Wilsonian Effective Action and Entanglement Entropy

This is a continuation of our previous works on entanglement entropy (EE) in interacting field theories. In arXiv:2103.05303, 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. As shown in the next paper arXiv:2105.02598, 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.

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, Wilsonian approach of renormalization [40][41][42][43] 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 [1,2] based on the Z M gauge theory on Feynman diagrams.In [1], 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 [2], 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 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 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 Sec.II, we first briefly summarize the notion of the Z M gauge theory on Feynman diagrams, two particular contributions to EE from propagators and vertices in the φ 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 Sec.III, 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 Sec.IV, 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 V.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.

II. SUMMARY OF PREVIOUS WORKS
In this section, we first summarize our previous works in [1] and [2], and then introduce a new concept of the generalized 1PI in order to unify the contributions to EE from propagators and vertices.
A. Z M gauge theory on Feynman diagrams Entanglement entropy of a subsystem A is defined by where ρ A = Tr Ā ρ tot is a reduced density matrix of ρ tot obtained by integrating out the complementary system Ā 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 → 1 limit of Rényi entropy S n = 1 1−n log Tr ρ n A [3]; by taking the replica parameter n = 1/M , we consider free energy of interacting quantum field theories on the orbifold R 2 /Z M × R d−1 denoted by F (M ) .Then, EE is written as Since a physical state of the Z M orbifold theory is invariant under the Z M projection operator, where ĝ is a 2π/M rotation operator around the origin, the orbifold theory can be interpreted as the 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(ĝ n i x, y) and summing over all independent configurations of such twists.Then, the notion of Z M gauge symmetry appears since we can rotate away some of the twists of propagators by the Z M gauge transformations on vertices of the Feynman diagram.
As a result, we can classify independent configurations of twists up to Z M transformations in terms of 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 Fig. 1.
A simple example of the Z M invariant configuration is given by Fig. 2. The 1-loop diagram has only one plaquette and the 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 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 Eq. (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 Apendix B. See also the discussions in Sec.IV.B and Sec.IV.C in our previous paper [2].

B. Propagator contributions to EE
In order to evaluate EE in Eq.( 2), we need to extract all the configurations of fluxes that do not vanish in the M → 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 Eq. (2), which assures the area law of EE1 .Among various configurations contributing to EE, we have focused on two particular configurations.The first type of configurations are given in Fig. 3 and can be interpreted as a twist of the propagator.The simplest configuration of this type is given by Fig. 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 [26] 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 FIG. 4: Twist of a vertex: three types of configurations can be attributed to a twist of the vertex.The dotted lines in the figures on the right-hand sides represent a virtual propagator that appears by opening the four-point vertex into a pair of three-point vertices by a delta function.The twist of a vertex is interpreted as a twist of the dotted propagator.
where G is the renormalized propagator2 and V d−1 is the volume of the boundary ∂A.The (d + 1)-dimensional momentum is written as (k, k ), where k is the two-dimensional components of time and the direction normal to the boundary and k is the (d − 1)-dimensional components parallel to the boundary.The UV cutoff is introduced.Writing the full inverse propagator as G −1 = (G −1 0 − Σ), the logarithm can be expanded as a sum, Then, it can be interpreted as a chain of free propagators connected by the self-energy Σ.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 Sec.II A. If we twisted every propagator in the chain, it would overcount the contributions to EE from the 1-loop Feynman diagram.

C. Vertex contributions to EE and generalized 1PI
The second type of configurations of fluxes we are going to focus on is given by Fig. 4 for the φ 4 scalar theory, L pot = λ 4 φ 4 /4.These configurations of fluxes are interpreted as twists of the interaction vertices, which in turn, 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 [2].
If composite operators propagating three channels are mixed like the φ 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 φ 4 -theory, it is given [2] by where In the following, the cutoff is not explicitly written for notational simplicity, as it can be recovered by dimensional analysis.The square brackets [O] represent the normal ordering of an operator O.The coefficient −3λ 4 /2 is a product of −λ 4 /4 and 6, where −λ 4 /4 is the coefficient in front of the interaction vertex and the coefficient 6 is a combinatorial factor for separating four φ(x)'s into a pair of φ 2 (x) and φ 2 (y).As shown in Fig. 5, the Green function of the composite operator can be written as where Σ (g) ] in a generalized sense.We call it g-1PI.Namely, the quantity with the superscript (g) does not contain a diagram like Fig. 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 beads diagrams are not included in g-1PI diagrams.
x 2 u S p 6 y P b V q 7 W H l r 9 D x 9 X I 8 b g D 9 g / g c x n c h t n H r r o 2 K 7 a m q f P g N u j e 2 9 A + c r k c e G 8 P X W R n j r H H D e w T y A F Z W 0 g a a L H j w H S J n I V F 1 6 5 W g 7 r z z 1 E 9 5 5 4 t I C 0 S n X o + 9 w Y 5 t N Y D z B v o X V 9 I e F P B V 4 / p q w B k p / L j c f A s 5 g w Z R V o A z 5 i v A f A W K M 6 8 S 1 N W F r w B j a 3 X l K l A g F u q q q M C G s v L 5 5 3 q t 7 P A 4 D a W q 2 6 4 E d X X n p 5 / y 6 s T n n + t S f R 3 3 z u w b 6 v 7 8 Y L Q S X N e P j s + a f A E X a r 4 l + V 5 e T z 7 s 6 V A 2 k m 9 P m m 2 p F w R 6 V z P 1 g k A f a q T e S 7 Z Y    By using Eq.( 9), we can rewrite Eq.( 7) as In the following equations including Eq.( 10), the argument (k = 0, k ) of the integrand for the k integral is implicit.Now we can write both the propagator and vertex contributions in Eq.( 5) and Eq.( 10) in a unified matrix form as 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 Eq.( 11) is convenient for a unified description in the following discussions, but it is always possible to go back to the form like Eq.( 7), where the vertex contributions are written in terms of the ordinary renormalized propagators without the superscript (g).Also, note that all the single twist contributions from a vertex can be written in the above 1-loop type formula, Eq.( 7) or Eq.( 12).In the case of the propagator contributions to EE, we have proved the statement by using the 2PI formalism in [1, 2].Here we use a diagrammatic method to prove it for the vertex contributions in Appendix B.

III. GENERAL VERTEX CONTRIBUTIONS TO EE
In this section, we extend the analysis of vertex contributions to EE from the φ4 interaction to more general cases.
A. φ 6 scalar field theory First, let us consider the φ 6 interaction, In this case, we have two types of vertex configurations3 as drawn in Fig. 7 and need to introduce three types of composite operators, φ 2 , φ4 , and φ 3 , to extract all the vertex contributions to EE.Since the theory has Z 2 invariance under φ → −φ, the Z 2 -even operators, . . . . . .

