Wilsonian Effective Action and Entanglement Entropy

Abstract

This is a continuation of our previous works on entanglement entropy (EE) in interacting field theories. In previous papers, we have proposed the notion of ${\mathbb{Z}}_M$ gauge theory on Feynman diagrams to calculate EE in quantum field theories and shown that EE consists of two particular contributions from propagators and vertices. We have also shown that the purely non-Gaussian contributions from interaction vertices can be interpreted as renormalized correlation functions of composite operators. In this paper, we will first provide a unified matrix form of EE containing both contributions from propagators and (classical) vertices, and then extract further non-Gaussian contributions based on the framework of the Wilsonian renormalization group. It is conjectured that the EE in the infrared is given by a sum of all the vertex contributions in the Wilsonian effective action.

1. Introduction

Entanglement entropy (EE) captures correlations for bipartite entangled states between two subspaces, in particular spatially separated regions, and has been widely investigated in conformal field theories (CFTs) [1,2,3,4,5], perturbations from CFTs [6,7,8] or in the context of AdS/CFT correspondence [9,10,11,12] (for review, [12,13]). In free quantum field theories (QFTs), where the vacuum wavefunctional is Gaussian, EE is well-understood [1,14] and we can perform explicit calculations [15,16,17,18,19]. On the other hand, we have little understanding of EE in general interacting QFTs, apart from exactly solvable cases [20] or some supersymmetric theories [12,13,21,22,23]. EE in interacting theories are discussed in perturbative [24,25], nonperturbative [26,27,28,29,30,31,32], lattice [33,34,35,36,37], or in terms of variational trial wave functions [28,29,30].

In interacting QFTs, EE is divergent and needs regularizations. First, EE is UV divergent because there are infinitely many degrees of freedom. This divergence occurs even in a free theory. Second, in interacting theories, the infinite degrees of freedom cause UV divergences in physical parameters and renormalizations are necessary to extract finite results. Finally, if a theory contains massless fields, it may cause IR divergence, but in this paper, we consider a massive theory so that the IR divergences are assumed to be absent.

The EE in the infrared limit, or its variation with respect to some parameters such as mass or coupling constants, is determined by correlations of the renormalized vacuum wave function and should be independent of the UV cutoff. In this sense, the IR part of EE must be determined by the IR fixed point of the renormalization group (RG). On the other hand, EE at a fixed scale will change along the RG flow on which a theory transmutes from one fixed point to another. In order to understand these behaviors of EE, the Wilsonian approach of renormalization [38,39,40,41] will be useful. The Wilsonian effective action (EA) describes a flow of effective actions at a given scale where all higher momentum fluctuations of fields are integrated out, together with rescaling of the momentum so as to make the UV cutoff back to the original one.

In this paper, we will investigate EE in interacting field theories based on the Wilsonian picture of renormalization combined with our previous works [42,43] based on the ${\mathbb{Z}}_M$ gauge theory on Feynman diagrams. In [42], we succeeded to extract two particular contributions to EE in interacting field theories: one is the Gaussian contributions written in terms of renormalized two-point correlation functions in the two-particle irreducible (2PI) formalism. Another set of important contributions comes from classical vertices, which reflects non-Gaussianity of the vacuum wave function. In [43], we showed that the vertex contributions can be interpreted as contributions from renormalized two-point correlation functions of composite operators. These results are obtained by evaluating EE in the notion of the ${\mathbb{Z}}_M$ gauge theory on Feynman diagrams, whose picture is derived from the replica method of EE (the number of replicas n is replaced by $n=1/M$). In the formulation, EE is given by a sum of various configurations of ${\mathbb{Z}}_M$ fluxes on plaquettes in Feynman diagrams and the above two contributions to EE are described by two particular types of flux configurations. Thus, an important question left unanswered is how to extract other contributions described by other configurations of fluxes. In this paper, we address this question in the framework of the Wilsonian RG, where a variety of quantum vertices appears as the energy scale decreases.

For this purpose, we first generalize our previous results to describe operator mixings. We also give a natural, unified description of the contributions to EE from propagators and vertices in a matrix form. This unified description is one of the main results of the present paper. By using this generalized expression of EE, we conjecture that the IR part of EE is given by a sum of the propagator and vertex contributions in the Wilsonian EA.

The paper is organized as follows: in Section 2, we first briefly summarize the notion of the ${\mathbb{Z}}_M$ gauge theory on Feynman diagrams, two particular contributions to EE from propagators and vertices in the ${\varphi }^4$ scalar theory, and an interpretation of the vertex contribution in terms of a correlator of a composite operator. In addition, we give a unified description of both contributions in a matrix form. In Section 3, we generalize it when various operators are mixed with each other and also when the composite operators have spins in the two-dimensional spacetime normal to the boundary. In Section 4, we discuss the IR behavior of EE in the framework of the Wilsonian RG and give a conjecture that EE in the IR is given by a sum of the propagator and vertex contributions in the Wilsonian EA. Finally, we give conclusions in Section 5. In Appendix A, we prove the area law for Rényi entropy and the capacity of entanglement [44,45]. In Appendix B, we give a proof that all the single twist contributions from vertices are written in the 1-loop type expression of composite operators. This is a generalization of the proof for the propagator contributions based on the 2PI formalism.

2. Summary of Previous Works

In this section, we first summarize our previous works in [42,43], and then introduce a new concept of the generalized 1PI in order to unify the contributions to EE from propagators and vertices.

