1. Anomalous quark spectra QCD critical points --- effects of soft modes and van Hove singularity --- Anomalous quark spectra around QCD critical points --- effects of soft modes and van Hove singularity --- T. Kunihiro NFQCD2013 Nov. 27, 2013, YTP Based on works done in collaboration with M.Kitazawa, Y. Nemoto, T. Koide, K.Mitsutani.

2. Thoretical (half-conjectured) Phase Diagram of QCD LHC/RHIC T  CEP Tc 2Tc chiral sym. broken (antiquark-quark condensate) confinement chiral sym. restored deconfinement quark-quark condensate GSI,J-PARC compact stars QGP ? Cross over

3. QCD EoS QCD @ T>0 QCD @ T>0 T Tc*Tc* Success of Hadron Resonance Gas model for higher-order cumulants below Tc: Bazavov et al (HotQCD coll.) (‘13)

4. QCD @ T>0 QCD @ T>0 T Tc*Tc* What is the physical picture of the elementary Excitations in the crossover region? How do quarks and gluons disappear?

5. Quarks at Extremely High T Quarks at Extremely High T Klimov ’82, Weldon ’83 Braaten, Pisarski ’89  2 collective excitations having “thermal mass” ~gT  width ~g 2 T  Minimum of the plasmino mode at nonzero p “plasmino” p / m T  / m T

6. Thoretical (half-conjectured) Phase Diagram of QCD LHC/RHIC T  CEP Tc 2Tc chiral sym. broken (antiquark-quark condensate) confinement chiral sym. restored deconfinement quark-quark condensate GSI,J-PARC compact stars diquark fluctuations chiral fluctuations Phase transitions large fluctuations owing to strong coupling around the critical points. There may exist also other low-lying (hadronic) excitations in the QGP phase. density fluctuations Here we explore how they affect quark quasi-particle picture! QGP

7. ContentsContents 1.Introduction 2. diquark fluctuations and pseudogap in quark specral function in hot and dense quark matter --- a lesson from condensed matter physics --- 3.Soft modes of chiral transition and anomalous quasi-quark spectrum 4. Effects of para-pion with non-hyperbolic dispersion rel. and van Hove singularity 4.Summary and concluding remarks

8. 2. Diquark fluctuations and pseudogap in quark spectral function in hot and dense quark matter

9. The mechanism of the pseudogap in High-TcSC is still controversial, but see, Y. Yanase et al, Phys. Rep. 387 (2003),1, where the essential role of pair fluc. is shown. :Anomalous depression of the density of states near the Fermi surface in the normal phase. Pseudogap Phase diagram of cuprates to be high-Tc superconductor Renner et al.(‘96) Lesson from condensed matter physics on strong correlations (hole doping) A typical non-Fermi liq. behavior! Fermi energy

10.  = 400 MeV Possible pseudogap formation in heated quark matter Fermi energy N(  )/10 4  M. Kitazawa, T. Koide, T. K. and Y. Nemoto Phys. Rev. D70, 956003(2004); Prog. Theor. Phys. 114, 205(2005), T  Pseudogap ? pseudogap ! CSC Pseudogap is formed above Tc of CSC in heated quark matter! How?

11. for  = 400 MeV T = 1.05T c Dynamical Structure Factorof diquark fluctuations sharp peak at the origin (= diffusive over-damping mode) Mechanism of the pseudogap formation 1. Development of precursory diquark fluctuations above Tc 2.Coupling of quarks with the diquark fluctuations Cherenkov-like emission of diquark mode around Fermi energy quark self-energy Depression of the quark spectral Function around the Fermi energy = 400 MeV, =0.01

12. G C =4.67GeV -2  k kFkF Janko, Maly, Levin, PRB56,R11407 (1995) Mixing between particles and holes owing to the Landau damp. by the collective diquark mode. level repulsion of energy level Level repulsion or Pseudo gap due to resonant scattering Level repulsion or Pseudo gap due to resonant scattering M. Kitazawa, T.K. and Y. Nemoto, Phys. Lett.B 631(2005),157 particle hole particle hole particle hole particle # particle # +1

13. 3. Soft modes of chiral transition and anomalous quasi-quark spectrum

14. Chiral Transition and the collective modes c.f. The sigma as the Higgs particle in QCD ; Higgs field Higgs particle 0 para sigma para pion The low mass sigma in vacuum is now established: pi-pi scattering; Colangero, Gasser, Leutwyler(’06) and many others Full lattice QCD ; SCALAR collaboration (’03) and others. q-qbar, tetra quark, glue balls, or their mixed st’s?

