Hamilton-Jacobi Approach to Holographic Renormalization of Massive Gravity

Recently, a practical approach to holographic renormalization has been developed based on the Hamilton-Jacobi formulation. Using a simple Einstein-scalar theory, we clarify that this approach does not conflict with the Hamiltonian constraint as it seems. Then we apply it to the holographic renormalization of massive gravity, where the selection of Gauss normal coordinates avoids unnecessary complication due to the breaking of diffeomorphism symmetry. We derive the complete counterterms up to the boundary dimensions d=4. Interestingly, the conformal anomaly can even occur in odd dimensions, which is different from the Einstein gravity.


Introduction
Anti-de Sitter/conformal field theory (AdS/CFT) correspondence provides a powerful tool to study the strongly coupled field theories [1]. Among others, the Gubser-Klebanov-Polyakov-Witten dictionary that identifies the generating functional of the field theory with the on-shell gravitational action plays an essential role in the calculation [2,3]. The most obvious technical obstacle to use the dictionary is the divergence involved on both sides of the duality [4]. According to the renormalization method to deal with the UV divergence in the field theory, the called holographic renormalization is developed to remove the IR divergence in the gravity.
There are different approaches to holographic renormalization. The first systematic one was presented in [5][6][7], which is usually called as the standard approach [8]. Its main procedure includes: a) solving the second-order equation of motion in the Fefferman-Graham (FG) coordinates to obtain the asymptotic expansion of the dynamical fields [9]; b) calculating the regularized on-shell action on the boundary to separate the divergent terms; c) reversing the FG expansion to express the divergent terms by the local fields on the boundary. The standard approach is strict, conceptually simple, and universal for diverse situations. However, the FG expansion and its reverse are technically tedious. So it is natural to expect an alternative approach which always respects the local field expression.
Actually, such approach was put forward by de Boer, Verlinde, and Verlinde (dBVV) based on the Hamiltonian formulation of gravity [10], see Ref. [11] for a nice review. To proceed in dBVV's approach, one writes down the most general ansatz for the covariant counterterms, organizes it by the derivative expansion, and specifies it by solving a series of descent equations induced from the Hamiltonian constraint, where the canonical momenta are replaced by the variations of the on-shell action with respect to boundary fields. Comparing the standard and dBVV's approaches, one can find that the latter is usually more simple than the former, mainly because the latter solves the algebraic descent equations instead of the second-order differential equations, and determines the counterterms directly on the cutoff surface without performing the FG expansion and reversion. The main drawbacks of dBVV's approach are [4,12]: a) the solution of some descent equations is not unique; b) the logarithmic counterterms have not been explicitly obtained; c) the ansatz may include many unnecessary terms; d) sometimes the sufficient ansatz is difficult to be figured out. In Ref. [12,13], Kalkkinen, Martelli and Muck removed the ambiguities in the descent equations by comparison with free field calculations. They also isolated the logarithmic counterterms, which are related to the breakdown of the recursion of descent equations.
Subsequently, Papadimitriou and Skenderis [8,14] developed the previous approaches where the key difference is that the covariant expansion is organized according to the eigenvalues of the dilatation operator. Interestingly, this approach does not rely on the ansatz and can be applied to more general backgrounds [15][16][17]. However, it is technically involved partially because the recursion equations may be the functional differential equations.
Besides the standard and Hamiltonian approaches, Brown and York in the early days proposed to remove the divergence of the stress tensor by subtracting the contribution from the reference spacetime [18]. This requires that a boundary with intrinsic metric is embedded in the reference spacetime, which is often not possible [19]. Moreover, instead of selecting the Dirichlet boundary conditions, the Kounterterm approach is developed where the variational principle is associated with the fixed extrinsic curvature on the boundary [20,21]. Other attempt based the dimensional renormalization can be found in [22].
As explicitly pointed out in dBVV's work [10], the Hamiltonian constraint ensures the invariance under the diffeomorphism along the radial direction. This implies that the Hamilton-Jacobi (HJ) equation is equivalent to the Hamiltonian constraint for any holographic theories with the radial diffeomorphism symmetry [15]. On the contrary, by focusing on the complete HJ equation rather than the Hamiltonian constraint, a new approach to holographic renormalization has been presented recently [23]. This approach, which we will refer as the HJ approach 1 , is partially motivated by [24], where the interesting point captured by [23] is that the HJ equation is used to isolate the infrared divergences of scalar fields in a fixed de Sitter background. In [23], it has been exhibited in several Einstein-scalar theories that the HJ approach is practical. Here we emphasize that it is tailored to handle the systems with conformal anomalies, because the derivations of the logarithmic and power counterterms are equivalently fluent and have nothing different such as the breakdown of descent equations. However, the reason why the HJ approach does not conflict with the Hamiltonian constraint has not been clarified 2 . In this paper, one of two aims is to address this problem.
Another aim of this paper is to apply the HJ approach to the massive gravity with different dimensions. The research on massive gravity has a long history [25,26]. The two main motivations include finding a self-consistent theory with massive spin-2 graviton and modifying the Einstein gravity at long distance for self-accelerated expansion of the Universe [27]. Massive gravity has obtained revived interest since de Rham, Gabadadze, and Tolley (dRGT) proposed a covariant non-linear theory where the well-known Boulware-Deser ghost can be excluded [28,29]. Recently, massive gravity has been applied to the AdS/CFT correspondence, where the reference metric can imitate the mean-field disorder in realistic materials [30][31][32][33]. The holographic renormalization of massive gravity with boundary dimension d = 3 has been studied previously using the standard approach [34].
However, the resultant counterterms are not general, since some additional conditions are imposed on the characteristic tensor of massive gravity. Moreover, we will show that the conformal anomalies can occur in both odd and even dimensions, which are missed in [34].
As we have emphasized, this indicates that the HJ approach is particularly suitable for massive gravity.
The rest part of this paper is arranged as follows. In Section 2, we will decompose the HJ equation and construct an equation that is actually used by holographic renormalization.
In Section 3, we will apply the HJ approach to the massive gravity with different dimensions.
The conclusion of this paper will be given in Section 4. In Appendix A, we will review the HJ approach to the holographic renormalization of the Einstein gravity with massive scalars.
In Appendix B and C, we will provide some calculation details and basic formulas.

