Thermodynamics of the parity-doublet model: Symmetric nuclear matter and the chiral transition

We present a detailed discussion of the thermodynamics of the parity-doublet nucleon-meson model within a mean-field theory, at finite temperature and baryon-chemical potential, with special emphasis on the chiral transition at large baryon densities and vanishing temperature. We consider isospin-symmetric matter. We systematically compare the parity-doublet model to a related singlet model obtained by disregarding the chiral partner of the nucleon. After studying the ground-state properties of nuclear matter, the nuclear liquid-gas transition, and the density modifications of the nucleon sigma term which govern the low-density regime, we give new insight into the underlying mechanisms of the zero-temperature chiral transition occurring at several times the nuclear saturation density. We show that the chiral transition is driven by a kind of symmetry energy that tends to equilibrate the populations of opposite-parity baryons. This symmetry energy dictates the composition of matter at large baryon densities, once the phase space for the appearance of the negative-parity partner is opened. We furthermore highlight the characteristic role within the thermodynamics of the chiral-invariant mass of the parity-doublet model. We include the chiral limit in all of our discussions in order to provide a complete picture of the chiral transition.

Figure 1

Fermion mass $M_{\pm }$ and singlet mass $y\sigma$ as a function of the $\sigma$ field. The mass $M_{+}$ features a minimum at ${\sigma }_{\min }$ (cf. also Refs. [34, 35]).

Figure 2

Thermodynamic potential in vacuum ($T=0, {\mu }_{\mathcal{B}}=0$), i.e., $\mathrm{\Omega }\equiv U$.

Figure 3

Graphical solution of the gap equation. The green curve represents $dU/d\sigma$ as a function of $\sigma$. This curve is independent of $n$. The black curves give the contribution from the scalar density $-y{n}_{\mathrm{s}}$ as a function of $\sigma$ for given density $n$ (solid for $n=n_0$, dotted for $n=n_0/2$, dashed for $n=3n_0$, dash-dotted for $n=6n_0$). Note that, in the range of densities and values of $\sigma$ relevant to the present discussion, $n_{\mathrm{s}}\simeq n$. The intersection point between the green curve and a black one gives the solution ${\sigma }_n$ of the gap equation for the corresponding values of $n$.

Figure 4

Saturation curve, $E/A=\mathcal{E}\left(n\right)/n-M_N$. This is obtained by solving the gap equation with the full potential, with in the right-hand side either $-y{n}_{\mathrm{s}}$ or $-yn=-y{n}_{\mathcal{B}}$. The third curve shows the nonrelativistic approximation to the kinetic energy.

Figure 5

Saturation curves for both the singlet and doublet models. Only the part between $n=0$ and $n=n_0$ contributes to the surface energy. The small difference between the blue and green curves in the region $0\le n\le n_0$ entails a significant difference in the estimate of the surface tension, that correlates to the different values of the compression modulus in the two models.

Figure 6

Parameter bands for the parity-doublet model in the $m_{\sigma }-m_0$ plane matching experimental data (within error bars) of Table 4 (and including the compression modulus drawn here as the band $K=230\pm 20\mathrm{MeV}$). Shown is the result for the effective mass (a) $M_{*}/M_N=0.91$, (b) 0.93, and (c) 0.95. The gray-shaded area indicates parameter sets leading to $T_c>30\mathrm{MeV}$ for the liquid-gas transition.

Figure 7

Parameter bands for the singlet model in the $m_{\sigma }-M_{*}/M_N$-plane matching experimental data (within error bars) of Table 4 (and including the compression modulus $K=230\pm 20\mathrm{MeV}$). The region called “initialization fail” consists of parameters for which the initialization conditions were not met, e.g., the solution of the gap Eq. (23) corresponds to a local maximum, not a minimum.

Figure 8

Equation of state (pressure $P$ as a function of baryon density $n_{\mathcal{B}}$) for various temperatures.

Figure 9

The nucleon mass $M_{+}$, as a function of $\widehat{h}$, in both the singlet and doublet models.

