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Chiral symmetry breaking and Chiral Magnetic Effect in lattice gluodynamics with background magnetic field

P.V.Buividovich (ITEP, Moscow, Russia and JIPNR "Sosny" Minsk, Belarus), M.N.Chernodub (LMPT, Tours University, France and ITEP, Moscow), E.V.Luschevskaya (ITEP, Moscow, Russia), M.I.Polikarpov (ITEP, Moscow, Russia)

FOURTEENTH LOMONOSOV CONFERENCE ON ELEMENTARY PARTICLE PHYSICS Moscow, 19 25 August, 2009

I use a lot of slides made by M.N. Chernodub and P.V. Buividovich and some made by D.E. Kharzeev

Plan
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Strong magnetic (not chromo-magnetic!) fields Lattice simulations with magnetic fields Chiral symmetry breaking: Dirac eigenmodes and their localization Magnetization of the vacuum: first lattice results Chiral Magnetic Effect? - a new effect at RHIC? Conclusions

n

n n

Magnetic fields in non-central collisions

The medium is filled with electrically charged particles Large orbital momentum, perpendicular to the reaction plane Large magnetic field along the direction of the orbital momentum

Another estimation of magnetic fields, now in «eV»

Effects observed at very strong magnetic fields on the lattice
1. Enhancement of the chiral condensate
(arXiv:0812.1740)

2. Magnetization of the QCD vacuum
(arXiv:0906.0488)

3. Chiral Magnetic Effect (arXiv:0907.0494)
[signatures of "CP violation" at non-zero topological charge first discussed by Fukushima, Kharzeev, Warringa, McLerran '07-'08 + data from RHIC:
Conference "Quark Matter 2009, 30 March - 4 April" in Knoxville, USA]

Magnetic fields in our lattice simulations

1) We use the overlap Dirac operator on the lattice 2) We perform simulations in the quenched limit

We calculate

< >; = 1, ,

in the external magnetic field and in the presence of the vacuum gluon fields

H external magnetic field

3 2 1 0

T

Quenched vacuum, overlap Dirac operator, external magnetic field

2qk eB = 2 ; eB 250 Mev L

Free massless fermions in (2+1) dimensions
eB < >= - 2
V.P.Gusynin, V.A.Miransky, I.A.Shovkovy, hep-ph/9405262I Physics: strong magnetic fields reduce the system to D = (0 + 1) !!! Physics: magnetic field is the only dimensionful parameter!!!

Free massless fermions in (3+1) dimensions
eB < >= - 2 m ln 4 m
2 UV 2

V.P.Gusynin, V.A.Miransky, I.A.Shovkovy, hep-ph/9412257 Physics: strong magnetic fields reduce the system to D = (1 + 1) !!! Physics: the poles of fermionic propagator are at Landau levels

Chiral condensate in QCD

= - < >
1997]

[A. Gorsky, A. Zayakin 2008]

How to calculate the chiral condensate?

Evolution of the eigenmodes with increase of the magnetic field

Q=1

Q=0

Features: 1) The stronger the field the denser near-zero eigemodes 2) The exact zero eigenmode is insensitive to the field!

Chiral condensate vs. field strength

We are in agreement with the chiral perturbation theory: the chiral condensate is a linear function of the strength of the magnetic field!

Lattice results vs. chiral perturbation theory

Localization of Dirac Eigenmodes Typical densities of the chiral eigenmodes vs. the strength of the external magnetic field

Localization of Dirac Eigenmodes

Zero fields, B=0
no clear visible localization
[but in fact the localization exists on 2D surfaces]

Non-zero fields, B ~ a few GeV2
Localization appears at Lowest Landau Level

The magnetic susceptibility of the Yang-Mills vacuum

Magnetization of the vacuum as a function of the magnetic field

Magnetic susceptibility - results

= - cN c (8 2 f2 )

Magnetic susceptibility - results

< > = - 46(3) Mev our result < > -50 Mev QCD sum rules (I. I. Balitsky,1985, P. Ball, 2003.)

