Wilson network expansion for four-point contact and exchange scalar Feynman diagrams in AdS$_2$

Abstract

We derive new integral identities for AdS propagators and further develop the Wilson network expansion for AdS Feynman diagrams. In particular, we demonstrate that four-point contact and exchange scalar diagrams in two dimensions can be expanded into several infinite series of matrix elements of Wilson line network operators with running conformal weights. Each series is characterized by specific multi-trace operators associated with the external and intermediate edges of the corresponding graphs. The resulting expansions near the conformal boundary reproduce the well-known decompositions of the corresponding four-point Witten diagrams into conformal blocks.

Figures (12)

• Figure 1: Conformal block expansions of a four-point contact (a) and exchange (b) Witten diagrams. On the left-hand side, four scalars with masses $m^2_i = h_i(h_i-1)$ are located in the boundary points $z_i$, $i=1,2,3,4$; 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 is the 4-point scalar conformal block with specified intermediate conformal weights $\tilde{h}$ and $h_{ij|n} = h_i+h_j+2n$.
• Figure 2: The contour $C_i(\varepsilon)$ in the complex $z$-plane, $i=1,...,4$. The red crosses denote poles of the modified propagator $\widehat{G}({\bf x},{\bf x}_i,w_i)$.
• Figure 3: Integration contours $C_u$ and $C_z$ in the complex $u$- and $z$-planes, respectively. Red crosses and lines denote the branch points and cuts of the integrand in \ref{['transition_id']}.
• Figure 4: Contour $C$ in the complex $z$-plane. Red crosses and lines denote the branch points and cuts of the integrand in \ref{['cont_mod']}.
• Figure 5: Decomposition of a four-point contact AdS$_2\;$ Feynman diagram into diagrams with the modified propagators \ref{['G_hat']} and \ref{['G_tilde']}. On the left-hand side, four scalars with masses $m^2_i = h_i(h_i-1)$ are located in the bulk points ${\bf x}_i$, $i=1,2,3,4$; the central dot denotes the AdS integration, and the solid lines represent the standard bulk-to-bulk scalar propagators. On the right-hand side, the red lines denote the modified propagators $\widetilde{G}_h({\bf x},{\bf x}')$, while the dashed lines denote the modified propagators $\widehat{G}_h({\bf x},{\bf x}',w)$. The first diagram in this decomposition produces two terms in the final expansion \ref{['vert_decomp']}, whereas the other diagrams produce only one.