ˆ ⌃(g)
< l a t e x i t s h a 1 _ b a s e 6 4 = " A 1 T 2 e J 7 N N e i 5 g j w y y y W z e L t b 6 l 6 s V K e φ 2 and φ 4 , are mixed with themselves while the Z 2 -odd operator φ 3 is mixed with the fundamental field φ.Therefore, the propagator contribution in Eq.( 5) needs a modification.
First, let us consider the modified propagator contributions in the φ6 theory.Such contributions come from 1-loop type diagrams of mixed correlations of φ and φ 3 operators.They are given by (Fig. 8) It is a natural generalization of Eq.( 12) including an operator mixing.The diagonal component of Ĝ0 is the bare propagators of φ and φ 3 operators, respectively.λ is a matrix whose matrix element represents the coefficients of opening the φ 6 vertex.The coefficient for φ 3 to it consists of 1PI diagrams that do not contain beads diagrams shown in Fig. 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 Fig. 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 Z 2even operators, φ 2 and φ 4 .This case is simpler because the bare Green function is unity; . Then, we have the same matrix form where, in this case, matrices are given by λ = The coefficient comes from 5/2 = 1/6 × 6 C 2 .It is a 2 × 2 matrix generalization of Eq.( 7).
The g-1PI self-energy Σ(g) does not contain beads diagrams, especially diagrams connected by the φ 6 vertex decomposed into φ 2 and φ 4 .
Note that EE of Eqs.( 14) and ( 16) written in terms of the g-1PI functions can be rewritten in terms of the renormalized correlation functions as in the φ 4 case of Eqs.( 4) and ( 9).The only difference is that we now have operator mixings and the relationship becomes more complicated.Let us explicitly check it for the Z 2 -odd case of Eq.( 14).It is rewritten as Writing the inside of the parenthesis as G, its matrix elements are given by We can explicitly see that the sum of g-1PI's in each matrix element is combined into the ordinary 1PI functions Σ's, and hence can be written by the renormalized correlation functions as As a result, Eq.( 14) can be summarized as where The same discussion can be applied to Eq.( 16).This gives an alternative, unified formula for EE in terms of the renormalized Green functions.
B. φ 4 + φ 6 theory and further generalizations Let us generalize a bit more and consider a case when the Lagrangian contains two interaction terms As in the φ 6 theory, we need to consider three composite operators, φ 2 , φ 4 , and φ 3 , in order to take into account contributions to EE from these vertices.Again, we have Z 2 invariance and EE is a sum of Z 2 -even and odd contributions.The Z 2 -odd contribution is given by where Ĝ0 is the same as in Eq.( 15) while Z 2 -even contribution is given by Now a generalization to e.g.φ 2n vertices with higher n is evident.The propagator and vertex contributions to EE are unified to be written in a matrix form as Eq.( 14): 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.Ĝ0 is a diagonal matrix whose entry is mostly 1 except the fundamental field.λ represents a mixing among operators via vertices while Σ(g) represents amputated correlators of all the fundamental and composite operators.
The notion of the g-1PI is also extended to exclude all the beads 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 Eq.( 32) in terms of the renormalized correlation functions in the same manner as in Eq.( 27) as Here, Ĩ = diag(0, 1, • • • , 1), Ĝ 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 φ.The size of the matrices is finite as far as there is a finite number of vertices.In the φ n -theory, we need to consider only the composite operators [φ j ] with j ≤ n − 2, which appear to open vertices.

C. 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, In this case, the two types of scalar composite operators, [φ 2 ] and [(∂φ) 2 ], as well as a spin-1 operator [φ∂ µ φ] appear from an opened vertex.Since the spin-1 operator does not mix with either φ or [φ 2 ] or [(∂φ) 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 Eq.( 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 Z M twist and hence an extra phase appears in evaluating EE [47].The operator J µ := [φ∂ µ φ] 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 Eq.( 32).On the other hand, the contribution to EE from the 2-dimensional vector J is modified.From Eq.(2.21) in [47], the coefficient of EE is proportional to for a bosonic field with spin s.This coefficient c eff replaces the coefficient of 1/6 in front of Eq.( 5).Thus for (d + 1)-dimensional vector J µ , 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 is given by either of the following two forms, where 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 2-dimensional rotation with spin s.
n.This means that at least perturbatively, higher-dimensional composite operators tend to contribute less to EE.
Another important point to note in Eq.( 33), particularly for its vertex part, is that if some composite operators in λ Ĝ dominates 1 in the logarithm in a strong coupling region, the contribution from the composite operator can be approximated as tr log 1 + λ Ĝ ∼ tr log Ĝ (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.

B. 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 [42,43]. 5In the Wilsonian RG, we first divide the momentum domain into low and high regimes.Schematically, with t > 0 and then, integrate quantum fluctuations over the high regimes.Then, we rescale the momentum k → k = e t k so that k ∈ [0, Λ].In this procedure, the original parameters in the action are renormalized, e.g., In addition, new interaction terms appear, e.g., in the φ 4 theory in (3 + 1) dimensions, 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 By integrating the high momentum region, nothing happens except fluctuations of that region are discarded: Then, we rescale the momentum as k = e t k to obtain Of course, for a free field theory, it is equal to Eq.( 44) with the integration range [0, e −t Λ].
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 Λ = m.Then, EE is given by Eq.( 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 where Λ is proportional to the UV cutoff as Λ = Λ exp Φ(−1, 1, d+1 2 )/2 / √ 2. Φ(z, s, α) ≡ ∞ n=0 z n /(n + α) s is the Lerch transcendent.For example, in d = 3, it is given by Λ = e 1/2 Λ/2.Eq.( 48) coincides with the ordinary universal term in even spacetime dimensions.
V d−1 is the area of the boundary and is the effective number of degrees of freedom that can contribute to EE in the IR.The result of Eq.( 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 ξ = 1/m of the system.
S IR EE (m) becomes larger for smaller masses m.

C. 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 Eq. (10).The correlator G φ 2 φ 2 is graphically given by the upper figure of Fig. 9 and the first term is given by Eq.( 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.φ 2 , are drastically simplified in the IR limit in which we can replace the Green function G φ 2 φ 2 by the leading diagrams as shown in the lower figure of Fig. 9.After all, the vertex contribution in Eq. (10) becomes in the IR limit.The coupling constant λ 4 is the renormalized one since it is a coefficient of Wilsonian EA in the IR limit. 6As 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 ∈ [0, Λ] to [0, m].Instead, we may integrate up to 1/ξ φ 2 where ξ φ 2 is the correlation length of the operator [φ 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 ξ φ 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 Z M gauge theory on Feynman diagrams, vertex contributions are only a part of all the contributions to EE.But, 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].
V. CONCLUSIONS This is the third paper in a series of our investigations on EE in interacting field theories based on the notion of the Z M gauge theory on Feynman diagrams, proposed in [1] and extended in [2].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 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.
at least one nonzero twist.As a result, the argument of the delta function in Eq.(A2) is always nonzero and it combined with I carries a nontrivial dependence in M after the summation over twists.Unless an explicit calculation is done, we do not know the precise M dependence of Eq.(A2) or S 1/M .Nevertheless, since terms contributing to S 1/M always have nonzero arguments of the delta function, there is no more volume factor other than If M can be analytically continued to M = 1/n, this completes the proof of the area law for Rényi entropy S n .
The proof above only depends on the technique of Feynman diagrams and thus the area law for Rényi entropy is proven for any locally interacting QFTs, given a half space as a subregion.
It is worthwhile to note that the area law for Rényi entropy immediately implies the area law for the capacity of entanglement [44,45], as well as EE since C A is linear in Rényi entropy.Since C A is alternatively written as the fluctuation of the modular Hamiltonian − log ρ A , it is more sensitive to the change of dominant contributions in the replicated geometry and recently discussed in the context of the black hole evaporation [53][54][55].It is interesting if we can compute such quantities in interacting theories and follow the behavior of higher orders in M .
Although the area law itself is intuitive for physicists as entanglement across the boundary ∂A should be dominant for any local QFTs, the proof of this is difficult; a general proof is known only for gapped systems in (1 + 1) dimensions [56].It is remarkable that we can show the area law of both EE and Rényi entropy in any locally interacting theories.
As a further generalization, it is intriguing to relax several assumptions and see how the EE and Rényi entropy deviates from the area law.In our setup, ∂A is smooth, the interactions are local, and the system is translationally invariant.Some cases are known where the above features are not satisfied and the area law is violated.For example, when the entangling surface ∂A has a singular geometry, a logarithmic correction appears (see [57] for example).For (non-)Fermi liquid theories [58], another logarithmic violation to the area law is known.For nonlocal [59] or non-translationally invariant [60,61] systems, the volume law instead of the area law of EE has been confirmed.To see the transition from the area law to the volume law, Lifshitz theories [62][63][64] might be an interesting playground as it possesses nonlocal feature in some limit.
+ • • • < l a t e x i t s h a 1 _ b a s e 6 4 = " f i I q / 1 9 u R Q g D n C X K z 5 J J 9 u 9 S F z s = " > A A A l A 3 i c 7 V n N j y N H F e 8 s X 2 E g J A s X J C S 2 x M 7 A j O h p 6 r u 6 d 7 C U g J A 4 R F G S 3 U 0 i e S a j d r t s t 9 Z u m + 7 2 z M 5 a P n L h H + D A i U g g o d w Q V 0 5 c + A c 4 5 E 9 A H I P E h Q O v q q t 7 7 P H Y 0 e z u r H b F 2 v K 4 6 l W 9 j 3 o f v 3 7 l 6 U y G a V F i / N k r N 7 7 0 5 a 9 8 9 W u v f n 3 r G 9 9 8 7 V u v v 3 H z 2 x 8 U 4 2 m e 6 P v J e D j O P V e 9 7 3 g 9 A O / G U 9 6 b 3 S + 9 d 7 7 6 X 3 H r t F r v 1 0 1 u t 7 V 9 v f 7 r 9 5 + 2 / V F t v v O J 4 v u M t v b b / + j + F y 7 R 9 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " f i I q / 1 9 u R Q g D n C X K z 5 J J 9 u 9 S F z s = " > A A A l A 3 i c 7 V n N j y N H F e 8 s X 2 E g J A s X J C S 2 x M 7 A j O h p 6 r u 6 d 7 C U g J A 4 R F G S 3 U 0 i e S a j d r t s t 9 Z u m + 7 2 z M 5 a P n L h H + D A i U g g o d w Q V 0 5 c + A c 4 5 E 9 A H I P E h Q O v q q t 7 7 P H Y 0 e z u r H b F 2 v K 4 6 l W 9 j 3 o f v 3 7 l 6 U y G a V F i / N k r N 7 7 0 5 a 9 8 9 W u v f n 3 r G 9 9 8 7 V u v v 3 H z 2 x 8 U 4 2 m e 6 P v J e D j O P        l G s Y 4 q t X 0 1 p 4 J J q N V x N V 2 X G 6 l l A L 0 V p Y q p n g H V z o v h g q L c s B g X B 1 q y n T g n B 5 E H q K d t p J w z R A b r a X h The correct formula must take the redundancies caused by Z M gauge invariance into account.
Such redundancies occur in the above expansion of Eq.(B6) when there are more than one Σ (g) φ k φ k as shown in Fig. 10.The coefficients of these terms in Eq.(B6) overcount the effects of the twist.
The resolution to avoid the overcounting is simple.For the term consisting of m g-1PI parts in Eq.(B4), we should divide it by m.Consequently, by replacing G φ k φ k in the naive estimation Eq.(B5) with Σ (g) we get the correct contributions to EE as This is the result of Eq.( 10).In the 2PI formalism, the result is interpreted that only the 1loop diagram provides a single twist contributions of propagators and all the other diagrams cancel each other.In the above discussions, we did not separate diagrams into 1-loop and others, but instead used the very basic relation of Eq.(B3).Then, using the property of the Z M redundancy, the logarithmic factor for the 1-loop diagram naturally appears.
The above discussion can be straightforwardly generalized to more general composite operators with operator mixings.When we have a set of operators {O a } by opening vertices, we consider g-1PI self-energies Σ (g) OaO b and a matrix generalization of the nodal structure of (λ n /n) × C (n) ab .It is also straightforward when the fundamental fields are mixed with other operators; it is sufficient to consider Σ(g) Ĝ0 in the formulation.As a result, we arrive at the unified form of Eq.( 32).

