Holographic self-tuning of the cosmological constant

Journal of High Energy Physics, Sep 2017

We propose a brane-world setup based on gauge/gravity duality in which the four-dimensional cosmological constant is set to zero by a dynamical self-adjustment mechanism. The bulk contains Einstein gravity and a scalar field. We study holographic RG flow solutions, with the standard model brane separating an infinite volume UV region and an IR region of finite volume. For generic values of the brane vacuum energy, regular solutions exist such that the four-dimensional brane is flat. Its position in the bulk is determined dynamically by the junction conditions. Analysis of linear fluctuations shows that a regime of 4-dimensional gravity is possible at large distances, due to the presence of an induced gravity term. The graviton acquires an effective mass, and a five-dimensional regime may exist at large and/or small scales. We show that, for a broad choice of potentials, flat-brane solutions are manifestly stable and free of ghosts. We compute the scalar contribution to the force between brane-localized sources and show that, in certain models, the vDVZ discontinuity is absent and the effective interaction at short distances is mediated by two transverse graviton helicities.

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Holographic self-tuning of the cosmological constant

Received: May Holographic self-tuning of the cosmological constant Christos Charmousis 0 2 4 7 9 Elias Kiritsis 0 2 4 5 6 9 Francesco Nitti 0 2 4 5 9 0 Department of Physics, University of Crete 71003 Heraklion , Greece 1 Institute for Theoretical and Computational Physics 2 Sorbonne Paris Cite , B 3 CNRS/IN2P3, CEA/IRFU , Obs. de Paris 4 Universite Paris-Saclay , 91405 Orsay , France 5 APC, Universite Paris 7 6 Crete Center for Theoretical Physics 7 Laboratoire de Physique Theorique , CNRS, Univ. Paris-Sud 8 atiment Condorcet , F-75205, Paris Cedex 13 , France 9 We compute the We propose a brane-world setup based on gauge/gravity duality in which the four-dimensional cosmological constant is set to zero by a dynamical self-adjustment mechanism. The bulk contains Einstein gravity and a scalar eld. We study holographic RG ow solutions, with the standard model brane separating an in nite volume UV region and an IR region of nite volume. For generic values of the brane vacuum energy, regular solutions exist such that the four-dimensional brane is determined dynamically by the junction conditions. Analysis of linear uctuations shows that a regime of 4-dimensional gravity is possible at large distances, due to the presence of an induced gravity term. The graviton acquires an e ective mass, and a ve-dimensional regime may exist at large and/or small scales. We show that, for a broad choice of poat-brane solutions are manifestly stable and free of ghosts. tentials 1 Introduction and summary 2 The self-tuning theory Emergent gravity and the brane-world Results and outlook Field equations and matching conditions The Poincare-invariant ansatz Holographic interlude 2.3.1 2.3.2 UV region IR region The self-adjustment mechanism Consistent self-tuning solutions Concrete examples 2.6.1 2.6.2 Case study I: an IR-regular model Case study II: a class of stable self-tuning models 3 Linear perturbations around at solutions Bulk perturbations Brane perturbations and rst junction condition Gauge xing and second junction condition 1.1 1.2 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 4.1 4.2 5.1 5.2 5.3 5.4 5.5 1 B The holographic parameters and the integration constants C Avoiding Weinberg's no-go theorem { i { D Linearized bulk equations and matching conditions D.1 Perturbed bulk equations D.2 Brane perturbations and linearized junction conditions D.3 Tensor junction conditions D.4 Scalar junction conditions D.5 Gauge-invariant action for scalar modes E The bulk propagator for tensor modes E.1 Large-p behavior E.2 Perturbation expansion for small-p E.3 Regularity of the small-p expansion 68 70 eld theories for low-energy interactions are a general framework addressing observable physics from particle physics to cosmology. While typically successful, they have so far failed to address the cosmological constant problem, [1] (see also [2, 3] and [4] for an updated review and references within). Indeed our main dynamical theory underlying cosmology, General Relativity (GR), and those of particle physics, namely quantum eld theories in at space-time, seem to be incompatible when it comes to vacuum energy. Experiments (such as the Lamb shift [5] or the Casimir e ect [6]) indicate that any particle will give zero-point energy contributions to the vacuum energy, [7]. These contributions scale with the fourth power of the cut-o , which can be as high as the Planck scale, the generically assumed UV cut-o of any QFT. On the other hand, vacuum energy couples to gravity as an e ective cosmological constant, which by Einstein's equations gives rise to a non-zero space-time curvature. If we assume the existence of supersymmetry broken at some scale SUSY , then the cosmological constant is expected to be of order O( 4SUSY ). Experiment states that such a scale must be quite larger than a TeV and therefore supersymmetry cannot solve the cosmological constant conundrum. For illustration purposes, we may simply consider the contributions to zero point energy due to the electron: this provides a contribution to the vacuum energy of order O(me4). According to the principle of equivalence for GR any form of energy gravitates. Due to covariance, the vacuum energy gravitates as a cosmological constant. Gravitationally, a positive cosmological constant will seed a de Sitter space-time with a nite distance (curvature) scale, the de Sitter horizon scale1 (in the static frame). This scale is inversely proportional to the square root of the cosmological constant. Putting in the numbers for the vacuum energy due to the electron, would tell us that the size of our Universe is comparable to the earth-moon distance, as Pauli was amused to note back 1A negative cosmological constant instead would give nite life-time for the universe. { 1 { in 1920 (see references within [2]). Needless to say that the Universe will become a lot smaller if we allow for heavier particles and higher UV scales or phase transitions in the Universe (which will also provide a cosmological constant due to the energies of the broken symmetry phase). The experimental prediction, measured via gravity and cosmology, is the observed size of the accelerating universe, which gives a di erent answer. Given the present size of our observed universe, the observed vacuum energy is of order O((10 3eV )4). We are allowed to change/renormalize the value of the cosmological constant by a bare gravitational cosmological constant (cc) which can be added to the GR action. For this to work, our bare cc must be such that it exactly switches o QFT contributions to a renormalized value, the observed value of the cc. This involves an enormous ne-tuning which is the ( rst) cosmological constant problem in its \classical" formulation. This ne-tuning has to be done throughout the later history of the universe, for each time the vacuum energy appears, a bare value should be there to switch it o almost exactly. To this embarrassing ne-tuning between theory and experiment one has to add the second problem of radiative instability of the vacuum: the cosmological constant will receive higher loop corrections to each order spoiling the ne tuning undertaken for the rst problem. In many respects, this is a harder problem-one which has to be solved not only in the gravitational but also in the QFT sector (for recent progress see the sequestering proposals by [8{10]). The cosmological constant problem may be also pointing to a shortcoming of GR and there has been some e ort to approach the problem from the viewpoint of modi ed gravity theories [15] in four space-time dimensions. One idea which has been proposed is to introduce in the gravity sector some new degree of freedom, usually a scalar eld, which can absorb vacuum energy contributions throughout the later evolution of the universe leaving space-time curvature unchanged. Any mechanism by which the cosmological constant is adjusted dynamically by some extra degree of freedom is what is generally referred to as self-tuning (or self-adjustment ) of the cosmological constant. More generically, we will refer to a model as self-tuning if at four-dimensional space time is a solution to the gravitational eld equations for generic values of the vacuum energy.2 Most recently, the idea of self-tuning has been formulated in a subset of four-dimensional scalar-tensor theories [13, 14]. In this setup, named Fab Four, the scalar eld can eat up any cosmological constant without xing any of the parameters of the theory whilst spacetime curvature remains at. For a cosmological setup for example, the scalar eld is time dependent for a locally at Milne space-time. The presence of integration constants allows zero curvature solutions whatever the value of the cosmological constant and without any xing of coupling constants of the theory. In other words, the cosmological constant is xed by the scalar eld solution and not the theory, thereby realizing the self-tuning idea or self-adjustment mechanism. 