2.1. ${\mathbb{Z}}_M$ Gauge Theory on Feynman Diagrams

Entanglement entropy of a subsystem A is defined by

$$\begin{array}{c}\hfill S_{EE}=-{\mathrm{Tr}}_A{\rho }_{A}log{\rho }_A,\end{array}$$

(1)

where ${\rho }_A={\mathrm{Tr}}_{\overline{A}}{\rho }_{\mathrm{tot}}$ is a reduced density matrix of ${\rho }_{\mathrm{tot}}$ obtained by integrating out the complementary system $\overline{A}$ in the Hilbert space. In this paper, we take A as a half space on a time slice in a $(d+1)$-dimensional spacetime and utilize the orbifold method [46,47] to calculate $S_{EE}$. This method is a variation of the replica trick for EE in which EE is given through the $n\to 1$ limit of Rényi entropy $S_n=\frac{1}{1-n}log\mathrm{Tr}{\rho }_A^n$ [1]; by taking the replica parameter $n=1/M$, we consider free energy of interacting quantum field theories on the orbifold ${\mathbb{R}}^2/{\mathbb{Z}}_M\times {\mathbb{R}}^{d-1}$ denoted by $F^{(M)}$. Then, EE is written as

$$\begin{array}{c}\hfill S_{EE}=-\frac{\partial \left(M{F}^{(M)}\right)}{\partial M}{|}_{M\to 1}.\end{array}$$

(2)

Since a physical state of the ${\mathbb{Z}}_M$ orbifold theory is invariant under the ${\mathbb{Z}}_M$ projection operator,

$$\begin{array}{c}\hfill \widehat{P}=\sum _{n=0}^{M-1}\frac{{\widehat{g}}^n}{M}\prime \end{array}$$

(3)

where $\widehat{g}$ is a $2\pi /M$ rotation operator around the origin, and the orbifold theory can be interpreted as the ${\mathbb{Z}}_M$ gauge theory on Feynman diagrams. Namely, each propagator in a Feynman diagram is sandwiched by the projection operators; this corresponds to assigning a twist $n_i$ on the i-th propagator $G({\widehat{g}}^{n_i}x,y)$ and summing over all independent configurations of such twists. Then, the notion of ${\mathbb{Z}}_M$ gauge symmetry appears since we can rotate away some of the twists of propagators by the ${\mathbb{Z}}_M$ gauge transformations on vertices of the Feynman diagram. As a result, we can classify independent configurations of twists up to ${\mathbb{Z}}_M$ transformations in terms of ${\mathbb{Z}}_M$ fluxes of twists on plaquettes. Here, a flux of twists on a plaquette is defined by a sum of twists around the plaquette as shown in Figure 1.

A simple example of the ${\mathbb{Z}}_M$ invariant configuration is given by Figure 2. The 1-loop diagram has only one plaquette and the ${\mathbb{Z}}_M$ invariant twist is given by the flux m. The number of independent flux is always 1 even if we divide the propagator into several connecting pieces. One may think that each propagator can be twisted separately, but such multiplicities are removed by the ${\mathbb{Z}}_M$ transformations on vertices connecting the divided propagators. Thus, there is only one independent twist in the 1-loop diagram. This property is essential to prove our main result of Equation (32) and responsible for the fact that EE can be written as a sum of 1-loop type diagrams of various composite operators. See the proof in Appendix B. See also the discussions in Section 4B and Section 4C in our previous paper [43].

2.2. Propagator Contributions to EE

In order to evaluate EE in Equation (2), we need to extract all the configurations of fluxes that do not vanish in the $M\to 1$ limit. In the previous papers, we have shown that, if all the fluxes of twists are zero, they do not contribute to EE in Equation (2), which assures the area law of EE (It is easy to see that this property also holds for the Rényi entropy or entanglement capacity, and so does the area law. For more details, see Appendix A). Among various configurations contributing to EE, we have focused on two particular configurations.

The first type of configurations are given in Figure 3 and can be interpreted as a twist of the propagator. The simplest configuration of this type is given by Figure 2 and already present in a free field theory. Interactions induce renormalization of physical quantities, such as a mass, appearing in the EE. A seminal calculation was studied by Hertzberg [24] and completed in our previous papers based on the two-particle irreducible formalism (2PI) where we have shown that the propagator contributions are exactly given by

$$\begin{array}{cc}\hfill S_{\mathrm{propagator}}& =-\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}log\left[G^{-1}(\mathbf{0};k_{\Vert }){ϵ}^2\right]\hfill \end{array}$$

(4)

$$\begin{array}{cc}& =-\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}log\left[(G_0^{-1}-\Sigma )(\mathbf{0};k_{\Vert }){ϵ}^2\right],\hfill \end{array}$$

(5)

where G is the renormalized propagator (The same symbol G is used to represent the Green function in coordinate and momentum spaces for notational simplicity. They are distinguished by their arguments if necessary.) and $V_{d-1}$ is the volume of the boundary $\partial A$. The $(d+1)$-dimensional momentum is written as $(\mathit{k},k_{\Vert })$, where $\mathit{k}$ is the two-dimensional components of time and the direction normal to the boundary and $k_{\Vert }$ is the $(d-1)$-dimensional components parallel to the boundary. The UV cutoff $ϵ$ is introduced. Writing the full inverse propagator as $G^{-1}=(G_0^{-1}-\Sigma )$, the logarithm can be expanded as a sum,

$$\begin{array}{c}\hfill -log{G}^{-1}=-log{G}_0^{-1}+\sum _{n=1}\frac{{(G_0\Sigma )}^n}{n}.\end{array}$$

(6)

Then, it can be interpreted as a chain of free propagators connected by the self-energy $\Sigma$. On the other hand, the full propagator G itself is expanded similarly but without the $1/n$ factor. This $1/n$ factor in the EE comes from the redundancy of twists: twisting n propagators in the chain is not independent. There is only a single twist in the plaquette as explained at the end of Section 2.1. If we twisted every propagator in the chain, it would over count the contributions to EE from the 1-loop Feynman diagram.

2.3. Vertex Contributions to EE and Generalized 1PI

The second type of configurations of fluxes we are going to focus on is given by Figure 4 for the ${\varphi }^4$ scalar theory, ${\mathcal{L}}_{\mathrm{pot}}={\lambda }_4{\varphi }^4/4$. These configurations of fluxes are interpreted as twists of the interaction vertices, which, in turn, are regarded as twists of the corresponding composite operators. This interpretation is obtained by opening a 4-point vertex into two 3-point vertices and assign the twist to the propagator connecting the two 3-point vertices. Corresponding to three different channels of the opening, s, t, and u, there are three different configurations of fluxes and contributions to EE, respectively. If different quantum numbers are assigned to these three channels, we can utilize a method of auxiliary fields and EE is given by a sum of propagator contributions of three different auxiliary fields as shown in [43]. If composite operators propagating three channels are mixed like the ${\varphi }^4$-theory, we cannot use the method of auxiliary fields, but from diagrammatic analysis (see Appendix B), we can show that it is written in terms of a correlation function of composite operators. In the case of the ${\varphi }^4$-theory, it is given [43] by

$$\begin{array}{c}\hfill S_{\mathrm{vertex}}=\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\log \left[1-\frac{3}{2}{\lambda }_4{G}_{{\varphi }^2{\varphi }^2}(\mathbf{0},k_{\Vert })\right],\end{array}$$

