an inner product of N component vectors such that , µ 2 is the bare scalar mass parameter, and u is the coupling constant of the ϕ 4 interaction, whose canonical dimension is 4 − d. While it is consistent to take u as N independent, as will be seen later, the mass parameter µ 2 is expanded as where µ 2 i is cut-off dependent in order to make the physical mass finite order by order in the 1/N expansion. This model describes the free massive scalar at u = 0, while it is equivalent to the nonlinear σ model (NLSM) in the u → ∞ limit, whose action is obtained from eq. (1) as with the replacement Some regularization which preserves O(N) symmetry is assumed in this paper, so that we can always make formal manipulations without worrying about divergences. 1 Calculations of 2-pt and 4-pt functions at the next-to-leading order (NLO) of the 1/N expansion in d dimensions will be given in appendix A.

Flow equation in the 1/N expansion
In this paper, we consider the flow equation, given by where µ 2 f and u f can be different from µ 2 and u in the original d dimensional theory. As in the case of d dimensions, u f is kept fixed and N independent, whereas µ 2 f is adjusted as where m f is a renormalized mass. The flow with µ f = µ and u f = u is called gradient flow, as it is given by the gradient of the original action.
In the case of the free flow (u f = 0), the solution is easily given by We therefore consider the interacting flow (u f = 0) hereafter unless otherwise stated. 1 We will call the infinite cutoff (Λ → ∞) limit the 'continuum limit'.

4
The above flow equation leads to the Schwinger-Dyson equation (SDE) [9] as where z = (t, x), O is an arbitrary operator and the expectation value O should be calculated in d dimensions as (z, z, z, z 1 , · · · , z 2n−1 ), (10) where Γ n is the n-point function, defined by which is analogous to the d dimensional counterpart in eq. (A3). We consider only the symmetric phase in this paper, where Γ 2n−1 = 0 for all positive integers n.
We consider the next-to-leading order of the 1/N expansion, so that the following two SDE's need to be considered.

Solutions to the flowed SDE at NLO
The solutions to the SDE at NLO are summarized below. Details of calculations can be found in appendix B. At the NLO, the 2-pt function is given by where ζ(t) is defined in eq. (B7), and the NLO contribution G 1 (t 1 , t 2 |p) is given in appendix B.3.2. In the continuum limit, ζ(t) approaches to ζ 0 (t) and is finite as long as t > 0, where with the incomplete gamma function Γ(a, x), while Z(m f ) diverges at d > 1.
The leading contribution of the connected 4-pt function appearing at the NLO of the 1/N expansion can be obtained as Here the variables to the left of the vertical line refer to flow times and those to the right refer to momenta. Explicitly we have in the continuum or NLSM limits and thus B(0|Q) = B(Q 2 ), defined in appendix A. Here ψ and ω satisfy where The derivation of these results is given in appendix B.
3 Running coupling from flowed fields

Definitions
Using the connected 4-pt functions g ≡δĝ for the flow fields given in eq. (25), we define the t-dependent dimensionless coupling as where {p} sym is given by p 2 i t = 3∆/4 (i = 1 ∼ 4) and p 2 12 t = p 2 34 t = ∆ (p ij = p i + p j ), which is the symmetric point for d > 2, and t 2−d/2 is introduced to make the coupling dimensionless. Here ∆ is an arbitrary dimensionless constant but we can take ∆ = 1 without loss of 7 generality by the rescaling t → ∆t. Explicitly we havê where we removeδ by defining O =δÔ for O = g, X, Y, Z, and

d = 2
In 2-dimensions, we obtain which behaves in the UV limit (t → 0) and IR limit (t → ∞) as In the massless limit m 2 → 0, we have

d = 3
At d = 3, the running coupling is given by which behaves as In the massless limit, we have which correspond to the asymptotic free UV fixed point and the Wilson-Fisher IR fixed point, respectively.

d ≥ 4
Since B(Q 2 ) diverges as Λ d−4 (log Λ at d = 4) at d ≥ 4, the running coupling vanishes as the cut-off is removed (Λ → ∞). Thus the theory is trivial in the continuum limit at d ≥ 4.

Massless limit
We next consider the interacting flow case, where we need to evaluateX andŶ , which are difficult to calculate in general. We therefore consider the massless limit. 2 In this limit, the kernel function is reduced to where and we regard D ≡ Q 2 = ∆/t as an independent variable. Here the z integral is convergent for d > 2 while the bubble integral B(0|Q) is finite for d < 4. We thus concentrate on the d = 3 case hereafter.