2For some ideas associated to a quasi-spontaneous breaking of conformal invariance see [11, 12]. { 2 { In this work we propose a framework which implements the self-tuning mechanism using the brane-world idea, i.e. higher dimensions [15{19] and also, crucially, holography. In the brane-world scenario [20{23], our four-dimensional universe (brane) is embedded in a higher dimensional bulk. Ordinary matter and gauge elds are constraint to propagate only on the brane, but gravity propagates in all dimensions and the brane interacts gravitationally with the higher-dimensional degrees of freedom. The extra dimensions may remain undetectable from present day experiments, if for example their size is su ciently small or the bulk is su ciently curved. In the original brane-world model of Randall and Sundrum (RS) [23] with a single brane, the latter was embedded in a ve-dimensional anti de Sitter space-time. In order to have a at brane solution to the eld equations, a ne tuning was necessary between the brane tension, interpreted as world-volume vacuum energy, and the (negative) bulk cosmological constant. This was the brane-world version of the cosmological constant problem- at solutions are not generic in the presence of vacuum energy. They are on the contrary very nely tuned. Brane-worlds were generalized in various directions and such generalizations are reviewed in [15{19, 24{26]. It was a natural step to try and implement a brane-world version of the self-tuning mechanism: the idea was that the brane vacuum energy due to matter may curve the bulk, but leave the four-dimensional brane (our universe) at. It was initially noted [27, 28] (see also [29, 30]) that a non-trivial bulk scalar eld could indeed relax the ne-tuning for the cosmological constant on the brane in a 5 dimensional brane-world setup. This idea was also implemented in 6 dimensional space-times, or co-dimension 2 [31{37], for more generic gravity theories [38{40] or both [41]. Indeed the dynamical nature of the scalar introduced integration constant(s) that did allow for a at brane solution without ne tuning of the brane tension. The 5 dimensional self-tuning solutions though had an important shortcoming: they had a naked singularity in the bulk space-time at a nite distance from the brane [27{30]. When this did not happen, the gravitational interaction on the brane was not four-dimensional, [30]. Various other related setups were analyzed, leading eventually to instabilities or hidden ne tunings [31{40]. It was also realized that various brane-words in AdS space-time have a holographic interpretation, [42{45]. This opened a new perspective on the relevant physics as it is mapped into QFT dynamics. The holographic correspondence provides a nontrivial map between gravity/string theory dynamics in the bulk and QFT dynamics at the boundary. Moreover it can be considered as a UV-complete de nition of quantum gravity, [56]. The study of holography for 20 years has revealed many novel features of QFT especially in the strongly-coupled regime, as well as novel features of gravity and its connection to QFT thermodynamics and hydrodynamics. The rules of the game have been understood in many more contexts than the original N=4 sYM theory example and numerous successful checks have been done. When it comes especially to cosmology, holography suggests several intriguing dual views encapsulated in the several versions of the de Sitter/(p)CFT correspondence, [46{50] which has been also extended to general cosmological ows, [51]. These look di erent from { 3 { the brane-world cosmology that is driven by moving branes3 and rolling vevs, [52, 55] , but they may have a deeper connection. In the rest of the introduction we describe the structure and holographic motivation for the brane-world self-tuning setup we study, and we present a summary of our results. Emergent gravity and the brane-world An important realization of the self-tuning setup is suggested by the holographic ideas on emergent gravity. This is a setup where the interactions of the Standard Model (SM), including gravity, are generated by 4d conventional QFTs. For this to work, we need in the simplest setup three ingredients, [56] The gauge theories and other interactions of the SM. A large-N, strongly coupled and stable 4d QFTN that will generate the gravitational sector (this may be a non-abelian gauge theory where N is the number of colours). A theory of bifundamental \messengers" that will couple the QFTN to the SM by renormalizable interactions. Therefore the messengers must be charged under both gauge group of the SM as well as the QFTN gauge group. They must have large masses, of order (the UV cuto scale). At energy scales E the messengers can be integrated out and the SM is directly coupled to the operators of QFTN . These operators involve the universal conserved stress tensor of QFTN as well as many other operators. An appropriate linear combination of the stress tensors becomes the universal metric that will couple to the SM elds, and di eomorphism invariance will be an emergent feature. This is were gauge/gravity duality comes into play. Rather than using the four-dimensional description above, we will now assume the existence of a holographically dual version of the strongly coupled QFTN in terms of classical gravity and other interactions in a 5-dimensional bulk space with a (UV) nearAdS boundary. In this language, four-dimensional di eomorphism invariance is manifest and is a consequence of the overall energy conservation. The SM is weakly coupled at E = and therefore its coupling to QFTN follows the semi-holographic setup: it can be represented by a 4-brane embedded in the bulk geometry at the position corresponding to the cuto scale induced by the messenger mass. Therefore, in the gravitational description the setup is that of a SM-brane embedded in the QFTN bulk gravitational theory. The bulk elds of the gravitational sector couple to the SM elds on the brane. An important ingredient of this coupling is the induced action for the bulk elds on the brane. This is generated by the SM quantum e ects that will induce a non-trivial action for the bulk elds. Since the SM elds are localized on the brane, the same applies to this induced gravitational action. In general, bulk operators that are not protected by symmetries will obtain brane potentials that will scale as the fourth power of the cuto scale . For the bulk operators 3The (equivalent) time dependent brane world perspective was undertaken in [53] and the connection to [52] was explained via Birkho ' s theorem in the presence of branes in [54]. { 4 { that are protected by symmetries, like the graviton, possible conserved currents (giving rise to graviphotons) and the universal instanton density (giving rise to the universal axion) the corrections start at two derivatives, and scale as The framework of emergent gravity from 4d QFTs described above therefore can be modeled in the gravitational picture with a 4-d SM brane embedded in the bulk space-time generated by the QFTN . In this paper we will simplify this e ective description by keeping track of two basic bulk elds: the metric as well as a single scalar. With this eld content the action we will consider, up to two derivatives reads, where S = Sbulk + Sbrane; 1 2 WB(') ZB(') (1.1) (1.2) (1.3) where gab is the bulk metric, R is its associated Ricci scalar and the induced metric and intrinsic curvature of the brane. We have kept the dimension d above general although our main concern is for d = 4. We expect Mpd 1 N 2. The above action is the most general two-derivative action in Einstein-scalar theory which preserves the full group of bulk di eomorphisms (including those transverse to the brane, since the latter is allowed to uctuate). All we assume initially for the bulk potential is that it has a maximum supporting a (stable) AdS solution. We will be interested in (fully backreacted) solutions in which the scalar eld evolves in the bulk radial direction (transverse to the brane), interpolating between an in nite volume asymptotic AdS boundary region where ' approaches the maximum of V ('), and a region with asymptotically vanishing volume element, with the brane separating the two. In the dual eld theory language, the scalar corresponds to a relevant operator of the QFTN , and the solution to a renormalization group (RG) ow driven by this operator. The large and small volume regions correspond respectively to the UV and IR of the RG ow. This structure is represented in gure 1. Although the overall volume of the bulk is in nite, our model allows regimes in which gravity behaves as four-dimensional, as an e ect of the localized Ricci scalar term on the brane in equation (1.3). This gives rise to a quasi-localized graviton resonance as in the DGP model [57]. As we will eventually conclude, in this framework, and with the insights from the holographic perspective, it is possible to avoid all the drawbacks of previous brane-world self-tuning constructions. Holography provides an important guideline in organizing the space of solutions. Furthermore, the IR endpoint of the RG- ow can be singularity-free if the scalar eld approaches another AdS extremum (in this case a minimum of the potential). Moreover, some mild singularities are acceptable because they can be resolved. Furthermore, the holographic interpretation naturally requires an in nite volume region in the UV. As we will see, this is crucial for the self-tuning mechanism: any solution which has nite volume on both sides of the brane must necessarily be ne-tuned. { 5 { u between an in nite volume UV region and a nite volume IR region. The coordinates x are world-volume four-dimensional coordinates on the brane-world, u represents the holographic radial direction. In general, self-tuning models are severely constrained by Weinberg's no-go theorem [1], which essentially states that in any local theory with dynamical gravity, preserving local Poincare invariance, and whose solutions are determined by a local action principle, selftuning cannot work.4 The framework we present here avoids this theorem, and this has a clear interpretation in view of holography: each solution contains quantities (the \vevs" of the dual operators) which are not determined by extremizing a local action, but rather by a regularity condition which relates the UV and the IR, and has no classical analog in local eld theories. Given this input from holography it is now instructive to check our action ingredients from the brane world perspective. For a start, our brane will be an asymmetric one, separating an in nite volume UV region and a nite volume IR region. Asymmetric selftuning models were studied early on by [30], but these did not include an induced gravity term. Secondly, given that the overall volume of our brane model is in nite, there will not be a localized zero mode graviton uctuation on the brane (as it is the case instead for the classic RS model [23]). This is in turn where the induced gravity term plays an important positive role for the phenomenology of our model providing a quasi-localized graviton zero mode in the tensor uctuations, by