1. Masayasu Harada (Nagoya Univ.) @KEK 理論センター研究会 「原子核・ハドロン物 理」 (August 11, 2009) based on M.H. and C.Sasaki, arXiv:0902.3608 M.H., C.Sasaki and W.Weise, Phys. Rev. D 78, 114003 (2008) see also M.H. and C.Sasaki, PRD74, 114006 (2006) Effects of vector – axial-vector mixing to dilepton spectrum in hot and/or dense matter

2. Origin of Mass ? of Hadrons of Us One of the Interesting problems of QCD =

3. Origin of Mass = quark condensate Spontaneous Chiral Symmetry Breaking

4. ☆ QCD under extreme conditions ・ Hot and/or Dense QCD ・ large flavor QCD ◎ Chiral symmetry restoration T critical ～ 170 – 200 MeV  critical ～ a few times of normal nuclear matter density Nf critical ～ 5 – 12 (still asymptotically free) Change of Hadron masses ?

5. Masses of mesons become lighter due to chiral restoration ☆ Dropping mass of hadrons ◎ Brown-Rho scaling G.E.Brown and M.Rho, PRL 66, 2720 (1991) for T → T critical and/or ρ → ρ critical ◎ NJL model T.Hatsuda and T.Kunihiro, PLB185, 304 (1987) ◎ QCD sum rule : T.Hatsuda and S.H.Lee, PRC46, R34 (1992) T.Hatsuda, Quark Matter 91 [NPA544, 27 (1992)] ◎ Vector Manifestation M.H. and K.Yamawaki, PRL86, 757 (2001) M.H. and C.Sasaki, PLB537, 280 (2002) M.H., Y.Kim and M.Rho, PRD66, 016003 (2002)

6. ☆ Di-lepton data consistent with dropping mass Analysis : R.Rapp and J.Wambach, ANP 25,1 (2000) Exp: G.Agakishiev et al. [CERES], PRL75, 1272 (1995) ◎ KEK-PS/E325 experiment m  = m 0 (1 -   /  0 ) for  = 0.09 m  = m 0 (1 -   /  0 ) for  = 0.03 K.Ozawa et al., PRL86, 5019 (2001) M.Naruki et al., PRL96, 092301 (2006) R.Muto et al., PRL98, 042501 (2007) F.Sakuma et al., PRL98, 152302 (2007) ☆ Di-lepton data (NA60) consistent with NO dropping mass H.v.Hees and R.Rapp, NPA806, 339 (2008) All P T J.Ruppert, C.Gale, T.Renk, P.Lichard and J.I.Kapusta, PRL100, 162301 (2008)

7. Outline 1. Introduction 2. Effect of Vector-Axialvector mixing in hot matter 3. Effect of V-A mixing in dense baryonic matter 4. Summary

8. ☆ A model including π, ρ, A1 based on the generalized hidden local symmetry ・ Generalized HLS M.Bando, T.Kugo and K.Yamawaki, NPB 259, 493 (1985) M.Bando, T.Fujiwara and K.Yamawaki, PTP 79, 1140 (1988) ・ Loop effect in the G-HLS MH, C.Sasaki, PRD 73, 036001 (2006) ◎ Dropping A 1 without dropping ρ A1(1260) becomes light, while ρ does not Effect of dropping A1 to di-lepton spectrum (Standard Scenario for Chiral Restoration) ◎ Dropping A 1 with dropping ρ Both A1(1260) and ρ become light. (Hybrid Scenario for Chiral Restoration) 2. Effect of Vector-Axialvector mixing in hot matter MH, C.Sasaki and W.Weise, Phys. Rev. D 78, 114003 (2008)

9. ☆ T – dependence of ρ and A1 meson masses Hadronic many body effects (  and A1 loop corrections) included Note : m  = 0 (chiral limit) ◎ Dropping A 1 without dropping ρ

10. A1 meson A 1 meson effects ・ Im Gv around the ρ pole is suppressed (collisional broadening) ・ Im Gv is enhanced around the A 1 -π threshold e-e- e+e+ A1 π + ρ e-e- e+e+ π ρ

11. ◎ Effect of pion mass e-e- e+e+ A1 π + ρ e-e- e+e+ π ρ Effects of pion mass ・ Enhancement around s 1/2 = m a – m π ・ Cusp structure around s 1/2 = m a + m π

12. ◎ V-A mixing → small near Tc

13. ☆ Dropping A1 with dropping ρ (m π = 0) T/Tc = 0.8 Effect of A1 meson s 1/2 = 2 m ρ Effect of dropping A 1 ・ Spectrum around the ρ pole is suppressed ・ Enhancement around A 1 -π threshold ・ Cusp structure around s 1/2 = 2m ρ

14. All P T J.Ruppert, C.Gale, T.Renk, P.Lichard and J.I.Kapusta, PRL100, 162301 (2008) H.v.Hees and R.Rapp, NPA806, 339 (2008) ☆ Dropping A1 was seen ? may need detailed analysis ?

