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Spectral Properties of Quarks at High Temperatures: Thermal Mass, Dispersion Relation, and Self-Energy

This study examines the behavior of quarks in the region above the critical temperature in Quark-Gluon Plasma. It analyzes spectral properties, thermal mass, dispersion relation, and self-energy using lattice QCD and perturbation theory.

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Spectral Properties of Quarks at High Temperatures: Thermal Mass, Dispersion Relation, and Self-Energy

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  1. LATTICE2010, June 18, 2010 Spectral Properties of Quarks above Tc ~Thermal mass, Dispersion Relation and Self-Energy~ Masakiyo Kitazawa (Osaka U.) Olaf Kaczmarek, Frithjof Karsch, Wolfgang Soeldner F. Karsch and MK, PLB658,45 (2007); PRD80,056001(2009); Kaczmarek, Karsch, MK, Soeldner, in preparation.

  2. QCD @ T>0 RHIC LHC Lattice QCD perturbation How does the matter behave in this region? How do hadrons cease to exist? How do quarks and gluons disappear? Tc* T

  3. Quarks at Extremely High T Klimov ’82, Weldon ’83 Braaten, Pisarski ’89 • Hard Thermal Loop approx. ( p, w, mq<<T ) • 1-loop (g<<1) • Gauge independent spectrum • 2 collective excitations • having a “thermal mass” ~ gT w / mT “plasmino” • width ~g2T • The plasmino mode has • a minimum at finite p. p / mT

  4. Decomposition of Quark Propagator w / mT w / m p / mT p / m HTL ( high T limit ) Free quark with mass m

  5. Z2 Z1 w -E2 E1 Lattice Study of Quarks above Tc Karsch, MK, ’07, ’09 • quenched approximation • clover improved Wilson • Landau gauge fixing 2-pole ansatz for r+(w). 4-parameter fit E1, E2, Z1, Z2 m/T=0.01 r+(w,p=0) 0.1 0.3 0.45 0.8 w/T

  6. Lattice Study of Quarks above Tc Karsch, MK, ’07, ’09 • success of 2-pole ansatz • emergence of thermal mass mT ~0.8T • dispersion relation (p dependence) • bare quark mass (m0) dependence r+(w) m0 ,p=0 r+(w) w p r+(w) w -mT mT w m0

  7. Spectral Function Z1 Z2 w -E2 E1 T = 3Tc 643x16 (b = 7.459) T=3Tc E2 E / T w = m0 pole of free quark E1 Z2 / (Z1+Z2) m0 / T Z2 Z1 w -E2 E1

  8. Spectral Function T = 3Tc 643x16 (b = 7.459) T=3Tc E2 E / T w = m0 pole of free quark E1 Z2 / (Z1+Z2) m0 / T • Limiting behaviors forare as expected. • Quark propagator approaches the chiral symmetric one near m0=0. • E2>E1 : qualitatively different from the 1-loop result.

  9. Spatial Volume Dependence of mT • Strong spatial volume dependence of mT. • Results with Ns/Nt=4 still have ~15% deviation. 483x16 mT/T 643x16

  10. Spatial Volume Dependence of mT • Strong spatial volume dependence of mT. • Results with Ns/Nt=4 still have ~15% deviation. 483x16 mT/T 643x16 Discretization of p m=0 k2nB(Ek) Nx/Nt=4 m=T • Nx/Nt=4 pmin~1.57T • Nx/Nt=8 pmin~0.79T k/T

  11. Result on 1283x16 Latttice New analysis Extrapolation of mT T=3Tc mT/T 483x16 643x16 1283x16 mT/T=0.725(14) • Result on 1283x16 lattice seems to converge. • Much larger lattice is needed to understand V dependence.

  12. HTL(1-loop) Quark Dispersion on 1283x16 Lattice T=3Tc • E2 has a minimum at p>0. • HTL spectrum has a continuum in space-like region, which is not included in the 2-pole ansatz.

  13. Reliability of the Pole Ansatz Caution: Lattice correlator is insensitive to the structure of r (w) at low energy |w|<T.