FIG. 1 :FIG. 2 :
FIG. 1: The figure shows a Z M twist configuration on a Feynman diagram.Given twists {n i } assigned on propagators, a flux of twists m = 0, • • • M − 1 is defined on a plaquette by a sum of twists around the plaquette m = i n i mod M .The flux is invariant under Z M gauge transformations on vertices.

FIG. 3 :
FIG. 3: Twist of a propagator: if the fluxes of plaquettes straddling a shared propagator are given by m and −m, such a configuration is interpreted as a twist of the shared propagator.The upper figures show a relevant part in a general Feynman diagram that twists the propagator.The lower figures are an example of such a configuration in a 3-loop diagram.
S 2 m / t o 2 U G j g 9 U V N o l 4 3 u h + e B W i I e 4 8 / 1 z D g O v r 5 2 6 S y u 4 z W j w 6 B Y Y e 9 h a T K E 2 P n X n 7 J + 0 O u t M 7 z a L B k q 2 c / T S a 9 I f W J 7 7 q Z f Z M P b j C A 8 Z k l 8 n r q x w r b y n P w Y b P k E g u + A s w S n D 3 m W 1 e c / 0 E o y 8 g c K 1 f f O D G d r r n b D u 9 w X C W L 5 C F 7 s X T Y e O / n k 5 e v 4 V D 7 F 5 o c 0 D 8 4 F b g X + / M b r z 8 c d A L h s E s G A R n w S S I g 2 m Q w z g N o m A B P 0 c B C X A w B 9 l x s A R Z B q P E z c f B K t g D 3 T N Y F c O K C K S P 4 P c I n o 6 8 d A r P F n P h t A d g J Y V 3 B p o o 2 M d / x X / A n + I / 4 z / i v + H / X Y m 1 d B j W l w v 4 7 B e 6 8 f z k t V 9 / + / 6 / d 2 p N 4 D M P x p V W q 8 9 5 c B p o 5 2 s C v s + d x E Y x K P Q f P / 3 N p / f v v r e / / D 7 + G P 8 d / P 8 d / g T / C S K S 2 m / t o 2 U G j g 9 U V N o l 4 3 u h + e B W i I e 4 8 / 1 z D g O v r 5 2 6 S y u 4 z W j w 6 B Y Y e 9 h a T K E 2 P n X n 7 J + 0 O u t M 7 z a L B k q 2 c / T S a 9 I f W J 7 7 q Z f Z M P b j C A 8 Z k l 8 n r q x w r b y n P w Y b P k E g u + A s w S n D 3 m W 1 e c / 0 E o y 8 g c K 1 f f O D G d r r n b D u 9 w X C W L 5 C F 7 s X T Y e O / n k 5 e v 4 V D 7 F 5 o c 0 D 8 4 F b g X + / M b r z 8 c d A L h s E s G A R n w S S I g 2 m Q w z g N o m A B P 0 c B C X A w B 9 l x s A R Z B q P E z c f B K t g D 3 T N Y F c O K C K S P 4 P c I u b 5 h s 3 P i Z k e g f w j R h Y F o a D x 2 u S p 6 y P b V q 7 W H l r 9 D x 9 X I 8 b g D 9 g / g c x n c h t n H r r o 2 K 7 a m q f P g N u j e 2 9 A + c r k c e G 8 P X W R n j r H H D e w T y A F Z W 0 g a a L H j wJ U 5 h b 5 L L n f Z z c H 7 N I Q r N 3 m X 3 3 r j J 7 4 D v x r c D L 4 H 1 k m g g j e C n w X v B A + D w c 0 H N 5 / e / O X N X 6 G / o H + i / 6 D / F k t f f s n r f C t o v L 7 7 y v 8 B N Z F 1 D A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D l l a t f L n Y b H 6 y / g 9 W V G f R G f b u I A = " > A A A h S n i c x V l N j y N H G e 6 E AG H 4 S B Y u i L 0 U 7 A 5 4 h L e p 7 4 9 d j R S I I j i g K M n u J p H G k 1 H b 7 r F b 2 / 6 g 3 b O z s 5 Z P c O I P c O B E J A 6 I n 8 G F A 1 c O + Q k I c S F I C M G B t 6 q r 3 d 1 j T 1 u 7 G W U 9 6 3 H 1 W / U + 7 9 d T b 5 V n + / M 0 W e Q Y f / L S y 1 9 4 5 Y t f + v K r X 9 n 7 6 t e + / o 3 X X r / x z f c X s 7 N s E D 8 c z N J Z 9 m E / W s R p M o 0 f 5 k m e x h / O s z i a 9 N P 4 g / 6 j N + 3 8 B 4 / j b J H M p g / y i 3 l 8 P I S 2 m / t o 2 U G j g 9 U V N o l 4 3 u h + e B W i I e 4 8 / 1 z D g O v r 5 2 6 S y u 4 z W j w 6 B Y Y e 9 h a T K E 2 P n X n 7 J + 0 O u t M 7 z a L B k q 2 c / T S a 9 I f W J 7 7 q Z f Z M P b j C A 8 Z k l 8 n r q x w r b y n P w Y b P k E g u + A s w S n D 3 m W 1 e c / 0 E o y 8 g c K 1 f f O D G d r r n b D u 9 w X C W L 5 C F 7 s X T Y e O / n k 5 e v 4 V D 7 F 5 o c 0 D 8 4 F b g X + / M b r z 8 c d A L h s E s G A R n w S S I g 2 m Q w z g N o m A B P 0 c B C X A w B 9 l x s A R Z B q P E z c f B K t g D 3 T N Y F c O K C K S P 4 P c I FIG.5:A Green function of a composite operator can be written in terms of the generalized self-energy Σ (g) φ 2 φ 2 , which is 1PI with respect to the propagator of the composite operator at the vertex as well as the fundamental field.

