Thermodynamics and phase transition of the $O(N)$ model from the two-loop $\Phi$-derivable approximation

We discuss the thermodynamics of the $O(N)$ model across the corresponding phase transition using the two-loop $\Phi$-derivable approximation of the effective potential and compare our results to those obtained in the literature within the Hartree-Fock approximation. In particular, we find that in the chiral limit the transition is of the second order, whereas it was found to be of the first order in the Hartree-Fock case. These features are manifest at the level of the thermodynamical observables. We also compute the thermal sigma and pion masses from the curvature of the effective potential. In the chiral limit, this guarantees that Goldstone’s theorem is obeyed in the broken phase. A realistic parametrization of the model in the $N=4$ case, based on the vacuum values of the curvature masses, shows that a sigma mass of around 450 MeV can be obtained. The equations are renormalized after extending our previous results for the $N=1$ case by means of the general procedure described in Ref. [8]. When restricted to the Hartree-Fock approximation, our approach reveals that certain problems raised in the literature concerning the renormalization are completely lifted. Finally, we introduce a new type of $\Phi$-derivable approximation in which the gap equation is not solved at the same level of accuracy as the accuracy at which the potential is computed. We discuss the consistency and applicability of these types of “hybrid” approximations and illustrate them in the two-loop case by showing that the corresponding effective potential is renormalizable and that the transition remains of the second order.

Figure 1

Parametrization in the chiral limit ($h=0$). The scanned region is bounded by the ${\Lambda }_{\mathrm{p}}/T_{\star }=50$ (upper) and ${\overline{T}}_c=0$ (lower) curves. The ${\overline{\varphi }}_{c,0}$ curve is only present in the hybrid approximation, in which case the grey region is excluded for a reason explained in the text. The points which form vertical lines are obtained in the hybrid approximation, while the squares denote the solution of the full two-loop approximation. The iso-$T_{\mathrm{c}}$ and the iso-${\widehat{M}}_{\mathrm{L},0}$ curves are obtained in the hybrid case. The palette shows the value of the renormalization scale $T_{\star }$. The inset shows the variation of $T_{\mathrm{c}}$ with $T_{\star }$ along iso-${\widehat{M}}_{\mathrm{L},0}$ curves.

Figure 2

Parametrization at $h\ne 0$ in the hybrid case. The location of the investigated points relative to the characteristic curves is indicated with smaller size points in the (${\lambda }_{\star }$, $m_{\star }^2/T_{\star }^2$) plane of the figure in the left panel. The points having bigger size indicates the parameters used in Fig. 3 to compare the result of the two-loop and hybrid approximations. The smaller size points satisfy the two criteria ${\widehat{M}}_{\mathrm{T},0}=138\pm 1.38\mathrm{MeV}$ and ${\widehat{M}}_{\mathrm{L},0}\ge 2{\widehat{M}}_{\mathrm{T},0}$. The value of the renormalization scale $T_{\star }$ is indicated on the figure in the right panel. The two palettes show the values of ${\widehat{M}}_{\mathrm{L},0}$ and $T_{\mathrm{pc}}$, respectively.

Figure 3

Comparison between the two-loop and hybrid approximations based on the relative change of the $T=0$ curvature masses used for parametrization and the pseudocritical temperature [$\Delta Q[\%]=100(Q_{2-\mathrm{loop}}/Q_{\mathrm{hybrid}}-1)$]. Different point types denote different values of $m_{\star }^2/T_{\star }^2$ and for each corresponding value of ${\lambda }_{\star }$ there are points at three values of $h/T_{\star }^3\mathrm{\text{:}}0.5$, 1.5, 2.5, which generally increase from bottom to top.

Figure 4

The dependence of the pseudocritical temperature $T_{\mathrm{pc}}$ on the renormalization scale $T_{\star }$ in the hybrid approximation at $h\ne 0$ along different lines of constant physics specified by the value of ${\widehat{M}}_{\mathrm{L},0}$.

