Abstract
We study the O(N) and Gross-Neveu models at large N on AdSd+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 frac{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 AdS2 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.
Highlights
Developing non-perturbative approaches to strongly interacting quantum field theories is undoubtedly an important subject in theoretical physics
In the Gross-Neveu model we show the existence of an extra operator which corresponds to a bound state in flat space, while in the O(N ) vector model in the symmetry-breaking phase we find a distinctive pattern of the anomalous dimensions of the operators which can be thought of as the AdS analogue of the resonance phenomenon
It is equivalent to a boundary conformal field theory (BCFT) with d + 1 dimensional bulk, which has spacetime symmetry SO(d + 1, 1)
Summary
Developing non-perturbative approaches to strongly interacting quantum field theories is undoubtedly an important subject in theoretical physics. We will see another use of this idea which is more analytical; namely we compute resummed loop diagrams by imposing the consistency of the operator product expansion in the boundary conformal theory and take the flat-space limit to reproduce the S-matrix. (iii) We can compute observables in the large-N expansion using ordinary Feynman diagrams, with the Feynman rules induced by (2.5), namely: the propagator for δσ is the resummed one in eq (2.12), the propagator for δφi is just a free-field massive propagator √with mass-squared M 2, there is a cubic vertex between δσ and two δφi’s of order 1/ N , and there are self-interactions of δσ induced by the 1-loop 1PI effective action In this perturbation theory we do not include diagrams containing as a subdiagram any 1PI 1-loop n-point function of δσ, because those are already accounted for by the full propagator and self-interactions of δσ. In AdS we have to specify boundary conditions
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