# Presentation: 665097

## Spectral Properties of Quarks

- (Osaka Univ.)

- BNL, Apr., 25, 2008

- at Finite Temperature in Quenched Lattice QCD

- F. Karsch and MK, PL**B658**,45 (2007) [arXiv:0708.0299];
- in preparation.

- with
- Frithjof Karsch

##  Why Quarks ?  

- The quark may be one of the basic d.o.f. of the QGP.
- 	— success of the recombination model
- 	— fluctuations in full lattice QCD
- It is a gauge dependent quantity.
- 	— but, poles should be gauge independent
- The quarks at high *T* have collective natures.

- Exploring the quark spectral function as functions of
- 	— bare quark mass
- 	— temperature (above and below *T*c)
- 	— momentum

##  Quarks at Extremely High T  

- Hard Thermal Loop approx. ( ***p***, **, *m**q*<<*T* )
- 1-loop (*g*<<1)

- Klimov ’82, Weldon ’83
- Braaten, Pisarski ’89

- “plasmino”

*p / m**T*

***/ m**T*

- Gauge invariant spectrum

- 2 collective excitations
-  having a “thermal mass” ~ *gT*

- The plasmino mode has
-  a minimum at finite ***p***.

-  width ~*g*2*T*

##  Decomposition of Quark Propagator  

-  Decomposition of Quark Propagator

- Free quark with mass *m*

- HTL ( high *T* limit )

##  Bare quark mass dependence   at zero momentum for T=3Tc  

##  Quark Spectrum as a function of m0  

-  Quark Spectrum as a function of *m*0

- Quark propagator in hot medium at *T *>>*T*c

- - as a function of bare scalar mass *m*0

##  Fermion Spectrum in QED & Yukawa Model  

- Baym, Blaizot, Svetisky, ‘92

- Yukawa model:

- 1-loop approx.:

- Spectral Function for* g *=1 , *T *=1

- thermal mass *m**T*=*gT*/4

- single peak at *m*0

- Plasmino peak disappears
- as *m*0* /T* becomes larger.

- cf.) massless fermion + massive boson
- MK, Kunihiro, Nemoto,’06

##  Simulation Setup  

**quenched** approximation
**clover** improved **Wilson**
**Landau** gauge fixing

| T |  | size | conf. |
| --- | --- | --- | --- |
| 3Tc | 7.45 | 643x16 | 51 |
|  |  | 483x16 | 51 |
|  | 7.19 | 483x12 | 51 |
| 1.5Tc | 6.87 | 643x16 | 44 |
|  |  | 483x16 | 51 |
|  | 6.64 | 483x12 | 51 |
| 1.25Tc | 6.72 | 643x16 | 41 |
|  |  | 483x16 | 50 |
| 0.9Tc | 6.42 | 483x16 | 41 |
| 0.55Tc | 6.13 | 483x16 | 57 |

- configurations generated
- by Bielefeld collaboration

- 2-pole approx. for **+(**,**p**=**0**)
- wall source

##  Correlator and Spectral Function  

- observable
- in lattice

- dynamical
- information

##  Choice of Source  

-  Choice of Source

- Wall source, instead of point source

- point:

- wall :

- same (or, less) numerical cost
- quite effective to reduce noise!!

- the larger spatial volume,
- the more effective!

*t*

*t*

- _What’s the source?_

- point

- wall

##  Dirac Structure of ()   

- quark propagator

**p**=**0**

- even
- odd

##  Correlation Function  

-  Correlation Function

- We neglect 4 points near the source from the fit.
- 2-pole ansatz works quite well!! ( ** 2*/*dof.~2 in correlated fit)

- 643x16, **= 7.459, **= 0.1337, 51confs.

***/T*

##  Spectral Function  

-  Spectral Function

*T* = 3*T**c* 643x16 (**= 7.459)

*E*2

*E*1

**= *m*0
- pole of free quark

*m**0 **/ T*

*E / T*

*Z*2 */ *(*Z*1*+Z*2)

*T*=3*T*c

##  Spectral Function  

-  Spectral Function

*T* = 3*T**c* 643x16 (**= 7.459)

*E*2

*E*1

**= *m*0
- pole of free quark

*m**0 **/ T*

*E / T*

*Z*2 */ *(*Z*1*+Z*2)

- Limiting behaviors for                             are as expected.
- Quark propagator approaches the chiral symmetric one near *m*0=0.
*E*2>*E*1 : qualitatively different from the 1-loop result.

