Thermodynamics of a gas of hadrons with attractive and repulsive interactions within an $S$-matrix formalism

We report the effect of including repulsive interactions on various thermodynamic observables calculated using an $S$-matrix based hadron resonance gas (HRG) model applied to already available corresponding results with only attractive interactions [A. Dash, S. Samanta, and B. Mohanty, Phys. Rev. C 97, 055208 (2018)]. The attractive part of the interaction is calculated by parametrizing the two-body phase shifts using the $K$-matrix formalism, while the repulsive part is included by fitting to the experimental phase shifts which carry the information about the nature of the interaction. We find that the bulk thermodynamic variables for a gas of hadrons, such as energy density, pressure, entropy density, speed of sound, and specific heat, are suppressed by the inclusion of repulsive interactions and are more pronounced for second- and higher-order correlations and fluctuations, particularly for the observables ${\chi }_Q^2, {\chi }_B^2-{\chi }_B^4$, and $C_{BS}$ in the present model. We find a good agreement between lattice QCD simulations and the present model for $C_{BS}$. We have also computed two leading-order Fourier coefficients of the imaginary part of the first-order baryonic susceptibility at imaginary baryon chemical potential within this model and compared them with the corresponding lattice results. Additionally, assuming that the value of interacting pressure versus temperature for a gas of hadrons calculated in the $S$-matrix formalism is the same as that from a van der Waals HRG (VDWHRG) model, we have quantified the attractive and repulsive interactions in our model in terms of attractive and repulsive parameters used in the VDWHRG model. The values of parameters thus obtained are $a=1.54\pm 0.064$ GeV ${\text{fm}}^3$ and $r=0.81\pm 0.014$ fm.

Figure 1

Energy dependence of $NN$ scattering phase shifts taken from SAID partial-wave analysis [60]. The notation to specify $NN$ scattering channels is ${}^{2S+1}l_J$, where $l,S,J$ correspond to orbital, spin, and total angular momentum respectively.

Figure 2

Temperature dependence of various thermodynamic quantities [(a) $P/T^4$, (b) $\varepsilon /T^4$, (c) $s/T^3$, (d) $(\varepsilon -3P)/T^4$, (e) $C_V/T^3$, and (f) $c_S^2]$ at zero chemical potential. “Total” contains both the attractive and repulsive interactions whereas KM contains only the attractive part. IDHRG 1 corresponds to results of the ideal HRG model, with same number of particles as used in the KM formalism. IDHRG (PDG 2016) in (a) corresponds to results of the ideal HRG model for all the hadrons and resonances listed in PDG 2016 [68]. Results are compared with lattice QCD data of Refs. [21] (WB) and [22] (HotQCD).

Figure 3

Temperature dependence of secondorder susceptibilities [(a) ${\chi }_B^2$, (b) ${\chi }_Q^2$, (c) ${\chi }_{BS}^{11}$, and (d) ${\chi }_{BQ}^{11}]$ at zero chemical potential. “Total” contains both the attractive and repulsive interactions whereas KM contains only the attractive part. IDHRG 1 corresponds to results of the ideal HRG model, with same number of particles as used in the KM formalism. Results are compared with lattice QCD data of Refs. [17] (WB), [19] (HotQCD), and [23] (Lattice).

Figure 4

The left panel shows the variation of Fourier coefficients $b_j\left(T\right)$ with temperature, computed using the $S$-matrix formalism, compared to the lattice results from Ref. [84]. Open symbols (circles and triangles) denote the result from lattice QCD. Solid (black) line denotes the result of Fourier coefficient $b_1\left(T\right)$. The dot-dashed purple line and the dotted red line represent the positive and negative parts of the Fourier coefficient $b_2\left(T\right)$ respectively. The right panel shows the variation of ${\chi }_B^2-{\chi }_B^4$ with temperature at zero chemical potential. Nonzero values come mainly due the $NN$ interaction, which is shown by the dashed blue and solid purple lines, denoting the positive and negative parts assuming FD statistics in the ideal part. The dot-dashed purple line and the dotted red line represent the positive and negative parts of the Fourier coefficient $6b_2\left(T\right)$ respectively (see text). Results are compared with lattice QCD data of Ref. [23] (Lattice), with open red and solid green symbols denoting the positive and negative parts.

Figure 5

The temperature dependence of $C_{BS}$ at zero chemical potential calculated in the current work (Total). IDHRG 1 corresponds to results of the ideal HRG model, with the same number of particles as used in the KM or $S$-matrix formalism. IDHRG (PDG 2016) corresponds to results of the ideal HRG model using the hadronic spectrum of PDG 2016 [68]. Results of the ideal HRG model with additional resonances which are not yet confirmed are also shown [IDHRG (PDG 2016+)]. Lattice QCD data of $C_{BS}$ are taken from Refs. [17] (WB) and [19] (HotQCD).

Figure 6

Variation of ${\overline{P}}_{\text{int}}/n^2$ with temperature at zero chemical potential. “Total” corresponds to both the attractive and repulsive interactions in the HRG model from the current work. The curve is fitted with the straight line $-TB\left(T\right)=-bT+a$ [see Eq. (25)].