Thermodynamic geometry of Nambu–Jona Lasinio model

Abstract

The formalism of Riemannian geometry is applied to study the phase transitions in Nambu–Jona Lasinio (NJL) model. Thermodynamic geometry reliably describes the phase diagram, both in the chiral limit and for finite quark masses. The different thermodynamic geometrical behavior of NJL model and of (2+1) Quantum Chromodynamics at high temperature and small baryon density gives some hints on the connection between chiral symmetry restoration/breaking and deconfinement/confinement regimes.

Appendices

NJL model with two quarks

To evaluate the scalar curvature R, one needs the derivatives of the potential $\phi$, up to third order, which can be written in terms of the dynamical generated mass M. Therefore, the solution of the GAP equation uniquely determines all those functions. Indeed, after a straightforward calculation, one gets (a comma indicates partial derivative)

$$\begin{aligned}&\displaystyle M_{,\beta } =\frac{b_1\;M}{1-f_1-f_2\;M^2}, \end{aligned}$$

(26)

$$\begin{aligned}&\displaystyle M_{,\gamma } =\frac{g_1\;M}{1-f_1-f_2\;M^2}, \end{aligned}$$

(27)

$$\begin{aligned}&\displaystyle M_{,\beta \beta } = d\; \left[ b_3\;M+\left( b_2+f_{1,\beta }\right) M_{,\beta } + \left( b_4+2\,T\,f_2\right) M^2\,M_{,\beta }+f_3\;M\,M_{,\beta }^2 \right] , \end{aligned}$$

(28)

$$\begin{aligned}&\displaystyle M_{,\gamma \gamma } = d\;\left[ g_3\;M+\left( g_2+f_{1,\gamma }\right) M_{,\gamma } + g_4\,M^2\,M_{,\gamma }+ +f_3\;M\,M_{,\gamma }^2 \right] , \end{aligned}$$

(29)

$$\begin{aligned}&\displaystyle M_{,\beta \gamma } = d\;\left[ g_5\;M+b_2\,M_{,\gamma } + f_{1,\gamma } M_{,\beta } + g_4\,M^2\,M_{,\beta }+ f_2\;T\,M^2\,M_{,\gamma }+f_3\;M\,M_{,\beta } M_{,\gamma } \right] \nonumber \\ \end{aligned}$$

(30)

with

$$\begin{aligned}&\displaystyle d = \left( 1-f_1-f_2\;M^2\right) ^{-1}, \end{aligned}$$

(31)

$$\begin{aligned}&\displaystyle f_{1}= \kappa _M\; \int _0^\varLambda \mathrm{d}p\;\frac{p^4\;\varPsi }{E^3}, \end{aligned}$$

(32)

$$\begin{aligned}&\displaystyle f_2= \kappa _M\,\int _0^\varLambda \mathrm{d}p\;p^2\;\frac{n_-\left( 1-n_-\right) +n_+ \left( 1-n_+\right) }{T\;E^2}, \end{aligned}$$

(33)

$$\begin{aligned}&\displaystyle f_3=\kappa _M\,\int _0^\varLambda \mathrm{d}p\;p^4\;\frac{n_-\left( 1-n_-\right) +n_+ \left( 1-n_+\right) }{T\;E^4}, \end{aligned}$$

(34)

$$\begin{aligned}&\displaystyle b_1 = \kappa _M\,\int _0^\varLambda \mathrm{d}p\;p^2\;\left[ n_-\left( 1-n_-\right) +n_+ \left( 1-n_+\right) \right] , \end{aligned}$$

(35)

$$\begin{aligned}&\displaystyle b_2 = \kappa _M\,\int _0^\varLambda \mathrm{d}p\;p^4\;\frac{\varPsi _{\beta }}{E^3}, \end{aligned}$$

(36)

$$\begin{aligned}&\displaystyle b_3= \kappa _M\int _0^\varLambda \mathrm{d}p p^2\;\left[ n_{-,\beta }\left( 1-2n_-\right) +n_{+,\beta } \left( 1-2n_+\right) \right] , \end{aligned}$$

