## Abstract

We compute a family of scalar loop diagrams in AdS. We use the spectral representation to derive various bulk vertex/propagator identities, and these identities enable to reduce certain loop bubble diagrams to lower loop diagrams, and often to tree- level exchange or contact diagrams. An important example is the computation of the finite coupling 4-point function of the large-N conformal O(N) model on AdS_{3}. Remarkably, the re-summation of bubble diagrams is equal to a certain contact diagram: the D¯1,1,32,32(zz¯) function. Another example is a scalar with ϕ^{4} or ϕ^{3} coupling in AdS_{3}: we compute various 4-point (and higher point) loop bubble diagrams with alternating integer and half- integer scaling dimensions in terms of a finite sum of contact diagrams and tree-level exchange diagrams. The 4-point function with external scaling dimensions differences obeying ∆_{12} = 0 and ∆_{34} = 1 enjoys significant simplicity which enables us to compute in quite generality. For integer or half-integer scaling dimensions, we show that the M -loop bubble diagram can be written in terms of Lerch transcendent functions of the cross- ratios z and z¯. Finally, we compute 2-point bulk bubble diagrams with endpoints in the bulk, and the result can be written in terms of Lerch transcendent functions of the AdS chordal distance. We show that the similarity of the latter two computations is not a coincidence, but arises from a vertex identity between the bulk 2-point function and the double-discontinuity of the boundary 4-point function.

Original language | English |
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Article number | 49 |

Journal | Journal of High Energy Physics |

Volume | 2020 |

Issue number | 6 |

DOIs | |

State | Published - 1 Jun 2020 |

Externally published | Yes |

### Bibliographical note

Funding Information:This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

Publisher Copyright:

© 2020, The Author(s).

## Keywords

- AdS-CFT Correspondence
- Conformal Field Theory

## ASJC Scopus subject areas

- Nuclear and High Energy Physics