A note on the resolution of the entropy discrepancy

It was found by Hung, Myers and Smolkin that there is entropy discrepancy for the CFTs in 6-dimensional space-time, between the field theoretical and the holographic analysis. Recently, two different resolutions to this puzzle have been proposed. One of them suggests to utilize the anomaly-like entropy and the generalized Wald entropy to resolve the HMS puzzle, while the other one initiates to use the entanglement entropy which arises from total derivative terms in the Weyl anomaly to explain the HMS mismatch. We investigate these two proposals carefully in this note. By studying the CFTs dual to Einstein gravity, we find that the second proposal can not solve the HMS puzzle. Moreover, the Wald entropy formula is not well-defined on horizon with extrinsic curvatures, in the sense that, in general, it gives different results for equivalent actions.


Introduction
Hung, Myers and Smolkin (HMS) find that the logarithmic term of entanglement entropy derived from the field theoretical approach does not match the holographic result for 6d CFTs [1]. Recently, two different approaches are proposed to resolve this puzzle. One proposes to use the anomaly of entropy and the generalized Wald entropy from the Weyl anomaly to solve this puzzle [2]. While the other one suggests to use the entropy from total derivatives in the Weyl anomaly to explain the HMS mismatch [3,4]. However, the results of [3,4] are highly based on the FPS regularization [5]. By applying the LM regularization [6] or Dong's regularization [7] instead, it is found that the entropy of covariant total derivatives is indeed trivial [8]. This implies that the proposal of [3,4] is unreliable.
In this note, we provide further evidence that the HMS puzzle can not be solved by the approach of [3,4].
It is really counterintuitive that total derivatives could contribute to non-trivial entropy. If so, the logarithmic term of entanglement entropy (EE) for CFTs would lose conformal invariance and depend on the approaches of regularization [3,4]. That is completely unacceptable. Entropy is physical and thus should be independent of the regularizations one choose. The authors of [3,4] argue that this is not a problem for 4d CFTs. That is because no total derivatives appear in the holographic Weyl anomaly in 4d spacetime [9]. However, total derivatives do appear in the holographic Weyl anomaly in 6d spacetime. In the holographic analysis, these total derivatives are fixed. The authors of [3,4] propose to use the entropy from these total derivatives to explain the HMS mismatch. They did not consider all of the total derivatives but only choose some of them to explain the HMS mismatch [4].
In this note, we apply the method of [3,4] to investigate the logarithmic term of EE for 6d CFTs dual to Einstein gravity. We consider all of the total derivatives in the holographic Weyl anomaly and find that the field theoretical result does not match the holographic one. Thus, the proposal of [3,4] does not resolve the HMS puzzle [1].
The paper is organized as follows. In Sect. 2, we briefly review the HMS mismatch [1] and two possible resolutions [2,3,4]. In Sect. 3, we use the methods of [3,4] to calculate the logarithmic term of EE for 6d CFTs dual to Einstein gravity. It turns out their proposal can not explain the HMS mismatch. In Sect. 4, we apply the methods of [2] to study the logarithmic term of EE for 6d CFTs dual to Einstein gravity. We find the proposal of [2] can indeed resolve the HMS puzzle. In Sect.5, we provide further evidence that it is the proposal of [2] rather than those of [3,4] that can resolve the HMS puzzle. In sect.6, we show that there is arbitrariness in the derivations of Wald entropy. And only the total entropy is well-defined. We conclude in Sect. 7.

