Physics Overview

The Hadron Resonance Gas Model

The Hadron Resonance Gas (HRG) model describes strongly-interacting matter in the hadronic phase by assuming hadrons to be the fundamental degrees of freedom in a thermodynamic system. Beginning from a system of interacting hadrons in their ground state, the HRG model derives thermodynamic quantities by creating larger and larger, non-interacting resonances of those hadrons. This non-kinetic usage of energy—which increases the number of species and their masses, rather than the momentum per particle reproduces experimental observations.

The partition function of the HRG model is summed over all known hadron species and resonances categorized by the Particle Data Group (PDG), having the following form:

\[\begin{split}\begin{aligned} \ln \mathcal{Z}^{\text{HRG}} (T,V,\mathbf{\mu}) & = \sum_{i \in \text{PDG}} \ln \mathcal{Z}_i (T,V,\mathbf{\mu}) \\ & = \sum_{i \in \text{PDG}} \pm \frac{V g_i}{2\pi^2} \int p^2 dp \, \ln \big[ 1 \pm \lambda_i (T, \mathbf{\mu}) \exp(-\beta \epsilon_i) \big] \end{aligned}\end{split}\]

where

$\pm$ for fermions and bosons, respectfully
$\epsilon_i = \sqrt{ \mathbf{p}^2 + m_i^2}$ is the rest energy
$g_i$ is the spin degeneracy factor
$\lambda_i (T, \mathbf{\mu}) = \exp\!\Big[ (B_i \mu_B + S_i \mu_S + Q_i \mu_Q)/T \Big]$ is the fugacity, with baryon ($B$), strange ($S$) and electric ($Q$) charge.

For our analysis, the partition function can be simplified through the expansion of the logarithm with respect to the fugacity:

\[\ln \mathcal{Z}_i (T,V,\mathbf{\mu}) = \frac{V g_i}{2\pi^2} \sum_{N = 1}^{\infty} \frac{(\pm 1)^{N+1} \lambda_i^N}{N} \int p^2 dp \, \Big[ \exp(-\beta \epsilon_i) \big]^N\]

Performing the momentum integral, the partition function is expressed with the Bessel functions:

\[\ln \mathcal{Z}_i (T,V,\mathbf{\mu}) = \frac{V g_i}{2\pi^2} \sum_{N = 1}^{\infty} \frac{(\pm 1)^{N+1} \lambda_i^N}{N} m_i^2 K_2\!\left( N \frac{m_i}{T}\right)\]

This expression of the partition function allows for an analytical form for thermodynamic quantities, but it can be further simplified by considering the sum of both the hadron and anti-hadron (all charge signs flipped):

\[\begin{split}\begin{aligned} \ln \mathcal{Z}_i (T,V,\mathbf{\mu}) + \ln & \mathcal{Z}_{\overline{i}} (T,V,\mathbf{\mu}) = \frac{V g_i}{2\pi^2} \sum_{N = 1}^{\infty} (\pm 1)^{N+1} \frac{m_i^2 K_2\!\left( N \frac{m_i}{T} \right)}{N} \Big[ \lambda_i^N + \lambda_i^{-N} \Big] \\ &= \frac{V g_i}{2\pi^2} \sum_{N = 1}^{\infty} (\pm 1)^{N+1} m_i^2 K_2\!\left( N \frac{m_i}{T} \right) \cosh\!\left( N \frac{B_i \mu_B + S_i \mu_S + Q_i \mu_Q}{T} \right) \end{aligned}\end{split}\]

This form allows us to isolate different hadron contributions to the full pressure by distinguishing them by the quantum numbers baryon ($B$), strange ($S$) and electric ($Q$) charge. We consider the first time of the expansion, $N = 1$, corresponding to the Boltzmann approximation, where higher terms are suppressed as long as $m_N-\mu_B \leq T$.

The proportionality to the cosh functions and their behaviour with respect to the partial derivatives will prove useful in finding partial pressures. To summarize, the behaviour of interests follows:

\[\begin{split}\begin{aligned} \frac{\partial^n \big( P / T^4 \big)}{\partial (\mu_i / T)^n} &= F_n (T) \cosh(\mu_i / T) \hspace{24pt} \text{for even $n$} \\ \frac{\partial^n \big( P / T^4 \big)}{\partial (\mu_i / T)^n} &= F_n (T) \sinh(\mu_i / T) \hspace{24pt} \text{for odd $n$} \end{aligned}\end{split}\]

Susceptibilities

Within the analysis of thermodynamic quantities of hadron, sensitivity to charge fluctuations plays an important role comparing experimental results or contrasting theoretical models. We quantify charge fluctuations through the susceptibilities of charges:

\[\chi_{mnk}^{BSQ}=\frac{\partial^{(m+n+k)}(P/T^4)}{\partial(\mu_B/T)^m \partial(\mu_S/T)^n \partial(\mu_Q/T)^k} \Biggr|_{{\mu}_{BQS}=0}\]

This may be rewritten in terms of the dimensionless chemical potential $\hat{\mu}_i = \mu_i / T$ :

\[\chi_{mnk}^{BSQ}=\frac{\partial^{(m+n+k)}(P/T^4)}{\partial\hat{\mu}_B^m \partial\hat{\mu}_S^n \partial\hat{\mu}_Q^k} \Biggr|_{{\mu}_{BQS}=0}\]

Under the Boltzmann approximation of the HRG pressure, odd powers of susceptibilities vanishes as $\mu_i = 0$.

