Beth-Uhlenbeck approach for repulsive interactions between baryons in a hadron gas

The quantum mechanical Beth-Uhlenbeck (BU) approach for repulsive hard-core interactions between baryons is applied to the thermodynamics of a hadron gas. The second virial coefficient $a_2$—the “excluded volume” parameter—calculated within the BU approach is found to be temperature dependent, and it differs dramatically from the classical excluded volume (EV) model result. At temperatures $T=100–200$ MeV, the widely used classical EV model underestimates the EV parameter for nucleons at a given value of the nucleon hard-core radius by large factors of 3–4. Previous studies, which employed the hard-core radii of hadrons as an input into the classical EV model, have to be re-evaluated using the appropriately rescaled EV parameters. The BU approach is used to model the repulsive baryonic interactions in the hadron resonance gas (HRG) model. Lattice data for the second- and fourth-order net baryon susceptibilities are described fairly well when the temperature dependent BU baryonic excluded volume parameter corresponds to nucleon hard-core radii of $r_c=0.25–0.3$ fm. Role of the attractive baryonic interactions is also considered. It is argued that HRG model with a constant baryon-baryon EV parameter $v_{NN}\simeq 1{\mathrm{fm}}^3$ provides a simple yet efficient description of baryon-baryon interaction in the crossover temperature region.

Figure 1

The temperature dependence of the nucleon-nucleon excluded volume parameter $v_{NN}$ (solid black line), the proton-proton excluded volume parameter $v_{pp}$ (dashed red line), and the proton-neutron excluded volume parameter $v_{pn}\equiv 2v_{NN}-v_{pp}$ (dashed red line), as calculated within the relativistic Beth-Uhlenbeck approach for a hard-core potential with the nucleon hard-core radius of $r_c=0.3$ fm. The dashed horizontal line shows the prediction of the classical EV model (6) with the same value of $r_c=0.3$ fm.

Figure 2

The temperature dependence of the nucleon-nucleon excluded volume parameter $v_{NN}$ calculated using the non-relativistic (solid black line) and relativistic (dash-dotted red line) dispersion relations in the Beth-Uhlenbeck approach for hard core interaction potential, shown on the logarithmic temperature scale. Nucleon hard-core radius of $r_c=0.3$ fm is assumed. The dashed horizontal line shows the prediction of the classical EV model (6) with the same value of $r_c=0.3$ fm.

Figure 3

The temperature dependence of the second virial coefficient $a_2\left(T\right)$ of nucleon-nucleon interaction, calculated within different approaches. The calculations within the relativistic Beth-Uhlenbeck approach for the system of nucleons with a hard-core interaction are depicted by the blue band, which results from the variation of the nucleon hard-core radius in the range $0.25<r_c<0.30$ fm. The calculations of Ref. [45] within the $S$-matrix formalism, employing the empirical phase shifts of $NN$ scattering, are depicted by the yellow line with a band. The red line depicts the second virial coefficient of nucleon-nucleon interaction in the quantum van der Waals model of nuclear matter [61]. Lattice QCD results for the second virial coefficient of “baryon-baryon interaction” [46], obtained from simulations at an imaginary baryochemical potential, are depicted by black symbols with error bars.

Figure 4

The temperature dependence of (a) ${\chi }_2^B$ and (b) ${\chi }_4^B/{\chi }_2^B$ net baryon number susceptibilities at ${\mu }_B=0$, as calculated within the I-HRG model (dashed black lines), the BU-HRG model (red bands), and the EV-HRG model (blue bands) with the temperature dependent baryon excluded volume parameter, using for all (anti)baryons the Beth-Uhlenbeck value for nucleons. The bands result from the variation of the nucleon hard-core radius in the range $r_c\simeq 0.25–0.3$ fm. The lattice QCD results of the Wuppertal-Budapest [4, 6] and HotQCD [68, 69] collaborations are shown by the full and open symbols, respectively. Solid lines correspond to the EV-HRG model with $v_{BB}=1{\mathrm{fm}}^3$.