FIG. 7 :
FIG. 7: Two different composite operators appear by opening the φ 6 vertex.Each flux configuration corresponds to twisting the propagator of the respective composite operator.
FIG. 8: A Schwinger-Dyson type diagram to represent mixings between different operators.All possible composite operators are assigned to each dotted part.λ is a matrix-valued vertex and Σ (g) is a generalized 1PI (g-1PI) self-energy with respect to the composite operators.
s U 9 5 J + / C R L s l 7 8 y T C N 2 6 e H v f j p 4 b M H b v z p e Z y O k k H / c X Y 5 j P d P 2 8 f 9 5 C j p t D N 0 H d y 8 9 f 7 G x l 6 W P H s x i r N x j J m X 4 T D p Z G d p / I N o l F 3 2 4 p 2 x g w 4 v k m 5 2 s k r 7 X 6 y X D U Z y D D p l h n B t K B I d s U P C d o O 4 e J b 3 e z j g 9 P m y F T M p 7 a d x t h e R + e J z G c d + 3 D n y 5 I z / F w z 3 4 n 6 3 8 d / a B 2 / d J h H x P + F s g x a N 2 0 H x 8 + H g 5 o 1 P g 7 2 g G w y C T n A W n A Z x 0 A 8 y t H t B O x j h d z e g A Q m G 6 N s P x u h L 0 U r 8 e B x c B R t Y e 4 Z Z M W a 0 0 f s M f 4 / x a b f o 7 e O z w x z 5 1 R 1 Y 6 e G V Y m U Y b J I / k 9 + R z 8 m f y O / J 3 8 m / F 2 K N P Y b j c o n 3 w 3 x t P D x 4 8 5 f f f v S v l a t O 8 Z 4 F J 5 N V S z l n w V F g P N c E 3 I e + x 3 n R y d e f v / j V 5 4 / u P d w c f 4 9 8 S v 4 B / r 8 h n 5 E / w oP + + T 8 7 v / 0 o f v j r J X w O w W X Z j i W w 1 c b + u N 3 t 4 u 8 R f h 2 T I d h f Y o X r H y 1 B G B d c R 8 U O b 6 I 9 Q F / o 9 / 3 U z 3 a x O / S x C 4 M L b / E E r V M f 3 w R I L / B p K 3 g c f I D 3 J v r 2 E r v l H m e w 4 y K 8 n K X z x F l 0 L C 6 8 V 8 9 W e J U z c x G Y z C p 7 e 9 6 j 1 G f m p U c / Q X v o c 6 7 l e 1 N w u v C f h m i N v N c J W n 0 f 4 2 M / 6 8 x z z / 1 o o + e w 4 H D s V 5 9 h X t c j d H y E O m j F G G k V e 5 r 5 T / 2 l f u 8 2 o r C D U x X 5 F / E x L r 2 5 i 5 j H 8 K P v u f Q 9 3 u a S f R / W d i e f e Y i R Y + / Z B H X o d z k D S l q c W j e z n i 1 j / L 7 n W b W w y / W R g + o M 7 w T f D 9 7 F y 8 3 o + R k H 2 G s 3 J 4 G N g d c N t z s H f v Q Y 6 3 f g X R T I x o p z v 1 e 5 h s y f f V U x d Pn b 9 b n a 9 n m 8 B f v b e B 8 H d z B 6 7 q P r d s X F t O c Z 3 M H a + z O r d / 1 e d g q 2 O 9 6 z M 5 + x + w 3 s A + w B r S w k D b T Y 5 8 D C P Y X u 0 m m V n W 1 8 z C J K I v q R u P 3 O D w s F f i P 4 T v B d W K e B D t 4 J f h J 8 G D w J O r f + c O s v t / 5 6 6 2 + b W 5 s f b D 7 Z f J p P v f F a s e Z b Q e N n 8 + f / A U T P c 8 c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 o i O y S Y 0 w 7 H K p i 5 l z D 8 U A z y A l a s U 9 5 J + / C R L s l 7 8 y T C N 2 6 e H v f j p 4 b M H b v z p e Z y O k k H / c X Y 5 j P d P 2 8 f 9 5 C j p t D N 0 H d y 8 9 f 7 G x l 6 W P H s x i r N x j J m X 4 T D p Z G d p / I N o l F 3 2 4 p 2 x g w 4 v k m 5 2 s k r 7 X 6 y X D U Z y D D p l h n B t K B I d s U P C d o O 4 e J b 3 e z j g 9 P m y F T M p 7 a d x t h e R + e J z G c d + 3 D n y 5 I z / F w z 3 4 n 6 3 8 d / a B 2 / d J h H x P + F s g x a N 2 0 H x 8 + H g 5 o 1 P g 7 2 g G w y C T n A W n A Z x 0 A 8 y t H t B O x j h d z e g A Q m G 6 N s P x u h L 0 U r 8 e B x c B R t Y e 4 Z Z M W a 0 0 f s M f 4 / x a b f o 7 e O z w x z 5 1 R 1 Y 6 e G V Y m U Y b J I / k 9 + R z 8 m f y O / J 3 8 m / F 2 K N P Y b j c o n 3 w 3 x t P D x 4 8 5 f f f v S v l a t O 8 Z 4 F J 5 N V S z l n w V F g P N c E 3 I e + x 3 n R y d e f v / j V 5 4 / u P d w c f 4 9 8 S v 4 B / r 8 h n 5 E / w o P 1 e 5 h s y f f V U x d P n b 9 b n a 9 n m 8 B f v b e B 8 H d z B 6 7 q P r d s X F t O c Z 3 M H a + z O r d / 1 e d g q 2 O 9 6 z M 5 + x + w 3 s A + w B r S w k D b T Y 5 8 D C P Y X u 0 m m V n W 1 8 z C J K I v q R u P 3 O D w s F f i P 4 T v B d W K e B D t 4 J f h J 8 G D w J O r f + c O s v t / 5 6 6 2 + b W 5 s f b D 7 Z f J p P v f F a s e Z b Q e N n 8 + f / A U T P c 8 c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 o i O y S Y 0 w 7 H K p i 5 l z D 8 U A z y A l a s U 9 5 J + / C R L s l 7 8 y T C N 2 6 e H v f j p 4 b M H b v z p e Z y O k k H / c X Y 5 j P d P 2 8 f 9 5 C j p t D N 0 H d y 8 9 f 7 G x l 6 W P H s x i r N x j J m X 4 T D p Z G d p / I N o l F 3 2 4 p 2 x g w 4 v k m 5 2 s k r 7 X 6 y X D U Z y D D p l h n B t K B I d s U P C d o O 4 e J b 3 e z j g 9 P m y F T M p 7 a d x t h e R + e J z G c d + 3 D n y 5 I z / F w z 3 4 n 6 3 8 d / a B 2 / d J h H x P + F s g x a N 2 0 H x 8 + H g 5 o 1 P g 7 2 g G w y C T n A W n A Z x 0 A 8 y t H t B O x j h d z e g A Q m G 6 N s P x u h L 0 U r 8 e B x c B R t Y e 4 Z Z M W a 0 0 f s M f 4 / x a b f o 7 e O z w x z 5 1 R 1 Y 6 e G V Y m U Y b J I / k 9 + R z 8 m f y O / J 3 8 m / F 2 K N P Y b j c o n 3 w 3 x t P D x 4 8 5 f f f v S v l a t O 8 Z 4 F J 5 N V S z l n w V F g P N c E 3 I e + x 3 n R y d e f v / j V 5 4 / u P d w c f 4 9 8 S v 4 B / r 8 h n 5 E / w o P