Partial Pressure

Hadrons can be categorized by the $B,Q,S$ charge number contents. This allows us to rewrite the HRG pressure in terms of pressure based on charge number contents [Insert Reff] :

\[\begin{split}P\left(\hat{\mu}_{B}, \hat{\mu}_{S}, \hat{\mu}_{Q}\right) &=P_{000}+P_{00|1|} \cosh \left(\hat{\mu}_{Q}\right)+P_{100} \cosh \left(\hat{\mu}_{B}\right) \\ &+P_{101} \cosh \left(\hat{\mu}_{B}+\hat{\mu}_{Q}\right)+P_{10-1} \cosh \left(\hat{\mu}_{B}-\hat{\mu}_{Q}\right) \\ &+P_{102} \cosh \left(\hat{\mu}_{B}+2 \hat{\mu}_{Q}\right)+P_{0|1| 0} \cosh \left(\hat{\mu}_{S}\right) \\ &+P_{0|1||1|} \cosh \left(\hat{\mu}_{S}+\hat{\mu}_{Q}\right)+P_{1|1| 0} \cosh \left(\hat{\mu}_{B}-\hat{\mu}_{S}\right) \\ &+P_{1|1| 1} \cosh \left(\hat{\mu}_{B}-\hat{\mu}_{S}+\hat{\mu}_{Q}\right) \\ &+P_{1|1|-1} \cosh \left(\hat{\mu}_{B}-\hat{\mu}_{S}-\hat{\mu}_{Q}\right) \\ &+P_{1|2| 0} \cosh \left(\hat{\mu}_{B}-2 \hat{\mu}_{S}\right) \\ &+P_{1|2||1|} \cosh \left(\hat{\mu}_{B}-2 \hat{\mu}_{S}-\hat{\mu}_{Q}\right) \\ &+P_{1|3||1|} \cosh \left(\hat{\mu}_{B}-3 \hat{\mu}_{S}-\hat{\mu}_{Q}\right)\end{split}\]

where $P_{ijk} \equiv P_{ijk} (T)$ are the partial pressures we want to obtain.

Method to Determine the Partial Pressure From Susceptibilities

Applying the expression of the pressure above to the definition of the susceptibilities, we can obtain a linear relationship to express the susceptibilities in terms of the partial pressures. This is expressed in the form:

\[\mathcal{X} = M \mathcal{P}\]

we will consider each partial pressures to be a linear combination of the susceptibilities:

\[p_i = \sum^{21}_{j = 1} \alpha_j \chi_j\]

where $\alpha_j$ are constants to be solved. This form allows us to re-express the linear system into an equivalent form:

\[M^T \pmb{\alpha} = \mathbf{e}_i\]

where $i$ denoted the index of a particular partial pressure.

One additional implementation we apply to solving the equivalent linear system is to consider restricting $M$ into a subset of the susceptibilities, which is repeated for all $2^{21}$ subsets.

Selected Linear Combinations

The following expressions for each partial pressures are obtained by comparing the solutions of susceptibilities from the ideal-HRG model and the recent continuum lattice QCD data []. To obtain the partial pressures from the large list of possible solutions, we require the following criteria:

Partial pressures calculated through HRG susceptibilities must be non-negative for all temperatures
Partial pressures calculated with HRG and Lattice susceptibilities must coincide within the overlapping temperature range (125-175 MeV).
Partial pressures through lattice susceptibilities should not become excessively negative at high temperatures.

\[\displaystyle P_{001} / T^4 = - \dfrac{1}{6} \chi^{S}_{2} + \chi^{Q}_{2} + \dfrac{1}{6} \chi^{S}_{4} - \chi^{BS}_{31} - \chi^{SQ}_{31} - \chi^{BS}_{22} - \chi^{BQ}_{22} - 2 \chi^{BSQ}_{211} - 3 \chi^{BSQ}_{121}\]

\[\displaystyle P_{010} / T^4 = \chi^{S}_{4} + \dfrac{2}{3} \chi^{BS}_{11} - \dfrac{4}{3} \chi^{SQ}_{11} + \dfrac{1}{3} \chi^{BS}_{13} + \dfrac{1}{3} \chi^{SQ}_{31} +2 \chi^{BSQ}_{211} + \chi^{BSQ}_{121}\]