(7)

where

$$\begin{array}{cc}\hfill G_{{\varphi }^2{\varphi }^2}(x-y):=& 〈[{\varphi }^2](x)[{\varphi }^2](y)〉.\hfill \end{array}$$

(8)

In the following, the cutoff $ϵ$ is not explicitly written for notational simplicity, as it can be recovered by dimensional analysis. The square brackets $[\mathcal{O}]$ represent the normal ordering of an operator $\mathcal{O}$. The coefficient $-3{\lambda }_4/2$ is a product of $-{\lambda }_4/4$ and 6, where $-{\lambda }_4/4$ is the coefficient in front of the interaction vertex and the coefficient 6 is a combinatorial factor for separating four $\varphi (x)$’s into a pair of ${\varphi }^2(x)$ and ${\varphi }^2(y)$. As shown in Figure 5, the Green function of the composite operator can be written as

$$\begin{array}{ccc}\hfill G_{{\varphi }^2{\varphi }^2}=& {\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)}+\left(\frac{-3{\lambda }_4}{2}\right){({\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)})}^2+{\left(\frac{-3{\lambda }_4}{2}\right)}^2{({\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)})}^3+\cdots =\hfill & \hfill \frac{{\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)}}{1-\left(\frac{-3{\lambda }_4}{2}\right){\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)}},\end{array}$$

(9)

where ${\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)}$ is the 1PI self-energy of $[{\varphi }^2]$ in a generalized sense. We call it g-1PI. Namely, the quantity with the superscript $(g)$ does not contain a diagram like Figure 6 that is separable by cutting a vertex in the middle. We call such a diagram a beads diagram: 1PI in the ordinary sense but not in the generalized sense. Thus, these bead diagrams are not included in g-1PI diagrams.

By using Equation (9), we can rewrite Equation (7) as

$$\begin{array}{c}\hfill S_{\mathrm{vertex}}=-\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\log \left[1-\left(-\frac{3}{2}{\lambda }_4\right){\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)}\right].\end{array}$$

(10)

In the following equations including Equation (10), the argument $(\mathit{k}=\mathbf{0},k_{\Vert })$ of the integrand for the $k_{\Vert }$ integral is implicit. Now, we can write both the propagator and vertex contributions in Equations (5) and (10) in a unified matrix form as

$$\begin{array}{c}\hfill S_{EE}^{({\varphi }^4)}=-\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}\log \left[{\widehat{G}}_0^{-1}-\widehat{\lambda }{\widehat{\Sigma }}^{(g)}\right],\end{array}$$

(11)

where

$$\begin{array}{cc}\hfill {\widehat{G}}_0=& \left(\begin{array}{cc}{G}_0& 0\\ 0& 1\end{array}\right),\widehat{\lambda }=\left(\begin{array}{cc}1& 0\\ 0& -3{\lambda }_4/2\end{array}\right),{\widehat{\Sigma }}^{(g)}=\left(\begin{array}{cc}{\Sigma }^{(g)}& 0\\ 0& {\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)}\end{array}\right).\hfill \end{array}$$

(12)

In the following, we first generalize these results to include higher-point vertices whose composite operators are mixed in a complicated way. Then, we apply the concept of Wilsonian effective action to extract further contributions to the IR part of the EE. It is important to note that the form of Equation (11) is convenient for a unified description in the following discussions, but it is always possible to go back to the form like Equation (7), where the vertex contributions are written in terms of the ordinary renormalized propagators without the superscript $(g)$. In addition, note that all the single twist contributions from a vertex can be written in the above 1-loop type formula, Equation (7) or Equation (12). In the case of the propagator contributions to EE, we have proved the statement by using the 2PI formalism in [42,43]. Here, we use a diagrammatic method to prove it for the vertex contributions in Appendix B.

3. General Vertex Contributions to EE

In this section, we extend the analysis of vertex contributions to EE from the ${\varphi }^4$ interaction to more general cases.

3.1. ${\varphi }^6$ Scalar Field Theory

First, let us consider the ${\varphi }^6$ interaction,

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{pot}}=\frac{{\lambda }_6}{6}{\varphi }^6.\end{array}$$

(13)

In this case, we have two types of vertex configurations (The ${\varphi }^6$ interaction will induce ${\varphi }^4$ interaction by contracting two $\varphi$’s, but, in this section, we simply set it zero by renormalization and do not consider contributions to EE from such diagrams as the vertex contributions at this stage. A model containing both ${\varphi }^4$ and ${\varphi }^6$ interaction vertices are studied in the next section.) as drawn in Figure 7, and three types of composite operators need to be introduced, ${\varphi }^2$, ${\varphi }^4$, and ${\varphi }^3$, to extract all the vertex contributions to EE. Since the theory has ${\mathbb{Z}}_2$ invariance under $\varphi \to -\varphi$, the ${\mathbb{Z}}_2$-even operators, ${\varphi }^2$ and ${\varphi }^4$, are mixed with themselves while the ${\mathbb{Z}}_2$-odd operator ${\varphi }^3$ is mixed with the fundamental field $\varphi$. Therefore, the propagator contribution in Equation (5) needs a modification.