(37)

$$\begin{aligned}&\displaystyle b_4= \kappa _M\int _0^\varLambda \mathrm{d}p p^2\;\frac{n_{-,\beta }\left( 1-2n_-\right) +n_{+,\beta } \left( 1-2n_+\right) }{T\;E^2}, \end{aligned}$$

(38)

$$\begin{aligned}&\displaystyle g_1 = \kappa _M\,\int _0^\varLambda \mathrm{d}p\;p^2\;\frac{n_- \left( 1-n_-\right) - n_+\left( 1-n_+\right) }{E}, \end{aligned}$$

(39)

$$\begin{aligned}&\displaystyle g_2 = \kappa _M\,\int _0^\varLambda \mathrm{d}p\;p^4\;\frac{\varPsi _{\gamma }}{E^3}, \end{aligned}$$

(40)

$$\begin{aligned}&\displaystyle g_3= \kappa _M\int _0^\varLambda \mathrm{d}p p^2\;\frac{n_{-,\gamma }\left( 1-2n_-\right) -n_{+,\gamma } \left( 1-2n_+\right) }{E}, \end{aligned}$$

(41)

$$\begin{aligned}&\displaystyle g_4= \kappa _M\int _0^\varLambda \mathrm{d}p p^2\;\frac{n_{-,\gamma }\left( 1-2n_-\right) +n_{+,\gamma } \left( 1-2n_+\right) }{T\;E^2}, \end{aligned}$$

(42)

$$\begin{aligned}&\displaystyle g_5= \kappa _M\int _0^\varLambda \mathrm{d}p p^2\;\left[ n_{-,\gamma }\left( 1-2n_-\right) +n_{+,\gamma } \left( 1-2n_+\right) \right] , \end{aligned}$$

(43)

$$\begin{aligned}&\displaystyle \kappa _M = 2\;G\;\frac{N_\mathrm{c}\,N_\mathrm{f}}{\pi ^2} \end{aligned}$$

(44)

and $n{\pm }$ in Eq. (16).

By deriving Eqs. (24) and (11) and defining

$$\begin{aligned} \kappa _\varOmega = \frac{\kappa _M}{2\,G}, \end{aligned}$$

(45)

one gets

$$\begin{aligned}&\displaystyle \phi _{,\beta } = \kappa _\varOmega \int _0^\varLambda \mathrm{d}p\,p^2\,E\,\varPsi -\frac{\left( M-m\right) ^2}{4\,G}, \end{aligned}$$

(46)

$$\begin{aligned}&\displaystyle \phi _{,\gamma } = \kappa _\varOmega \int _0^\varLambda \mathrm{d}p\,p^2\,\left( n_+ - n_-\right) . \end{aligned}$$

(47)

The calculation of second- and third-order derivatives is straightforward.

Finally, the two-flavor chiral susceptibility, $\chi$, is defined as [38]

$$\begin{aligned} \chi ^{2f} = \frac{\partial M}{\partial m} = \frac{1}{1 - f_1 - f_2\;M^2} = \frac{M_{,\beta }}{b_1\;M} = \frac{M_{,\gamma }}{g_1\;M}. \end{aligned}$$

(48)

Three flavors

In three-flavor systems, the derivatives of the dynamically generated mass $M_u=M_d$ and $M_s$ are

$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle M_{u,\beta }\left( \delta - b_u\,M_u\,\epsilon \right) = a_u\,\epsilon - M_{s,\beta }\,\zeta \\ \\ \displaystyle M_{s,\beta } = \frac{\left( a_s\,\theta - a_u\,\lambda \right) \left( \delta - b_u\,M_u\,\epsilon \right) - a_u\,\epsilon \,b_u\,\lambda \,M_u}{\left( \ \eta -b_s\,M_s\,\theta \right) \left( \delta - b_u\,M_u\,\epsilon \right) - b_u\,\lambda \,M_u\,\zeta } \end{array}\right. }, \end{aligned}$$