Decomposition of HJ equation
The bulk dynamics of a holographic theory can be formulated as a Hamiltonian system. As usual, the Hamiltonian and on-shell action obey the HJ equation One feature of the holographic Hamiltonian system is that the radial coordinate r is identified as the time. The other is that the Hamiltonian constraint H = 0 should be imposed, provided that the on-shell action is diffeomorphism-invariant along the radial direction.
In Ref. [23], the Hamiltonian constraint is not imposed at the beginning. In other words, it is assumed that ∂S on−shell /∂r = 0. Then the coefficients in the action ansatz are allowed to depend on the radial coordinate and the HJ equation induces the one-order differential equations of the coefficients which can be solved unambiguously near the boundary. One can find that this approach to the holographic renormalization is practical indeed but its legitimacy has not been clearly stated. Here we will address this problem.
Consider that the on-shell action can be separated into the renormalized part and the divergent (non-renormalized) part S on−shell = S ren + S non−ren .
Then the HJ equation can be decomposed into where H ren denotes the part of H relevant to S ren and H non−ren is defined as We point out that what is actually used to implement the holographic renormalization in [23] is 3 The HJ approach based on Eq. (5) does not conflict with the Hamiltonian constraint H = 0 since it does not impose H non−ren = 0 in general. In fact, we emphersize that each term in Eq.
(3) should include the finite terms if there are conformal anomalies. This subtlety implies that Eq. (5) is not simply the leading orders of Eq. (1). Therefore, whether it is correct or not requires proof. In the following, we will illustrate Eq. (5) using the Einstein gravity with massive scalars. The extension to other theories should be straightforward. Note that for convenience, we will refer Eq. (5) as the non-renormalized HJ (NRHJ) equation.
Suppose that the system is described by the action where G IJ is a metric on the scalar manifold, g µν is the bulk metric, γ ij is the metric on the boundary, and K is its extrinsic curvature. Adopting the Arnowitt-Deser-Misner (ADM) decomposition 4 and selecting the usual gauge where N is the lapse and N i is the shift, the Hamiltonian is given by 5 where and R is the scalar curvature on the boundary. The canonical momenta are defined by According to the standard classical mechanics [35], they should be equal to the variations of the on-shell action with respect to boundary fields where and the bracket {S a , S b } is defined through Keep in mind the finiteness of S ren and the asymptotic behavior of the fields 6 whereγ ij andΦ I are the sources on the field theory and ∆ I = d 2 + d 2 4 + m 2 I is the conformal dimension. One can see immediately that {S ren , S ren } vanishes as r → ∞. Furthermore, at leading order, we have 5 In Appendix A.1, we have reviewed briefly the Hamiltonian formalism of the Einstein-scalar theory. 6 When m 2 I = − d 2 4 , the leading behaviour of Φ I is given by [7,12] Φ I ≃Φ I re − 1 2 dr instead of Eq. (17). Nevertheless, the remaining derivation of the NRHJ equation is still effective.
Substituting them into Eq. (14) gives 7 It exactly cancels that is, Thus, the complete HJ equation (1) has been reduced to the NRHJ equation (5).
To compare what we have done with previous references, some remarks are in order.
First, the separation of the on-shell action (2) is different from dBVV's approach [10]. Our S non−ren includes all the divergent terms but S loc in Eq. (14) of [10] does not involve the logarithmic divergences. Second, in Eqs. (18) and Eq. (19), we have used the equality between two forms of canonical momenta at leading order. Equation (19) is nothing but the step 2 of the algorithm in [23], which is taken as a shortcut to fix some coefficients of the ansatz. Third, the NRHJ equation (5) is not a completely new result. In fact, a similar equation 8 has been given by Eq. (27) in [4] using the Hamiltonian formulation of the renormalization group of local quantum field theories. Our contribution here is to provide a direct illustration by holography and point out that it can be taken as a master equation to implement the holographic renormalization.

