# Repulsive interactions and their effects on the thermodynamics of hadron gas

## TL;DR

This paper compares two models of repulsive interactions in hadron gases, revealing that a single phenomenological parameter cannot universally describe repulsion across different interaction channels, and highlights the utility of the S-matrix approach.

## Contribution

It demonstrates the limitations of the excluded volume approximation and advocates for the S-matrix formalism to accurately model hadronic repulsive interactions.

## Key findings

- Excluded volume approach conflicts with phase shift analysis for pion-nucleon interactions.
- Repulsive forces are channel-dependent and cannot be captured by a single parameter.
- S-matrix formalism provides estimates of effective eigenvolumes for hadrons.

## Abstract

We compare two approaches in modeling repulsive interactions among hadrons: the excluded volume approximation and the S-matrix formalism. These are applied to study the thermodynamics of the $\pi N \Delta$ system. It is shown that the introduction of an extraneous repulsion between pions and nucleons via the excluded volume approach, in addition to the interaction that generates the $\Delta$-resonance, is incompatible with the analysis based on the physical phase shift of pion-nucleon scattering in the $P_{33}$ channel. This finding suggests that the repulsive force between hadrons is interaction-channel dependent and is hence unlikely to be captured by a single phenomenological parameter. The S-matrix approach employed here can be used, however, to provide useful estimates of the magnitude of the effective eigenvolume.