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Hamilton-Jacobi renormalization for Lifshitz spacetime

Baggio, M.; de Boer, J.; Holsheimer, K. DOI 10.1007/JHEP01(2012)058 Publication date 2012 Document Version Final published version Published in The Journal of High Energy Physics

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Citation for published version (APA): Baggio, M., de Boer, J., & Holsheimer, K. (2012). Hamilton-Jacobi renormalization for Lifshitz spacetime. The Journal of High Energy Physics, 2012(1), 058. https://doi.org/10.1007/JHEP01(2012)058

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Download date:26 Sep 2021 JHEP01(2012)058 Springer atter physics January 13, 2012 December 20, 2011 : September 30, 2011 terterms in order : : hese counterterms using 10.1007/JHEP01(2012)058 nalogue of the conformal Published conditions from the start, Accepted Received es the asymptotic behavior hitz spacetimes are dual to erlands doi: les that this procedure indeed at divergences can be canceled m, Published for SISSA by [email protected] , [email protected] , 1107.5562 Gauge-gravity correspondence, Holography and condensed m Just like AdS spacetimes, Lifshitz spacetimes require coun [email protected] Science Park 904, Postbus 94485, 1090E-mail: GL Amsterdam, The Neth Institute for Theoretical Physics, University of Amsterda Hamilton-Jacobi renormalization for Lifshitz spacetime Open Access ArXiv ePrint: (AdS/CMT) Abstract: Marco Baggio, Jan de Boer and Kristian Holsheimer well-behaved field theories.anomaly As for a Lifshitz byproduct, spacetimes. we will find theKeywords: a the Hamilton-Jacobi method. Ratherwe than will derive imposing suitable boundary boundaryusing only conditions local by counterterms. requiring We th leads will to demonstrate in a examp finite bulkof action the while fields. at the This same puts time more it substance determin to the belief that Lifs to make the on-shell value of the bulk action finite. We study t JHEP01(2012)058 3 5 6 8 8 1 2 11 12 15 15 16 16 17 19 21 12 AdS spacetimes, rgences that appear till are confusing are its echniques can be applied n this case than using the ations coming for example maly tractable. Lifshitz”, the nature of the the effective action using the e holographic dual descriptions – 1 – φ ] and have since appeared in many different setups, 2 ) and 0 , 1 α ]. Moreover, they have appeared as solutions of string ] that one needs non-local counterterms to remove diver- 3 − 11 = 2 α z ] and although they are not yet at the same footing as ordinary 10 – 4 Motivated by this we decided to explore the nature of the dive Certain features of Lifshitz spacetimes that have been and s 3.4 Boundary conditions 3.5 The special case of 3.1 Perturbative analysis3.2 of the Einstein-Proca On-shell equations action 3.3 Vector and tensor perturbations 2.1 Lifshitz spacetime2.2 and the Einstein-Proca Hamilton-Jacobi equation2.3 action and the Lifshitz-scaling Initial ano conditions 2.4 Radial behavior2.5 of ( Non-derivative counterterms 2.6 Origin of the continuous ambiguities from holography Schr¨odinger [ global causal structure,boundary the conditions absence on of the a metric version and of other “global fields, and indic gences in the on-shell value of the action. theory [ it is worthwhile to exploreto to Lifshitz what spacetimes extend as the well. usual AdS/CFT t for example as IR geometries [ 1 Introduction Lifshitz spacetimes wereof originally non-relativistic introduced field as theories possibl [ B Higher-derivative counterterms 4 Conclusions A The Einstein-Proca Hamiltonian 3 Renormalized on-shell action Contents 1 Introduction 2 Holographic renormalization Fefferman-Graham expansion, which rapidly becomes quite in in Lifshitz spacetimes whenHamilton-Jacobi computing method, the which on-shell turns value out of to be more efficient i JHEP01(2012)058 ] 13 he paper [ eviously proposed , and issues related z the Hamilton-Jacobi description of the ex- plicitly computing the ry effective action. All he boundary of Lifshitz. agrangians all power law owever, follow a different der, and find the analogue scussions of several issues s of the counterterms have the counterterm action. We ergences should be canceled ergences, qualitative depen- , one needs to first say some- fic bulk action we shall use. using different methods. xponent arginal deformations, exactly eled using local counterterms. deed rendered finite. that appear and the existence l-defined UV completion. terms. This strongly suggests annot be canceled using local covariant quantities made out raphic renormalization and we . o the Lifshitz scaling anomaly. auge field to second order. We ticular boundary conditions for icitly compute the counterterms e for the analogue of the conformal – 2 – As we were preparing this paper for submission to the ArXiv, t ] are insufficient to cancel divergences beyond the leading or Various subtleties, such as the presence of logarithmic div The appendices contain some background material and a brief In section 3 we perform a non-trivial consistency check by ex The outline of this paper is as follows. In section 2 we review Normally, in order to perform holographic renormalization Along the way, we will show that counterterms that had been pr In addition, certain ambiguities that appear in the analysi 12 dence of the answers on the value of the so-called dynamical e to the boundary conditions are discussed in the conclusions tension of our methods to the termsNote containing added. derivatives. on-shell action for scalarwill perturbations find of that the with metric our and counterterms g the on-shell action is in power-law divergent terms in the effectiveSometimes, action logarithmically can divergent be terms canc counterterms, appear and it which is c precisely theseanomaly. that We are also responsibl describe theof relation marginal between deformations. ambiguities method and apply it to the non-derivative terms in the bounda After that, we describe the Hamilton-Jacobiintroduce method the of ‘Lifshitz holog scalingat anomaly’. the Finally, level we of expl We no carry spacetime out derivatives and the contributionsparticularly analysis t transparent. adding a scalar field, which makes di 2 Holographic renormalization In this section we setbegin up with the general a framework brief for review computing of Lifshitz spacetime and the speci appeared, which reaches similar conclusions as we do though of the conformal anomaly for Lifshitz spacetimes. in [ With this approach, wedivergences will can also indeed show be thatthat canceled for Lifshitz using a spacetimes only class are local of dual counter to bulk field L theories witha a natural wel interpretation in theas dual was field the theory case in terms for of AdS/CFT. m thing about the boundaryand novel conditions approach. for As the weby will fields. local show, counterterms, if We we this will,the require will h fields. that automatically all More enforce div precisely, par of we the will bulk find fields that have particular to local scale in a specific way as we approach t JHEP01(2012)058 ) t, 2.1 (2.7) (2.3) (2.2) (2.5) (2.4) (2.6) (2.1) . , 1 ν = 3), but we keep z − ]. We have chosen A 2 d z dt. ). Unlike so-called 14 , 2 m λ , zr −  1 e = K, µ 0 , log 2 = ], which consists of time A α µν  0 µ γ 15 − − ) ution to the Einstein-Proca F A ] that we consider consists of φ − r √ µ 2 ( d theory at a UV fixed point 1 √ ra [ see e.g. [ ∇ m V = ξ 1 2 d 7→ invariant under Galilean boosts irect couplings between the scalar n order to find Lifshitz spacetime − d r − φ Z µ . not nd anisotropic scaling invariance ( µν z, α φ∂ ! F , which is presumed to be known. It is in t µ z ∂ µν λ x λ = 2 2Λ) + ... 2 1 F ,A 2

