Abstract
We consider 5-point functions in conformal field theories in d > 2 dimensions. Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal blocks appearing in 5-point functions of scalar operators, reducing them to a linear combination of blocks with scalars exchanged. We additionally derive recursion relations for the conformal blocks which appear when one of the external operators in the 5-point function has spin 1 or 2. Our results allow us to formulate positivity constraints using 5-point functions which describe the expectation value of the energy operator in bilocal states created by two scalars.
Highlights
Conformal field theories (CFTs) are remarkable quantum field theories (QFTs) endowed with an enhanced symmetry under the conformal group
Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal blocks appearing in 5-point functions of scalar operators, reducing them to a linear combination of blocks with scalars exchanged
We detail how to encode the corresponding blocks for symmetric traceless tensor exchange in terms of appropriate combinations of weight-shifting operators acting on lower-spin blocks for a 5-point function of purely scalar external operators
Summary
Conformal field theories (CFTs) are remarkable quantum field theories (QFTs) endowed with an enhanced symmetry under the conformal group. We detail how to encode the corresponding blocks for symmetric traceless tensor exchange in terms of appropriate combinations of weight-shifting operators acting on lower-spin blocks for a 5-point function of purely scalar external operators The latter objects act as the seed blocks and may in turn be computed with the aid of the recursion relations described in the previous section. We turn to the convenient and elegant framework afforded by the weight-shifting operator formalism [40] This approach empowers us to derive a set of efficient recursion relations for generating the 5-point conformal blocks for the case of arbitrary symmetric traceless exchange. We give a brief overview of this formalism, highlighting some of its essential features and laying out the basic method for obtaining recursion relations for conformal blocks
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