FIG. 9 :
FIG. 9: The upper figure shows Feynman diagrams constituting G φ 2 φ 2 .Only the leading diagram survives in the IR limit.
T 9 L e 2 k S l 0 A 6 v v n 9 P 2 5 t H Z b p g 0 e F L m c a d p 6 h S Z q U 0 1 z / J C j K s 6 F u z Y x o d J p 2 y 0 E L B 0 p M y v k c 7 c B 2 1 N W 9 e D o s 0 f k G V I 5 R t Q e 5 1 9 Z h R / f T b G Z U O M H z 9 k M n y D 9 r B k U S g 6 5 9 4 j + s R u R o a 2 e a J e P R S G c l S n s o 1 7 + a p r m + g w 4 n M S j a x T 5 h e O / A W Z L G / T w e o a H u l a g o 4 7 x E c Y m w O s 7 a g Z z I 3 M w n I n o G U I n x w 4 k b e D / 4 H / h D / H f 8 e f 4 n / i / 6 6 V N b M y j C 1 n 8 N 2 p e P X k + P X f f P f u f 7 6 Q a w T f p T c 4 5 9 p o c + n 1 v N D a m o L t E 0 s x p 0 g q / p N H v / 3 8 7 p 3 3 d 2 Y / x J / g f 4 H 9 v 8 e f 4 b / B C b K T f y d / e E + / / 7 s N 9 n T A l k 0 e S 0 F X D P 4 x 3 u 3 C 3 x 6 8 j S U T s P 4 M O A y 9 2 C B h 5 m w t n I d 3 Y D w T 9 L e 2 k S l 0 A 6 v v n 9 P 2 5 t H Z b p g 0 e F L m c a d p 6 h S Z q U 0 1 z / J C j K s 6 F u z Y x o d J p 2 y 0 E L B 0 p M y v k c 7 c B 2 1 N W 9 e D o s 0 f k G V I 5 R t Q e 5 1 9 Z h R / f T b G Z U O M H z 9 k M n y D 9 r B k U S g 6 5 9 4 j + s R u R o a 2 e a J e P R S G c l S n s o 1 7 + a p r m + g w 4 n M S j a x T 5 h e O / A W Z L G / T w e o a H u l a g o 4 7 x E c Y m wj + K s i w Z x g Q Y 6 7 Q 9 K N O 4 h 4 N r a u T u I J y D q r T x B 7 b R 7 p z u h X P G Q Y c Y i S n E k m D w C y 7 t 5 f I p Q 2 3 6 N J 3 G S l m D v E d o l U S A i X 6 o A 8 z 0 U B C g Z Z 2 U + H h a w I g I h f M k C H u 5 Z 3 b u E + S I M q L A b 7 Y w G n F x k Y z 4 n g Y w c D 4 W x z 1 j F w m h A S T 0 7 Z + E 0 C J k l W x Z B A t 4 I M W w V 4 T J l b i U M m H L M P A y U M l a r i p f z Q I T m f F T u o f 3 9 2 g Y n D A j J W T L U 6 A D V H r J p 2 Jr l / Y 6 P q B B 3 c t 3 1 E T 5 A / V z r z I 4 6 w 6 k 2 a x j N E f K X P E r Q / 5 t L D y A B y z y 2 D H F a B W 7 m 6 3 R C 1 w 2 9 T 8 Q p g D R T 7 C k + T i 6 H T B u x R R u p a O Y r k M l I W C + + y J B J K a N w d 1 D w W I o k C 8 n G O z Z X z C X P S v f D 4 e F s C p m e d z / c d A u m R u r u x x H I a g t S r X B 8 3 v 1 w J U 1 1 u t 6 H h 9 I V 8 s X e h 0 f K F X L d + w g s b C H j p v d x l M s U 1 0 v q v P e B n t s W M q 1 7 H x 6 F r p D r R t O Z 4 w R e C 2 a+ d P U G z F Q h I Z g q g E w c Y v w U b 5 / 2 r g m e N 5 i p m j Z z O R a W Y v D T Y u a m N p M q R r n g V J I Q A J 6 x q 2 I m l 9 x g p K x b O L g R N v N 1 O g E n I 4 Y V h j B y w a A J v P a r e W R x s D G S h 6 S Z r 4 A m 4 E y 9 + A K B J r q n H 5 b o n X F X u / P s U l N 7 v v n x K w M q a s d Z M g D f Z u O 8 H K B T X Z R H 7 T T L d I 4 K P X G / p h p z e l A M r c O 3 4 7 y v / a t G Y y l L Z 9 u j 7 f n l l n F z I 9 o n p k Y f w 7 b h d d p G M F z X m p / k r m K d k d p a J x W q k 5 p g P O a R r z U c B B J e m X g 8 h 9 G Q I m B + q E w H f N V Y H N 5 N + 6 M Y H c 8 O J 4 M U f T y j c 9 S M 5 h / P d l F / b 7 4 u Q a H x i + x j Q D 1 G E v x 4 j V R G C d w g f H 7 l 0 z y L J G A A F 8 I W 5 e M Z d 6 1 p Q O H q b 6 8 t V 6 y e J 0 w D Z h L g 2 a v l u M 6 + x 4 G g d d n H l X L Z 9 x x C E F e h z b 7 n 8 I H A S Q R J w O G G / E y T Q B B R q W X P N v d M A + l H 9 B m r F f K J U v 4 w 6 Y 7 L A h n h h z r r L v 3 P + P i N 2 9 B p 2 R d a H R A 3 u O 2 5 1 7 v j m z c + 8 Q 6 9 r j f 2 E m / q j T z t Z V 4 J 4 6 E X e w W 8 2 x 7 x s D c B 2 p E 3 A 1 o O o 9 S u a 2 / u b Q H v F H Z p 2 B E D 9 Q H 8 7 cO s 7 a g Z z I 3 M w n I n o G U I n x w 4 k b e D / 4 H / h D / H f 8 e f 4 n / i / 6 6 V N b M y j C 1 n 8 N 2 p e P X k + P X f f P f u f 7 6 Q a w T f p T c 4 5 9 p o c + n 1 v N D a m o L t E 0 s x p 0 g q / p N H v / 3 8 7 p 3 3 d 2 Y / x J / g f 4 H 9 v 8 e f 4 b / B C b K T f y d / e E + / / 7 s N 9 n T A l k 0 e S 0 F X D P 4 x 3 u 3 C 3 x 6 8 j S U T s P 4 M O A y 9 2 C B h 5 m w t n I d 3 Y D wG G r J + H 9 n d J n Y d G z v k n V q N A x i N b H x T k P Q I Z r v e P e 8 d + F 6 W v r d B b + 3 j E v S Y C G + 2 0 p z E a D R W n N p T P f i C U 1 W W m Q i c 7 6 q p Q 3 u i 3 G b m m Z U + g P H E 5 p x v q T n Y d G p n E x g V 9 t Q p j D I b 4 7 7 d N b W 2 V + e I g d J x N v Q t 9 x T 2 d a 2 E x E Y o g Z G G F d / 5 t L S z b O O 5 2 0 t R a E F V B f a D b Y z r 0 + x D z D W c I 7 O 2 Z F b e z g a / T x a 8 U + 3 s w E r f n u x c 6 s R 6 u Q Q p u a t a s 3 M x W 2 b w / o W 1 y g c v L 6 4 c N z X c 8 n 7 k v Q U f s 2 N o d x y D r 8 2 e F H S M L W 4 Y 7 x z b 1 T 7 w t + B 0 g S e W O E 6 s r y o M u X z 3 v L H Q 5 G / X 5 m p s 8 3 g X 9 O / B 9 8 z b h t U T G 1 3 j F R P T o b V g G 3 g P V r j b 1 p e J s 7 Z l T z a 1 G X u 0 J P s Y f E A a D e m S N G 1 z Y K 1 P A X f J R Z R d H X x A A w I X r v f 4 7 T d / 5 h D 4 Ve 9 7 3 g 9 A O / G U 9 6 b 3 S + 9 d 7 7 6 X 3 H r t F r v 1 0 1 u t 7 V 9 v f 7 r 9 5 + 2 / V F t v v O J 4 v u M t v b b / + j + F y 7 R 9 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " f i I q / 1 9 u R Q g D n C X K z 5 J J 9 u 9 S F z s = " > A A A l A 3 i c 7 V n N j y N H F e 8 s X 2 E g J A s X J C S 2 x M 7 A j O h p 6 r u 6 d 7 C U g J A 4 R F G S 3 U 0 i e S a j d r t s t 9 Z u m + 7 2 z M 5 a P n L h H + D A i U g g o d w Q V 0 5 c + A c 4 5 E 9 A H I P E h Q O v q q t 7 7 P H Y 0 e z u r H b F 2 v K 4 6 l W 9 j 3 o f v 3 7 l 6 U y G a V F i / N k r N 7 7 0 5 a 9 8 9 W u v f n 3 r G 9 9 8 7 V u v v 3 H z 2 x 8 U 4 2 m e 6 P v J e D j O P + r E h R 6 m m b 5 f p u V Q f z T J d T z q D P W H n Q c / N + s f n u i 8 S M f Z v f J s o o 9 G c T 9 L e 2 k S l 0 A 6 v v n 9 P 2 5 t H Z b p g 0 e F L m c a d p 6 h S Z q U 0 1 z / J C j K s 6 F u z Y x o d J p 2 y 0 E L B 0 p M y v k c 7 c B 2 1 N W 9 e D o s 0 f k G V I 5 R t Q e 5 1 9 Z h R / f T b G Z U O M H z 9 k M n y D 9 r B k U S g 6 5 9 4 j + s R u R o a 2 e a J e P R S G c l S n s o 1 7 + a p r m + g w 4 n M S j a x T 5 h e O / A W Z L G / T w e o a H u l a g o 4 7 x E c Y m w j + K s i w Z x g Q Y 6 7 Q 9 K N O 4 h 4 N r a u T u I J y D q r T x B 7 b R 7 p z u h X P G Q Y c Y i S n E k m D w C y 7 t 5 f I p Q 2 3 6 N J 3 G S l m D v E d o l U S A i X 6 o A 8 z 0 U B C g Z Z 2 U + H h a w I g I h f M k C H u 5 Z 3 b u E + S I M q L A b 7 Y w G n F x k Y z 4 n g Y w c D 4 W x z 1 j F w m h A S T 0 7 Z + E 0 C J k l W x Z B A t 4 I M W w V 4 T J l b i U M m H L M P A y U M l a r i p f z Q I T m f F T u o f 3 9 2 g Y n D A j J W T L U 6 A D V H r J p 2 J r l / Y 6 P q B B 3 c t 3 1 E T 5 A / V z r z I 4 6 w 6 k 2 a x j N E f K X P E r Q / 5 t L D y A B y z y 2 D H F a B W 7 m 6 3 R C 1 w 2 9 T 8 Q p g D R T 7 C k + T i 6 H T B u x R R u p a O Y r k M l I W C + + y J B J K a N w d 1 D w W I o k C 8 n G O z Z X z C X P S v f D 4 e F s C p m e d z / c d A u m R u r u x x H I a g t S r X B 8 3 v 1 w J U 1 1 u t 6 H h 9 I V 8 s X e h 0 f K F X L d + w g s b C H j p v d x l M s U 1 0 v q v P e B n t s W M q 1 7 H x 6 F r p D r R t O Z 4 w R e C 2 a+ d P U G z F Q h I Z g q g E w c Y v w U b 5 / 2 r g m e N 5 i p m j Z z O R a W Y v D T Y u a m N p M q R r n g V J I Q A J 6 x q 2 I m l 9 x g p K x b O L g R N v N 1 O g E n I 4 Y V h j B y w a A J v P a r e W R x s D G S h 6 S Z r 4 A m 4 E y 9 + A K B J r q n H 5 b o n X F X u / P s U l N 7 v v n x K w M q a s d Z M g D f Z u O 8 H K B T X Z R H 7 T T L d I 4 K P X G / p h p z e l A M r c O 3 4 7 y v / a t G Y y l L Z 9 u j 7 f n l l n F z I 9 o n p k Y f w 7 b h d d p G M F z X m p / k r m K d k d p a J x W q k 5 p g P O a R r z U c B B J e m X g 8 h 9 G Q I m B + q E w H f N V Y H N 5 N + 6 M Y H c 8 O J 4 M U f T y j c 9 S M 5 h / P d l F / b 7 4 u Q a H x i + x j Q D 1 G E v x 4 j V R G C d w g f H 7 l 0 z y L J G A A F 8 I W 5 e M Z d 6 1 p Q O H q b 6 8 t V 6 y e J 0 w D Z h L g 2 a v l u M 6 + x 4 G g d d n H l X L Z 9 x x C E F e h z b 7 n 8 I H A S Q R J w O G G / E y T Q B B R q W X P N v d M A + l H 9 B m r F f K J U v 4 w 6 Y 7 L A h n h h z r r L v 3 P + P i N 2 9 B p 2 R d a H R A 3 u O 2 5 1 7 v j m z c + 8 Q 6 9 r j f 2 E m / q j T z t Z V 4 J 4 6 E X e w W 8 2 x 7 x s D c B 2 p E 3 A 1 o O o 9 S u a 2 / u b Q H v F H Z p 2 B E D 9 Q H 8 7 cO s 7 a g Z z I 3 M w n I n o G U I n x w 4 k b e D / 4 H / h D / H f 8 e f 4 n / i / 6 6 V N b M y j C 1 n 8 N 2 p e P X k + P X f f P f u f 7 6 Q a w T f p T c 4 5 9 p o c + n 1 v N D a m o L t E 0 s x p 0 g q / p N H v / 3 8 7 p 3 3 d 2 Y / x J / g f 4 H 9 v 8 e f 4 b / B C b K T f y d / e E + / / 7 s N 9 n T A l k 0 e S 0 F X D P 4 x 3 u 3 C 3 x 6 8 j S U T s P 4 M O A y 9 2 C B h 5 m w t n I d 3 Y D wG G r J + H 9 n d J n Y d G z v k n V q N A x i N b H x T k P Q I Z r v e P e 8 d + F 6 W v r d B b + 3 j E v S Y C G + 2 0 p z E a D R W n N p T P f i C U 1 W W m Q i c 7 6 q p Q 3 u i 3 G b m m Z U + g P H E 5 p x v q T n Y d G p n E x g V 9 t Q p j D I b 4 7 7 d N b W 2 V + e I g d J x N v Q t 9 x T 2 d a 2 E x E Y o g Z G G F d / 5 t L S z b O O 5 2 0 t R a E F V B f a D b Y z r 0 + x D z D W c I 7 O 2 Z F b e z g a / T x a 8 U + 3 s w E r f n u x c 6 s R 6 u Q Q p u a t a s 3 M x W 2 b w / o W 1 y g c v L 6 4 c N z X c 8 n 7 k v Q U f s 2 N o d x y D r 8 2 e F H S M L W 4 Y 7 x z b 1 T 7 w t + B 0 g S e W O E 6 s r y o M u X z 3 v L H Q 5 G / X 5 m p s 8 3 g X 9 O / B 9 8 z b h t U T G 1 3 j F R P T o b V g G 3 g P V r j b 1 p e J s 7 Z l T z a 1 G X u 0 J P s Y f E A a D e m S N G 1 z Y K 1 P A X f J R Z R d H X x A A w I X r v f 4 7 T d / 5 h D 4 V e 9 7 3 g 9 A O / G U 9 6 b 3 S + 9 d 7 7 6 X 3 H r t F r v 1 0 1 u t 7 V 9 v f 7 r 9 5 + 2 / V F t v v O J 4 v u M t v b b / + j + F y 7 R 9 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H 3 f o 0 k L 9 y J L B G W c P S F / q O E y n K M M = " > A A A j k n i c t V l L b y R X F a 6 E V z C P Z I A F E h q p h G O Y k X q K + 3 7 M y F L C K B K L C O W d S J 6 R 1 W 6 X 7 d K 0 u 5 v u 8 k w 8 1 v w B / g A L 2 B C J B e J f w A Y J t i z y E x D L I L F h w X c f V V 2 v b g 9 Sx p b t q n v v O d 8 5 5 5 7 v 3 H P b R 4 t p s S o J + f y l l 7 / y 1 a 9 9 / R u v f H P n W 9 / + z n d f f e 3 G 9 z 5 a z S + W k / z D y X w 6 X 3 5 y N F 7 l 0 2 K W f 1 g W 5 T T / Z L H M x + d H 0 / z j o 0 f 3 3 f z H j / P l q p j P P i g v F / n D 8 / H p r D g p J u M S Q 4 c 3 b s q d n Q d l 8 e j p K i + v c q y 8 T B f F p L x Y 5 j / L V u X l N N + / c q r T J 8 V x e b Z P M i 0 X 5 b N n 6 R 6 W p 8 f 5 y f h i W q b r B W k 5 T 8 O a N H 7 t P D j K T 4 v Z l Y O I i p 8 d f B o V j S 7 r h 9 V k D K w 7 d P R p e K I P d / Y u Z p P 5 + X k + K 9 P i J F 3 m v 7 o o l v n d 9 M F i D K B b Z E S 5 v n 0 v W l K M T 5 f j 8 3 S a n 5 T p q h w v y 3 R c p m S U j m f H 6 d l 4 l Z 7 l x e l Z m c 5 P U k j t 7 L 1 / N l 5 A 1 f 1 i O Z n m 6 U F x f P d 4 I a k S 3 C r O h F S a K / U Q t h 8 v x 0 / S 9 O C k m E 7 3 r 5 a n R 6 O U S X l 3 m R + P U i r l v f R 0 m e e z + H w 0 v c j 9 Y / o s T U d O J J 0 v x p O i d E 5 K l T 5 M b w m a K T 2 y N o P l a Z a l k / m s X M 6 n q 2 r G S C y 8 7 Y 2 + J V l m 1 E i L a u 0 t p d y S 9 c B a 2 J B M 1 D N e 2 D b V

