Second Berkeley School on Collective Dynamics, May 21-25, 2007 Tetsuo 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 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-guides L -1 0 (thermodynamics limit) : finite size scaling a 0 (continuum limit) : asymptotic freedom m 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 techniques
76763131 5 0.05 Example of improvement: Number of floating-point operations To collect 100 config. on 2LxL 3 lattice with DDHMC algorithm: 1 year = 3 x 10 7 sec HNC DDHMC Del debbio, Giusti, Luscher, Petronzio, Tantalo, hep-lat/
To collect 1000 indep. gauge conf. on 24 3 x40, a=0.08 fm lattice (T=0) Clark, hep-lat/
QCD FNAL Tsukuba RBRC-Columbia Rome KEK
time space r M ∞ E 0 = 2M + V(r) Heavy quark potential time space M = finite E 0 = ground state mass Meson mass Typical measurement of mass : QQ pair
Examples in quenched QCD R 0.5 fm1.0 fm Linear confining string Bali, Phys. Rep. 343 (’01) 1 Charmoniums CP-PACS, Phys. Rev. D65 (’02) S+1 L J
Examples in full QCD string breaking N f = 2, Wilson sea-quarks, 24 3 x40 a= fm, L= 2 fm, m p /m r = SESAM Coll., Phys.Rev.D71 (2005) fm0.5fm 1.5fm [ V(r) - 2m HL ] a Charmoniums MILC Coll., PoS (LAT2005) 203 [hep- lat/ ] N f = 2+1, staggered sea-quarks, 16 3 x48, 20 3 x64, 28 3 x96 a = 0.18, 0.12, 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/ (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 force nucleus
Modern Nuclear Force from NN scatt. data One-pion exchange by Yukawa (1935) repulsive core Repulsive core by Jastrow (1950,1951) ... Multi-pions & heavy mesons
Machleidt and Entem, nucl-th/ High precision NN potentials
2.Maximum mass of neutron stars CAS A remnant Nuclear force Nuclear 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 ) ρ 0 = 0.16 fm -3 3ρ03ρ0 5ρ05ρ0 Akmal, Pandharipande & Ravenhall, PRC58 (’98) State-of-the-art nuclear EoS E/A (MeV) Nuclear Equation of State
Mass-Radius relation of neutron star in Akmal-Pandharipande-Ravenhall EoS PSR Neutron star binary Vela-X1Cyg-X2 X-ray binaries J Neutron star - WD binary EXO (X-ray bursts) (ρ max ～ 6ρ 0 )
How to extract (bare) NN force in QCD ? unrealistic fundamental difficulty (i) Born-Oppenheimer potential r Takahashi, Doi & Suganuma, hep-lat/ (ii) NN “wave function” NN potential Ishii, Aoki & Hatsuda, hep-lat/ similar in spirit with phenomenological potentials (phase shift data NN potential)
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 E p /E th LS equation : Ishii, Aoki & Hatsuda, hep-lat/ ＋ paper in preparation
time space r M ∞ E 0 = 2M + V(r) Heavy quark potential time space M = finite E 0 = ground state mass Meson mass Typical measurement of mass : QQ pair
Measurement of (r) (s-wave) time space x y J y J y + all possible combinations NN potential:
a = fm L = 4.4 fm KEK Simulation details 32 4 lattice Quenched QCD Plaquette gauge action Wilson fermion Periodic (Dirichlet) B.C. for spatial (temporal) direction m (GeV) N conf as of today m = 0.89 GeV m N = 1.34 GeV m = 0.84 GeV m N = 1.18 GeV
BS amplitude (r) for m =0.53 GeV 2s+1 L J Ishii, Aoki & Hatsuda, hep-lat/
Yukawa tail mid-range attraction repulsive core 1 S 0 channel 3 S 1 channel NN central potential V c (r) for m =0.53 GeV 2s+1 L J Ishii, Aoki & Hatsuda, hep-lat/
1 S 0 channel 3 S 1 channel NN central potential V c (r) for m =0.53 GeV 2s+1 L J Ishii, Aoki & Hatsuda, hep-lat/
Pion exchange attraction for 1 S 0 & 3 S 1 + ghost exchange (quenched artifact) attraction for 1 S 0 repulsion for 3 S 1 Beane & Savage, PLB535 (2002)
Quark mass dependence (preliminary) Ishii, Aoki & Hatsuda, in preparation
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 3 S D 1
N Z LQCD GFMC AMD MCSM Nuclear chart Nuclear force : bridge between one and many