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Kazuya Mitsutani （ YITP ） in collaboration with Masakiyo Kitazawa （ Osaka Univ. ）

QCD School @ Les Houches 2008. A quasi-particle picture of quarks coupled with a massive boson at finite temperature ~ mass effect and complex pole ~. Kazuya Mitsutani （ YITP ） in collaboration with Masakiyo Kitazawa （ Osaka Univ. ） Teiji Kunihiro （ YITP ） Yukio Nemoto （ Nagoya Univ. ）.

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Kazuya Mitsutani （ YITP ） in collaboration with Masakiyo Kitazawa （ Osaka Univ. ）

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  1. QCD School @ Les Houches 2008 A quasi-particle picture of quarks coupled with a massive boson at finite temperature ~ mass effect and complex pole ~ Kazuya Mitsutani（YITP） in collaboration with Masakiyo Kitazawa（Osaka Univ.） Teiji Kunihiro（YITP） Yukio Nemoto（Nagoya Univ.）

  2. So then how quark behave above and near Tc ? • sQGP picture • success of perfect fluid model in the analysis of the RHIC data • J/Psi, eta_c near and above Tc on the other hand… • several result suggest the quasi particle with quark quantum number • success of recomb. model in RHIC phenomenology R.J.Fries, B.Muller, C.Nonaka and S.A.Bass(2003); V.Greco, C.M.Ko and P. Levai (2003); S.A.Voloshin(2003); D.Molnar and S.A.Volshin(2003) • In some non-perturbative calculations suggest quark spectral function may have several sharp peaks near Tc (SD eq., lQCD) F.Karsch and M.Kitazawa (2007) ,M.Harada, Y.Nemoto and S.Yoshimoto (2007)

  3. What is important near Tc is fluctuation of the order parameter, i.e., chiral soft mode, for the 2nd or nearly 2nd order P. T.（T.Hatsuda and T.Kunihiro, PRL55, 158(‘85)） Understood form Level Mixing • Quark Spectrum Near And Above Tc 3-peak structure In NJL Model mq = 0 ( chiral limit) （M.Kitazawa, et.al. Phys.Lett. B633 (2006) 269） • Softening of Mesonic Mode • Chiral Soft Mode • But real quarks have finite masses • (constituent quark mass, thermal mass, current mass) We investigate the effect of mass in a Yukawa model

  4. P－K K P Formulation • Self Energy (1-loop) Yukawa coupling of scalar filed and fermion • Spectral Function projection Deal with only positive number component cf : parity property p = 0 only for simplicity

  5. Spectral functions

  6. Pole structure of the propagator Poles are found by solving The residue at pole which indicate the strength of the excitation The residues and the sum rule of the spectral function :The sum rule for the fermion spectral function If

  7. Pole Structure Of Propagators( mf= 0 ) (A) Real • Poles of the retarded functions in the lower half plane z (C) (B) infinite #s of poles Imaginary - 2T z: complex energy variable

  8. The case that mf = 0 Pole position the pole distribution is symmetric with respect to the imaginary axis because mf = m = 0 T-dependence of the residues T↑ The sum of the three residues approximately satisfy the sum rule The three residues comparable at T ~ mb which support 3-peak structure

  9. The case that mf is finite • The pole at T=0 (red) moves toward the origin as T rise • The pole at w < 0 have large imaginary part comparing with the pole in the positive w > 0 at same T. mf / mb = 0.1 • The residue at the pole in the negative w region is depressed at T ~ mb . it support the depression of three-peak structure • The sum of the residues approximately satisfy the sum rule also in this case.