the same mechanism well-known in a at bulk [57]. The particular role of asymmetry [58, 59], combined with the induced gravity term were realized in [60] where dark energy models were constructed (but this time, without 4In the case of Fab Four, the scalar breaks Poincare invariance thus evading Weinberg's no-go theorem [2]. { 6 { a bulk scalar). In the latter paper it was also realized, however, that the positive role played by the induced gravity term in the tensor uctuations, was negative5 in the scalar uctuations: there, it was found that the induced gravity term contributed to a scalar ghost whenever a spin 2 zero mode was not present in the spectrum. This was because without a bulk scalar eld, dynamical scalar uctuations only existed on the brane, but not in the bulk. This is where the bulk scalar in our model plays an essential role: it also contributes in the scalar sector allowing, as we will see, for the absence of scalar (but also tensor) uctuation pathologies. In this setup we consider solutions to the classical equations of motion for g and ' that correspond to Lorentz-invariant saddle points of the dual QFTN , as described by the action Sbulk. The presence of the SM brane in the geometry is taken into account by the Israel matching conditions. Our goal in this paper is to rst examine the existence of solutions to the bulk equations which are holographically acceptable (either with regular bulk geometries or with good IR singularities) having a at induced metric on the brane. This is the essence of the selftuning mechanism: although there is a non-trivial vacuum energy (or cosmological term) on the brane, the metric of the brane universe is at. We nd that holographically acceptable solutions generically exist. In these solutions, the brane is placed at a speci c equilibrium position '0 in the bulk, which is determined dynamically by solving Israel's junction conditions. We show that one can generically nd an acceptable equilibrium solution in the vicinity of a zero of WB, for a generic bulk potential V ('). Thus, the existence of self-tuning solutions is generic in this framework. The next question we investigate is: is this equilibrium position stable? More specifically, are the uctuations around this solution regular (not ghost-like) and stable (not tachyonic)? Connected to this question is also the following: what kind of interactions such uctuations mediate on the brane world? Is gravity similar to observable gravity? Is the equivalence principle upheld? We derive the uctuation equations around the equilibrium brane position, for general bulk and brane potentials. There are two sets of propagating modes. One is a spin-two mode associated to the 5d graviton. We nd that the equations it satis es are similar to the DGP scenario, [57] with the important di erence that in our case the bulk geometry is non-trivial. We calculate the propagator that controls the interaction of sources on the SM-brane. This propagator is DGP-like at short enough distances but is a massive propagator at long distances6 The reason for this is the behavior of the bulk to bulk propagator on the brane. At short enough distances it vanishes, with the same behavior as in at space. But at long enough distances it asymptotes to a constant that is determined by the bulk geometry. It is this di erent behavior that is responsible for the mass at long distances. 5literally opposite in sign! 6This behavior was seen before in a DGP framework with a codimension higher than 1 thick brane, [61, 62]. { 7 { The framework presented here has a rich gravitational phenomenology, displaying several di erent potential signals of long- and short-distance modi ed gravity. The graviton propagation is four-dimensional at both short and long distances, and also has a mass. Depending on parameters, a ve-dimensional phase may appear at intermediate distances. The e ective four-dimensional gravitational coupling constant is controlled by the induced Einstein term on the brane, and the mass of the graviton is controlled by the same quantity and by the geometry around the equilibrium position. We lay out the conditions for constructing speci c models in which the modi ed gravity regime falls outside the scales probed by current observations. This includes having an arbitrarily light graviton in a technically natural way. The analysis of the scalar uctuations is more involved.7 There is a single gauge invariant scalar uctuation in the bulk, but two invariant ones on the brane. We derive the dynamics of the scalar uctuations and we formulate it as matrix Sturm-Liouville problem. This formulation enables us to derive su cient conditions for the uctuations to be manifestly regular (not-ghostlike) and stable (not tachyons). We also construct the brane-to-brane scalar propagator, which takes the form of a matrix coupling two kinds of sources: the trace of the stress tensor, and the scalar \charge". We do not address here a full discussion of the phenomenology of the scalar sector. This is an important aspect, because it leads to constraints from fth-force and violation of the equivalence principle. Moreover, it is important to investigate how the non-linearieties of the theory modify the gravitational e ects beyond one-graviton exchange, as these can lead to stringent constraints on scalar-tensor theories (as was analyzed by [66] in the context of \Fab Four"-like theories). However neither the linearized scalar-mediated interaction nor the non-linear e ects are universal, and they can manifest themselves at di erent scales in a model-dependent way. Thus this discussion must be carried out in speci c models and is beyond the scope of the present paper. Our results are encouraging but constitute only the tip of the iceberg. There are several further tasks that must be accomplished before this setup is physically acceptable. A detailed analysis on the dependence of the observable parameters (4d Planck scale, mass of the graviton) from the inputs (nature of bulk QFT, UV couplings and the induced brane cosmological constant) must be made in order to assess which ingredients provide a physical answer. The massive graviton has, generically, a vDVZ discontinuity, [64, 65]. Finding the associated Vainshtein scale, [67, 68], is important in order to understand the viability of the setup. It is important to note that the theory of the massive graviton is an e ective theory near the equilibrium position and for this reason is not subject to the standard constraints on massive graviton theories. Such constraints are stringent if the theory only a contains a massive 4d graviton and no other gravitational degrees of freedom, [69]. On the other hand, consistent theories containing massive gravitons like KK theory and string theory/holography have appropriate couplings to avoid 7For an earlier discussion of scalar uctuations in brane-worlds with a bulk scalar, see [63]. { 8 { such direct constraints, [48, 70]. Similar considerations have also been addressed in scalar-tensor theories of the \Fab Four" type in [66]. In that work it was shown that, requiring non-linearities to screen extra scalar modes around spherically symmetric solutions, together with the validity of e ective eld theory at the observed scales, puts non-trivial constraints. In the present context, to answer the same questions one would have to analyze solutions with spherically symmetric brane sources, and investigate how the non-linear scale interplays with the other bulk and brane scales, and it is not easy to \guess" whether the constraints will invalidate the framework. This is an important but complex study, and will be left for a future work. It is interesting that this setup always provides for a massive graviton on the brane. It has been observed that the cosmological evolution driven by a massive graviton is similar to an e ective cosmological constant MP2 m20MP2 , [17, 18] which is the right size to explain the observable cosmological constant. Whether there is a connection between these two observations remains to be seen by analyzing the full cosmology of the theory. Although the conditions for \healthy" scalar uctuations have been derived, more details need to be known about the forces mediated by the scalar excitations. The fact that there are two possible scalar excitations on the brane indicate that there are generically two charges associated to the scalar interaction. The nature of the scalar force, its range and its couplings to observable matter must be elucidated, as a function of the inputs: the localized action and the bulk dynamics. The existence of a at 4d-space-time solution which accommodates a large brane vacuum energy while allowing for reasonable gravitational interactions, does not fully solve the problem. One should investigate how one arrives at such a solution dynamically. For this one rst needs to investigate alternative solutions with maximal symmetry but where the induced brane-world metric is positively or negatively curved. The nal step is to derive the full time-dependent evolution equations for the system brane+bulk. The issue of radiative corrections to the framework we discuss in this paper is important. The bulk gravitational theory has both higher derivative corrections (that are controllable at strong coupling according to AdS/CFT intuition) and loop corrections that are controllable at large N . The induced brane action for the bulk elds is expected to be generated by brane- eld quantum e ects and all such e ects are assumed to be included in the brane potential two-derivative terms. There can be higher derivative corrections that we have neglected here. They will provide corrections to the matching conditions that are not IR relevant. In the worst case scenario they can a ect the scale m4 that controls the onset of massive brane gravity. The full time-dependent dynamics of the system must be derived and analyzed. This is tantamount to analysing the cosmological evolution of the setup. In particular this is important in order to verify the naive expectation where the brane starts at { 9 { HJEP09(217)3 early times in a bulk position near the boundary and far away from the \equilibrium" position '0. The ensuing evolution towards this equilibrium position