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

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 School @ 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 dat

a– 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 Modelmq = 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 2nd or nearly 2nd 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 componentcf : parity property

p = 0 only for simplicity

projection

K

P － 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 ( mf = 0 )

• Poles of the retarded functions in the lower half plane z

RealIm

agin

ary

- 2T

z: complex energy variable

infinite #s of poles

(A)

(B)(C)

The case that mf = 0

The three residues comparable at T ~ mb which support 3-peak structure

the pole distribution is symmetric with respect to the imaginary axis because mf = = 0

The sum of the three residues approximately satisfy the sum rule

Pole position

T-dependence of the residuesT↑

The case that mf is finite•The pole at T=0 (red) moves toward the origin as T rise

•The pole at < 0 have large imaginary part comparing with the pole in the positive > 0 at same T.

•The residue at the pole in the negative 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.

mf / mb = 0.1

Structural change of the pole behaviormf / mb = 0.3mf / mb = 0.2

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- region as T rise.This behavior is qualitatively different from the other cases.

we find that mfcrit/mb ~ 0.21

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 < 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 ~ 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

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 atT, p, mf

“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) ）(1/8)(1/8) （ ×CF ）

Hard Thermal Approximation

The Spectral Function

The spectral function

projection op.

Deal with only positive number componentcf : 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 ~ mb

• 2 peak structure at higher T consistent with HTL app.

mf = 0Quark 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 mf / mb = 0.1

• suppression of the peak in the region < 0• 3-peak structure barely remain

The spectral function for mf / mb = 0.2

The spectral function for mf / mb = 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 ( mf = 0 )

• Poles of the retarded functions in the lower half plane z

RealIm

agin

ary

- 2T

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 approximationsT/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

The residues

• Residues have similar values at T ~ mb : consistent with 3-peak structure !

Sum rule is satisfied approximately

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)

Breit-Wigner approximations

• The spectral function is well described by the three poles we picked up !

T/mb = 0.5 T/mb = 1.0

T/mb = 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 ~ mb

Sum rule is approximately satisfied

Structural change in the pole behaviormf / mb = 0.2 mf / mb = 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.mf / mb = 0.2 mf / mb = 0.3

critical mass is mf / mb ~ 0.21

The behavior at high temperature

• Im z / Re z is small at high T• mT of HTL well approximate the real part at hig

h T

ex) T/mb = 20.0

mT =gT/4

Q = mb

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-qth 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

| |

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

(Ef, k)

(,0) (Eb, k) (,0)

(Eb, k)

(Ef, k)

0 < < |mb-mf| 0 > > - |mb-mf|

The energy of the state which mixed with the original (free) state

> = Eb – Ef (>0)< = Ef – Eb (<0)

Eb = sqrt(mb2 + k2)

Ef = sqrt(mf2 + k2)

In the discussions above, the momentum of the boson is set to zero.Here I consider general values of the absorbed/emitted boson.

M - = mb - mf

>

<

>

<

Im prop. to the decay rateThis suggest that the mixing processes occur most often at two energy value →effectively the mixing b/w three states→three peak structure

competition b/w1)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 downin energy at high T

mf < m* mf > m*

mf = 0,0.1,0.2 mf = 0.3

Coupling with pseudo scalar boson

mf / mb = 0.2

•qualitatively the same•the peak near the origin is enhanced•critical mass for the pole structure becomes mPS

* ~ 0.23

Subtracted Dispersion RelationKramers-Kroenig Relation for f(x)

Finiteness of Rerequire | 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 p0 > 0 upper , for p0 < 0 lower ）

Mass Shell RenormalizationMass Renorm.

Wv. Fnc. Renorm.

In terms of

•Finite temperature part converge → 　 disp. rel. without subtraction.

（ dbl sign : for p0 > 0 upper , for p0 < 0 lower ）

Im and decay processes

（ I ） （ II ）

（ IV ）

（ I）（ II ）（ III ）（ IV ）

External line have positive quark number

（ III ）

Landau Damping

time

Allowed Energy Region For The Processes

mf < mb

mf > mb

（ I ）

（ I ）

（ II ）

（ II ）

（ III ）

（ III ）

（ IV ）

（ IV ）

|mb-mf|-|mb-mf| (mb+mf)-(mb+mf)

(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 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

mf / mb = 0.1 mf / mb = 0.2 mf / mb = 0.3

T/mb = 0.5 at which T=0 pole start to move notably

minima exist at < mf minima exist at > mf

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