First, let us consider the modified propagator contributions in the ${\varphi }^6$ theory. Such contributions come from 1-loop type diagrams of mixed correlations of $\varphi$ and ${\varphi }^3$ operators. They are given by (Figure 8)

$$\begin{array}{cc}\hfill S_{{\mathbb{Z}}_2-\mathrm{odd}}=& -\frac{V_{d-1}}{6}\int \frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}log\left[{\widehat{G}}_0^{-1}-\widehat{\lambda }{\widehat{\Sigma }}^{(g)}\right],\hfill \end{array}$$

(14)

where

$$\begin{array}{cc}\hfill {\widehat{G}}_0=& \left(\begin{array}{cc}{G}_0& 0\\ 0& 1\end{array}\right),\widehat{\lambda }=\left(\begin{array}{cc}1& 0\\ 0& -10{\lambda }_6/3\end{array}\right),{\widehat{\Sigma }}^{(g)}=\left(\begin{array}{cc}{\Sigma }^{(g)}& {\Sigma }_{\varphi {\varphi }^3}^{(g)}\\ {\Sigma }_{{\varphi }^3\varphi }^{(g)}& {\Sigma }_{{\varphi }^3{\varphi }^3}^{(g)}\end{array}\right).\hfill \end{array}$$

(15)

It is a natural generalization of Equation (12) including an operator mixing. The diagonal component of ${\widehat{G}}_0$ is the bare propagators of $\varphi$ and ${\varphi }^3$ operators, respectively. $\widehat{\lambda }$ is a matrix whose matrix element represents the coefficients of opening the ${\varphi }^6$ vertex. The coefficient for ${\varphi }^3$ to ${\varphi }^3$ in $\widehat{\lambda }$ is given by $1/6{\times }_6{C}_3$. ${\widehat{\Sigma }}_{}^{(g)}={\widehat{\Sigma }}_{}^{(g)}(\mathit{k}=\mathbf{0},k_{\Vert })$ is the ${\mathbb{Z}}_2$-odd g-1PI function (In the ${\mathbb{Z}}_2-\mathrm{odd}$ set of operators, and the $[{\varphi }^5]$ operator does not appear in the mixing, though the ${\varphi }^6$ vertex can be decomposed into $\varphi$ and ${\varphi }^5$. It is because a diagram with $\langle \varphi [{\varphi }^5]\rangle$ is not 1PI while the g-1PI is 1PI as well in the ordinary sense). Namely, it consists of 1PI diagrams that do not contain bead diagrams shown in Figure 6. Such a generalization of the 1PI concept is mandatory since, in calculating the vertex contributions to EE, we need to open a vertex to take account of various channel contributions, and special care of the beads diagram in Figure 6 is necessary. This is the reason why we have generalized the concept of 1PI.

The above discussions can be straightforwardly extended to the contributions from ${\mathbb{Z}}_2$-even operators, ${\varphi }^2$ and ${\varphi }^4$. This case is simpler because the bare Green function is unity; $G^{(g)}=1$. Then, we have the same matrix form

$$\begin{array}{cc}\hfill S_{{\mathbb{Z}}_2-\mathrm{even}}=& -\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}log(\widehat{1}-\widehat{\lambda }{\widehat{\Sigma }}^{(g)}),\hfill \end{array}$$

(16)

where, in this case, matrices are given by

$$\begin{array}{cc}\hfill \widehat{\lambda }=& \left(\begin{array}{cc}0& -5{\lambda }_6/2\\ -5{\lambda }_6/2& 0\end{array}\right),{\widehat{\Sigma }}^{(g)}=\left(\begin{array}{cc}{\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)}& {\Sigma }_{{\varphi }^2{\varphi }^4}^{(g)}\\ {\Sigma }_{{\varphi }^4{\varphi }^2}^{(g)}& {\Sigma }_{{\varphi }^4{\varphi }^4}^{(g)}\end{array}\right).\hfill \end{array}$$

(17)

The coefficient comes from $5/2=1/6\times {}_6{C}_2$. It is a $2\times 2$ matrix generalization of Equation (7). The g-1PI self-energy ${\widehat{\Sigma }}^{(g)}$ does not contain bead diagrams, especially diagrams connected by the ${\varphi }^6$ vertex decomposed into ${\varphi }^2$ and ${\varphi }^4$.

Note that EE of Equations (14) and (16) written in terms of the g-1PI functions can be rewritten in terms of the renormalized correlation functions as in the ${\varphi }^4$ case of Equations (4) and (9). The only difference is that we now have operator mixings and the relationship becomes more complicated. Let us explicitly check for the ${\mathbb{Z}}_2$-odd case of Equation (14). It is rewritten as

$$\begin{array}{cc}\hfill S_{{\mathbb{Z}}_2-\mathrm{odd}}& =\frac{V_{d-1}}{6}{\int }^{1/\epsilon }\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}\ln \left({\widehat{G}}_0\frac{1}{\widehat{1}-{\widehat{\lambda }}_{}{\widehat{\Sigma }}_{}^{(g)}{\widehat{G}}_0}\right)\hfill \\ & =\frac{V_{d-1}}{6}{\int }^{1/\epsilon }\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}\ln \left({\widehat{G}}_0+{\widehat{G}}_0{\widehat{\lambda }}_{}{\widehat{\Sigma }}_{}^{(g)}{\widehat{G}}_0+{\widehat{G}}_0{\widehat{\lambda }}_{}{\widehat{\Sigma }}_{}^{(g)}{\widehat{G}}_0{\widehat{\lambda }}_{}{\widehat{\Sigma }}_{}^{(g)}{\widehat{G}}_0+\cdots \right).\hfill \end{array}$$

(18)

Writing the inside of the parenthesis as $\tilde{G}$, its matrix elements are given by

$$\begin{array}{cc}\hfill {(\tilde{G})}_{11}& =G_0+G_0{\Sigma }^{(g)}{G}_0+G_0{\Sigma }_{\varphi {\varphi }^3}^{(g)}\left(-\frac{10}{3}{\lambda }_6\right){\Sigma }_{{\varphi }^3\varphi }^{(g)}{G}_0+G_0{\Sigma }^{(g)}{G}_0{\Sigma }^{(g)}{G}_0+\cdots ,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {(\tilde{G})}_{12}& =G_0{\Sigma }_{\varphi {\varphi }^3}^{(g)}+G_0{\Sigma }^{(g)}{G}_0{\Sigma }_{\varphi {\varphi }^3}^{(g)}+G_0{\Sigma }_{\varphi {\varphi }^3}^{(g)}\left(-\frac{10}{3}{\lambda }_6\right){\Sigma }_{{\varphi }^3{\varphi }^3}^{(g)}+\cdots ,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {(\tilde{G})}_{21}& =\left(-\frac{10}{3}{\lambda }_6\right)\left({\Sigma }_{{\varphi }^3\varphi }^{(g)}{G}_0+{\Sigma }_{{\varphi }^3\varphi }^{(g)}{G}_0{\Sigma }^{(g)}{G}_0+{\Sigma }_{{\varphi }^3{\varphi }^3}^{(g)}\left(-\frac{10}{3}{\lambda }_6\right){\Sigma }_{{\varphi }^3\varphi }^{(g)}{G}_0+\cdots \right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {(\tilde{G})}_{22}& =1+\left(-\frac{10}{3}{\lambda }_6\right)\left({\Sigma }_{{\varphi }^3{\varphi }^3}^{(g)}+{\Sigma }_{{\varphi }^3\varphi }^{(g)}{G}_0{\Sigma }_{\varphi {\varphi }^3}^{(g)}+{\Sigma }_{{\varphi }^3{\varphi }^3}^{(g)}\left(-\frac{10}{3}{\lambda }_6\right){\Sigma }_{{\varphi }^3{\varphi }^3}^{(g)}+\cdots \right).\hfill \end{array}$$

