1. Teiji Kunihiro (YITP, Kyoto) Dubna round table session July 7-9, 2005 JINR, Dubna, Russia Status of the chiral symmetry restoration and sigma-meson physics

2. Contents  Introduction : vacuum structure v.s. elementary excitations  The sigma meson  Chiral restoration and the sigma meson  Chiral restoration as seen in other channels including chiral anomaly  Summary

3. A condensed matter physics of vacuum (Y. Nambu; 1960)

4. Gauge invariance Axial gauge (chiral) symm. (Bogoliubov-Anderson)

5. Inter-deterministic property of the matter and vacuum in QFT ; vacuum In the definition of vacuum in QFT, the definition of the particle picture is pre-requisite. Equivalence between what is the vacuum and what are the particles to be observed. Change of the vaccumChange of the particle picture Eg. Superconductivity The definition of vacuum:

6. 常伝導超伝導 gap Change of vacuum Change in the elementary excitations Dispersion relation of electron-quasi particle in the normal metal and superconducting matterial and Bogoliubov-Anderson mode

7. Chiral Transformation Chirality: For, the chiral transformation forms (left handed) (right handed)

8. In the chiral limit (m=0), ; Chiral invariant ; Chiral invariant! Ｘ Chiral Invariance of Classical QCD Lagrangian in the chiral limit (m=0) invariant! not invarinat!Dirac mass term order parameter

9. the generators of a continuous transformation ; Noether current eg. Chiral transformation for The two modes of symmetry realization in the vacuum : a. Wigner mode b. Nambu-Goldstone mode The notion of Spontaneous Symmetry Breaking Notice; The symmetry is spontaneously broken. Now, Chiral symmetry is spontaneously broken!

10. QCD 真空の非摂動的性質 （補） Gell-Mann-Oakes-Renner using We have QCD sum rules for heavy-quark systems, Non-perturbative properties of QCD vacuum: condensates

11. Confinement and chiral transion in Lattice QCD

14. CS C T   SB QGP QCD critical point meson condensation? ?  0 CFL QCD phase diagram H matter? ~150MeV

15. The sigma meson Real or not? If real, what is it?

16. The significance of the  meson in low energy hadron physics and QCD 1. The pole in this mass range observed in the pi-pi S-matrix. As a compilation of the pole positions of the  obatined in the modern analyses: Significance of respecting chiral symmetry,unitarity and crossing symmetry to reproduce the phase shifts both in the  (s)- and , (t)-channels with a low mass  pole;(Igi and Hikasa(1999)). 2. Seen in decay processes from heavy particles; E. M. Aitala et al, Phys. Rev. Lett. (86), 770 (2001) 3. Responsible for the intermediate range attraction in the nuclear force. 4. Accounts for  I=1/2 enhancement in K ->2  compared with K + ->    . E.P. Shabalin (1988); T. Morozumi, C.S. Lim and I. Sanda (1990).  -N sigma term 40-60 MeV (naively » 15 MeV) enhanced by the collectiveness of the  (.T.Hatsuda and T.K.(1990)) ; see the next slide.  6. The  of the chiral order parameter The Higgs particle in the WSG model

17. Chiral Transition and the collective modes 0 c.f. Higgs particle in WSH model ; Higgs field Higgs particle

18. K. Igi and K. Hikasa, Phys. Rev. D59, 034005(1999) The phase shifts in the sigma and rho channel in the N/D Method; resp. chiral symm., crossing symm and so on. No  but  in the t-channel and the  in the t-channel Both with the  in the s-

19. The poles of the S matrix in the complex mass plane for the sigma meson channel: complied in Z. Xiao and H.Z. Zheng (2001) G.Colangero, J. Gasser and Leutwyler (2001)

20. E. M. Aitala et al, Phys. Rev. Lett. (86), 770 (2001) Without sigma pole With a sigma pole: 24 23 42 40

21. The numbers in (, ) are those in the naive quark model. (T.K. and T. Hatsuda, Phys. Lett. B240 (1990) 209) The quark content (or the scalar charge of the quarks) is enhanced by the collective  mode in the scalar channel! flavor mixing

22. Issues with the low-mass  meson in QCD In the constituent quark model; the mass in the 1.2 --- 1.6 GeV region. Some mechanism needed to down the mass with ~ 600 MeV; (i) Color magnetic interaction between the di- quarks? (Jaffe; 1977) (ii) The collectiveness of the scalar mode as the ps mode; a superposition of states. Chiral symmetry (NJL) (iii) The - molecule as suggested in scatt.. (vi) a mixed state of scalar glue ball and states

23. I=0 ; I=1; a 0 = 0 ++, a 1 = 1 ++, a 2 =2 ++, b 1 = 1 +- 3 P 0 3 P 1 3 P 2 1 P 1 We need some nontrivial dynamics to down the mass as low as 500 – 600 MeV! in the constituent quark model They all should be in the same mass range 1.2 – 1.6 GeV