15. Digression: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) Softening ? See also, I. Caprini, G. Colangero and H. Leutwyler, PRL(2006); H. Leutwyler, hep-ph/0608218 ; M_sigma=441 – i 272 MeV

16.   the softening of the  with increasing T T  T Fluctuations of chiral order parameter around Tc in Lattice QCD

17. The spectral function of the degenerate ``para-pion” and the ``para-sigma” at T>Tc for the chiral transition: Tc=164 MeV T. Hatsuda and T.K. (1985) T  T (NJL model cal.)

18. PNJL model S. Rößner, T. Hell, C. Ratti, W. Weise.arXiv:0712.3152 [hep-ph] K. Fukushima (2004) C. Ratti, M. Thaler, W. Weise (2006) Para-pion and para-sigma modes are still seen in PNJL model P=Polyakov-loop coupled W. Weise, talk at NFQCD2008 at YITP, Kyoto.

19. Quark Self-energy at finite T =0 Quark Spectral function particle antiparticle c.f: for a free quark at zero density Quark Spectral Function =0 (chiral limit)

20. We incorporate the fluctuation mode into a single particle Green function of a quark through a self-energy. Non self-consistent T-approximation (1-loop of the fluctuation mode) Quarks coupled to chiral soft modes near Tc N.B. This is a complicated multiple integral owing to the compositeness of the para-sigma and para-pion modes.

21. k [MeV]  [MeV]  + ( ,k)  - ( ,k) k [MeV]  = 0.05 Spectral Function of Quark Spectral Function of Quark Kitazawa, Nemoto and T.K., Phys. Lett. B633, 269 (2006), Quasi-dispersion relation for eye-guide; Three-peak structure emerges. The peak around the origin is the sharpest.

22. Digression: Quarks at very high T (T>>Tc) ---- physical origin of plasmino and thermal mass --- 1-loop (g<<1) + HTL approx. T CSC hadron QGP  Tc  p quasi-q anti- plasmino plasmino quasi-anti-q p plasmino quasi-anti-q quarkanti-quark quark anti-q th hole quark Thermally excited anti-quarks = Klimov, Weldon(’82), Pisarski(’89), A.Schaefer, Thoma (’99)

23. quark anti-q hole quark anti-q quark hole anti-q quark part: anti-quark part: The level crossing is shifted by the mass of the fluctuation modes. Mechanism of the 3-peak fromation p [MeV]  [MeV] p [MeV]  - (,p)  + (,p) level repulsion : the HTL result only with the normal quark and plasmino.

24. Fluctuations of the chiral codensate sharp peak in time-like region  -mode Spectrum of the fluctuations p  propagating mode T = 1.1Tc  =0 c.f.: diffusion-like mode in diquark fluctuations CSC T Tc m p para

25. quark + massive boson. Near the critical point, the soft-modes may be represented by an elementary boson. Yukawa model! Quark Spectrum in Yukawa models Quark Spectrum in Yukawa models T=1.2m B  The 3-peak structure emerges irrespective of the type of the boson at. Kitazawa, Nemoto and T.K., Prog. Theor. Phys.117, 103(2007), g=1, T/m=1.5 Massie scalar/pseudoscalar boson Massie vector/axial vector boson Remark: Bosonic excitations in QGP may include , , , J/ , … / glue balls and even the W/Z bosons in the electroweak theory. (at one-loop)

26. Neutrino spectral density at electroweak-scale temperature K. Miura, Y. Hidaka, D. Satow, TK, PRD 88 (2013), 065024 See talk presented by K. Miura at this workshop, Nov.20,2013.

27. The complex quasi-quark pole The three residues comparable at T ~ m b which support the 3-peak structure There are three poles corresponding to the three peaks in the spectral function; the pole distribution is symmetric with respect to the imaginary axis because m f =  = 0 The sum of the three residues approximately satisfy the sum rule Pole position T-dependence of the residues T↑ Mitsutani, Kitazawa, Nemoto, T.K. Phys. Rev. D77, 045034 (2008)

