Global Conformal Invariance and Bilocal Fields with Rational Correlation Functions

Abstract

The singular part of the \textit{operator product expansion} (OPE) of a pair of \textit{globally conformal invariant} (GCI) scalar fields $\phi$ of (integer) dimension $d$ can be written as a sum of the 2-point function of $\phi$ and $d-1$ bilocal conformal fields $V_{\nu}(x_1, x_2)$ of dimension $(\nu, \nu)$, $\nu = 1, ..., d-1$. As the correlation functions of $\phi(x)$ are proven to be rational [6], we argue that the correlation functions of $V_{\nu}$ can also be assumed rational. Each $V_{\nu}(x_1, x_2)$ is expanded into local symmetric tensor fields of \textit{twist} (dimension minus rank) $2\nu$. The case $d=2$, considered previously [5], is briefly reviewed and current work on the $d=4$ case (of a Lagrangean density in 4 space--time dimensions) is previewed.