Entanglement entropy in scalar field theory and $\mathbb{Z}_M$ gauge theory on Feynman diagrams

Entanglement entropy (EE) in interacting field theories has two important issues: renormalization of UV divergences and non-Gaussianity of the vacuum. In this letter, we investigate them in the framework of the two-particle irreducible formalism. In particular, we consider EE of a half space in an interacting scalar field theory. It is formulated as $\mathbb{Z}_M$ gauge theory on Feynman diagrams: $\mathbb{Z}_M$ fluxes are assigned on plaquettes and summed to obtain EE. Some configurations of fluxes are interpreted as twists of propagators and vertices. The former gives a Gaussian part of EE written in terms of a renormalized 2-point function while the latter reflects non-Gaussianity of the vacuum.


I. INTRODUCTION
Entanglement entropy (EE) provides important information of a given state, in particular, correlations in a ground state between two spatially separated regions and has been widely discussed in quantum information, condensed matter physics and, even in quantum gravity, cosmology, and high energy physics [1][2][3][4][5][6][7][8][9][10][11][12]. Despite its importance, the practical computation of EE in field theories is not an easy task and consequently, much of the works have focused on Gaussian states [13][14][15][16][17][18][19], lowenergy sectors of conformal field theories (CFTs) [5,[20][21][22] or holographic CFTs [6,7,23]. For Gaussian states, both the so-called real-time approach and the imaginary time approach as known as the replica trick [24] are applicable. The computations make use of its Gaussianity; the reduced density matrix is still Gaussian [25]. For CFTs, EE of quite general shapes of subregions can be studied while the conformal symmetry plays an important role in reducing the problem simpler and tractable. For theories with holographic duals, EE can be computed in a simple, easy manner as a geometric quantity while the existence of the AdS/CFT correspondence [26][27][28] is obviously necessary. Many features of EE were clarified, but there are only a few studies on EE in interacting theories: a perturbation from free theories [25,29] or CFTs [30], the renormalization group flow given fixed point CFTs [31], and large N expansions [32,33]. There are also some nonperturbative studies [34][35][36][37][38][39][40][41][42], but their analytical evaluations are difficult. Our goal in this paper is to provide a field theoretic, systematic way to explore EE in a massive interacting theory, which is neither free nor conformally invariant and the existence of its holographic dual is not assumed.
Besides computability, EE has an obvious problem specific to field theories. Since field theories contain infinitely many degrees of freedom, EE suffers from ultraviolet divergences and an appropriate regularization and renormalization are necessary to obtain finite results. For free theories, the UV-divergent EE can be regularized by suitably renormalizing parameters in the background gravity [43][44][45][46][47][48]. There are additional UV divergences in interacting field theories, which should be dealt with the usual flat space renormalization. A perturbative treatment of this renormalization was discussed [29].
In this paper, we give a systematic study of EE in interacting field theories. We consider a scalar field theory with φ 4 interactions in a simple geometrical setup, a half space being traced over. It is formulated as a Z M gauge theory on Feynman diagrams: We perturbatively evaluate EE in the two-particle irreducible (2PI) formalism and obtain a generalized 1-loop type expression of EE in terms of renormalized propagators. Moreover, we show that the non-Gaussian nature of the vacuum wave function gives further corrections to EE associated with 4-point vertex functions.
For the Hilbert space composed of two subsystems on a time slice, H tot = H A ⊗ HĀ, the EE for A is defined as S A = − Tr A ρ A log ρ A , where ρ A = TrĀ ρ tot is a reduced density matrix of the total one, ρ tot . In this paper, we choose the subregion A as a half space specified by A = {x 0 = 0, x ⊥ ≥ 0, ∀x } andĀ as its complement, where x ⊥ and x are the normal and parallel directions to ∂A respectively. A standard method to calculate EE S A is known as the replica method [5,49], where Let us define an unnormalized density matrixρ tot by ρ tot =ρ tot /Z 1 , where Z 1 is a partition function on R d+1 as a Euclidean path integral. Then, usingρ A = Tr Aρtot , an unnormalized reduced density matrix, Trρ n A can be viewed as a partition function on Σ n ×R d−1 , where Σ n is an n-folded cover of a two-dimensional plane and, thus, a two-dimensional cone with deficit angle 2π(1 − n). The EE can be rewritten in terms of the free energy F n ≡ − log Z n ≡ − log Tr Aρ n A as S A = ∂F n /∂n| n→1 − F 1 .