4 1 2 + − − , 4 −  – 3 – 7→ R φ d~x µν (  g 4 r T 2 v ! g g − e t 1 2 x , but for simplicity we will restrict our attention to − − + √ 2

3 + x √ √ = = 1. It should be noted that we could also have chosen A φ 2 ], x x 2 3 +1 φ µν v dt , m d 14 +1 +1 g πG d d d zr + 2 d d 2 e Z + 4) φ + Λ Z Z 2 z − = µ µν 2 = = φ + 1 2 S 2 +1)-dimensional Lifshitz spacetime [ dr A is the Proca stress tensor. We wish to add a scalar at some poin Rg z , with S d 1 2 ( grav = ) = A A 1 2 µν S ). 2 φ S − δS ( δg − ds g + 2.7 V µν − 2 R √ Λ = grav S = . We will eventually work in 3+1 bulk spacetime dimensions ( = µν vt S T + x arbitrary for as long as possible. These fields comprise a sol 7→ the simple case ( principle possible to consider differentfield setups, and e.g. the with d vector field such as with a potential This metric is invariant under the so-called Lifshitz algeb The equations of motion are where we used the convention 16 a different action that has the Lifshitz metric as a solution, Schr¨odinger spacetimes, the Lifshitz spacetime is the Einstein-Maxwell theory foras its a relative solution, simplicity. we I must pick our parameters to be translations, spatial translations, spatial rotations,(with a a simultaneous shift in the radial coordinate the following metric and vector [ so let us give the scalar action as well, x where The configuration of ( Lifshitz spacetime is awith proposed anisotropic gravitational (Lifshitz-like) scaling dual symmetry, to a fiel 2.1 Lifshitz spacetime and the Einstein-Proca action d action JHEP01(2012)058 l H (2.9) (2.8) ) and (2.11) (2.10) φ (and , α = 0, where at most logarith- = 0 H are the usual lapse ient to consider the t the on-shell action a econdary importance e may find anomalous − L , obi equation and their ifshitz background, i.e. 2 N for relative comparison. ) per we will assume that a = 0 cise in the following. As amilton-Jacobi method to φ iate class of asymptotically E φ he form and a a , k. The reason for including the ) is the Lagrangian restricted rmations in a simple setting. D ) nomaly. The second reason for φ N oca action, which is derived in ( a π∂ ( 1 2 H V + 1 m a b 2 rgence that must be removed by adding − N E φ − , , while + ab a 2 φ r F π H φ∂ 1 2 + Γ + a N ∂ b ( − 1 2 loc E γ a b S − − – 4 – E D a = a a √ respectively. The Hamiltonian constraint is A ) diverges on-shell, and it is necessary to introduce x E cl A a d φ 2 1 S ]. This will be discussed in more detail in due time below. d 2 there will be a mode that acts as a source for an irrelevant − A 2.4 are the canonical momenta dual to the induced metric r 2 16 − , we include the scalar field Σ φ ab m by imposing the Hamilton constraint α Z π π  z > 1 2 b 2 = π loc D − . In order to have a nice intuitive understanding of the radia ) and ( 1 2 b S and H ab A − 1 a − 2.3 a F 2; when is devoted to the dicussion of the radial scaling of d E = A ab , and the scalar a ab − F ab 2.4 4 1 a γ H π ab A < z < π = − ab α π 2Λ  − − in our discussion is that we can illustrate explicitly that w ; it is given by contains the local power-law divergent terms and Γ diverges R explains possible ambiguities in solving the Hamilton-Jac is a hypersurface of large but constant A φ = nonetheless is twofold. First, we shall find that it is conven = r loc 2.6 H S φ L . r Our aim is to construct a finite on-shell action for an appropr At this point it should be stressed that adding a scalar is of s This is true for 1 , induced vector 1 ab In particular, section without the scalar field, which will be explored inLifshitz future spacetimes, wor whereusual, this the notion action given will in be ( made more pre breaking of the symmetryWe under also expect anisotropic such scaling anomalous transfo symmetry breaking in the pure L section scaling behavior of the quantity operator in thenon-local field counterterms theory along the and lines could of in [ principle give a dive where Σ such counterterms are localdetermine in their the form. fields, We and expect we the will on-shell use action the to H be of t a set of counterterms to remove these divergences. In this pa relation to an anisotropic versionincluding of the holographic Weyl a when it comes tocan the be main renormalized goal byscalar of adding this local work, counterterms which alone. is to show tha composite scalar and shift functions. The momentum constraint is given by where to Σ γ where the quantities where is the radial Hamiltonianappendix corresponding to the Einstein-Pr mically. We will determine JHEP01(2012)058 and . b (2.17) (2.14) (2.18) (2.13) (2.19) (2.12) (2.15) (2.16) cl S δG t δA b ∂ ... = 0, while ..., . − D ) + r + H a ( =   δF cl i δA ˆ φ Γ . The divergent ) (for simplicity Hamilton-Jacobi φ H a δφ δ δ δS , which will be our ˆ D φ hO t contain power-law γ 2 ˆ φ 2.16 his procedure possi- − φ the loc 1 − 1 λ m and is generically de- } − L S − φ 2 √ λ n ( + loc . n principle, this equation ticle are − t + } rem ,S ˆ Γ i Γ b ) = A δ H , r δ A . Therefore we can use the simply becomes loc . ( etic’ part of the Hamiltonian t δG Γ δA φ S G  ˆ { hO A a t Γ z = 0 ˆ + δφ δ A . δF ˙
≡ { δA z + cd } φ and cl Γ δG ab , π + ij δγ δφ + , Γ ) γ loc δS ˆ F γ i δ r a , i 1 2 δ } − L ab loc ( H i Γ a cl cl a S δF ij T δ cl − δγ δA h { γ δA δS ,S a δA δS , is the action evaluated on the classical path  ˙ + cl A δφ δG cl , but it is far too difficult to solve. Since we are γ + 2 ab + 2ˆ i cd S cl t cl γ { + t – 5 – δS 1 − δγ S tt S Γ δφ T ; δF ab ˆ γ √ h = δ ab γ δ Γ 1 2 z , φ δ 1 } − L , which we denote by tt  δγ H a ˆ − γ 1 ) = , we can recast the problem in a more tractable form. ˆ − γ loc loc ab z r ,A ˙ 2 γ d − ( loc H 0 = ,S a ab   S p γ − loc x x x d d d S bd = 0 determines the divergent terms in H { d d d γ ac ,E Z Z Z ) γ r 0 =  ( = = = loc,div cl } ab is symmetric and bilinear in H Γ δS δγ , + Γ to write: } γ } ≡ − loc loc ): 1 − S S , where the on-shell action F,G = 0), { √ { γ F,G = ti should vanish by itself because the non-local part shouldn’ { 2.11 γ − /∂q cl 2 cl ) ) = loc √ S r γ x ( H ∂S d − d ab √ π We will see that, in the presence of marginal deformations, t ( Z ) = t 2 ( with given initial and final conditions. The first HJ equation the second one is generalized to It is useful toconstraint ( introduce the following notation for the ‘kin we assume The HJ equation isdetermines a the form functional of PDE the on-shell for action the on-shell action. I such that the Hamiltonian constraint is simply The Hamilton-Jacobi (HJ) equations of motion for a point par 2.2 Hamilton-Jacobi equation and the Lifshitz-scaling anomaly p interested only in the local part The two HJequation, equations may be combined into what is known as The bracket part of splitting We define the “local part” of this expression as divergences. Solving bly leaves a finite remainder in termined unambiguously. Let us now rewrite the other piece i counterterms. JHEP01(2012)058 - c ). d he 2.5 ; it is (2.20) (2.21) (2.23) (2.22) r , cf. ( ], such that J equations /z 20 . 1) . – ˆ limit. Plugging e metric on φ Γ scussion of this − δ δ 18 r rem z ˆ γ H ]. More specifically, 2( r 1 − ish the leading-order − . 12 [ √ +2) = r z ( = 1) 0 e . We thus see that there is i − α the most general covariant . z φ ( focus will be on this level of
