3.17 Comments on the entropy of interacting fields

So far we have considered free fields in a fixed gravitational (black hole) background. The interaction can be included by adding a potential term ∫ ddxV (ψ) to the classical action, here ψ = {ψi , i = 1, ..,N } is the set of fields in question. In the one-loop approximation, one splits ψ = ψc + ψq, where ψc is the classical background field and ψ q is the quantum field. The integration over ψ q then reduces to the calculation of the functional determinant of operator 2 š’Ÿ + M (ψc ), where 2 2 M ij = ∂ijV(ψc). The fields ψc representing the classical background are, in general, functions on the curved spacetime. In some cases these fields are constants that minimize the potential V(ψ ). The matrix Mij (x) plays the role of an x-dependent mass matrix. In the approximation, when one can neglect the derivatives of the matrix M, the heat kernel of operator 2 š’Ÿ + M is presented as the product −sš’Ÿ − sM2 Tre ⋅ Tre. Using the already calculated trace of the heat kernel −sš’Ÿ Tre on space with a conical singularity one obtains at one-loop the entanglement entropy of the interacting fields. In d dimensions one obtains [205] (see also [175] for a related discussion)
-----1---- ∫ d− 2 d- 2 2 S(d) = d−2 ΣTr [M Γ (1 − 2,M šœ– )], (152 ) 12 (4π) 2
where we used that ∫ šœ–∞2 s−dāˆ•2e−M2s = Γ (1 − d,M 2šœ–2) 2. Using the asymptotic behavior Γ (− α, x) = α− 1x −α + .., we find that the leading UV divergence of the entropy (152View Equation) is again (multiplied by N) (81View Equation) and is thus not affected by the presence of the interaction in the action. However, the interaction shows up in the sub-leading UV divergent and the UV finite terms. For instance, in four dimensions on a flat background we find
A(Σ )N 1 ∫ ( ) S = ------2-+ ---- (γ − 1 )TrM 2 + TrM 2ln (šœ–2M 2) . (153 ) 48 π šœ– 48 π Σ
We see that the leading UV divergent term proportional to the area is not modified by the presence of the self-interaction. The mass matrix M 2 is a function of the background field ψc. Thus, the result (153View Equation) indicates that at tree-level the entropy should contain terms additional to those of the standard area law, which depend on the value of field ψ at the horizon. In order to illustrate this point consider a 4 ψ model of a single field (a two-dimensional model of this type was considered in [151Jump To The Next Citation Point], in four dimensions the role of self-interaction was discussed in [46])
1 ∫ √ --( λ ) W [ψ] = -- d4x g (∇ ψ )2 + ξR ψ2 + --ψ4 , (154 ) 2 6
where we included the term with the non-minimal coupling. In fact, if we had not included this term, it would have been generated by the quantum corrections due to the self-interaction of the field ψ4. This is a well-known fact, established in [182]. The renormalized non-minimal coupling in the model (154View Equation) is
( ) -λ-- 1- ξren = ξ − 8π2 6 − ξ ln šœ–, (155 )
where we omit the terms of higher order in λ. Splitting the field ψ into classical and quantum parts in Eq. (154View Equation) we find M 2 = ξR + λ ψ2 c. Suppose for simplicity that the background metric is flat. Then to leading order in λ the entanglement entropy (omitting the UV finite terms) is
A λ ∫ Sdiv = -----2 + ---- ψ2c ln šœ–. (156 ) 48π šœ– 24π Σ
This entropy should be considered as a quantum correction to the tree-level entropy
∫ Stree = -A---− 2π ξ ψ2 , (157 ) 4GN Σ c
which follows from the action W + W [ψ ] gr. We see that the logarithmic divergences in Eqs. (156View Equation) and (155View Equation) agree if ξ = 0. On the other hand, the renormalization of ξ (155View Equation) does not make the total entropy Stree + Sdiv completely UV finite. This is yet another manifestation of the puzzling behavior of non-minimal coupling.

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