Heavy Mesonic Spectral Functions at Finite Temperature and Finite Momentum

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1 Heavy Mesonic Spectral Functions at Finite Temperature and Finite Momentum Masayuki Asakawa Department of Physics, Osaka University September EMMI Workshop

2 QCD Phase Diagram T LHC RHIC QGP (quark-gluon plasma) MeV crossover CP(critical point) 1st order Hadron Phase chiral symmetry breaking confinement order? CSC (color superconductivity) 5-10ρ 0 µ B

3 Momentum At finite momentum, S and PS channels: ρω (, p ) V and AV channels: ρ ( ω, p) = ρ ( ω, p)( P ) + ρ ( ω, p)( P ) µν T T µν L L µν P T (P L ): transverse (longitudinal) projection operator finite temperature and finite momentum, potential model? collisions with gluons and quarks (hot wind effect) dispersion relation difference between ρ T and ρ L p T dependence of J/ψ yield etc. dilepton spectra (ρ T and ρ L ) and photon spectra (ρ T, on-shell)

4 1st Principle Calculation of SPF What s measured on the Lattice is Imaginary Time Correlation Function D(τ ) 3 D( τ) O( τ, xo ) ' (0,0) dx = D( τ ) and A( ω) A( ω,0) However, 0 are related by D() τ = K(, τω) A( ω) dω Measured in Imaginary Time Measured at a Finite Number of discrete points Noisy Data Monte Carlo Method K(τ,ω): Known Kernel χ 2 -fitting : inconclusive! Direct Inversion: ill-posed!

5 Similar Difficulties in Many Areas Lattice Analytic Continuation to Imaginary Time is measured Measured at a Finite Number of discrete points Noisy Data X-ray Diffraction Measurement in Crystallography Fourier Transformed images are measured Measured at a Finite Number of data points Noisy Data Observational Astronomy Smeared Images due to Finite Resolution are measured Measured by a Finite Number of Pixels Noisy Data

6 MEM Maximum Entropy Method a method to infer the most statistically probable image (such as A(ω)) given data, instead of solving the (ill-posed) inversion problem D() τ = K(, τω) A( ω) dω 0 Theoretical Basis: Bayes Theorem In Lattice QCD PXY [ ] = Bayes' Theorem PYX [ ] P[ X] PY [ ] PXY [ ]: Probability of Xgiven Y D: Lattice Data (Average, Variance, Correlation etc. ) H: All definitions and prior knowledge such as A( ω) 0 P[ A DH] P[ D AH] P[ A H]

7 Ingredients of MEM P[ D AH ] = χ 2 -likelihood function P[ D AH ] = exp( L)/ ZL P[ AH ] given by Shannon-Jaynes Entropy P[ AH αm] = S exp( αs) Z S A( ω) = A ( ω) m( ω) A( ω)log m( ω) α S Z = e [ da], α R S dω max at A( ω) = m( ω) Default Model m( ω) R: Prior knowledge about A( ω) For further details, Y. Nakahara, and T. Hatsuda, and M. A., Prog. Part. Nucl. Phys. 46 (2001) 459 such as semi-positivity, perturbative asymptotic value, operator renormalization etc.

8 Result of Mock Data Analysis N(# of data points)-b(noise level) dependence

9 Result of Mock Data Analysis (cont d) N(# of data points)-b(noise level) dependence

10 Stat. and Syst. Error Analyses in MEM Generally, The Larger the Number of Data Points and the Lower the Noise Level The Closer the Result is to the Original Image Need to do the following: Put Error Bars and Make Sure Observed Structures are Statistically Significant Change the Number of Data Points and Make Sure the Result does not Change Statistical Systematic in any MEM analysis

11 Importance of Error Analysis 1 In MEM analysis, output image itself has little significance Only average is of significance Unfortunately, still many people do MEM analyses without error bars

12 Importance of Error Analysis 2 From our old paper Ave. ±1σ T = 1.62T c T = 1.70T c V channel (J/ψ) N σ = 32

13 Error Analysis in MEM (Statistical) MEM is based on Bayesian Probability Theory In MEM, Errors can be and must be assigned This procedure is essential in MEM Analysis For example, Error Bars can be put to 1 ω2 Average of Spectral Function in I = [ ω1, ω2], Aα = A ( ) d I ω α ω ω ω ω 1 1 δa [ da] dωdωδ A( ω) δa( ω ) P[ A DHαm] I ( ω ω ) I I ( ) 2 α = 2 l= δa( ω) = A( ω) A ( ω) QA ( ) = αs L [ da] = 2 1 δ QA ( ) dωdω 2 ( ω2 ω1) I I δa( ω) δa( ω ) Nω da l A l α 1 A= Aα 2 1 Gaussian approximation For further details, Y. Nakahara, and T. Hatsuda, and M. A., Prog. Part. Nucl. Phys. 46 (2001) 459

14 Statistical Error Analysis Jackknife Lattice fluctuation: included in χ 2, i.e. P[ D AH ] On the other hand, MEM calculates P[ A DH] P[ D AH] P[ A H] Statistical error analysis in MEM includes more than Lattice fluctuation Jackknife Method misses P[A H] part Statistical Error with Jackknife Method Ding et al. (2010)