loc e →∞ 2 so used hO t is covariant [ r H g the HJ equation thus reduces We shall find that this happens = r d~x to impose covariance on Σ = lim c i + φ 2 . It should be noted that the quantity is given by hO , 2 dt for a solution of our choice. Henceforth, ˆ t φ µ 2 r is the leading radial scaling of the scalar α ˆ Γ cl c A − φ δ δ S − φ is compared to the so-called stress tensor − λ ) + (derivative terms) λ ˆ b γ , + r a 2 1 − tt , and in order to solve them it is necessary to T α, φ e i γ – 6 – ( √ for a discussion of the higher-derivative levels. cl A zr = S 2 B = b hO γ U − t i e ˆ − dx A A a z √ hO x →∞ + dx d r , is subleading and vanishes in the large- ) i d -dependent speed of light i } r i r r ( Γ T Σ , h ab Z = lim ) we immediately arrive at the Lifshitz analogue of the Weyl Γ γ limit and where we used the hatted notation for the asymptoti , ]. + { = tt that the speed of light becomes infinite. is the only scalar that one can construct from the metric and t r i ab 13 γ Γ t ), ˆ a 2.16 t γ δ , loc δ T A S ] in which the tensor h a 12 ˆ γ z 2.16 → ∞ 21 A ) in an expansion about the Lifshitz background. In fact the H 2 − was defined to be the finite remainder in r ≡ ≡ √ ]: z α, φ α rem A ( 20 = – H i U the stress tensor of the (proposed) dual field theory; for a di 17 ab defined in [ = 3. Recall that the background value of T h not d is As an aside, notice that the induced metric may be viewed as th Before we move on to solving the HJ equation, we need to establ b a we set provide the initial conditions, that is the value of complex are functional differential equations for dimensional flat space with an a possibility that the Lifshitzfor scaling some specific symmetry values is of broken. the scalar mass-squared to solving the HamiltonianAnsatz one constraint. can take In is the present case, 2.3 Initial conditions As we mentioned earlier, one typically choosesthe an momentum Ansatz constraint tha is automatically satisfied. Solvin anomaly [ behavior of The final piece ineverything ( back into ( subtlety, see e.g. [ which holds in the large- Recall that Therefore, heuristically speaking, it seems quite natural vector field containing nono derivatives. spacetime At derivatives; see present, appendix our main T only in the limit The quantity field and the dots represent subleading contributions. We al values of the fields, e.g. ˆ JHEP01(2012)058 r = ij γ (2.33) (2.31) (2.27) (2.26) (2.24) (2.32) (2.29) (2.30) (2.28) (2.34) (2.25) r ∂ , tt zγ . = 2 tt = 0 γ r 01 ∂ , ) around the background b ution to the equations of , E b ∂α ∂U α, φ nta are given (via the second . D ( kground b a n in order to fix the leading-order π, U A φ D , a ab m 2 γ A ) 1 0 m ∂α ∂U + , u α b + 2 2 z + ∂H/∂p ab − A . a π ab − a φ = 2 α γ E π A ( q = − − ,  − − 10 mn = = = u ∂α U ∂α ∂U ∂U ) , – 7 – ) , ) a a ab ab α φ γ A m,n X ∂α ∂U ∂φ 2 1 ∂U − . ). We evaluate the above Hamilton equation on the γ π γ E γ π a = 0 in the background) into the values for the first δH U δH = 2 = = − − − δH A ) = ∂φ ∂U 1 2 2 ) around the Lifshitz background as a φ φ α, φ √ √ √ ab ( ( ( ( π  − − E π δ δ δ α, φ U ( α, φ = = = ( = = = U a = 0 ( , u φ ab U a r φ denotes only the local part of the momentum (and similarly fo A ab γ r ∂ φ r A r γ ∂ r r ∂ r ∂ ab + 1) loc ∂ ∂ ∂ z δγ δS γ 1 − ). = 2( )) by: √ , and t 00 = α, φ u zA 2.12 ( ab U π = ), so that we find t φ π A r ∂ , and ij a γ Lifshitz background, so that we find few coefficients in the function 2 We can then translate the radial behavior of the Lifshitz bac At the level of noHJ spacetime equation derivatives, ( the canonical mome E Let us expand the function where now have been determined bymotion. imposing that the background is a sol We shall use the Hamilton equation of the type ˙ Therefore, the first three coefficients in the expansion for behavior of JHEP01(2012)058 ) , and ) 2.32 φ (2.36) (2.40) (2.38) (2.39) (2.37) (2.35) (2.41) ( 20 u V tenlohner- + as discussed zα rem H ..., + 4) + . The radial behavior z φ ) + pacetime derivatives by ore, this comes down to 0 + . α 2 z  ( − 2 φ. roca field equations would be n the Lifshitz background, 2 µ α . − g the Hamilton equation ( µ ( 2 α 1) ) spoils these relations. Later on, + 4 =  λ ) − 2 ]. φ = z b α, φ ∂φ ∂U + (derivatives) and we expanded the ( ( ∂ ... 14 2.31 a 3 2  , + 2) U ∂ + ∂α ∂U 1 1 2 z − ) γ U φ ( ab α − − γ 20 02 p u √ u ) and ( + φ + 4 = 0, so this term is in fact absent. x ± 2 ∂α 1) ∂U d – 8 – φ 2 − r d 11 ) yields α − 2 2.30 ∂ r ( u is dictated by the value of the coefficients R = z αU 2 + 2) φ 4, see also [ z 1 2 − φ φ / λ ) and z = r 2 ( 0 + 2) 8( 2, which agrees with the answer provided by the equa- is a composite field containing both the vector and the − ∂  / z α 2 loc α U α φ = 2 1 S  1 2 λ + 2) − ) exp( α ) and + ( − − z 0 − λ ( α φ ∂α α ∂U = 2 r = ∂ φ  − t, x, y ) = λ 02 ( ≥ − 0  α u χ α α 2 µ indicates that there might be a finite remainder = − ). Notice that there is also the Lifshitz analogue of the Brei + 2 φ α ∼ = 2 ( r α 2.35 } − L ∂ ), we similarly find via ( 2 1 0 loc  α . Remember that − ,S − 2.2 2 loc α U S 3 8 { = ∼ = One may wonder whether the bilinear term in For ( Let us start with the more familiar example of the scalar field respectively. We will now show how this comes about. 0 02 tions of motion ( Alternatively, one can find this radial behavior by expandin Freedman bound, i.e. to second order, such that can be obtained directly from the free scalar field equation i The radial behavior of ( with in section In this case findingconsiderably the more radial difficult, behavior since using the Einstein-P 2.4 Radial behavior of ( Later, we will find that We shall now set outsolving to the find local the partsolving counterterms of the at local the the part HJ level of of equation. the no Hamiltonian s As constraint, we mentioned bef metric. 2.5 Non-derivative counterterms Taking the Ansatz u where the symbol we will find that the HJ equation gives JHEP01(2012)058 . 