Massive Gravity
We will study the massive gravity where the only dynamical field is the spacetime metric and the boundary is supposed to be the AdS at infinity. We will show that the NRHJ equation 7 Here and below, we have considered that Sren can be taken as the functionals of (γ kl ,Φ I ) and (γ kl , Φ I , r) from the viewpoints of the field theory and its gravity dual, respectively. can be applied to the holographic renormalization of massive gravity. Our target boundary dimensions are the most interesting cases: d = 2, 3, 4. The renormalization procedure for massive gravity is only slightly different from the one for the Einstein-scalar theory, which is given in Appendix A. We recommend reading it first since we will neglect some similar details here.

Hamilton-Jacobi formalism
Consider the massive gravity with the action [30] The mass terms are constructed subtly to avoid the Boulware-Deser ghost, where β n are constants and we will reparameterize them by α n = m 2 β n . The characteristic tensor X µ ν is defined as the square root of g µλ f λν . Here g µλ and f λν are the dynamical and reference metric, respectively. e n (X ) are symmetric polynomials of the eigenvalues of the (d + 1) × where we denote [X ] = X µ µ .
When the Hamiltonian formulation is implemented in massive gravity, one might encounter an unnecessary complication. Massive gravity explicitly breaks the diffeomorphism symmetry, which indicates that one cannot fix the gauge (8) in the whole bulk spacetime.
These extra degrees of freedom, if involved, would complicate the gravitational Hamiltonian and the sequent holographic renormalization. Fortunately, because the HJ approach only invokes the leading asymptotic behavior of the fields, the mentioned complication can be avoided, provided that we select the Gauss normal coordinates (GNC) near the boundary Note that the GNC always can be well-defined in some region including the boundary but it will fail eventually when the geodesics along the normal vector of the boundary focus and intersect [36].
The reference metric can have various forms. Here we focus on with f ti = 0, which is popular in the application of holography [30][31][32][33]. We would like to rewrite the mass terms by the boundary metric γ ij . For this aim, let's define Due to Eqs. (25) and (26), we have [X n ] = [X n ] and thereby e n (X ) = e n (X).
We proceed to study the NRHJ equation for massive gravity. Similar to the derivation of Eq. (A.7) in Appendix A, one can obtain the Hamiltonian for massive gravity by a Legendre transformation of the Lagrangian where Note that we have been working in the GNC. Using the Hamiltonian density (30), the NRHJ equation (5) for massive gravity can be built up following the same procedure in Section 2.
Furthermore, it can be changed into the form similar to Eq. (A.23): where

