1. Effects of vector – axial-vector mixing to dilepton spectrum in hot and/or dense matter Masayasu Harada (Nagoya Univ.) @ Heavy Ion Meeting 2010-12 at Yonsei (December 11, 2010) based on M.H. and C.Sasaki, PRD74, 114006 (2006) M.H. and C.Sasaki, Phys. Rev. C 80, 054912 (2009) M.H., C.Sasaki and W.Weise, Phys. Rev. D 78, 114003 (2008) see also M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 82, 076010 (2010) M.H. and M.Rho, arXiv:1010.1971 [hep-ph]

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 Chiral symmetry restoration T critical 170 – 200 MeV critical a few times of normal nuclear matter density Change of Hadron masses ?

5. Masses of mesons become light 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 vector meson mass 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 consistent with NO dropping vector meson mass Analysis : H.v.Hees and R.Rapp, NPA806, 339 (2008) All P T Analysis : J.Ruppert, C.Gale, T.Renk, P.Lichard and J.I.Kapusta, PRL100, 162301 (2008) NA60 CLAS R. Nasseripour et al. PRL99, 262302 (2007). M.H.Wood et al. PRC78, 015201 (2008)

7. Quark Structure and Chiral representation coupling to currents and densities (S. Weinberg, 69) longitudinal components Note: ρ and A 1 π and σ are chiral partners These are partners …

8. Chiral Restoration linear sigma model vector manifestation Hybrid Scenario (π ρ, A1, σ degenerate) : m σ = m A1 = m ρ = m π π σ degenerate: m σ m π ρ, A1 degenerate: m A1 m ρ π, ρ degenerate: m ρ m π σ, A1 degenerate : m σ m A1 Either of 3 is expected to happen dropping mass

9. Signal of chiral symmetry restoration axial-vector current (A 1 couples) vector current (ρ couples) These must agree with each other e-e- e+e+ vector mesons (ρ etc.) e ν axial-vector mesons Impossible experimentally But, in medium, this might be seen through the vector – axial-vector mixing ! How these mixing effects are seen in the vector spectral function ? ρ etc etc e+e+ e-e-

10. Outline 1. Introduction 2. Di-lepton spectrum from the dropping ρ in the vector manifestation (Effect of the violation of meson dominance) 3. Effect of Vector-Axial-vector mixing in hot matter 4. Effect of V-A mixing in dense baryonic matter 5. Summary

11. 2. Di-lepton spectrum from the dropping ρ in the vector manifestation (Effect of the violation of vector dominance)

12. Chiral Restoration linear sigma model vector manifestation Hybrid Scenario (π ρ, A1, σ degenerate) : m σ = m A1 = m ρ = m π π σ degenerate: m σ m π ρ, A1 degenerate: m A1 m ρ π, ρ degenerate: m ρ m π σ, A1 degenerate : m σ m A1 only and are taken into account : Dropping mass

13. View of the VM in Hot Matter Assumptions Relevant d.o.f until near T c -ε only π and ρ Other mesons (A 1, σ,...) still heavy Partial chiral restoration already at T c -ε Formulation of the VM

14. M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL 54 1215 (1985) M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988) H.Georgi, PRL 63, 1917 (1989); NPB 331, 311 (1990): M.H. and K.Yamawaki, PLB297, 151 (1992); M.Tanabashi, PLB 316, 534 (1993): M.H. and K.Yamawaki, Physics Reports 381, 1 (2003) Systematic low-energy expansion including dynamical Hidden Local Symmetry EFT for and based on chiral symmetry of QCD = gauge boson of the HLS massive through the Higgs mechanism loop expansion derivative expansion Formulation of the VM VM is protected by the VM fixed point stable against the quantum corrections M.H. and K.Yamawaki, Phys. Rev. Lett. 86, 757 (2001) stable against the thermal corrections stable against the density corrections M.H. and C.Sasaki, Phys. Lett. B 537, 280 (2002) M.H., Y. Kim and M. Rho, Phys. Rev. D 66, 016003 (2002).

