1. Masakiyo Kitazawa (Osaka Univ.) HQ2008, Aug. 19, 2008 Hot Quarks in Lattice QCD

2. Masakiyo Kitazawa (Osaka Univ.) HQ2008, Aug. 19, 2008 Lattice QCD and Hot Quarks 1. Introduction to Lattice QCD 2. Hot quarks in lattice QCD 3. Discussions

3. WHY we study Lattice QCD? WHY we study Lattice QCD? Lattice QCD provides a “first principle” calculation of QCD. Lattice results justify QCD as well as lattice itself. inputs for the physics beyond the standard model. hadron mass spectrum PACS-CS collab. 2007 reproduces experiments quite well! Will lattice QCD take over heavy-ion experiments?

4. Path Integral – Quantum Mechanics Path Integral – Quantum Mechanics transition amplitude in Feynman’s path-integral n-dimensional integral; With fixed n, this amplitude is numerically calculated in principle. t tItI t1t1 t2t2 t3t3 tntn tFtF

5. Path Integral – Field Theory Path Integral – Field Theory infinite degrees of freedom for each t : discretize space-time and sum up all field configurations Lattice QCD is formulated in the path integral formalism. Note: t x y t

6. Systematic Errors Lattice action : discrete QCD action approaches QCD action in a  0 limit various choices different results for different actions quarks actions: Wilson staggard (KS) Domain wall Ginsparg-Wilson in numerical simulations, a : lattice spacing V : lattice volume m : quark mass a  0 (continuum limit) V  infinite m  m phys (chiral extrapolation) in the real world heavy numeciral cost Lattice2007, Karsch critical temp.

7. Dynamical Quarks Example : Meson propagator neglect quark-antiquark loops ~10 3 times faster than full calc. full QCD quenched QCD quenched (N f =0) N f =2 (two light quarks) N f =2+1 (two-light + strange) Simulation settings heavier calculation M(x)M(x)M(y)M(y)M(y)M(y) M(x)M(x)

8. Lattice QCD at T>0 Lattice QCD at T>0 : Partition function : Expectation value of O Lattice is not the real-time simulation. Lattice can deal with only the equilibrium system. Statistical mechanics in equilibrium Note: imaginary-time action periodicity at  =  =1/T

9. Bulk Thermodynamics Bulk Thermodynamics Partition function: thermodynamic quantities: actually, we calculate s, susceptibilities, etc… energy density  pressure p sudden increase of  at T~190MeV Cheng, et al., 2007

10. Correlation Function (Propagator) Correlation Function (Propagator) Imaginary-time propagator (Correlation function) observables on the lattice Spectral function Expectation values: F.T.  Real-time propagator dynamical propagation F.T.  discrete and noisy continuous function analytic continuation Ill-posed problem Note: Only the Euclidean propagator is calculated on the Lattice.

11. Maximum Entropy Method (MEM) method to infer the most probable image with the lattice data and a set of prior information Asakawa,Hatsuda, Nakahara, 2001 Charmonium spectral function above T c charmonium survives even above T c up to 1.5~2T c. Datta, et al. 2004

12. Summary for the First Part Lattice QCD at finite T is formulated based on the quantum statistical mechanics, with path integral in the Euclidean space. It treats the equilibrium physics. The propagator calculated on the lattice is the imaginary-time function. Analytic continuation is needed to extract dynamical information. We need ideas to measure observables on the lattice. topics not mentioned here: finite density / viscosities / Polyakov loop / etc.

13. Hot Quarks in Lattice QCD Karsch, Kitazawa, PLB658,45 (2007); in preparation.

14. Hot Quarks in sQGP Hot Quarks in sQGP Success of recombination model suggests the existence of quark quasi-particles in sQGP. Lattice simulations do not tell us physics under observables.

