QCD Phase Transitions and Quark Quasi-particle Picture

Transcription

1 QCD Phase Transitions and Quark Quasi-particle Picture Teiji Kunihiro (YITP, Kyoto) YITP workshop New Developments on Nuclear Self-consistent Mean-field Theories May 30 June 1, 2005 YITP, Kyoto

2 1.Introduction

3 QCD phase diagram and quasi-particles T T >> T c plasmon, plasmino T c QCD CEP QGP χ ρ 0? BCS-BEC? superconducting H matter? phases meson condensation? Various phases CFL µ

4 Abuki, Kitazawa and T.K.(hep-ph/ ; P.L.B615,102(2005))

5 Color Superconductivity T 0 strong coupling! ~ 100MeV in electric SC / E F ~ / E F ~ Very Short coherence length ξ. µ weak coupling Mean field approx. works well. There exist large fluctuations of pair field. Large pair fluctuations can invalidate MFA. cause precursory phenomena of CSC. cf.) Bosonization of Cooper pairs Matsuzaki, PRD62, (2000) Abuki, Hatsuda, Itakura, PRD 65, (2002)

6 Y.Nishida and H. Abuki (hep-ph/ ) 4-body contact interaction with massive fermion fermion mass Attractive interaction in color 3, flavor 1 fermion chemical potential and J P =0 + diquark channel Fermion pair correlation in normal phase <Mean field + Gauss fluctuation> mu-rho eq. with a fixed charge density Ntotal Ntotal =Nfermion+Nboson =Nbound +Nunstable gap eq. at critical temperature Tc (Thouless criterion) S a de Melo et al., PRL 71 (1993) 3202 Kitazawa et al.,prd 70 (2004)

7 BCS-BEC transition in QM 1 TBEC_RL Y.Nishida and H. Abuki, hep-ph/ Tmf T/E F TBEC G/G 0

8 Gauge Fields in a Plasma In the high T limit, ( p, ω, << T) m e (hard thermal loop (HTL) approximation) thermal mass (= pole mass) (= plasma frequency ) ω / m g 2 mg = 1 e 2 T 3 2 plasmon In QCD 2 1 m g = N 6 f g 2 T 2 p / m g

9 Fermions in a Plasma 1-loop (g<<1) + HTL approx. ( p, ω, << T) m q Σ ( ω, p) = thermal masses 2 1 m e 2 T 2 f = QED 2 m f = g 2 T 2 QCD chiral invariant ω / m f E p Weldon 1989 E h plasmino p / m f

10 quark distribution anti-quark like E anti-q distribution particle(q)-like E n (E) n(e) Plasmino excitation

11 How about when the temperature Is lowered close to T? c

12 The wisdom of many-body theory tells us: If a phase transition is of 2nd order or weak 1st order, soft modes ; the fluctuations of the order parameter eg. softening of 2+ phonon - quadrupole deformation Gamow-Teller GR ; a soft mode of pion condensation (T.K., 1981) Chiral Transition = a phase transition of QCD vacuum, being the order parameter. Lattice QCD; There can be hadronic excitations (para pion and sigma) as the soft mode of the chiral transition in the ``QGP phase. T. Hatsuda and T. K., Phys. Rev. Lett.55( 85)158; PLB71( 84),1332 Prog. Theor. Phys 74 (1985), 765: Cf. T<T_c ; the σ meson becomes the soft mode of chiral restoration at

13 QCD phase diagram and quasi-particles T 2.precursory hadronic modes? 3. free quark spectra? QCD CEP QGP 1. preformed pair fields? quark spectra modified? χ ρ 0? H matter? CFL µ meson condensation?

14 2. Precursory Phenomena of Color Superconductivity in Heated Quark Matter Ref. M. Kitazawa, T. Koide, T. K. and Y. Nemoto, Phys. Rev. D65, (2002); D70, (2004)

15 QCD phase diagram T QCD CEP QGP 1. preformed pair fields? quark spectra modified? χ ρ 0? H matter? CFL µ meson condensation?

16 Color Superconductivity; diquark condensation Dense Quark Matter: quark (fermion) system with attractive channel in one-gluon exchange interaction. Cooper instability at sufficiently low T SU (3) c gauge symmetry is broken! p q q [ ] c [ ] c [ ] c [ ] c Attractive! Fermi surface -p T ~100MeV at moderate density µ q ~ 400MeV confinement chiral symm. broken Color Superconductivity(CSC) µ

17 Nature of CSC in Intermediate Density (µq ~500MeV) gap ~ 100MeV at T=0 T order of T c ~ 50MeV / E F ~ 0.1 in electric SC / E F ~ χ µ owing to Strong coupling nature of QCD implies, Short correlation length of Cooper pairs Large fluctuation of the pair field is expected even at T>0! may be relevant to newly born neutron stars or intermediate states in heavy-ion collisions (GSI, J-PARC)