  10. Structural change of the pole behavior mf / mb = 0.2 mf / mb = 0.3 The pole at T=0 moves toward the origin as T rise. This behavior is qualitatively same as in the case that mf / mb = 0.0,0.1 The pole at T=0 moves to large-w region as T rise. This behavior is qualitatively different from the other cases. we find that mfcrit/mb ~ 0.21

  11. Summary • Spectral function • In the case that mf = 0 the spectral function have a three-peak structure at T ~ mb. • mf depress the peak at w < 0 • Pole structure • there are poles corresponding the peaks in r(w) • the residues at the three poles approximately satisfy the sum rule • pole approx. to r(w) is quite valid at T ~ mb so that the quasi-particle picture also good. • there are structural change of the quasi-particle picture • The behaviors can be understood by level mixing

  12. Future work • effect on the observable • Dilepton production rate (W.I.P) • finite chemical potential • C.E.P (T.C.P)

  13. Back Up

  14. k p << T p - k “plasmino” H.A.Weldon, PRD40(1989)2410 k ~ T がループ積分に おいて支配的 Quark at high T Hard Thermal Approximation (E. Braaten and R. D. Pisarski) available atT >> w, p, mf • Two dispersion relation • A dispersion relation have minimum • -plasmino • Thermal mass : prop. to T • Chiral symmetric mass （×CF） （First appear in V. V. Klimov, Yad. Fiz 33, 1734 (1981)）

  15. The Spectral Function

  16. The spectral function projection op. Deal with only positive number component cf : parity property となる wで Im S+ (w) が小さければピークがある

  17. The spectral function Quark spectrum in Yukawa models was shown in Kitazawa et al, Prog.Theor.Phys.117, 103 (2007) mf= 0 • Only a peak for free quark at T = 0 • 3-peak structure at T ~ mb • 2 peak structure at higher T consistent with HTL app.

  18. Under standing from the self-energy Imag. T （Kramers - Kronig relation） for Peaks in ImS → heaves in ReS Real • New peaks at higher temperature T cf : “quasi”-disp. rel. ＊This Analysis in Ch. Lim. Still Shown by Kitazawa, et. al. （M.Kitazawa, et.al. Phys.Lett. B633 (2006) 269）

  19. The spectral function for mf/ mb = 0.1 • suppression of the peak in the region w < 0 • 3-peak structure barely remain

  20. The spectral function for mf / mb = 0.2

  21. The spectral function for mf / mb = 0.3 The suppression of the peak in the negative energy region become strong

  22. Under standing from the self-energy Imag. T • Mass lower the y-int. 　（⇔ free particle peak is at w > 0） Quasi Dispersion Relation : Real T The peak with negative energy require higher temp.

  23. The Pole Structure Of The Quark Propagator

  24. Poles of the quark propagator Poles are found by solving The residue at pole which indicate the strength of the excitation Pole approximation of the spectral function A sum rule for the fermion spectral function If pole approximation is good

  25. Pole Structure Of Propagators( mf= 0 ) (A) Real • Poles of the retarded functions in the lower half plane z (C) (B) infinite #s of poles Imaginary - 2T z: complex energy variable

  26. How the poles move (A) • pole (A) stay at the origin irrespective of T • Imag. parts of the poles (B) and (C) become smaller as T is raised (C) (B)

  27. Breit-Wigner approximations T/mb= 0.5 T/mb= 1.0 T/mb= 1.5 The pole approximation describe the spectral function except near the origin : the residues at many poles near the origin have negative real parts

  28. The residues Sum rule is satisfied approximately • Residues have similar values at T ~ mb: consistent with 3-peak structure !

  29. How the poles move (mf /mb=0.1) • Pole (A) moves toward the origin as T is raised • Pole (C) have larger imaginary part than pole (B) (A) (B) (C)

  30. Breit-Wigner approximations T/mb= 0.5 T/mb= 1.0 • The spectral function is well described by the three poles we picked up ! T/mb= 1.5

  31. The residues and a sum rule Sum rule is approximately satisfied • Residue at pole (A) decrease at high T • Residue at pole (B) is larger than that for pole (C) at T ~ mb

  32. Structural change in the pole behavior mf / mb = 0.3 mf / mb = 0.2 • The T-dependence of the poles qualitatively change (A) (A) (B) (B) (C) (C)

  33. Residues • It seems that behavior of poles A and B is exchanged. mf / mb = 0.3 mf / mb = 0.2 critical mass is mf / mb~ 0.21

  34. The behavior at high temperature wQ = mb phase space for decays vanish • Imz / Rez is small at high T • mTof HTL well approximate the real part at high T ex) T/mb = 20.0 mT =gT/4