2 µ and = 0, , and (2.48) (2.43) (2.44) (2.46) (2.45) (2.47) (2.42) 20 10 11 u u H . We can , ) remains n = 20 02 φ u 01 m , u . This special ) 0 H 11 02 α u = , u where both modes are such that both − 20 2 00 are determine the radial 01 , u , α µ ( u H 2  φ uch a remainder would z 02 gn in 32 , 5 ormalizable mode for the u mn 2 u µ . ree unknowns , − + 20 resent case, for this specific n , and 1 2 rder. mainder in the Hamiltonian z 11 φ l Hamiltonian constraint, 2 20 and 10 m,n u + u m 20 u ) cally, i.e. ) and/or 1  P , ), which is obtained by requiring 0 20 2 02 0 . − z 2 u α u α 00 − 2  z 2 u + 7 2 2 z − − ) = z 9 µ z ), namely the scalar mass-squared − α α z ( p − 02 )(2 + 3 α, φ + 4 + 8 u ( z ± 02 2 mn 20 2), one finds U u z − H 2 u 2 − + 2) 5) + 2) (2 − − – 9 – z z 4 1 m,n X z 5 respectively. The choice of the sign in − ( 20 −  1)( = u z + ( there is a special window for φ p 1 < − (2 1) 2 11 φ 2 ± 2 z − u − − z µ 2 1 that the coefficients 2 in principle. 2 z z The above values for 2 11 2 z + 2 ) and ≤ − 8 u 6= 2( z 0 20 z 2 2 2.4 2 1  2 α u z , the minus-sign root will correspond to the non-normalized 16(  loc,non-deriv µ , 1 2 − 4 1 02 − − H u + 2) = = 2 = α 2), however, one of the three coefficients ( = = 0 = z ( 11 02 20 − 1 4 20 11 02 and H H H u u u z − 20 1)( u The constraints at second order in ( − z about the Lifshitz background as we find no such remainder. 2 U µ = 2( 2 µ Remember from section correspond to choosing the either the normalizable or non-n . There is also one free parameter (apart from 02 02 window is given by contribute to a Lifshitzvalue scaling of anomaly. Although, in the p undetermined. In principle,constraint it that is cannot possible be that set there to is zero a by re tuning a coefficient. S The generic case, for which the order-zero and the order-one constraints vanish identi u Order two. For even though they depend on are normalizable and one has the freedom to choose the plus si that the Klein-Gordon norm is finite. do a similar expansion of the non-derivative part of the loca behavior for the fields ( which is a system of three coupled equations in terms of the th function Order zero and order one. This allows us to solve the Hamiltonian constraint order by o modes. Though, for the scalar field fields. For both u JHEP01(2012)058 , the 2 (2.50) (2.52) (2.49) (2.55) (2.54) (2.51) (2.53)
+ 20 and + 2) 0 as long as z z z β ( >
20 1 4 z = 2. γ ... − − z 2 2 + + 39 ) an constraint that z z z  9 β 2 , √ 30 φ
− ) remains undetermined. z 0 ady at third order; this − = γ . α 2 + 2 , we see that
z z − it a continuous ambiguity). 2 z − β anishes for z µ β ( 15 ... α 1 ( 16 + 2 r + z =  + 1) +2)
3 + 2 + 2 z z z ( φ 170 + µ z β e r ] from these equations for generic values + 1)  − depends on
32( z z z 22 +2) 03 z 6 + β z γ , u ( →∞ lim 64( e r − 20  z 6 +
is the third order coefficient in the potential + 223 z z , and − 2 β , there is a contribution to the Lifshitz-scaling 3 . Since , which is above the BF bound, the coefficient – 10 – →∞ lim z 3 v z 12 z 2 . This is the same contribution to the anomaly r + φ , z , u 3 z 3 3 , so we have a contribution to the Lifshitz anomaly, , v 21 12 160 v 2 v − u u 21 + 2) + 2 + µ −

= + u , = z z z z = 3 , z ( γ β + 1) 30 z 03 2 9
2 2 30 A 12 = 1), see e.g. [ u u z z u rem 35
−
β − − z H z z H Recall that z 2 z 64( γ β = γ β 3 3 1) 6 + 2 2 . − − − − µ 2 − − 3 µ 2 z z z + 2 + 2 + 2 + 2 is undetermined and there is also a remainder, namely For z ), + 4 z z z z We find a second ambiguity when 2 − is positive for any value of 12 128( 2 2 2 2 1 1 1 1 The constraints at third order are given by u z 2.21 = β − − − − + z . + 2) 2 = = = = A z ( 12 03 30 21 remains undetermined. There is a remainder in the Hamiltoni + 2) p H H H H z ( 03 4 1 anomaly ( cannot be set to zero, Ambiguity 1. u Interestingly, this contribution to the Lifshitz anomaly v Thus, for this specific value of Ambiguity 2. coefficient which is present for all values of that was found in AdS ( . There are two special values for which one of the coefficients − ) = ). The first thing one sees is that the equations decouple alre 2 The quantity • • µ φ 2 > µ ( ( z 2 of continues to hold at higherOne order determines the (for coefficients as long as one does not h V where we introduced the abbreviations for the square-roots γ Order three. µ JHEP01(2012)058 . g is 12 − α u 2 λ φ ) (2.59) (2.58) (2.57) (2.61) (2.62) (2.56) (2.60) or = ) with ) from 0 0 r α − φ 03 α . r λ u − − α +2) α z ( t large e or ( . . ∼ 3 02 φ , γ u , 2 + 2) . − 2)  − z √ 2) ( . For generic values of 2) = , we indeed see that − ± 2 nation − φ 2) that we found at second φ ginal deformation. Namely, z > = < z < his case. This ambiguity is z ( sence of a marginal operator, − further study. f a term mixing ( + φ + 2) (1 z 1)( λ . In the second case, i.e. For and . z 2 ( − + 1)( µ  2 9 ± 2 z ), where ) − φ − µ − 0 r z . α φ = λ + 8( + 4 − 2 2 2 + 2) coming from by giving linear contributions to the right + 2) µ = 2( α z z 2 φ exp( ( , λ µ + 2) + 2) − ∼ 1) z z ( ( – 11 – and φ = − 0 , , p p z − φ α ( λ is marginal. 3 2 ± ± 3 − describes the radial behavior of the non-normalizable , λ 2 ) and + 2) + 2) − r φ α z z ) ( ( α , we find that + 2 + 2 α,φ − 0 20 + 2) 2 λ λ − − α z z u At third order, we found two ambiguities, which comes from ). This suggests the possible presence of a one-parameter z   ( = = − 1) 1 2 2 1 − + 2) exp( The critical value α − φ + φ that of the normalizable modes (vevs). In terms of the scalin − 2 2.38 − − z λ λ = is simply a result of tuning the scalar mass such that z ( 2 ∼ 4 1 z = = ) . Another way of saying this is that either the coefficient + α φ + α,φ 0 r + 2 + 2 λ 8( λ ± φ ± α − α , which mixes − α − α λ λ 2 + + 2) cancels against the +( +2) = ) ) and ( λ λ − z 11 z , we see that ( z − α is marginal for this specific value of α u ( ) and α β µ − λ λ (1). In order to see this, let us recall the radial behavior (a 0 3 e − 2.37 − α O φ − , i.e. ( + 2 α z ( 2.4 1 16 = becomes of 2 of order one in coefficients, we always have Note that this hand side of ( Let us refer to the correspondingmodes modes (sources) as and ( φ remains undetermined. In the first case, i.e. for family of allowed boundary conditions, but weOrder-three leave ambiguities. this for the freedom to tune the scalar mass such that either the combi the scalar mass Order-two ambiguity. The continuous ambiguities arise when the radial behavior o 2.6 Origin of the continuous ambiguities In this case, the operator ( When we plug in the values we found from the HJ equation, we get We should stress that thistherefore ambiguity is it not is related not toparametrized the by surprising pre that there is no remainder in t section µ This continuous ambiguity comes from thethe appearance of operator a mar order in ( JHEP01(2012)058 , 3.3 (3.3) (3.6) (3.1) (3.5) (3.2) (3.4) (2.63) . For r  j, . ...  + 8) + 2 z K ... ) 2 0 + α γ + 6
ur space of solutions. 2 − ial coordinate ) − z r √ ( α j. ξ ˜ ( f d ) 4) 1 2 1)(4 rd-order on-shell action are d erturbations to section ary conditions. z utions is to perform a non- we set the scalar field to its − β − Z do not receive any corrections. 1) s in the constant perturbation z ) + ms that are of higher order in z j, xplicit form of the renormalized ] and then we set out to solve ct our analysis of the linearized r ( + − ( rµ 5 + . Again, we focus on the non- 1) j ˜ g 2 12 −  1)(2 − z ′ − µ 2.5 j 2 z − z A ε 16( z µ , (2 , 2 +  A  + 2) z 2
