Lovelock theories, holography and the fate of the viscosity bound

Xain O. Camanho 2 Joes D. Edelstein 0 2 Miguel F. Paulos 1 Open Access 0 Centro de Estudios Cienctos , Valdivia, Chile 1 Laboratoire de Physique Tehorique et Hautes Energies , CNRS UMR 7589, Universiet Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France 2 Department of Particle Physics and IGFAE, University of Santiago de Compostela , E-15782 Santiago de Compostela, Spain We consider Lovelock theories of gravity in the context of AdS/CFT. We show that, for these theories, causality violation on a black hole background can occur well in the interior of the geometry, thus posing more stringent constraints than were previously found in the literature. Also, we nd that instabilities of the geometry can appear for certain parameter values at any point in the geometry, as well in the bulk as close to the horizon. These new sources of causality violation and instability should be related to CFT features that do not depend on the UV behavior. They solve a puzzle found previously concerning unphysical negative values for the shear viscosity that are not ruled out solely by causality restrictions. We nd that, contrary to previous expectations, causality violation is not always related to positivity of energy. Furthermore, we compute the bound for the shear viscosity to entropy density ratio of supersymmetric conformal eld theories from d = 4 till d = 10 | i.e., up to quartic Lovelock theory {, and nd that it behaves smoothly as a function of d. We propose an approximate formula that nicely ts these values and has a nice asymptotic behavior when d ! 1 for any Lovelock gravity. We discuss in some detail the latter limit. We nally argue that it is possible to obtain increasingly lower values for =s by the inclusion of more Lovelock terms. Contents 1 Introduction 2 Lovelock gravity 2.1 AdS vacua and black hole solutions 3 Holographic dictionary 3.1 Calculation of CT 3.2 Three-point function 4 Shear viscosity and nite temperature instabilities 4.1 Graviton potentials and causality 4.2 Plasma instabilities 5 Bulk causality and stability 5.1 The cubic theory in higher dimensions 5.2 Quartic Lovelock theory 5.3 Expansions at =s = 0 6 The =s ratio in higher order Lovelock theories 6.1 The minimum ratio in quartic theory 6.2 No dimension-independent =s bound 7 Discussion A Three-point function parameters B A curve through the parameter space of Lovelock gravities B.1 Stability analysis B.1.1 Stability at the horizon B.1.2 Full stability B.2 Causality analysis 1 Introduction The AdS/CFT correspondence has provided a tool to study hydrodynamical aspects of quantum eld theories at strong coupling. This was particularly timely due to the advent of experiments that prompted the exploration of QCD in a region of phase space where it displays such behavior. One of the most striking predictions of AdS/CFT had to do with the shear viscosity to entropy density ratio, =s , of strongly coupled plasmas. Interestingly enough, it was found that there is a universal value for this ratio, =s = 1=4 , for theories whose gravity dual is governed by the Einstein-Hilbert action [1], regardless of the matter content, the number of supersymmetries, the existence or not of a conformal symmetry, and even the spacetime dimensionality. On the other hand, all measured values for this ratio in any quantum relativistic system are above this value. This led to speculations that =s 1=4 might be an exact statement in quantum relativistic systems, the so-called KSS bound conjecture [2]. Indeed, it is possible to provide a hand waving argument, relying on a quasiparticle description of the plasma, that links the KSS bound to the Heisenberg uncertainty principle. This argument is questionable, though, since there are convincing hints supporting a non-quasiparticle description of strongly interacting plasmas. The KSS bound conjecture has been thoroughly scrutinized for many years (see, for example, [3] for a recent review). It turned out to be the case, however, that when quantum corrections are included in the gravitational action, under the form of curvature squared terms, the =s ratio can be smaller than the KSS bound [4, 5]. In particular, there are string theory constructions where this is the case [5, 6]. On general grounds, the coecient of the curvature squared terms must be small. However, for the particular combination given by the Gauss-Bonnet (GB) term, one may consider nite values of the coecient, as the gravitational theory is then two-derivative. In this case the ratio is modied to =s = 1=4 (1 4 ), being the appropriately normalized GB coupling, and the KSS bound is violated whenever is positive [4, 5]. Even if a priori the addition of a GB term would lead to an arbitrary violation of the KSS bound, it turns out that causality constraints arise preventing the possibility of going to arbitrarily low values of =s [7]. In the case of 4d CFTs, for instance, this imposes the constraint 9=100, which reduces the minimum value of =s by a factor of 16=25.1 The study of a possible bound is interesting both from a theoretical standpoint as well as from a phenomenological one | it has been found experimentally that the quarkgluon plasma created in relativistic heavy ion collisions appears to have a very low shear viscosity to entropy density ratio [9]. A similar result was also found recently in a radically dierent context: that of strongly correlated ultracold atomic Fermi gasses in the so-called unitarity limit [10]. Both systems have measured values of =s compatible with 1=4 . On the other hand, from a theoretical standpoint, the causality constraints coming from the behavior of the geometry near the boundary, that rule the attainability of lower values of =s , have a beautiful holographic dual in the CFT side: they arise from positivity of energy conditions [11{13]. The perfect matching of these two quite fundamental restrictions on a physically sensible theory constitutes a striking check of the AdS/CFT correspondence. Whether this is still valid for higher-dimensional CFTs became a natural question that was subsequently answered in a series of papers [14{16]. Summarizing, it turned out that GB theories lead to a violation of the KSS bound in any spacetime dimensionality, and causality constraints exactly match positivity of energy bounds on the CFT side. The fact that this matching is valid regardless of the dimensionality is puzzling and seems to provide 1It is interesting to mention at this point that the link between causality violation and the viscosity bound does not apply for thermal theories undergoing a low temperature phase transition [8]. clues on possible non-stringy versions of AdS/CFT. Our knowledge of higher-dimensional CFTs is, however, too poor yet to push these arguments forward. In higher dimensions one has in general the choice of including other Lovelock terms in the action. These are higher order in curvature that still lead to second order equations o (...truncated)