28. Finite quark mass effects There still exist three poles. The pole at T=0 (red) moves toward the origin as T is raised. The pole in the  < 0-region has a larger imaginary part than that in the positive-  region for the same T. The residue at the pole in the negative  region is suppressed at T ~ m b, corresponding to the suppression of the peak in the negative-energy region. The sum of the residues approximately satisfy the sum rule also in this case. m f / m b = 0.1 Mitsutani, Kitazawa, Nemoto, T.K. Phys. Rev. D77, 045034 (2008) Re Z

29. m f / m b = 0.3m f / m b = 0.2 The pole at T=0 moves toward the origin as T is raised. This behavior is qualitatively the same as in the case of lower masses. The pole at T=0 moves toward the large-  region as T is raised. This behavior is qualitatively different from that in the smaller mass cases.. Structure change of the pole behavior The physics contents of the three poles change at a critical mass. We find Level crossing in the complex energy plane

30. Beyond one-loop Schwinger-Dyson approach for lin. sigma model; Harada-Nemoto(’08) The three peak structure in the quark spectral function is still there for small momenta, although the central peak gets to have a width owing to multiple scattering. See also, S. –x Qin et al, PRD 84, (2011), 014017; H. Nakkagawa et al, PRD 85 (2012) 031902. Possible confirmation in Lattice QCD Unquenched lattice simulation with hopefully chiral fermion action on a large lattice is necessary for accommodate the possible chiral fluctuations with energy comparable to MeV. Nevertheless, see F. Karsch and M. Kitazawa, PLB 685 (2007), 45; PRD 80 (2009), 056001; O. Kaczmarek, Karsch and Kitazawa, PRD86 (2012) 036006. Harada, Nemoto,0803.3257(hep-ph)

31. Introducing Nonzero m 0 Introducing Nonzero m 0 2-flavor NJL model What’s NEW!  Effect of m 0  Phase transition becomes crossover.  Constituent quarks stay massive.  Soft modes do not become massless.  The critical point appears, where the phonon mode (density fluctuation) becomes the soft mode. M.Kitazawa, Y.Nemoto and TK, in prep.

32. Stable  modes above T PC Stable  modes above T PC  modes can be stable even above T PC for m  <2M

33. Pion Dispersion Relation Pion Dispersion Relation Continuum threshold Dispersion for a rela. free particle (hyperbolic curve)  Dispersion relation of stable pion deviates from Lorentz form.  Pionic modes become unstable at nonzero p. T=206MeV

34. Anomalous pion dispersion relation in hot and dense hadronic matter A.B. Migdal(‘78), T.Ericson and F. Myhre (‘78),C.Gale and J. Kapusta(‘87), G.Bertsch et al (‘88), G. E/ Brown, E. Oset, M. Vicente Vacas, and W. Weise (‘89), E.Shuryak (‘90), R.Pisarski and M.Tytgat (‘96) and many others. G. Bertsch et al, NPA 490 (1988), 745. C. Gale and J. Kapusta, PRC35 (1987), 2107 Density of states/ (inverse) group velocity G.E. Brown et al, NPA 505 (1989), 823

35. Quark self-energy with composite bosons at finte T (Joint) Density of states (DOS) : difference of the group velocities : Hilbert tr. of the imaginary part A singularity in Im. Part affect the real part.

36. van Hove Singularity van Hove Singularity  Zeros of group velocity Divergences of joint DoS van Hove singularity Relative group velocity of quarks and pions

37. Quark Spectrum Quark Spectrum T=206 MeV, p=0 sharp peak at broad peak at  Strong modification of the quark spectrum  Appearance of sharp peak at low energy  [MeV] M

38.  There exist divergences in Im  + (  )! T=206 MeV, p=0 Real part

39. T Dependence of Quark Spectrum T Dependence of Quark Spectrum

40. 5. Summary and concluding remarks In the fermion-boson system with m F <<m B, the fermion spectral function has a 3-peak structure at 1-loop approximation at T ~ m B. The physical origin of the 3-peak structure is the Landau damping of quarks and anti-quarks owing to the thermally excited massive boson, which induces a mixing between quarks and anti-quark hole, If the chiral transition is close to a second order, quarks may have a 3-peak structure in the QGP phase near T c. If a QCD phase transition is of a second order or close to that, there should exist specific soft modes, which may be easily thermally excited. The boson may be vector-type or glueballs. The logic to produce the three-peak structure is rather robust/universal. Thus it would be interesting to explore the fermion spectrum coupled with bosonic excitations at nonzero temperature, e.g. electro-weak theory and condensed matter.