Figure 10

The squared pion mass ${\widehat{m}}_{\pi }^2$ as a function of $\widehat{h}$, illustrating its nonlinear behavior in the doublet model.

Figure 11

Density dependence of the chiral condensate in the parity-doublet model. The full solution to the gap equation (solid line) is successively approximated for small baryon density $n_{\mathcal{B}}$ by a linear function proportional to ${\sigma }_N^{\left(3\right)}$ (black dotted line) and higher Taylor polynomials of $O\left(n_{\mathcal{B}}^2\right)$ (dash-double-dotted line), $O\left(n_{\mathcal{B}}^5\right)$ (dash-dotted line), and $O\left(n_{\mathcal{B}}^6\right)$ (dashed line). The gray dotted line indicates the difference to the full solution when approximating $n_{\mathrm{s}}^{+}$ by $n_{\mathcal{B}}$, which can be seen to be an excellent approximation in this range of densities.

Figure 12

Density dependence of the chiral condensate in the singlet model. Analogous plot to Fig. 11.

Figure 13

Phase diagram of the singlet model: Isoscalar condensate $\sigma$ as a function of temperature $T$ and baryon chemical potential ${\mu }_{\mathcal{B}}$. Contours of constant $\sigma$ are given every $10\mathrm{MeV}$.

Figure 14

Phase diagram of the doublet model: Isoscalar condensate $\sigma$ as a function of temperature $T$ and baryon chemical potential ${\mu }_{\mathcal{B}}$. Contours of constant $\sigma$ are given every $10\mathrm{MeV}$.

Figure 15

First-order chiral transition in the parity-doublet model, as a function of ${\mu }_{\mathcal{B}}$ and ${\widehat{m}}_{\pi }$. The pion mass ${\widehat{m}}_{\pi }$ varies continuously between the chiral limit (${\widehat{m}}_{\pi }=0$) and the physical value $m_{\pi }=138\mathrm{MeV}$. In the chiral limit, the first-order transition line (black) ends in a tricritical point (black dot) at $T_{\mathrm{tri}}\simeq 33\mathrm{MeV}$ and ${\mu }_{\mathrm{tri}}\simeq 1460\mathrm{MeV}$. Above $T_{\mathrm{tri}}$, the transition line is of second order. Values of $(T,{\mu }_{\mathcal{B}})$ on the red surface correspond to a first-order transition. The first-order transition line (red) for physical pion mass terminates in the critical endpoint (CEP) at $(T_c,{\mu }_c)\simeq (8.5\mathrm{MeV},1580\mathrm{MeV})$.

Figure 16

Phase coexistence in the $T-n_{\mathcal{B}}$ plane (for physical pion mass), where the density is measured in unit of the nuclear matter density $n_0$. The points $\mathrm{A}$ and $\mathrm{B}$ represent the two coexisting phases with identical pressure and baryon chemical potential at a given temperature. In the low-density phase $\mathrm{A}$, chiral symmetry is still broken ($\sigma >0$), while in the high-density phase $\mathrm{B}$ chiral symmetry is restored ($\sigma$ approaches zero). With increasing temperature, the critical chemical potential decreases (see Fig. 15) and the coexistence region terminates at a critical endpoint (CEP) characterized by $(T_c,n_c)\simeq (8.5\mathrm{MeV},2.01{\mathrm{fm}}^{-3})$.

Figure 17

Phase coexistence in the $T-n_{\mathcal{B}}$ plane (in the chiral limit). Analogous plot to Fig. 16. The black dot denoted “CP” corresponds to the tricritical point in Fig. 15.

Figure 18

The average $\sigma$ field at zero temperature as a function of the baryon chemical potential ${\mu }_{\mathcal{B}}$ in the singlet model (dark green: physical pion mass; light green: chiral limit).