+ ( + 4)( z ) ′ ... m ... r j z ], the first order field equations for constant ] for which the Lifshitz geometry is perturbed 1 2 ( ) + 0 1) f 1)(4 12 12 ) + ) + − 2 1 α r – 12 – r − + 2( ( − ( ′ ˜ − µν z f ˜ k z j ) + ( 2 α F 2 r ( ε ε ( − µν 2 z + 2( j ′ + 6) F ′ f ) + ) + − z f 1 4 ε r r ( ( (4 − to keep track of the order in the perturbative expansion. We + 1) − + 1) ε f 1 + ε k + 1) = 0. ε ′ z  2Λ z z f φ As was noted in [ zr − 1 + 2( 1 + e + (  + 3(   0 ′ R + 1) ij ′′ zr k α g  δ 2 z f r − − e ( g 2 − e √ √ − − + 1) x + 1) √ ′′ z = = = j x z +1 t tt ij d A γ +1 γ d d d Z 0 = 2( 0 = 2 0 = ( Z − . 0 α We adopt the same parametrization as [ Γ = − First-order solution. work in radial gauge, which means that the components We use the small parameter simplicity, we postpone the treatment of vector and tensor p perturbations reduce to the following three equations. as follows. derivative sector throughout thisfield paper, so equations we to shall constant restri modes whichand only here depend we focus onbackground only the value on as rad the well, scalar subsector.Furthermore, the second-order equations.trivial The check purpose of of finding the these counterterms sol that we found in The ellipses denote higher-orderα derivatives as well as ter 3.1 Perturbative analysisFirst, of the we Einstein-Proca discuss equations the first-order solution obtained in [ Up to this pointIn we this did section we notsector will impose up analyze any to the second boundary Einstein-Proca orderindeed conditions equation and removed on using show o that our divergences formalism in for the a thi large class of bound 3 Renormalized on-shell action For future reference, let uson-shell conclude action this at section with the the constant e level, The on-shell action JHEP01(2012)058 , . 4 5 c c (3.9) (3.7) (3.8) + + (3.10) r r 2. The ) ) z z = 1 (just β β . It is also z − − z > +2 +2 3.5 z z ( ( mode diverges. , 2 2 1 1 r suggested by the ) 3 garithmic modes. − − z c 2 and e e β 3 3 − c c ]. Additional evidence ) ) +2 z z z z z ( 2 the 23 β β β β ld equations, notice that 1 2 < z < , = 2 (and similarly for the ird one is first order. This − − − . The solution is given by z e tric modes when 14 2 + 4 + 5 unterterms could modify the ause the ADM mass is com- z > 3 c − − + 2 + 2 1 z z z z e in [ . − . , . . . , c ) + 1) z 2(5 2(3 1 1 2 . i z c c ε ; when ( ∂ ( − + i 2) at linear order. In fact it is easy to see O r r + x ) ) 5 z z 5 r − 2 c + 2 ) β β c → ∞ z ) + ). z z β 5 r r + ( c 4( +2+ +2+ t and ε z z εj ( ( +2+ – 13 – 4 2 2 1 1 t ∂ z = ( c and = 4 − − 2 1 2 c e e 0 4 − c 2 2 α e = c c ADM 2 ) ) − correspond to linearized diffeomorphisms, generated by ξ c z z = 2 shall be discussed later on, in section z z M 1 5 β β α β β z c − − − + 1) z 2 4 z and shift and ( − − + 2 + + 2 + 3 4 , z z c z z − 2 , r 1 c 2(5 2(3 +2) 1) z + − ( 2, all the modes decay as − modes have the same radial behavior when r r − e z 2 ( c +2) +2) 1 z z c ( ( 1 → − − modes). The case of 3 e e and < z < , − 2 + 1) 5 = 2 must be treated separately because of the appearance of lo , 1 , z 1 1 1 is a scalar, it cannot depend on 4 c z mode should be related to the mass of a black hole solution, as z c c c c ( + 2 + 2 4 2 α 1 z z − c For 1 and = = = 3 j k f asymptotically Lifshitz black holes considered for exampl case of that, at this order The To see that thesee.g. logarithmic the modes are needed to solve the fie This solution holds for the range of the dynamical exponent 1 for this is provided by the ADM mass: Since interesting to see that the vector modes decouple from the me Notice however that thisputed computation by is means a of bit backgroundanswer. subtraction, suspicious and bec additional co means that there must be five integration constants: The first two of these equations are second order, while the th Notice in particular that the vector field rescale c JHEP01(2012)058 3. he , 2 ′ , (3.12) (3.11) (3.15) (3.13) (3.14) (3.16) ff = 1 appear at ′′ i ′′ 1 ˜ k + 1) ff − not z 1 1 ( z z z z , with − 4 − 4 i 6 4 , 4 z c ′ z 4 k f j k 9. + ′ − ′ + + + f + r ′′ r r ′′ as well as the first and r r ˜ ) r 1 ) f ,..., ) β β z z jj pear in the second-order z β +2) kk +2) 1 − − +2) 4 − z z − z z 1 = 5 z 2( ′ ′ − 2( 4 +2 + 8 +2 i . z 2( , +2 , z − z ′ − − r z eld equations and which ones r thus only depend on the first- 4 ( z − ( r + modes that did stants. Another thing one can ) e ) e ( ) ′′ 1 2 1 2 e z β β 2 1 2 z jj z jj 7 β 7 ′ − 7 t-order differential equation and − . − − − k f , − − ′ ′ kk k j z jf e + e ′′ ) are related by ˜ ) with e ˜ ˜ k k ′ i 3 i 1 + 8 + 8 1 3 + ˜ + k + 3 +2) +2) k k ′ ′ +2) f z z z j r ′ z r , k + 4 z r , k 1 ) − − i ) 8 2 ) i f 2 1 1 + β + β z ′ (3( + β (3( z z , f (3( − + 2) + 1) 8 1 − 1 2 r j i − , f 1 2 r z jf z jf − 1 2 r ) − − i ) j ) z z z z − − + − j β β − ( ( +2 − β +2 z z 2 e +2 e ′ z e 2 z z z + 4 z ′ + 4 + 4 + ( z ( 9 9 ( 4 8 ′ 9 ′ k − 2 2 k − f and ( kk +2+ j +2+ − ′ ′ f +2+ ˜ k e z e − z ′ 1 e j j z 4 ( − + . Instead of listing the coefficients explicitly, we ( ( + j 6 + j 6 + 1 2 2 1 j 6 ′ ′ z 2 1 ′ – 14 – 2 1 z k f − r c j r j j 1 ˜ ˜ r 2 k − ) ) − ) − − + 4 + 4 + 1) z e z z β β e β 24 + + e ′ ′ + + 4 1 − z z 2 2 r 2 r f f r z ( 2 z − ) ′ ′ k and ˜ ) f ′ ) j − 2 z + 8 + 8 ′ β j j β + β f j +2)+ 8 +2)+ ′ ′ +2)+ z ′ + + c + z z 1 ˜ ˜ z j ˜ f f 1 kk r r r − + +2+ +2+ z + 4 + 4 +2+ (3( (3( − (3( 1 The coefficients ( ′ z z − ′ z 2 1 1 2 z 2 1 ( ( 2 2 ( +2) +2) z ′ ′ +2) 1 1 z 3 depend on the dynamical exponent z − − − z − − z − − 3 . f f + 8 kk + 2) + 1) ( ( 24 ( e e i e e e e i The second order field equations are given by z ′ − − z fj − c − − z z 8 5 8 5 k ˜ 5 8 + + + 4 f e e ( ( e z z f f j j k k − 2 2 2 z z 1 1 1 z 1 ′ −


4 4 + 2) j f k + + + + + + ′ j j j − z 2 2 2 − + 4 + ( are products of the modes we had already found at first order. T , and z i ff z fj z ) = ) = ) = j j j ˜ ˜ ˜ − − − f ˜ k 4 4 r r r , ( ( ( f f f 2) i ˜ j − ˜ ˜ k 1 f − j ′ ′ + 3) + 2) + 2) − and 2) − z z z z ˜ ( ( ( ( z f − + 2) + 2) z z z z z fj z ff , 2 z z z 4 2 j ˜ ( ( ( z z z + 8 + 8 + 8 − − − 0 = 0 = 0 = The coefficients we are talking about here are 3 shall discuss which coefficients areare fully related determined by integration by constants. the fi The coefficients of the second-order integration constants solution is thus given by first order are entirely determinedorder by integration the constants field equations and Just like the first-order equations,two second these order consist ones, of so oneread again firs there off are from five these integrationfunctions con equations is that the only modes that can ap The coefficients Second-order solution. JHEP01(2012)058 4 2 5 c c c = ˜ d are (3.18) (3.17) (3.22) (3.19) (3.20) (3.23) (3.21) 4 and f 4 c sources , , 3 3 1 c c c + ˜
8 − β 2) 1) ], we have: l action at higher orders. unction of the integration omputation shows that the − − 12 nfinite when one uses those ly third-order corrections to z z ations sector. The remaining ivergences. rce/state, where ( in [ , , ) 2)( ε ... . we find is should be sufficient to compute r , at the second order on-shell action ( ε − + ! i + 1) 2 ) z v ( ) z r (2) r + 1) ( ( r . Therefore we identify 2 , S ( d 1 3 z ) e t o 2 c c r t ]. 2 ε ( − i c 2)( + 2 1 24 ) ) + ) 2 + v z r r r − , ( ( ( + ) zr i z o d (1) 1 e r t t ( ( – 15 – v , which cannot be determined by the asymptotic 1 sources 0 1 c ε S