41. The quark spectrum near but above T PC coupled with the would-be soft modes off the chiral limit. Effects of nonzero m 0 nonvanishing masses of constituent quarks would-be soft modes pionic modes: stable even above TPC non-hyperbolic dispersion  Quark spectrum is significantly modified by the van Hove singularity induced by the scattering of quarks and para- pions.  Quark spectrum near TPC can have a sharp peak with a far small energy.

42. Future problems: Full self-consistent calculation Confirmatin in the lattice QCD experimental observables ; eg. Lepton-pair production (PHENIX?) transport coefficients Soft mode (density-fluctuations) at the CP(‘s)* and quark spectrum; M. Kitazawa, Y. Nemoto and TK, in preparation. Incorporation of CSC with/without inhomogenious condensates Density(phonon)/entropy fluctuations Implications to condensed matter physics: eg. Graphen at finite temperature, in particular searching for possible van Hove singularity.

43. Vector and Anomaly terms: 1. Effects of mismatched Fermi sphere by charge-neutrality 2. Then effect of G_V comes in to make ph. tr. at low T cross over. Z. Zhang and T.K., Phys.Rev. D83 (2011) 114003

44. S.Carignano, D. Nickel and M. Buballa, arXiv:1007.1397 E. Nakano and T.Tatsumi, PRD71 (2005) Interplay between G_V and Polyakov loop is not incorporated; see also P. Buescher, Mater thesis submitted by Darmstadt University,where Ginzburg-Levanyuk analysys shows also an existence of Lifschitz point at finite G_V. Spatial dependence of Polyakov loop should be considered explicitly.

45. The same universality class; Z2 Fluctuations of conserved quantities such as the number and energy are the soft mode of QCD critical point! The sigma mode is a slaving mode of the density. H. Fujii, PRD 67 (03) 094018;H. Fujii and M.Ohtani,Phys.Rev.D70(2004) Dam. T. Son and M. A. Stephanov, PRD70 (’04) 056001 T P Solid Liq. Classical Liq.-Gas Critical point Triple.P gas Plausible QCD phase diagram:

46. What is the soft mode at CP? Sigma meson has still a non-zero mass at CP. This is because the chiral symmetry is explicitly broken. What is the soft mode at CP? T-dependence (  =  CP )  -mode Space-like region (the soft modes) T>Tc (non-soft mode) Phonon mode in the space-like region softens at CP. H. Fujii (2003) H. Fujii and M.Ohtani(2004) Spectral function of the chiral condensate It couples to hydrodynamical modes, See also, D. T. Son and M. Stephanov (2004) does not affect particle creation in the time-like region. leading to interesting dynamical critical phenomena. At finite density, scalar-vector mixing is present. P=40 MeV

47. Spectral function of density fluctuations in the Landau frame In the long-wave length limit, k→0 Long. Dynamical ：： specific heat ratio ： sound velocity thermal expansion rate ： Rel. effects appear only in the sound mode. Rel. effects appear only in the width of the peaks. rate of isothermal exp. Notice: As approaching the critical point, the ratio of specific heats diverges! The strength of the sound modes vanishes out at the critical point. enthalpy sound modes thermal mode Y. Minami and T.K., Prog. Theor. Phys.122 (2009), 881.

48. Spectral function of density fluctuation at CP The sound mode (Brillouin) disappears Only an enhanced thermal mode remains. Furthermore, the Rayleigh peak is enhanced, meaning the large energy dissipation. Suggesting interesting critical phenomena related to sound mode. Spectral function at CP 0.4 The soft mode around QCD CP is thermally induced density fluctuations, but not the usual sound mode. Y. Minami,T.K., Prog. Theor. Phys.122 (2009), 881.

49. Back Ups

50. Quarks at Extremely High T Quarks at Extremely High T Klimov ’82, Weldon ’83 Braaten, Pisarski ’89  2 collective excitations having “thermal mass” ~gT  width ~g 2 T  Minimum of the plasmino mode at nonzero p  / m T normal plasmino

51. Quark Self-Energy at p=0 Quark Self-Energy at p=0 Im  ( ,0) T =1 T =1.5 T =2 T  m B Re  ( ,0)  m B  Two peaks develop at low energy.  Two “oscillating behavior” grows as T is raised.

52. Landau Damping Landau Damping Im  ( ,0) T =1 T =1.5 T =2 T  m B Landau damping   boson quark

53.  [MeV] Im  Im   There exist divergences in Im  + (  )! T=206 MeV, p=0