The HMS mismatch
In this section, we briefly review the HMS mismatch [1]. It was found by Hung, Myers and Smolkin find that the logarithmic term of entanglement entropy derived from the field theoretical approach does not match the holographic result for 6d CFTs [1]. For simplicity, HMS focus on the cases with zero extrinsic curvature.
In the field theoretical approach, the logarithmic term of EE can be derived as the entropy of the Weyl anomaly [1,5]. In six dimensions, the trace anomaly takes the following form where E 6 is the Euler density, ∇ i J i are total derivatives and I i are conformal invariants defined by For entangling surfaces with the rotational symmetry, only Wald entropy contributes to holographic entanglement entropy. Thus, we have where The logarithmic term of EE can also be derived from the holographic entanglement entropy. We call this method as the holographic approach. Take Einstein gravity as an example. The logarithmic term of EE is given by [1] S HEE = 2π log(ℓ/δ)L The mismatch between holographic result eq.(6) and field theoretical result eq.(3) becomes This is the famous HMS mismatch. Note that the above equations are derived in the case of zero extrinsic curvatures.
It is proposed to use the anomaly of entropy and the generalized Wald entropy to explain the HMS mismatch in [2]. When the extrinsic curvatures vanish, only C 2 ijkl C ijkl ≃ −∇ m C ijkl ∇ m C ijkl in I 3 contributes to non-zero anomaly of entropy. Take into account these contributions, the field theoretical and the holographic results match exactly. Note that the entropy of total derivatives vanishes by applying the LM regularization [6] or Dong's regularization [8]. However, the authors of [3,4] also contribute to the logarithmic terms of EE. They find that the entropy of total derivatives is non-zero by applying the FPS regularization [5]. And they propose to use the entropy from total derivatives to explain the HMS puzzle [3,4].

The APS proposal
In this section, we use the method of [3,4] to calculate the entropy from total derivatives carefully.
It turns out that the field theoretical result does not match the holographic one. Thus, the proposal of [3,4] does not solve the HMS puzzle.
In the field theoretical approach, the logarithmic term of EE can be derived from the entropy of the Weyl anomaly [1,5]. For Einstein gravity, the holographic Weyl anomaly is given by [10,11].
where we have set the Newton's constant G = 1 16π and the AdS radius l = 1. C 5 = 1 2 C ijkl C ijkl and the total derivative is given by [11] with C k defined as In the case of zero extrinsic curvatures, the entropy of E 6 , I 1 , I 2 reduces to Wald entropy. Thus, the HMS mismatch can only come from (I 3 − C 5 ) and ∇ i J i . Interestingly, although [2] and [4] use different approaches of regularizations, they both find that the total entropy minus the Wald entropy of (I 3 − C 5 ) can explain the HMS mismatch. The LM regularization [6] or Dong's regularization [7] is used in [2]. As a result the entropy of total derivatives is trivial in the approach of [2]. While a different regularization [5] is used in [4]. It turns out that the entropy of total derivatives is non-zero in their approach [4]. To resolve the HMS puzzle in the approach of [4], it is necessary to prove that the entropy from the total derivative ∇ i J i eq. (9) vanishes. This is, however, not the case. Now let us focus on the FPS regularization [5]. For simplicity, we use the following regularized conical metric where f n = r 2 +b 2 n 2 r 2 +b 2 and τ ∼ τ + 2nπ. Following the approaches of [3,4], we obtain where we have replaced r by bx in the above derivations. Note that [3,4] choose to drop the contribution at r = 0 (x = 0). Thus c k are irrelevant to the final results. From eq.(12), we can derive the entropy of eq.(9) as which is non-zero. Note that eq.(13) works in the Lorentzian signature, which differs from its Euclidean form by a minus sign. Now it is clear that the proposal of [3,4] can not solve the HMS puzzle [1]. In other words, the field theoretical and the holographic results of the logarithmic term of EE does not match in the approach of [3,4].

The MG proposal
In this section, we prove that the entropy of the total derivative eq.(9) indeed vanishes in the approach of [2]. Thus, the proposal of [2] does resolve the HMS puzzle [1]. We apply Dong's regularization [7] instead of the FPS regularization [5] in this section.
Let us focus the following regularized conical metric with τ ∼ τ + 2π. For the total derivative eq.(9), we firstly expand it in powers ofH and then do the τ integral. It turns out that only theH 2 terms contribute to the entropy. The other terms are either in higher order O(n − 1) 2 or vanishing in the limit b → 0. Focus on theH 2 terms, we have where r = bx and q 2k are given by p 2k are irrelevant to the entropy. We find the following formulas are useful 1 From the above equations together with b 4− 4 n = 1 + O(n − 1), we can derive the entropy of the total derivative eq.( 9) as which indeed vanishes. Thus the proposal of [2] does solve the HMS puzzle [1].

