Lattice QCD in China

Jianbo ZhangDepartment of Physics, Zhejiang University

Sept. 20, 2006

Outline

1. China Lattice QCD Collaboration

2. Selected Topics Charmonium Spectrum

Pion-Pion Scattering Phase Shift

3. Future Perspectives

1. China Lattice QCD Collaboration

An Introduction

• Members Faculty Members: Ying Chen Institute of High Energy Physics, CAS

Chuan Liu Peking UniversityYubin Liu Nankai University

Xiang-Qian Luo Zhongshan University Jian-Ping Ma Institute of Theoretical Physics, CAS

Jianbo Zhang Zhejiang University

Graduate Students (not complete):Ming Gong (PKU), Xin Li (PKU), Ji-Yuan Liu (PKU),

Xiang-Fei Meng (NKU), Gang Li (IHEP), Yuan-Jiang Zhang (IHEP) , etc.

Deepcomp 6800 Dawning 4000A NKStars

Speed 4.2Tflop

(Linpack)

10.2 Tflops(peak)

8 Tflops (Linpack)

4.7 Tflops (peak)

3.2 Tflops (Linpack)

Node Numbers 197 Comp. node

4 I/O, 1 Console

512 Comp. node

16 I/O, 4 Console

384 Compu. node

12 I/O, 4 Console

Processors 4 CPU/node

(1.3GHz Intel

Itanium, 8/16GB)

4 CPU/node

(2.4 GHz AMD Opereron, 8GB)

2 CPU/node

(IBM Xserver, 2GB)

Network Globus, MPI-G;

Oracle 10G

Myrinet 2000 Myrinet

Hard Disk 80TB 20TB 54TB

• Computers available

At present, Roughly 0.6-1.0 million CPUhours are allocated per year.

• Forthcoming Machines

Supercomputing Center of CAS (SCCAS)

A 100 Tflops new computer is planned and expected to be available in 2008.

Shanghai supercomputer Center (SSC)

A 100 Tflops new computer is expected to be available in 2008.

• Projects in progress

Charmonium spectrum ( excited states, hybrids, etc.)

Pion-pion scattering phase shift

Lattice QCD at finite temperature and finite density

2. Selected Topics

Charmonium Spectrum

• Motivation A series of heavy meson states of open-charm and closed-

charm have been observed recently.

X(3872) (most likely , but refuses to fit into the 2P state predictions of non-relativistic quark models ).

Y(4260) (likely a hybrid charmonium?)

Many model-dependent theoretical interpretation of the

newly observed meson states. ( Review: Swanson, Phys. Rep. 429 (2006) 243-305 )

1

1

• Lattice Calculations (Quenched)

Tadpole improved Symanzik’s gauge action. Tadpole improved Clover fermions action. Anisotropic lattices.

(fm) (fm) #config

2.4 0.222(1) 1.78 200

2.6 0.176(1) 2.11 200

2.8 0.139(1) 2.22 200

TL 3sa sLa

4083

64123

80163

• Lattice interpolation field operators

The operators are constructed by quark bilinear sandwiched with Gamma matrices and color fields.

When calculating the two-point functions, the disconnected diagrams are neglected by assuming the OZI suppression.

• Data analysis ---Sequential Empirical Bayes Method

(Y. Chen et al., hep-lat/0405001)

Bayes: constrained-curve fitting prior

Empirical: priors are derived from part of data

**(‘prior’ means the prior information of parameters)

Sequential: states fitted one by one from low to high.

tm

ii

ieWtC )(

An Example of SEB• Here is an example for the fitting

procedure in the vector channel (four-mass-term fit), where t_max =68, t_1=50, t_2=30, t_3=15

• Fit model

• The mass terms are fitted one by one from low to high

1. Choose the t_max. 2. Varying t_1 towards small value, one-

mass -term fitting in the interval [t_1,t_max], until the χ2/dof blows up ;

3. Add the second mass term at t_2=t_1-1. Varying t_2, two-mass-term fitting in the interval [t_2,t_max] with the first state constrained by the fitted parameters in Step 2.

4. When χ2/dof blows up again at the last t_2, and add the third state at t_3=t_2-1.

5. Repeat. 6. The latest state is generally taken as a

garbage can and therefore not a realistic state.

tm

ii

ieWtC )(

0++

Three-mass-term fitting procedure in 0++ channel

Red points are data from the simulation, the blue curve is the plot of fit model with fitted parameters.