  14. Reliability of the Pole Ansatz Caution: Lattice correlator is insensitive to the structure of r (w) at low energy |w|<T. Quark correlator at p=0, m0=0 2-pole mT~0.8T w S0(t) fit / S0(t) latt Gauss G~1.2T w Completely different r (w) can similarly reproduce S(t)!

  15. Moment Expansion Taylor expand coshw(t-1/2T) If r (w) vanishes for |w|>L, higher order terms behave

  16. Moment Expansion Taylor expand coshw(t-1/2T) If r (w) vanishes for |w|>L, higher order terms behave Quark correlator for p=0, m0=0 divide by (t -1/2T)2 midpoint subtracted S0(t,0) S0(t,0)/ (t -1/2T)2 tT tT

  17. Quark Quasi-Particle Peaks p=0 2-pole ansatz well reproduces S(t) over wide ranges of m0 & p. Gaussian ansatz does not. S0(t,0)/ (t -1/2T)2 p=2pmin MEM analysis also supports the existence of quasi-particle peaks. r+(w); MEM • Quark spectrum near but above Tc most probably has two sharp peaks (normal & plasmino). • Analysis with the 2-pole ansatz would grasp qualitative behavior of the spectrum. k =0.1340 k =0.1337

  18. Quark Self-Energy MK, et al., in preparation.

  19. Quark Self-Energy Im. part Self-energy (ImS) • decay rate of quasi-quark @pole Spectral function • Theoretically, ImS is useful to • understand spectral properties. • experimental observable

  20. Our strategy Correlator & Self-Energy lattice observable nonlinear What we want…

  21. F.T. S(t) S(iwn) An algorithm: inversion F.T. S(t) S(iwn) Schwinger-Dyson eq.: Inverse matrix of lattice correlator (in t space) vanish, unless t=0 Note: S(t) is not the fermion matrix. Inverse, after statistical average.

  22. Numerical Result: S(t) T=3Tc, p=0, chiral limit S0(t)/T 643x16 483x12 1283x16 Ns/Nt=4 2 poles (HTL) Ns/Nt=8 tT • S(t) is consistent with the values expected from the pole ansatz (HTL self-energy) for t T~1/2. • Strong deviation near the source.

  23. Summary We analyzed spectral properties of quarks above Tc on the quenched lattice of size up to 1283x16. Success of 2-pole ansatz for wide ranges of m0 & p • Quark spectrum above but near Tc would contain sharp • quasi-particle peaks. • The analysis of r (w) at low energy is difficult. Progress in extrapolation of mT to infinite volume limit: • Old: mT/T=0.771(20) New: mT/T=0.725(14) for T=3Tc The self-energy contains more information on spectral properties. The first analysis of S(t) looks promising.

  24. Exceptional Configurations Exceptional behavior manifests itself only in The result indicates the existence of a topological object on gauge conf.

  25. Choice of Source What’s the source? Wall source, instead of point source point: wall : point t • same (or, less) numerical cost • quite effective to reduce error!! Quality of data on 1283x16 lattice is about 3 times better than on 643x16. wall t

  26. Check 3 : Correlator & Fitting Result 1283x16, 643x16, T=3Tc, chiral limit, Ndata=9 for S0(t)fit S0(t)latt / S0(t)fit tT It seems that there exists a contribution of a pole with an energy of order a -1 and with a negative residue. Due to this lattice artifact, datapoints near the source cannot be used. Much finer lattice is needed to improve Ndata.

  27. Scalar term SS(t) for p>0 • Scalar term is small, but has a momentum dependence.

  28. Quark correlator below Tc • is concave upward. • indicates a negative norm. • does not approach the chiral • symm. one in the chiral limit. Below Tc for 483x16 lattices T/Tc= 3, 1.5, 1.25, 0.93, 0.55.

  29. Dirac Structure of Spectral Function even odd odd • positive definite • not even nor odd • even only with chiral symm. zero momentum chirally symmetric

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