Further support
In this section, we provide further support that it is the proposal of [2] rather than those of [3,4] that can resolve the HMS puzzle. We calculate the entropy for all of the terms in the Weyl anomaly eq.(8) by using the methods of [2] and [3,4], respectively. It turns out that only the methods of [2] can yield consistent results with the holographic ones.
1 In principle, one should firstly intrgrate x from 0 to (r 0 /b) and then subtract off the contributions from the singular cone with b = 0. The detailed approach can be found in [8]. In general, there are non-universal terms which depend on r 0 and the universal terms in the integral. Only the universal terms survive once we subtract off the contributions from the singular cone. Here we use a simpler method. We integate x from 0 to ∞ for some suitable range of n, and then do the analytic continuation for n. It turns out that only the universal terms appear in the results. Thus the method here produces the same results as the one of [8].
In the holographic approach, the universal terms of EE for 6d CFTs dual to Einstein gravity is given by [1] S HEE = π log(ℓ/δ) The above formula applies to the case with zero extrinsic curvatures. For the general case, please see [12]. For the conical metrics eqs. (11,14) with b = 0 and n = 1, the above equation becomes Let us rewrite the holographic Weyl anomaly eq.(21) in the initial form of [10] Note that the curvature in our notation is different from the one of [10] by a minus sign.
Using the methods of [3,4,5] together with the metric eq.(11), we obtain the total entropy of eq.(21) in the Lorentzian signature as which does not match the holographic result eq.(20) at all.
Applying the methods of [2,7] with the conical metric eq.(14), we derive the total entropy of eq.(21) in the Lorentzian signature as This can be regarded as a check of our calculations.
Now it is clear that it is the proposal of [2] rather than those of [3,4] that can solve the HMS puzzle.

The arbitrariness of Wald entropy
In this section, we show that Wald entropy is not a well-defined physical quantity. In general, it is inconsistent with the Bianchi identities. This is not surprising. In addition to Wald entropy, the anomaly of entropy [5,7,13] and the generalized Wald entropy [2] also contribute to the total entropy.
It does not matter as long as the total entropy is well-defined. This is indeed the case. It should be mentioned that the arbitrariness of Wald entropy does not affect our above discussions, since we always focus on the total entropy in this note.
Let us take an example to illustrate the arbitrariness of Wald entropy We work in Euclidean signature in this section. Thus we have ǫ ij ǫ ij = 2, ǫ im ǫ j m =g ⊥ij . From the Bianchi identities, we have The Wald entropy of the left hand side of eq.(25) is given by And the Wald entropy of the right hand side of eq.(25) is given by where D i are the intrinsic convariant derivatives and k a = k a ij g ij are the traces of the extrinsic curvatures. Clearly, eq.(26) and eq.(27) are different for the cases with non-zero extrinsic curvatures.
This implies that, in general, the Wald entropy is not a well-defined physical quantity. It should be mentioned Wald entropy works well for entangling surfaces Σ with the rotational symmetry. Thus nothing goes wrong in the initial work of Wald [14]. For entangling surfaces Σ with the rotational symmetry, Wald entropy becomes the total entropy and thus must be well-defined.
The total entropy of left hand side and the right hand side of eq.(25) can be calculated by using the methods of the appendix. Clearly, both sides give the same results. That is because eq.(25) is an identity. Thus the left hand side and the right hand side of eq.(25) make no differences in the approach of the appendix. This implies only the total entropy is well-defined. On the other hand, there is arbitrariness in the derivations of the Wald entropy. The Wald entropy changes when one rewrite the action into an equivalent form by using the Bianchi identities.
Let us consider another example. Let us rewrite the total derivative C 6 eq.(10) into two equivalent expressions. The first one is and the second one is [15] After some calculations, we derive the Wald entropy ofC 6 andĈ 6 as Remarkably, although the total derivativesC 6 andĈ 6 are equivalent, they give different Wald entropy 2 . This clearly shows that Wald entropy is not a well-defined physical quantity. There is too much arbitrariness in its derivations. On the other hand, the total entropy is indeed well defined. One can check that the total entropy ofĈ 6 andC 6 is both zero by using the MG approach [2]. By applying the APS approach [3,4] instead, the total entropy ofĈ 6 andC 6 is non-zero, but still the same.
In conclusion, the Wald entropy itself makes no sence. There is too much arbitrariness in its derivations. Instead, only the total entropy consisted of Wald entropy [14], the generalized Wald entropy [2] and the anomaly of entropy [5,7,13] is well-defined.