\[\begin{split}\begin{aligned} \displaystyle P_{100} / T^4 = \dfrac{1}{24} \chi^{S}_{2} + \dfrac{1}{2} \chi^{BQ}_{11} + \dfrac{1}{2} \chi^{SQ}_{11} + \chi^{B}_{4} &- \dfrac{1}{24} \chi^{S}_{4} - \dfrac{1}{4} \chi^{BS}_{13} + \dfrac{1}{2} \chi^{BQ}_{13} + \dfrac{5}{4} \chi^{BS}_{31} \\ &- \chi^{BQ}_{22} - \dfrac{1}{2} \chi^{SQ}_{22} - \dfrac{1}{2} \chi^{BSQ}_{211} - \dfrac{3}{2} \chi^{BSQ}_{112} \end{aligned}\end{split}\]

\[\displaystyle P_{011} / T^4 = \frac{2}{3} \chi^{SQ}_{11} + \frac{1}{3} \chi^{BS}_{13} + \frac{2}{3} \chi^{BS}_{31} + \frac{1}{3} \chi^{SQ}_{31} + \chi^{BS}_{22}+ \chi^{BSQ}_{121}\]

\[\displaystyle P_{110} / T^4 = \frac{1}{6} \chi^{S}_{2} - \chi^{BS}_{11} - \chi^{SQ}_{11} - \frac{1}{6} \chi^{S}_{4} + \chi^{SQ}_{31} + 2 \chi^{BSQ}_{211} + 2 \chi^{BSQ}_{121} + \chi^{BSQ}_{112}\]

\[\begin{split}\begin{aligned} \displaystyle P_{101} / T^4 = \frac{1}{24} \chi^{S}_{2} + \frac{1}{12} \chi^{BS}_{11} + \chi^{BQ}_{11} - \frac{1}{24} \chi^{S}_{4} - \frac{1}{12} \chi^{BS}_{13} &- \frac{1}{2} \chi^{BQ}_{13} + \frac{1}{3} \chi^{SQ}_{13} \\ &+ \frac{1}{6} \chi^{SQ}_{31} + \frac{1}{2} \chi^{BQ}_{22} - \frac{1}{2} \chi^{SQ}_{22} \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} \displaystyle P_{10-1} / T^4 = \frac{1}{8} \chi^{S}_{2} &+ \frac{1}{4} \chi^{BS}_{11} - \frac{1}{8} \chi^{S}_{4} - \frac{1}{4} \chi^{BS}_{13} - \frac{1}{6} \chi^{BQ}_{13} + \frac{1}{6} \chi^{SQ}_{13} \\ &- \frac{1}{3} \chi^{BQ}_{31} + \frac{1}{3} \chi^{SQ}_{31} + \frac{1}{2} \chi^{BQ}_{22} - \frac{1}{2} \chi^{SQ}_{22} - \chi^{BSQ}_{211} \end{aligned}\end{split}\]

\[\displaystyle P_{111} / T^4 = -\frac{1}{2} \chi^{BSQ}_{211} -\frac{1}{2} \chi^{BSQ}_{112}\]

\[\displaystyle P_{11-1} / T^4 = -\frac{1}{12} \chi^{S}_{2} - \frac{1}{2} \chi^{BS}_{11} + \frac{1}{12} \chi^{S}_{4} - \frac{1}{2} \chi^{SQ}_{13} - \frac{1}{2} \chi^{BS}_{22} + \frac{1}{2} \chi^{SQ}_{22} + \frac{5}{2} \chi^{BSQ}_{211} + \frac{3}{2} \chi^{BSQ}_{121}\]

\[\displaystyle P_{102} / T^4 = \frac{1}{6} \chi^{BQ}_{13} - \frac{1}{6} \chi^{BQ}_{31}\]

\[\displaystyle P_{120} / T^4 = \frac{1}{36} \chi^{S}_{2} - \frac{1}{36} \chi^{S}_{4} - \frac{1}{3} \chi^{BS}_{13} + \frac{1}{6} \chi^{SQ}_{13} - \frac{1}{6} \chi^{SQ}_{31} - \frac{1}{3} \chi^{BS}_{22}\]

\[\displaystyle P_{121} / T^4 = \frac{1}{9} \chi^{S}_{2} - \frac{1}{6} \chi^{SQ}_{11} - \frac{1}{9} \chi^{S}_{4} + \frac{2}{3} \chi^{BS}_{31} + \frac{1}{6} \chi^{SQ}_{31} + \frac{2}{3} \chi^{BS}_{22}\]

\[\displaystyle P_{131} / T^4 = \frac{1}{12} \chi^{SQ}_{13} + \frac{1}{6} \chi^{SQ}_{31} - \frac{1}{4} \chi^{SQ}_{22} + \frac{1}{4} \chi^{BSQ}_{121} - \frac{1}{4} \chi^{BSQ}_{112}\]

Do note that the expressions selected, and the resulting temperature-dependent behaviours for each partial pressure depend heavily on the specific criterion selections and the sensitivity of lattice data sampling. Hence, the expressions above are not unique, and should be considered as one of the possible solutions to obtain the partial pressures from a given susceptibility dataset.