ij zr α c 2 5 k 2 + 1) c + r − = 0, while the first-order term is given by e 2 z = ) = ( e − √ + 2 r (0) z (0) ( The second order solutions should be sufficient to check S ]. At second order in (1) and z = = = 5 S ] had been devised to cancel first-order divergences only, ij c i S 4 ti k 1 ij 12 12 c c A γ γ 2 Γ = is finite with our counterterms. This provides an additional + (3) do not enter the field equations at all, so let us call them S 4 1) k are then naturally identified with the state of the system, an − 2 z ( c + 2 and 4 z 4 c 1 f c and 2 1 for consistency of notation. c − 5 c = is a second-order correction to 1 = ˜ (2) c 4 S k Since the counterterms in [ coming from the third order solutions). In fact an explicit c 1 which reproduces the result of [ The leading-order term vanishes, and 3.2 On-shell action These solutions allow usconstants to up to compute second the order on-shell in action the expansion as parameter a f where is finite. Furthermore we recognize a familiar structure sou analysis and is therefore arbitrary. It is pleasing to see th finiteness of the on-shellc action at third order (we expect on one does not expectAt second them order, to onecounterterms. properly finds indeed renormalize that the the on-shel on-shell actionThird-order is on-shell i action. where ˜ The coefficients determined by the initial and final conditions [ 3.3 Vector andFor tensor vector perturbations and tensor perturbations, thethe linearized second-order analys on-shell action. Using again the notation non-trivial check that our counterterms indeed remove the d while a linear combination of with the boundary conditionsparameters in the constant scalar perturb third-order contribution JHEP01(2012)058 . 2 o t (3.24) (3.29) (3.28) (3.26) (3.25) (3.27) (3.30) and ) term is should be 2 ε d ng term in 1 t o t , i 2 c . , and i i 4 3 1 c d c = 2. The first-order . . i . t i i 2 , 4 ) z 2 , c , 1 i c 5 c 4 ) c /c 8) c 4 i + ) c 4 , + r i c − 1 r + c z 3 3.17 +2) r 4 , . c y change the boundary values z . . 5 3 tions for 4) 2 ( and 2 ) 4, since it leads to a divergent 2 c c o − n ( 2 − i c , − d − 2 3 e , , /t 4 t i r z r 1 c z 1 3 /c o , ( d c z > + (4 + ( i axis. t t z 1 +2) 1)( +2) zr r r − c z z 3 4, and its second-order (in 4 4 x z ( + ( + 2 − − − − 3 − − 2 e and e e , o z z e e i 3 t 2 3 2 ) ) 2 c 1 d z < o c 2( d r r + o t t t 2 2 + r /t + + c c + + 1 r 4) r i d 4 1 1 3 – 16 – t − d o c − z + 4 + 4 t t i ( + 1)( +2) e 1 2 2 e z c z i ( ) c c 2 r 5 − c ) = ) = e 2 r r i c − + 5 ( ( 2 4) 4 = 2( + 1) o d c 1 1 t is problematic when t + c c z − − i 1 + (2) = 2 2 z (4 (4 2 c + 2) i S ( 1)( ]; it is given by c z ( 1 z z z c 1 1 Finally the tensor modes, given by: ( 12 24 The vector sector is parametrized by − − 12 ) converges only for z ) = ) = 4( ) = ) = ) = r r ( ( r r r 3.17 i i ( ( ( = 2 1 j k f v v (2) S is a source for the massive vector field and is fixed by the leadi 3 c 4 . The corresponding expectation value is given by 0 . The expectation values of the dual operators are given by α We note that the mode ij For simplicity, we take the perturbation to lie along the γ 4 − 3.5 The special case of on-shell action. solution was computed in [ Let us repeat the analysis of the constant (scalar) perturba The mode α interpreted as the sourcesof in the metric sector, because the 3.4 Boundary conditions At this point we can draw some conclusions: the modes given by The source/state structure is in this case lead to the second-order contribution to the on-shell actio Once again we recognize the source/state structure The on-shell action ( Tensor perturbations. Vector perturbations. JHEP01(2012)058 4 ). k by 6= 2 and 3.9 z 4 (3.32) (3.33) (3.31) c and , while a ( 1 4 = ˜ c f y, but let . 4 , r f r 8 , 8 r − − 8 e = 2. Although , e − 2
z e
c 2 2 r
r + ˜ 2 9 9 r 5 ] for k are non-normalizable. f c 9 2 j 5 ntributions to the ‘Lif- 25 + rly renormalize the on- c + c r + r obtained by squaring the determining the boundary 8 12 19 8 r k f 8 . The coefficients n for asymptotically Lifshitz + pear. The coefficients of the j at the counterterms are inde- 3 , and l fully determined by the field + 9. y + c 5 4 , es, e.g. [ 7 c c + is given by 8 7 ency check for the counterterms terterm action we find is a local 6= 2, for instance the ADM mass 1 k , , f 7 ε c 7 3 j z , 6= 2 case, we call them c 6 + + , + z 4 + 2 for 2 c r 2 r 1 2 = 3 6 r c , c 6 i k 6 f 2 j c 3 24 25 + 2 + + r r − r ) with = 5 5 4 i ) are related by ˜ k 5 f c 5 – 17 – j , k 1 (1) i + c + , k S + 1 6 5 , f 4 ) are related by the integration constant ˜ 4 i 4 k 1 f j j , f . We also checked whether the on-shell action is finite − = 2 at first order in 4 2 , k + + 3 j + c 1 z ]. At second order, we find r c depend on the first-order and second-order integration r r 4 and ( 2 4 4 , f are normalizable, while i c 12 − − 5 1 − k j e 2 e j e c 12 25 ) for ,

i
, and ( 2 2 f 2 2 + should be seen as the sources, while the vevs are represented r r The boundary conditions are essentially the same as in the = 2. r c , 3 and 3 3.17 3 5 i 3 c z k c f j 1 j 1 c , c + 4 + + respectively. We shall not list these coefficients explicitl can be interpreted as the source for the vector’s mass term vi c r r r , j 3 2 = 3 2 mode takes on a similar role as 3 2 c c k f j c 2 3 when c (2) + + / + S 2 1 1 1 and ˜ c The coefficients ( j f k j c 5 = 4 is a correction to the vev . Again, ) are related by ˜ 2 2 for consistency of notation. ) = ) = ) = 2 c r r r c 5 , k ( ( ( c 2 ˜ j ˜ ADM ˜ k f The on-shell action ( To clarify, these coefficients are , f = ˜ and 5 M 2 4 1 j equations. k us mention where themodes second-order that integration were not constants present ap in the first-order solution are al do not enter the field equations at all, so just like the The possible modes infirst-order modes. the second-order The second-order solution solution can is again thus be given b at third order and we find thatBoundary it conditions is. ( case. The modes constants obtained via this method.shell action even We for saw higher-orderfunctional that perturbations. of our the The fields counterterms coun bypendent prope construction. of the Moreover, radial we cut-off, find unlike th some previous approach We have found a new andconditions systematic on method the for one simultaneously hand,spacetimes and on finding the the counterterm other actio shitz hand. scaling anomaly’. This We method performed allowed a us non-trivial consist to find co 4 Conclusions c which reproduces the result in [ is where ˜ Notice that the In this case, the modes Again, the coefficients JHEP01(2012)058 ]. 