Running coupling and β function
Using the above results, the running coupling at d = 3 is given by where µ = 1/ √ t and With the numerical values given above we obtain G 1 = 21.0378(1) and G 2 = 2.00105(1) at ∆ = 1. 3 3 It turns out that G 2 (∆) has only one zero at ∆ = 0.36228(1).
We then calculate the β function for g 0 (µ) as which becomes zero at g 0 (µ) = G 1 and g 0 (µ) = G 1 + G 2 . The coupling g 0 (µ) near G 1 behaves as approaching to the UV fixed point from above, while near G 1 + G 2 we have the IR fixed point as where the coupling approaches from below to the Wilson-Fisher fixed point in the 3 dimensional scalar theory. Note that the derivative of the β function with respect to g 0 at the fixed point becomes which should be compared with the same quantities calculated for the standard running coupling in the 3 dimensional massless theory in Ref. [10], where β ′ (0) = −1 (UV) and β ′ (48) = 1 (IR). The derivative of the β function at the fixed point gives the anomalous dimension of the operator conjugate to the coupling in the conformal theory at the fixed point, and thus is universal. Our flow coupling indeed satisfies this condition and the derivatives at the two fixed points agree with those for the conventional definition of the coupling.
This establishes that our flow coupling gives a good definition of the running coupling of the theory. The scaling dimension γ of the operator conjugate to the running coupling g 0 is given by γ = d + β ′ (g 0 ), so that γ UV = 2 and γ IR = 4 in this model. Interestingly γ UV = 2 corresponds to the canonical dimension of the ϕ 4 operator in 3 dimensions, which is the interaction term in our theory.
By the redefinition of the coupling as g(µ) ≡ (g 0 (µ) − G 1 )/G 2 , the corresponding β function is simplified as 11 4 NLO corrections to the induced metric In Ref. [1], the induced metric has been calculated from the flowed scalar field in the large N limit. It has been shown that the metric from the massive scalar field describes a space which becomes the Euclidean AdS space asymptotically both in UV and IR limits, where the radius R IR in the IR is larger than the radius R UV in UV as while the metric describes the whole AdS space in the massless limit with the radius R UV .
In this section, we consider the NLO correction to the induced metric in the 1/N expansion as another application of the NLO calculation, in particular, in the massless case at d = 3, in order to see whether the space remains AdS or not and how the radius changes at the NLO.

Induced metric at NLO
The VEV of the induced metric is defined from the normalized flowed field as [1] with some length scale R 0 , where z = (τ = 2 √ t, x) and µ, ν = 0, 1, · · · , d. Here σ a (z) is the normalized flowed field such that σ 2 (z) = 1, and the corresponding 2-point function is explicitly given at NLO as where After some algebra (see appendix D), we obtain where and A 1 (t) in general is a very complicated function given in appendix D.

Induced metric in the massless limit at d = 3
In the massless limit at d = 3, the metric at the LO is given by which describes the AdS space for all τ .
At the NLO, A 1 (t) is given by which leads to where h total (Q 2 ) is a function given in appendix D.

