Entanglement Entropy in Scalar Quantum Electrodynamics

We find the entanglement entropy of a subregion of the vacuum state in scalar quantum electrodynamics, working perturbatively to the 2-loops level. Doing so leads us to derive the Maxwell-Proca propagator in conical Euclidean space. The area law of entanglement entropy is recovered in both the massive and massless limits of the theory, as is expected. These results yield the renormalisation group flow of entanglement entropy, and we find that loop contributions suppress entanglement entropy. We highlight these results in the light of the renormalization group flow of couplings and correlators, which are increased in scalar quantum electrodynamics, so that the potential tension between the increase in correlations between two points of spacetime and the decrease in entanglement entropy between two regions of spacetime with energy is discussed. We indeed show that the vacuum of a subregion of spacetime purifies with energy in scalar quantum electrodynamics, which is related to the concept of screening.

In the context of particle physics, neutrino oscillations are being actively investigated using tools from quantum information theory [30][31][32][33][34], where such methods can provide numerous insights into exotic mechanisms in relativistic settings.Moreover, a systematic inquiry of the entanglement generated in 2 → 2 tree-level scatterings in spinor QED between helicity/polarisation degrees of freedom has been undertaken, both for pure initial states [35] and for general (potentially mixed) states [10], where the generation of Von Neumann entropy was also discussed.Here, we shall go beyond tree-level processes, allowing us to delve into the renormalization of such quantum information-theoretic properties and examine vacuum fluctuations rather than scattering processes.
The dependence of entanglement entropy on the partition size of the system in pure Maxwell theory (a conformal field theory) has been the subject of some debate [36,37].The behaviour of geometric entropy in the context of lattice gauge theories has also been explored [1,[38][39][40][41][42][43].Interestingly, the properties of entanglement entropy in φ 3 and φ 4 theories have been studied by Hertzberg [4] up to two-loop orders, and a tentative renormalization procedure has been undertaken.Significant work has been conducted to understand the renormalization of entropy in conformal field theory [44] and in φ 4 theory [45], as well as the regularization of entropy in more general QFT settings [46].A generalised 2PI formalism has also been used to approach similar topics nonperturbatively, and the non-Gaussianity of entanglement entropy induced by the Wilson renormalization group procedure has been analysed [47][48][49].
In this paper, we shall pursue such efforts by computing the ground state entanglement entropy in scalar QED perturbatively up to two-loop order.Doing so requires us to work in conical Euclidean space -where the so-called replica trick, detailed in Section II below, is used to recover the flat spacetime limit [50].Since the entropy depends on the propagators of the theory of interest in conical space, we shall derive in Section III the Maxwell-Proca propagators in such a setting.In Section IV we will show that the area law of entanglement entropy is indeed obtained, as can be expected, and we shall explore in Section V the renormalization of vacuum entropy in such a context.We will further consider the interplay between the renormalization of couplings and correlators with that of entanglement entropy.

II. SETTING AND PRELIMINARIES
We work in units = 1, Feynman gauge ξ = 1, and Euclidean space with negative signature.We consider a QFT (in our case, scalar QED), which lies on an infinitely large and flat D = (1 + d)-dimensional spacetime Ω. Going to Euclidean time through a Wick rotation, we divide Ω into two regions Ω = A ∪ Ā through an (arbitrary) cut on the real negative axis, such that the sub-spaces A and Ā have a flat dividing boundary of dimension d ⊥ = d − 1.The density matrix of the ground state of the QFT on the sub-region of interest A is obtained by tracing out the degrees of freedom in the region Ā, that is, ρ A = Tr Ā(ρ).The associated entanglement entropy of the correlations between vacuum fluctuations is given by the Von Neumann entropy of the reduced density matrix We can rewrite this using the replica trick which involves n copies of ρ A .In general, for a thermal bath with Hamiltonian Ĥ and temperature T, we have ρ ∼ e − Ĥ/T and partition function Z = Tr(ρ).Associating Euclidean time with temperature, we can then relate periods of Euclidean time to the partition function on a Riemann surface, and, more generally, on an n-sheeted Riemann surface, we have, in the ground state [1,4] Tr( In a general QFT setting, the entanglement entropy is that of the correlations between the vacua of A and Ā and their fluctuations, i.e. each order of the expansion in the relevant couplings contributes to the entropy via vacuum diagrams.We can then determine the entanglement entropy as an expansion in powers of the couplings as so that equations (2) and (3) now give In practice, this is the flat space limit of the geometric entropy [50,51] in conical space with deficit angle δ = 2π(1−n).
In coordinates x = {r, θ, x ⊥ }, the conical space metric is [52]: where r ≥ 0, θ ∈ [0, 2π), and the x ⊥ are the usual Cartesian coordinates on the d ⊥ -dimensional transverse space.We will then want to expand the partition function perturbatively in powers of the couplings and determine each one individually to determine the entanglement entropy of vacuum fluctuations order by order.The story here is as follows.We take the reduced density matrix of a subregion of spacetime of the vacuum state of the QFT.This exists in itself, and has a well-defined non-perturbative Von Neumann entropy.As we expand this Von Neumann entropy in powers of the couplings, what we are doing is that we are setting a scale at which we are looking at this vacuum: at tree-level we are looking at ρ A at smaller energies (larger length scales) than at 2-loops level, etc.This means that looking at loop levels of the entropy gives us how it runs with energy scales, which is the behaviour that one expects when one considers the renormalization group flow.