15 Stat. and Syst. Error Analyses in MEM Generally, The Larger the Number of Data Points and the Lower the Noise Level The Closer the Result is to the Original Image Need to do the following: Put Error Bars and Make Sure Observed Structures are Statistically Significant Change the Number of Data Points and Make Sure the Result does not Change Statistical Systematic in any MEM analysis

16 Importance of Error Analysis lattice β = 6.47, 150 confs. isotropic lattice (T<T c ) Necessity to use large β (=6/g 2 ) to have many point in τ Necessity to change N data

17 Importance of Error Analysis 4 N τ = 46 (T = 1.62T c ) V channel (J/ψ) N σ = 32 N τ = 40 (T = 1.87T c ) V channel (J/ψ) N σ = 32

18 Lattice Parameters 1. Lattice Sizes 64 3 N τ (previously 32 3 N τ ) 2. β = 7.0, ξ 0 = 3.5 ξ = a σ /a τ = 4.0 (anisotropic) 3. a τ = fm L σ = 2.5 fm 4. Standard Plaquette Action 5. Wilson Fermion 6. Heatbath : Overrelaxation = 1 : sweeps between measurements 7. Quenched Approximation 8. Gauge Unfixed 9. Machine: of Tsukuba Nagoya University

19 Lattice Parameters and # of Conf. anisotropic lattice: ξ = a σ /a τ = 4 β = 7.0, a τ = fm large spatial volume: N σ = 64 (L σ =2.5fm) P min ~ 0.5 GeV (previously ~ 1 GeV) Large spatial volume: to have small enough P min to study finite T effects Large β = small a τ : to have large enough N τ for MEM analysis N τ (T/T c ) 96 (0.78) 54 (1.38) 46 (1.62) 44 (1.70) 42 (1.78) 40 (1.87) 32 (2.33) # of conf

20 Our Old Result (N σ =32) for PS channel (η c ) PS channel Lattice Artifact η c (p = 0) disappears between 1.62T c and 1.70T c Asakawa and Hatsuda, PRL 2004

21 PS channel with N σ =64 N τ (T/T c ) 96 (0.78) 54 (1.38) 46 (1.62) 44 (1.70) 42 (1.78) 40 (1.87) 32 (2.33)

22 PS channel with N σ =64 N τ (T/T c ) 96 (0.78) 54 (1.38) 46 (1.62) 44 (1.70) 42 (1.78) 40 (1.87) 32 (2.33)

23 PS channel with N σ =64 N τ (T/T c ) 96 (0.78) 54 (1.38) 46 (1.62) 44 (1.70) 42 (1.78) 40 (1.87) 32 (2.33)

24 PS channel with N σ =64 N τ (T/T c ) 96 (0.78) 54 (1.38) 46 (1.62) 44 (1.70) 42 (1.78) 40 (1.87) 32 (2.33) η c melts between 1.62T c and 1.70T c

25 PS channel with N σ =64 N τ (T/T c ) 96 (0.78) 54 (1.38) 46 (1.62) 44 (1.70) 42 (1.78) 40 (1.87) 32 (2.33)

26 PS channel with N σ =64 N τ (T/T c ) 96 (0.78) 54 (1.38) 46 (1.62) 44 (1.70) 42 (1.78) 40 (1.87) 32 (2.33)

27 Data Point Number Dependence not used in the analysis note: anisotropy = 4 3 correlator 3 max. 39

28 N data Dependence of the Peak Shape 1st peak Larger N data : sharper signal Average value of ρ is almost the same. Smaller N data : larger error It is necessary to have enough number of N data in MEM analyses

29 Default Model Dependence N data =39 m 0 =1.15 pqcd + Operator Renormalization (as in Asakawa and Hatsuda (2004))

30 Error Analysis must also be done Peak shape changes as default model changes But error size changes as well Error size takes minimum at m 0 among these four choices of default model m 0 is best choice among these

31 Temperature Dependence of η c Mass ω (GeV) Asakawa and Hatsuda (2004) In the previous calculation, peak shift was not evident low statistics, large p min Now, peak shift is clear, it is upward

32 η c at Non-zero Momenta P min ~ 0.5 GeV

33 η c at Non-zero Momenta T=1.62T c η c exists at non-zero momentum (at least up to ~ 4 GeV) Apparently, the peak is lowered at finite momenta Default effect (residue decrease) + additional medium effect?

34 Dispersion Relation below T c T=0.78T c The dispersion relation at T=0.78T c on the lattice is consistent with that in the vacuum

35 Despersion Relation in Finite T p=1 Above about p=3gev, the modification of the dispersion relation becomes evident Clue to the structure of the resonance in medium? p=6 p=7 p=8

36 Threshold Enhancement? Pole mass of charm barely changes as T increases E 2 : plasmino E 1 : normal mode m p = bare (free) quark mass Karsch and Kitazawa, PRD (2009) Threshold enhancement cannot explain peak shift of η c

37 Summary Charmonia at Finite Temperature Spectral functions with large spatial volume (finite T effect) with many data points in τ (needed for MEM analysis) At p=0 η c survives up to T=1.62T c (confirms previous results) Peak position ( mass ) moves up as T increases This cannot be explained by threshold enhancement At p 0 (T=0.78T c and 1.62T c ) η c exists at least up to p=4gev Peak height decreases as p increases Dispersion relation is modified above T c (validity of potential model?) To be Finished Soon Separation of the two spectral functions in V channel Calculation with smaller quark masses