15. Quarks at Extremely High T 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

16. p / m  / m Decomposition of Quark Propagator Decomposition of Quark Propagator Free quark with mass mHTL ( high T limit ) p / m T  / m T

17. Quark Spectrum as a function of m 0 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 How is the interpolating behavior? How does the plasmino excitation emerge as m 0  0 ? m 0 << gT  m 0 >> gT  We know two gauge-independent limits: m0m0 mTmT -m T +(,p=0)+(,p=0) +(,p=0)+(,p=0)

18. Fermion Spectrum in QED & Yukawa Model Fermion Spectrum in QED & Yukawa Model Baym, Blaizot, Svetisky, ‘92 Yukawa model: 1-loop approx.: m/T=0.01 0.8 0.45 0.3 0.1 +(,p=0)+(,p=0)  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 M.K., Kunihiro, Nemoto,’06

19. Simulation Setup Simulation Setup quenched approximation clover improved Wilson Landau gauge fixing T  size# of conf. 3Tc3Tc 7.4564 3 x1651 (0) 48 3 x1651 (0) 7.1948 3 x1251 (0) 1.5T c 6.8764 3 x1651 (7) 48 3 x1651 (0) 6.6448 3 x1251 (0) 1.25T c 6.7264 3 x1671 (31) configurations generated by Bielefeld collaboration vary bare quark mass m 0

20. Correlator and Spectral Function Correlator and Spectral Function  E1E1 E2E2 Z1Z1 Z2Z2 observable in lattice dynamical information 2-pole structure may be a good assumption for  + (  ). 4-parameter fit E 1, E 2, Z 1, Z 2

21. 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) 64 3 x16,  = 7.459,  = 0.1337, 51confs.  /T Fitting result

22. Spectral Function Spectral Function  E1E1 E2E2 Z1Z1 Z2Z2  E1E1 E2E2 Z1Z1 Z2Z2 T = 3T c 64 3 x16 (  = 7.459) E2E2 E1E1  = m 0 pole of free quark m 0 / T E / T Z 2 / (Z 1 +Z 2 ) T=3T c

23. Spectral Function Spectral Function T = 3T c 64 3 x16 (  = 7.459) E2E2 E1E1  = 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=3T c

24. 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 = 3T c T = 1.5T c minimum of E 1 E2E2 E1E1 m 0 / T E / T Z 2 / (Z 1 +Z 2 ) 1-loop result might be realized for high T. 64 3 x16 T = 1.25T c

25. Lattice Spacing Dependence Lattice Spacing Dependence 64 3 x16 (  = 7.459) 48 3 x12 (  = 7.192) E / T E2E2 E1E1 m 0 / T same physical volume with different a. No lattice spacing dependence within statistical error. T=3T c

26. Spatial Volume Dependence Spatial Volume Dependence E2E2 E1E1 m 0 / T E / T T=3T c 64 3 x16 (  = 7.459) 48 3 x16 (  = 7.459) same lattice spacing with different aspect ratio. Excitation spectra have clear volume dependence even for N  /N  =4.

27. 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=1.5T c T=3T c m T /T = 0.800(15) m T = 322(6)MeV m T /T = 0.771(18) m T = 625(15)MeV 48 3 x16 64 3 x16 T=1.25T c m T /T = 0.816(20) m T = 274(8)MeV

28. Pole Structure for p>0 E 2 <E 1 ; consistent with the HTL result. E 1 approaches the light cone for large momentum. HTL(1-loop) 2-pole approx. works well again.

29. Discussions ?

30. 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 2m c. mcmc preliminary threshold 2m c T=1.5T c

31. Role of Thermal Mass Does chiral symmetry breaking take place even with m T ? Does thermal mass contribute to the stability of mesons? + + Interaction: Hidaka, MK, PRD75, 011901(R) (2007)  YES  NO. Mesons are unstable even for  <2m T.

32. Away Side Particle Distribution Quark mass ~T  Partons have position dependent mass. orbit of light in medium slow fast orbit of quarks in sQGP light heavy in very progress… high low T

33. Summary Summary Quarks seem to behave as a good quasi-particles. Thermal gluon field gives rise to the thermal mass in the light quark spectra. The plasmino mode disappears for heavy quarks. The ratio m T /T is insensitive to T near T c. Lattice simulations provide us many information about the sructure of quark propagator successfully. Information about the quark propagator will used for phenomenological studies of the QGP. Future Work Future Work full QCD / gauge dependence / volume dependence / …

34. Effect of Dynamical Quarks 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