III. GREEN'S FUNCTIONS IN CONICAL SPACE
We consider scalar QED, that is, the theory of charged scalar particles.The corresponding action in Euclidean space in the Feynman gauge is [53] where the covariant derivative is D µ ≡ ∂ µ − ieA µ .The partition function on the cone is up to an overall normalisation factor which we ignore.In the free theory, the interaction terms vanish so that the only contributions come from the photonic and scalar propagators.

A. Massive Scalar Field
We have that the partition function at 0 temperature on the cone of a single massive scalar field with Euclidean action S E [φ] is is the partition function for the free theory [54].Thus where In Euclidean space, the scalar propagator satisfies the equation where in flat space whilst in conical space this is solved by [1] G n (x, x ′ ) = 1 2πn where d 0 = 1, d k≥1 = 2 and J is the Bessel function of the first kind.Using the Euler-Maclaurin formula where B 2j are the Bernouilli numbers, we find that, in the coincidence limit x ′ → x, this becomes where We have and the same behaviour occurs for generic x and x ′ : as can be expected: fields and propagators vanish at infinity.

B. Photon and Proca Fields
For a Proca field -a massive spin-1 field -the partition function of the free theory is which reduces to ln Z γ,(0) n,0 = − ln(det[−∆]) [54] in the smooth massless limit -note the factor of 2 difference with the scalar case.In Euclidean space, the Proca propagator in Feynman gauge is which is solved by Fourier transform in flat space as Since the additional degree of freedom introduced by the mass term completely decouples with the transverse degrees of freedom in the massless limit, Maxwell-Proca theory has a smooth limit into QED.Thus, we can deduce the partition function in conical space for the free photon as It must be noted that this is not necessarily true for non-abelian QFTs, where Goldstone bosons associated to the extra degrees of freedom form a non-linear sigma model.In the case of perturbative quantum gravity, this is exemplified by the van Dam-Veltman-Zakharov discontinuity [55,56], where massive quantum gravity disagrees with massless quantum gravity even in the massless limit.We thus have where, as for the scalar case, we have, in the coincidence limit where f n (r) is given in equation (19).Likewise, for the photon propagator,

IV. ENTANGLEMENT ENTROPY IN THE FREE THEORY
For the scalar field contribution to the entropy, we follow the work done by Hertzberg [4] to get where Likewise, for the Proca field contribution to the entropy, we use (27) to get where we do not have a factor of 1 2 for the partition function and the Proca and scalar propagators differ by a minus sign, so that the photonic contribution to the tree-level ground state entropy is obtained in the massless limit and yields Hence, in the free scalar QED theory, the leading contribution to the vacuum entropy is

