Equations from Non-Linear Chiral Transformations

TL;DR

This paper demonstrates that non-linear chiral transformations can derive integral equations for propagators, revealing complex radiative effects and connecting to Dyson-Schwinger and Bethe-Salpeter equations, with applications beyond LEP energies.

Contribution

It introduces a novel approach using non-linear chiral transformations to derive exact integral equations for propagators in the Nambu-Jona-Lasinio model, capturing richer physical effects.

Findings

Derived exact integral equations from non-linear chiral transformations.

Connected these equations to Dyson-Schwinger and Bethe-Salpeter equations.

Applicable to high-energy events beyond LEP energies.

Abstract

In comparison with the $WT$ chiral identity which is indispensable for renormalization theory, relations deduced from the non-linear chiral transformation have a totally different physical significance. We wish to show that non-linear chiral transformations are powerful tools to deduce useful integral equations for propagators. In contrast to the case of linear chiral transformations, identities derived from non-linear ones contain more involved radiative effects and are rich in physical content. To demonstrate this fact we apply the simplest non-linear chiral transformation to the Nambu-Jona-Lasinio model, and show how our identity is related to the Dyson-Schwinger equation and Bethe-Salpeter amplitudes of the Higgs and $\pi$. Unlike equations obtained from the effective potential, our resultant equation is exact and can be used for events beyond the LEP energy.