1. Compact stars in the QCD phase diagram II,2009
2019/4/30
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, (2009) .
2019/4/30
CSQCD II,Peking U.
Compact stars in the QCD phase diagram II,Peking University

2. Pion Superfluidity in the mean field approx.
NJL model:
Chiral and pion condensates:
Thermodynamic potential in MF:
where:
where:
2019/4/30
CSQCD II,Peking U.

3. Pion Superfluidity in the mean field approx.
2019/4/30
Pion Superfluidity in the mean field approx.
Gap equations in MF:
Isospin denstiy
Meson mass (RPA):
normal
Sarma
BEC
BCS
phase diagram
He et al. Phys.Rev.D71,116001(2005),
ibid,74,036005(2006)
2019/4/30
CSQCD II,Peking U.
Compact stars in the QCD phase diagram II,Peking University

4. Why do we go beyond mean field?
2019/4/30
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 the isospin chemical potential increases. Pions are bound states in the BEC regime, we have to include the meson fluctuation effect.
2019/4/30
CSQCD II,Peking U.
Compact stars in the QCD phase diagram II,Peking University

5. Pion Superfluidity beyond mean field approx.
Total thermodynamic potential:
Meson fluctuation part:
pole approximation: if
where
Gap equations beyond MF:
2019/4/30
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:
2019/4/30
CSQCD II,Peking U.

7. Numerical results Normal Pion superfluidity
Fig.1: The pion condensates in mean field approximation (thin line) and including meson fluctuations (thick line) as functions of isospin density.
2019/4/30
CSQCD II,Peking U.

8. Numerical results beyond the mean field approx.
BEC
BCS
normal
Sarma
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.
2019/4/30
CSQCD II,Peking U.

9. Numerical results beyond the mean field approx.
BEC
BCS
normal
Fig.3: Phase diagram of pion superfluidity in nB-nI plane
2019/4/30
CSQCD II,Peking U.

10. Numerical results beyond the mean field approx.
BEC
BCS
Fig.4: The effective chemical potential as a
function of isospin density . BEC region is strongly
shrunk.
2019/4/30
CSQCD II,Peking U.

11. Numerical results beyond the mean field approx.
BEC
BCS
Fig.5: The meson mass (solid line) and quark mass m (dashed line) as functions of isospin density.
2019/4/30
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.
2019/4/30
CSQCD II,Peking U.

13. Thank you!
2019/4/30
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.
2019/4/30
CSQCD II,Peking U.