Abstract
We elaborate on some general aspects of the crossing symmetric approach of Polyakov to the conformal bootstrap, as recently formulated in Mellin space. This approach uses, as building blocks, Witten diagrams in AdS. We show the necessity for having contact Witten diagrams, in addition to the exchange ones, in two different contexts: a) the large c expansion of the holographic bootstrap b) in the ϵ expansion at subleading orders to the ones studied already. In doing so, we use alternate simplified representations of the Witten diagrams in Mellin space. This enables us to also obtain compact, explicit expressions (in terms of a 7F6 hypergeometric function!) for the analogue of the crossing kernel for Witten diagrams i.e., the decomposition into s-channel partial waves of crossed channel exchange diagrams.
Highlights
Relate an expansion of the s-channel amplitude, in conformal blocks, with the t-channel one [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]
The log pieces in the former were interpreted as corrections from anomalous dimensions and the holographic bootstrap confirmed that these matched with the contact Witten diagram contributions
In this paper we have elaborated on the crossing symmetric formalism introduced by Polyakov and recast in Mellin space in [21, 22]
Summary
We describe a couple of different useful forms in which we can cast Witten exchange diagrams in Mellin space (with identical external scalars for simplicity). These results will play a direct role in our considerations of the following sections. We will systematise the discussion of the contact Witten diagrams in Mellin space. The measure factor (due to Mack) Γ2(−t)Γ2(s + t)Γ2(∆φ − s) will be denoted as ρ∆φ(s, t). Under this transform, various position space entities have their corresponding Mellin space representation. In the following we will describe various useful forms for M∆(s,)l(s, t)
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