1. TopologyT. Onogi1 Should we change the topology at all? Tetsuya Onogi (YITP, Kyoto Univ.) for JLQCD collaboration RBRC Workshp: “Domain Wall Fermions at Ten Years” March 16 2007 at BNL 1. 1.Topology in unquenched simulation 2. 2.QCD vacuum 3. 3. and Q dependence of the observables 4. 4.Topological susceptibility 5. 5.Summary

2. TopologyT. Onogi2 Members of Dynamical Overlap project JLQCD +TWQCD+.. KEK: S.Hashimoto, T.Kaneko, H.Matsufuru, J. Noaki, M.Okamoto, N.Yamada Riken: H.Fukaya Tsukuba: S.Aoki, N.Ishizuka, K.Kanaya, Y.Kuramashi, Y.Taniguchi, A.Ukawa, T.Yoshie Hiroshima: K.Ishikawa, M.Okawa Kyoto: T.O. Taiwan: T-W. Chiu, K.Ogawa,…

3. TopologyT. Onogi3 The aim of this talk Topology change in unquenched QCD is a serious problem. One should carefully think which strategy should be taken: Enforce the topology change ? or fix the topology? Review the theoretical understanding of QCD in vacuum and QCD at fixed topology. Claim that the fixed Q effect is a finite size effect, which can be removed in large volume or correctly estimated. Give a proposal to measure topological susceptibility at fixed topology. Talk by T-W. Chiu

4. TopologyT. Onogi4 1. Topology in unquenched simulation Topological charge evolution in HMC becomes slower towards weaker coupling, smaller sea quark mass, and better chirality Staggered: Bernard et al 2003 Domain wall: RBC, Antonio et al. 2006 Evolution of the topology in 2+1 dynamical domain-wall. Figiure from RBC hep-lat/0612005 Better chirality: Iwasaki < DBW2 < C (Plaq+Rect) Topology change: Iwasaki > DBW2 > C (Plaq+Rec)

5. TopologyT. Onogi5 Additional problem in dynamical overlap fermion: appearance of low eigenmodes of Fodor, Katz, Szabo; Cundy et al. ; deGrand, Shafer Reflection/Refraction Huge numerical cost Our strategy: Fix the topology to avoid the problem Low mode spectrum of Hw

6. TopologyT. Onogi6 How to extract physics from fixed topologies? Intuitively, fixing the topology should not affect physics for large enough volume for. But, finite volume effects should be estimated. If large instanton contributes the vacuum, the finite size effect may be large. Is local fluctuation of the topology sufficiently active without the topology change through dislocation? Measuring topological susceptibility is imporant.

7. TopologyT. Onogi7 2. QCD vacuum Witten’s picture Solution to U(1) problem leads to the picture that large Nc expansion gives a good approximation to theta dependence of QCD vacuum. E. Witten Nucl.Phys.B156(1979)269 Instanton (= classical (anti)self-dual configuration) contribtuion is NOT a major component. Local fluctuation of density is the dominant contribution the vacuum. E. Witten Nucl.Phys. B149(1979) Good news suggesting small finite size effect.

8. TopologyT. Onogi8 Is there a numerical test of this picture ? Yes, studies of local chirality of the low mode of Dov. But still controvertial. DeGrand, Hasenfratz : instanton contribution Horvath et al. : no major instanton contribution However, it seems that local topological charge density fluctuation dominates. Correlation length of topological charge is small. Finite size effect can be small. The vacuum can consist of huge number of relatively independent topological lumps of positive and negative charge.

9. TopologyT. Onogi9 Local chirality of low modes of Dov for Nf=2 JLQCD nf=2 results for local chirality with The distribution has peaks at maximum positive and negative chirality. There are local topological lumps even for Q=0.

10. TopologyT. Onogi10 Chiral Lagrangian for vacuum Dashen’s phenomena

11. TopologyT. Onogi11 3.and Q dependence of the observables Brower, Chandrasekaran, Negele, Wiese, Phys.Lett.B560(2003)64 +discussions with S.Aoki, H. Fukaya and S. Hashimoto : partition function in vacuum : partition function at fixed Q : observable in vacuum : observable at fixed Q The partition function and observable at fixed Q can be obtained from those in vacuum using saddle point approximation for large V (volume)

12. TopologyT. Onogi12 Saddle point approximation Parameterize the vacuum energy as Then, the partition function at fixed topology is Changing variables as If is satisfied

13. TopologyT. Onogi13 Parameterizing the vacuum energy as one obtains “(n)” means n-th derivative in Difference of observables with fixed Q and in vacuum can be estimated as 1/V correction and higher order. Topological susceptibility as well as higher moments are the key quantities. One can also obtain the dependence of CP-odd observable. EDM can be obtained.

14. TopologyT. Onogi14 Example: Q dependence of the meson mass dependence estimated from ChPT The correction from fixing the topology is 3%-1% for with (2fm)^4

15. TopologyT. Onogi15 Other hadrons (nucleon, ) ChPT prediction or Q dependent correction only comes through as subleading corrections. If pion mass is under control, other hadronic quantities are safe.

16. TopologyT. Onogi16 4. Topological susceptibility Measure the topological susceptibility –check thermal equilibrium in topology –Useful for estimate the finite size effects Definitions – Giusti, Rossi, Testa Phys.Lett.B587(2004)157 disconnected loop – Luescher, Phys.Lett.B(2004)296 n-point function without div. – Asymptotic value Fukaya, T.O. Phys.Rev.D70(2004)054508

17. TopologyT. Onogi17 (1) Ward-Takahashi identity (2) Cluster Property Q distribution (1)&(2) Topological suscpetibility can be measured indirectly from asymptopic values of Pseudoscalar 2-pt ftn Intuitive proof

18. TopologyT. Onogi18 More sytematic proof Aoki, Fukaya, Hashimoto, Onogi in progress Consider the topological charge density correlator. Using formula where Using the clustering property

19. TopologyT. Onogi19 One can use arbitrary function to define the topological charge density up to total divergence. Examples

20. TopologyT. Onogi20 Schwinger model case (fixed topology simulation) There is indeed a nonzero constant for This constant gives topological susceptibility consisitent with direct measurement

21. TopologyT. Onogi21 5. Summary The effect of fixing the topology is a finite size effect, which can be removed in large volume or correctly estimated by the topological susceptibility and suitable effective theory. Fortunately, the pion mass receives the largest correction but other quantities receives only subleading corrections through pion mass. The theta dependence of CP-odd observable can also be extracted from fixed topology simulation. Topological susceptibility can be measured by the asymptotic values of single pseudoscalar 2pt function at fixed topology. Talk by T-W. Chiu Systematic study of next-leading order (partially quenched ) ChPT is needed..

22. TopologyT. Onogi22 Back up slides

23. TopologyT. Onogi23

24. TopologyT. Onogi24 Low mode and topology change Zeros of Hw(m) arise when the topology changes through localized modes. Edwards, Heller, Narayanan Nucl.Phys.B535(1998)403. Spectral flow of Hw Localization size of the crossing mode