# Presentation: 128360

## Charmonia in the Deconfined Plasma from Lattice QCD

-  **Masayuki Asakawa**
- Osaka University

*Intuitive and Quick Introduction to Lattice QCD*
*and*

##  Warm Up: Quantum Mechanics

-  In (Undergraduate) Quantum Mechanics,
-    Transition amplitude is given by

-  The same quantity has another expression,

##  AB Effect and Path Integral

-  Famous AB(Aharonov-Bohm) effect  (*Modern Quantum Mechanics,* J.J.Sakurai)

##  Path Integral in Field Theory

-  Field Theory : Quantum Mechanics with Infinite Degrees of Freedom

- degree of freedom distributed at each point

##  Discretization, a.k.a. Lattice

- So far, no approximation has been done

- In order to calculate path integral numerically,
- Space-Time is discretized

- Also, let us consider field theory in Euclidean space-time

##  QCD on the Lattice

-  Gauge Fields on the Lattice

-  In order to retain Gauge Invariance,
-   Gauge Fields are distributed between lattice points (=link),
-   while Fermion Fields still live at lattice points

- continuum gauge field action

- color field
- matrix

##  Why Monte Carlo Method?

-  Expectation value of QCD observable (on the lattice)

##  Why Reweighting?

-  This is the case at =0, non-zero  (SU(2)),
-    but not the case at non-zero  (QCD).

- Generate Gauge Field Configurations with weight
- exp(-Sp) or exp(-Sp+log(detMq)) *if* exp(-Sp) or exp(-Sp+log(detMq)) is Real

- With some applicability limit

##  Systematic and Statistical Errors

-  Lattice Calculation is carried out with
-    Finite Lattice Spacing *a*, Finite Lattice Volume *V*, quark mass *m* ≠ *m**phys** *

- as in usual integral

##  Things not to do

- van Leeuwen for NA49, QM2002

##  Order of Phase Transition at = 0

- 3d O(4) scaling

- tricritical point

- 3d Ising scaling

- 2nd order line:
- mud    (ms*-ms)5/2
- near the
- tricritical point

- 3d Z(3) Potts

- Kanaya, QM2002

##  Critical Endpoint in Effective Models

- Compilation by Stephanov

##  What is the evidence of strongly int. QGP?

-  *e* rises quickly, while *p* increases slowly

- Is this due to interesting physics?

-  Models with Massive Quarks and Gluons...etc.
-  Some Hadronization Models before RHIC

*e* - 3*p*: interaction measure

## What is the “normal” behavior of e, p, e-3p?

- Even if *s* approaches the Stefan-Boltzmann value just above *T**c*,
*p* does not approach *p**SB*
- In addition, *e *- 3*p* never vanishes!

##  Example

- /*T**c *= 0.05

- Hatsuda and M.A., 1997

*Interaction Measure does not measure interaction !*

##  PLAN of the Second Part

- What is Spectral Function? Why Spectral Function?
- Necessity of MEM (*Maximum Entropy Method)*
- Models at Finite T/
- MEM Outline
- Importance of Error Analysis
- Finite Temperature Results for *J/*** and ***c*
- Error Analysis
- Statistical
- Systematic

- Hatsuda and M.A., PRL 2004

##  Spectral Function

* **Spectral Function*

-  Pretty important function to understand QCD

- Dilepton production rate, Real Photon production rate, ...etc.

*holds regardless of states, either in Hadron phase or QGP*

-  Definition of Spectral Function

##  CERES(NA45) AB data

##  Hadron Modification in HI Collisions? 

*Comparison with Theory *
*(with no hadron modification)*

*Experimental*  *Data*

- Mass Shift ? Broadening ? or Both ?
- or More Complex Structure ?

##  Why Theoretically Unsettled

* **Mass Shift*
*(Partial Chiral Symmetry Restoration)*

* **Spectrum Broadening*
*(Collisional Broadening)*

## Mass shift or Coll. broadening or -hole or...

-  QCDSR

-  Assumption for the shape of the spectral function
-  Strongly depends on 4-quark condensates

-  Conventional Many Body Approach

-  Model dependence
-  How is the effect of chiral symmetry restoration taken into account?