Action ansatz and variation
The main difference that we mentioned at the beginning of this section resides in the inverse metric expansion of U 9 . The definition of X suggests that the counterterms in massive gravity may contain the terms with half-integer inverse metrics, that is, where U (2k) contains k inverse metrics. The sufficient ansatz for each order is where "· · · " denote the terms which can be related to the existed terms by total derivatives ) or which turns out to have the vanishing coefficients finally (like the term ∼ X ij ∇ j ∇ k X ki ). We will explain this issue later.
Taking the variation of the action ansatz with respect to the boundary metric, we can obtain each term in the expansion of K which is relevant to the non-renormalized momenta. The detail of computation is presented in Appendix B. Here we write down the results

Solution of NRHJ equation
We proceed to solve the NRHJ equation (31) iteratively to determine the unknown coeffi- • The order 0 descent equation is which has the solution We only keep the leading term. By power counting, one can see that the subleading term is not divergent.
• With the order 0 result, one is able to solve the order 1 descent equation The solution about [X] is So U (2k) does contain the term with half-integer k.
• It is turned to deal with the order 2 descent equation which is needed when d ≥ 2. Here we have introduced the sign function which is used to emphasize the polynomial e d (X) = 0 under the choice f tµ = 0, as presented in (C.10). The independence of boundary conditions results in HereB is defined as a constant, denoting the solved but unfixed coefficient B. Later notations aboutC i andD i are similar. The above equations have the solutions: • So far we have determined all the divergent terms for d = 2 but not enough for d = 3, 4.
The next is the order 3 descent equation Collecting various functional terms gives where we have used C 3 = −C 2 . The solutions are Specifically, one can read D 2 = −2D 1 , D 3 = 2D 5 , D 4 = −3D 5 .
• Now the case d = 3 is completed. Let us deal with the order 4 descent equation Demanding the coefficient of each independent term to vanish in (48) gives a series of Although there are so many equations, their solutions are still simple. When d = 4, they are Particularly we notice the simplification Obviously, the number of divergent terms increases quickly when the spacetime dimension increases. Here we give a remark that is useful to avoid neglecting certain divergent terms. Suppose that there should be a real divergent term labeled by a(r)A (k) in the ansatz U (k) and A (k) does not appear in B, where B contains every term in H non−ren except K.
Then we write the order k NRHJ equation as: where We proceed to present an assumption that will be falsified in the end: A (k) only appears in U (k) in Eq. (55). Setting m (or n) = 0 and using Y (k) = k 2 U (k) + total derivatives, we have The solution is a = O(e (k−d)r ), which is impossible for a real divergent term because: the scaling of √ γa(r)A (k) is e dr · e (k−d)r · e −kr = O(1). Thus, our previous assumption is invalid, that is, the terms other than U (k) and B (k) in Eq. (55) must contain A (k) whose coefficient is nonvanishing. Note that all these terms can be worked out with the pre-solved U (m) , where m < k.
Put it another way, suppose that one has accidentally neglected a real divergent term a(r)A (k) in the ansatz U (k) . When organizing the kth order NRHJ equation, one then will obtain an ill-defined algebraic equation about the potentially divergent term A (k) . This is implied by the above analysis. Thus, the NRHJ equation can remind one to add a(r)A (k) which makes the ansatz sufficient.
Keeping this remark in mind, we can explain quickly why the term like E 16 X ij ∇ j ∇ k X ki in the ansatz U (4) is not necessary. This is because in the 4th order NRHJ equation, X ij ∇ j ∇ k X ki only appears in U (4) .
Finally, we turn back to present the counterterm action by collecting above results. It can be written as The first two terms have the uniform But other terms depend on the dimensions, which will be listed as follows.
• d = 2 Here we have used e d = 0. Then the counterterm action is The counterterm action is This result is the same as Eq. (3.15) in [34] up to the last logarithmic terms. Note that the logarithmic terms vanish precisely if one takes the metric (B7) in [34].
• d = 4 The counterterm action is