UV and IR limits
The above expression in the UV limit (τ → 0) leads to which shows that the NLO correction is less singular than the LO contribution. Therefore the space becomes asymptotically AdS in the UV limit at the NLO whose AdS radius is equal to that at the LO.
We cannot naively take the τ → ∞ limit in eqs. (75) and (76), on the other hand, due to the enhancement of the UV contribution of the Q integrals. Careful evaluations of these Q integrals in appendix D give where r = −0.41869(1). 4 Therefore, the space becomes asymptotically AdS again in the IR limit, whose radius, however, is smaller than that in the UV limit. 5 The induced metric at the NLO describes a 4 dimensional space which is asymptotically AdS in both UV and IR regions with different radii but non-AdS in-between.
It is clear that the NLO correction to the AdS radius in the IR limit is related to the Wilson-Fisher fixed point in the original 3 dimensional scalar theory, since the eqs. (75) and (76) can be written as where µ = 1/ √ t = 2/τ , and β(g(x)) is the β function for the running coupling g(x) from the free flowed field defined in the previous section with ∆ = 1 as As an application of the NLO calculation, we investigated the running coupling defined from the connected 4-pt function of flowed fields. In particular at d = 3 in the massless limit, we showed that the running coupling has two fixed points, the asymptotic free one in the UV region and the Wilson-Fisher one in the IR region. We also derived the corresponding β function. Our study suggests that the flow equation can be interpreted as a renormalization group transformation.
We also calculated the NLO correction to the d + 1 dimensional metric induced from the massless scalar field theory at d = 3. In the massless limit, the whole 4-dimensional space becomes AdS at the LO of the 1/N expansion [1]. We found that the NLO corrections give small perturbations to the metric, which make the space only asymptotically AdS in both UV (t = 0) and IR (t = ∞) limits. In addition, while the NLO corrections do not change the AdS radius at the LO in the UV limit, the AdS radius is reduced by the NLO correction in the IR limit, which corresponds to the Wilson-Fisher IR fixed point of the original theory. The nontrivial fixed point in the field theory appears as a change of the AdS radius at the NLO. The induced metric at NLO describes a 4-dimensional space which connects one asymptotically AdS space at UV to the other asymptotically AdS space at IR.
This paper contains two important messages. One is that the flow equation can provide an alternative method to define a renormalization group transformation. The other is that the massless scalar field in d dimensions plus the extra dimension from the RG scale not only generates a d + 1 dimensional AdS space at LO [1] but also gives a NLO correction, which makes the d + 1 dimensional space asymptotically AdS only in UV and IR limits at d = 3.
In particular, the AdS radius in the IR limit, which corresponds to the Wilson-Fisher fixed point, becomes smaller than that in the UV limit, which is equal to the radius at the LO. Although the relation found in this paper between the massless scalar field theory and the induced geometry is very suggestive, further studies will be needed to establish an alternative interpretation of AdS/CFT correspondences proposed in Ref. [2] in terms of field theories.

A The 1/N expansion in the d dimensional theory
In this appendix, we consider the 1/N expansion in the d dimensional theory.

A.1 Schwinger-Dyson equation(SDE)
In order to perform the 1/N expansion, we consider the SDE of this model, which can be written compactly as where δ a x ϕ b (y) = δ ab δ (d) (x − y)ǫ with a small parameter ǫ, so that Here the vacuum expectation value of an operator O is defined in eq. (9). We define 2n-point functions Γ 2n 6 as and so on. As mentioned in the main text, we assume we are working in a phase where O(N) symmetry is not broken. We therefore do not add the external source term hϕ(x) to the action, so that the action has the symmetry under ϕ → −ϕ, which implies Γ 2n−1 = 0 for all positive integers n.
In terms of these, the SDE for X(ϕ) = ϕ a 2 (x 2 ) becomes where δ 12 ≡ δ a 1 a 2 δ (d) (x 1 − x 2 ) and x b = x 1 , so that b in the summation runs over the O (N) indices only.  We finally obtain and

A.2 The leading order in the 1/N expansion
We introduce the 1/N expansion as and so on, together with At the leading order (LO) of the 1/N expansion, the eq. (A12) in momentum space becomes which can easily be solved as where m ≥ 0 is the renormalized mass and Z(m) is given in eq. (6). Thus the 2-pt function at the LO becomes Eq. (A13) at the LO leads to Introducing a function G 0 (p 1 , p 2 , p 3 , p 4 ) to rewrite K 0 (x 1 , x 2 , x 3 , x 4 ) as where p 12 = p 1 + p 2 , and This agrees with the previous result obtained by a different method [10]. We here specify the way we introduce the cut-off Λ for the case where B(Q 2 ) diverges.

A.4 Renormalization
Let us now consider the renormalization of the theory.
Our renormalization condition for the renormalized 2-pt function Γ R is given in momentum space as where m is interpreted as the renormalized mass, which is independent of both N and the cut-off. Relating the bare field to the renormalized field by the renormalization constant Z R as Z 1/2 R ϕ R = ϕ, we explicitly obtain where At the LO of the 1/N expansion, the above condition implies where Z(m) is potentially divergent at d > 1. We therefore introduce the momentum cutoff Λ to regulate the integral, and µ 2 0 is tuned to cancel the effect of Z(m) including such divergences, in order to keep the renormalized mass m finite and constant. The lattice regularization or dimensional regularization is more consistent than the momentum cut-off, but 20 calculations become much more complicated in the lattice regularization or power divergences are difficult to deal with in the dimensional regularization. Since the momentum cut-off is enough to see the leading divergences, we adopt it in this paper.
At the NLO, the renormalization condition implies where The renormalization condition for the coupling, which first appears at the NLO of the 1/N expansion, is given by G 0 (Q 2 = s) = −u r (s)/3, so that u r (s) is regarded as the renormalized coupling at the scale s. Eq. (A20) thus leads to where B(Q 2 ) is divergent at d ≥ 4. Therefore the renormalized coupling goes to zero as at d ≥ 4. This indicates the triviality of the ϕ 4 theory at d ≥ 4.