18 Collective Mode in CSC Response Function of Pair Field Linear Response external field: ind C ( exψ γ5τ2λψ 2 h.c. ) Hex = d x i + expectation value of induced pair field: [ ] C ψ( x ) i γ τ λψ ( x ) = i ds H ( s ), O ( x, t ) C R ( x) = 2 G ψ( x) iγ τ λψ ( x) = dt ' d x D ( xx, ') ( x' ) C R D ( x,) t 2 ψ ( ) γτλψ ( ), ψ(0) iγτλψ (0) θ( t) C C = GC x i x Fourier transformation with Matsubara formalism ex ex t t ex ex Retarded Green function RPA approx.: = + + GQ C ( k, ωn) 1 + GQ( k, ω ) C n with Q( k, ω ) = n

19 After analytic continuation to real time, Equivalent with the gap equation (Thouless criterion)

20 Precursory Mode in CSC (Kitazawa, Koide, Nemoto and T.K., PRD 65, (2002)) Spectral function of the pair field at T>0 ε = at k=0 T T T C C As T is lowered toward T C, The peak of ρ becomes sharp. (Soft mode) Pole behavior The peak survives up to ~ electric SC ~ 0.005

21 The pair fluctuation as the soft mode; --- movement of the pole of the precursory mode--- ω

22 How does the soft mode affect the quark spectra? ---- formation of pseudogap ---- Ref. M. Kitazawa, T. Koide, T. K. and Y. Nemoto Phys. Rev. D70, (2004); hep-ph/ ; Prog. Theor. Phys. in press, M.Kitazawa, T.K. and Y. Nemoto, hep-ph/

23 Pseudogap :Anomalous depression of the density of state near the Fermi surface in the normal phase. Conceptual phase diagram of HTSC cuprates Renner et al.( 96) The origin of the pseudogap in HTSC is still controversial.

24 Density of State in BCS theory Quasi-particle energy: ω = sgn( k µ ) ( k µ ) ω µ k Density of State: 2 dk N( ω) k dω dε k µ = dk ( k µ ) N ( ω ) 2 µ ω The gap on the Fermi surface becomes smaller as T is increased, and it closes at T c.

25 Density of State N(ω) d k N ω = ρ ω (2 π ) 3 0 ( ) ( k, ) 3 N = 3 0 d x ψγ ψ 1 ( k, ) = Tr Im (, ) 4 k 0 0 R ρ ω γ G ω T-matrix Approximation G( k, ω ) = Σ Σ ( k, ) = = ω n Soft mode q, iω m k+ q +, iω n iω m d q = T Ξ + + n 1 (, ) (, ω ) 0 G k iω n Σ k i 3 0 ( k q, ω ) (, ) 3 n ωm G q ωm m (2 π ) G ( k, iωn) = ( iω n + µ ) γ γ k = n :free progagator

26 11-Particle -Particle Spectral Spectral Function Function quasi-particle peak of anti-particle, ω = k µ µ= 400 MeV ε=0.01 quasi-particle peak, ω =ω (k)~ k µ position of peaks ω k kf=400mev k Fermi energy quasi-particle peaks at ω =ω (k)~ k µ and ω = k µ. Quasi-particle peak has a depression around the Fermi energy due to resonant scattering. scattering µ = 400MeV 0 ω k, k µ : on-shell 0, 0 : the soft mode

27 Density of state of quarks in heated quark matter µ = 400 MeV N(ω)/10 4 Pseudogap structure manifests itself in N(ω). The pseudogap survives up to ε ~ 0.05 ( 5% above T C ). ω Fermi energy

28 Diquark Coupling Dependence stronger diquark coupling G C µ= 400 MeV ε=0.01 G C

29 Resonance Scattering of Quarks G C =4.67GeV -2 Mixing between quarks and holes ω (M.Kitazawa, Y. Nemoto, T.K. hep-ph/ ) k

30 Summary of this section There may exist a wide T region where the precursory soft mode of CSC has a large strength. The soft mode induces the pseudogap, Typical Non-Fermi liquid behavior Future problems: resonant scattering effects of the soft mode on. H-I coll. & proto neutron stars anomalous eg.1) lepton pair production collective mode: eg.2) (ω,k) ; cooling of newly born stars (M.Kitazawa and T.K in progress.)

31 3. Precursory Hadronic Mode and Single Quark Spectrum above Chiral Phase Transition

32 QCD phase diagram and quasi-particles T precursory hadronic modes? free quark spectrum? QCD CEP QGP χ ρ 0? H matter? CFL µ meson condensation?