(49)

$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle M_{u,\gamma }\left( \delta - b_u\,M_u\,\epsilon \right) = c_u\,\epsilon - M_{s,\gamma }\,\zeta \\ \\ \displaystyle M_{s,\gamma } = \frac{\left( c_s\,\theta - c_u\,\lambda \right) \left( \delta - b_u\,M_u\,\epsilon \right) - c_u\,\epsilon \,b_u\,\lambda \,M_u}{\left( \ \eta -b_s\,M_s\,\theta \right) \left( \delta - b_u\,M_u\,\epsilon \right) - b_u\,\lambda \,M_u\,\zeta } \end{array}\right. }, \end{aligned}$$

(50)

$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle M_{u,\beta \beta }\left( \delta - b_u\,M_u\,\epsilon \right) = d_u\,\epsilon + A_{u,\beta }\,\epsilon _{,\beta }- \left( M_{u,\beta }\delta _{,\beta }+M_{s,\beta }\zeta _{,\beta }+\epsilon \,D_{u,\beta }\,M_u\,M_{u,\beta }\right) - M_{s,\beta \beta }\,\zeta \\ \\ \displaystyle \begin{aligned} M_{s,\beta \beta } = &{} \frac{\left( d_s\,\theta -d_u\,\lambda + A_{s,\beta }\,\theta _{,\beta }-A_{u,\beta }\,\lambda _{,\beta } + \lambda \,D_{u,\beta }\,M_u\,M_{u,\beta } - \theta \,D_{s,\beta }\,M_s\,M_{s,\beta }\right) \left( \delta - b_u\,M_u\,\epsilon \right) }{ \left( \eta - b_s\,M_s\,\theta \right) \left( \delta - b_u\,M_u\,\epsilon \right) -\zeta \,\lambda \,b_u\,M_u}\\ &{}- \lambda \,b_u\,M_u\,\frac{\left[ d_u\,\epsilon + A_{u,\beta }\,\epsilon _{,\beta }- \left( M_{u,\beta }\delta _{,\beta }+M_{s,\beta }\zeta _{,\beta }+\epsilon \,D_{u,\beta }\,M_u\,M_{u,\beta }\right) \right] }{ \left( \eta - b_s\,M_s\,\theta \right) \left( \delta - b_u\,M_u\,\epsilon \right) -\zeta \,\lambda \,b_u\,M_u } \end{aligned} \end{array}\right. } , \end{aligned}$$

(51)

$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle M_{u,\gamma \gamma }\left( \delta - b_u\,M_u\,\epsilon \right) = e_u\,\epsilon + A_{u,\gamma }\,\epsilon _{,\gamma }- \left( M_{u,\gamma }\delta _{,\gamma }+M_{s,\gamma }\zeta _{,\gamma }+\epsilon \,D_{u,\gamma }\,M_u\,M_{u,\gamma }\right) - M_{s,\gamma \gamma }\,\zeta \\ \\ \displaystyle \begin{aligned} M_{s,\gamma \gamma } = &{} \frac{\left( e_s\,\theta -e_u\,\lambda + A_{s,\gamma }\,\theta _{,\gamma }-A_{u,\gamma }\,\lambda _{,\gamma } + \lambda \,D_{u,\gamma }\,M_u\,M_{u,\gamma } - \theta \,D_{s,\gamma }\,M_s\,M_{s,\gamma }\right) \left( \delta - b_u\,M_u\,\epsilon \right) }{ \left( \eta - b_s\,M_s\,\theta \right) \left( \delta - b_u\,M_u\,\epsilon \right) -\zeta \,\lambda \,b_u\,M_u} \\ &{}- \lambda \,b_u\,M_u\,\frac{\left[ e_u\,\epsilon + A_{u,\gamma }\,\epsilon _{,\gamma }- \left( M_{u,\gamma }\delta _{,\gamma }+M_{s,\gamma }\zeta _{,\gamma }+\epsilon \,D_{u,\gamma }\,M_u\,M_{u,\gamma }\right) \right] }{ \left( \eta - b_s\,M_s\,\theta \right) \left( \delta - b_u\,M_u\,\epsilon \right) -\zeta \,\lambda \,b_u\,M_u } \end{aligned} \end{array}\right. }\nonumber \\ \end{aligned}$$

