Towards holographic higher-spin interactions: four-point functions and higher-spin exchange

Received: December Towards holographic higher-spin interactions: four-point functions and higher-spin exchange X. Bekaert 0 1 2 4 J. Erdmenger 0 1 2 4 D. Ponomarev 0 1 2 3 4 C. Sleight 0 1 2 4 Open Access 0 1 2 4 c The Authors. 0 1 2 4 0 F ohringer Ring 6, D-80805 Munich , Germany 1 Parc de Grandmont , 37200 Tours , France 2 F ed eration de Recherche 2964 Denis Poisson Country, Universit e Fran cois Rabelais 3 Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians University Munich 4 Theresienstr. 37, D-80333 Munich , Germany Within holography, we calculate the contribution of an arbitrary spin-s gauge boson exchange in AdSd+1 to the four-point function with scalar operators on the boundary. As an important ingredient, we first compute the complete bulk-to-bulk propagators for massless bosonic higher-spin fields in the metric-like formulation, in any dimension and in various gauges. The split representation of the bulk-to-bulk propagators in terms of bulk-to-boundary propagators allows to present the higher-spin exchange diagram in the form of a conformal partial wave expansion. Our results provide a step towards the larger goal of the holographic reconstruction of bulk interactions, and of clarifying bulk locality. bMax-Planck-Institut fur Physik (Werner-Heisenberg-Institut) 1 Introduction 2 Ambient space formalism 2.1 Ambient AdS tensors 2.2 Ambient boundary tensors 3.1 3.2 3.3 Free massless higher-spin fields in AdS Basis of harmonic functions de Donder gauge Manifest trace gauge Bulk-to-boundary propagators 3.7 Split representation of bulk-to-bulk propagators 4 Four-point exchange Manifest trace gauge 4.5 On improvements 5 Conclusion and outlook A Operations with ambient tensors B Exchange computation in traceless gauge C Single trace of the currents D Multiple traces of the currents Massless higher-spin bulk-to-bulk propagators Bulk locality remains one of the most important and elusive properties of the anti-de Sitter/ conformal field theory (AdS/CFT) correspondence [14]. This property is expected to hold in the usual regime where the duality is tested: when the AdS radius is large compared to the Planck and string lengths which, on the CFT side, corresponds to a large-N expansion and a gap in the spectrum of anomalous dimensions. The latter two properties were argued to provide necessary and sufficient conditions for a CFT to possess a local bulk dual [57] (a third condition was added in [8]). On the other hand, the conjectured duality [9, 10] between Vasilievs higher-spin gravity [1113] and a vector model at a free or critical fixed point provides a convenient playground for probing deep issues, such as bulk locality, in holography. This is because it holds in a regime where in principle both sides are calculable, in contrast to that of the standard AdS/CFT correspondence described above. A particular example that should be tractable is the explicit computation of the quartic vertex in higherspin gravity, matching the four-point correlator of the free CFT. The result may shed some light on the issue of bulk locality in higher-spin holography, and the present paper aims to prepare the technical tools for attacking the above concrete match. In the recent years, there has been great progress connecting CFT correlation functions to scattering processes in AdS spacetime [1417]. This progress was based on the technology of Mellin amplitudes [1821]. The programme of reconstructing bulk theories from their dual CFTs via the rewriting of Mellin amplitudes as Witten diagrams appears to apply to a large class of strongly-coupled CFTs. Unfortunately it does not directly1 apply to the simplest example of weakly-coupled CFTs: free scalar fields. Nevertheless, a putative bulk dual appear to exist in the form of Vasiliev higher-spin gravity. At a conceptual level, the fact that free CFTs fall outside the scope of the above holographic reconstruction programme can be seen by inspecting the set of necessary and sufficient conditions, proposed in [8], for a CFT to possess an AdS effective field theory dual: these conditions are not satisfied by free large-N CFTs. Firstly, the spectra of free CFTs contain a gapless infinite set of single-trace primary operators. Secondly and more importantly, their Mellin amplitudes may not be bounded by a polynomial of Mellin space variables, actually they may even not be defined at all (cf. footnote 1). In bulk terms, this translates into the fact that, first, the spectrum of bulk fields contains an infinite tower of massless fields with unbounded spin and, second, that the bulk theory does not admit an expansion in (non-negative) powers of the cosmological constant. These properties are perfectly consistent with key features of Vasiliev theory (see e.g. [22] for a nontechnical review), and explain which assumptions of the Weinberg low-energy theorem2 generalised to AdS [8] are circumvented by higher-spin gravity. Nevertheless, a tantalising open question remains: what is precisely the status of locality in higher-spin theories? Clarifying this issue is of fundamental importance, since locality is one of the core properties of fundamental field theories. But even in the familiar setting of QFT, the 1Technically, this can be seen by computing the correlation function of four single-trace scalar operators in the free O(N ) vector model via Wick contraction. This gives depending on the two cross ratios that is not fixed by conformal symmetry in the four-scalar correlation function. Following the prescription [18], the corresponding 4-point Mellin amplitude should be proportional to the Mellin transform of F (u, v) over both variables. However, the former is not well-defined since the function (1.1) is a sum of products of powers of u and v, while power functions do not admit a Mellin 2Which prevents long-range higher-spin exchanges. issue is a subtle one since for instance the Wilsonian view on QFT is based on effective field theories, which are quasilocal. This is in the sense that they possess a perturbative expansion (in powers of fields and their derivatives) where each individual term in the total Lagrangian is local, though the total number of derivatives may be unbounded in the full series.3 Indeed, a natural candidate of a suitably enlarged definition of AdS effective field theory (in order to possibly contain Vasiliev theory as a paradigmatic example of bulk dual to a free CFT) is quasilocality. An immediate proviso is the fact, often emphasised by Vasiliev, that there is no well-defined derivative expansion around (A)dS background. More precisely, the expansion in the number of (covariant) derivatives mixes with the expansion in powers of the cosmological constant, since the commutator of two background covariant derivatives is of the same order as the cosmological constant. A second proviso is the fact that higher-spin interactions are weighted by powers of the AdS length. This property is responsible fo (...truncated)