Physics Background

The pressure is given by

\[P = -\Omega^{\mathrm{tot}} \equiv - \sum_{f=u,d,s} \left( \Omega_f^{(0)}(m_f,\mu_f) + \Omega_f^{(1)}(m_f,\mu_f) \right),\]

where the sum runs over the quark flavors \(f\).

The total thermodynamic potential is taken from arXiv:1507.08941 [hep-ph]:

\[\Omega_f^{(0)} -\frac{N_c}{12\pi^2}\left[\mu_f k_f\left(\mu_f^2 - \frac{5}{2}m_f^2\right) + \frac{3}{2}m_f^4\ln\left(\frac{\mu_f+k_f}{m_f}\right)\right]\]
\[\begin{split}\begin{aligned} \Omega_f^{(1)} &= \frac{\alpha_s N_G}{16\pi^3}\Bigg\{3\Biggr[m_f^2 \ln\left(\frac{\mu_f+k_f}{m_f}\right) - \mu_f k_f\bigg]^2- 2k_f^4 \\ & +m_f^2\left[ 6\ln\left(\frac{\Lambda}{m_f}\right)+4\right]\left[ \mu_f k_f- m_f^2 \ln\left(\frac{\mu_f+k_f}{m_f}\right)\right]\Biggr\} \end{aligned}\end{split}\]

Here,

  • \(N_c\) is the color degeneracy,

  • \(N_G = N_c^2 - 1\) is the number of gluons,

  • \(\Lambda\) is the renormalization scale,

  • \(\Omega^{(0)}\) is the free-gas contribution,

  • \(\Omega^{(1)}\) is the next-to-leading-order contribution,

  • \(k_f \equiv \sqrt{\mu_f^2 - m_f^2}\) for each flavor \(f\).

The running coupling is

\[\alpha_s(\Lambda)=\frac{4\pi}{\beta_0 \bar{L}}\left[1- 2\frac{\beta_1}{\beta_0^2}\frac{\ln \bar{L}}{\bar{L}}\right]\]

The strange-quark mass as a function of the strong coupling is

\[m_s(\Lambda) = \hat{m}_s \left(\frac{\alpha_s}{\pi}\right)^{4/9} \left[ 1 + 0.895062\frac{\alpha_s}{\pi} \right]\]

where

\[\bar{L} = 2\ln\left(\frac{\Lambda}{\Lambda_{\overline{\mathrm{MS}}}}\right), \qquad \beta_0 = 11 - \frac{2}{3}N_f, \qquad \beta_1 = 51 - \frac{19}{3}N_f.\]

The scale \(\Lambda_{\overline{\mathrm{MS}}}\) and the invariant mass \(\hat{m}_s\) are fixed according to the same definitions used in arXiv:1507.08941 [hep-ph].

The renormalization scale is chosen as

\[\Lambda = \sqrt{ (2 \pi T)^2 + \left( C_B \mu_B+ C_Q\mu_Q + C_S \mu_S \right)^2 }.\]

where the :math: C’s are constants that can be set by the user. The default values are \(C_B = 1/3\), \(C_Q = 1/6\), and \(C_S = 0\). The energy density is given by the Gibbs-Duhem relation at vanishing temperature:

\[\epsilon = n_B\mu_B + n_S\mu_S + n_Q\mu_Q - P.\]

The isospin asymmetry coefficient is defined as

\[\delta_I = 1 - 2Y_Q + Y_S.\]

In terms of the quark densities, the charge fraction can be written as

\[Y_Q = \frac{2n_u - n_d - n_s}{n_u + n_d + n_s}.\]