15. ◎ V-A mixing from the current algebra analysis in the low density region B.Krippa, PLB427 (1998) 3. Effect of V-A mixing in dense baryonic matter ・ This is obtained at loop level in a field theoretic sense. ・ Is there more direct V-A mixing at Lagrangian level ? such as ￡ ～ V  A  ? ・・・ impossible in hot matter due to parity and charge conjugation invariance ・・・ possible in dense baryonic matter since charge conjugation is violated but be careful since parity is not violated [M.H. and C.Sasaki, arXiv:0902.3608]

16. ☆ A possible V-A mixing term violates charge conjugation but conserves parity generates a mixing between transverse  and A 1 ex : for p  = (p 0, 0, 0, p) no mixing between V 0,3 and A 0,3 (longitudinal modes) mixing between V 1 and A 2, V 2 and A 1 (transverse modes) S.K.Domokos, J.A.Harvey, PRL99 (2007) ◎ Dispersion relations for transverse  and A 1 + sign ・・・ transverse A 1 [p 0 = m a1 at rest (p = 0)] - sign ・・・ transverse  [p 0 = m  at rest (p = 0)]

17. ☆ Determination of mixing strength C ◎ An estimation from  dominance ・  A 1  interaction term an empirical value : (cf: N.Kaiser,U.G.Meissner, NPA519,671(1990)) ・  NN interaction provides the  condensation in dense baryonic matter an empirical value : ・ Mixing term from  dominance

18. ◎ An estimation in a holographic QCD (AdS/QCD) model ・ Infinite tower of vector mesons in AdS/QCD models ,  ’,  ”, … ・ These infinite  mesons can generate V-A mixing ・ This summation was done in an AdS/QCD model S.K.Domokos, J.A.Harvey, PRL99 (2007)

19. ☆ Dispersion relations  meson A 1 meson ・ C = 0.5 GeV : small changes for  and A 1 mesons ・ C = 1 GeV : small change for A 1 meson substantial change in  meson ・ C ～ 1.1 GeV : p 0 2 < 0 for  → vector meson condensation ? [S.K.Domokos, J.A.Harvey, PRL99 (2007)]

20. ☆ Integrated vector spectrum for C = 1 GeV 2 m  note :   = 0 for √s < 2 m  ◎ low p region ・ longitudinal mode : ordinary  peak ・ transverse mode : an enhancement for √s < m  and no clear  peak a gentle peak corresponding to A 1 meson ・ spin average (Im G L + 2 Im G T )/3 : 2 peaks corresponding to  and A 1 ◎ high p region ・ longitudinal mode : ordinary  peak ・ transverse mode : 2 small bumps and a gentle A 1 peak ・ spin averaged : 2 peaks for  and A 1 ; Broadening of  peak

21. ☆ Di-lepton spectrum at T = 0.1 GeV with C = 1 GeV 2 m  ・ A large enhancement in low √s region → result in a strong spectral broadening ・・・ might be observed in with low-momentum binning at J-PARC, GSI/FAIR and RHIC low-energy running note :   = 0 for √s < 2 m 

22. ☆ Effects of V-A mixing for  and  mesons ・ Assumption of nonet structure → common mixing strength C for  -A 1,  -f 1 (1285) and  -f 1 (1420) ・ Vector current correlator note : we used the following meson widths

23. ◎  meson spectral function spin averaged, integrated over 0 < p < 1 GeV ・ C = 1 GeV : suppression of  peak (broadening) ・ C = 0.3 GeV : suppression for √s > m  enhancement for √s < m 

24. ☆ Integrated rate with ,  and  mesons for C = 0.3, 0.5, 1 GeV ・ An enhancement for √s < m , m  (reduced for decreasing C) ・ An enhancement for √s < m  from  -f 1 (1420) mixing → a broadening of  width (at T = 0.1GeV)

25. 4. Summary ◎ Effect of vector - axial-vector mixing in hot matter ・ Dropping A1 in the standard scenario Through the V-A mixing, we may see the dropping A1. (Note: might be difficult since V-A mixing becomes small at T=Tc ?) ・ Dropping A1 with dropping ρ, effect of A1 suppress the di-lepton spectrum ◎ Effect of V-A mixing in dense baryonic matter for  -A 1,  -f 1 (1285) and  -f 1 (1420) → ・ modification of dispersion relations ・ change in vector spectral function (broadening) ○ Large C ? : ・ If C = 0.1GeV, then this mixing will be irrelevant. ・ If C > 0.3GeV, then this mixing will be important. ・ We need more analysis for estimation.