The generating functional of correlation functions as a high-momentum limit of a Wilson action

The generating functional of correlation functions as a high-momentum limit of a Wilson action


Introduction
A Wilson action is usually thought of as a functional integral half done: the field with momenta below the ultraviolet (UV) cutoff still needs to be integrated [1]. To obtain full correlation functions from a Wilson action, we have two ways. We can compute the correlation functions by functional integration over the exponentiated Wilson action. Thanks to the UV momentum cutoff incorporated into the action, this functional integration is well defined. Alternatively, we can lower the momentum cutoff all the way to zero, where the Wilson action becomes the generating functional of the connected correlation functions. The two ways are equivalent, and neither is easy.
The exception is given by scale-invariant theories. In the dimensionless convention of the renormalization group (RG), where the momentum cutoff stays fixed, the scale-invariant theories correspond to fixed points under the RG transformation. Given a fixed-point Wilson action (getting it is actually the hard part), we can switch to the dimensionful convention, where the momentum cutoff decreases under the RG flow. The Wilson action now depends on , but the dependence is given by simple scaling. It is trivial to take to zero, obtaining the correlation functions. Transcribing the vanishing cutoff limit into the dimensionless convention, the correlation functions appear as a high-momentum limit of the Wilson action because any finite momentum in units of the vanishing cutoff becomes large.
Considering how simple the idea is, the reader may find the paper too long or even unnecessary. Our excuse is that the purpose of the paper is to provide a technically robust derivation to justify the idea. We use the formalism of the exact renormalization group (ERG) for generic real scalar theories in D-dimensional Euclidean space (Sect. 11 of Ref. [1]).
To reach a wide range of readers, including those who have not been much exposed to the ERG formalism, we have provided plenty of background material. In fact, most of what is written here can be considered a review. To derive the main result of the paper, which is Eq. (65), all we have PTEP 2017, 123B01 H. Sonoda to do is to collect the right background material and present it in the right order. It is helpful if the reader is familiar with the idea of ERG through reading the first third [up to Eq. (19)] of Ref. [2].
We organize the paper as follows. In Sect. 2 we review ERG by following the perturbative treatment of Ref. [2]. The goal of this section is to introduce the idea of a generating functional W with an infrared (IR) cutoff , and to show that it becomes the generating functional W of the connected correlation functions in the limit that goes to zero. In Sects. 3 and 4 we generalize the ERG formalism just enough for the discussion of fixed points in Sect. 5, where we derive the main result (65) that gives the connected correlation functions of a fixed-point theory as a high-momentum limit of its Wilson action. Section 5 is followed by two short sections: in Sect. 6 we check the consistency of Eq. (65) with potential conformal invariance, and in Sect. 7 we extend Eq. (65) to massive theories. We conclude the paper in Sect. 8. We have prepared three appendices. In Appendix A we show how to derive the diffusion equation satisfied by the generating functional with an IR cutoff, starting from the ERG differential equation of the corresponding Wilson action. In Appendix B we give details of conversion between the dimensionless and dimensionful conventions. In Appendix C we rewrite Eq. (65) for the effective action. Throughout the paper we use shorthand notation such as