V. ENTANGLEMENT ENTROPY IN SCALAR QED
Let us now consider the full interacting theory of scalar QED, given by the action (8) and corresponding partition function (9).We then have with ln T r(ρ n (1) ) = ln(Z n,1 ) − n ln(Z 1,1 ) so that We start with diagram 1a, where it is a priori nontrivial to apply the momentum-space Feynman rules (given in 0 by Gupta-Bleuler This is a total derivative of propagators which vanish at infinity as was seen in equation ( 21), so the surface term contribution will be finite and thus subleading to the order of interest at infinity.Thus, we only need to consider the coincidence limit x ′ → x for r → 0, i.e. the only boundary contribution at this order is that at the tip of the cone.By the divergence theorem, and since the flux is n µ = δ r µ i.e. we go away from the tip of the cone, we have where we integrated at coincidence, so that ln as g rr = −1.At r → ∞, propagators go to 0 faster than linearly so that contribution vanishes, and note that lim r→0 rf n (r) ∼ lim r→0 r log n (r) = 0 so the first term vanishes.The non-vanishing contribution comes from the integral term.Thus, ln Z The first term is just the flat space contribution, and the contributions to the entropy of the terms with (2πn) 2 f n (r) k vanish when we take the derivative with respect to n and set n = 1 unless k = 1.Moreover, ∂ ∂n (2πn) where, importantly, the area law is explicit and the overall contribution is negative.Furthermore, for diagram 1b, ln This last term vanishes when we differentiate with respect to n and set n = 1, and For the φ 4 2-loop diagram 1c, we have [4] ln where the factor of 3 comes from Wick's theorem.We have The second term vanishes when we differentiate with respect to n and set n=1.Thus, the φ 4 2-loop contribution to the entropy is Hence, putting everything together, we have where the area law is explicit at every order, as expected.From this, one may straightforwardly deduce the ǫ divergences.In particular, in D = 4 dimensions (d = 3, d ⊥ = 2), the momentum integral leads to a (subleading) logarithmic divergence, and the Green's functions at coincidence are quadratically divergent in momentum ∼ Λ 2 where Λ is a momentum cutoff, so regulating in position space we get G 1 (0) ∼ 1 ǫ 2 , with S ∼ A((e 2 + λ)/ǫ 2 ).Importantly, we see that entanglement entropy is cutoff-dependent, i.e. it depends on the details of the microphysics.In particular, putting this theory on a lattice would make entanglement entropy dependent on lattice spacing and configurations.Furthermore, we see that these loop contributions necessarily reduce the entropy.How do we make sense of this in light of the flow of the couplings?Indeed, to leading order in λ and e, we have [53] β e (e, λ) = e 3 48π 2 (73) Both β functions are positive, so couplings increase with energy (and so decrease with distance).In practice, this means that the 2-point correlation functions of scalar QED ought to increase with energy, i.e. as we "zoom in" to smaller and smaller distances.In particular, this can be done at the boundary between A and Ā, i.e. the correlations between both regions increase as we "zoom in" -we expect this to diverge at the Landau pole.However, the Von Neumann entropy of region A -i.e. the entanglement entropy -seems to decrease as we "zoom in".On the one hand, we seem to have that correlation between both regions increase at the boundary; on the other hand, the reduced density matrix ρ A becomes more and more pure, so entanglement between the two regions shrinks.How do we solve this apparent contradiction?
There are several ways to see this.The particle physics way is to consider that at higher energies, the contributions to the local physics are getting increasingly local, i.e. long-distance contributions are subleading to local fluctuations -this is reminiscent of the concept of screening in QED.As one probes the vacuum to higher and higher energies (smaller and smaller distances), one gets more and more vacuum fluctuations, so the physics will look more and more local.At the Landau pole, the local physics completely dominates, and A is entirely independent of Ā with no entanglement between the two regions.Thus, the local vacuum purifies with increasing energy.From the point of view of quantum information theory, this is in agreement with the general trait of entanglement "monogamy" [57,58] -whereby a subsystem strongly interacting and thus entangled with a second subsystem cannot be strongly entangled with a third one.

VI. CONCLUSIONS AND OUTLOOK
For the usual spinor QED, we expect to see the same behaviour.For perturbative (asymptotically safe) quantum gravity, we also expect this.For Quantum ChromoDynamics (QCD), however, it would be interesting to explore how asymptotic freedom and the negative beta functions influence the flow of entanglement entropy and whether antiscreening has the opposite effect on entropy at high energies.We indeed conjecture that the beta function and RG flow of couplings dictate the flow of the entanglement entropy.Thus, we expect the entanglement entropy to be scale invariant in supersymmetric theories where loop contributions cancel out precisely and beta functions are zero.This work may also be straightforwardly expanded to consider entropy and entanglement renormalization within NLQFT, as has recently been done for φ 4 theory [28], where entanglement entropy was shown to be free of UV divergences.This result is expected to carry on to scalar QED.
The methods and results obtained may also be contrasted to those obtained in condensed matter systems.Indeed, there has recently been a lot of interest in understanding the area law of entanglement entropy in such contexts -e.g.see [59,60] and references therein.For instance, (spinor) QED in D = 1 + 2 dimensions is related to the spin-1/2 Heisenberg antiferromagnet, and the RG flow of couplings in such a discretised system has been studied [61], where it was found that for generic disorder the flow led to strong couplings.It may thus be insightful to analyse the flow of entanglement entropy in such systems, which may be readily compared with our results by regularising the quantum field theory, and see whether entanglement entropy also follows an opposite behaviour to that of couplings under the renormalization group.

Figure 1 :
Figure 1: Vacuum contributions to the entropy from scalar QED to O(e 2 )