Pion-Nucleon Scattering in Kadyshevsky Formalism: I Meson Exchange Sector

TL;DR

This paper develops a formalism for pion-nucleon scattering using Kadyshevsky approach, focusing on meson exchange diagrams, and introduces methods for handling complex couplings and the second quantization of quasi particles.

Contribution

It introduces a comprehensive treatment of general couplings, including derivatives and higher spins, within the Kadyshevsky formalism, and presents new methods for practical calculations.

Findings

Application of Takahashi-Umezawa and Gross-Jackiw methods to Kadyshevsky formalism

Introduction of the $ar{P}$ method for practical calculations

Derivation of the Takahashi-Umezawa method from Bogoliubov theory

Abstract

In a series of two papers we present the theoretical results of $\pi N$/meson-baryon scattering in the Kadyshevsky formalism. In this paper the results are given for meson exchange diagrams. On the formal side we show, by means of an example, how general couplings, i.e. couplings containing multiple derivatives and/or higher spin fields, should be treated. We do this by introducing and applying the Takahashi-Umezawa and the Gross-Jackiw method. For practical purposes we introduce the $\bar{P}$ method. We also show how the Takashashi-Umezawa method can be derived using the theory of Bogoliubov and collaborators and the Gross-Jackiw method is also used to study the $n$-dependence of the Kadyshevsky integral equation. Last but not least we present the second quantization procedure of the quasi particle in Kadyshevsky formalism.