Chiral Magnetic Effect [Fukushima, Kharzeev, Warringa, McLerran '07-'08]
Electric current appears at regions 1. with non-zero topological charge density 2. exposed to external magnetic field

Experimentally observed at RHIC : charge asymmetry of produced particles at heavy ion collisions

Chiral Magnetic Effect by Fukushima, Kharzeev, Warringa, McLerran

Theoretically: electric dipole moment at regions 1. with non-zero topological charge density 2. exposed to external magnetic field. Experimentally: leads to charge asymmetry of produced particles at heavy ion collisions (currently at RHIC)

Red: momentum Blue: spin u-quark: q=+2/3 d-quark: q= - 1/3 Effect of topology: uL uR dL d R

Chiral Magnetic Effect on the lattice, qualitative picture

Density of the electric charge vs. magnetic field

Chiral Magnetic Effect on the lattice, qualitative picture

Non-zero field, subsequent time slices

Chiral Magnetic Effect on the lattice, qualitative picture

Effect of field increasing

Chiral Magnetic Effect on the lattice, numerical results
300 250 (IR)1/6, MeV 200 150 100 50 0 0 0.5 1 1.5 2 H, GeV2 2.5 3 3.5 4 a a a a = = = = 0.103 0.103 0.103 0.089 fm, fm, fm, fm, 14 14 16 16
4 4 4 4

, , , ,

24 14 24 24

ev. ev. ev. ev.

Regularized electric current:

< j > IR =< j ( H , T ) > - < j (0,0) >,
2 3 2 3 2 3

j3 = 3

Chiral Magnetic Effect on the lattice, numerical results

T =0
2 < j12 >=
(IR)1/6, MeV

F 0 12 < j >=
2 3 2 0

260 240 220 200 180 160 140 120 100 80 60 40 0 0.5 1

T >0 F 0 12
2 < j12 >=
j1, j2, j0, j3, j1, j2, T = 350 j3, T = 350 j0, T = 350 1.5 2 H, GeV
2

T=0 T=0 MeV MeV MeV 2.5 3 3.5 4

2 2 < j3 >

Regularized electric current:

< j > IR =< j ( H , T ) > - < j (0,0) >,
2 i 2 i 2 i

ji = i

Chiral Magnetic Effect, EXPERIMENT VS LATTICE DATA (Au+Au)
3 2.5

Lattice data STAR data (prelim.)

103 (a++ + a-- - 2 a+-)

2 1.5 1 0.5 0 -0.5 0 0.1 0.2 0.3 0.4 0.5 frac. most central 0.6 0.7

Chiral Magnetic Effect, EXPERIMENT VS LATTICE DATA

experiment

R 5 fm
our fit D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys. A 803, 227 (2008),

0 . 2 fm 1 fm

our lattice data at T=350 Mev

Chiral Magnetic Effect on the lattice, RESULTS and QUESTIONS
We see rather weak correlation between topological charge density and fluctuations of the density of the electric currents (What is the origin of CME effect?) [1] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D 78, 074033 (2008),
URL http://arxiv.org/abs/0808.3382. [2] D. Kharzeev, R. D. Pisarski, and M. H. G.Tytgat, Phys. Rev. Lett. 81, 512 (1998), URL http://arxiv.org/abs/hep-ph/9804221. [3] D. Kharzeev, Phys. Lett. B 633, 260 (2006), URL http://arxiv.org/abs/hep-ph/0406125. [4] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys. A 803, 227 (2008), URL http://arxiv.org/abs/0711.0950.
1

y

0.8 c(52, j2) 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 q B, GeV2 j1 , j3 , j1, T = 350 j3, T = 350 j0, T = 350 T=0 T=0 MeV MeV MeV

Chiral Magnetic Effect on the lattice, RESULTS and QUESTIONS

D. Leinweber Topological charge density after vacuum cooling

S.Morozov, F.Gubarev, M.Polikarpov, V. Zakharov Density of eigenfunctions of the Dirac equation in not cooled vacuum

Chiral Magnetic Effect on the lattice, RESULTS and QUESTIONS
y
Large correlation between square of the electric dipole moment and chirality

0i = i [ 0 , i ]
1.2 1 0.8 0.6 0.4 0.2 0 0

5 = 5

03, 02, 12, 03, T = 350 02, T = 350 12, T = 350

c(5, )

T=0 T=0 T=0 MeV MeV MeV

0.5

1

1.5

2 H, GeV
2

2.5

3

3.5

4

Conclusions
1. We observe that the chiral condensate is proportional to the strength of the magnetic field, the coefficient of the proportionality agrees with Chiral Perturbation Theory. Microscopic mechanism for the chiral enhancement is the localization of fermion modes in the vacuum (arXiv:0812.1740) 2. The calculated vacuum magnetization is in a qualitative agreement with model calculations (arXiv:0906.0488) 3. We obsereve signatures of the Chiral Magnetic Effect, but the physics may differ from the model of Kharzeev, McLerran and Warringa (arXiv:0907.0494) 4. We observe very large correlation between electric dipole moment and chirality (not yet published)