A Closed Integral Form for the Background Gauge Connection

TL;DR

This paper derives a closed integral form for the background gauge connection using Fock-Schwinger gauge properties, enabling exact calculations of one-loop effective Lagrangians in quantum field theory across various dimensions and gauge groups.

Contribution

It introduces a novel closed integral form of the background gauge connection, simplifying the calculation of one-loop effective Lagrangians for arbitrary orders of covariant derivatives.

Findings

Derived the closed integral form of the gauge connection.

Validated the form by reproducing known results for real boson fields up to 8 mass dimensions.

Applicable to quantum field theories in arbitrary dimensions and gauge groups.

Abstract

By the appropriate use of the Fock-Schwinger gauge properties, we derive the closed integral form of the `point-split' non-local background gauge connection originally expressed as a finite sum. This is achieved in the limit when the finite sum becomes infinite. With this closed integral form of the connection, we obtain the same exact results in the calculation of one-loop effective Lagrangian accommodating arbitrary orders of covariant field derivatives in quantum field theory of arbitrary spacetime dimensions and of arbitrary gauge group. Particularly, we display the one-loop effective Lagrangian for real boson fields up to 8 mass dimensions-the same result obtained when the connection was yet in the finite sum form.