Second Berkeley School on Collective Dynamics, May 21-25, 2007Tetsuo Hatsuda, Univ. Tokyo

PHYSICS is FUN 　 LATTICE is FUN 　

[1] Lattice QCD basics　　 [2] Nuclear force on the lattice ( dense QCD)

[3] In-medium hadrons on the lattice ( hot QCD)

[4] Summary

I

II

Why lattice ?

• well defined QM (finite a and L) • gauge invariant • fully non-perturbative

• hadron mass, coupling, form factor etc • scattering (phase shift, potential etc) • hot plasma 　

What one can do

• cold plasma • far from equilibrium system

What one cannot do (at present) quarks q(n) on the sites gluons U(n)

on the links

Lattice QCD Basics

QCD partition function QCD partition function

1/T

a

L

• Zero temperature : 1/T ~ L• Finite temperature : 1/T << L

quenched QCD : det F=1 (exploratory studies)full QCD : det F≠1 (precision studies)

n n+

n++ n+

Wilson gauge action

plaquette

link variable

Important limits and theory-guidesImportant limits and theory-guides

L-1 0 (thermodynamics limit) : finite size scalinga 0 (continuum limit) : asymptotic freedomm 0 (chiral limit) : chiral pert. theory

L-1

a

m

Improved actions: asqtad, 　 p4, stout, clover … different way of reducing the discretization error

Fermions: staggered, Wilson, Domain-wall, Overlap

different way of handling chiral symmetry

Modern algorithms: RHMC, DDHMC … techniques to make the simulations fast and reliable

Simulation techniquesSimulation techniques

76315 0.05

Example of improvement:

Number of floating-point operationsTo collect 100 config. on 2LxL3 lattice with DDHMC algorithm:

1 year = 3 x 107 sec

HNCDDHMC

Del debbio, Giusti, Luscher, Petronzio, Tantalo, hep-lat/0610059

To collect 1000 indep. gauge conf.on 243x40, a=0.08 fm lattice (T=0)

Clark, hep-lat/0610048.

QCD Cluster @ FNAL PACS-CS @ Tsukuba

QCDOC @ RBRC-ColumbiaApeNEXT @ Rome

BlueGene/L @ KEK

time

space

r

　　　　 M ∞

E0 = 2M + V(r)Heavy quark potential

time

space

　 M = finite

E0 = ground state mass Meson mass

Typical measurement of mass : QQ pair Typical measurement of mass : QQ pair

Examples in quenched QCDExamples in quenched QCD

R

0.5 fm 1.0 fm

Linear confining string

Bali, Phys. Rep. 343 (’01) 1

Charmoniums

CP-PACS, Phys. Rev. D65 (’02) 094508

2S+1LJ

Examples in full QCDExamples in full QCD

string breaking

Nf= 2, Wilson sea-quarks, 243x40a= 0.083 fm, L= 2 fm, mp/mr= 0.704

SESAM Coll., Phys.Rev.D71 (2005) 114513

1fm0.5fm 1.5fm

[ V

(r)

- 2m

HL ]

a

Charmoniums

MILC Coll., PoS (LAT2005) 203 [hep-lat/0510072]

Nf= 2+1, staggered sea-quarks, 163x48, 203x64, 283x96a = 0.18, 0.12, 0.086 fm, L= 2.8, 2.4, 2.4 fm

spin ave. 1S energy

• light hadron spectroscopy • heavy hadron spectroscopy• exotic hadrons• various “charges”• form factors• weak matrix elements• etc

Many applications

One of the latest developments

The nuclear force

Ishii, Aoki & Hatsuda,hep-lat/0611096 (to appear in Phys. Rev. Lett.)

Nuclear Force

• Why the nuclear force important now?• How to extract the nuclear force from QCD ?

H. Yukawa, “On the Interaction of Elementary Particles, I”, Proc. Phys. Math. Soc. Japan (1935)

H. Bethe, “What holds the Nucleus Together?”, Scientific American (1953)

F. Wilczek, “Hard-core revelations”, Nature (2007)

Nuclear forcenucleus

Modern Nuclear Force from NN scatt. dataModern Nuclear Force from NN scatt. data

One-pion exchange by Yukawa (1935)

repulsivecore

Repulsive core by Jastrow (1950,1951)

...

Multi-pions & heavy mesons

Machleidt and Entem, nucl-th/0503025

High precision NN potentialsHigh precision NN potentials

2. Maximum mass of neutron stars 　

CAS A remnant

Nuclear force

Nuclear repulsive coreNuclear repulsive core

Origin of RC is not known ….

