1. Compact stars in the QCD phase diagram II,2009 Pion Superfluidity beyond Mean Field Approximation In the Nambu-Jona-Lasinio Model Chengfu Mu Tsinghua University, Beijing Collaborator: Professor Pengfei Zhuang Ref: Phys.Rev.D79, 094006(2009) . CSQCD II,Peking U.

2. Pion Superfluidity in the mean field approx. NJL model: Chiral and pion condensates: Thermodynamic potential in MF: where: where: CSQCD II,Peking U.

3. normal Sarma normal BEC BCS Pion Superfluidity in the mean field approx. Gap equations in MF: Isospin denstiy Meson mass (RPA): phase diagram He et al. Phys.Rev.D71,116001(2005), ibid,74,036005(2006) CSQCD II,Peking U.

4. Why do we go beyond mean field? • The mean field result is qualitatively correct for the phase transitions, we have to go beyond mean field approximation to gain the quantitative ones. • The Sarma phase can exist when the isospin density is not very high in the MF approximation, but it sticks up in the phase diagram. • The pion condensation undergoes a BEC-BCS crossover when theisospin chemical potential increases. Pions are bound states in the BEC regime, we have to include the meson fluctuation effect. CSQCD II,Peking U.

5. Pion Superfluidity beyond mean field approx. Total thermodynamic potential: Meson fluctuation part: pole approximation: if where Gap equations beyond MF: CSQCD II,Peking U.

6. Pion Superfluidity beyond mean field approx. Expand fluctuation part around the mean field values : Keep only first order, final gap equations: Isospin denstiy The effective coupling constants: CSQCD II,Peking U.

7. Normal Pion superfluidity Numerical results Fig.1: The pion condensates in mean field approximation (thin line) and including meson fluctuations (thick line) as functions of isospin density. CSQCD II,Peking U.

8. normal Sarma BEC BCS Numerical results beyond the mean field approx. Fig.2: mean field approximation: thin lines, including meson fluctuations: thick lines. The solid lines are the phase transition lines and the dashed lines are the BEC-BCS crossover lines. CSQCD II,Peking U.

9. normal BCS BEC Numerical results beyond the mean field approx. Fig.3: Phase diagram of pion superfluidity in nB-nIplane CSQCD II,Peking U.

10. BCS BEC Numerical results beyond the mean field approx. Fig.4: The effective chemical potential as a function of isospin density . BEC region is strongly shrunk. CSQCD II,Peking U.

11. BEC BCS Numerical results beyond the mean field approx. Fig.5: The meson mass (solid line) and quark mass m (dashed line) as functions of isospin density. CSQCD II,Peking U.

12. Numerical results beyond the mean field approx. Summary From our numerical calculations, the main effects of meson fluctuations on the phase structure are: 1) the critical temperature of pion superfluidity is highly suppressed and the Sarma phase which exists at low isospin chemical potential in mean field approximation is fully washed away. 2) the BEC region at low isospin density is significantly shrunk. CSQCD II,Peking U.

13. Thank you! CSQCD II,Peking U.

14. backups Considering the fact that mesons, in particular pions because of their low mass, dominate the thermodynamics of a quark-hadron system at low temperature, the mesonic fluctuations should be significant for the phase structure of pion superfluidity. By recalculating the minimum of the thermodynamic potential including meson contribution, we derived a new gap equation for the pion condensate which is similar to the mean field form but with a medium dependent coupling constant. CSQCD II,Peking U.