Figure 5

The cutoff dependence of the relative change of $\overline{\varphi }$ (upper raw) and that of ${\widehat{M}}_{\mathrm{L}}^2$ (lower raw) obtained in the two-loop approximation at different temperatures: $T=0$, $T=T_{\mathrm{c}}$ and $T=2T_{\mathrm{c}}$. The different parameter sets are (a) $m_{\star }^2/T_{\star }^2=0.124$, ${\lambda }_{\star }=22.28$, $h/T_{\star }^3=1.775$ for which ${\widehat{M}}_{\mathrm{L},0}\approx 280\mathrm{MeV}$, ${\Lambda }_{\mathrm{p}}\approx 186\mathrm{GeV}$ and $T_{\star }\approx 101\mathrm{MeV}$; (b) $m_{\star }^2/T_{\star }^2=0.04$, ${\lambda }_{\star }=17.39$, $h/T_{\star }^3=0.6$, for which ${\widehat{M}}_{\mathrm{L},0}\approx 360\mathrm{MeV}$, ${\Lambda }_{\mathrm{p}}\approx 16.2\mathrm{GeV}$ and $T_{\star }\approx 146\mathrm{MeV}$; (c) $m_{\star }^2/T_{\star }^2=0.04$, ${\lambda }_{\star }=19.2$, $h/T_{\star }^3=0.38$ for which ${\widehat{M}}_{\mathrm{L},0}\approx 465\mathrm{MeV}$, ${\Lambda }_{\mathrm{p}}\approx 3.35\mathrm{GeV}$ and $T_{\star }\approx 167\mathrm{MeV}$. The given ${\widehat{M}}_{\mathrm{L},0}$ and $T_{\star }$ values correspond to the largest $\Lambda$ point of each set. The discretization is characterized by $N_{\tau }=512$ and $N_s=3\times 2^{10}$ except for the points of set (a) at $T=0$ for $\Lambda /{\Lambda }_{\mathrm{p}}>0.04$ where $N_{\tau }=3\times 512$ was used. The step $\Delta L/T_{\star }$ was 5 for the cases (a) and (b) and 1 for case (c).

Figure 6

Illustration of the second order nature of the phase transition in the chiral limit within the two-loop approximation. The inset shows the convergence of the static critical exponents to the mean-field values, as the upper boundary of the fitting range is decreased. The parameters are $m_{\star }^2/T_{\star }^2=0.124$, ${\lambda }_{\star }=22.277$, while the discretization is characterized by $\Lambda /T_{\star }=100$, $N_s=3\times 2^{10}$ and $N_{\tau }=512$ for $T>77\mathrm{MeV}$ and $N_{\tau }=2\times 2^{10}$ for $T\le 77\mathrm{MeV}$. We also defined ${\overline{M}}_{\mathrm{L}/\mathrm{T}}\equiv {\overline{M}}_{\mathrm{L}/\mathrm{T}}(0,\Delta k)$.

Figure 7

The temperature evolution of the order parameter, the curvature and gap masses in the two-loop approximation where ${\overline{M}}_{\mathrm{L}/\mathrm{T}}\equiv {\overline{M}}_{\mathrm{L}/\mathrm{T}}(0,\Delta k)$. For $\overline{\varphi }$ and ${\widehat{M}}_{\mathrm{L}}$ we also show for comparison the curves obtained in the hybrid approximation, in which case we show the critical value of the field below which, at a given temperature, the effective potential is not accessible. The parameters are $m_{\star }^2/T_{\star }^2=0.04$, ${\lambda }_{\star }=17.39$, $h/T_{\star }^3=0.6$. The discretization used in the two-loop case is characterized by $\Lambda /T_{\star }=55$, $N_s=3\times 2^{10}$ and $N_{\tau }$ was increased for decreasing temperature from 512 used at $T\ge 40\mathrm{MeV}$ to $2\times 2^{10}$ for $T[25,40]\mathrm{MeV}$ and $4\times 2^{10}$ for $T\le 25\mathrm{MeV}$.

Figure 8

The temperature dependence of the pressure, energy density and entropy density normalized to the corresponding Stefan-Boltzmann limit obtained for the same parameter set of Fig. 7 in the the two-loop (continuous lines) and hybrid approximations (dashed lines). The inset shows the variation of the interaction measure with the temperature.

Figure 9

The square of the speed of sound $c_{\mathrm{s}}^2$ and the heat capacity $C$ obtained for the same parameter set of Fig. 7 in the two-loop (continuous lines) and hybrid approximations (dashed lines). The left and right axes of the plot correspond to $c_{\mathrm{s}}^2$ and $C$, respectively.

Figure 10

The two real branches of the Lambert $\mathcal{W}$ function. The upper branch ${\mathcal{W}}_0(x)$ (dashed) is defined for $x\in [-1/e,\infty )$ and the lower one ${\mathcal{W}}_{-1}(x)$ (solid) for $x\in [-1/e,0)$.