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. ） QCD Les Houches 2008

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)

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

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

Spectral functions

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

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

The case that m f = 0 The three residues comparable at T ~ m b which support 3-peak structure 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↑

The case that m f is finite The pole at T=0 (red) moves toward the origin as T rise The pole at  0 at same T. The residue at the pole in the negative  region is depressed at T ~ m b. it support the depression of three-peak structure The sum of the residues approximately satisfy the sum rule also in this case. m f / m b = 0.1

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

Summary Spectral function –In the case that m f = 0 the spectral function have a three-peak structure at T ~ m b. –m f depress the peak at  < 0 Pole structure –there are poles corresponding the peaks in  –the residues at the three poles approximately satisfy the sum rule –pole approx. to  is quite valid at T ~ m b 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

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

Back Up

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

The Spectral Function

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

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

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

The spectral function for m f / m b = 0.1 suppression of the peak in the region  < 0 3-peak structure barely remain

The spectral function for m f / m b = 0.2

The spectral function for m f / m b = 0.3 The suppression of the peak in the negative energy region become strong

Mass lower the y-int. （⇔ free particle peak is at  ） The peak with negative energy require higher temp. Quasi Dispersion Relation : T T Imag. Real Under standing from the self-energy

The Pole Structure Of The Quark Propagator

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

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

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

Breit-Wigner approximations T/m b = 0.5T/m b = 1.0 T/m b = 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

The residues Residues have similar values at T ~ m b : consistent with 3-peak structure ! Sum rule is satisfied approximately

How the poles move (m f /m b =0.1) Pole (A) moves toward the origin as T is raised Pole (C) have larger imaginary part than pole (B) (A) (B) (C)

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

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

Structural change in the pole behavior m f / m b = 0.2 m f / m b = 0.3 The T -dependence of the poles qualitatively change (A) (B) (C) (A) (B) (C)

Residues It seems that behavior of poles A and B is exchanged. m f / m b = 0.2 m f / m b = 0.3 critical mass is m f / m b ~ 0.21

The behavior at high temperature Im z / Re z is small at high T m T of HTL well approximate the real part at high T ex) T/mb = 20.0 m T =gT/4  Q = m b phase space for decays vanish

The Level Mixing The Physical Origin Of Multi Peak Structures

Level Mixing ~ massless fermions coupled with a massless boson ~  + ( ,k) quark anti-q th hole quark | （ H.A.Weldon, PRD40(1989)2410 ）    k)  k

Level Mixing ~ massless fermions coupled with a massive boson ~ quark anti-q hole quark mm mm |  k  + ( ,k)    k) Kitazawa et al, Prog.Theor.Phys.117, 103 (2007) M.Kitazawa, et.al. Phys.Lett. B633, 269 (2006)

Energies of the mixed levels (E f, k) (,0)(,0) (E b, k) (,0)(,0) (E f, k) 0 <  < |m b -m f | 0 >  > - |m b -m f | The energy of the state which mixed with the original (free) state  > = E b – E f (>0)  < = E f – E b (<0) E b = sqrt(m b 2 + k 2 ) E f = sqrt(m f 2 + k 2 ) In the discussions above, the momentum of the boson is set to zero. Here I consider general values of the absorbed/emitted boson.

M - = m b - m f >> << >> << Im  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 competition b/w 1)phase volume ∝ k 2 2)dist. func. n(k), f(k) suppression of the peak with negative energy

 k 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  k the structural change from the aspect of level mixing Effective mixed level original level is pushed up in energy at high T original level is pushed down in energy at high T m f < m* m f > m* m f = 0,0.1,0.2m f = 0.3

Coupling with pseudo scalar boson m f / m b = 0.2 qualitatively the same the peak near the origin is enhanced critical mass for the pole structure becomes m PS * ~ 0.23

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

Explicit expression of T = 0 part Diverge !

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 : Even if above expression diverge, following expression sometimes converge. This expression is called “twice-subtracted” dispersion relation （ Dispersion Relation ）

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

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

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

Im  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

Im  のピークの位置と m f の比較 we use the minima in the imaginary parts of the self-energies for rough indication for effective energy of intermediate states m f / m b = 0.1m f / m b = 0.2m f / m b = 0.3 T/m b = 0.5 at which T=0 pole start to move notably minima exist at  < m f minima exist at  > m f

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