26 rder. In our for an example 4 which remove B z > e set of equations of in “coordinates plus nceled by local coun- be able to remove all n-Jacobi analysis can ptotic behavior of the lographic Weyl anomaly ations to black hole so- e course. tion which seems related ilton-Jacobi equations or aling at the linear level). ld reveal that one of the , in the computations in liminary analysis seems to ppendix 2, and therefore we expect ons for ormalizable modes. his works for all reasonable e leave this for future work. rder in the sources to make , as was emphasized in [ es of the dynamical exponent the radius. The counterterms t is also unclear whether this in our case this cut-off depen- nding constraints will end up oundary fields, the coefficients s in more detail. Furthermore, nd we have only shown that it z > z dual or hold for a more general strategy we have been employing, d, and the precise nature of these sion of our work to Schr¨odinger t would be interesting to explore this ]. 16 = 3. Though in principle straightforward, the d – 18 – 4, divergences seem to appear which cannot be z > = 2 and z , for 3 for b A a A ab R ] only managed to make the on-shell action finite up to linear o 12 , five free parameters appeared, whereas for a non-degenerat 3 There are various qualitative differences for different valu Although we focused on constant perturbations, the Hamilto Another thing which would be interesting to compute is the ho It should perhaps be emphasized that the idea that one should There are many further directions to explore, such as applic One may wonder whether the fact that all divergences can be ca . As mentioned in section z be used to find higher-derivative counterterms as well. See a we would be forced to impose more stringent boundary conditi constraint would somehow followrequire from separate the input. analysis of the Ham canceled using local counterterms. If we blindly follow the five parameters can beto removed a by bulk a diffeomorphism suitableHowever, (which a gauge turns preliminary transforma out analysis tobeing suggests non-linear be once that a higher the Lifshitz ordernon-linear correspo corrections resc boundary are include conditions remains to be determined. I fields, there aresection still some puzzlesmotion that one remain. would expectmomenta” to For or find example equivalently an inOne even sources would number and expect that expectation can that values be a split canonical analysis in this sector wou divergences using only localworks counterterms in is particular a examples. conjecturebulk a A Lagrangians and full to and all general orders proof is that lacking. t of such a calculation. Though our method determines the asym case, the on-shell action remains finite when we turn on non-n these divergences and it wouldthe be source interesting for to the explore massive thi vector field is irrelevant when that appear depend explicitly ondence is the only radial implicit cut-off, through whereas thefound dependence in of [ the fields on in that paper the counterterms are local functionals of the b that non-local counterterms arethe needed on-shell at action finite, sufficiently as high pointed o out in [ class of non-relativistic scale-invariant theories,question and directly i in field theory. terterms is a special feature of field theories with a Lifshit in a curved backgroundinvolve for terms the like Lifshitz case, which from a pre lutions, applications tospacetimes, correlation etc, functions, and we the hope exten to turn back to some of these in du relevant computations turn out to be extremely tedious and w JHEP01(2012)058 = , e.g. ab write (A.5) (A.4) (A.3) (A.7) (A.2) µ (A.1) (A.6) γ ξ n . Let us nto nor- L ab 1 2 γ it should be respectively. ν b p = a µ a N p ab . ) + K ν ν n n , ν µ ) b ely ∇ n ) as follows. µ eful discussions and Dim- search on Matter (FOM), and shift ab en by dx n = + 1)-dimensional vector , such that orm of a parametric relation ace. Thus, the projector is , γ + d N − to the Einstein-Proca action, ly given by V a project. This work is part of µν Research (NWO). a µ g e either intrinsic or extrinsic to ∂ dr , n b a ν − ome ( N,N . K N ∇ a a a , ν )( A µ a . K n a dx n ( 2 p µ a a µ V , dx L p + 2 the surface of constant radial coordinate µ ∇ a N + + = r + ab a + a ∂x , so that we need not worry about possible ∂X F + 2 dr A µ dr K a a ∅ ab µ ab = – 19 – A F r N . Nn µ a = K ( n and p = = µ = ab ab r = ··· µ µ γ = 0. The cotangent basis is spanned by µ 1 Σ K µν µ µ A A µ a ∂ r + T F µ µ p − ξ dX 2 n A 2 µ µν L n dr F = n n K µ a 2 p V . We denote by Σ + N ··· R 2.1 = 1 1 µ a 2 = p is the induced metric or first fundamental form. In order to re ds +1) d µν ( g R ν b can be rewritten in terms of the fields ( p points along the radial flow, which does not necessarily mean ). It is useful to define the projector µ a Similarly, the Maxwell term and the mass term can be split up i a µν µ p g r 6 . r, x = ( ν , we really mean µ n ab n µ a γ X ··· ∇ 1 = a By the Lie derivative of a tangential object with respect to s ν b 6 , we use a projected form of the Gauss-Codazzi equations, nam T p µ r ξ µ a . We assume for simplicity that The extrinsic curvature, or second fundamental form, is giv the gravitational Lagrangian in termsΣ of quantities that ar where p L boundary terms later on.X The foliation can be written in the f where the normal and tangent pieces are given by the lapse r proportional to the unit normal. The flow vector is generical The metric mal and tangential pieces using the completeness relation The vector use the short-hand notation orthogonal to the unit normal, which projects onto the directions tangent to the hypersurf A The Einstein-Proca Hamiltonian In this section we shallwhich compute is the given Hamiltonian in associated section We would like to thankitrios Geoffrey Balt Korres Comp`ere, van for Rees, collaboration forthe us at research a programme very ofwhich early the is stage Foundation part for of of Fundamental this the Re Netherlands Organisation for Scientific Acknowledgments JHEP01(2012)058 . a H  (A.8) (A.9)
(A.18) (A.16) (A.14) (A.13) (A.17) (A.12) (A.15) (A.19) (A.10) (A.11)  ab a π b E . a , via which D  2 2 A D V . − 2 − V dr L m a 2 − L 2 1 R N V 2 . This will be taken into ) , − 2 a . + = a  ) 2 a  b a m E A
). γ E K , a A S H a ) b rmation. Before we do so, − + , 2Λ D a K ab √ a ( N A.16  − 1 2 N 2 A and ab + a R 1 avitational action. m + − and K γ 2 A K V 2 a 2 ab H − ) and ( 1 κ ab grav N A − 2 ( m a ab K γ π , N and the momentum constraint a a 1 2 ) A −  ∂ − b A.14 E K  − dr L 2 ( a H γ  √
