A study of quantum field theories in AdS at finite coupling

Abstract

We study the O(N ) and Gross-Neveu models at large N on AdS d+1 background.Thanks to the isometries of AdS, the observables in these theories are constrained by the SO(d, 2) conformal group even in the presence of mass deformations, as was discussed by Callan and Wilczek [1], and provide an interesting two-parameter family of quantities which interpolate between the S-matrices in flat space and the correlators in CFT with a boundary. For the actual computation, we judiciously use the spectral representation to resum loop diagrams in the bulk. After the resummation, the AdS 4-particle scattering amplitude is given in terms of a single unknown function of the spectral parameter. We then "bootstrap" the unknown function by requiring the absence of double-trace operators in the boundary OPE. Our results are at leading nontrivial order in 1 N , and include the full dependence on the quartic coupling, the mass parameters, and the AdS radius. In the bosonic O(N ) model we study both the massive phase and the symmetry-breaking phase, which exists even in AdS 2 evading Coleman's theorem, and identify the AdS analogue of a resonance in flat space. We then propose that symmetry breaking in AdS implies the existence of a conformal manifold in the boundary conformal theory. We also provide evidence for the existence of a critical point with bulk conformal symmetry, matching existing results and finding new ones for the conformal boundary conditions of the critical theories. For the Gross-Neveu model we find a bound state, which interpolates between the familiar bound state in flat space and the displacement operator at the critical point.

On marginal operators in boundary conformal field theory

“…The unitarity bound ∆ > d−3 2 in the AdS/CFT literature corresponds to our bound µ > −1 that we arrived at through normalizability considerations. The expression (5.10) has an interesting consequence for the boundary operator product expansion, one that could have been anticipated given the relation to AdS/CFT [2]. In a boundary CFT, we expect to be able to decompose any bulk operator into a sum over boundary operators.…”

“…In order for Y to be above the unitarity bound, one should take ∆ ≤ d+2 2 (or ∆ = d when Y is the identity). 2 Translations in the normal direction are already broken explicitly due to the presence of the boundary. Nevertheless, the current is conserved locally in the bulk and can be thought of as an approximate symmetry of the UV, that is when we probe distances which are much shorter than the distance to the boundary.…”

Unitarity methods in AdS/CFT

“…Thus we first study the five-point tree and its crossing properties. 28 Let us emphasize: while some properties are special to the double-ladder and five-point tree, there is nothing fundamentally unique about these diagrams in the application of our unitarity methods. 27 For example, the leading log comes from a term (γ (1) n, ) 3 in dDisc(G 2−loop ).…”

Spinor vertices and bubbles in the old conformal bootstrap in AdS/CFT and the fermion mass hierarchy problem: The Yukawa model

“…In [37], [38] spinor-spinor-scalar vertexes were calculated in formalism of embedding space and also in [38] spectral representation of the bulk spinor Green function is presented. Bulk fermion loop of scalar field was first calculated in [39] also in the formalism of embedding space, whereas one-loop self-energy of Fermi field on AdS was not calculated earlier, as to our knowledge. Here we don't use formalism of embedding space and perform calculations in physical AdS d+1 .…”

“…Like in the case of vertex of type I expression for M Fermionic bubble diagram of scalar field on AdS was first calculated in [39] in formalism of embedding space. This one-loop contribution to the two-point correlator of scalar field φ(Z) is generated by its bulk coupling g φ(Z)ψ(Z)χ(Z) with two spinor fields; it is formed by the bulk-to-boundary propagators of scalar field K φ (6) and bulk Green functions of spinor fields…”