1. Helmholtz International Summer School, Dubna, 24 July 2008 Pion decay constant in Dense Skyrmion Matter Hee-Jung Lee ( Chungbuk Nat’l Univ.) Collaborators : Byung-Yoon Park (Chungnam Nat’l Univ.) Mannque Rho (Saclay) Vicente Vento (Univ. of Valencia) Dong-Pil Min (Seoul Nat’l Univ.)

2. Contents Overview SU(2) Skyrme Model Pion decay constant in Skyrmion Matter Results Summary and discussion

3. Properties of Pion in free space : Properties of Pion in Nuclear Matter : Overview

4. 1. Chiral Perturbation Theory - H. Yabu et al, PRD 50 (1994) 3549. - V. Thorsson et al, NPA 589 (1995) 633. - T. S. Park et al, PLB 409 (1997) 26. - U. Meissner et al, Ann.Phys297(2002)27. : Fermi Gas & Mean Field Approx.

5. 2. Optical Potential Theory - T. Waas et al, PLB 405 (1997) - T. Yamazaki et al, PLB 418 (1998) : S-wave pion- nucleon potential of the deeply bound states in Pb nucleus observed at GSI Pion mass increases - a 7% at the center Pb nucleus - a 3% in symmetric nuclear matt.

6. SU(2) Skymion Model Approach - Baryons are appeared as static solitons of the Skyrme Lagrangian. - The pion-matter interactions are generated from the same Skyrme Lagrangian as classical potentials. - The results contain higher orders in density.

7. SU(2) Skymion Model 1. The Model Lagrangian : 2. Hedgehog Ansatz : Single Baryon by topology Baryon number density

8. 3. Skyrmion Matter : Skyrmion Crystal - I. Klevanov, NPB 262(1985) 133; - M. Kugler et al, PRD 40 (1989) 3421.

12. 4. Pions in Skyrmion Matter : The dynamics of the pion in dense skyrmion matter. - S. Saito et al, Prog. Theor. Phys. 75 (1986) 68.

13. - Lagrangian :

14. Classical potentials

17. - The Pion Effective Mass ( ) Pole equation of the Pion Propagator :

18. PCAC Relation : - The in medium decay constant ( ) :

19. - Results :

20. At high den., the Chiral symm. is restored. Trace (Scale) Anomaly in QCD Introduce a scalar field into the Skyme theory

21. QCD Trace Anomaly The classical QCD action is scale-invariant in the chiral limit under At quantum level, this invariance becomes broken even for the chiral limit. The dilatation current is not conserved :

22. Modified Skyrme Lagrangian - J. Ellis and J. Lanik, PLB150 (1985) 289 - G. E. Brown and M. Rho, PRL 66 (1991) 2720 The scalar field ( ) with unity scale dim. and the potential energy generate the trace anomaly in this effective Lagrangian : - Furnstahl, et al, PRC 52 (1995) 1368

23. 1. Set A : R. J. Furnstahl, et al, PRC 52 (1995) 1368 2. Set B : C. Song, et al, PRC 56 (1997) 2244 3. Set C : M. S. Shifman, et al, NPB 147 (1979) 385 ! If one interpret the scalar field as a pure glueball, the gluon condensation gives the mass of the scalar field as:

25. Energy per baryon Potential Energy for the scalar field :

26. Energy per single skyrmion as a function of the scalar field X for a given lattice size with the parameter sets B.

27. Phases of skyrmion matter

28. Pions and Scalar in Skyrmion Matter Fluctuations on top of the static skyrmion crystal : Lagrangian for pions and the scalar field : Lagrangian for the static background skyrmion matter

29. Lagrangian for pions on the matter : Lagrangian for the scalar field on the matter : Lagrangian for the interaction between pions and the scalar field on the matter :

30. : External axial vector current Pion decay constants : Axial vector current and vector current of SM

34. Pion decay constants

35. 0th-order propagator Pion-BG interactions Pion velocity

36. =  + + ….

37. = + +  Pion propagator in medium  + Pion velocity in medium : for massless pion

38. Pion velocity in medium H.-J. Lee, B.-Y. Park, M. Rho, V. Vento, Nucl. Phys. A (2004)

39. Relation between two kinds of constants For massless pion

40. Summary and ….. 1.Phase transition with respect to chiral symmetry happens with density in skyrmion matter. 2.Mass and decay constants of pion decrease with density of skyrmion matter. 3. Velocity of pion decreases to minimum and returns the value in free space when chiral sym. is restored. 4. Relation in two kinds of constants for massive pion? 5.This approach would be too simple to be a real physical situation. : need to quantize skyrmions in skyrmion matter. 6. Heating skyrmion matter, what happens? Thank you!!