Check on the background
When d = 3, the finiteness of the on-shell action appended with the counterterms has been checked in a black-hole background [34]. We will check whether this is the case for d = 4.
In massive gravity, a static five-dimensional black-hole solution takes the form [37] ds where z is related to r via z = e r and h ab is a maximally symmetric 3-dimensional metric.
The blackening factor is where m 0 is the mass parameter and the curvature parameter k = 0, ±1. With the following reference metric one has The renormalized action is defined as where z 0 denotes the radial cutoff. Using the background metric, the counterterms (67) can be reduced to h and h is the determinant of the metric h ab . One can see that the logarithmic divergence vanishes precisely in this static black-hole solution. Combining the counterterms with the on-shell action from (23), we have As shown, all divergent terms in the on-shell action have been canceled out. Moreover, one can find that the Hawking temperature T = f ′ (z + )/ (4π), the Bekenstein entropy S = 4πz 3 + V / 2κ 2 , and the grand potential Ω = −T S ren obey the thermodynamical formula ∂Ω/∂T = S. This is a self-consistent check of our results.

Conclusion
As part of the foundations of AdS/CFT correspondence, holographic renormalization is a systematic procedure to remove the divergences by appending the local boundary counterterms to the on-shell action. Among several approaches to holographic renormalization, the one based on the Hamiltonian formalism has been developed recently. The new approach starts from the HJ equation and has been argued to be practical [23]. However, it seems to conflict with the Hamiltonian constraint which should be respected by any theories of gravity that are invariant under the diffeomorphism along the radial direction. In this paper, we divide the HJ equation into two parts and point out that only one part is actually used to execute the holographic renormalization, hence avoiding conflicts with the Hamiltonian constraint.
Then we apply the HJ approach to massive gravity. Previously, the standard approach with additional conditions was used to build up the counterterms [34]. Those results are not applicable in a general spacetime beyond the background level. On the contrary, our results are general. They should be useful for the holographic calculation of thermodynamics and transports in the strongly coupled field theories dual to massive gravity. Moreover, we have found that the conformal anomalies appear in both odd and even dimensions. This is different from the (pure) Einstein gravity: it is well-known that there are no conformal anomalies in odd boundary dimensions [5,6]. It would be interesting to study whether it has some profound implications on the renormalization group flow.
Our work suggests that the HJ approach is a practical approach to holographic renormalization, especially for the theories with conformal anomalies. This is because the logarithmic divergences can be identified by the same fluent procedure as the power divergences.

A Einstein-Scalar theory
We will give a brief review on the HJ approach to the holographic renormalization of the Einstein gravity with massive scalars. More details can be found in [23]. One can find here that the master equation is the NRHJ equation and the procedure can be conveniently split into three steps, which are corresponding to three subsections.

A.1 Hamilton-Jacobi formalism
Consider the bulk action (6) and the ADM metric (7). The scalar Gauss relation is The extrinsic curvature is given by where the dot denotes a derivative with respect to r, and ∇ i is the covariant derivative compatible with the induced metric. The unit normal vector n µ is spacelike: σ = n µ n µ = 1.
The bulk action can be rewritten by separating the radial coordinate and the boundary coordinates. This leads to the Lagrangian Note that the contribution from the total derivative term on the RHS of Eq. (A.1) is precisely cancelled out by the Gibbons-Hawking term. The canonical momenta conjugate to the fields are given by Since Eq. (A.3) involves neitherṄ norṄ i , the shift and lapse are Lagrangian multipliers which lead to the primary constraints The Hamiltonian can be defined by a Legendre transformation of Lagrangian An important feature of H and H i is that they are independent with N and N i . Thus, the Hamilton's equations for N and N i impose the secondary constraints which are called the Hamiltonian constraint and the momentum constraint, respectively.
Furthermore, due to the diffeomorphism symmetry, one can fix the gauge Then the bulk metric is simply and the Hamiltonian is reduced to Consider that the canonical momenta in the Hamiltonian formalism can be replaced by [35] π ij = δS on−shell δγ ij , π I = δS on−shell δΦ I .
where Σ is the hypersurface at finite radial cutoff near the boundary. Accordingly, we can define the non-renormalized momenta Now the non-renormalized Hamiltonian (15) can be written by and for convenience we have defined the operation: Finally, the NRHJ equation takes the form 10 One can find that this equation is nothing but the master equation (2.15) in [23]. Here we have shown that it should be understood as the NRHJ equation instead of the complete HJ equation.