can be mostly driven by the brane cosmological constant giving therefore a period of brane in ation. Approaching '0 the e ective cosmological constant becomes smaller and smaller and the brane evolution is driven more and more by the energy densities on the brane. These expectations are reasonable and should be veri ed. An interesting open problem is to assess what can act as dark energy in this setup. Several possibilities can be investigated already within this framework, due to the presence of scalar modes (including the brane position) which may act as quintessence or leave a residual cosmological constant if the brane is slightly displaced from its equilibrium position. It is important to stress that the brane cosmological constant is not a xed potential of the bulk elds but also depends in general on brane- eld order parameters (examples for the SM are the Higgs eld or chiral symmetry condensates). This intertwines interestingly with the self-tuning mechanism and in principle allows both an accommodation of phase transitions into the relaxation mechanism but also the possibility that the solutions to the CC Problem and the Electroweak hierarchy problem are intimately connected. The fact that gravity is generically 5 dimensional o the brane world indicates that there may be a period in the evolution of the (brane) universe where there is an exchange of energy between the SM-brane and the bulk, [62]. Such an e ect can a ect the cosmology on the brane.8 This paper is structured as follows. Section 2 presents the model, the vacuum solutions, and the self-tuning mechanism arising from Israel's junction conditions. We give a review of the geometry of holographic RG ows and what makes for holographically acceptable singularities. We show that selftuning junctions are generically present for a wide variety of brane and bulk potentials, and we give concrete examples with and without an IR singularity. In section 3 we lay the ground for the analysis of linear perturbations around vacuum solutions, and identify the relevant bulk and brane perturbations, as well as their gauge transformations. After xing the gauge we derive the linearized bulk eld equations and junction conditions for physical scalar and tensor perturbations. Section 4 is dedicated to the analysis of tensor modes, and in particular to the calculation of the tensor-mediated interaction mediated between sources localized on the brane. We compute the tensor brane-to-brane propagator and discuss its di erent regimes, and discuss the associated phenomenology at di erent scales. In section 5 we analyze scalar perturbations. We write the gauge- xed linearized junction conditions in terms of a single scalar perturbation, and show that the bulk equation plus junction conditions can be written in terms of a vector-valued Sturm-Liouville problem with Robin boundary conditions. We discuss the stability of the background solutions and 8This phenomenon has been investigated in phenomenological brane setups in [45, 80, 81]. consider reads, where, 1 2 g R V (') + SGH ; WB(') ZB(') ; (2.3) give su cient conditions for the absence of ghosts/tachyons. We compute the scalar braneto-brane propagator which enter the scalar-mediated interaction between brane-localize sources, and we speculate on a class of models free of the vDVZ problem. Several technical details are left to the appendix. In appendix A we give a classi cation of the di erent possible types of junctions; appendix B relates the boundary values of the elds at the brane with the asymptotic behavior in the UV, in particular the UV coupling for the relevant operator driving the ow in the dual eld theory. In appendix C we review Weinberg's no-go theorem and describe how it is avoided in our framework. Appendix D contains the technical details of the linear perturbations around the vacuum. Finally, in appendix E we give details about the large- and small- momentum asymptotics of the bulk Green's function, which is one of the ingredients entering in the brane-to-brane propagator. 2 The self-tuning theory We consider a scalar-tensor Einstein theory in a d + 1-dimensional bulk space-time parametrized by coordinates x a (u; x ). We consider a d-dimensional brane embedded in the bulk parametrized by coordinates x . The most general 2-derivative action to (2.1) (2.2) 4 (2.4) (2.5) where gab is the bulk metric, R is its associated Ricci scalar and the induced metric and intrinsic curvature of the brane while V (') is some bulk scalar potential. SGH is the Gibbons-Hawking term at the space-time boundary (e.g. the UV boundary if the bulk is asymptotically AdS). The ellipsis in the brane action involves higher derivative terms of the gravitational sector elds ( ; ) as well as the action of the brane-localized elds (the \Standard Model" (SM), in the case of interest to us). WB('); ZB(') and UB(') are scalar potentials which are generated by the quantum corrections of the brane-localized elds (that couple to the bulk elds, see [56]). As such, they are localized on the brane. In particular, WB(') contains the brane vacuum energy, which takes contributions from the brane matter elds. All of WB('); ZB(') and UB(') are cuto dependent and generically, WB(') is the UV cuto of the brane physics as described here. Its ZB(') UB(') 2 where origin was motivated in subsection 1.1. 2.1 Field equations and matching conditions The bulk eld equations depend only on V (') and are given by: Rab 1 2 gabR = 1 2 p 2. Discontinuity of the extrinsic curvature and normal derivative of ': h K K iIR UV = p 1 Sbrane ; where K is the extrinsic curvature of the brane, K = unit normal vector to the brane, oriented towards the IR. Sbrane ; ' its trace, and na a Using the form of the brane action, equations (2.7) are given explicitly by: h K K WB(') + UB(')G( ) r ( ) ( ) r r ( ) ( ) UB(') r dUB R( ) + 2 d ; (2.9) '0(x) side of the brane, and by junction conditions are: h iIR X UV The brane, being codimension-1, separates the bulk in two parts, denoted by \U V " (which contains the conformal AdS boundary region or more generally, in non-asymptotically AdS solutions, the region where the volume form becomes in nite ) and \IR" (where the volume form eventually vanishes, and may contain the AdS Poincare horizon, or a (good) singularity, or a black hole horizon etc, as we will discuss in section 2.3.2). We will take the coordinate u to increase towards the IR region. Denoting gaUbV; gaIRb and 'UV; 'IR the solutions for the metric and scalar eld on each the jump of a quantity X across the brane, Israel's 1. Continuity of the metric and scalar eld: h iUV gab IR = 0; h iIR ' UV d du = _ ; p K 1 1 2 ; '0(x) 1 p (2.6) (2.7) (2.8) (2.10) (2.11) where '0(x ) is the scalar eld on the brane. 2.2 The Poincare-invariant ansatz We consider the case where the bulk space-time has d-dimensional Poincare invariance, so that the solution would be dual to the ground state of a Lorentz-Invariant QFT. The brane will be embedded at speci c radial distance u0 so that the induced metric is a at d-dimensional Minkowski metric. In the domain-wall (or Fe erman-Graham) gauge, the metric and scalar eld (on each side of the brane) are: ds2 = du2 + e2A(u) dx dx ; ' = '(u); We use the notation: and we denote by (AUV(u); 'UV(u)) and (AIR(u); 'IR(u)) the bulk solution in the UV and IR regions, respectively. The brane sits at a xed value u0 and we de ne: A0 A(u0); '0 '(u0): (2.12) Only '0 (not u0) is a gauge-invariant quantity.9 The induced metric on the brane is = e2A0 . With the ansatz (2.10), the eld equations (2.4){(2.5) become equivalent to: 1 2 d(d 1)(A_ )2 ('_ )2 = V ('); A = (2.13) These equations can be cast in a rst order form by de ning UV and IR superpotential functions WUV(') and WIR(') [29], such that: A_ UV(u) = A_ IR(u) = 2(d 2(d 1 1 1) 1) WUV('(u)); WIR('(u)); '_ UV(u) = '_ IR(u) = d' d' dWUV ('(u)) ; dWIR ('(u)) : The scalar functions WUV;IR are both solutions to the (gauge-invariant) superpotential equation: d 4(d 1) W 2 + 1 2 dW d' 2 = V: The choice of W determines the geometry on each side, up to the choice of an initial condition (A ; ' ) , which only a ects a shift in A (i.e. an overall choice of scale of the d-dimensional theory). These boundary conditions have a clear interpretation in the boundary QFT dual to the bulk gravitational theory: A sets the scale of the boundary Minkowski metric of the dual QFT while ' determines the UV coupling constant of the scalar operator O(x) dual to '. On the other hand W is invariant under bulk di eomorphisms. The superpotential equation (2.16) implies an inequality jW ( )j B( ) 2 r d 1 d V ( ) which also de nes the function B( ) that acts as a lower bound on the space of solutions of the superpotential equation, [73]. across the brane: The continuity conditions (2.6){(2.7) simply state that A(u) and '(u) are continuous AUV(u0) = AIR(u0) = A0; 'UV(u0) = 'IR(u0) = '0: (2.18) Therefore, only one initial condition (A ; ' ) must be imposed, for example in the UV. The interpretation of these initial conditions is holographically clear and it will be discussed in greater detail in subsection B. 9By gauge invariant we mean invariant under bulk di eomorphisms. (2.14) (2.15) (2.16) (2.17) The non-trivial matching conditions are the ones imposed on the rst derivatives, (2.8){ (2.9). Indeed, the extrinsic curvature and normal derivatives of ' are given by: K = A_ e2A ; K K = (d 1)A_ e2A ; (2.19) The junction conditions can be cast in a gauge-invariant form using the superpotentials W on each side of the brane: making use of the expressions (2.14){(2.15) for A_ and '_ , as well as (2.19), equations (2.8){(2.9) simply become statements about the jump in the superpotential and its derivative across the brane [30]: WIR WUVj'0 = WB('0) dWIR d' dWUV d' '0 = d dWB ('0); (2.20) (2.21) To summarize, the full system of bulk and brane eld equations boils down to two bulk equations, (2.16), relating the two super potentials WIR and WUV to the bulk potential V , and two matching conditions relating the bulk and brane superpotentials together, (2.20), (2.21). Before explaining the logic we will use in picking the relevant solutions to these equations (which will be the subject of section 2.4), we make a digression on holography and the properties and interpretation of the UV and IR parts of the geometry. 