15. Key 1 : Strong Violation of meson dominance in the VM e+e+ e-e- meson dominance at T = 0 a = 2 vector dominance a /2 1 – a /2 long standing problem not clearly explained in QCD ! dominance at T > 0 ? e+e+ e-e- a = 2 kept fixed in several analyses (No T-dependence on a) 0 1 1 1/2

16. Key 2 : Dropping ρ occurs only for T > Tf ~ 0.7 Tc G.E.Brown, C.H.Lee and M.Rho, PRC74, 024906 (2006); NPA747, 530 (2005) (H.v.Hees and R.Rapp, hep-ph/0604269) Suppression by the violation of VD When the meson coming from the region T < Tf gives dominant contribution to the dilepton spectrum, the dropping scenario may not be excluded by NA60 data cf : Hadronic freedom leads to the small interactions for T > Tf, so that the from T > Tf is highly suppressed. G.E.Brown, MH, J.W.Holts, M.Rho, C.Sasaki, PTP 121, 1209 (2009)

17. T dependence of with dropping mass for T > Tf = 0.7 Tc Tf/TcTf/Tc ρ massm ρ 0 VM VM with VD T = 0.75 T c VM ~ VM with VD /1.8 VM ~ VM with VD / 2 T = 0.8 T c violation of Vector meson dominance Large suppression near Tc We need to include other hadrons such as A1 to make a comparison with experiment.

18. 3. Effect of Vector-Axialvector mixing in hot matter MH, C.Sasaki and W.Weise, Phys. Rev. D 78, 114003 (2008) Using an Effective field theory including,, A1 based on the generalized hidden local symmetry

19. Hybrid Scenario (π ρ, A1, σ degenerate) : m σ = m A1 = m ρ = m π Chiral Restoration linear sigma model vector manifestation π σ degenerate: m σ m π ρ, A1 degenerate: m A1 m ρ π, ρ degenerate: m ρ m π σ, A1 degenerate: m σ m A1 ρ mass is invariant dropping A1 & σ masses

20. T – dependence of ρ and A1 meson masses A simple assumption for the parameters of the GHLS Lagrangian no-droppingdropping

21. Vector spectral function at T/Tc = 0.8 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

22. V-A mixing small near Tc T-dependence of the mixing effect

23. Chiral Restoration linear sigma model vector manifestation Hybrid Scenario (π ρ, A1, σ degenerate) : m σ = m A1 = m ρ = m π π σ degenerate: m σ m π ρ, A1 degenerate: m A1 m ρ π, ρ degenerate: m ρ m π σ, A1 degenerate : m σ m A1 Either of 3 is expected to happen dropping mass

24. Dropping A1 with dropping ρ 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

25. Vanishing V-A mixing (g a1 = 0) at Tc ? In quark level A1 Left chirality L L R We need to flip chirality once in a1- - coupling g a1 0 for T Tc Vector – axial-vector mixing vanishes at T = Tc !

26. 4. Vector – Axial-vector mixing in dense baryonic matter based on M.H. and C.Sasaki, Phys. Rev. C 80, 054912 (2009) see also M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 82, 076010 (2010) M.H. and M.Rho, arXiv:1010.1971 [hep-ph]

27. V-A mixing from the current algebra analysis in the low density region B.Krippa, PLB427 (1998) 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

28. 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) 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)]

29. Determination of mixing strength C An estimation from dominance e-e- e+e+ A1 ρ + C 0.1 GeV × (n B / n 0 ) n 0 : normal nuclear matter density An estimation in a holographic QCD (AdS/QCD) model Infinite tower of vector mesons (,,, …) in AdS/QCD models These effects of infinite mesons can generate V-A mixing This summation was done in an AdS/QCD model S.K.Domokos, J.A.Harvey, PRL99 (2007) a very rough estimation

30. Can infinite tower of mesons contribute ? This is related to a long-standing problem of QCD not clearly understood: Why does the / meson dominance work well ? a1, K K K* a1 K K,, K K,, K K K* Example 1: vector form factor in -> K K bar at CLEO CLEO, PRL92, 232001 (2004) : 1/(1+ + ) = 1.27 : /(1+ + ) = - 0.40 : /(1+ + ) = 0.13 CLEO results :