+ N 2 2 + b 2 R a m E − π ( 1 κ ab K E 1 2 1 2 . ba , the Hamiltonian constraint function is 1 2 √ b = D a F F V x 1 − b − − − D d ab d a d + 2 N −K F  ab = grav 2Λ + A r 2 − 1 4 – 20 – S F ab + Σ = π − − ab Z a ab 1 + 1)-dimensional space such that the total-divergence term + ab a b π grav A ˙ F γ R ˙ K d a 1 = 7 A − E ab δ NK 1 4  δ E δL δL π d N A ab 8 a A  − γ γ grav F H E = 2 2 = − L γ N 1 1 L − −  2 1 κ a a + ab − ab − 2 √ √ − − γ A N π a √ − r r  γ N ˙ = = ab  x ab A + grav ˙ L L γ d − a N π a ab d H N ab √   E = = π r  x 2 = γ E γ Σ a d ab κ ˙ γ π γ Z d − − ˙ A γ 2 H − − r √ √ − dr Σ √ √ x x Z d d x x = Z d d d d d d 2 dr r r to be the boundary of our ( H 1 r r κ Σ Σ r 2 Σ Σ Z Z Z Z Z ======A A ) precisely cancels against the Gibbons-Hawking term in the gr S H grav grav S We take Σ Note that, strictly speaking, the canonical momenta are A.5 7 8 H After integrating out the non-dynamical field where we introduced the Hamiltonian constraint account in the Legendre transformations to be performed in ( Now, we are ready to perform the Legendre transformation We can also combine the above two Hamiltonians as and similarly for the vector field The actions from before thus become in ( we may obtain the Hamiltonianshowever, by we means must of define a our Legendre generalized transfo velocities first. Let us define the Lagrangians such that Finally, the canonical momenta are JHEP01(2012)058 , (B.4) (B.3) (B.2) (B.1) (B.7) (B.8) (B.6) (B.5)  ) (A.20) (A.21) d cal mo- to total A c D d , so let us go A 2 c A ( ... ab + γ , R + r ) . . c α a  lly solve the local part of A b A ) , b a . b gher-derivative terms in the A b = 0, as it will not be more terms will suffice. We assume a A red in section A D + Φ( E a a 2 b D b ( φ es (thus finding the local higher- b D m A A D b A a c 1 2 a . A A . D A − a b − a 2 ab α, ) A − A A 2 F ) ab − A a b 2 α · ab ) m F A ( E F ) A ab D D − ab 1 4 · α ( F ( F a + 4 1 − D D 2Λ A ... ( a + b – 21 –  − − R + A +
A ab a a a = = π , R b D A ) ) A 2Λ) ) a α − D D  (0) (2) α α In the following we are interested in deformations that ( − ′ 2 D ( L L  ) C C U ′ − R 2 α ( ( = Φ( = = = 2 C 1 κ = a 2 (2) loc loc (1) (0) loc D − C a and the massive vector H ) + L L L 2 1 = (1) α ab (  γ E L U = = At level one we have only two possible structures, The canoni ab (1) loc γ loc S L γ , so that we need to specify the possible counterterms only up ∅ 1 − √ = = is the Lagrangian density restricted to the hypersurface Σ r Σ ab L ∂ (1) π and derivatives. We perform a derivative expansion, Of course, there areAnsatz, other but for two-derivative our terms purposethat as of well illustrating as our method hi these and menta are given by directly to the level of one spacetimeOne derivative. derivative. The momentum constraint function is given by The non-derivative level (level zero) has already been cove illuminating in this specific discussion. Local on-shell action Ansatz. involve only the metric B Higher-derivative counterterms In this section, wethe Hamiltonian briefly constraint mention at the howderivative level one counterterms). of could higher derivativ systematica We shall put the scalar field Here, JHEP01(2012)058 . (B.9) = 0 (B.16) (B.18) (B.11) (B.17) (B.15) (B.13) (B.14) (B.10) (B.12) n. The or a not ... + R  + 1 ′ U ′ )Φ 4 A r order coefficients. lution is a polynomial, we ..., , , + ..., ) al part of the on-shell action, 2 2 a A + range that we were able to de- A A neral solution of the differential ... Φ + 2 C . = ) ( ab ) + ). The function Φ satisfies a first- 0 a z α ). R ) α term does not mix with the other β D α ) + (4 term. Only terms with ... R ′ ab − ( − z B.15 R = φ + U β R α (Φ δγ R 2 + b ( b a δ Φ . 2 ′ A . − A b a ) where − C 2 + 1 z − a z 0 RA ′ D A U α − b ′ = ′ (2) loc ) + = 2 0 z Φ A ... 0 L − – 22 – 5 a (Φ b α + 1)( D 2 ab − + α = 2Φ A Since the Φ z ′ γ − A a a R C 2 1 2 1 4( α Φ ( (2) RA 1 = + ′ ab = + 2 E b ). We obtain g term in the final expression: 1 a ab U 2 1 b + A term. R Φ 0 (2) a yields = 2Φ = 1 4 b B.15 π D 1 R a − is just a total derivative, b C (2) ab Φ uniquely, which amounts to specifying the initial conditio  (2) π Φ = 0 loc (1) E R b L = (2) can produce a µν } − L R (2) loc ,S (0) loc Let us briefly discuss an important feature of ( S { 2 There does not seem to be a continuous ambiguity for the highe and we plug this result into ( reason for this is that,we since are we want using to a compute power-series theequation expansion. polynomi might Nevertheless, not the ge be polynomial, so by requiring that our so order differential equation, thereforetermine it the seems coefficient somewhat st A similar computation for Again, we expand Φ in power series in ( The Hamilton constraint can be solved if Therefore we have The resulting term in and can be discarded. Two derivatives: the contracted two-derivative terms, we can consistently solve for We now want to compute the coefficient of the JHEP01(2012)058 ed: s (B.19) (B.21) (B.20) ommons , (2011) 013 , gauged 05 , = 2 . ]. N supergravity JHEP , ]. is undetermined. This = 11 SPIRE A rterms. IN t ]. d by using the power-series mmercial use, distribution, D SPIRE Φ + 1 = 0 r(s) and source are credited. ][ ]. ] IN ].  Lifshitz solutions in supergravity ][ and , U SPIRE 1 4 trate this phenomenon with a toy . IN SPIRE SPIRE = 0. Nevertheless, if the coefficient SPIRE − ][ , then the solution is not polynomial . This feature is very general and it = 10 IN IN ′ IN A ][ ][ , D ]. ][ 2.3 a ,... αU 2 − 1 2 ) + 1 = 0 −  , x Ax SPIRE ( arXiv:0712.1136 1 [ + Non-Relativistic solutions of IN + ′ − af arXiv:1009.3445 , – 23 – ][ Φ [ 1 a Lifshitz spacetimes from AdS null and cosmological  − ) + 6= 0 Gravity duals of Lifshitz-like fixed points x arXiv:1102.5740 , and stability required that this coefficient vanishes [ a ( αU ′ α arXiv:0808.1725 arXiv:1008.2828 1 2 r arXiv:1005.3291 [ [ (2008) 104 [ Electron stars for holographic metallic criticality ∂ xf Lifshitz solutions of + On existence of self-tuning solutions in static braneworld − ]. (2010) 047 ′ 02 Constructing Lifshitz solutions from AdS U ) 12 α (2011) 041 SPIRE arXiv:1008.2062 [ JHEP IN (2010) 014 + 4 08 , 2 ][ JHEP is simply (2008) 106005 (2011) 046003 α 08 , ′ ) can be cast in a form similar to the toy model we just consider ( This article is distributed under the terms of the Creative C  JHEP , D 78 D 83 B.15 (2010) 002 JHEP , 12 , as we explained at the end of section 0 is an arbitrary constant. If α A arXiv:1102.5344 → supergravity and string theory [ solutions JHEP Phys. Rev. without singularities Phys. Rev. Equation ( 6= 0 this has the following general solution: α a is a negative integer, the solution is indeed a polynomial bu [7] D. Cassani and A.F. Faedo, [8] N. Halmagyi, M. Petrini and A. Zaffaroni, [4] K. Balasubramanian and K. Narayan, [5] A. Donos and J.P. Gauntlett, [6] R. Gregory, S.L. Parameswaran, G. Tasinato and I. Zavala [2] P. Koroteev and M. Libanov, [3] S.A. Hartnoll and A. Tavanfar, [1] S. Kachru, X. Liu and M. Mulligan, If explains why the HJ method is ableOpen to fix Access. the derivative counte as Attribution Noncommercial License whichand permits reproduction any in any nonco medium, provided the original autho References The coefficient of Φ example. Consider the differential equation: are effectively determining the initial condition. We illus where method. amounts precisely to a continuous ambiguity that we would fin and using a Taylor expansiona amounts to choosing JHEP01(2012)058 , , ]. ]. , , , SPIRE (1998) 023 , IN SPIRE n ][ (2011) 004 IN 07 multi-trace ]. , ][ 07 (2001) 339 ]. ]. ]. , 49 JHEP SPIRE s , ]. IN ]. JHEP ]. ][ , SPIRE SPIRE SPIRE IN IN IN SPIRE [ effects in non-relativistic ][ ][ SPIRE hep-th/0205061 IN [ SPIRE IN Holography for Schr¨odinger ][ IN /N ][ arXiv:0812.2909 [ ][ Fortsch. Phys. ]. ]. ]. ]. , ]. ]. 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