##  Lattice? But there was difficulty ...

-  What’s measured on Lattice is Correlation Function *D*(**)

##  Way out ?

-  Example of 2-fitting failure

##  Difficulty on Lattice

- Thus, what we have is
- Inversion Problem

##  MEM

*M*aximum *E*ntropy *M*ethod

- successful in crystallography, astrophysics, ...etc.

##  Principle of MEM

* **Principle of MEM*

-  Theoretical Basis: Bayes’ Theorem

*In MEM, Statistical Error can be put to the Obtained Image*

-  MEM

-  In Lattice QCD

*In MEM, basically Most Probable Spectral Function is calculated*

##  Ingredients of MEM

- given by Shannon-Jaynes Entropy

- For further details,
- Y. Nakahara, and T. Hatsuda, and M. A., Prog. Part. Nucl. Phys. 46 (2001) 459

##  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

##  Uniqueness of MEM Solution

***Unique if it exists !***

- T. Hatsuda, Y. Nakahara, M.A., Prog. Part. Nucl. Phys. 46 (2001) 459

##  Result of Mock Data Analysis 

- N(# of data points)-b(noise level) dependence

- Nakahara, Hatsuda, and M.A.
- Prog. Nucl. Theor. Phys., 2001

* **Application of MEM to Lattice Data (T=0)*

*Resonance Physics has become possible on Lattice*

##  Parameters on Lattice at finite T

* **Parameters on Lattice at finite T*

- Lattice Sizes  323  32  (*T* = 2.33*T**c*)           40  (*T* = 1.87*T**c*)           42  (*T* = 1.78*T**c*)           44  (*T* = 1.70*T**c*)           46  (*T* = 1.62*T**c*)           54  (*T* = 1.38*T**c*)           72  (*T* = 1.04*T**c*)           80  (*T* = 0.93*T**c*)           96  (*T* = 0.78*T**c*)
-  ** = 7.0,  **0 = 3.5 ** = *a*/*a* = 4.0 (anisotropic)
-  *a* = 9.75  10-3 fm *L* = 1.25 fm
- Standard Plaquette Action

- Wilson Fermion
- Heatbath : Overrelaxation1      :       41000 sweeps between measurements
- Quenched Approximation
- Gauge Unfixed
-  **p **= **0**  Projection
- Machine: CP-PACS

##  Result for V channel (J/)

* **Result for V channel (J/****)*

*J*/** (**p ****0**) disappears
- between 1.62*T**c* and 1.70*T**c*

*A*(**)  ***2***(**)

##  Result for PS channel (c)

* **Result for PS channel (****c**)*

***c* (**p ** **0**) also disappears
- between 1.62*T**c* and 1.70*T**c*

*A*(**)  ***2***(**)

##  Statistical Significance Analysis for J/

- Statistical Significance Analysis = Statistical Error Putting

*T* = 1.62*T*c

*T* = 1.70*T**c*

##  Statistical Significance Analysis for c

- Statistical Significance Analysis = Statistical Error Putting

*T* = 1.62*T**c*

*T* = 1.70*T**c*

##  Dependence on Data Point Number (1)

* **Dependence on Data Point Number (1)*

*N***= 46 (*T* = 1.62*T**c*)
- V channel (*J*/**)

- Data Point # Dependence Analysis = Systematic Error Estimate

##  Dependence on Data Point Number (2)

* **Dependence on Data Point Number (2)*

*N***= 40 (*T* = 1.87*T**c*)
- V channel (*J*/**)

- Data Point # Dependence Analysis = Systematic Error Estimate

##  Summary and Perspectives (1)

-  Spectral Functions in QGP Phase were obtained
-    for heavy quark systems at **p **= **0** on *large* lattices at several *T*

-  Both *Statistical and Systematic Error Estimates*   have been carefully carried out

##  Summary and Perspectives (2) 

*Further study needed**for better understanding of QGP *
*and Hadronic Modes in QGP !*