Global Conformal Invariance and Bilocal Fields with Rational Correlation Functions

TL;DR

This paper explores the structure of bilocal conformal fields in globally conformal invariant theories, demonstrating their rational correlation functions and expanding on the case of four-dimensional fields relevant to Lagrangian densities.

Contribution

It introduces the assumption that bilocal conformal fields have rational correlation functions and analyzes their expansion into local tensor fields, extending previous work from two to four dimensions.

Findings

Correlation functions of bilocal fields are rational.

Bilocal fields expand into local tensor fields of specific twist.

Extension from 2D to 4D conformal field theories.

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.