Exploring Real-time Functions on the Lattice with Inverse Propagator and Self-Energy Masakiyo Kitazawa (Osaka U.) 22/Sep./2011 Lunch BNL

T>0 T>0 T RHIC LHC perturbation Lattice QCD How does the matter behave in this region? Tc*Tc*

Quantum Statistical Mechanics Quantum Statistical Mechanics  Static expectation value  Dynamical response EoS chiral condensate susceptibilities screening mass … Spectral func:

Spectral Functions at T>0 Spectral Functions at T>0 quasi-particle excitation width ~ decay rate  transport coefficients  ( ,p)  peaks Kubo formulae slope at the origin shear viscosity : T 12 bulk viscosity : T  electric conductivity : J ii

Analytic Continuation Analytic Continuation analytic continuation Retarded (real-time) propagator Spectral function Imaginary-time (Matsubara) propagator Lattice Dynamics

Analytic Continuation Analytic Continuation MEM analysis of  (  ) most probable image estimated by lattice data + prior knowledge Asakawa, Hatsuda, Nakahara, 1999 qualitative structure of  (  ) Asakawa, Hatsuda, 2004; Datta, et al., 2004;… Lattice Dynamics

Analytic Continuation Analytic Continuation  Questions: Are there other useful formulas to relate real-time and Euclidean functions? Is  (  ) only a real-time function worth analyzing? We consider the inverse propagator. cf.)sum rules Kharzeev, Tuchin; Karsch, Kharzeev, Tuchin, 2008 Romatchke, Son, 2009; Meyer, 2010; …

A Difficulty in Analyzing Low Energy Spectrum A Difficulty in Analyzing Low Energy Spectrum

Moment Expansion Moment Expansion Taylor expand cosh  (  -1/2T) moment of “thermal” spectrum C n  n-th moment of  ’(  ). Aarts, et al., 2002 Petreczky, Teany, 2006 Ding, 2010

Low Energy Spectrum Low Energy Spectrum   Contribution of higher order moments of low energy part is severely suppressed. For  =T, the ratio is 2.1x10 -5 for n=6.

Inverse Propagator Inverse Propagator and Analytic Continuation and Analytic Continuation Inverse Propagator Inverse Propagator and Analytic Continuation and Analytic Continuation

Inverse Propagator Inverse Propagator Lattice observable Dynamical information analytic inverse analytic inverse F.T. inverse

Numerical Procedure Numerical Procedure Lattice observable Dynamical information inverse Standard analysis Present study Translational symmetry reduces costs for the inverse. [S] -1 (  ) is not the statistical average of fermion matrix. F.T.Inv. F.T.

Quark Self-Energy decay rate of Theoretically, Im  is useful to understand spectral properties. experimental observable Spectral function Self-energy (Im  ) Im. part

To Interpret Physics behind the SPC physical interpretation via optical theorem Disp. Rel.   

Self-Energy for Discrete Spectrum Self-Energy for Discrete Spectrum

Poles of G R (  ) and G R (  ) -1 are staggered on real-axis.

Numerical Results for Numerical Results for Inverse Quark Correlator Inverse Quark Correlator Numerical Results for Numerical Results for Inverse Quark Correlator Inverse Quark Correlator

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 independent spectrum 2 collective excitations having a “thermal mass” ~ gT The plasmino mode has a minimum at finite p. width ~g 2 T

Simulation Setup Simulation Setup quenched approximation Landau gauge fixing clover improved Wilson T/T c  sizeN conf x x x x x x x x x x1660 Karsch, MK, ’07; ’09 MK, et al., in prep.

Ansatz for Spectral Function Ansatz for Spectral Function 2-pole structure for  + (  ). 4-parameter fit E 1, E 2, Z 1, Z 2

Correlation Function Correlation Function We neglect 7 points near the source from the fit. 2-pole ansatz works quite well!! (  2 /dof. is of order 1 in correlated fit) 64 3 x16,  = 7.459,  = , 51confs.  /T Fitting result  /T

Quark Dispersion on x16 Lattice Quark Dispersion on x16 Lattice T=3T c HTL(1-loop) E 2 has a minimum at p>0 Existence of the plasmino minimum is strongly indicated. E 2, however, is not the position of plasmino pole. MK, et al., in preparation

Quark Correlator at k=0, m=0 Quark Correlator at k=0, m=0 Quark Correlator Inverse Correlator Lines: fits by 2-pole ansatzGood agreement for 0.4<  <0.7 2-pole structure of quark spectrum supported

Quark Correlator at k>0, m>0 Quark Correlator at k>0, m>0 Quark Correlator Inverse Correlator  Difference in correlators which behave similarly can become clear in terms of the inverse correlator.  2-pole inverse correlator is inconsistent with the lattice one x16, T=3T c

Deviation near the Source Deviation near the Source Free Wilson propagator: Continuum: Lattice:

Deviation near the Source 2 Deviation near the Source 2 Deviation from the 2-pole prediction scales in lattice unit

Summary Summary  A correlator which is consistent with a lattice result can be inconsistent with the inverse correlator.  Inverse correlator can further constrain real-time functions!  Analysis of inverse quark correlator supports the existence of normal and plasmino modes in quark spectrum near Tc. Future Work Future Work  MEM analysis for the inverse correlator  Analysis of meson propagator  Comparison with analytic studies in terms of self-energy  etc.

To Obtain a Better Image of  (  ) Decay width might be evaluated with a reasonable statistical error. For isolated peaks, ImD(  ) at the peak represents the decay width of the quasi-particle excitation. typical errorbars in MEM analysis; insufficient to determine the width