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Wilson network decomposition of AdS Feynman diagrams in two dimensions

K. B. Alkalaev, V. S. Khiteev

TL;DR

This work develops a Wilson-network framework for AdS$_2$ Feynman diagrams with all endpoints in the bulk, showing that such diagrams decompose into an infinite series of Wilson line network matrix elements analogous to conformal block decompositions. Central to the construction are three Euclidean AdS$_2$ propagator identities (conversion, superposition, splitting) that rewrite standard bulk propagators in terms of modified propagators, enabling a clear separation into single-trace and double-trace contributions. The authors then formulate Wilson line networks as basis functions, deriving a concrete 3-point AdS$_2$ decomposition where the leading term corresponds to the single-trace AdS vertex function and the remaining terms are double-trace AdS vertex functions, mirroring the boundary conformal-block structure. This provides a principled bulk analogue of conformal-block expansions and suggests a path to generalize to $n$-point and higher-spin diagrams, with potential wide-ranging implications for holographic decompositions and bulk-coding of CFT data.

Abstract

We show that Feynman diagrams in AdS$_2$ space can be decomposed into infinite series of matrix elements of Wilson line network operators. The case of the 3-point scalar Feynman diagram with endpoints in the bulk is studied in detail. The resulting decomposition is similar to the conformal block decomposition of Witten diagrams, i.e. it comprises a single-trace term and infinite sums of double-trace terms. We derive a number of AdS propagator identities which relate the standard bulk-to-bulk propagators with the modified bulk-to-bulk propagators of two different types responsible for extracting single-trace and double-trace terms.

Wilson network decomposition of AdS Feynman diagrams in two dimensions

TL;DR

This work develops a Wilson-network framework for AdS Feynman diagrams with all endpoints in the bulk, showing that such diagrams decompose into an infinite series of Wilson line network matrix elements analogous to conformal block decompositions. Central to the construction are three Euclidean AdS propagator identities (conversion, superposition, splitting) that rewrite standard bulk propagators in terms of modified propagators, enabling a clear separation into single-trace and double-trace contributions. The authors then formulate Wilson line networks as basis functions, deriving a concrete 3-point AdS decomposition where the leading term corresponds to the single-trace AdS vertex function and the remaining terms are double-trace AdS vertex functions, mirroring the boundary conformal-block structure. This provides a principled bulk analogue of conformal-block expansions and suggests a path to generalize to -point and higher-spin diagrams, with potential wide-ranging implications for holographic decompositions and bulk-coding of CFT data.

Abstract

We show that Feynman diagrams in AdS space can be decomposed into infinite series of matrix elements of Wilson line network operators. The case of the 3-point scalar Feynman diagram with endpoints in the bulk is studied in detail. The resulting decomposition is similar to the conformal block decomposition of Witten diagrams, i.e. it comprises a single-trace term and infinite sums of double-trace terms. We derive a number of AdS propagator identities which relate the standard bulk-to-bulk propagators with the modified bulk-to-bulk propagators of two different types responsible for extracting single-trace and double-trace terms.

Paper Structure

This paper contains 27 sections, 6 theorems, 117 equations, 10 figures.

Table of Contents

  1. Introduction
  2. HKLL reconstruction and Witten diagrams
  3. Reconstruction formula
  4. Conformal block decomposition
  5. AdS integral identities and modified propagators
  6. Three identities
  7. Modified AdS Feynman diagrams
  8. Decomposition.
  9. Equations of motion.
  10. Boundary asymptotics.
  11. Contact diagrams.
  12. Commentary
  13. Wilson network decomposition of AdS Feynman diagrams
  14. Wilson networks as basis functions
  15. Deriving the Wilson network decomposition
  16. ...and 12 more sections

Key Result

Lemma 1

The AdS$_2\;$ bulk-to-bulk scalar propagator has the following integral representation: where ${\bf x} = (u,z)$, ${\bf x}' = (u',z')$ and $u,u' \in \mathbb{R}$ and $w, z, z'\in \mathbb{C}$.

Figures (10)

  • Figure 1: The conformal block decomposition of the 4-point exchange Witten diagram. On the left-hand side: there are four scalars of masses $m_i^2 = h_i(h_i-1)$, $i=1,...,4$, and one exchange scalar of mass ${\tilde{m}}^2 = \tilde{h} (\tilde{h}-1)$; the boundary points $z_i$; the two central dots denote the AdS integrations; the straight lines are the standard bulk-to-bulk and bulk-to-boundary scalar propagators. On the right-hand side: each term in this sum is the 4-point scalar conformal block with specified intermediate conformal dimensions $\tilde{h}$ and $h_{ij|n} = h_i+h_j+2n$, the $c$-coefficients are given in section \ref{['sec:wit']}.
  • Figure 2: The Wilson line network decomposition of the 3-point AdS$_2\;$ Feynman diagram. On the left-hand side: there are three scalars of masses $m^2_i = h_i(h_i-1)$ located in the bulk points $x_i$, $i=1,2,3$; the central dot denotes the AdS integration; the straight lines are the standard bulk-to-bulk scalar propagators. On the right-hand side: 3-valent graphs denote matrix elements of Wilson line networks which are Wilson lines (wavy lines) of three different weights connected by $sl(2,\mathbb{R})$ intertwiners; weights $h_{ij|n} = h_i+h_j+2n$, the $\kappa$-coefficients are given in section \ref{['sec:wilson_basis']}.
  • Figure 3: The exchange Witten diagram \ref{['4pt_expansion']} of four scalars of mass $m_i^2 = h_i(h_i-1)$ and one exchange scalar of mass ${\tilde{m}}^2 = \tilde{h} (\tilde{h}-1)$. The conformal boundary $\partial$AdS$_2\;$ is shown by the circle, the two bulk points are ${\bf x}$ and ${\bf x}'$. Each term in this decomposition is a geodesic Witten diagram obtained by restricting the integration domain of the exchange Witten diagram to geodesics (dashed lines) connecting boundary points $z_1, z_2$ and $z_3, z_4$. Each geodesic Witten diagram computes the conformal block with particular intermediate conformal weight Hijano:2015zsa.
  • Figure 4: (a) The contour $C$ on the complex $z$-plane $\mathbb{C}$. The red crosses are poles of the analytically continued propagator $\widehat{G}_h({\bf x}(u,z),{\bf x}'(u',z'),w)$. (b) The Pochhammer contour $P$ in $\mathbb{C}\ \{w-iu,w+iu\}$.
  • Figure 5: The contour $C$ on the complex $z$-plane. The red crosses are poles of the integrand \ref{['3ptVertexInterm']}; the regularization parameter $\varepsilon \to 0$.
  • ...and 5 more figures

Theorems & Definitions (6)

  • Lemma 1: the conversion identity
  • Lemma 2
  • Lemma 3: the splitting identity
  • Corollary 1
  • Lemma 4: the superposition identity
  • Lemma 5: the multipoint superposition identity