(22)

We can explicitly see that the sum of g-1PI’s in each matrix element is combined into the ordinary 1PI functions $\Sigma$’s, and hence can be written by the renormalized correlation functions as

$$\begin{array}{cc}\hfill {(\tilde{G})}_{11}& =G_0+G_0\Sigma G_0+G_0\Sigma G_0\Sigma G_0+\cdots =G,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {(\tilde{G})}_{12}& =(G_0+G_0\Sigma G_0+G_0\Sigma G_0\Sigma G_0+\cdots ){\Sigma }_{\varphi {\varphi }^3}=G_{\varphi {\varphi }^3},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {(\tilde{G})}_{21}& =\left(-\frac{10}{3}{\lambda }_6\right){\Sigma }_{{\varphi }^3\varphi }(G_0+G_0\Sigma G_0+G_0\Sigma G_0\Sigma G_0+\cdots )=-\frac{10}{3}{\lambda }_6{G}_{{\varphi }^3\varphi },\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {(\tilde{G})}_{22}& =1+\left(-\frac{10}{3}{\lambda }_6\right)\left({\Sigma }_{{\varphi }^3{\varphi }^3}+{\Sigma }_{{\varphi }^3\varphi }G{\Sigma }_{\varphi {\varphi }^3}\right)=1-\frac{10}{3}{\lambda }_6{G}_{{\varphi }^3{\varphi }^3}.\hfill \end{array}$$

(26)

As a result, Equation (14) can be summarized as

$$\begin{array}{cc}\hfill S_{{\mathbb{Z}}_2-\mathrm{odd}}=& \frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}log\left[\tilde{I}+{\widehat{\lambda }}_{}\widehat{G}\right],\hfill \end{array}$$

(27)

where

$$\begin{array}{c}\hfill \tilde{I}=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right),\widehat{\lambda }=\left(\begin{array}{cc}1& 0\\ 0& -10{\lambda }_6/3\end{array}\right),\widehat{G}=\left(\begin{array}{cc}G& G_{\varphi {\varphi }^3}\\ G_{{\varphi }^3\varphi }& G_{{\varphi }^3{\varphi }^3}\end{array}\right).\end{array}$$

(28)

The same discussion can be applied to Equation (16). This gives an alternative, unified formula for EE in terms of the renormalized Green functions.

3.2. ${\varphi }^4+{\varphi }^6$ Theory and Further Generalizations

Let us generalize a bit more and consider a case when the Lagrangian contains two interaction terms

$$\begin{array}{c}\hfill {\mathcal{L}}_{pot}=\frac{{\lambda }_4}{4}{\varphi }^4+\frac{{\lambda }_6}{6}{\varphi }^6.\end{array}$$

(29)

As in the ${\varphi }^6$ theory, we need to consider three composite operators, ${\varphi }^2$, ${\varphi }^4$, and ${\varphi }^3$, in order to take into account contributions to EE from these vertices. Again, we have ${\mathbb{Z}}_2$ invariance, and EE is a sum of ${\mathbb{Z}}_2$-even and odd contributions. The ${\mathbb{Z}}_2$-odd contribution is given by

$$\begin{array}{cc}\hfill S_{{\mathbb{Z}}_2-\mathrm{odd}}=& -\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}log\left[{\widehat{G}}_0^{-1}-\left(\begin{array}{cc}1& 0\\ 0& -10{\lambda }_6/3\end{array}\right)\left(\begin{array}{cc}{\Sigma }_{}^{(g)}& {\Sigma }_{\varphi {\varphi }^3}^{(g)}\\ {\Sigma }_{{\varphi }^3\varphi }^{(g)}& {\Sigma }_{{\varphi }^3{\varphi }^3}^{(g)}\end{array}\right)\right],\hfill \end{array}$$

(30)

where ${\widehat{G}}_0$ is the same as in Equation (15) while ${\mathbb{Z}}_2$-even contribution is given by

$$\begin{array}{cc}\hfill S_{{\mathbb{Z}}_2-\mathrm{even}}=& -\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}log\left[\widehat{1}-\left(\begin{array}{cc}-3{\lambda }_4/2& -5{\lambda }_6/2\\ -5{\lambda }_6/2& 0\end{array}\right)\left(\begin{array}{cc}{\Sigma }_{{\varphi }^2{\varphi }^2}^{(g)}& {\Sigma }_{{\varphi }^2{\varphi }^4}^{(g)}\\ {\Sigma }_{{\varphi }^4{\varphi }^2}^{(g)}& {\Sigma }_{{\varphi }^4{\varphi }^4}^{(g)}\end{array}\right)\right].\hfill \end{array}$$

(31)

Now, a generalization to e.g., ${\varphi }^{2n}$ vertices with higher n is evident. The propagator and vertex contributions to EE are unified to be written in a matrix form as Equation (14):

$$\begin{array}{cc}\hfill S_{EE}=& -\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}log\left[({\widehat{G}}_0^{-1}-\widehat{\lambda }{\widehat{\Sigma }}^{(g)})(\mathit{k}=0,k_{\Vert })\right].\hfill \end{array}$$

(32)

The size of matrices becomes larger as a larger number of operators are mixed, and each set of mixed operators forms a block diagonal component. ${\widehat{G}}_0$ is a diagonal matrix whose entry is mostly 1 except the fundamental field. $\widehat{\lambda }$ represents a mixing among operators via vertices while ${\widehat{\Sigma }}^{(g)}$ represents amputated correlators of all the fundamental and composite operators. The notion of the g-1PI is also extended to exclude all the bead diagrams constructed by all the vertices along with the ordinary non-1PI diagrams. This form of EE contains all the contributions from the propagators and the vertices. We provide the derivation in Appendix B.