)1(

)(ln

1

tC

tC

ameff

1++

Three-mass-term fitting procedure in 1++ channel

Red points are data from the simulation, the blue curve is the plot of fit model with fitted parameters.

1+-

Three-mass-term fitting procedure in 1+- channel

Red points are data from the simulation, the blue curve is the plot of fit model with fitted parameters.

• 1S, 2S and 1P, 2P, 3P states

This figure and table on next page illustrate the results from the finest lattice. The continuum extrapolation will be taken later

• 2P states and X(3872)

BGS represents the predictions of Swanson et al quark model. It is difficult to change quark model, as it can reproduce precisely the masses of almost all the known charmonium states (Swanson, hep-ph/0601110).

For 2P states, earlier (quenched) lattice QCD predictions (CP-PACS and Chen) of their masses are roughly 100 MeV larger than QM prediction. This may be attributed to their two-mass-term fitting where the contamination of higher states to the first excited states cannot be neglected.

Our result for 2P(1++) is consistent with X(3872) in mass.

• Continuum limit extrapolation performed.

• It is smaller than the experiment value.• The result is in

agreement with previous (quenched) works

• Hyperfine splitting )()/( cMJMM

MeVM )3(83

• hybrids (with exotic quantum numbers)cgc

•These results are in agreement with previous quenched lattice QCD results.

GeVM )7(30.4)1(

GeVM )17(67.4)0(

GeVM )15(88.5)0(

• Non-exotic hybrids and conventional charmoniumcgc

• Masses from the four-mass-term SEB fitting of hybrid-hybrid (HH) and meson- meson (MM) correlation functions in 1-- and 0-+ channels. • It is understandable that the masses of the ground states are almost the same and the masses of the first excited states are consistent with each other, because the operators with the same quantum numbers can overlap to the same hadron states.• The masses of the second excited state of HH are very different from those of MM• No convincing results of masses of non-exotic hybrids can be derived in our work.

• A summary to the charmonium spectrum study

1. With SEB, the masses of the first excited states (even the second excited states in some channel) can be reliably derived from charmonium two-point functions.

2. The masses of 2S charmonium states agrees well with experimental data.

3. The masses of 2P charmonium states obtained in this work are 3.798( 70), 3.827(50), and 3.799(60) for 0++, 1++ and 1+- states, respectively. Given 1++ for X(3872), 2P(1++) is consistent with X(3872) in mass.

4. Masses of hybrid charmonia with exotic quantum numbers can be derived more soundly, since there are no admixtures of conventional charmonia. However for hybrid charmonia with no-exotic quantum numbers, it still a tough task to separate them from conventional charmonia unambiguously in the present lattice study.

5. Specifically, we have not observed a clear hybrid states with mass around 4260 MeV in the vector channel.

Pion-Pion Scattering Phase Shift• The study of hadron-hadron scattering is helpful for understanding the low-energy

structure of QCD.

• Luscher’s formula relates the pion-pion phase shift in the continuum to the energy of two-pion system in a finite cubic box,

);1()(tan

200

2/3

qZ

qk

);1( 200 qZ

222 kmE kL

q2

where is modified zeta-function and can be numerically calculated. q and k are related to the two-pion

energy by and

• at different k can be calculated directly on lattice. )(kE

• We work on the spatially asymmetric lattices , so that there are more non-degenerate non-zero three momentum modes.

. Luscher’s formula in a finite cubic box needs to be modified. For lattice of spatial size (η1Lxη2LxL), with η1 and η2 equal or great than 1

. Z00(1, q2,η1,η2) represents the modified zeta function for the asymmetric box, and can be calculated numerically.

3/ 21 2

200 1 2

tan ( )(1; , , )

qk

Z q

1 2( )L L L

1 2( )L L L

1(1 2( )L L L

Here is the preliminary results obtained at two beta values. More work is in progress

3. Future Perspectives

• Numerical study of lattice QCD with dynamical fermions.

• Lattice QCD at finite temperature and finite density.

How far we can go depends on the computing resource available.

• A way out ---- International collaboration

Thank You!

谢谢 !