2.3 Holographic interlude In a bulk theory allowing for a holographic interpretation, not all bulk geometries are on the same footing. Below we summarize the structure of the \UV" (i.e. large volume) and \IR" (small volume) regions of the bulk geometry, and their interpretation in the holographic dictionary. 2.3.1 UV region First of all, we will consider only UV-complete solutions, i.e. those containing a region which extends all the way to an asymptotically AdS boundary where eA e u=` The presence of such a region on one side of the interface is a crucial ingredient of the self-tuning mechanism: this is because, as we will discuss below in more detail, generic solutions in this region ow to the UV xed point independently of the particular value ! +1. WUV('0) which solves equations (2.20){(2.21). Therefore, we do not need to ne-tune the UV side of the solution. The asymptotic UV region usually corresponds to the scalar eld asymptoting to a maximum of the scalar potential. A given potential may allow for several UV xed points, but one can restrict the boundary conditions of the gravitational theory to pick one of them: indeed, the choice of UV boundary conditions is part of the de nition of the holographic theory. This includes not only the choice of the UV extremum, but also the boundary conditions on scalar elds, the boundary induced metric, etc. Even with these restrictions, there always exist an in nite number of solutions to the superpotential equation in the UV, which satisfy all the correct boundary conditions at leading order in a near-boundary expansion, and di er by subleading terms. In fact, there exists an arbitrary integration constant CUV to the superpotential equation which parametrizes a continuous family of nonequivalent solutions which get closer and closer to each other as one approaches the extremal point of V . For a recent detailed discussion of the solutions to the superpotential equation, see [73]. All these solutions asymptote the same AdS geometry, and they are all regular close to the boundary of AdS. To be more explicit, such a \UV" AdS solution is realized near a maximum of bulk potential (say at ' = 0), The constant term xes the asymptotic AdS length `, and the mass term xes the dimension of the corresponding operator10 by: V (') ' d(d 1) `2 + m2 2 '2 + : : : = d 2 r superpotentials corresponding to (2.23) all have the form, for where the dots indicate analytic higher order terms, and CUV is an undetermined constant. Solving for the scalar eld and scale factor via equations (2.14), one nds: eA(u) ' e u=`; '(u) ' g0 ` eu=` d + (2 CUV` d) 0 g (d ) ` eu=` ; u ! 1; (2.25) where g0 is one more integration constants which, importantly, does not appear in the superpotential. In the equation above, the factors of ` are inserted to absorbe the appropriate mass dimensionality of g0 and CUV (d and one, respectively) while keeping '(u) dimensionless. We now describe how the solution above is interpreted in the holographic dictionary. The scale factor diverges as ' ! 0, signaling an asymptotically AdS region with conformal boundary at u = 1, where the scalar reaches the \UV xed point" Both leading and subleading terms in the scalar eld vanish as we go to the boundary (u ! 1), signaling that the xed point is an attractor. The constant g0 controlling the leading term in the scalar eld near-boundary expansion represents the coupling of the dual operator O(x) associated to ' in the dual 10We assume here what is called \standard" form of the holographic dictionary. For operator dimensions such that d=2 1 < < d=2, there is an \alternative" de nition of the theory which is obtained by replacing $ (d ) in equation (2.23). We will not discuss this case further. (2.22) HJEP09(217)3 (2.23) eld theory in the far UV. In other words the UV CFT is deformed by a term of the form: eld theory). Z Ssource = ddx g0O(x): Notice that g0 does not appear in the superpotential, but rather is generated by the boundary conditions one imposes at the AdS boundary (extreme UV limit of the dual Similarly, one has to x asymptotically the boundary conditions for the leading term in the metric. With a generic ansatz of the form, u ! 1; (0) represents the metric of the space where the UV CFT is de ned , and it is also xed by boundary conditions at the AdS boundary. In particular, in the solution we are considering, the CFT lives in at space-time with Minkowski metric. It is crucial that neither g0 nor (0) are elds with respect to which we have to extremize the gravitational action, but they are part of the de nition of the dual eld theory. The subleading term in the near-boundary expansion of ' in equation (2.25) is controlled by CUV. In the dual eld theory, this term corresponds to the vacuum expectation value of the operator O: (2.26) (2.27) (2.28) hOi = CUV` g0(d ) We see explicitly from equations (2.24) and (2.25) that the integration constant CUV enters only at subleading order, and all superpotentials get closer and closer to each other as ' ! 0, independently of CUV. In other words, no matter what initial conditions we pick for WUV away from ' = 0, the solution is attracted to the same asymptotically AdS boundary at ' = 0. This also implies that the initial condition W (0) = 2(d 1)=` is ill-de ned because it does not x the solution. 2.3.2 IR region The situation is conceptually di erent in the IR. The di erence between the UV and the IR is that not all solutions reaching the IR are regular, and not all of them are acceptable, but only those obeying some restrictions. The others are to be considered as \spurious" solutions of Einstein's equations which are unphysical from the holographic point of view, i.e. they do not correspond to a true state (saddle-point) in the dual eld theory. More speci cally, a solution is acceptable in the IR if it belongs to one of the two classes below: IR-regular solutions; \Good" IR-singular solutions (near the boundary of eld space). Below we will explain what characterizes these two classes. The crucial point is that, as we will explain, independently of the choice of V ( ), there is at most a nite number of IR-acceptable solutions of the superpotential equation.11 Before we delve into the classi cation of IR solutions we note that, on the IR side of the brane, eventually eA ! 0. Indeed, by de nition, going towards the IR the scale factor is decreasing. We will discard the presence of an IR brane (or \hard wall" in the holographic lore) cutting o the small volume part of the geometry, and at which A reaches a nite limit. This case su ers from the same problems as in non-computable singularities that we will discuss below (i.e. it is neither regular nor acceptable). On the other hand, assuming there is no hard wall, the scale factor cannot approach a non-vanishing constant asymptotically, without the theory violating the null energy condition (this can be seen using A = These are solutions in which the curvature invariants are all nite as eA ! 0. In practice, in a co-dimension-one setup, the only asymptotic behavior compatible with regularity is: eA(u) e u=`IR u ! +1 where we approach the Poincare horizon of an AdS space-time with curvature radius `IR. This corresponds to both V ( ) and W ( ) approaching a nite constant: (2.29) (2.30) V ! d(d 2 `IR 1) ; W ! 2(d 2 `IR 1) : Since the asymptotic geometry approaches AdS, we are again in the presence of an asymptotically conformal theory. However, now this is in the interior of the space, where the scale factor approaches zero asymptotically. Therefore this CFT is found in the IR limit. The actual IR limit of the scalar eld, IR, may be nite (in which case the dual theory ows to an IR conformal xed point at a nite value of the coupling), or it can be in nite ( runaway AdS behavior, [71, 72]). An important di erence with respect to the case of a UV xed point is that solutions W ( ) reaching an IR xed point are isolated points in the space of solutions of the superpotential equations and are not part of a continuous family. In other words, an innitesimal deformation (in the space of solutions) leads to missing the xed point and owing elsewhere while typically becoming singular in the process. Acceptable singular solutions. Putting aside the AdS IR asymptotics described above, all other cases eA ! 0, lead to an IR naked singularity where we have additionally that ! +1. However, some singularities may be acceptable from holographic arguments or gravitationally if there exists a way of resolving them. The presence of a (classical curvature) singularity in holography may be interpreted as follows: 1. The solution does not describe a semiclassical saddle point. These are what are customarily called \bad" singularities in holography, [74]. 