2 7 l 6 C
N 4 w i L q 2 I N p p t 4 p A m D N k T 4 K A w D S k 0 c z Q 1 h 0 v Y + 7 k L 2 8 Y H s c x X e Y e o G p g l m W D 9 U I / u r + 3 D 7 h H N T c G B a a x T S m b W L K I 5 h d E + q t 9 P E w 9 h 1 U c M m 0 7 p 0 X 4 O p c Q 8 y F H c / K V C + C e 9 A Y o M p 6 W M K a H Q n B L F N d 0 0 Y G w I e b N 0 0 c 5 9 h M t z I r P 9 s 8 c v r 3 I T x X Y v t f 0 x S z d C M q t p w 3 w m 5 p G 4 a a E o H U 2 W 2 B g / + B H 1 B u u l 8 g 3 Y P 8 t l x 6 1 + Y h 6 / t k o z 4 r 7 T / Q O P D b h K / 3 p n f e P m z 5 E F y n M y T S X K R n C d 5 M k tK P E + T c b L C 9 0 F C E 5 I s M P Y w u c L Y E k + F n 8 + T Z 8 k O Z C + w K s e K M U Y f 4 f c p 3 g 7 i 6 A z v T u f K S 0 + A M s X P E p J p s k f + Q f 5 I v i B / J X 8 i / y T / 3 a j r y u t w t l z i 7 1 G Q z R e H r / 7 6 h + / / 5 1 q p c / w t k 7 O 1 1 F a b y + Q k M d 7 W A r Y v / I j z Y h L k H z / 9 z R f v 3 3 1 v 7 + o n 5 D P y L 9 j / e / I 5 + Q s 8 m D 3 + 9 + Q P 7 + b v / X a L P U e w Z V v E C m C N E R 8 X 3 W P 8 P s G 3 s 2 Q B 6 y 8 h 4 c Z X W z R c R V t X Mc J 7 e J 5 j L P V x P / e r 3 d 4 d + b 1 L k y c e 8 Q x P 5 3 5 / C 2 h 6 i r d b y Q f J L / G 3 r f 3 2 F t w q x i V w 3 A 5 v t 9 J 5 4 h C d F U + 8 V 4 + u 8 S p Y 5 n Z g v a o a n X q P l j 4 z L 7 3 2 M z w v f M 6 N / O g S N j 3 x b w s 8 r b z X B Z 5 m f o 9 P / a o L b 3 v w Y 4 y R o 2 j D q Z e + w L p j r 2 H i d 2 i C p x w z o x j T 0 r / N t v p 9 0 N q F f b A q 8 z / E 7 3 H l z R 3 s e Q 4 / Z t 6 W m d e 3 t y X u i 0 Z 0 w s o j z J x 6 z 9 Z a F z 7K J b Q s I 2 v d y m a 2 X O H 7 L W / V C F F u z h z W H N 5 P f p q 8 i R + 3 Y u p X H C L W b k 0 B j L m v G y 4 6 h 3 7 2 F P L 7 8 C 5 L Z E v i s Y 9 V q C HD q 5 / V F r r 8 P f a 5 O v Z 5 f A v 4 t / H 3 K n k d s 4 / 9 7 r q o u D 2 d e g t e h + y 9 n v S B j + U k W r v v P b v w G f u w p f s Q M a A 1 Q t H S l v s c 2 B h T 1 F 3 a r b L 9 h 4 9 Y h m a X v i t 2 3 / h 5 r M C v J D 9 K f g x 0 m u j k j e Q X y T v J h 8 n k 5 u 9 u / v n m 3 2 7 + f f c H u 3 d 3 3 9 y 9 H 5 a + / F K U + X 7 S + t p 9 + 3 9 6 o 3 X v < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H 3 f o 0 k L 9 y J L B G W c P S F / q O