An essential point is that we can rewrite Equation (32) in terms of the renormalized correlation functions in the same manner as in Equation (27) as

$$\begin{array}{c}\hfill S_{EE}=\frac{V_{d-1}}{6}{\int }^{1/\epsilon }\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}\ln \left(\tilde{I}+\widehat{\lambda }\widehat{G}\right).\end{array}$$

(33)

Here, $\tilde{I}=\mathrm{diag}(0,1,\cdots ,1)$, $\widehat{G}$ is the matrix form of the correlators of operators, and we have arranged the elements of the matrices so that the first line and first column involve the fundamental field $\varphi$. The size of the matrices is finite as far as there is a finite number of vertices. In the ${\varphi }^n$-theory, we need to consider only the composite operators $[{\varphi }^j]$ with $j\le n-2$, which appear to open vertices.

3.3. Derivative Interactions

Special care is necessary for generalizations with derivative interactions since composite operators with Lorentz indices appear. Let us consider the following interaction as an example:

$$\begin{array}{c}\hfill {\mathcal{L}}_{pot}=\frac{{\lambda }_{\partial }}{4}{(\varphi \partial \varphi )}^2.\end{array}$$

(34)

In this case, the two types of scalar composite operators, $[{\varphi }^2]$ and $[{(\partial \varphi )}^2]$, as well as a spin-1 operator $[\varphi {\partial }_{\mu }\varphi ]$ appear from an opened vertex. Since the spin-1 operator does not mix with either $\varphi$ or $[{\varphi }^2]$ or $[{(\partial \varphi )}^2]$, we can separately study its contribution to EE. Thus, we have three block-diagonal sectors.

The spin-0 sectors can be treated as before. Thus, let us focus on the spin-1 sector. The formula Equation (32) gets a bit modified since EE of a spinning field is different from that of a scalar field due to the rotation of the internal spin induced by ${\mathbb{Z}}_M$ twist, and hence an extra phase appears in evaluating EE [47]. The operator $J_{\mu }:=[\varphi {\partial }_{\mu }\varphi ]$ is decomposed into its two-dimensional part $J$ and $(d-1)$-dimensional part $J_i$. The latter is a scalar on the two-dimensional spacetime normal to the boundary and can be treated as in Equation (32). On the other hand, the contribution to EE from the two-dimensional vector $J$ is modified. From Equation (2.21) in [47], the coefficient of EE is proportional to

$$\begin{array}{c}\hfill c_{\mathrm{eff}}^{\mathrm{boson}}(s)=\frac{1}{4}\frac{\partial J(s,M)}{\partial M}{|}_{M=1}=\frac{1}{6}-\frac{|s|}{2}\end{array}$$

(35)

for a bosonic field with spin s. This coefficient $c_{\mathrm{eff}}$ replaces the coefficient of $1/6$ in front of Equation (5). Thus, for $(d+1)$-dimensional vector $J_{\mu }$, the total coefficient is given by $(d-1)/6+2(1/6-1/2)=(d-5)/6.$ Therefore, the propagator and vertex contributions to EE with this derivative interaction are given by either of the following two forms:

$$\begin{array}{cc}\hfill S_{EE}=& -\frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}\left(Slog\left[{\widehat{G}}_0^{-1}-\widehat{\lambda }{\widehat{\Sigma }}^{(g)}\right]\right)\hfill \\ \hfill =& \frac{V_{d-1}}{6}{\int }^{1/ϵ}\frac{d^{d-1}{k}_{\Vert }}{2{(2\pi )}^{d-1}}\mathrm{tr}\left(Slog\left[\tilde{I}+\widehat{\lambda }\widehat{G}\right]\right),\hfill \end{array}$$

(36)

where

$$\begin{array}{c}S=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& (d-5)\end{array}\right),\tilde{I}=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),\widehat{\lambda }=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& -{\lambda }_{\partial }& 0\\ 0& -{\lambda }_{\partial }& 0& 0\\ 0& 0& 0& -{\lambda }_{\partial }/2\end{array}\right),\hfill \\ {\widehat{G}}^{}=\left(\begin{array}{cccc}G& 0& 0& 0\\ 0& G_{{\varphi }^2{\varphi }^2}& G_{{\varphi }^2{(\partial \varphi )}^2}& 0\\ 0& G_{{(\partial \varphi )}^2{\varphi }^2}& G_{{(\partial \varphi )}^2{(\partial \varphi )}^2}& 0\\ 0& 0& 0& G_{(\varphi {\partial }_{\mu }\varphi )(\varphi {\partial }^{\mu }\varphi )}\end{array}\right).\hfill \end{array}$$

(37)

S is an additional coefficient due to the spin. Here, we have summed over $(d+1)$-dimensional vector contributions, but, generally speaking, it is more convenient to write a matrix corresponding to each irreducible representation of the two-dimensional rotation with spin s. According to [47], the coefficient c for fermions with odd half-integer spin s is given by $c_{\mathrm{eff}}^{\mathrm{fermion}}(s)=-1/3$. Thus, if we treat each two-dimensional spin component as an independent field, the diagonal component of the matrix $S/6$ is given by $c_{\mathrm{eff}}^{\mathrm{boson}/\mathrm{fermion}}(s)$ for each spin s field.

4. IR Behavior of EE and Wilsonian Effective Action

Thus far, we have succeeded to extract contributions to EE, particularly from propagators of fundamental fields and vertices, and its most general form is given by the unified matrix form in Equation (36). Thus, if we can calculate correlators ${\widehat{\Sigma }}^{(g)}$ or $\widehat{G}$, we can obtain their contributions to EE. These contributions to EE are, however, a part of the whole EE in the framework of the ${\mathbb{Z}}_M$ gauge theory on Feynman diagrams, and we wonder how we can extract the other contributions to EE. In this section, we give a conjecture that the IR part of EE is exhausted by summing all the vertex contributions (together with propagator contributions) constructed from the IR Wilsonian effective action.