11We exclude from the discussion here the case of at directions in the dual QFT. 2. The singularity appears because we have not included all possible relevant degrees of freedom. If we include them then the singularity is resolved. Examples of such resolutions exist both by re-including KK states of the bulk theory, [85, 86], or in more complicated contexts stringy states, [87, 88]. Such resolvable singularities are usually called \good singularities" in holography, [74]. A criterion for a \good" (i.e. resolvable) singularity was proposed by Gubser in [74]. It postulates that the solution admits an in nitesimally small deformation which may cloak the singularity behind a regular black hole horizon. In this way, \heating up" slightly the theory, in principle, cloaks the naked singularity without a drastic change to the solution. For a concrete example, take the case of our bulk action (2.2) with a speci c Liouville potential V ( ) exp b . In this case the general solution with the relevant planar (d 1)symmetry is known [82]. It is found that black hole solutions exist only for b < otherwise solutions have an uncloaked naked singularity. This agrees with the postulated q 2d , d 1 criterion (2.33) as we will see in a moment. There is additional evidence concerning solutions with \good" singularities. The calculation of correlators involves the solution of the uctuation equations with appropriate boundary conditions. There are two possibilities: The behavior of correlators at nite energy does not depend on the resolution of the singularity. This case is realized if in the associated Sturm-Liouville problem only one of the two linearly independent solutions is normalizable at the singularity. In this case the boundary condition (normalizability) xes the correlator uniquely. In mathematical terms, an equivalent statement is that the corresponding radial Hamiltonian is essentially self-adjoint. We will call such \good" singularities computable or IR-complete. Such a singularity resolution was encountered early on in higher co-dimension brane world models [75], in [76], and in the holographic context in [77]. The behavior of correlators at nite energy does depend on the resolution of the singularity. This case is realized if in the associated Sturm-Liouville problem both linearly independent solutions are normalizable at the singularity. In this case one needs an extra boundary condition, (which is supplied by the singularity resolution). In mathematical terms, an equivalent statement is that the corresponding radial Hamiltonian has an in nity of self-adjoint extensions determined by extra boundary conditions at the singularity. Therefore, without an explicit resolution of the singularity the correlators cannot be computed. We will call such \good" singularities non-computable or IR-incomplete.Examples of such cases are described in detail in [71{73]. There are many examples where IR-completeness fails, but the most straightforward is the example of a hard wall, i.e. an IR-brane that cuts-o the small volume region of the geometry: in this case, all solutions of bulk wave equation are trivially normalizable at the wall, but di erent boundary conditions lead to very di erent spectra. IR-incompleteness does not necessarily mean that the holographic model is unphysical: rather it hints that with the present ingredients we do not have enough information to compute any observable The counterterms do not depend on the solution (C.8) . After renormalization is carried out, one is left with [83]: S0 = Z d x 4 q (0)CUVg0d=d = Z d x 4 q (0)hOig0; (C.11) where in the second equality we have used equation (2.28). Notice that this contribution does not depend on A0 nor 0 . One may have expected also a boundary term from the far IR, but this always vanishes if the solution is IR-regular or has an acceptable IR singularity. HJEP09(217)3 So far we have \integrated out" the bulk but we have not yet solved the eld equations for the metric and scalar eld at the interface. The e ective 4d action for these variables is the sum of the terms in equations (C.10) and (C.11), plus the world-volume action (2.3): Se [A0; '0; CUV] = Z + Z d x 4 q (0)CUVg0d=d d x 4 q (0)e4A0hWIR('0) WUV('0; CUV) WB('0) (C.12) i This action depends on the dynamical variables ('(u0); A(u0)); on the xed quantities g0 and (0) which are part of the de nition of the UV CFT; and on the extra free parameter CUV. Notice that we should not vary the e ective action with respect to g0 nor (0) nor CUV (in particular the rst line in equation (C.12) is a constant, independent of the dynamical variables. Extremizing the action with respect to the dynamical variables ('0; A0) gives back the matching conditions, (2.20){(2.21), as expected. We can now compare the action (C.12) with the one assumed in the no-go theorem, (C.1). First, notice that the A0 equation of motion is essentially the same as (C.7): Ve (A0; '0; CUV) e4A0hWIR('0) WUV('0; CUV) WB('0) = 0: (C.13) i Contrary to equation (C.7) however, this equation determines the extra parameter CUV (which does not appear in the full de nition of the model, neither in the bulk nor on the brane nor on the boundary) and does not require ne-tuning between the model parameters. This is where the no-go theorem fails: it assumed that the action depends only on dynamical variables, determined by their own eld equations, and that there are no extra free parameters. This is true for weakly coupled eld theories. Here however the quantity CUV is not a dynamical variable but it is determined in a di erent way: on the gravity side, by insisting that the UV solution, through the matching conditions at the brane, glues correctly to the xed IR-regular solution; in the dual eld theory language, it is the strong coupling dynamics which determines the value of the VEV of the operator in the UV. These are a ected also by the low energy degrees of freedom. Indeed, it is natural that the presence of the brane-world degrees of freedom at intermediate energies a ect the UV value of the VEVs and the running of couplings, but not the bare UV coupling g0. Linearized bulk equations and matching conditions In this appendix we derive the perturbed equations and matching conditions for the tensor and scalar modes. We restrict to the physically interesting case of a ve-dimensional bulk, i.e. from now on we set d = 4. We use conformal coordinates in the bulk, such that the unperturbed metric and scalar eld are: ds2 = a(r) dr2 + dx dx ; ' = '(r); where = diag( ; +; +; +). We denote derivatives with respect to r by a prime. The background Einstein equations are, in these coordinates: a or in terms of the superpotential: The brane is located at the equilibrium position r0. All quantities with a subscript 0 are evaluated at r0 (e.g. a0 a(r0) etc). We write the perturbed 5-d metric and scalar eld as: ds2 = a2(r) (1 + 2 )dr2 + 2A dx dr + ( + h )dx ; dx ; ' = '(r) + ; where the quantities ; A ; h ; are functions of r; x and will be treated as small perturbations around the r-dependent homogeneous background. We further decompose the metric perturbations in a scalar-tensor decomposition:37 h = 2 A where the tensor perturbation h^ is transverse and traceless: @ h^ = h = 0. Unless explicitly stated, all indices are raised and lowered with the at Minkowski metric . D.1 Perturbed bulk equations In the bulk, the system contains one tensor perturbation h^ and (before gauge- xing) ve scalar perturbations ( ; ; W; E; ). The components of the linearized Einstein tensor are: G(r1r) = 12 G( 1 ) = a0 a 1 2 a 0 a0 a a0 a 0 a 0 + 2 0 a00 a (W E0)0 3 (W E0) : 37We set to zero the transverse vector modes AT and V T appearing in the general decomposition (3.3), since there is no physical vector in the bulk, and these modes decouple. + 3 a0 a (D.1) (D.2) (D.3) (D.4) (D.5) (D.6) 2 a00 a (D.7) (D.8) The linearized Einstein equations are then: G(a1b) = M 3 1 pg Sbulk[g; ] ( 1 ) gab ; where the right hand side is the linearized matter stress tensor obtained from the variation of the matter bulk action in equation (2.2). At linear order, Einstein equations do not couple tensor and scalar modes and we can discuss the two sectors separately. Tensor modes. Since there are is no tensor-like matter, the transverse-traceless part of the right hand side of equation (D.9) is identically zero (this can be easily checked explicitly). Therefore, the linearized eld equation for tensor modes h^ is obtained by setting to zero the rst square bracket in equation (D.8), and it reads: = 0: Scalar modes. to linear order: Keeping only scalar modes, the perturbed Einstein equations (D.9) are, '0 6 0 a2 dV 6 d' (D.9) (D.10) (D.11) (D.12) (D.13) (D.14) (rr) (r ) ( 6= ) ( = ) 4 a0 a a0 a 2 1 '0 0 6 0 = (W 0 a0 a ('0)2 6 1 '0 6 a0 a + a a2 dV 6 d' 0 + 2 1 3 a0 a a00 a 2 ('0)2 E0)0 3 (W E0) = 0 (KG) a 2 3 a + a2V = a 2 d2V d'2 where the right hand sides are the explicit form of the linearized matter stress tensor appearing in equation (D.9). We also have the perturbed Klein-Gordon equation (which is not independent of equations (D.11){(D.14) , but it can be useful to work with): 2a2 dV d' '0 0 + 4'0 0 (D.15) These equations contain ve scalar perturbations, but we can impose two scalar gauge conditions plus two scalar constraints (this will be discussed in detail in appendix D.4). These leave one physical scalar bulk uctuation, which can be taken to be the gaugeinvariant combination: (r; x ) (r; x ) (r; x ); z (D.16) a'0 a0 1 z(r) expression in equation (D.15). the bulk. From equations (D.11){(D.14) one can obtain a single second order equation for the (r; x ), which reads: = 0: (D.17) Regardless of the gauge xing, one can arrive at equation (D.16) by solving equation (D.12) for E0) in favor of and , and inserting their Equations (D.10) and (D.17) describe the full system of linearized perturbations in HJEP09(217)3 D.2 Brane perturbations and linearized junction conditions In order to write the linearized Israel matching conditions (2.8){(2.9) we need to write the perturbed induced metric, normal vector, and extrinsic curvature, to linear order, in terms of the metric perturbations (D.4){(D.6), plus the brane-bending mode (x ). The latter is de ned by perturbing the embedding equation: r(x ) = r0 + (x ); where r0 is the unperturbed equilibrium position. perturbations: The normal vector nA and induced metric AB nAnB are, to rst order in n A = a 1(r0 + ) (1 ; A + h )dx dx ] ; : It is convenient to explicitly expand to linear order in the prefactor a(r0 + ) in equation (D.20), and to write the perturbed induced metric as: = a02 + h~ ; h~ h a0 : The scalar eld perturbation at the (perturbed) brane position is: '(r(x )) = '0 + + '00 In equations (D.21){(D.22) all quantities are evaluated at r0, the unperturbed equilibrium position. order: From equation (D.21){(D.22) we can deduce the continuity conditions (2.6) to linear h h UV = 0; h'00 + = 0: Notice that the bulk metric and scalar eld perturbations are not continuous at the brane, unless one chooses a gauge where = 0. This is not the most convenient choice to deal with bulk perturbations, however. We will come back to the gauge xing problem in appendix D.4 when we discuss in detail the matching conditions in the scalar sector. (D.18) (D.19) (D.20) (D.21) (D.22) (D.23) The linearized junction conditions are given by (2.8){(2.9). On the right hand side, in addition to the brane action Sbrane in equation (2.3), we allow the possibility of some localized matter: Sloc = Sbrane[ ; '] + Sm; Sm Z d4xp Lm( i; '0) The localized matter elds i (which may include the Standard Model elds), are taken to be trivial in the vacuum. We assume the matter elds are minimally coupled to the induced metric but may have a direct