Effective degrees of freedom, trace anomaly and c-theorem like condition in the hadron resonance gas model

Abstract

The relation between the effective degrees of freedom (EDOF) and the trace anomaly is studied in the hadron resonance gas (HRG) model. If we regard the thermodynamical relation as the evolution equation and define the EDOF as P/T^4, where P and T are the pressure and the temperature, respectively, we obtain the equation which relates to the trace anomaly. The structure of the equation resembles that of the so called c-theorem in the two dimensional conformal field theory which asserts that the EDOF should not increase as the energy scale parameter decreases. There is a stationary point where the trace anomaly (modified trace anomaly) vanishes, and the scale symmetry is restored. To investigate the limiting temperature of the HRG model with the excluded volume effects, we consider two types of the c-theorem like conditions for the EDOF. The first condition requires that the EDOF should not decrease when T increases. This condition is equivalent to the condition that the trace anomaly (modified trace anomaly) should not be negative. The second condition requires that the EDOF should be convex downwards as a function of T. It is found that the first condition gives the limiting temperature of the HRG model with the excluded volume effect which is much higher than the crossover transition temperature obtained by the lattice QCD calculation and, at zero baryon number density, is close to the transition temperature in the pure gluonic theory, while the second one gives the limiting temperature which almost coincides with the one obtained by using the normalized baryon number fluctuation in the previous work and is consistent with the critical point predicted by the lattice QCD calculation.

Figures (17)

• Figure 1: The solid (dash-dotted) line shows the $T$-dependence of the dimensionless baryon number fluctuation $\chi_2^{\rm B}/T^2$ at $\mu =0$ in the HRG model with EVE (without EVE). For the detailed description of the model, see Sec. \ref{['HRGEVE']}. The dotted line shows result in the ideal massless three flavor QGP. The squares with error bar show the LQCD results in Ref. Borsanyi:2011sw.
• Figure 2: The $T$-dependence of the effective degrees of freedom $g_{\mu}(T)=P/T^4$ at $\mu =0$. The squares with error bar show the LQCD results in Ref. Borsanyi:2012cr. The upper (lower) dotted line shows result in the ideal massless three flavor QGP with (without) the gluon contributions. The dashed and dash-dot-dotted lines show the results in the HRG model with and without EVE, respectively. The solid, and dash-dotted lines show the baryon contribution $g_{\mu, B}$ in the HRG model with and without EVE, respectively. See Sec. \ref{['HRGEVE']} for the detailed description of the HRG model and see Sec. \ref{['HRGEDOF']} for the detail explanation of the lines.
• Figure 3: The solid line shows Schematic diagram of the effective degree of freedom in the transition from the low temperature phase to the high temperature phase. The solid circle shows the inflection point.
• Figure 4: The $T$-dependence of the normalized trace anomaly $\Delta/T^4$ at $\mu =0$. The dashed and dash-dot-dotted lines show the results in the HRG model with and without EVE, respectively. The solid, and dash-dotted lines show the baryon contribution $\Delta_{\rm B}/T^5$ in the HRG model with and without EVE, respectively. The dotted line shows the result in the ideal massless three flavor QGP with (without) the gluon contributions. The squares with error bar show the LQCD results in Ref. Borsanyi:2012cr.
• Figure 5: The $T$-dependence of the $T^5$-scaled trace anomaly $\Delta/T^5$ at $\mu =0$. The dashed and dash-dot-dotted lines show the results in the HRG model with and without EVE, respectively. The solid and dash-dotted lines show the baryon contribution $\Delta_{\rm B} /T^5$ in the HRG model with and without EVE, respectively. The dotted line shows result in the ideal massless three flavor QGP with (without) the gluon contributions. The squares with error bar show the LQCD results in Ref. Borsanyi:2012cr.