[Paper Review] Global Conformal Invariance and Bilocal Fields with Rational Correlation Functions

Nikolay Nikolov, Yassen S. Stanev+1 more|ArXiv.org|Nov 12, 2002

Black Holes and Theoretical Physics6 references3 citations

TL;DR

This paper proposes a global conformal invariant (GCI) framework for scalar fields in 4D Minkowski space, introducing bilocal conformal fields $V_\nu(x_1,x_2)$ in the operator product expansion (OPE) of two scalar fields. It establishes that correlation functions of these bilocal fields are rational and that $V_1$ decomposes into an infinite series of twist-2 conserved symmetric traceless tensors, providing a non-perturbative construction of GCI correlation functions in non-abelian gauge theories.

ABSTRACT

The singular part of the extit{operator product expansion} (OPE) of a pair of extit{globally conformal invariant} (GCI) scalar fields $ϕ$ of (integer) dimension $d$ can be written as a sum of the 2-point function of $ϕ$ and $d-1$ bilocal conformal fields $V_ν(x_1, x_2)$ of dimension $(ν, ν)$, $ν= 1, ..., d-1$. As the correlation functions of $ϕ(x)$ are proven to be rational [6], we argue that the correlation functions of $V_ν$ can also be assumed rational. Each $V_ν(x_1, x_2)$ is expanded into local symmetric tensor fields of extit{twist} (dimension minus rank) $2ν$. 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.

Motivation & Objective

• To generalize the OPE structure of globally conformal invariant (GCI) scalar fields in 4D Minkowski space beyond the $d=2$ case.
• To establish that correlation functions of bilocal fields $V_\nu$ arising in the OPE are rational, extending results from the $d=2$ case.
• To analyze the structure of $V_1$ as an infinite series of twist-2 conserved symmetric traceless tensor fields.
• To construct a 4-parameter family of truncated 4-point functions for a $d=4$ scalar field $\mathcal{L}(x)$, with no lower-dimensional fields contributing.
• To provide a first step toward a non-perturbative construction of GCI correlation functions in non-abelian gauge theories.

Proposed method

• The OPE of two GCI scalar fields $\phi(x_1)\phi(x_2)$ is decomposed into a 2-point function, $d-1$ bilocal conformal fields $V_\nu(x_1,x_2)$ of dimension $(\nu,\nu)$, and a normal-ordered product.
• Bilocal fields $V_\nu$ are expanded into local symmetric traceless tensor fields of twist $2\nu$, with universal coefficients independent of the specific theory.
• Crossing symmetry and conformal invariance are used to derive that the vacuum expectation value $\langle 0|V_\nu(x_1,x_2)V_\nu(x_3,x_4)|0\rangle$ is a rational function of conformally invariant cross ratios $s$ and $t$, satisfying $s_{12}$-symmetry.
• The d’Alembert equation $\Box_1 V_1 = \Box_2 V_1 = 0$ is derived from 3-point functions, enabling the computation of the function $f_d(s,t)$ in the 4-point function.
• A 4-parameter family of truncated 4-point functions $\mathcal{W}_4^t$ is constructed for a $d=4$ scalar field $\mathcal{L}(x)$, with $F_1$ and $F_2$ contributions from twist-2 and twist-4 fields.
• The OPE of $V_2$ is shown to involve $\mathcal{L}$ when $2b_1 + b_2 \neq 0$, indicating non-vanishing 3-point functions and a non-abelian structure.

Experimental results

Research questions

• RQ1 Can the OPE of two globally conformal invariant scalar fields in 4D be systematically extended beyond the $d=2$ case using bilocal conformal fields?
• RQ2 Are the correlation functions of the bilocal fields $V_\nu$ rational, given that the scalar field $\phi$ has rational correlation functions?
• RQ3 How does the bilocal field $V_1$ decompose into local conserved symmetric traceless tensor fields, and what is the role of conformal invariance in this decomposition?
• RQ4 What is the structure of the truncated 4-point function $\mathcal{W}_4^t$ for a $d=4$ scalar field, and how many independent parameters does it depend on?
• RQ5 Under what conditions does the OPE of $V_2$ involve the scalar field $\mathcal{L}$, and what does this imply for the underlying gauge theory?

Key findings

• The bilocal field $V_1(x_1,x_2)$ satisfies the d’Alembert equation $\Box_1 V_1 = \Box_2 V_1 = 0$, which follows from the conformal invariance of its 3-point functions with local tensor fields.
• The vacuum expectation value $\langle 0|V_1(x_1,x_2)V_1(x_3,x_4)|0\rangle$ is expressed as $[(13)(24) + (14)(23)]f_d(s,t)$, where $f_d(s,t)$ is a rational function of cross ratios $s$ and $t$ satisfying $s_{12}$-symmetry.
• The function $f_d(s,t)$ is computed explicitly for the $d=4$ case, showing that $V_1$ is an infinite sum of twist-2 conserved symmetric traceless tensor fields.
• A 4-parameter family of truncated 4-point functions $\mathcal{W}_4^t$ is constructed for a $d=4$ scalar field $\mathcal{L}(x)$, with no contributions from lower-dimensional fields.
• Only a 1-parameter subset of this family corresponds to a free abelian gauge theory, indicating that the general solution describes a non-abelian gauge theory.
• The OPE of $V_2(x_1,x_2)$ involves $\mathcal{L}(x)$ when $2b_1 + b_2 \neq 0$, implying a non-vanishing 3-point function $\langle V_2 \mathcal{L} \rangle$, which signals a non-abelian structure.