1. Chiral Magnetic Effect on the Lattice Seminar @ Komaba, June 13, 2012 Arata Yamamoto (RIKEN) AY, Phys. Rev. Lett. 107, 031601 (2011) AY, Phys. Rev. D 84, 114504 (2011) AY, Lect. Notes of Phys., in press

2. Chiral Magnetic Effect [D.E.Kharzeev, L.D.McLerran, H.J.Warringa (2007)] Early Universe [from NASA’s web page][from BNL’s web page] heavy-ion collision (RHIC&LHC) [from KEK’s web page] chiral magnetic effect: charge separation induced by a strong magnetic field via the axial anomaly, i.e., nontrivial topology

3. cf.) permanent magnet ~ 10 2 eV 2 magnetar ~ 10 MeV 2 magnetic field ~ 10 4 MeV 2 non-central collision of heavy ions beam

4. magnetic field electric current If L = R, the net current is zero. If L R, the net current is nonzero.

5. the index theorem: Globally, Locally, topological fluctuation in lattice QCD [from D.Leinweber’s web page]

6. topological fluctuation beam magnetic field beam “event-by-event” charge separation electric current

7. [STAR Collaboration (2009)(2010)] Experiments Some asymmetry was observed, but what is it? charged-particle correlation in RHIC & LHC magnetic field reaction plane emission

8. [K.Fukushima, D.E.Kharzeev, H.J.Warringa (2008)] Chiral chemical potential produces a chirally imbalanced matter. right-handed Fermi sea left-handed Fermi sea Chiral Chemical Potential

9. magnetic field electric current positive helicitynegative helicity

10. [K.Fukushima, D.E.Kharzeev, H.J.Warringa (2008)] the Dirac equation coupled with a background magnetic field Induced current magnetic field electric current induced electric current

11. “sign problem” In lattice QCD at finite density, For small chemical potential, reweighting, Taylor expansion, canonical ensemble, imaginary chemical potential, density of states, … two-color QCD, isospin chemical potential, chiral chemical potential For large chemical potential, Sign problem

12. Wilson-Dirac operator NO sign problem !!

13. continuum QCD: discretization uncountable infinite functional integral countable infinite (finite) multiple integral Lattice simulation is powerful in nonperturbative QCD !! lattice QCD: Lattice QCD Simulation

14. magnetic field vector current L R magnetic field Q 0 + - Chiral magnetic effect in lattice QCD topological charge: chiral chemical potential: by A.Y.by Connecticut and ITEP

15. 2+1 flavor QCD+QED with the domain-wall fermion [M. Abramczyk, T. Blum, G. Petropoulos, R. Zhou (2009)] Lattice QCD with a fixed-topology

16. SU(2) quenched QCD with the overlap fermion [P.V.Buividovich, M.N.Chernodub, E.V.Luschevskaya, M.I.Polikarpov (2009)] Lattice QCD with a background topology

17. Why can we obtain nonzero current? Lattice QCD at : Q=2 gauge configuration [M.Garcia Perez, A.Gonzalez Arroyo, A.Montero, P.van Baal (1999)]

18. the Wilson gauge action + the Wilson fermion action flavor: lattice size: lattice spacing: fm pion/rho-meson mass: deconfinement phase Lattice QCD with a chiral chemical potential

19. Chiral charge density

20. Induced current

22. [K.Fukushima, D.E.Kharzeev, H.J.Warringa (2008)] by fitting the lattice data from the Dirac equation Induced current lattice artifacts e.g. dielectric correction [K.Fukushima, M.Ruggieri (2010)] e.g. renormalization physical effects

23. Systematic Analysis quenched QCD simulation lattice spacing dependence volume dependence quark mass dependence of

24. Renormalization renormalization factor: cf.) nonperturbative renormalization [L.Maiani, G.Martinelli (1986)] The local vector current is renormalization-group variant on the lattice. discretization artifact: In the continuum limit,

25. Lattice spacing The induced current depends on the lattice spacing.

26. Spatial volumeQuark mass Independent of volume, quark mass, and temperature chiral limit

27. P and its susceptibility is independent of the spatial volume. crossover confinement deconfinement Phase Diagram

28. crossover 1.0 ? isospin chemical potential [J.B.Kogut, D.K.Sinclair (2004)] For a first-order transition, confinement deconfinement

29. Summary We have performed a lattice QCD simulation with the chiral chemical potential. By applying an external magnetic field, we have obtained the induced current by the chiral magnetic effect. The continuum extrapolation is quantitatively important. chiral symmetry ?