  35. The Level Mixing The Physical Origin Of Multi Peak Structures

  36. quark anti-qth hole quark | | Level Mixing ~ massless fermions coupled with a massless boson ~ （H.A.Weldon, PRD40(1989)2410） w r+(w,k) r-(w,k) k

  37. M.Kitazawa, et.al. Phys.Lett. B633, 269 (2006) ms ms quark anti-q hole quark | | Level Mixing~ massless fermions coupled with a massive boson ~ Kitazawa et al, Prog.Theor.Phys.117, 103 (2007) w r+(w,k) r-(w,k) k

  38. Energies of the mixed levels In the discussions above, the momentum of the boson is set to zero. Here I consider general values of the absorbed/emitted boson. 0 < w < |mb-mf| 0 > w > - |mb-mf| (Eb, k) (Ef, k) (Ef, k) (w,0) (Eb, k) (w,0) The energy of the state which mixed with the original (free) state w> = Eb – Ef(>0) w< = Ef – Eb(<0) Eb = sqrt(mb2 + k2) Ef = sqrt(mf2 + k2)

  39. Im S prop. to the decay rate This suggest that the mixing processes occur most often at two energy value →effectively the mixing b/w three states →three peak structure suppression of the peak with negative energy competition b/w 1)phase volume ∝k2 2)dist. func. n(k), f(k) w> w> w< w< M - = mb - mf

  40. the structural change from the aspect of level mixing • Intermediated states are thermally excited • at low T effect of level mixing is weak • graphs are schematic picture at enough high T so that level mixing become effective and pole (A) start to move original level is pushed up in energy at high T original level is pushed down in energy at high T mf > m* w mf < m* w k k mf= 0,0.1,0.2 mf= 0.3 Effective mixed level

  41. Coupling with pseudo scalar boson • qualitatively the same • the peak near the origin is enhanced • critical mass for the pole structure becomes mPS* ~ 0.23 mf / mb = 0.2

  42. Subtracted Dispersion Relation Kramers-Kroenig Relation for f(x) Finiteness of ReS require | f(z)| to converge as z → \ . Else one should use Those are called once “subtracted” and twice “subtracted” dispersion relation respectively.

  43. Explicit expression of T = 0 part Diverge !

  44. Renormalization Of T = 0 Part • We use twice subtracted disp. rel. for regularization of integral General complex function f(x) obey to the following relation : （Dispersion Relation） Even if above expression diverge, following expression sometimes converge. This expression is called “twice-subtracted” dispersion relation

  45. Mass Shell Renormalization In terms of S Mass Renorm. Wv. Fnc. Renorm. Renormalization Of T = 0 Part （dbl sign : for p0 > 0 upper , for p0 < 0 lower） （dbl sign : for p0 > 0 upper , for p0 < 0 lower） • Finite temperature part converge →disp. rel. without subtraction.

  46. Im S and decay processes External line have positive quark number （I） Landau Damping （II） （III） （IV） （I） （II） （III） （IV） time

  47. 0 (II) （III） Allowed Energy Region For The Processes -(mb+mf) -|mb-mf| |mb-mf| (mb+mf) mf < mb （IV） （III） （II） （I） mf > mb （IV） （II） （III） （I） Thermal excitation Landau damping processes -including thermally excited particles in the initial state

  48. Im S and decay processes finite T ランダウ減衰 ボゾン質量が有限であることを反映して2ピーク

  49. ImSのピークの位置と mf の比較 we use the minima in the imaginary parts of the self-energies for rough indication for effective energy of intermediate states T/mb = 0.5 at which T=0 pole start to move notably mf / mb = 0.1 mf / mb = 0.2 mf / mb = 0.3 minima exist at w < mf minima exist at w > mf

  50. 擬スカラーの場合：自己エネルギーからの解釈擬スカラーの場合：自己エネルギーからの解釈 散乱の行列要素が変わるのが原因 正エネルギー領域における実部のうねりがスカラーの場合よりも大きくなり下位の位置に変化

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