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

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 $\pi N\mathrm{\Delta }$ 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 $\mathrm{\Delta }$-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.

Figure 1

The interaction contribution to the pressure and baryon number susceptibility for the $\pi N\mathrm{\Delta }$ system under various schemes, all scaled to dimensionless units, versus temperature at zero chemical potential. Free gas corresponds to an ideal gas of pions, nucleons, and $\mathrm{\Delta }$ baryons. $S$ matrix ${\delta }_{\mathrm{\Delta }}$ makes use of the empirical $\pi N$($P_{33}$) phase shift data. The scheme ${\delta }_{\mathrm{\Delta }}+{\delta }_{rc}$ is a similar $S$-matrix treatment with an additional $P$-wave hard-core phase shift imposed. Model A of the excluded volume approach assigns a universal eigenvolume of radius $r_0=0.3$ fm to all hadrons, while model B is a similar implementation but imposing repulsion only between pions and nucleons. Details of the schemes are discussed in the text.

Figure 2

Left-hand figure: The $\pi N$ phase shifts in the $P_{33}$ channel, as a function of center-of-mass energy $M$. The ${\delta }_{\mathrm{\Delta }}$ is from Eq. (14) and ${\delta }_{rc}$ is from Eq. (18). Right-hand figure: The corresponding effective spectral functions $\mathcal{B}\left(M\right)$ calculated from Eq. (12), as a function of $M$. The symbols, GW WI08, are the empirical phase shift values from Ref. [19].