A.5 Renormalization constants
We here explicitly evaluate the renormalization constants.

A.5.1 d = 1
At d = 1, µ 2 0 is finite as is finite, and the coupling is also finite and nonzero since has a finite limit as Λ → ∞.

21
The most divergent part of Z 1 is given by which shows that Z 1 is finite for all u including u = ∞. Eqs. (A36) and (A37) thus tell us that µ 2 1 is also finite for all u including u = ∞, and therefore, there is no divergence at d = 1 up to the NLO.

A.5.2 d = 2
At d = 2, µ 2 0 is logarithmically divergent as On the other hand, B(Q 2 ) is finite as so that the renormalized coupling becomes u r (s) = 6u The most singular term of Z 1 for u = ∞ becomes which is manifestly finite, while at u = ∞, we have which diverges as Z 1 ≃ log log Λ 2 .
The most divergent part of µ 2 1 is given by 22

A.5.3 d = 3
At d = 3, µ 2 0 is linearly divergent as while B(Q 2 ) is finite as and the renormalized coupling becomes u r (s) = 6u The most singular term of Z 1 for u = ∞ becomes which is manifestly finite at d = 3. On the other hand, at u = ∞, we have whose divergent part becomes The most divergent part of µ 2 1 becomes 23
where F (q 2 ) = 2(q 2 + 2) Let us now consider the continuum limit of Z 1 . By rescaling the momentum, we have As α 2 → 0 in the Λ → ∞ limit, we have where the second term is finite in this limit, while the first term is bounded from above (− log α 2 ) r r r! + (finite terms) , (A66) so that Z 1 in eq. (A64) vanishes as α 2 → 0.
The most divergent part of µ 2 1 becomes where δ 1 is finite, but is not universal as it depends on how we regulate the integral.
We also write from which we obtain whereB . As in the case at d = 4, u r (s) = 0 in the limit that Λ → ∞.
By the change of variable Q = Λq in eq. (A31) and then taking the limit Λ → ∞, we obtain The fact thatB(0, 0) = 0 establishes that Z 1 is finite at d > 4.

25
The most divergent part of µ 2 1 is given by where with the change of variables as q 2 = Q 2 /Λ 2 . It is easy to show that δ 1 is finite.
B Solving the SED for the flow equation In this appendix we explicitly solve the SDE in d + 1 dimensions, in order to obtain the 2-pt and 4-pt functions for the flow fields at the NLO.

B.1 Solution for Γ 0
We first solve the equation at the LO for Γ 0 . If we introduce one unknown function F (t, p) as with the initial condition F (0, p) = 1, we have whereḞ means a t-derivative of F . Then, the SDE (20) becomeṡ which tells us that F (t, p) is independent of p, so we put F (t, p) = F (t). The above equation is thus reduced toḞ where ζ 0 (t) is defined in eq. (24), whose solution is given by where m f is defined in eq. (6) and In the case of the interacting flow with u f > 0, µ 2 f negatively diverges as Z(m f ) → +∞ in the continuum limit at d > 1 or as u f → +∞ in the NLSM limit. In these limits, ∆(t) vanishes as for t > 0. In the case of free flow (u f = 0), we simply have F (t) = 1.
We then obtain

B.2.1 Solution for Y
Terms which depend only on t 1 in eq. (B15) can be written as where ρ(t|34) is defined in eq.
satisfying eq. (B18). Eq. (B21) is reduced to which shows ψ does not depend on p 1 , p 2 , where K is defined in eq. (34). Since u f F 2 (t)e −2tµ 2 f = u f Z(m f )/ζ(t) goes to infinity in the continuum limit at t > 0 and d > 1 or in the NLSM limit u f → ∞, eq. (32) must hold in either of the two limits.