33 Chiral Transformation (left handed) (right handed) Chirality: For N f = 3, the chiral transformation forms

34 Chiral Invariance of Classical QCD Lagrangian in the chiral limit (m=0) In the chiral limit (m=0), µ q γ Dµ q ; Chiral invariant = q γ µ q invariant! ;Chiral invariant!

35 The notion of Spontaneous Symmetry Breaking a Q the generators of a continuous transformation µ a = 0 j a a a µ (x) Noether current Q = dx j 0 ( x) j µ ; eg. Chiral transformation for Q 0 a a b ab = dx qγ γ / 2q [ iq q( x) iγ τ q( x) ] = δ q( x) q( ) a 5 5τ SU L Notice; ( 2) SU R(2) 5, 5 x The two modes of symmetry realization in the vacuum : a. Wigner mode a Q 0 = 0 a b. Nambu-Goldstone mode Q a a The symmetry is spontaneously broken. a a Now, qq 0 = 0 [ Q, qγ τ ] q 0 qq 0 0 a Q Chiral symmetry is spontaneously broken!

36 Chiral invariant forms : = 2 N f transformation: Def. : :, Any function of ; Invariant! eg.1 eg.2 (Linear sigma model) (Nambu-Jona-Lasinio model)

37 Chiral Transition and the collective modes 0 para sigma para pion c.f. Higgs particle in WSH model ; Higgs field Higgs particle

38 Hadronic Modes in the QGP Phase The `para-sigma and `para-pion T. Hatsuda and T. K.,(1985) The driving force leading to the phase transition should be strong enough to form the collective modes even at T > Tc T. Hatsuda and T. K., Phys. Rev. Lett.55( 85)158; PLB71( 84),1332 ; Prog. Theor. Phys 74 (1985), 765. Large T

39 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) The idea is rediscovered and revived : E. Shuryak and I. Zahed, hep-ph/307267, G.E. Brown,C.H. Lee, M. Rho and E. Shuryak, hep-ph/ ; M.Gyulassy and L.McLerran, Nucl-th/

40 How does the soft mode affect a single quark spectrum near Tc? Y. Nemoto, M. Kitazawa,T. K. (in preapration)

41 Method low-energy effective theory of QCD 4-Fermi type interaction (Nambu-Jona-Lasinio with 2-flavor) 2 2 L = qiγ q + GS[( qq) + ( qiγ 5τ q) ] τ SU(2) Pauli matrices 6-2 m m = chiral limit G = GeV, Λ = 631MeV 0 S u = d The parameters are determined so as to reproduce m π and f π in the chiral lim. Chiral phase transition takes place at Tc=193.5 MeV(2 nd order). Self-energy of m π a quark (above Tc) 3 d q Σ( ωn, p) = T D( ω, p q) G0(, q) 3 n ωm ωm (2π ) D( ω n, p) = m p ) = Σ( ωn, p) iω = ω+ iε D( ω ω, p q) n m G ( ω m, ) 0 q = + + scalar and pseudoscalar parts R Σ ( ω, : imaginary time real time n +

42 Self-Energy and Spectral Func. self-energy dispersion relation for a quark ω p Re Σ ( ω, p) = 0 spectral function three peaks: one is at ω>0 and the others are at ω<0. Antiquark sector is similar.

43 Dispersion Relations of Quarks T=196 MeV (1.01 Tc) T=200 MeV (1.03Tc) T=210 MeV (1.09Tc) T=220 MeV (1.14Tc)

44 Dispersion Relations T=230 MeV (1.19 Tc) T=240 MeV (1.24 Tc) T=250 MeV (1.29 Tc) T=300 MeV (1.55 Tc)

45 Dispersion Relations T=400 MeV (2.07 Tc) T=500 MeV (2.58 Tc)

46 Spectral function of the quarks

47 Spectral Function of Quarks T=210 MeV, p=50 MeV

48 Summary of this section Near (above) Tc, the quark spectrum at long-frequency and long wave-length is modified drastically by the soft mode for the chiral condensate, qq. The many-peak structure of the spectral function can be understood in terms of two resonant scatterings at small ω and p of a quark and an antiquark. CSC : The Fermi surface is significant. Chiral: Antiquarks are significant. (antiquark holes) Future finite quark mass effects. (2 nd order crossover) finite density (tricritical point, critical end-point) phenomenological applications

49 From E.V.Shyryak and I.Zahed, PRC70,021901( 04) zero-binding lines bound states in QGP

50 Summary of the Talk T 2.precursory hadronic modes? strongly modified quark spectra χ ρ 0 QCD CEP QGP? QCD phase diagram 1. preformed pair fields? quark spectra modified? `QGP itself seems surprisingly rich in physics! Condensed matter physics of strongly CFL coupled Quark-Gluon systems will µ constitute a new field of fundamental physics. H matter?