Figure 19

Condensate $\sigma$ at zero temperature as a function of the baryon density $n_{\mathcal{B}}$ in the singlet model. Results for physical pion mass are given in dark green, those of the chiral limit in light green. The black dashed line (“crit.”) represents the analytic solution (112) valid near $n_c$. The dash-dotted line shows the linear density approximation proportional to the sigma term ${\sigma }_N^{\left(3\right)}$. See also Fig. 28.

Figure 20

The integral $I\left(z\right)$ defined in Eq. (125) and related functions.

Figure 21

Condensate $\sigma$ at zero baryon chemical potential as a function of temperature $T$ in the singlet model. The dashed line (“crit.”) shows the critical behavior (129).

Figure 22

Condensate $\sigma$ at zero baryon chemical potential as a function of temperature $T$ in the parity-doublet model (dark blue shows physical pion mass, light blue shows chiral limit). The dashed line (“crit.”) represents the approximate solution (139).

Figure 23

Condensate $\sigma$ at zero temperature as a function of baryon chemical potential ${\mu }_{\mathcal{B}}$ in the parity-doublet model (dark blue shows physical pion mass, light blue shows chiral limit).

Figure 24

The densities $n_{\mathcal{B}}^{+}$ and $n_{\mathcal{B}}^{-}$ as functions of the baryon-chemical potential. The coloring (physical pion mass or chiral limit) coincides with the one of Fig. 23. The respective total baryon density $n_{\mathcal{B}}=n_{\mathcal{B}}^{+}+n_{\mathcal{B}}^{-}$ is given in black or gray.

Figure 25

Gap equation in the parity-doublet model. Note that given the large value of $m_0$, the scalar densities coincide with the respective baryon density; that is, $n_{\mathrm{s}}^{\pm }\simeq n_{\mathcal{B}}^{\pm }$.

Figure 26

Effective mass $M_{*}$ as a function of density (dark blue shows physical pion mass, light blue shows chiral limit). The black and gray lines represent the mass $M_{-}$ of the negative-parity baryons (up to threshold) for physical pion mass and in the chiral limit, respectively.

Figure 27

Condensate $\sigma$ at zero temperature as a function of baryon density $n_{\mathcal{B}}$ in the parity-doublet model (for physical pion mass). The blue dashed line represents the result obtained by continuously integrating the differential Eq. (151).

Figure 28

Condensate $\sigma$ at zero temperature as a function of baryon density $n_{\mathcal{B}}$ in the parity-doublet model. The solutions to the gap equation were computed in discrete steps of the baryon chemical potential (between ${\mu }_{\mathcal{B}}=0$ and ${\mu }_{\mathcal{B}}=1.8\mathrm{GeV}$; cf. Fig. 23). Results for physical pion mass are given in dark blue, those of the chiral limit in light blue. The dash-dotted line shows the linear density approximation proportional to the sigma term ${\sigma }_N^{\left(3\right)}$.

Figure 29

Energy density as a function of $\alpha$ for different $n_{\mathcal{B}}$ (for physical pion mass). Dots indicate the respective minima. Gray lines from left to right indicate $n_{\mathcal{B}}=12n_0, 13n_0, 14n_0$, and $15n_0$; magenta line indicates the last value before the transition ($n_{\mathcal{B}}\approx 12.44n_0$); orange line indicates the first value after the transition ($n_{\mathcal{B}}\approx 12.77n_0$).

Figure 30

Energy density as a function of $\alpha$ for different $n_{\mathcal{B}}$ (in the chiral limit). Color code equivalent to Fig. 29. Gray lines from left to right indicate $n_{\mathcal{B}}=11n_0$ (dashed), $11.6n_0$ (dash-dotted), $11.75n_0$ (long-dashed), $11.8n_0$ (dotted), and $12.5n_0$ (dash-double-dotted); magenta line indicates $n_{\mathcal{B}}\approx 11.41n_0$; orange line indicates $n_{\mathcal{B}}\approx 12.04n_0$. Note that the last two gray dots (for $n_{\mathcal{B}}=11.8n_0$ and $12.5n_0$) coincide with the orange dot. The lines demonstrate in particular how the original minimum at small $\alpha$ becomes a saddle point for increasing density (around $11.8n_0$).