A.2 Action ansatz and variation
For simplicity, we will only involve a single massive scalar below. Then the action is Since we assume the AdS boundary, the leading asymptotic behavior of the induced metric gives √ γ ∼ e dr γ, (A. 25) whereγ ij is the source of the boundary stress energy tensor. This implies that the ansatz for U can be organized into the expansion where U (2k) contains k inverse metrics (or 2k derivatives) and ⌊d/2⌋ denotes the integer no more than d/2. For the Einstein-scalar theory, the potentially divergent terms in U (2k) are made of the scalar field Φ and boundary metric γ ij . Using the leading asymptotic behavior of the scalar is the conformal dimension of the dual operator, one can figure out the maximal number of the scalar that can be included in a potential divergent term.
The ansatz for the first two order is Note that any terms are considered as equivalent if they are related by a total derivative.
In terms of the action ansatz, we can calculate the non-renormalized momenta by variations. The relevant quantities are where

A.3 Solution of NRHJ equation
By inserting the ansatz into the NRHJ equation (A.23) and using the momentum-relevant quantities calculated above, one can solve the NRHJ equation order by order. We start with the order 0 equation Collecting the non-functional terms, we have The solution is The subleading terms give only finite contribution and can be discarded directly 11 . The coefficients of Φ 2 can be organized into another differential equation The mass of the scalar field is restricted by Breitenlohner-Freedman bound [38] The solutions of Eq. (A.38) rely on the value of mass We will use the solution (A.41) to continue the renormalization procedure. Another branch is similar. Thus, U (0) has been specified We turn to the order 2 equation For d = 2, the solution for B 0 is In addition, the solutions for B 2 and B 3 indicate that they are not relevant to the divergent terms. Then 11 We notice that the integral constant happens to be at the subleading order. Otherwise additional boundary conditions are needed to determine the integral constant, which can complicate or even invalidate the HJ approach. This situation is interesting and can be traced back to the fact that the integral constant in the solution of the HJ equation is exactly an additive constant tacked on to the on-shell action [35,39].
For d > 2, the solutions are which implies One can further deal with higher order descent equations if needed. Finally, as its definition, the negative non-renormalized action is equal to the counterterm action, that is This result agrees with the one given by the standard approach [6,40].

B The details of computation
Here we present the details when dealing with K term in the NRHJ equation of massive gravity. The basic formulas of functional variations with respect to the boundary metric are given in Appendix C.
The following equations have been used when we calculate the terms Y (m)ij Y ij (n) : One can simplify the computation by utilizing the relation between U (k) and Y (k) . We state it from the beginning, when k = 1: Similarly, when k = 2,

· · ·
This directly gives Y (4) = γ ijδ U (4) δγ ij = 2U (4) + total derivatives. (B.11) One can find that up to the total derivatives, the relation between U (k) and Y (k) looks like the Euler's homogeneous function theorem. It would be interesting to give a general proof in the future.

C Some basic formulas
Here we present some basic formulas that we have used. They arẽ where we have definedXδ Note that Eqs. (C.1)-(C.4) have originally been listed in [23] and Eq. (C.6) was proved in [41]. The rest part of this appendix is our demonstration for Eqs. (C.7) and (C.8).

C.1 Variation of the matrix X
The variation of a square root matrix with respect to the metric was studied by Bernard, etc. The result is presented in (4.18) in [42], where S is X in our notation. Now we multiply X ρσ on each side of that equation and then make use of Cayley-Hamilton theorem, which gives This equation takes the same form as (C.7). Nevertheless, (4.18) in [42] is unsuitable to our case. The main reason is following. The calculation in [42] is applicable only if the matrix e 3 I + e 1 X 2 is invertible. In consideration of the gauge f tµ = 0 that we have adopted, however, one has e D−1 (X ) = e D−1 (X) = det(X) = 0, (C.10) Here we have used e n = 0 for any n > d. This is just (C.7). The proof of an odd d and (C.8) can be given in the same way.

C.2 Auxiliary formulas
• For an even n, X n can be written as This gives • Equation (2.20) in [42] gives