Abstract
In this work we present a closed form expression for Polyakov blocks in Mellin space for arbitrary spin and scaling dimensions. We provide a prescription to fix the contact term ambiguity uniquely by reducing the problem to that of fixing the contact term ambiguity at the level of cyclic exchange amplitudes — defining cyclic Polyakov blocks — in terms of which any fully crossing symmetric correlator can be decomposed. We also give another, equivalent, prescription which does not rely on a decomposition into cyclic amplitudes. We extract the OPE data of double-twist operators in the direct channel expansion of the cyclic Polyakov blocks using and extending the analysis of [1, 2] to include contributions that are non-analytic in spin. The relation between cyclic Polyakov blocks and analytic Bootstrap functionals is underlined.
Highlights
The following, this prescription allows to uniquely fix the entire contact term ambiguity in a simple manner
In this work we present a closed form expression for Polyakov blocks in Mellin space for arbitrary spin and scaling dimensions
We provide a prescription to fix the contact term ambiguity uniquely by reducing the problem to that of fixing the contact term ambiguity at the level of cyclic exchange amplitudes — defining cyclic Polyakov blocks — in terms of which any fully crossing symmetric correlator can be decomposed
Summary
Starting from the Mellin representation (1.9) of an s-channel exchange, one can generate the corresponding crossing symmetric solution by adding up the different channels. Any other choice of contact term would violate the condition (2.11), which makes clear that it is a universal contribution to exchange amplitudes With this prescription, the Polyakov block for identical external legs is given unambiguously by (2.5). 3 we shall determine the conformal block expansion of the cyclicly ordered Polyakov blocks (1.16) in the direct channel To this end, for each cyclic exchange amplitude (1.15) it is convenient to separate the pole and polynomial parts in s as we did for the exchange amplitude (2.5) above which, respectively, identify the terms which. In Eτ(,14|m23)(s, t) the only polynomial contribution in s is entirely proportional to p −1(X, Y ) This implies that the contact term ambiguity p −1(X, Y ) parameterises the non-analyticity in spin of the conformal block decomposition of Eτ(,14|m23)(s, t) in the direct-channel.
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