b g 9 S
x p b t q n v v O d 8 5 5 5 7 v 3 H P b R 4 t p s S o J + f y l l 7 / y 1 a 9 9 / R u v f H P n W 9 / + z n d f f e 3 G 9 z 5 a z S + W k / z D y X w 6 X 3 5 y N F 7 l 0 2 K W f 1 g W 5 T T / Z L H M x + d H 0 / z j o 0 f 3 3 f z H j / P l q p j P P i g v F / n D 8 / H p r D g p J u M S Q 4 c 3 b s q d n Q d l 8 e j p K i + v c q y 8 T B f F p L x Y 5 j / L V u X l N N + / c q r T J 8 V x e b Z P M i 0 X 5 b N n 6 R 6 W p 8 f 5 y f h i W q b r B W k 5 T 8 O a N H 7 t P D j K T 4 v Z l Y O I i p 8 d f B o V j S 7 r h 9 V k D K w 7 d P R p e K I P d / Y u Z p P 5 + X k + K 9 P i J F 3 m v 7 o o l v n d 9 M F i D K B b Z E S 5 v n 0 v W l K M T 5 f j 8 3 S a n 5 T p q h w v y 3 R c p m S U j m f H 6 d l 4 l Z 7 l x e l Z m c 5 P U k j t 7 L 1 / N l 5 A 1 f 1 i O Z n m 6 U F x f P d 4 I a k S 3 C r O h F S a K / U Q t h 8 v x 0 / S 9 O C k m E 7 3 r 5 a n R 6 O U S X l 3 m R + P U i r l v f R 0 m e e z + H w 0 v c j 9 Y / o s T U d O J J 0 v x p O i d E 5 K l T 5 M b w m a K T 2 y N o P l a Z a l k / m s X M 6 n q 2 r G S C y 8 7 Y 2 + J V l m 1 E i L a u 0 t p d y S 9 c B a 2 J B M 1 D N e 2 D b V

