3.12 The statement on the renormalization of the entropy

As explained in Sections 3.9 and 3.10, the entanglement entropy is a UV divergent quantity. The other well-known quantity, which possesses UV divergences is the effective action. The standard way to handle the UV divergences in the action is to absorb them into a redefinition of the couplings, which appear in the gravitational action. In four dimensions the gravitational action should also include the terms quadratic in the Riemann curvature. The renormalization procedure is well studied and is described in textbooks (see for instance [22Jump To The Next Citation Point]). The idea now is that exactly the same procedure renormalizes the UV divergences in the entropy. In order to demonstrate this statement, consider a minimally-coupled scalar field. For simplicity suppose that the mass of the field vanishes. The bare (tree-level) gravitational action in four dimensions is the sum of the Einstein–Hilbert term and all possible combinations quadratic in the Riemann curvature,
∫ √ --4 ( 1 2 2 2 ) Wgr = gd x − -------(R + 2ΛB ) + c1,BR + c2,BR μν + c3,BR μναβ , (112 ) 16πGB
where GB, ΛB, c1,B, c2,B, c3,B are the bare coupling constants in the gravitational action.

The UV divergences of the gravitational action are computed by the heat kernel method using the small s expansion (69View Equation). For a minimal massless field (X = 0 in the scalar field equation) one finds

( ) --1---∫ ----1---∫ --1--∫ --1- 2 -1-- 2 -1- 2 Wdiv (𝜖) = − 64π2𝜖4 E 1 − 192 π2𝜖2 E R + 16π2 E 180 R αβμν − 180R αβ + 72 R ln 𝜖. (113 )
These divergences are removed by standard renormalization of the gravitational couplings in the bare gravitational action
Wgr(GB, ci,B, ΛB ) + Wdiv (𝜖) = Wgr (Gren,ci,ren,Λren), (114 )
where G ren and c i,ren are the renormalized couplings expressed in terms of the bare ones and the UV parameter 𝜖;
1 1 1 1 1 G----= G---+ 12π𝜖2-, c1,ren = c1,B + 32-π236-ln𝜖 , ren B c2,ren = c2,B − -1---1-ln 𝜖, c3,ren = c3,B + -1---1-ln 𝜖. (115 ) 32π2 90 32π2 90

The tree-level entropy can be obtained by means of the same replica trick, considered in Sections 3.8, 3.9 and 3.10, upon introduction of the conical singularity with a small angle deficit 2π (1 − α), S (GB, ci,B) = (α∂ α − 1)Wgr(α ). The conical singularity at the horizon Σ manifests itself in that a part of the Riemann tensor for such a manifold Eα behaves as a distribution having support on the surface Σ. Using formulas (56View Equation) – (59View Equation) one finds for the tree-level entropy

1 ∫ S(GB, ci,B) = ----A (Σ) − (8 πc1,BR + 4πc2,BRii + 8πc3,BRijij) . (116 ) 4GB Σ
Thus, the Bekenstein–Hawking entropy S = 41GA (Σ ) is modified due to the presence of R2-terms in the action (112View Equation). It should be noted that Eq. (116View Equation) exactly coincides with the entropy computed by the Noether charge method of Wald [221Jump To The Next Citation Point, 144] (the relation between Wald’s method and the method of conical singularity is discussed in [140]).

The UV divergent part of the entanglement entropy of a black hole has already been calculated, see Eq. (82View Equation). For a minimal massless scalar, one has

∫ ( ) A-(Σ)- -1--- 1- Sdiv = 48π 𝜖2 − 144π Σ R − 5(Rii − 2Rijij) ln 𝜖. (117 )
The main point now is that the sum of the UV divergent part (117View Equation) of the entanglement entropy and the tree-level entropy (116View Equation)
S (GB, ci,B) + Sdiv(𝜖) = S (Gren,ci,ren) (118 )
takes again the tree-level form (116View Equation) if expressed in terms of the renormalized coupling constants Gren, ci,ren defined in Eq. (115View Equation). Thus, the UV divergences in entanglement entropy can be handled by the standard renormalization of the gravitational couplings, so that no separate renormalization procedure for the entropy is required.

It should be noted that the proof of the renormalization statement is based on a nice property of the heat kernel coefficients an (68View Equation) on space with conical singularity. Namely, up to (1 − α)2 terms the exact coefficient an = areg + aΣ n n on conical space E α is equal to the regular volume coefficient areg n expressed in terms of the complete curvature, regular part plus a delta-like contribution, using relations (55View Equation)

reg Σ reg sing 2 an(Eα ) = a n (¯R) + aN = an (R¯+ R ) + O ((1 − α ) ). (119 )
The terms quadratic in Rsing are not well defined. However, these terms are proportional to (1 − α )2 and do not affect the entropy calculation. Thus, neglecting terms of order 2 (1 − α) in the calculation of entropy, the renormalization of entropy (118View Equation) directly follows from the renormalization of the effective action (114View Equation).

That the leading 1∕𝜖2 divergence in the entropy can be handled by the standard renormalization of Newton’s constant G has been suggested by Susskind and Uglum [213Jump To The Next Citation Point] and by Jacobson [141Jump To The Next Citation Point]. That one also has to renormalize the higher curvature couplings in the gravitational action in order to remove all divergences in the entropy of the Schwarzschild black hole was suggested by Solodukhin [196Jump To The Next Citation Point]. For a generic static black hole the renormalization statement was proven by Fursaev and Solodukhin in [112Jump To The Next Citation Point]. In a different approach based on ’t Hooft’s “brick-wall model” the renormalization was verified for the Reissner–Nordström black hole by Demers, Lafrance and Myers [62Jump To The Next Citation Point]. For the rotating black hole described by the Kerr–Newman metric the renormalization of the entropy was demonstrated by Mann and Solodukhin [170]. The non-equilibrium aspect (as defining the rate in a semiclassical decay of hot flat space by black hole nucleation) of the black hole entropy and the renormalization was discussed by Barbon and Emparan [12].

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