(52)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle M_{u,\beta \gamma }\left( \delta - b_u\,M_u\,\epsilon \right) = f_u\,\epsilon + A_{u,\beta }\,\epsilon _{,\gamma }- \left( M_{u,\beta }\delta _{,\gamma }+M_{s,\beta }\zeta _{,\gamma }+\epsilon \,D_{u,\beta }\,M_u\,M_{u,\gamma }\right) - M_{s,\beta \gamma }\,\zeta \\ \\ \displaystyle \begin{aligned} M_{s,\beta \gamma } = &{} \frac{\left( f_s\,\theta -f_u\,\lambda + A_{s,\beta }\,\theta _{,\gamma }-A_{u,\beta }\,\lambda _{,\gamma } + \lambda \,D_{u,\beta }\,M_u\,M_{u,\gamma } - \theta \,D_{s,\beta }\,M_s\,M_{s,\gamma }\right) \left( \delta - b_u\,M_u\,\epsilon \right) }{ \left( \eta - b_s\,M_s\,\theta \right) \left( \delta - b_u\,M_u\,\epsilon \right) -\zeta \,\lambda \,b_u\,M_u} \\ &{}- \lambda \,b_u\,M_u\,\frac{\left[ f_u\,\epsilon + A_{u,\beta }\,\epsilon _{,\gamma }- \left( M_{u,\beta }\delta _{,\gamma }+M_{s,\beta }\zeta _{,\gamma }+\epsilon \,D_{u,\beta }\,M_u\,M_{u,\gamma }\right) \right] }{ \left( \eta - b_s\,M_s\,\theta \right) \left( \delta - b_u\,M_u\,\epsilon \right) -\zeta \,\lambda \,b_u\,M_u } \end{aligned} \end{array}\right. } ,\nonumber \\ \end{aligned}$$

(53)

where

$$\begin{aligned}&\displaystyle a_f = \frac{N_\mathrm{c}}{\pi ^2} \int _0^\varLambda \mathrm{d}p\, p^2 \left[ n_{-f}\left( 1-n_{-f} \right) + n_{+f}\left( 1-n_{+f} \right) \right] , \end{aligned}$$

(54)

$$\begin{aligned}&\displaystyle b_f = \frac{N_\mathrm{c}}{\pi ^2} \int _0^\varLambda \mathrm{d}p\, p^2 \frac{ n_{-f}\left( 1-n_{-f} \right) + n_{+f}\left( 1-n_{+f} \right) }{T\;E^2_f}, \end{aligned}$$

(55)

$$\begin{aligned}&\displaystyle c_f = \frac{N_\mathrm{c}}{\pi ^2} \int _0^\varLambda \mathrm{d}p\, p^2 \frac{ n_{-f}\left( 1-n_{-f} \right) - n_{+f}\left( 1-n_{+f} \right) }{E_f}, \end{aligned}$$

(56)

$$\begin{aligned} d_f= & {} \frac{N_\mathrm{c}}{\pi ^2} \int _0^\varLambda \mathrm{d}p\; p^2 \left\{ \left( \frac{2\,M\,M_{,\beta }}{E^2} +p^2\;\frac{M^2_{,\beta }}{T\,E^4}\right) \right. \nonumber \\&\times \left[ n_{-f}\left( 1-n_{-f} \right) + n_{+f}\left( 1-n_{+f} \right) \right] \nonumber \\&+\left. \left( 1 +\frac{M\,M_{,\beta }}{T\,E^2}\right) \left[ \left( 1-2\,n_{-f} \right) n_{-f,\beta } + \left( 1-2\,n_{+f} \right) n_{+f,\beta } \right] \right\} ,\end{aligned}$$