4.1. More Properties of Vertex Contributions to EE

First, note that the leading order term of the vertex contribution corresponding to composite operators $\{{\mathcal{O}}_n\}$ is perturbatively given by expanding the logarithm as

$$\begin{array}{ccc}\hfill S_{\mathrm{vertex}}& =\frac{V_{d-1}}{12}\int \frac{d^{d-1}{k}_{\Vert }}{{(2\pi )}^{d-1}}\mathrm{tr}\widehat{\lambda }{\widehat{\Sigma }}^{(g)}(\mathit{k}=0,k_{\Vert })\hfill & \hfill =\frac{V_{d-1}}{12}\int d^2\mathit{r}\sum _{m,n}{(\widehat{\lambda })}_{mn}{\langle {\mathcal{O}}_n(-\frac{\mathit{r}}{2},x_{\Vert }=0){\mathcal{O}}_m(\frac{\mathit{r}}{2},x_{\Vert }=0)\rangle }^{(g)}.\end{array}$$

(38)

The integral in the second line reflects the property of a twisted propagator that its center coordinate is pinned at the boundary $x_{\Vert }=0$ with two loose ends. For more detailed discussions, see [42,43]. For instance, when $\mathcal{O}=[{\varphi }^2]$, the leading perturbative term is given by using the renormalized propagator G of the fundamental field $\varphi$ as

$$\begin{array}{c}\hfill S_{\mathrm{vertex}}\sim \frac{V_{d-1}}{6}\int d^2\mathit{r}\left(-\frac{3{\lambda }_4}{2}\right)G{(\mathit{r},0)}^2.\end{array}$$

(39)

$-3{\lambda }_4/2$ is the component of $\widehat{\lambda }$ which associates $[{\varphi }^2]$ to $[{\varphi }^2]$. If we consider an operator such as $\mathcal{O}=[{\varphi }^n]$, the integrand is proportional to $G{(\mathit{r},0)}^n$ and decays faster for a larger n. This means that, at least perturbatively, higher-dimensional composite operators tend to contribute less to EE.

Another important point to note in Equation (33), particularly for its vertex part, is that, if some composite operators in $\widehat{\lambda }\widehat{G}$ dominate 1 in the logarithm in a strong coupling region, the contribution from the composite operator can be approximated as

$$\begin{array}{c}\hfill \mathrm{tr}log(\widehat{1}+\widehat{\lambda }\widehat{G})\sim \mathrm{tr}log(\widehat{G})\end{array}$$

(40)

up to a constant depending on the coupling constant. Then, EE can be written as a logarithm of renormalized correlators similar to the fundamental field. There is no explicit dependence on the coupling constant other than the overall factor and its dependence is only given through the renormalization of correlators.

4.2. Wilsonian RG and EE: Free Field Theories

Now, we discuss the issue of other contributions to EE besides the propagators and vertices. For this purpose, it is convenient to utilize the concept of the Wilsonian renormalization group (RG) to the effective field theory in the IR region [40,41] (A modern approach for the Wilsonian RG is given by the functional RG method [48,49]). In the Wilsonian RG, we first divide the momentum domain into low and high regimes. Schematically,

$$\begin{array}{c}\hfill k\in [0,\Lambda ]=[0,e^{-t}\Lambda ]+[e^{-t}\Lambda ,\Lambda ]\end{array}$$

(41)

with $t>0$ and then it integrates quantum fluctuations over the high regimes. Then, we rescale the momentum $k\to k^{\prime }=e^{t}k$ so that $k^{\prime }\in [0,\Lambda ]$. In this procedure, the original parameters in the action are renormalized, e.g.,

$$\begin{array}{c}\hfill m\to m^{\prime },{\lambda }_4\to {\lambda }_4^{\prime }.\end{array}$$

(42)

In addition, new interaction terms appear, e.g., in the ${\varphi }^4$ theory in $(3+1)$ dimensions,

$$\begin{array}{c}\hfill {\lambda }_6\frac{{\varphi }^6}{{\Lambda }^2},{\lambda }_{\partial }\frac{{(\varphi \partial \varphi )}^2}{{\Lambda }^2},{\lambda }_8\frac{{\varphi }^8}{{\Lambda }^4}\cdots .\end{array}$$

(43)

First, let us look at what happens for a free theory. For a free scalar field with a mass m, EE is simply given by

$$\begin{array}{cc}\hfill S_{\mathrm{EE}}(\Lambda )& =-\frac{V_{d-1}}{12}{\int }^{\Lambda }\frac{d^{d-1}{k}_{\Vert }}{{(2\pi )}^{d-1}}log\left[(k_{\Vert }^2+m^2)/{\Lambda }^2\right].\hfill \end{array}$$

(44)

By integrating the high momentum region, nothing happens except fluctuations of that region are discarded:

$$\begin{array}{cc}\hfill S_{\mathrm{EE}}(e^{-t}\Lambda )& =-\frac{V_{d-1}}{12}{\int }^{e^{-t}\Lambda }\frac{d^{d-1}{k}_{\Vert }}{{(2\pi )}^{d-1}}log\left[e^{2t}(k_{\Vert }^2+m^2)/{\Lambda }^2\right].\hfill \end{array}$$

(45)

Then, we rescale the momentum as $k^{\prime }=e^{t}k$ to obtain

$$\begin{array}{c}\hfill S_{\mathrm{EE}}^{\prime }(\Lambda )=-\frac{V_{d-1}}{12}{\int }^{\Lambda }\frac{d^{d-1}{k}_{\Vert }^{\prime }}{e^{(d-1)t}\times {(2\pi )}^{d-1}}log\left[(k_{\Vert }^{\prime 2}+e^{2t}{m}^2)/{\Lambda }^2\right].\end{array}$$

(46)

Of course, for a free field theory, it is equal to Equation (44) with the integration range $[0,e^{-t}\Lambda ]$. For an interacting theory, it is different since high and low momentum modes are entangled. We continue the integration over high momentum modes until $e^{-t}\Lambda =m$. Then, EE is given by Equation (44) with the integration range $[0,m]$. It gives the IR part of the EE at the scale m, and the discarded parts in higher momentum are UV cut-off dependent. By performing the momentum integration, the EE at the scale m is now given by

$$\begin{array}{cc}\hfill S_{\mathrm{EE}}^{\mathrm{IR}}(m)& \equiv -\frac{V_{d-1}}{12}{\int }^m\frac{d^{d-1}{k}_{\Vert }}{{(2\pi )}^{d-1}}log\left[(k_{\Vert }^2+m^2)/{\Lambda }^2\right]\hfill \end{array}$$