II. AREA LAW OF EE IN ORBIFOLD METHOD
We first show the area law of EE. For this purpose, the orbifolding method [50,51] is convenient. We consider a space R 2 /Z M instead of the n-folded space, Σ n . Since M arXiv:2103.05303v3 [hep-th] 20 May 2021 can be interpreted as n = 1/M , the vacuum EE on the Z M orbifold is given in terms of the free energy on the Z M orbifold F (M ) = F 1/n as provided M ∈ Z >1 can be analytically continued to 1. A state on the orbifold can be obtained by acting the Z M projection operator,P = n /M on a state in an ordinary two-dimensional plane, whereĝ is a 2π/M rotation operator around the origin.
In this paper, we call a Z M rotationĝ n as an n (∈ Z mod M ) twist operation.
Let us consider, for simplicity, a scalar field theory on the Z M orbifold without a nonminimal coupling to the curvature. Since scalar fields have no spin and are singlet under the spatial rotation, the Z M actionĝ on the internal space of the fields is trivial. Its explicit action is given as follows: where the two-dimensional coordinates on Z M is given by x or equivalently by the complex coordinates (x ⊥ ,x ⊥ ). The remaining codimension-two coordinates parallel to the subregion boundary are given by x . The total (d+1)dimensional coordinates are denoted by x. The action for the scalar field theory on Z M orbifold is given in terms of the field φ(x) on a flat space R 2 × R d−1 as In the following, we consider λφ 4 /4 potential for simplicity. However, this particular choice of the potential is only for simplicity and generalizations to the other form of potentials such as cubic or higher orders are straightforward. From the action Eq.(3), the inverse propagator can be read off asĜ Since the propagator is its inverse on the Z M orbifold, it satisfies the relation Thus, the propagator on the orbifold is written as where The projection operator on y is eliminated by a rotation of the momentum p. Since p ·ĝ n x =ĝ −n p · x, we see that the flow-in momentum from the propagator at a vertex x is given by the twisted momentum,ĝ −n p. Twists of coordinates are equivalent to the inverse twists of the corresponding momenta.
For the calculation of EE, we need to compute the free energy, which is minus the sum of the all possible connected bubble diagrams. Consider a Feynman diagram with N V vertices, N P propagators, and L loops. At each vertex, there is a factor where 1/M comes from the integration measure in Eq.(3). Thus, an overall M dependence seems to be given by But it is not correct since N V − 1 of the projection operators in the N P propagators can be further eliminated by rotations of coordinates at the vertices. It can be understood as follows. If we particularly pay attention to a propagator G 0 (ĝ n x, y) and a vertex x, the twistĝ n in the propagator can be eliminated by changing the integration variable x. Thus the summation of the twist (n = 0, · · · M − 1) eliminates the 1/M factor at the vertex. This procedure can be continued only up to N V − 1 vertices. The last integration of the coordinates of a vertex cannot absorb a twist of propagators. In ordinary flat space without twists, due to the translational invariance, the integration gives the volume of the space-time, (2π) d δ d (0) = V d . In our case with twists, reflecting the absence of the translational invariance on the orbifold, the last x-integration instead . This procedure of eliminating redundant twists is depicted in Fig.1 is being replaced by the area V d−1 of the boundary of the subregion times an additional factor δ 2 ( L l=1 (1 −ĝ −n l )k l ) in the momentum integrations. Note that the additional factor gives the two-dimensional volume V 2 only when all n l = 0. Due to the overall 1/M factor, V d+1 -proportional terms in F (M ) , i.e., all n l = 0 (l = 1 · · · L), are canceled in Eq.(1), and do not contribute to EE, while the other terms, such that some of {n l } are nonvanishing, are proportional to the area V d−1 and contribute to EE. This analysis holds to all orders in the perturbation theory.

III. ZM GAUGE THEORY ON FEYNMAN DIAGRAMS
The orbifold field theory can be regarded as Z M gauge theory on Feynman diagrams. On a Z M orbifold, each Let us begin with a 1-loop diagram. In the following, we write (d + 1)-dimensional momenta and coordinates as (k, k ) and (x, x ). For a 1-loop diagram, there is a single twist n (Fig.3). The free energy with twist n is easily calculated [50] by noting that k|ĝ n |k = (2π) 2 δ 2 (k)/4 sin 2 (nπ/M ) for n = 0. Thus, we have .
