# Presentation: 683029

## Lattice QCD calculations of  Tc and Equation of State 

**Rajan Gupta**
**T-8**
** ****Los Alamos National Laboratory**

## Heavy quarks flow in a medium defined by gluons and light quarks. Quantities needed to model the hydrodynamic flow of QGP are

**T****c**
**Order of the transition**
**EOS (pressure and energy density)**

## What we know

**The underlying microscopic theory of hadronic interactions is Quantum Chromodynamics (QCD)**
**Matter consists of a plasma of quarks and gluons in the central collision region **
**QCD is strongly interacting at T = 200-500 MeV ****(T < gT < g****2****T)**
**Phase diagram – Qualitative. Need non-perturbative calculations for T****c****, EOS and transport coefficients**

## The QCD phase diagram

- B

- QGP

- Nuclei

- Neutron stars

- T

- Hadronic
- matter

- Critical endpoint ?

- 1st order line ?

- Quark-
- Gluon
- Plasma

- Tc

## Gauge action on a lattice

***S    =  *****action for the gluons**
***U******xy***** =  the 3 space-space plaquettes**
***U******xt***** =  the 3 space-time   plaquettes**
******  ****  ****=  6/g****2**** (gauge coupling on the lattice) **
**a****s,t**** =  lattice spacing in space, time direction**

***a***

## Lattice QCD at Finite Temperature 

- Nt= # of points in time direction		Ns= # of points in space direction
- at = lattice spacing in time direction	as = lattice spacing in space direction

## Parameters we tune

**Lattice action (to reduce discretization  errors and improve chiral symmetry)**
**Values of light (up and down) quark masses**
**N********  ****(continuum limit: N************, a********0, aN******** ****fixed)**
**Lattice asymmetry a****s****/a******
**Number of ******** (=T) values for each N******** ****and m****q**
**Monte Carlo Update algorithm**

## Order of the transitions and Tc

- Look for discontinuities, singularities, peaks in thermodynamics quantities
- Polyakov line and its susceptibility
- Chiral condensate           and its susceptibility
- Quark number susceptibility
- Finite size scaling analysis

**For 2+1 light quark flavors the **
**transition is a seen to be a crossover**

## A ~20MeV uncertainty in Tc

## Hydro dynamical analysis gives Tfreeze-out

## Fodor etal: different observables give different Tc

## EOS from Lattice QCD

- Z = Partition function		V = spatial volume  = (Ns a)3
-  = energy density		p  = pressure density

## Equation of State (NT=4, 2000)

- Latent heat

- One can test many kinds of EoS in hydrodynamics.

- Lattice QCD predicted a cross over in the vicinity of Tc ~ 170 MeV with rapid increase in energy density

- Lattice QCD simulations

- Typical EoS in hydro model

- H: resonance gas(RG)

- p=e/3

- Q: QGP+RG

- F.Karsch *et al.* (’00)

- P.Kolb and U.Heinz(’03)

## Issues with previous calculations

**Calculations with N****T****=4 are on very coarse lattices (a ~ 0.2 fermi). **
**Simulate  N****T****=4,6,8, ... and do a********0 extrapolation**
**T****c**** with different actions differ by ~ 20 MeV**
**Reduce uncertainty to ~ 5 MeV**** **
**Need simulations with 2+1 flavors with realistic light (up, down) and strange quark masses**
- Mlight / ms = 0.1

## Goal: Reliable continuum limit results

**Reduce discretization errors (compare 3 different improved actions)**
**Asqtad Staggered ****(MILC collaboration)**
**p4 staggered  ****(RBC-Bielefeld collaboration)**
**Domain wall fermions**
**New simulations at N****T****=8**** ****with 2+1 flavors (m****light**** ******** 0.1 m****s****)**
**Combine with ongoing/previous N****T ****= 4,6 simulations to perform a********0 extrapolation**

