Meson spectral function: an application of the Maximum Entropy Method

Transcription

1 Meson spectral function: an application of the Maimum Entropy Method Péter Petreczky Physics Department and RIKEN-BNL Meson correlations on lattice and the spectral functions Maimum Entropy Method in lattice QCD Numerical results : charmonium spectral functions at zero and finite temperature Conclusions and Outlook 4 th International Workshop on Numerical Analysis and Lattice QCD QCDNA May

2 00 3 J J J e d p G p i τ ψ τ ψ τ τ τ Γ = = + Meson correlators and spectral functions µ γ µ γ γ γ = Γ τ τ i D G = > = 0 /2 sinh 1/2 cosh d G ω ω τ ω ωσ τ Im 1 2 ω σ ω π π ω ω = = < > D R D D Lattice QCD is formulated in imaginary time Physical processes take place in real time t J J e d p t D J t J e d p t D p i p i < > = = if fit the large distance behavior of the lattice correlation functions his is not possible for and in the case of resonances e.g.

3 Reconstruction of the spectral functions : MEM data and degrees of freedom to reconstruct Bayesian techniques: find which maimizes Bayes theorem data prior probability Prior knowledge Maimum Entropy Method MEM: Asakawa atsuda Nakahara PRD Prog. Part. Nucl. Phys Likelihood function Shannon-Janes entropy: - default model -perturbation theory

4 Procedure for calculating the spectral functions ow to find numerically a global maimum in the parameters space of O100 dimenisons for fied α? Bryan Europ. Biophys. J Jakovác P.P. Petrov Velytsky PRD covariance matri ensemble average which can be done using Levenberg-Marquardt algorithm

5 Procedure for calculating the spectral functions cont d ow to deal with the α-dependence of the result?

6 Charmonium spectral functions at =0 Jakovác P.P. Petrov Velytsky PRD For the spectral function is sensitive to lattice cut-off ; good agreement with 2-eponential fit for peak position and amplitude

7 Charmonium spectral functions at =0 cont d

8 Charmonium spectral functions at =0 cont d What states can be resolved and what is the dependence on the default model? Reconstruction of an input spectral function : Lattice data in PS channel for: Ground states is well resolved no default model dependence; Ecited states are not resolved individually moderate dependence on the default model; Strong default model dependence in the continuum region

9 Comparison of MEM and 2-eponential fit

10 Charmonia spectral functions at >0 ground state peak is shifted ecited states are not resolved when become small no temperature dependence in the PS spectral functions within errors Jakovác P.P. Petrov Velytsky PRD

11 Charmonia spectral functions at >0 cont d

12 Conclusions MEM offers an alternative possibility to analyze meson correlation functions and we have seen that the meson spectral function etracted from lattice QCD have the epected form resonances + continuum Results obtained from MEM are compatible with multi-eponential fit however for charmonium correlators there is no real advantage of using MEM over the conventional method. Etracting width of resonances may be difficult through MEM Reliable calculation of meson properties at finite temperature using MEM is difficult with the lattice data available at present

13 What is the physics behind the 2 nd and higher peaks??

14 Using default model from the high energy part of the =0 spectral functions : resonances appears as small structures on top of the continuum almost no -dependence in the PS spectral functions till