Figure 31

Compressibilities in the chiral limit of the parity-doublet model.

Figure 32

The derivatives $d{n}_{\mathcal{B}}^{\pm }/d{n}_{\mathcal{B}}$ as functions of $n_{\mathcal{B}}$ for physical pion mass, together with the bounds (169). The derivatives are plotted with $n_{\mathcal{B}}$ starting at the ${\mathcal{B}}^{-}$ threshold.

Figure 33

Flow lines (a) $\sigma \left(n_{\mathcal{B}}\right)$ and (b) $n_{\mathcal{B}}^{+}\left(n_{\mathcal{B}}\right)$ obtained by solving the coupled Eqs. (172) and (173). The horizontal dashed line in panel (a) indicates the value ${\sigma }_{\min }$. The black line corresponds to the ${\mathcal{B}}^{-}$ threshold, $M_{*}=M_{-}$. Note that, for the bottom panel, this is just a straight line since at the ${\mathcal{B}}^{-}$ threshold, $d{n}_{\mathcal{B}}^{+}/d{n}_{\mathcal{B}}=1$.

Figure 34

Zero-temperature pressure in the parity-doublet model as a function of $n_{\mathcal{B}}$ (for physical pion mass). The black horizontal line represents the Maxwell construction corresponding to the chiral transition (the colored dots correspond to the densities of the colored lines in Fig. 29).

Figure 35

Zero-temperature pressure in the parity-doublet model as a function of $n_{\mathcal{B}}$ (in the chiral limit). The black horizontal line represents the Maxwell construction corresponding to the chiral transition (the colored dots correspond to the densities of the colored lines in Fig. 30). Approaching the transition from below, one can follow continuously the pressure as a function of increasing density up to the point where the sigma fields jumps to zero.

Figure 36

Illustration of the composition of coexisting phases. The points $\mathrm{A}$ and $\mathrm{B}$ correspond to the points $\mathrm{A}$ and $\mathrm{B}$ of Fig. 16. The concentration of negative-parity baryons is greater in phase $\mathrm{B}$ (chirally symmetric) than in phase $\mathrm{A}$. The $x$-axis labels correspond respectively to the two gray “blobs,” i.e., $n_{\mathcal{B}}^{-}/n_0$ to the left one and $n_{\mathcal{B}}^{+}/n_0$ to the right one, both data drawn on a single axis. This means that, e.g., the densities $n_{\mathcal{B}}^{+}$ and $n_{\mathcal{B}}^{-}$ of phase A add up to the total baryon density $n_{\mathcal{B}}=n_{\mathcal{B}}^{+}+n_{\mathcal{B}}^{-}$ of point A given in Fig. 16.

Figure 37

Same as Fig. 36 in the chiral limit and corresponding to Fig. 17. The two formerly separate gray blobs merge and coincide in the black middle line, as the populations of ${\mathcal{B}}^{+}$ and ${\mathcal{B}}^{-}$ are identical in phase B.

Figure 38

Bosonic potential in vacuum $\mathrm{\Omega }=U$ showing the magnitude of its various contributions. The orange curve shows the potential $V\left(\sigma \right)-h(\sigma -f_{\pi })$, without the fermion loop contribution. The magenta curve shows the potential including the renormalized fermion loop, but with ${\alpha }_3={\alpha }_4=0$. The gray curve shows the full potential $U\left(\sigma \right)$. The dark red curve shows the quartic approximation to $V\left(\sigma \right)-h(\sigma -f_{\pi })$.

Figure 39

Same as Fig. 38 but for the parity-doublet model.

Figure 40

Contour plot of the vacuum contribution ${\mathrm{\Omega }}_{\mathrm{vacuum}}$ to the thermodynamic potential of the parity-doublet model.

Figure 41

Showing the effect of a reduced mass for the parity partner.