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

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

cf.) permanent magnet ~ 102 eV2 magnetar ~ 10 MeV2

magnetic field ~ 104 MeV2

non-central collision of heavy ions

beam

beam

magnetic field

electric current electric current

If L = R, the net current is zero.If L R, the net current is nonzero.

the index theorem:

Globally,

Locally,

topological fluctuation in lattice QCD [from D.Leinweber’s web

page]

topological fluctuation

beam

magnetic field

beam

“event-by-event” charge separation

electric current

[STAR Collaboration (2009)(2010)]

Experiments

Some asymmetry was observed, but what is it?

charged-particle correlation in RHIC & LHC

magnetic field

reactio

n

plane

emiss

ion

[K.Fukushima, D.E.Kharzeev, H.J.Warringa (2008)]

Chiral chemical potential produces a chirally imbalanced matter.

right-handedFermi sea

left-handedFermi sea

Chiral Chemical Potential

magnetic field

electric current

positive helicity

negative helicity

[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

“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

Wilson-Dirac operator

NO sign problem !!

continuum QCD:

discretization

uncountable infinitefunctional integral

countable infinite (finite)multiple integral

Lattice simulation is powerful in nonperturbative QCD !!

lattice QCD:

Lattice QCD Simulation

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

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

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

Why can we obtain nonzero current?

Lattice QCD at :

Lattice QCD at :

Q=2 gauge configuration[M.Garcia Perez, A.Gonzalez Arroyo,

A.Montero, P.van Baal (1999)]

• 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

Chiral charge density

Induced current

Induced current

[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

Systematic Analysisquenched QCD simulation

lattice spacing dependencevolume dependencequark mass dependence

of

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 ,

Lattice spacing

The induced current depends on the lattice spacing.

Spatial volume Quark mass

Independent of volume, quark mass, and temperature

chiral limit

P and its susceptibility is independent of the spatial volume.

crossover

confinement

deconfinement

Phase Diagram

crossover

1.0

?

isospin chemical potential[J.B.Kogut, D.K.Sinclair (2004)]

For a first-order transition,

confinement

deconfinement

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 ?