6. 有限温度系への応用

1. 6. 有限温度系への応用

2. 6.1. 有限温度場の理論の 簡単な紹介

3. 6.1.1. Very Brief Review of Quantum Statistical Mechanics

4. ◎ micro canonical ensemble ・・・ isolated system E (energy), N (particle number), V (volume) ・・・ fixed ◎ canonical ensemble ・・・ a system in contact with a heat reservoir at temperature T T , N , V・・・ fixed 6.1.1.1. Ensemble ◎ grand canonical ensemble ・・・ system can exchange particles as well as energy with a reservoir T , V , μ (chemical potential) ・・・ fixed

5. ☆ Partition function : density matrix Ex) baryon number in QCD (number of baryons) – (number of antibaryons)

6. ☆ Thermodynamic properties ◎ pressure ◎ particle number ◎ entropy ◎ energy

7. 6.1.1.2. One bosonic degree of freedom ・ time-independent single-particle quantum mechanical state of bosons (Each boson has the same energy ω) ・ commutation relation ・・・ ◎ Hamiltonian and number operator Ignore the zero-point energy

8. The states are simultaneously number and energy eigenstates. → We can assign a chemical potential μ to the particles. ☆ Partition function ◎ Mean numeber ◎ Mean energy

9. 6.1.1.3. Free (identical) bosons in a box (cube) ◎ boundary condition ・・・Wave functions vanish at the surface of the box. ・ momenta

10. ◎ Hamiltonian and number operator ◎ Partition function

11. ◎ partition function ◎ pressure ◎ particle number ◎ energy

12. ☆ massless limit (μ= 0)

13. 6.1.2. Matsubara formalism

14. 6.1.2.1. Path integral in the quantum field theory ◎ ◎ Operators in the Hisenberg picture ・ Suppose the operators in two pictures agree with each other at t = 0, then ; ◎ Eigenstates ;

15. ☆ Transition amplitude for bosons (T=0)

16. imaginary time 6.1.2.2. Partition function for bosons in quantum statistical mechanics

18. 6.1.2.3. Neutral scalar field (μ= 0) ; ・・・ periodicity ◎ Lagrangian ◎ Fourier transformation of f ◎ Action ;

19. ☆ Partition function

20. zero-point energy same as the one obtained in the quantum statistical mechanics

21. 6.1.3. Interactions and Diagramatic Technique

22. We can use the methods used in the ordinary QFT to calculate and . 6.1.3.1. Thermal Green’s function and generating functional ◎ Thermal Green’s function ◎ Generating functional ・ perturbative expansion ・ Feynman diagrams

23. 6.1.3.2. Neutral scalar field(μ= 0) ; ☆ Feynman rules QFT FTFT ◎ propagator ◎ vertex ◎ integration

24. = = 6.1.3.3. 1-loop correction to propagator

25. ☆ Evaluation of Matsubara frequency sum ; ◎ contour C deformation

27. ☆ 1-loop correction ; = ・ same as the quantum correction at T=0 ・ includes the UV divergence ・ correction only for T>0 ・ does not include any UV divergences

28. ☆ renormalization ; ・・・ mass counter term ☆ effective mass Mass is changed at non-zero T !

29. 6.2. HLS in Hot Matter • M.H. and C.Sasaki, Phys. Lett. B 537, 280 (2002) • M.H., Y.Kim, M.Rho and C.Sasaki, Nucl. Phys. A 727, 437 (2003) • M.H. and C.Sasaki, hep-ph/0304282

30. + + + + + + + ☆ vector meson mass (propagator)-1 = (tree propagator)-1 + ◎ low temperature region r中間子はp中間子による遮蔽効果で重くなる

31. ◎ low temperature region ☆ Temporal and spatial pion decay constants parametric pion decay constant hadronic thermal correction consistent with low-temperature theorem difference appears already at one loop

32. ☆ pion velocity = 0 dispersion relation for p pion velocity ◎ low temperature region Pion velocity is smaller than the speed of light already at one loop

33. ☆ Parameter a and r meson dominance ◎Pion EM form factor (tree level at T = 0) rDominance ◎ low temperature region rdominance is well satisfied in the low temperature region.

34. The End