*T*=3*T*c

##  Lattice Spacing Dependence  

-  Lattice Spacing Dependence

- 643x16
- (**= 7.459)
- 483x12
- (**= 7.192)

*E / T*

*E*2

*E*1

*m**0 **/ T*

- same physical volume
- with different *a*.

- No lattice spacing dependence within statistical error.

*T*=3*T*c

##  Spatial Volume Dependence  

-  Spatial Volume Dependence

*E*2

*E*1

*m**0 **/ T*

*E / T*

*T*=3*T*c

- 643x16
- (**= 7.459)
- 483x16
- (**= 7.459)

- same lattice spacing
- with different aspect ratio.

- Excitation spectra have clear volume dependence
-   even for *N****/N***=4.

##  Temperature Dependence  

##  Temperature Dependence  

-  Temperature Dependence

*m**T **/T* is insensitive to *T.*
- The slope of *E*2 and minimum of *E*1 is much clearer at lower *T*.

*T** *= 3*T**c*

*T** *= 1.5*T**c*

*E*2

*E*1

*m**0 **/ T*

*E / T*

*Z*2 */ *(*Z*1*+Z*2)

- 643x16

*T** *= 1.25*T**c*

##  Extrapolation of Thermal Mass  

-  Extrapolation of Thermal Mass

- Extrapolation of thermal mass to infinite spatial volume limit:

- Small *T* dependence of *m**T**/T*,
- while it decreases slightly with increasing *T*.
- Simulation with much larger volume is desirable.

*m**T* */T*

*T*=1.5*T*c

*T*=3*T*c

* **m**T* */T* = 0.800(15)
-  *m**T* = 322(6)MeV

*m**T* */T* = 0.771(18)
*m**T* = 625(15)MeV

- 483x16

- 643x16

*T*=1.25*T*c

* **m**T* */T* = 0.816(20)
-  *m**T* = 274(8)MeV

##  Charm Quark  

** from Datta et al. PRD69,094507(2004).

- Charm quark is free-quark like, rather than HTL.
- The J/** peak in MEM seems to exist above 2*m*c.

*m*c

- preliminary

*T*=1.5*T*c

##  Below Tc  

- for 483x16 lattices

*T/T*c= 3, 1.5, 1.25,
- 0.9, 0.55.

##  Momentum Dependence  

##  Finite Momentum  

- Spectral function in each channel
-   is positive definite!

##  Pole Structure for p>0  

*E*2<*E*1; consistent with the HTL result.
*E*1 approaches the light cone for large momentum.

- 2-pole approx. works
-  well again.

##  Summary  

- We analyzed the quark spectral function at finite *T* in lattice QCD.

-  Future Work

- gauge dependence / volume dependence
- / full QCD / gluon propagator / ...

- Above *T**c*,

- The quark degrees of freedom have a simple quasi-particle
- picture similar to that in the high *T* limit even near *T**c*.
-      — Light quarks have the plasmino and thermal mass.
-      — The ratio *m**T**/T* is insensitive to *T* near *T**c*.

- The pole approximation fails completely.

- Below *T**c*,

##  Effect of Dynamical Quarks  

- Quark propagator
- in quench approximation:

-  screen gluon field  suppress *m**T*?

-  meson loop  will have strong effect
-                               if mesonic excitations exist

- In full QCD,

- massless fermion + massive boson
-  3 peaks in quark spectrum!
-                M.K., Kunihiro, Nemoto, ‘06

##  Beyond the Chiral Limit  

-  Beyond the Chiral Limit

*E*2

*E*1

*m**0 **/ T*

*E / T*

*Z*2 */ *(*Z*1*+Z*2)

*T*=3*T*c

- 643x16, *T*=3*T*c

- Inversion routine converges down to *m*0/*T*~-0.2 for 643x16, *T*=3*T*c.

- , *m**0 **/ T*

- Propagator has a symmetric properties for positive and negative mass.

##  What is the Plasmino?  

- A quark is scattered by a gluon.

- =

##  m0 Dependence of C+()  

** = 0.134

** = 0.132

** = 0.130

***/T*

- Shape of *C*+(**) changes from chiral symmetric
-   to single pole structures.

***c*=0.13390
- in vacuum

*m*0: small

*m*0: large

##  m0 Dependence of C+()  

** = 0.134

** = 0.132

** = 0.130

***/T*

- Shape of *C*+(**) changes from chiral symmetric
-   to single pole structures.

*m*0: small

*m*0: large

***c*=0.13390
- in vacuum