24. Scalar Mesons in the Di-quark picture (Jaffe(1977), Alford and Jaffe (2000))

25. The Scalar mesons on the Lattice The Scalar Collaboration: S. Muroya,A. Nakamura,C. Nonaka,M. Sekiguchi, H. Wada,T. K. (Phys. Rev. D70, 034504(2004)) ---- A full QCD calculation -----

26. Simulation parameters Lattice size : 8 3 × 16  = 4.8  = 0.1846, 0.1874, 0.1891 CP-PACS well established light meson with large lattice a = 0.197(2) fm,  c = 0.19286(14) ( CP - PACS, Phys. Rev. D60(1999)114508 ) Number of the Z 2 noise = 1000 Wilson Fermions & Plaquette gauge action Disconnected diagram

27. Propagator for  meson (2) Where Connected diagram Disconnected diagram - Vacuum contribution

28. m_m_ The meson masses

29. m/mm/mm/mm/m  m/mm/m m con/ m  0.18461.583±0.0982.400±0.018 0.18741.336±0.0712.436±0.025 0.18911.112±0.0602.481±0.031

30.  meson propagators Connected Part & Disconnected Parts (  = 0.1891 )

31. Chiral Transition = a phase transition of QCD vacuum, being the order parameter. Lattice QCD ; eg. F. Karsch, Nucl. Phys. Proc. Suppl. 83, 14 (2000). The wisdom of many-body theory tells us: If a phase transition is of 2nd order or weak 1st order, 9 soft modes » the fluctuations of the order parameter For chiral transition, The  meson becomes the soft mode of chiral transition at T. Hatsuda and T. K., Phys. Rev. Lett.; Prog. Theor. Phys (1985): It was also shown that hadronic excitations (para pion and sigma) exisit even in the ``QGP” phase.

32. Chiral Transition and the collective modes 0 c.f. Higgs particle in WSH model ; Higgs field Higgs particle

33. T. Hatsuda and T. K., Phys. Rev. Lett. 55 (1985), 158 T dependence of the (`para’) sigma and (`para’) pion masses Large T

34.   What is the significance of the  in hadron physics? the softening of the  with increasing T and

35. Wait! Is the pole observed in the pi-pi phase shift really the  as the quantum fluctuation of the order parameter of the chiral transition? A change of the environment a change of the mode coupled to the order parameter Production of the  -meson in a nuclear medium Useful for exploring the existence of the  and the possible restoration of chiral symmetry at finite density. ( T. K., Prog. Theor. Phys. Suppl. 120 (1994), 75) What is a good observables to see the softening in the sigma channel in nuclear medium? Notice: A particle might loose it identity when put in a medium. Need of a calculation of the Spectral function as seen by,2 and 4; lepton pairs at

36. The poles of the S matrix in the complex mass plane for the sigma meson channel: complied in Z. Xiao and H.Z. Zheng (2001) G.Colangero, J. Gasser and Leutwyler (2001) Softening !

37. T. Hatsuda, H. Shimizu, T.K., Phys. Rev. Lett. 82 (1999), 2840 Spectral function in the  channel

38. This ratio represents the net effect of nuclear matter on the interacting system. CB: Phys. Rev. Lett. 85, 5539 (2000). CHAOS:Phys. Rev. C60, 018201 (1999). P. Camerini et al, Phys. Rev. C64, 067601 (2001). (CHAOS coll.) A’=2 !208 CHAOS (1996)

39. C  ; A-dependence of  A Oset: Full model of the       process, standard nuclear effects discussed, P-wave pionic modes included and the  -meson dynamically generated. Oset and Vicente, PRC60(1999)064621 Muhlich: Model based on Oset’s developed for the       and       reactions, better treatment of FSI of pions with the nucleus, no medium modifications. Muhlich et al., PLB595(2004)216 TAPS CHAOS     I=0     I=1 E~420 MeV  ~ 2/3  0     I~0     I=2 E~420 MeV  ~ 1/3  0

40. Differential cross sections of the reaction A( ,     )A' J.G. Messchendorp et al, Phys. Rev. Lett. 89 (2002), 222302. ----- phase space TAPS experiment: L. Roca et al (2002) without softening

41.  A   o  o X  A   +/-  o X m(  o  o ) for isoscalar channel only: drops with increasing A consistent with isotropic angular distribution  o  o angular distribution E  = 400 - 460 MeV J.G.Messchendorp et al., PRL 89, 222302 L. Roca, et a l., PLB 541 (2002) 77, priv. comm. preliminary    S.Schadmand

42. P. Muelich, L. Alvarez-Ruso, O. Buss and U. Mosel, ( nucl-th/0401042).  FSI lowers the spectral function in the pi-pi invariant mass.

