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, 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 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.

Figures (13)

• Figure 1: Schematic pictures of RG flows of massive QFTs in flat space and AdS. (a) The RG flow in flat space. In flat space, the theory starts from the UV fixed point and flows either to a gapped phase or to a fixed point governed by CFT. (b) The RG flow in AdS for $\Lambda_{\rm AdS}\gg \Lambda_{\rm QFT}$. In this case, the theory starts seeing the AdS curvature as soon as it flows away from the UV fixed point, and simply flows to the gapped phase in AdS. (c) The RG flow in AdS for $\Lambda_{\rm AdS}\ll \Lambda_{\rm QFT}$. When $\Lambda_{\rm AdS}$ is small enough, the theory does not see the AdS curvature until it reaches the deep IR. Therefore there is a wide range of scales in which the physics can be well-approximated by the massive QFT in flat space.
• Figure 2: Examples of 1PI diagrams for the one- and two-point functions of $\sigma$ which are not suppressed by powers of $N$. The black lines are the propagators of $\phi^{i}$, and we also depicted how O($N$) indices are contracted.
• Figure 3: The large-$N$ effective potential $\frac{1}{N \lambda^3}V(\Phi^i)$ as a function of $|\Phi|$ in $d=2$ (i.e. in AdS$_3$), for $\lambda =1$ and various values of $m^2$. Dimensionful quantities are expressed in units of the AdS radius $L$. The position of the symmetry-preserving (-breaking) vacuum, when it exists, is indicated with a red (blue) dot. The line interrupts when there is no real solution to eq. \ref{['eq:minSAdS']}, i.e. when $\frac{m^2+1}{2} + \lambda(\Phi^i)^2 < 0$.
• Figure 4: The large-$N$ effective potential $\frac{1}{N \lambda}V(\Phi^i)$ as a function of $|\Phi|$ in $d=1$ (i.e. in AdS$_2$), for $\lambda = 0.5$ and various values of $m^2$. Dimensionful quantities are expressed in units of the AdS radius $L$. The position of the symmetry-preserving (-breaking) vacuum, when it exists, is indicated with a red (blue) dot. The line interrupts when there is no solution to eq. \ref{['eq:minSAdSd1']}, i.e. when $\frac{m^2+\frac{1}{4}}{2} + \lambda(\Phi^i)^2 +\frac{\lambda}{2\pi}(\gamma + \log(4\mu))< 0$.
• Figure 5: The resummation of the two-point function of $\delta\sigma$. The thick black lines are the bare propagators of $\sigma$ while the red curves are propagators of $\phi^{i}$.