Three easy exercises in off-shell string-inspired methods

TL;DR

This paper introduces three exercises demonstrating off-shell string-inspired methods (OSSIM) in QCD, extending gauge invariance, simplifying loop integral calculations, and applying non-perturbative techniques to string breaking.

Contribution

It presents novel extensions of OSSIM beyond Feynman gauge, simplifies loop integral expressions, and applies world-line techniques to non-perturbative string breaking phenomena.

Findings

Gauge invariance of OSSIM is demonstrated beyond Feynman gauge.

Loop integrals can be expressed using Feynman parameters with minimal effort.

Non-perturbative string breaking is modeled using world-line techniques.

Abstract

Off-shell string-inspired methods (OSSIM) calculate off-shell QCD Green's functions using Schwinger-Feynman proper-time techniques, always in the background field method (BFM) Feynman gauge for technical convenience, and so far only at one loop. We already know that these results are gauge-invariant, because this gauge realizes the prescriptions of the Pinch Technique (PT), a Feynman-graph formulation for any gauge, but the idea of the first exercise is to show this directly in OSSIM. In this exercise we extend proper-time OSSIM beyond the BFM Feynman gauge so that one can apply PT algorithms, and show that the intrinsic PT is equivalent to resolving ambiguities in OSSIM in other gauges. In the second exercise we use forty-year-old rules of the author and Tiktopoulos for expressing loop integrals with numerator momenta directly in terms of Feynman parameters after momentum integration (the goal of OSSIM) and show that these rules elegantly and with economy of effort give rise, at least at one loop, to standard OSSIM algorithms. In the third exercise we apply world-line techniques to the problem of the breaking of adjoint strings, requiring a non-perturbative treatment that in the end reduces to a variant of the Schwinger result for production of electron-positron pairs in an electric field. This generalizes OSSIM to non-perturbative processes.