But, it is intimately related to

１． Nuclear saturation 　

　 3. Ignition of Type II supernovae

Z=0

N=Z

ρ(fm-3)

ρ0

= 0.16 fm-3

3ρ0 5ρ0

Akmal, Pandharipande & Ravenhall, PRC58 (’98)

State-of-the-art nuclear EoS

E/A

(M

eV)

Nuclear Equation of State Nuclear Equation of State

Mass-Radius relation of neutron star

in Akmal-Pandharipande-Ravenhall EoS

Mass-Radius relation of neutron star

in Akmal-Pandharipande-Ravenhall EoS

PSR1913+16 Neutron starbinary

Vela-X1Cyg-X2 X-ray binaries

J0751+1807 Neutron star- WD binary

EXO0748-676(X-ray bursts)

(ρmax ～ 6ρ0)

How to extract (bare) NN force in QCD ? How to extract (bare) NN force in QCD ?

• unrealistic• fundamental difficulty

(i) Born-Oppenheimer potential r

Takahashi, Doi & Suganuma, hep-lat/0601006

(ii) NN “wave function” NN potentialIshii, Aoki & Hatsuda, hep-lat/0611096

similar in spirit with phenomenological potentials (phase shift data NN potential)

Equal time BS amplitude (r) Equal time BS amplitude (r)

Nucleon interpolating field:

Equal time BS amplitude:

Probability amplitude to find nucleonic three-quark cluster at point x 　　　　 and another nucleonic three-quark cluster at point y

cf: for π-πscattering,Lin, Martinelli, Sachradja & Testa, NP B169 (2001)CP-PACS Coll, Phys. Rev. D71 (2005)

+x

y

Local potential:

Non-local potential:

• asymptotic form of (r) (= the phase shift) determined by elastic pole interpolating operator independent

• inelastic contribution: interpolating operator dependent exponentially localized in space magnitude suppressed by Ep/Eth

LS equation :Ishii, Aoki & Hatsuda, hep-lat/0611096 　　　　　　 　　　　　＋ paper in preparation 　

time

space

r

　　　　 M ∞

E0 = 2M + V(r)Heavy quark potential

time

space

　 M = finite

E0 = ground state mass Meson mass

Typical measurement of mass : QQ pair Typical measurement of mass : QQ pair

Measurement of (r) (s-wave) Measurement of (r) (s-wave)

time

space

x

y

J y

J y

+ all possible combinations

NN potential:

a =

0.1

37 f

m

L = 4.4 fm

BlueGen

e/L @ KEK

Simulation details Simulation details

• 324 lattice• Quenched QCD• Plaquette gauge action• Wilson fermion• Periodic (Dirichlet) B.C. for spatial (temporal) direction

m(GeV) 0.37 0.53 0.73 0.99

Nconf 1093 1900 1000 1000

as of today

m = 0.89 GeV

mN= 1.34 GeV

m = 0.84 GeV

mN= 1.18 GeV

BS amplitude (r) for m=0.53 GeVBS amplitude (r) for m=0.53 GeV 2s+1LJ

Ishii, Aoki & Hatsuda, hep-lat/0611096

Yukawa tailmid-rangeattraction

repulsive core

1S0 channel

3S1 channel

NN central potential Vc(r) for m=0.53 GeV NN central potential Vc(r) for m=0.53 GeV 2s+1LJ

Ishii, Aoki & Hatsuda, hep-lat/0611096

1S0 channel

3S1 channel

NN central potential Vc(r) for m=0.53 GeV NN central potential Vc(r) for m=0.53 GeV 2s+1LJ

Ishii, Aoki & Hatsuda, hep-lat/0611096

Pion exchange Pion exchange

attraction for 1S0 & 3S1

+

ghost exchange (quenched artifact)ghost exchange (quenched artifact)

attraction for 1S0

repulsion for 3S1

Beane & Savage, PLB535 (2002)

Quark mass dependence (preliminary) Quark mass dependence (preliminary)

Ishii, Aoki & Hatsuda, in preparation

Remarks Remarks

4. Hyperons ? to be announced in two weeks (INPC2007)

3. Different Interpolating operators ? same phase shift but different V(r) at short distances

1. NN scattering length: fragile object in NN case

Luscher’s formula:

Luscher, CMP 105 (1986), NPB 354 (1991)

But situation is not that simple as “first Born” tells:

Born

2. Tensor force ? coupled channel 3S1-3D1

　　　　 N

　　

　　

Z

LQCD　

GFMCAMD

MCSM

Nuclear chart

Nuclear force : bridge between one and many Nuclear force : bridge between one and many