The volume factor proportional to V 2 vanishes in Eq.(1). By using the relation Here a UV cutoff scale is introduced. Note that EE decreases as the mass increases. The appearance of the area law can be interpreted as pinning of the propagator G 0 (x, y) at the origin of the orbifold as demonstrated below. A twisted propagator of G 0 (x, y) := G 0 (r; r ) is written as = G 0 (cos θ n r i + 2 sin θ n ki X k ; r ) = e cot θnR X /2 G 0 (2 sin θ n X; r ), whereR X = ij r i ∂ Xj , r = x − y, X = (x + y)/2 and θ n = nπ/M . Suppose that the twisted propagator is multiplied by a function F (r) of the relative coordinate r and integrated as I = dxdy G 0 (ĝ n x − y)F (r). Such integration appears when there are no more twists in the Feynman diagram. Then, due toR X F (r) = 0, the twisted propagator G 0 (ĝ n x − y) can be replaced by G 0 (2 sin θ n X; r ). For n = 0, by rescaling momentum p, it is written as where M 2 k := k 2 + m 2 and s n := sin θ n . Since ∂ X is set to zero via integration by parts in the X integration, the coordinate X = (x + y)/2 is pinned at the origin of the orbifold. It is straightforward to see that S 1-loop in Eq. (7) can be reproduced by using this pinned propagator. Note that, when there is another twist in the Feynman diagram, the function F depends on X and derivative terms in Eq. (8) cannot be dropped.
Next let us consider a figure-eight 2-loop diagram of Fig.4 with twists (m 1 , m 2 ). Its free energy is given by 0), correspond to a twist of each propagator (Fig.5) and renormalize the mass of the bare propagator in Eq.(7) [29]. The corresponding EE to first order in λ is given by This is nothing but S 1-loop of Eq. (7) with the mass replaced by m 2 + δm 2 , where δm 2 = 3λG 0 (0). Renormalization of propagators is one important aspect of EE in interacting field theories. There is another nontrivial contribution to EE from the twists (m, ±m) in Eq.(10), which is interpreted as twisting the 4-point vertex (Fig.6). By rewriting the integral of Eq.(10), for m 2 = −m 1 , as The same interpretation follows for m 2 = m 1 as From Eqs. (8) and (9), we can replace in the integral. Hence, the effect of twisting is interpreted as pinning of the position of the vertex at the origin. By the above replacements, we obtain the 2-loop contribution from twisting the vertex in the free energy (15) The 2-loop vertex correction to EE is then given by The vertex correction to EE is negative for repulsive (positive λ) interaction. In contrast to the twisting of propagators, it essentially originates from the non-Gaussianity of the vacuum. We also emphasize the importance of interpreting twisting in terms of Z M fluxes on plaquette. If we took a special gauge and assigned twists on particular links of Feynman diagrams, we could not find vertex corrections to EE since they are hidden in twisting multiple links. Now we wonder what contributions to EE come from the other twists of the figure-eight diagram; Fig.4 with m 1 and m 2 both nonzero and (m 1 , m 2 ) = (m, ±m). Performing the integration of Eq. (10), we have EE is obtained by the analytical continuation of M and

IV. EE IN 2PI FORMALISM
To study the renormalization of propagators systematically, we calculate EE in interacting field theories in the framework of the 2PI formalism [52,53]. The 2PI effective action is given, in addition to the classical action, by (18) where Γ 2 is (−1) times a collection of connected 2PI bubble diagrams, denoted by Φ in some literature, in which all propagators are the renormalized ones G. The 1PI effective action is given by solving the gap equation and substituting G into Γ. From the first logarithmic term, it is straightforward to see that we have whereG(k; k ) is a Fourier transform of the renormalized Green function, G(x; x ). Other contributions to EE follow from the second term in Eq. (18) and 2PI diagrams Γ 2 . On each plaquette, a flux of twist m i is assigned. Let us first focus on contributions to EE from twisting one of the renormalized propagators in Feynman diagrams. By taking a variation with respect to a propagator G and multiply a twisted propagator, these contributions are given by It is nothing but a twist of tr G −1 G /2, and gives a trivial result. Thus, only the logarithmic term of Eq. (20) provides the EE associated with a single twist of a propagator in the 2PI formalism: within the Gaussian approximation, this is a general result and consistent with the leading order of perturbative calculations in [25,29]. Among other contributions to