31. meson dominance ; Example 2: EM form factor In an effective field theory for and based on the Hidden Local Symmetry EM form factor is parameterized as In Sakai-Sugimoto model (AdS/QCD model), infinite tower of mesons do contribute k=1 : meson k=2 : meson k=3 : meson … T.Sakai, S.Sugimoto, PTP113, PTP114 = 1.31 + (-0.35) + (0.05) + (-0.01) + … a1(1260) (1450) PDG 776 1230 1465 SS 776(input) 1190 1607 Note: AdS/QCD model is for large Nc limit

32. Example 2: EM form factor meson dominance 2 /dof = 226/53=4.3 ; SS model : 2 /dof = 147/53=2.8 best fit in the HLS : 2 /dof=81/51=1.6 Exp data : NA7], NPB277, 168 (1996) J-lab F(pi), PRL86, 1713(2001) J-lab F(pi), PRC75, 055205 (2007) J-lab F(pi)-2, PRL97, 192001 (2006) Infinite tower works well as the meson dominance ! MH, S.Matsuzaki, K.Yamawaki, arXiv:1007.4715 cf : MH, K.Yamawaki, Phys.Rept 381, 1 (2003)

33. Example 3: transition form factor 1 best fit in the HLS : 2 /dof=24/30=0.8 Sakai-Sugimoto model : 2 /dof=45/31=1.5 Exp data: CMD-2, PLB613, 29 (2005) NA60, PLB677, 260 (2009) MH, S.Matsuzaki, K.Yamawaki, arXiv:1007.4715 cf : MH, K.Yamawaki, Phys.Rept 381, 1 (2003) meson dominance : 2 /dof=124/31=4.0

34. Example 4: Proton EM form factor M.H. and M.Rho, arXiv:1010.1971 [hep-ph] meson dominance : 2 /dof=187 best fit in the HLS : 2 /dof=1.5 a = 4.55 ; z = 0.55 Violation of / meson dominance may indicate existence of the contributions from the higher resonances. Contribution from heavier vector mesons actually exists in several physical processes even in the low-energy region Sakai-Sugimoto: Hong-Rho-Yi-Yee model : 2 /dof=20.2 a = 3.01 ; z = -0.042

35. 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 In the following, I take C = 0.1 - 1 GeV. Note that e.g. C = 0.5 GeV corresponds to C = 0.1 × (n B /n 0 ) at n B = 5 n 0 C = 0.5 × (n B /n 0 ) at n B = n 0 This may be too big, but we can expect some contributions from heavier,, …

36. 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

37. Vector spectral function for C = 1 GeV note : = 0 for s < 2 m low 3-momentum (p bar = 0.3 GeV) 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 3-momentum (p bar = 0.6 GeV) 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

38. 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

39. 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

40. 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

41. 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)

42. 5. Summary chiral symmetry restoration in hot and/or dense medium mass change of either ρ, A1 (or both) can be expected Dilepton spectrum from the dropping in the VM In the VM, strong violation of the vector (ρ) meson dominance is expected large suppression of dilepton spectrum Effect of axial-vector (A1) meson In the standard scenario, dropping A1 is expected. Through the V-A mixing, we may see the dropping A1. Note : The mixing becomes small associate with the chiral restoration. g a F 2 0 at T = Tc Observation of dropping A1 may be difficult In case of dropping A1 with dropping ρ, effect of A1 suppress the di-lepton spectrum cusp structure around s 1/2 = 2m

43. Effect of V-A mixing (violating charge conjugation) in dense baryonic matter for -A 1, -f 1 (1285) and -f 1 (1420) substantial modification of rho meson dispersion relation broadening of vector spectral function might be observed at J-PARC and GSI/FAIR 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 fixing C. Note : C = 0.3 GeV at n B = 3 n 0. This V-A mixing becomes relevant for n B > 3 n 0

44. Including dropping mass (especially dropping ) ? This mixing will not vanish at the restoration. (cf: V-A mixing in hot matter becomes small near Tc.) Future work Dropping may cause a vector meson condensation at high density. ex: m */m = ( 1 – 0.1 n B /n 0 ) suggested by KEK-E325 exp. C = 0.3 (n B /n 0 ) [GeV] just as an example Vector meson condensation ! vacuum longitudinal transverse p0p0 p n B /n 0 = 2 n B /n 0 = 3 n B /n 0 = 3.1