coupling to the dilaton ' evaluated on the brane. The localized matter stress tensor is de ned as: HJEP09(217)3 We also de ne the \dilaton charge operator" O of the localized matter as: T = 2 Sm O = p 1 Sm ' : With these ingredients, the perturbed matching conditions are derived by linearizing both sides of the two equations: In order to proceed, we need the components of the extrinsic curvature KAB = rAnB. They are: Krr = 0; K = a0(r0 + ) [(1 Notice that A appears only in the combination A + @ . ditions (D.27){(D.28) are, to linear order in the perturbations: Using the scalar-tensor decomposition (D.6), the left hand sides of the matching conK K 3a00 (D.24) (D.25) (D.26) (D.27) (D.28) (D.29) (D.30) (D.32) ; (D.31) = a0 1 '00 + 0 + '000 The right hand sides of equations (D.27){(D.28) are obtained by linearizing the expressions on the right hand side of equations (2.8){(2.9). For this, we need the linearized expressions of the brane Ricci tensor for the induced metric in equation (D.21): + 2 0 a0 a0 : (D.33) K ( 1 ) IR ( 1 ) IR UV UV = 1 p 1 p +a0 2 1 1 h^0 1 2 + 3 a00 0 a0 0 a0 0 '0 0 1 h0 2 '0 0 : h 1 2 + 0 a0 in tensor and scalar components as in equation (D.6), the above expres+ 0 a0 a0 + 0 a0 Notice that the longitudinal component E of the metric perturbation drops out of the Ricci tensor. hand sides of equations (D.27){(D.28): We can nally obtain, to linear order in the perturbations, the expressions on right = + h^ ' U0 2 2 0 +2 + 0 a0 + W00 W0 d2WB 0 + '00 + 0 a0 + '00 a0 + 0 a0 The brane matter stress tensor and dilaton charge appear as inhomogeneous source terms in equations (D.27){(D.28). In the following two subsections we will decompose the junction conditions in their tensor and scalar components, respectively. D.3 Tensor junction conditions Since tensor and scalar modes are decoupled at linear order, to study the tensor modes it is enough to set all the scalar modes to zero in the equations found in the previous subsection: Decomposing h sion becomes: whence: 1 p 1 p = W = = E = = 0; h = h^ : The continuity equation across the interface, equation (D.23) becomes simply i.e. tensor modes are continuous across the interface. h ^ iIR UV = 0; HJEP09(217)3 (D.34) (D.35) (D.36) (D.37) (D.38) (D.39) (D.40) The tensor (i.e. transverse and traceless) part of the second junction conditions is found by imposing (D.39) in equations (D.27) and (D.37) , and moreover by keeping only the transverse traceless component of the matter stress tensor, de ned by: T = T 1 3 T + (D.41) where T T in which case the expression above reduces to the rst three terms only: ^ T = T 1 3 T + (D.42) HJEP09(217)3 Setting all modes to zero except h^ T^ , equation (D.27) becomes: in equations (D.31) and (D.37) and replacing T by 3a00h^ 1 2 a0h^0 IR UV = 12 a20W0h^ 1 1 2M 3 T^ : Notice that the rst term on each side cancel thanks to the continuity of h^ the background matching condition since, in conformal coordinates and for d = 4, a0 = a2W=6, and [W ]IURV = W0 by equation (2.20). This leaves the simple jump condition for the rst derivative (plus the source term): a0 hh^0 iIR UV = M 3 1 T^ : D.4 Scalar junction conditions scalar modes, The relevant modes are de ned in equation (3.2), in which we keep only the bulk ; A = 2 plus the brane-bending mode (x) de ned in equation (3.10). Unlike the tensor modes, these elds are not gauge-invariant. Rather, they transform as follows under an in nitesi B = 0 (D.43) and to (D.44) (D.45) (D.46) (D.47) B = = = a0 5 0 '0 5; 5 = E = = 5(r0; x): ( 5)0 a0 5 a It is convenient to partially x the gauge: by an appropriate shift (x; r). This leaves a residual gauge freedom with parameters 5(x; r) and r-independent (x ): a 5-transformation can be compensated by an appropriate (r; x) to leave the condition B = 0 unchanged, and only 0 a ects B. Therefore we are still free to do radial gauge-transformations and r-independent space-time di eomorphisms and keep this gauge choice. In this gauge, setting the brane sources to zero,38 the rst matching conditions (2.6) IR h'(r0 + ) + iIR UV = 0 Expanding the scale factor and the background scalar eld pro le, these are equivalent to the following continuity conditions: where we have de ned the new bulk perturbations: h ^iUV IR = 0; h iUV ^ IR = 0; h iIR E UV = 0 ^(r; x) = + A0(r) (x); ^(r; x) = + '0(r) (x); where A0 = a0=a. new continues variables: The gauge-invariant scalar perturbation (3.6) has the same expression in terms of these (D.48) (D.49) (D.50) (D.51) (D.52) (D.53) (D.54) (D.55) (D.56) In general however (r; x) is not continuous across the brane, since the background quantity A0='0 jumps: Notice that this equation is gauge-invariant since, under a gauge transformation: therefore ^(r0) on the right hand side of equation (D.52) is invariant. It is convenient to x the remaining gauge freedom by imposing: = ^ A0 '0 ^: h iUV IR = A0 UV '0 IR ^(r0) ^(r; x) = 5(r; x) 5(r0; x) ; UV(x)'0UV(r0) = IR(x)'0IR(r0) To do this, one needs di erent di eomorphisms on the left and on the right of the brane, since '0 di ers on both sides. The continuity for ^ then becomes the condition: i.e. the brane pro le looks di erent from the left and from the right. This is not a problem, since equation (D.55) tells us how to connect the two sides given the background scalar eld pro le. In the gauge (D.47){(D.54) we have: = = ^ A0 ; ^(r0) = '0(r0) : 38The localized sources will be added back at the end of this section. This makes it simple to solve for using the bulk constraint equation (in particular, the r -component of the perturbed Einstein equation (D.12): = a0 a 0 = a0 a ^0 + a0 a00 where it is understood that this relation holds both on the UV and IR sides. In the gauge = B = 0, we can write the second matching conditions (D.27){(D.28), using equations (D.31){(D.32) and (D.37){(D.38): 3a0 2 ^ HJEP09(217)3 a0 '0 ^0 + a2(r0) WB('0) 2 6a0 d2WB '00 a'0 '0 IR UV a2 '0 + r0 dUB d' '0 2 a2(r0) dWB d' '0 '0 0 ; (D.58) 6 dUB a2 d' '0 Using the background matching conditions (2.20) and (2.21), as well as the de nitions (2.14){(2.15) in conformal coordinates, a0 a2 = 1 2(d 1) W; d' ; one can see that the rst two terms on each side of equation (D.58) cancel each other, and we are left with an equation that xes the matching condition for E0(r; x): hE0 UV a0 2 U0 ^(r0) 1 a0 dUB d' 0 '0 0 Equation (D.59) xes the discontinuity of ^0. It is convenient to write the equations for ^ and in the form: " '0a ^0 a0 h ^iIR UV #IR UV = 0 ; h'0 iIR UV = 0 ; Mb2 '0 a d' # r0 where we have de ned the brane mass: Using the background Einstein's equations (D.3) this can also be written as: 2 Mb d2Wb a(r0) d'2 '0 + ('0)2 a 6 a0 Mb2 = a0 a a00 IR d'2 '00 #IR '0 UV d2W IR ! d'2 UV : ; (D.57) (D.59) (D.60) (D.61) (D.62) (D.63) (D.64) (D.65) (to see this, use (D.2) and take a radial derivative of eq. (D.3) to write '00 = ad2W=d' + a0=a). We can eliminate E from equation (D.61) by acting with @ @ on both sides and using Einstein's equations (D.11) and (D.12), with = 0: 2E0 = 2 + a0 2 a2 a00 0 : Notice that the combination multiplying 0 can be written as (a=a0)('0)2=6 using (D.2). The bulk equation (3.9) for in this gauge) on both sides of the brane is: + 2 where z = '0a=a0. We can also write it in terms of ^ using (D.56). terms of : To summarize, we arrive at the following equations and matching conditions, either in a0 (D.66) (D.67) (D.68) (D.69) (D.70) (D.72) '0 ; (D.71) a2 '02 a02 6 a'0 a0 0 0 h iIR UV IR UV IR UV 2 00 + a0 a0 IR a'0 UV + 2 '0 ; 2U0 6 a0 h a iIR a0 UV 2 dUB d' 2 0 a'0 a0 ; h'0 iIR = 0 ; + UV a0 a0 ~ 2 Mb = a + ZB('0) 2 + 1 a0 d2WB d'2 dUB d' '02 ; 0 ~ 2 Mb d2W d'2 IR ! UV : Notice that these equations have 6 free parameters: 4 in the bulk (two integration constants for equation (D.68) in the UV, and two in the IR) and two brane parameters ( on each side). From these 6 we can subtract one: a rescaling of the solution, which is not a true parameter since the system is homogeneous in ( ; ). There is a total of 4 matching conditions, plus 2 normalizability conditions if the IR is con ning, or only one if it is not. Therefore,in the con ning case, we should nd a quantization condition for the mass spectrum, whereas in the non-con ning case the spectrum is continuous and the solution unique given the energy. The goal will be to show that such solutions exist only for positive values of m2, de ned as the eigenvalue of 2. To see this, one must go to the Schrodinger formulation. To put the matching conditions (D.69){(D.71) in a more useful form, it is convenient to eliminate L;R altogether using equations (D.69): a0 [ ]; ['0 ] = 0 (D.73) These can be solved to express the continuous quantities ^(r0) and '0 which appear on the r.h.s. of (D.70){(D.71) in terms of UV;IR only ([x] xUV for any quantity x): ^(r0) = : Using the above identi cations, equations (D.70){(D.71) become relation between the left and right functions and their derivatives: z 0 = 6 dUB a0 d' '0 2 Since the left hand side is in general non-degenerate, these equations can be solved to give L0 and R0 as linear combinations of L and R, i.e. one can put (D.75){(D.76) in the general form: U0V(r0) I0R(r0) where the matrices 1 and 2 are given by: (D.74) (D.75) (D.76) (D.77) Z0zI2R 1 A ; (D.78) (D.79) y(r0) 1 (r0) = a04M~ 2 [z ] [z] [1=z] 0 2 ! 