Conclusion
Recently, there appears two different proposals for the resolution of HMS puzzle. One proposes to use the entropy of total derivatives to explain the HMS mismatch. While the other one proposes to use the anomaly of entropy and the generalized Wald entropy to resolve the HMS puzzle. We investigate these two proposals carefully in this note. By studying the example of Einstein gravity, we find that it is the proposal of [2] rather than those of [3,4] that can resolve the HMS puzzle. This means that it is Dong's regularization [7] rather than the FPS regularization that yields the correct results for the entropy. It is a strong support of the work [8] that the covariant total derivatives do not contribute to non-trivial entropy. Finally, we find that there is arbitrariness in the derivations of Wald entropy. Thus, Wald entropy itself is not well-defined. It turns out that only the total entropy is well-defined. It should be mentioned that Wald entropy becomes the total entropy and thus is well-defined in stationary spacetime. Thus nothing goes wrong in the initial work of Wald [14].
order O(n − 1) 2 or vanishing in the limit b → 0. Focus on theH 2 terms, we have where r = bx and d 2k are given by Note that c 2k are irrelevant to the final result, so we do not list them. The first line of eq.(32) contribute to the Wald-like entropy. The second line of eq.(32) are the would-be logarithmic terms.
Naively, second line of eq.(32) is in order O(n − 1) 2 . It seems to be irrelevant to the entropy. However, it becomes in order O(n − 1) after the integral. The magic happens because the would-be logarithmic divergence gets a 1 n−1 enhancement. In general, we have two kind of would-be logarithmic terms. One is at x → 0 and the other one is It seems that the above two integrals could not be well-defined at the same time. Thus, the authors of [3,4] choose to drop the would-be logarithmic term at infinity eq.(35). However, as pointed out in [8], we actually do not need the condition n < 1 to derive eq.(35). Note also that the results after analytic continuation are both well defined for n < 1 and n > 1. So there is no reason to drop such term. However, since we are using the methods of [3,4], we adopt their choice in this paper. Note that the would-be logarithmic term at x → 0 vanishes in our case. In addition to eq.(34), we find the following formulas are useful Using eqs.(34,36) together with b 4n−4 = 1 + O(n − 1), we can derive drdτ dy 4 √ g < T i i > = (n − 1)π 5 dy 4 [trH 2 ] + O(n − 1) 2 (37) Now we get the entropy eq.(22) in Lorentzian signature Note that the entropy in Lorentzian signature differs from its Euclidean form by a minus sign. In the above derivations we have dropped the would-be logarithmic term at x → ∞ as [3,4]. Even if we recover this kind of term, the field theoretical result still does not match the holographic one. Now let us turn to derivation of eq.(23). The calculation is very similar to the above one. The only difference is that now we use Dong's regularization for the conical metric eq. (14). We obtain Now we obtain the entropy eq.(23) in the Lorentzian signature Note that Dong's formula of entropy (the first equality of eq.(43)) [7] differs from the one of FPS (the first equality of eq.(38)) [5] by a minus sign.