EE, the figure-eight diagram in Γ 2 gives the same form of EE as Eq. (16), with G 0 replaced by G. The next nontrivial contribution to EE comes from the 3-loop diagram in Fig.8. Single twists of propagators, as shown in Eq. (21), vanish in the 2PI formalism by using the gap equation. Some other configurations of twists are interpreted as twists of vertices. They are given by (0, m, 0) or (m, 0, −m) or (m, −m, m) in Fig.8. These configurations are regarded as s, t, uchannel for twisting the 4-point vertices. All of them give the same vertex correction. Each configuration of the twists can be interpreted as either twist of the upper or lower vertex (but not both). The corresponding EE is FIG. 8. A 3-loop diagram with twists (m1, m2, m3) (leftmost). A particular configuration (0, m, 0) corresponds to twisting a vertex, as well as (m, 0, −m) and (m, −m, m) (three diagrams on the right). These three diagrams are equivalent although they seem different. All of these three diagrams are a single twist of the delta function from x1 to x2.
given by Comparing it to Eq.(16), the delta function δ d−1 (r ), which follows twisting the bare 4-point function, is replaced by the square bracket in S vertex 3-loop . The integral including two Green functions might be interpreted as twisting a renormalized 4-point vertex function V 4 (x 1 , x 2 , x 3 , x 4 ) at 1-loop, as inferred from the right figure of Fig.8. To systematically formulate twisting of higher point functions, we need to evaluate, e.g., m =0 V 4 (ĝ m x 1 ,ĝ m x 2 , x 3 , x 4 ). We would like to come back to this issue in future investigations.

V. CONCLUSIONS AND DISCUSSIONS
We have calculated entanglement entropy (EE) of a scalar field theory with φ 4 -interactions in the 2PI formalism and showed that EE has two different kinds of contributions, one from propagators and another from vertices. The contributions from propagators are written in terms of renormalized 2-point Green functions. On the other hand, those from vertices reflect the non-Gaussian nature of the vacuum wave function. The calculations are performed by interpreting the free energy in terms of Z M (M → 1) gauge theory on Feynman diagrams; Z M fluxes are assigned on each plaquette. Special configurations of fluxes give the above two contributions. Due to the Z M twisting, center coordinates of propagators or positions of vertices are pinned at the origin of the Z M orbifold so that the area law of EE appears.
There are many issues to be solved. We have perturbatively calculated contributions from 4-point vertices up to 3-loops in the 2PI formalism. In contrast to the clear understanding of contributions from propagators, it is difficult to systematically understand vertex contributions in terms of fully renormalized 4-(and higher) point functions. Besides twisting a single propagator or a vertex, there are many other configurations of twists. The next simple configuration of twists will be twisting two separate propagators. We expect that it gives less dominant contributions to EE because two positions are simultaneously pinned at the origin due to the twisting, and the integration will be largely constrained in Feynman diagram integrals. This expectation is also plausible since, if two twists can be independently summed, each summation gives an (M 2 − 1) factor and in total (M 2 − 1) 2 . Then it does not contribute to EE. In general, they cannot be independent, but if we can introduce "distance" between twists, we could estimate their degrees of contributions to EE. For this, we need a deeper understanding of Z M gauge theory on Feynman diagrams.
Finally, we comment on the analytical continuation of M to M ∼ 1. The basic assumption of the orbifold method to calculate EE is an analytical continuation from an integer M to a real number. It is justified if there are no contributions to EE that vanish at integer M s. Then, the EE can be calculated by summing all the configurations of fluxes of twists on each Feynman diagram. In comparison, the heat kernel calculation of EE by Hertzberg [29] uses a propagator on a cone with an arbitrary deficit angle and no other modifications are made besides propagators. Our study indicates that in addition to the propagators, vertex functions also need to be modified on a cone. It is also interesting to see if some contributions to EE vanish for 1/M deficit angle corresponding to the orbifold case. This will give a justification (or a falsification) for our basic assumption of the analytical continuation.