0 [z ] 1 [ ] A ; [1=z] (D.80) 1 = 2 = a0M~ 2 where 2 zIR zIR 2 2 zUV zUV 2 ! + 0 + Z0zI2R 6zUV zzUIRV + 1 ddU'B '0 + 0 zzUIRV + Z0zU2V 6zIR zzUIRV + 1 ddU'B '0 12zUV ddU'B '0 0 zzUIRV d'2 '0 d2W d'2 0 = 6 6 WB WIRWUV '0 U0 : D.5 Gauge-invariant action for scalar modes Here we show that the action for the scalar perturbation, equation (5.9), can be written in a gauge-invariant form. To this end, we show that the action depends solely of the gauge-invariant bulk variable and gauge-invariant brane variables ^(r0); ^(r0). First, notice that equation (5.9) was obtained in the gauge = 0 in the bulk. In this gauge, the scalar quantity coincides with the gauge-invariant variable (see equation (3.6)). Therefore, the bulk part of the action ( rst line in equation (5.9)) can be written in a manifestly gauge-invariant fashion by replacing the 2-component object with Z ( IR; UV). the rst localized term: Next, we consider the localized terms in the second line of equation (5.9). Using the expressions for 1 and 2 in equation (D.78) and after some tedious algebra we obtain, for where 0 was de ned in equation (D.79). From equation (D.73) we observe that the components of the 2-vectors entering the above matrix products coincide, in our gauge = 0, with the gauge-invariant combinations: i.e. the gauge-invariant dilaton and metric trace on the brane. E The bulk propagator for tensor modes The bulk propagator D(p; r) is de ned by equation (4.11). It must satisfy normalizability conditions at the asymptotic AdS boundary (UV) and in the deep interior (IR). Here, normalizability is to be understood as square-integrability with respect to the appropriate integration measure, i.e. Z e(d 1)A j j2 < 1: The bulk propagator D(p; r) can then be written in terms of normalizable UV and IR eigenfunctions of the radial operator @re(d 1)A@r, with \energy" determined by p2: and for the second: [z ] [z] 0 6 ddU'B '0 [z] [1=z] (D.81) (D.82) (E.1) (E.2) (E.3) (E.4) (E.5) (E.6) where UV and IR satisfy the equations: and the matching conditions: D(p; r) = < 8 > > r < r0 r > r0 h h e(d 1)AUV(r)p2i (UpV) = 0 e(d 1)AIR(r)p2i I(Rp) = 0 I(Rp)(r0) = (UpV)(r0) h (p) i 1 The matching conditions (E.5){(E.6) follow by integrating equation (4.11) on a small interval across the interface. The mode functions in equation (E.2) are normalizable in the UV and IR, respectively. The solution therefore has four integration constants and four conditions (two normalizability conditions plus two matching conditions) that x the wave-functions uniquely (notice that the system is not homogeneous, and does not have a rescaling freedom). the derivatives of A(r), At large Euclidean p2, we can approximate the bulk equations as in at space, neglecting For small r, the AdS boundary acts as an in nite barrier and imposes a vanishing wavefunction at r = 0 (this is equivalent to normalizability in the UV). In the interior, assuming the IR is reached as r ! +1,39 the solutions for positive p2 are real exponentials, and for normalizability we require the solution to be vanishing as r ! +1. The solution satisfying appropriate boundary conditions (vanishing in the IR and for r ! 0) and matching condition at r0 is: For large pr0, we observe that: I(Rp) = sinh pr0 p e pr; (UpV) = 1 e pr0 p D(p; r0) ' 2p pr0 1 sinh pr; p pp2 D(0; r0) = d0 = e3A0 Z r0 e 3AUV(r0)dr0 0 39This is the case for example when the IR geometry asymptotes to an AdS interior. A full classi cation of the possible IR geometries in a general Einstein-dilaton theory, can be found in section 4 of [73]. Perturbation expansion for small-p For small p, the bulk propagator has the form of an expansion in p2: D(r0; p) = d0 + p2d2 + p4d4 + : : : where the coe cients di can be computed perturbatively in p2 solving equation (4.11) iteratively. We concentrate on the case d = 4. O(p0): setting p = 0 in equations (E.3) and (E.4), we can integrate them immediately e 3AUV(r0)dr0 + C2(0;U) V; I(R0) = C1(0;I)R Z r 0 e 3AUV(r0)dr0 + C2(0;I)R and nd: (U0V) = C1(0;U) V Z r 0 determine the values: and we nd: Normalizability implies C2(0;U) V = C1(0;I)R = 0. The matching conditions (E.5){(E.6) C1(0;U) V = e3A0 ; C2(0;I)R = e3A0 Z r0 0 e 3AUV(r0)dr0 Therefore, to lowest order in small p: (U0V)(r) = e3A0 Z r 0 e 3AUV(r0)dr0; I(R0)(r) = e3A0 Z r0 0 e 3AUV(r0)dr0; (E.7) (E.8) (E.9) (E.10) (E.11) (E.12) (E.13) (E.14) O(p2): to the next order, we write: (p) UV ' (U0V) + p2 (U2V); (p) IR ' I(R0) + p2 (2) IR The corrections to the wave-functions at order p2 satisfy the equations: = e3AUV(r) (0) UV = e3AIR(r) (0) IR r > r0: The matching conditions for (2) are: (E.15) (E.16) (E.17) (E.18) (E.19) (E.20) (E.21) (E.22) (E.24) (E.25) Z r 0 Z r r0 +CU(2V) Z r +CI(R2) Z r 0 0 0 r0 dr0e 3AUV(r0) dr0e 3AUV(r0) dr00 I(R0)(r00)e3AIR(r00) dr0 (U0V)(r0)e3AUV(r0); dr0e 3AUV(r0) CU(2V) + dr00 (U0V)(r00)e3AUV(r00) : I(R2)(r0) = as follows from equations (E.5){(E.6) and from the matching conditions at order p0. Integrating twice equations (E.16){(E.17), the general solution with normalizable homogeneous parts are: (U2V)(r) = I(R2)(r) = CU(2V) = CI(R2) = Z r0 0 Z r0 0 Z r0 0 r0 Imposing the continuity conditions (E.18) at r = r0 we nd: Inserting this result into equation (E.19) we nd: (U2V)(r) = Z r 0 dr0e 3AUV(r0) Z r0 dr00 (U0V)(r00)e3AUV(r00): Recall that d2 = tion (E.13), we obtain: d2 = e3A0 Z r0 0 (U2V)(r0): evaluating equation (E.22) at r = r0 and using equadr0e 3AUV(r0) Z r0 dr00e3AUV(r00) Z r00 dr000e 3AUV(r000): (E.23) 0 O(p2n) One can continue the above procedure iteratively: the wave-functions at order 2n satisfy the equations (2n) = e3AUV(r) (2n 2) UV (2n) = e3AIR(r) (2n 2) IR We will now extract the explicit dependence on A0 of the expansion coe cients d2n. This can be achieved by writing AUV as a function of ' as in equation (B.4), AUV(') = A0 + AUV('0; '); AUV('0; ') 1 Z ' WUV 2(d 1) '0 dWUV=d' and by changing variables to ' in all the integrals (4.16){(4.18), using the identity (valid for 0 < ' < '0 and 0 < r < r0): The result takes the form: d' dr = eAUV(r) dWUV : d' di = e A0Di('0) and must be continuous, with continuous derivative, at r0. This system of equation is identical to the one we have solved at order p2, and the solution is as follows: (U2Vn)(r) = Z r 0 r0 dr00 (U2Vn 2)(r00)e3AUV(r00): (E.26) The coe cient d2n is obtained by evaluating the above expression at r0 and consists of 2n + 1 alternating integrals: d2n = ( )ne3A0 Z r0 dr1e 3AUV(r1) Z r0 dr2e3AUV(r2) Z r2 dr3e 3AUV(r3) : : : 0 : : : Z r0 r2n 1 0 Z r2n 0 dr2n+1e 3AUV(r2n+1): (E.27) (E.28) (E.29) (E.30) (E.31) (E.32) (E.33) (E.34) (E.35) e AUV('0;') 0 1 ' ! 0 ' ! '0: R0 WUV('0); D2n('0) R0 1 2n+1 ; 0 ; As a consequence, the scale controlling Di is approximately the bulk curvature R at the interface, encoded in the superpotential factors in the denominators: and we have, roughly: where the coe cients Di('0) are independent of A0 but depend only on the superpotentials and the equilibrium position '0: Z '0 0 e 3AUV('0;'0) Z '0 e3AUV('0;'00) Z '000 W U0V('0) W U0V('00) d'000 e 3AUV('0;'000) W U0V('000) Z '0 0 e 3AUV('0;'0) W U0V('0) D0('0) = D2('0) = and similarly for D4('0). between zero and one, and: Therefore, the expansion coe cients of the bulk propagator at low momenta all scale as e A0 times a function that depends only on the bulk potentials. Notice that, for xed '0, the exponential e AUV appearing in the integrals, is bounded Regularity of the small-p expansion Here we discuss if and at which order the expansion in p2 used in equation (E.10) may break down. First, consider the case when the bulk theory has a con ning IR. In this case the spectrum of normalizable eigenmodes is discrete, and the bulk Green's function D(p; r) can be expanded in terms of the eigenfunctions of the bulk radial operator: D(p; r) = X n fn p2 + m2n where fn are some constants and mn are the \eigenvalues" for the radial kinetic operator, In this case, it is clear from equation (E.36) that the small momentum expansion is regular. Things are more subtle if the bulk theory has no gap, but rather it has a continuous spectrum starting at m = 0. This is the case either if the theory reaches a conformal xed point in the IR, or if ' reaches in nity but the superpotential grows slower than exp ' with 2 < 1=6 [71, 72]. In both cases, the IR is reached as the conformal coordinate r ! +1, where the scale factor behaves as: eAIR(r) 1 rz r ! +1; z 1 The constant z is related to the steepness of the bulk potential, with z = 1 corresponding to AdS asymptotics in the interior (thus to the case of a conformal IR xed point). For more details, the reader is referred to [71, 72]. The small-p behavior is expected to be governed by the far IR of the theory, i.e. by the behavior of the geometry as r ! 1. In this region, the bulk wave equation simpli es to: IR(r) ' cIR(p) np where function. 1; 2 and (E.36) (E.37) (E.38) (E.39) (E.40) (E.41) h00(r) r 3z h0(r) p2h(r) = 0 IR(r) = cIR(p)r 1+23z K 1+3z (pr) 2 This approximation is valid in the asymptotic region where the metric can be approximated by (E.38), and is independent of the value of p. The solution of equation (E.39) which is normalizable at in nity is: where K is the modi ed Bessel function which is exponentially vanishing at in nity, and cIR(p) is for the moment unknown. For xed large r, but for p 1=r, we can also expand equation (E.40) for small 1+3z 2 1 + 1r2p2 + O(r4p4) + p 3z2+1 r1+3z 1 + 2r2p2 + O(r4p4) o ; are some xed constants arising from the expansion of the Bessel We can compare equation (E.41) with the small-p expansion of the IR wave-function IR given in equation (E.15). To lowest order in this expansion the normalizable IR wavefunction is a constant (see equation (E.12): I(R0) = C(0): cIR(p) = C(0)p 3z2+1 : This is consistent with the dependence in cIR(p) is xed to be: xed r, p ! 0 limit of equation (E.41) if the momentum (E.42) (E.43) Inserting this expression back in equation (E.41) we nd, at small p and large r: IR(r) ' C0 1 + 1r2p2 + O(r4p4) + (pr)1+3z 1 + 2p2r2 + O(r4p4) (E.44) The only source of non-analyticity in the above expression is the p1+3z prefactor. Thus, we have a regular expansion in p at least up to the order 1 + 3z 4. The larger is z (and the faster the scale factor vanishes), the further the non-analytic terms arise in the expansion. The coe cients d2n are well-de ned and nite as long as 2n < 1 + 3z. The earliest the expansion can fail is at 2n = 4 for z = 1, with the appearance of terms p4 log p which are familiar for massless elds in asymptotically AdS space-times. Notice that we may evade the above argument, and have a singular limit of D(p; r0) as matching condition (E.6) to lowest order in p becomes (@r I(R0))(r0) = p ! 0, only if we somehow lose the constant solution (E.42). This is the case, for example, in the Randall-Sundrum type matching: if we impose Z2 symmetry at the brane, then the 1=2 which is not obeyed by the constant solution. This signals a singularity of the propagator as p ! 0, which indeed turns out to be the massless pole associated to the graviton zero mode in this theory. 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Christos Charmousis, Elias Kiritsis, Francesco Nitti. Holographic self-tuning of the cosmological constant, Journal of High Energy Physics, 2017, 31, DOI: 10.1007/JHEP09(2017)031