## Timeline

**Develop collaboration with Ron Soltz and formulate project for *****Blue Gene L***** at LLNL (Nov-Dec 2005) **
**Organized US wide meeting to explore doing RHIC phenomenology and Lattice QCD at LLNL (Feb 06)**
** ****Develop white paper for calculation of T****c**** and EOS on the *****Blue Gene L***** (Feb – May 06)**
**Get NNSA approval to run on *****Blue Gene L***** ( July 06)**
**Port Asqtad and p4fat codes to *****Blue Gene L *****(Aug-Sept 06)**
**Start N****T****=8 lattice QCD simulations Oct 06**
**Optimized code and started production runs (~30% done)**

*Blue Gene L* at LLNL represents a 4-6 Teraflop sustained resource

## P4 action N8 simulations: Status

| β | a mq | T
[MeV] | # iter | T = 0 lattices: size ( # iter) 
RBC-Bielefeld |
| --- | --- | --- | --- | --- |
| 3.460
3.490
3.510
3.540
3.570
3.600
3.630
3.660
3.690
3.760 | 0.00313
0.00290
0.00259
0.00240
0.00212
0.00192
0.00170
0.00170
0.00150
0.00139 | 154
169
179
194
209
225
241
256
271
313 | 4360
5790
7270
7440
7720
10000
10000
10000
10000
10000 | 163 X 32 (2500)
163 X 32 (4300)
163 X 32 (2300)
163 X 32 (4000)
243 X 32 (2400)
323 X 32 (1160), interpol. used
243 X 32 (3200)
323 X 32 (1100), interpol. used
243 X 32 (2300)
323 X 32 (1120), interpol. used |

- P4 action N8 simulations: Status

- Lattice Size 8323232
- Quark mass ml/ms = 0.1
- (real world ~0.04)

**Line of constant physics (LCP)**
**M****ss**** r****0**** = 1.58**
**M******** r****0****  ******** 0.52 ******** M******** ******** 220 MeV**

- Calculate Action, Polyakov loop, condensate & their susceptibilities

## Susceptibilities:  N = 4, 6, 8

- Susceptibilities:  N = 4, 6, 8

- Susceptibility for light (*l* = u,d) and strange (s) quarks:

## Slide 19

## Asqtad action data set as of 4/20/07

| ═6/g2 | t | Acceptance  | s exp(-s/2)>103 |  | # trajectories |
| --- | --- | --- | --- | --- | --- |
| 6.4580 | 0.01430 | 0.77 |   -13.0 +/- |   14.0 | 4245 |
| 6.5000 | 0.01430 | 0.79 | -2.6 | 11.0 | 6390 |
| 6.5500 | 0.01430 | 0.79 | -4.3 | 8.2 | 6210 |
| 6.6000 | 0.01430 | 0.80 | 0.6 | 7.6 | 6210 |
| 6.6500 | 0.01430 | 0.81 | 0.9 | 7.4 | 5915 |
| 6.7000 | 0.01250 | 0.84 | 18.0 | 7.7 | 5015 |
| 6.7600 | 0.01250 | 0.85 | 1.4 | 7.1 | 5595 |
| 6.8000 | 0.01250 | 0.87 | -5.1 | 5.7 | 5675 |
| 6.8500 | 0.01250 | 0.87 | 2.3 | 4.8 | 5645 |
| 6.9000 | 0.01250 | 0.87 | 9.6 | 4.9 | 5455 |
| 6.9500 | 0.01250 | 0.88 | -3.2 | 6.2 | 5795 |
| 7.0000 | 0.01250 | 0.89 | 4.0 | 4.0 | 6020 |
| 7.0800 | 0.01250 | 0.89 | -1.7 | 5.1 | 5750 |

**About 33% of target 15000 trajectories (statistics) achieved**

## Slide 21

## Future Timeline

**Present results on T****c**** and EOS at Lattice 2007 (July 31-Aug 4)**
**First Publication of T****c**** ~August 2007**
**Simulate T=0 as needed for EOS**
**Publish EOS results ~ Jan 2008**
**Plan for the calculation of spectral functions**
**Develop non-perturbative methods for calculating renormalization constants for transport coefficients **

## Extras

## Space-time picture of a R.H.I.C.

- Equilibration

- Hadronization

- Thermal freeze-out