(47)

$$\begin{array}{cc}& =\frac{N_{\mathrm{eff}}{V}_{d-1}}{12}{m}^{d-1}log\left[{\tilde{\Lambda }}^2/m^2\right],\hfill \end{array}$$

(48)

where $\tilde{\Lambda }$ is proportional to the UV cutoff as $\tilde{\Lambda }=\Lambda exp[\Phi (-1,1,\frac{d+1}{2})/2]/\sqrt{2}$. $\Phi (z,s,\alpha )\equiv {\sum }_{n=0}^{\infty }{z}^n/{(n+\alpha )}^s$ is the Lerch transcendent. For example, in $d=3$, it is given by $\tilde{\Lambda }=e^{1/2}\Lambda /2$. Equation (48) coincides with the ordinary universal term in even spacetime dimensions. $V_{d-1}$ is the area of the boundary and

$$\begin{array}{c}\hfill N_{\mathrm{eff}}={\left(\frac{1}{2}\right)}^{d-1}\frac{1}{{\pi }^{(d-1)/2}\Gamma ((d+1)/2)}\end{array}$$

(49)

is the effective number of degrees of freedom that can contribute to EE in the IR. The result of Equation (48) indicates that the universal part of EE originates in the quantum correlations of fields whose length scale is larger than the typical correlation length $\xi =1/m$ of the system. $S_{\mathrm{EE}}^{\mathrm{IR}}(m)$ becomes larger for smaller masses m.

4.3. Wilsonian RG and EE: Interacting Field Theories

In the free case, the Wilsonian RG can extract the IR behavior of EE that is independent of the UV cutoff. In the Wilsonian RG, quantization is gradually performed from high momentum to low, and in the IR limit, all fluctuations are integrated out so that all the loop effects are incorporated in the Wilsonian effective action (EA). The Wilsonian EA becomes more and more complicated as radiative corrections are gradually taken into account. Thus, we can expect that all the contributions to EE are encoded in the Wilsonian EA. We conjecture that EE is given by a sum of all the propagator and vertex contributions in the Wilsonian EA.

In the following, we focus on the IR limit of the Wilsonian EA. Let us recall a simple case in Equation (10). The correlator $G_{{\varphi }^2{\varphi }^2}$ is graphically given by the upper figure of Figure 9, and the first term is given by Equation (39). This diagram is present even in the IR limit where all the fluctuations are integrated out since it is simply connected by the propagators of the fundamental field. The other terms vanish in the IR limit of the Wilsonian RG since they are quantum corrections to the first classical term. After all the fluctuations are integrated out, further quantum corrections should be absent because such effects are already absorbed in the Wilsonian EA. Thus, we expect that the vertex contributions of the composite operator, e.g., ${\varphi }^2$, are drastically simplified in the IR limit in which we can replace the Green function $G_{{\varphi }^2{\varphi }^2}$ by the leading diagrams as shown in the lower figure of Figure 9. After all, the vertex contribution in Equation (10) becomes

$$\begin{array}{c}\hfill S_{\mathrm{vertex}}=-\frac{V_{d-1}}{12}{\int }^{\Lambda }\frac{d^{d-1}{k}_{\Vert }}{{(2\pi )}^{d-1}}\log \left[1+3{\lambda }_4\int \frac{d^{d+1}p}{{(2\pi )}^{d+1}}G(p)G(-\mathit{p},k_{\Vert }-p_{\Vert })\right]\end{array}$$

(50)

in the IR limit. The coupling constant ${\lambda }_4$ is the renormalized one since it is a coefficient of Wilsonian EA in the IR limit. ( We have already taken quantum fluctuations into account and eliminated the UV divergences in coupling constants and observables in the IR limit, but another UV divergences appear in the calculation of EE since we need to sum all the momentum modes. It is also necessary even for the free theory, and indeed we extracted the IR universal part by subtracting cutoff dependent terms). As in the free case, we separate the vertex contributions into IR and UV parts. The IR part is defined similarly by restricting the integration range from $k_{\Vert }\in [0,\Lambda ]$ to $[0,m]$. Instead, we may integrate up to $1/{\xi }_{{\varphi }^2}$, where ${\xi }_{{\varphi }^2}$ is the correlation length of the operator $[{\varphi }^2]$. The difference is a matter of definition of the IR universal part of EE, and we need a precise prescription to subtract the cutoff dependent terms in EE. For example, we may take a variation with respect to the mass m and then integrate to obtain the universal part of EE. In this definition, we need to know how ${\xi }_{{\varphi }^2}$ and m are related. This is under investigation in moment.

In general, of course, we need to take operator mixings into account, but the generalization is straightforward. We will investigate more detailed behaviors of EE in the infrared limit of the Wilsonian EA for a concrete model. The final question is whether there are contributions to EE other than the vertex contributions in the Wilsonian EA. In the formulation of EE based on the ${\mathbb{Z}}_M$ gauge theory on Feynman diagrams, vertex contributions are only a part of all the contributions to EE. However, in the IR limit of Wilsonian RG, all the quantum fluctuations are integrated out, and we do not need to evaluate loop diagrams: all the Feynman diagrams are tree diagrams. Thus, the vertex contributions, as well as the propagator contributions, to EE must suffice for the IR behavior of EE. We will investigate further issues of the RG flow of EE in a separate paper [50].

5. Conclusions

This is the third paper in a series of our investigations on EE in interacting field theories based on the notion of the ${\mathbb{Z}}_M$ gauge theory on Feynman diagrams, proposed in [42] and extended in [43]. In the previous papers, we have focused on two important contributions to EE: one from the propagators of the fundamental field and another from vertices which can be interpreted as correlations of composite operators. In this paper, we have further extended the results to include effects of mixings of various composite operators as well as the original fundamental fields. The final formula of EE is given in a unified matrix form. We then discuss an implication to the IR behavior of EE from the Wilsonian RG approach to effective field theories. We conjecture that the IR part of EE in interacting field theories is given by a sum of all vertex contributions in the Wilsonian effective action. In this context, it is interesting to look at the relation to the variational method of EE [28,29,30,51]. In this approach, EE of interacting field theories is expressed in terms of a non-Gaussian deformation of the Gaussian vacuum wave function. This deformation must be related to the vertex contributions we have found.