Review
We review Wilson's ERG (Sect. 11 of Ref. [1]) following the perturbative treatment by J. Polchinski [2]. We rely on perturbation theory for intuition, but the results we review below should be valid beyond perturbation theory. We consider the action where S I consists of interaction vertices. The free part of the action gives the propagator where K(p/ ) is a decreasing positive function of p 2 / 2 such as If K(p/ ) decays fast enough for large p 2 > 2 , and if the interaction part is reasonable, the theory defined by S is free of UV divergences. We can regard as the UV cutoff of the theory. Thus, we can assume that the correlation functions given by functional integrals are well defined. (Note that we use e S rather than e −S as the weight of integration). Now, how do we determine the -dependence of S I ? The short answer is that we give such -dependence of S I that compensates the -dependence of the propagator. Let us elaborate on this. When we lower the cutoff infinitesimally from to e − t < , the propagator changes by − t The functional integrals using the same interaction part S I change accordingly. If we wish to keep the same functional integrals, we must change the interaction part to compensate for the effect of Eq. (6). The required compensation comes in two types: two vertices connected by minus (6) and single vertices with a loop given by minus (6). This results in the differential equation Exponentiating S I , we can rewrite this as which is a functional generalization of the diffusion equation. Hence, as far as the internal propagators go, their cutoff dependence is compensated by the cutoff dependence of S I . But the external lines still depend on , and the two-point function and the connected part of the higher-point functions acquire the following -dependence: where both G 2 and G connected n>2 correspond to the sums of diagrams with the amputated external lines, and they are independent of the cutoff . (G 2 (p, q) is proportional to δ(p + q) in the absence of symmetry breaking.) To extract the -independent correlation functions, we must remove the cutoff functions from the external lines. For the connected part of the higher-point functions, we can do this simply by factoring out the cutoff functions: For the two-point function, we first subtract a high-momentum propagator to get Then, factoring out the cutoff function, we obtain a -independent two-point function: Incorporating the disconnected part, we can express the full correlation functions as To show how the exponentiated double differentiation works, we give an example of the four-point function: It is commonly taken for granted that only the low-momentum correlation functions are kept invariant under the exact renormalization group transformations, but we have shown more than that: via Eq. (13) we can recover the entire cutoff-independent correlation functions. (This was first pointed out in Ref. [3], and has been used extensively for the realization of symmetry in the ERG formalism [4].) Now, introducing a source J (p), and summing Eq. (13) over all n, we can express the generating functional W[J ] of the connected correlation functions as So far we have regarded as a UV cutoff because the momentum modes with p > are suppressed in the functional integration over e S . Let us note, however, that the interaction part S I results from 4 integrating over the modes with momentum higher than . So, if we regard S I as a consequence of functional integration, we may call an IR cutoff. Since the propagator of the high-momentum modes is the generating functional of the connected correlation functions with an IR cutoff is defined by where we have added the free part, and substituted into the interaction part. (The relation between W and S I was first pointed out in Ref. [5] and then derived carefully in Ref. [6].) Using the full action, we can rewrite W as where is a positive cutoff function that decays rapidly for p 2 > 2 . We note that Now, using W [J ] instead of S [φ] in Eq. (15), and using J instead of φ as integration variables, we obtain a simpler expression for W: It is straightforward to obtain the cutoff dependence of W [J ]. Since W [J ] is defined by Eq. (17), and the -dependence of S I is given by Eq. (7), we obtain Comparing this with Eq. (22) and using Eq. (21), we obtain This is the well-known equality referred to at the beginning of the abstract of the paper. Since W is directly related to S by Eq. (19), we can say that Eq. (25) gives the generating functional of the connected correlation functions as the zero-cutoff limit of the Wilson action.

Generalization
In the previous section we summarized Wilson's ERG following [2]. We introduced two types of generating functionals: W with an IR cutoff and W without. In this section we would like to generalize the formalism in two ways. So far, we have introduced only one cutoff function K(p/ ). Another cutoff function R (p) is given in terms of K(p/ ) by Eq. (20). Our first generalization follows Ref. [7], and we introduce K (p) and R (p) as two independent positive cutoff functions. (This is necessary not only for the second generalization, but also if we wish to include the original formulation of Ref. [1] under the same footing.) The second generalization, to be introduced in the next section, follows Ref. [8], and we introduce an anomalous dimension to the scalar field. Using two independent cutoff functions, we define the correlation functions by In the previous section we chose For Eq. (26) to be independent of , the Wilson action must satisfy For the choice (27), this reduces to which is (7) rewritten for the total action. We do not derive Eq. (29) here; we refer the interested reader to Ref. [7] for the derivation. Now, we define the generating functional with an IR cutoff in the same way as before by where (See Appendix A for the derivation.) The rest proceeds the same way as in the previous section. The generating functional of the connected correlation functions, defined by is given by Hence, we obtain the same result as Eq. (25):

Anomalous dimension
In this section, we introduce an anomalous dimension of the scalar field. A nonvanishing anomalous dimension is required by the nontrivial fixed point to be discussed in the next section. Let S be the Wilson action for which are independent of . We wish to construct -dependent Wilson actionsS so that Here, μ is a fixed reference scale chosen arbitrarily. For simplicity, we have chosen the anomalous dimension γ as a constant independent of . At = μ, the two actions agree: Unlike S , the correlation functions ofS are dependent, but the dependence is merely a change of normalization of the field. We wish to relateS to S in the following. We rewrite Eq. (37) as Since this is independent of ,S must satisfy . (40) (We obtain this from Eq. (29) by substituting μ γ K and μ 2γ R into K and R , respectively.) We define the generating functionalW with an IR cutoff forS , using the same cutoff functions as for S :W where Using Eq. (40), we can derive the cutoff dependence ofW as eW [J ] . (43) 8 we first rewrite the equation as This is solved by To relate W toW , we compare the above solution with which is obtained from the first line of Eq. (24). We easily obtain We could rewrite this as a relation between S andS , but we do not need it. We end this section by giving W[J ] as a limit ofW . We assume so that ( /μ) −2γ dominates over 1 as → 0+. If we assume further that which is a little stronger than Eq. (28), we obtain, from Eqs. (35) and (48),

