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Nf=2 lattice QCD & Random Matrix Theory in the ε-regime. Hidenori Fukaya (Riken Wako) for JLQCD collaboration HF et al, [JLQCD collaboration], hep-lat/0702003 (accepted by Phisical Review Letters ). 1. Introduction. JLQCD’s overlap fermion project (->Noaki’s talk)
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Nf=2 lattice QCD & Random Matrix Theory in the ε-regime Hidenori Fukaya (Riken Wako) for JLQCD collaboration HF et al, [JLQCD collaboration], hep-lat/0702003 (accepted by Phisical Review Letters )
1. Introduction • JLQCD’s overlap fermion project (->Noaki’s talk) On a 163 32 lattice with a ~ 1.6-1.8GeV (L ~ 1.8-2fm), we have achieved 2-flavor QCD simulations with the overlap quarks with the quark mass down to ~3MeV. NOTE m >50MeV with non-chiral fermion in previous JLQCD works. • Iwasaki (beta=2.3) + Stop(μ=0.2) gauge action • Overlap operator in Zolotarev expression • Quark masses ： ma=0.002(3MeV) – 0.1. • 1 samples per 10 trj of Hybrid Monte Carlo algorithm. • 5000 trj for each m are performed. • Q=0 topological sector (No topology change.)
1. Introduction • Systematic error from finite V and fixed Q Our test run on (~2fm)4 lattice is limited to a fixed topological sector (Q=0). Any observable is different from θ=0 results; Brower et al, Phys.Lett.B560(2003)64 where χ is topological susceptibility and f is an unknown function of Q. ⇒needs careful treatment of finite V and fixed Q . • Q=2, 4 runs are started. • 24348 (~3fm)4 lattice or larger are planned. • Check of ergodicity in fixed topological sector.
1. Introduction • Effective theory with finite V and fixed Q Due to the large mass gap between mπ and the other hadron masses, the pion should be most responsible for the finite V or Q effects. ⇒finite V and Q effects can be evaluated in pion effective theory ( ChPT or ChRMT) Examples where ⇒precise measurement of Σ, Fpi is important. Gasser & Leutsyler, 1987, Hansen, 1990, 1991, Damgaard et al, 2002, ……
1. Introduction • Dirac spectrum and ChRMT In particular, in the ε-regime, when m~0, s.t. chiral Random Matrix Theory (ChRMT) is helpful to evaluate the finite V scaling of the Dirac eigen spectrum; ChRMT ⇔ low-mode Dirac spectrum • Controlled by • Or by with chemical potential. ⇒precise measurement of Σ, Fπ and V effects… Shuryak & Verbaarschot, 1993, Damgaard & Nishigaki, 2001, Akemann, Damgaard, Osborn, Splitorff, 2006, etc.
2. QCD → RMT → ChPT Consider the QCD partition function at a fixed topology Q, • Weak coupling (λ >> ΛQCD) • Strong coupling (λ<<ΛQCD) ⇒ An assumption: for the low-modes with an unknown function V ⇒ ChRMT.
2. QCD → RMT → ChPT From the universality and symmetry of RMT, QCD should have the same low-mode spectrum with chiral unitary gaussian ensemble, up to overall factor In fact, • SU(Nf)*SU(Nf) -> SU(Nf) SSB. • Randomness -> kinetic term neglected. RMT predicts Dirac low-modes -> pion zero-mode !
3. Numerical results • Dirac spectrum and analytic prediction of ChRMT Nf=2 （m=3MeV) results • Lowest eigenvalue ⇒Σ=(251(7)(11)MeV)3 • Direct evidence of chiral SSB of QCD !! • Σ obtained without “chiral extrapolation”
3. Numerical results • Dirac spectrum with imaginary isospin chemical potential (preliminary) 2-point correlation function The eigenvalues of is predicted by Ch2-RMT. • Fπ ~ 70 MeV. See Akemann, Damgaard, Osborn, Splitorff, hep-th/0609059 for the details.
4. Summary and discussion The chiral limit is within our reach now! • On (~2fm)4 lattice, JLQCD have simulated Nf=2 dynamical overlap quarks with m~3MeV. • Finite V and Q dependences are important. • ChPT and ChRMT are helpful to estimate finite V and Q effects. • Comparing QCD in the ε-regime with RMT, • Direct evidence of chiral SSB from 1st principle. • ChRMT in the ε-regime ⇒Σ~(250 MeV)3. • Ch2-RMT in the ε-regime ⇒ Fπ~ 70MeV.
4. Summary and discussion To do • Precise measurement of hadron spectrum, started. • 2+1 flavor, started. • Different Q, started. • Larger lattices, prepared. • BK , started. • Non-perturbative renormalization, almost done. Future works • θ-vacuum • ρ→ππ decay • Finite temperature…
Fpi M2/m 3. JLQCD’s overlap fermion project • Numerical result (Preliminary) Both data confirm the exact chiral symmetry.
How to sum up the different topological sectors • Formally, • With an assumption, The ratio can be given by the topological susceptibility, if it has small Q and V’ dependences. • Parallel tempering + Fodor method may also be useful. V’ Z.Fodor et al. hep-lat/0510117
Initial configuration For topologically non-trivial initial configuration, we use a discretized version of instanton solution on 4D torus; which gives constant field strength with arbitrary Q. A.Gonzalez-Arroyo,hep-th/9807108, M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)
Topology dependence • If , any observable at a fixed topology in general theory (with θvacuum) can be written as Brower et al, Phys.Lett.B560(2003)64 • In QCD, ⇒ Unless ,（like NEDM） Q effects = V effects. Shintani et al,Phys.Rev.D72:014504,2005