2 7 l 6 C
N 4 w i L q 2 I N p p t 4 p A m D N k T 4 K A w D S k 0 c z Q 1 h 0 v Y + 7 k L 2 8 Y H s c x X e Y e o G p g l m W D 9 U I / u r + 3 D 7 h H N T c G B a a x T S m b W L K I 5 h d E + q t 9 P E w 9 h 1 U c M m 0 7 p 0 X 4 O p c Q 8 y F H c / K V C + C e 9 A Y o M p 6 W M K a H Q n B L F N d 0 0 Y G w I e b N 0 0 c 5 9 h M t z I r P 9 s 8 c v r 3 I T x X Y v t f 0 x S z d C M q t p w 3 w m 5 p G 4 a a E o H U 2 W 2 B g / + B H 1 B u u l 8 g 3 Y P 8 t l x 6 1 + Y h 6 / t k o z 4 r 7 T / Q O P D b h K / 3 p n f e P m z 5 E F y n M y T S X K R n C d 5 M k t K P E + T c b L C 9 0 F C E 5 I s M P Y w u c L Y E k + F n 8 + T Z 8 k O Z C + w K s e K M U Y f 4 f c p 3 g 7 i 6A z v T u f K S 0 + A M s X P E p J p s k f + Q f 5 I v i B / J X 8 i / y T / 3 a j r y u t w t l z i 7 1 G Q z R e H r / 7 6 h + / / 5 1 q p c / w t k7 O 1 1 F a b y + Q k M d 7 W A r Y v / I j z Y h L k H z / 9 z R f v 3 3 1 v 7 + o n 5 D P y L 9 j / e / I 5 + Q s 8 m D 3 + 9 + Q P 7 + b v / X a L P U e w Z V v E C m C N E R 8 X 3 W P 8 P s G 3 s 2 Q B 6 y 8 h 4 c Z X W z R c R V t X Mc J 7 e J 5 j L P V x P / e r 3 d 4 d + b 1 L k y c e 8 Q x P 5 3 5 / C 2 h 6 i r d b y Q f J L / G 3 r f 3 2 F t w q x i V w 3 A 5 v t 9 J 5 4 h C d F U + 8 V 4 + u 8 S p Y 5 n Z g v a o a n X q P l j 4 z L 7 3 2 M z w v f M 6 N / O g S N j 3 x b w s 8 r b z X B Z 5 m f o 9 P / a o L b 3 v w Y 4 y R o 2 j D q Z e + w L p j r 2 H i d 2 i C p x w z o x j T 0 r / N t v p 9 0 N q F f b A q 8 z / E 7 3 H l z R 3 s e Q 4 / Z t 6 W m d e 3 t y X u i 0 Z 0 w s o j z J x 6 z 9 Z a F z 7K J b Q s I 2 v d y m a 2 X O H 7 L W / V C F F u z h z W H N 5 P f p q 8 i R + 3 Y u p X H C L W b k 0 B j L m v G y 4 6 h 3 7 2 F P L 7 8 C 5 L Z E v i s Y 9 V q C H D q 5 / V F r r 8 P f a 5 O v Z 5 f A v 4 t / H 3 K n k d s 4 / 9 7 r q o u D 2 d e g t e h + y 9 n v S B j + U k W r v v P b v w G f u w p f s Q M a A 1 Q t H S l v s c 2 B h T 1 F 3 a r b L 9 h 4 9 Y h m a X v i t 2 3 / h 5 r M C v J D 9 K f g x 0 m u j k j e Q X y T v J h 8 n k 5 u 9 u / v n m 3 2 7 + f f c H u 3 d 3 3 9 y 9 H 5 a + / F K U + X 7 S + t p 9 + 3 9 6 o 3 X v < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H 3 f o 0 k L 9 y J L B G W c P S F / q O E y n K M M = " > A A A j k n i c t V l L b y R X F a 6 E V z C P Z I A F E h q p h G O Y k X q K + 3 7 M y F L C K B K L C O W d S J 6 R 1 W 6 X 7 d K 0 u 5 v u 8 k w 8 1 v w B / g A L 2 B C J B e J f w A Y J t i z y E x D L I L F h w X c f V V 2 v b g 9 Sx p b t q n v v O d 8 5 5 5 7 v 3 H P b R 4 t p s S o J + f y l l 7 / y 1 a 9 9 / R u v f H P n W 9 / + z n d f f e 3 G 9 z 5 a z S + W k / z D y X w 6 X 3 5 y N F 7 l 0 2 K W f 1 g W 5 T T / Z L H M x + d H 0 / z j o 0 f 3 3 f z H j / P l q p j P P i g v F / n D 8 / H p r D g p J u M S Q 4 c 3 b s q d n Q d l 8 e j p K i + v c q y 8 T B f F p L x Y 5 j / L V u X l N N + / c q r T J 8 V x e b Z P M i 0 X 5 b N n 6 R 6 W p 8 f 5 y f h i W q b r B W k 5 T 8 O a N H 7 t P D j K T 4 v Z l Y O I i p 8 d f B o V j S 7 r h 9 V k D K w 7 d P R p e K I P d / Y u Z p P 5 + X k + K 9 P i J F 3 m v 7 o o l v n d 9 M F i D K B b Z E S 5 v n 0 v W l K M T 5 f j 8 3 S a n 5 T p q h w v y 3 R c p m S U j m f H 6 d l 4 l Z 7 l x e l Z m c 5 PU k j t 7 L 1 / N l 5 A 1 f 1 i O Z n m 6 U F x f P d 4 I a k S 3 C r O h F S a K / U Q t h 8 v x 0 / S 9 O C k m E 7 3 r 5 a n R 6 O U S X l 3 m R + P U i r l v f R 0 m e e z + H w 0 v c j 9 Y / o s T U d O J J 0 v x p O i d E 5 K l T 5 M b w m a K T 2 y N o P l a Z a l k / m s X M 6 n q 2 r G S C y 8 7 Y 2 + J V l m 1 E i L a u 0 t p d y S 9 c B a 2 J B M 1 D N e 2 D b V O W G 7 E T n M U M o z W k k H f Z R 1 s R s j a / l g Z z X l 5 U V L p Z N v O X 7 n T j q 5 d H G / t 7 N 3 / 2 L 5 O E / f R g K t w j a M N Z N W U C K M k Z Y Y o 9 f b k M I y m 4 m R M Z n o G E G J d B M k G k C p z d j I m R T g j R l R n c n b w 4 h c c C G B R I 2 y V h H V R 6 S U Z a w L S R U w q G B Z t W N U y Q y R 4 A 7 o e l T C O S f U a C 6 Z Y p x T 2 Y I V P D N D j j K K c a 3 c u M f E O 3 M o N s Z Z b Y U 0 g s F P R a Q w V l g t 2 p D C W U / 7 f g K R c h M B 7 z D m 3 m n l Z B N x i F U I K r N U E s p B K s M s v 4 Z W W q x Z J U T F K s b U E K u Y c a y i F r v j s 0 t 2 k z N O g Q i 6 2 i V G e C Y 8s e J q x F C 7 R e u R R r g 5 z U g 9 F e R F U 6 W X F x v x 4 5 S j Q h 3 B q N M x p m N B Y 6 i h I t h b z Y V M s y 2 1 T k U 7 C N s p R i g V k h v C J S F U W 9 v c l N Q Z T V z q q a 4 h U o F h j g f B D W W R C M 6 q 6 A M 3 m c 8 F v T H 7 N E X O I w e 5 F Y Z K 0 w N 1 N O s l P E q P 8 T S r 4 s c D 8 X i m n x P Y G C s V N 0 a B c N b I N r + p F a G k d N 2

FIG. 10 :
FIG. 10: Graphical expression of Eqs.(B5) and (B6) for k = 2 in the φ 4 -theory.The gray blob on the left-hand side is the exact four-point correlation function.The dotted line denotes a twisted delta function to open the vertex.The twist of the diagram is given by the flux m in the center circle.On the right-hand side, the twist is made associated with the propagator of a composite operator in the opened vertex.If we would open all the vertices on the circled line and take all the contributions to EE, it would give an overcounting of EE.