43. The spectral enhancememnt in the nonlinear ｒ ealization D. Jido, T. Hatsuda and T. K.,Phys. Rev. D63} (2000), 011901(R). In the polar decomposition M=SU, In the heavy S-field limit, fixed ;

44. D. Jido et al (2000) Softening of the in-medium pi-pi cross section In the non-linear realization

45. The renormalization of the wave function Due to the new vertex: C.f. Importance of the w.f. renormalization in other physics: U. Meissner, J. Oller and A. Wirzba, Ann. Phys. 297 (2002) 27 E. Kolomeitzev, N. Kaiser and W. Weise, P.R.L. 90 (2003)092501 Deeply bound pionic nucleiw.f. renormalization E-dependece of opt. pot. c.f. Freidman and Gal (04)

46. Deeply-bound pionic nuclei and missing repulsion K. Suzuki et al., Phys. Rev. Lett. 92, 072302 ( ’ 04) Kolomeitsev, Kaiser, & W. Weise, Phys.Rev.Lett. 90 (’03) ○ LO+EE without w.f.r. ● LO+EE with w.f.r. △ NNLO with w.f.r. P.Kienle and T.Yamazaki,Prog. Part. Nucl. Phys. 52 (2004), 85

47. The movement of the sigma pole in the complex Energy plane in the N/D method with MFA K. Yokokawa, T. Hatsuda, A.Hayashigaki, And T.K.(2002) A:  model B:  model C:   model D:

48. The T matrix in the N/D method. The in-medium  -  cross sections in I=J=0 channel. The upper (lower) panel shows the case of small (large) restoration corresponding to K. Yokokawa, T. Hatsuda, A.Hayashigaki and T.K. (2002)

49. Vector Mesons QCD sum rules, effective theories as NJL model and Brown-Rho scaling suggest that The vector mesons mass/width, or more precisely, their spectral functions may change in hot and/or dense matter

50. The softening in the meson channel K. Yokokawa et al (2002)

51. Softening of the spectral function in the Vector channel

52. Softening of the vector mesons in Nuclear media (H. En’yo et al)

53. Experiment should be made and is being anallyzed: Spring-8 Vector mesons in nuclei as seen by photon KEK-PS (Naruki et al): show a mass drop! ELSA@Bonn (Metag); CB/TAPS 4 for

54. Meson: Related to Anomaly; otherwise ; ideal mixing realized

55. ; energy-momentum tensor of QCD Quantum effects! () Current divergences and Quantum Anomalies Dilatation Chiral Anomaly Dilatation(scale) Anomaly

56. Problem # of the generators 2x(8+1)=18 1+8=9 G= H= # of NG-bosons= dim G - dim H = 18 – 9 = 9 (?) Nambu-Goldstone Theorem # of the lightest pseudo-scalar mesons 3 + 4 + 1 = 8 ! (140)(500)(550)<< (958) Why is so massive ? ------ U A (1) Problem Anomaly Operator Equation! even in the chiral limit!

57. Also selective coupling ofwith N*(1535) chiral dynamics v.s. Chiral doublet (a la DeTar-Kunihiro( ‘ 89) )type? Near future experiment in GSI (Hayano et al) or chiral anomaly in the medium at finite T and T.Kunihiro (1989), Ohnishi et al(1998), Ruivo et al (00) Jido, Hirenzaki and Nagahiro

58. Summary The  meson as the quantum fluctuation of the order parameter of the chiral transition may account for various phenomena in hadron physics which otherwise remain mysterious. There have been accumulation of experimental evidence of the  pole in the pi-pi scattering matrix. ( chiral symmetry, analyticity and crossing symmetry. A full lattice QCD suggests the existence of the   Partial restoration of chiral symmetry in hot and dense medium as represented by the decreasing f  leads to a softening of the  and the  pole in the 2nd Riemann sheet in various chiral models.

59. Even a slight restoration of chiral symmetry in the hadronic matter leads to a peculiar enhancement in the spectral function in the  channel near the 2m  threshold. Such an enhancement might have been observed in the reaction The decrease of the of w.f. renormalization of the pion commonly seen in the deeply bound pionic nuclei, suggesting a strongly coupled system of the pion and nuclear medium. Other channels: simultaneous softening of the  and  c.f. KEK, CERES, STAR N * and parity doublets of other baryons (DeTar and T.K. (1989); Jido, Hosaka and Oka, Hirenzaki, Nagahiro …. ) QCD critical pointscalar – vector mode coupling

60. CS C T   SB QGP QCD c.p. meson condensation? ? precursory hadronic modes?  0 CFL QCD phase diagram H matter? (T. Hatsuda and T.K. (’84, ’85)