UV-finite “old” conformal bootstrap on AdS: scalar case

Abstract

The double-trace from UV to IR flow subtraction of infinities used earlier for the UV-convergent calculations of the Witten tadpole diagrams being applied to the bubble self-energy diagrams gives for them the amazingly simple expressions in case of the four-dimensional boundary space. For every N = 1...4 in the O(N) symmetric scalar fields model with the conformal Hubbard-Stratonovich field there are three roots of the "old" conformal bootstrap spectral equations that obey unitarity bound demand.

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

“…Equating of exact Green functions to the one-loop quantum contribution built of the same exact Green functions is the main postulate of the "old" conformal bootstrap. In the AdS/CFT context it is assumed in [1], [2] that X 1,2 , X, Y in (83) are the bulk coordinates in AdS d+1 and X 1,2 are sent to the horizon. Then LHS of (83) becomes conformal correlator of the boundary conformal theory ((7) for scalar and (13) for spinor), whereas G(X 1 , X), G(Y, X 2 ) in the RHS become the corresponding bulk-to-boundary propagators.…”

“…To extend the approach of [2] to physically more interesting spin 1/2 bulk fields is the goal of the present paper.…”

“…One of the candidates for the theory capable to fix masses of bulk fields may be the "old" conformal bootstrap in the AdS/CFT context [1], [2]. "Old" conformal bootstrap was proposed about 50 years ago in pioneer papers [14], [15] and developed in [16] - [21] (see e.g.…”

“…In [1] the values of bulk masses of scalar fields that are the roots of corresponding bootstrap spectral equations were found in certain models under the oversimplifying assumption of replacing of two "intermediate" Green functions in self-energy Witten diagrams (bubbles) by their harmonic counterparts. In [2] this assumption was abandoned, and UV divergence of the bubble was renormalized in the "double-trace from UV to IR flow" way used earlier in calculations of the UV-finite tadpoles and bulk vacuum quantum energies of scalar [23]- [28] and spinor [29]- [30] bulk fields. The sensible, that is obeying unitarity bound demand, values of conformal dimensions of scalar fields in O(N) symmetric model were obtained in [2] for N = 1...4 in case of 4 boundary dimensions.…”

“…In [2] this assumption was abandoned, and UV divergence of the bubble was renormalized in the "double-trace from UV to IR flow" way used earlier in calculations of the UV-finite tadpoles and bulk vacuum quantum energies of scalar [23]- [28] and spinor [29]- [30] bulk fields. The sensible, that is obeying unitarity bound demand, values of conformal dimensions of scalar fields in O(N) symmetric model were obtained in [2] for N = 1...4 in case of 4 boundary dimensions.…”