Loops in AdS: From the Spectral Representation to Position Space

TL;DR

This paper develops spectral representation techniques to compute scalar loop diagrams in AdS, simplifying complex loop diagrams into contact or exchange diagrams, with applications to conformal models and explicit results involving special functions.

Contribution

It introduces methods to reduce scalar loop diagrams in AdS to simpler diagrams using spectral identities, enabling explicit computations of 4-point functions and bubble diagrams.

Findings

Re-summation of bubble diagrams yields tree-level contact diagrams.

Explicit expressions for loop diagrams in terms of Lerch transcendent functions.

Identification of vertex identities linking bulk 2-point functions and boundary 4-point double-discontinuities.

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 tree-level contact diagram: the $\bar{D}_{1,1,\frac{3}{2},\frac{3}{2}} (z,\bar z)$ function. Another example is a scalar with $\phi^4$ or $\phi^3$ 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 $\Delta_{12}=0$ and $\Delta_{34}=1$ 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 $\bar z$. 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.