(57)

$$\begin{aligned} e_f= & {} \frac{N_\mathrm{c}}{\pi ^2} \int _0^\varLambda \mathrm{d}p\; p^2\left\{ p^2\;\frac{M^2_{,\gamma }}{T\,E^4}\left[ n_{-f}\left( 1-n_{-f} \right) + n_{+f}\left( 1-n_{+f} \right) \right] \right. \nonumber \\&+ \frac{M\,M_{,\gamma }}{T\,E^2}\left[ \left( 1-2\,n_{-f} \right) n_{-f,\gamma } + \left( 1-2\,n_{+f} \right) n_{+f,\gamma } \right] \nonumber \\&\left. + \frac{\left( 1-2\,n_{-f} \right) n_{-f,\gamma } - \left( 1-2\,n_{+f} \right) n_{+f,\gamma } }{E} \right\} , \end{aligned}$$

(58)

$$\begin{aligned} f_f= & {} \frac{N_\mathrm{c}}{\pi ^2} \int _0^\varLambda \mathrm{d}p\; p^2\left\{ \left( \frac{M\,M_{,\gamma }}{E^2} +p^2\;\frac{M_{,\beta }M_{,\gamma }}{T\,E^4}\right) \right. \nonumber \\&\times \left[ n_{-f}\left( 1-n_{-f} \right) + n_{+f}\left( 1-n_{+f} \right) \right] \nonumber \\&\quad + \left( 1 +\frac{M\,M_{,\beta }}{T\,E^2}\right) \left. \left[ \left( 1-2\,n_{-f} \right) n_{-f,\gamma } + \left( 1-2\,n_{+f} \right) n_{+f,\gamma } \right] \right\} , \end{aligned}$$

(59)

$$\begin{aligned}&\displaystyle A_{f,\beta }=a_f + b_f\;M_f\,M_{f,\beta }, \end{aligned}$$

(60)

$$\begin{aligned}&\displaystyle A_{f,\gamma }=c_f + b_f\;M_f\,M_{f,\gamma }, \end{aligned}$$

(61)

$$\begin{aligned}&\displaystyle C_{f,\beta \beta }=d_f + b_f\;M_f\,M_{f,\beta \beta }, \end{aligned}$$

(62)

$$\begin{aligned}&\displaystyle C_{f,\gamma \gamma }=e_f + b_f\;M_f\,M_{f,\gamma \gamma }, \end{aligned}$$

(63)

$$\begin{aligned}&\displaystyle C_{f,\beta \gamma }=f_f + b_f\;M_f\,M_{f,\beta \gamma }, \end{aligned}$$

(64)

$$\begin{aligned}&\displaystyle B(M_u,M_s) = 4\,G-2\,\frac{K^2}{G}\;u^2-2\,K\,s, \end{aligned}$$

(65)

$$\begin{aligned}&\displaystyle \delta (M_u,M_s) = \left( 1 - F_{1u}\,B\right) , \end{aligned}$$

(66)

$$\begin{aligned}&\displaystyle \zeta (M_u,M_s) = 2\,K\,u, \end{aligned}$$

(67)

$$\begin{aligned}&\displaystyle \epsilon (M_u,M_s)= B\,M_u, \end{aligned}$$

(68)

$$\begin{aligned}&\displaystyle \eta (M_u,M_s)=\left( 1-4GF_{1s}\right) \left( 1-F_{1u}B\right) - 8K^2u^2F_{1u}, \end{aligned}$$

(69)

$$\begin{aligned}&\displaystyle \theta (M_u,M_s) =4\,G\,\left( 1-F_{1u}B\right) M_s\,, \end{aligned}$$

(70)

$$\begin{aligned}&\displaystyle \lambda (M_u,M_s)= 4\,K\,u\,M_u, \end{aligned}$$

(71)

$$\begin{aligned}&\displaystyle F_{1f} = \frac{N_\mathrm{c}}{\pi ^2}\int _0^\varLambda \mathrm{d}p\,p^4\,\frac{ \varPsi _{f}}{E^3_f}, \end{aligned}$$

(72)

$$\begin{aligned}&\displaystyle n_{f\pm } =\frac{1}{1+\exp \left\{ \frac{\sqrt{p^2 + M^2_f} \pm \mu _f}{T} \right\} } \end{aligned}$$

(73)

and $u\equiv \left\langle {\overline{u}} u\right\rangle$, $s\equiv \left\langle {\overline{s}} s\right\rangle$.