Fixed points
The differential equation (40), or equivalently (43), does not have a fixed-point solution for an obvious reason: the cutoff keeps changing. We need to adopt the dimensionless convention in which we measure all the physical quantities in units of appropriate powers of the cutoff . We give a of conversion with the dimensionful quantities on the left, and the corresponding dimensionless quantities (with bars except for K, R) on the right: We assume that the dimensionless cutoff functions satisfy Hence, if the anomalous dimension satisfies we obtain Eq. (50). The correlation functions in the dimensionless convention are related to those in the dimensionful convention by (55a) Using Eq. (37), we can rewrite the right-hand side using -independent correlation functions as Hence, in the dimensionless convention the correlation functions satisfy the following scaling relation: Note that we are comparing the correlation functions for different sets of momenta at two different Wilson actions which are related by ERG. It is straightforward to obtain the ERG differential equations forS t andW t by rewriting the equations forS andW . For the rewriting we use We need only the equation forW t here (see Appendix B for the derivation): For this to have a fixed-point solution, we must choose γ appropriately. With γ = 0, we only get the Gaussian fixed point:W By choosing 0 < γ < 1 appropriately, we can obtain a nontrivial fixed pointW * [J ] that satisfies For the fixed point, the scaling relation (56) relates the correlation functions for the same fixed-point Wilson actionS * : φ (p 1 e t ) · · ·φ(p n e t ) Now we are ready to derive the main result of this paper. For a general theory, we get In Sect. 1 we briefly explained why we call Eq. (65) a high-momentum limit: any momentum in units of gets large as we take → 0+. To make this explanation more concrete, expandW * in powers ofJ : We then obtain This implies that 2 +D · w n (p 1 / , . . . , p n / ) δ(p 1 + · · · + p n ) .
Thus, the connected correlation functions are obtained as the high-momentum limit of w n (p 1 , . . . , p n ). Especially for n = 2, we obtain This implies that (70)

Conformal invariance
We would like to discuss the invariance properties ofW * and W. The fixed-point theory has scale invariance, and we expect W[J ] to have naive scale invariance where is the generator of scale transformation. This is a direct consequence of Eq. (65); the very existence of the limit implies (71).
The crucial difference of Eq. (84) from Eq. (65) is that the right-hand side is not the high-momentum limit of a fixedW * :W (ḡ) depends on the exponentially large parameterḡ. This is expected. Take a fixed momentum p for the left-hand side of Eq. (84). The mass scale is of order μ(g/μ 2 ) 1 y . Now, for the right-hand side, the corresponding dimensionless momentum is p/ . Since the ratio to the mass scale must be the same, we reproduce Eq. (80):ḡ To obtainW (ḡ) for largeḡ, we must solve the ERG equation for a wide range ofḡ. We have nothing to gain by switching to the dimensionless convention.

Conclusion
In this paper we have shown that the high-momentum limit of a fixed-point Wilson action contains the connected correlation functions of the corresponding massless theory. This is given explicitly by Eq. (65), where W is the generating functional of the connected correlation functions, andW * is the generating functional with an IR cutoff associated with the fixed-point Wilson actionS * .W * is directly related toS * byW * [J ] = 1 2 pJ where K, R are cutoff functions. In deriving Eq. (65), we have used two equivalent conventions for ERG: one with a dimensionful cutoff , and the other with a fixed dimensionless cutoff 1. In the dimensionful convention, the correlation functions are obtained from the Wilson action in the limit of the vanishing cutoff, as given by Eqs. (35) and (51). On the other hand, in the dimensionless convention, the correlation functions are obtained as the high-momentum limit of the Wilson action. We have used both conventions to derive Eq. (65).
Recently, in Ref. [14], a classical limit has been introduced as the limit of an infinite momentum cutoff where the naive scale and conformal invariance may be restored in the Wilson action. We have discussed the opposite limit of the vanishing cutoff in this paper.