About the thermodynamic potential $\phi =-\varOmega \,\beta$, one has

$$\begin{aligned} \phi _{,\beta }= & {} \sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2\,E_f\,\varPsi _f +2\,G\,s^2\nonumber \\&+ u\,\left( M_u-m_u\right) +s\,\left( M_s-m_s\right) \nonumber \\= & {} \sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2\,E_f\,\varPsi _f +K\,u^2\,s\nonumber \\&+ u\,\left( M_u-m_u\right) +\frac{s\,\left( M_s-m_s\right) }{2}, \end{aligned}$$

(74)

$$\begin{aligned} \phi _{,\gamma }= & {} \sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2\,\left( n_{+f}-n_{-f}\right) \end{aligned}$$

(75)

$$\begin{aligned} \phi _{,\beta \beta }= & {} -\sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2E_f\left( n_{+f,\beta }+n_{-f,\beta }\right) \end{aligned}$$

(76)

$$\begin{aligned} \phi _{,\beta \gamma }= & {} \sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2\,\left( n_{+f,\beta }-n_{-f,\beta }\right) \end{aligned}$$

(77)

$$\begin{aligned} \phi _{,\gamma \gamma }= & {} \sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2\,\left( n_{+f,\gamma }-n_{-f,\gamma }\right) \end{aligned}$$

(78)

$$\begin{aligned} \phi _{,\beta \beta \gamma }= & {} \sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2\,\left( n_{+f,\beta \beta }-n_{-f,\beta \beta }\right) \end{aligned}$$

(79)

$$\begin{aligned} \phi _{,\beta \gamma \gamma }= & {} \sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2\,\left( n_{+f,\beta \gamma }-n_{-f,\beta \gamma }\right) \end{aligned}$$

(80)

$$\begin{aligned} \phi _{,\gamma \gamma \gamma }= & {} \sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2\,\left( n_{+f,\gamma \gamma }-n_{-f,\gamma \gamma }\right) \end{aligned}$$

(81)

$$\begin{aligned} \phi _{,\beta \beta \beta }= & {} -\sum _{f=u,d,s} \frac{N_\mathrm{c}}{\pi ^2}\int ^\varLambda _0 \mathrm{d}p\,p^2E_f\left( n_{+f,\beta \beta }+n_{-f,\beta \beta }\right) \nonumber \\&+ \sum _{f=u,d,s}\left( a_f + b_f\,M_f\,M_{f,\beta }\right) M_f\,M_{f,\beta }. \end{aligned}$$

(82)

Finally, by defining

$$\begin{aligned} H_f = F_{1f} + b_f\;M^2_f, \end{aligned}$$

(83)

the chiral susceptibilities are

$$\begin{aligned} \chi _u=\chi _d = \frac{\partial M_u}{\partial m_u} = \frac{1 - 4\,G\,H_s}{1 - 4\,G(H_u+H_s)+4\,H_u\,H_s(4\,G^2-2\,K\,G\,s-K^2\,u^2)+2\,K\,s\,H_u}\nonumber \\ \end{aligned}$$

(84)

and

$$\begin{aligned} \chi _s = \frac{\partial M_s}{\partial m_s} = = \frac{1 - (4\,G-2\,K\,s)\,H_u}{1 - 4\,G(H_u+H_s)+4\,H_u\,H_s(4\,G^2-2\,K\,G\,s-2\,K^2\,u^2)+2\,K\,s\,H_u}.\nonumber \\ \end{aligned}$$

(85)